DIT - University of Trento

Transcription

1 PhD Dssertaton Internatonal Doctorate School n Informaton and Communcaton Technologes DIT - Unversty of Trento INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING st, Feburary, 2007 Advsor: Raffaele De Amcs Drector: Fondazone Graphtech Co-Advsor: Guseppe Cont Senor Scentst: Fondazone Graphtech February 2007

2

3 Acknowledgement Frst of all, I would lke to thank all the members of GT. Especally my advsor Raffaele De Amcs and Dr. Guseppe Cont, who have offered me many professonal advces and supports durng my three years PhD research. I would lke to thank all the members of the jury for the attenton they have pad to my work and for ther constructve remarks and questons. I also would lke to thank Jean Claude Leon, my cosupervsor n France, who has shared hs experence n the feld of free form surface deformaton and semantc shape operators despte a very busy tmetable. I cannot forget all the people wth whom I spend many hours, and that wll be more precous for enrchng my personal experence. Wthout the help of all of you I would not be what I am. Thank you, all the frends, and all the colleagues n Italy, Chna and France. And fnally, I partcularly would lke to thank, my husband, for hs endless love, patence, and support all along these years. I greatly apprecate my lttle son too, he s the most precous gft I have ever got n my lfe, and t s hm who has been gvng me such a great energy and motvaton to fnsh what I have done. st, Feb,

4 Abstract Conventonal geometrc surface modelng software offers the desgner shape nteracton and manpulaton through edtng the assocated control ponts, orders, knot vector and pont clouds. Such modelng by ndrect manpulaton of algebrac and geometrc parameters proves to be dffcult and tedous, especally for novce desgners. Researchers have nstead explored drect ways through addng physcal behavor to the tradtonal parametrc surface patches. Those constrants, encoded as user-appled sculptng forces that modfy the surface n predctable ways, mpose the physcal effects on the deformable models. However, to date, physcally-based manpulaton of B-splne represented shapes s not fully realzed. The goal of ths research s to address ths ssue through further mprovng the flexblty and effcency of B-splne surface modelng and manpulaton n a 3D sketchng envronment. Frstly, the system allows users to construct a B-splne surface along arbtrary curves wth respect to the user s desgn ntenton. Ths process hghlghts the method of the tradtonal llustraton for depctng 3D subjects, where the creaton of 3D objects s usually preceded by a sequence of drawng steps by usng splne strokes. Secondly, I am tryng to elmnate the need for the user to drectly manpulate B-splne parameters by provdng hgherlevel surface manpulaton tools based on physcal technques. In ths system, for each B-splne surface, a bar network s bult from ts control vertces. The physcal effects mposed by user s free splne sculptng deform the model accordng to the geometrc and parametrc constrants. The mnmzaton of the varaton of the external forces has been used; as a result the least possble ad-

5 justment to the control vertces s nvolved. Shape operators have been developed to correlate between B-splne shape parameters and the physcs-based sculptng framework; meanwhle, the research work has llustrated the surface propertes by defnng geometrc constrants (pont, tangency, curve and surface area so on). Moreover, the splne strokes are freely controlled by 3D dragger metaphor, whch wll produce a sequence of dynamc deformatons to facltate the user to acheve the desred models. Fnally, the mprovement from the lower geometrc dgtal shape control to a grammar-based shape manpulaton s presented, where the planar curve propertes are obtaned based on the analyss of the ntrnsc curvature extrema (Leyton Grammar). Fnally a set of semantc-based shape operators amng at dfferent curve propertes, encapsulated wth group of geometrc constrants, s defned to assst desgners for aesthetc curve control. Keywords: Sketch-based, Constrant-based surface modelng, Physcally-based deformaton, Varable model, Dynamcal control, Semantc-based shape handlng 5

6 6

7 Contents. INTRODUCTION..... MOTIVATION STRUCTURE OF THE THESIS STATE OF THE ART SKETCH-BASED GEOMETRIC MODELING Gestrue-based geometrc modelng Free form surface modelng Free-form surface modelng n VR Concluson FREE FORM SURFACE DEFORMATION TECHNIQUES Geometrc approaches Physcally-based approaches Concluson SKETCH-BASED MODELING PARADIGM SKETCH-BASED 3D MODELING PARADIGM Automatc sketches ndentfcaton Intellgent stoke nterpretaton Adaptve decson-makng for shape understandng FREE FROM SURFACE MANIPULATIONS D splne-based surface schetchng Hgher level surface manpulaton CURVE SKETCHING AND MODIFICATION D CURVE SKETCHING AND APPROXIMATION Dynamc threshold-based samplng method Adaptve B-splne approxmaton MODIFICATION BASED ON OVERSKETCHING SKETCH-BASED SURFACE MODELING... 92

8 5.. CONSTRAINT-BASED SURFACE SKETCHING D splne sketchng Mappng 2D strokes nto 3D splnes SPLINE-BASED SURFACE CONSTRUCTION Boundary-based rotaton surface Ral-based rotaton surface Splne-based surface sculptng PHYSICALLY-BASED SURFACE CONTROL CONSTRAINTS AND CURVE MANIPULATION USER-APPLIED CONSTRAINTS FOR SURFACE CONTROL SPLINE-BASED LOCAL DEFORMATION DYNAMIC 3D SENSOR SEMANTIC SHAPE OPERATORS LEYTON S SHAPE PROCESS GRAMMAR DEFORMATION OPERATORS BASED ON L-GRAMMAR DYNAMIC CURVATURE-BASED B-SPLINE MANIPULATION Cubc plannar B-splne curvatures Quanttatve parameters for curve analyss Aesthetc propertes of curve A MAPPING BETWEEN ASP AND GEOMETRIC GRAMMAR THE MANAGEMENT OF GEOMETRIC CONSTRAINTS CONCLUSION AND PERSPECTIVE CONCLUSION PERSPECTIVE...82 BIBLIOGRAPHY...84

9 Lst of Fgures Fgure 2- :... 9 Fgure 2-2:... Fgure 2-3:... 2 Fgure 2-4:... 3 Fgure 2-5:... 4 Fgure 2-6:... 6 Fgure 2-7: Fgure 2-8: Fgure 2-9: Fgure 2-0: Fgure 2- : Fgure 2-2: Fgure 2-3: Fgure 2-4: Fgure 2-5: Fgure 2-6: Fgure 2-7: Fgure 2-8: Fgure 2-9: Fgure 3- : Fgure 3-2: Fgure 3-3: Fgure 3-4: Fgure 3-5: Fgure 3-6: Fgure 3-7:... 6 Fgure 4- : Fgure 4-2: Fgure 4-3: Fgure 4-4:... 8 Fgure 4-5:... 82

10 Fgure 4-6:...85 Fgure 4-7:...87 Fgure 4-8:...89 Fgure 5- :...92 Fgure 5-2:...93 Fgure 5-3:...95 Fgure 5-4:...95 Fgure 5-5:...96 Fgure 5-6:...99 Fgure 5-7:...00 Fgure 5-8:...03 Fgure 5-9:...05 Fgure 5-0:...07 Fgure 5- :...08 Fgure 5-2:...09 Fgure 6- :... Fgure 6-2:...6 Fgure 6-3:...7 Fgure 6-4:...20 Fgure 6-5:...22 Fgure 6-6:...24 Fgure 6-7:...24 Fgure 6-8:...26 Fgure 6-9 :...28 Fgure 6-0:...29 Fgure 6- :...3 Fgure 6-2:...34 Fgure 6-3:...35 Fgure 6-4:...36 Fgure 6-5:...37 Fgure 6-6:...39 Fgure 6-7:...4 Fgure 7- :...43 v

11 Fgure 7-2: Fgure 7-3: Fgure 7-4: Fgure 7-5: Fgure 7-6:... 5 Fgure 7-7: Fgure 7-8: Fgure 7-9: Fgure 7-0: Fgure 7- : Fgure 7-2: Fgure 7-3: Fgure 7-4: Fgure 7-5: Fgure 7-6: Fgure 7-7: Fgure 7-8: Fgure 7-9: Fgure 7-20: Fgure 7-2: Fgure 7-22: Fgure 7-23: v

12

13 Chapter Introducton. Motvaton 3D Sketchng plays an mportant role durng the geometrc desgnng process, especally n the earler conceptual stylng stage, the most synthetcal, domnant and creatve stage n the whole desgn process; These frst sketches do not have to be very detaled or accurate, the mportant thng s to vsualze the result of the branstorms and refne them to represent alternatve desgn solutons. Therefore, the effectve mechansm to support quck llustraton and ntellgent manpulaton of these sketches wll boost the desgnng effcency and thus enhance creatvty. Especally, ncreasngly domnated by Splne-based free-form surfaces n CAS/CAD, ths hgh degree of freedom adds more dffcultes when creatng and manpulatng surfaces. For ths we need much more ntellgent sketch manpulaton systems to assstant stylst durng aesthetc desgn work. In the last two decades, more and more researchers have been pursung the challenge of developng the socalled sketch-based free drawng nterfaces, combnng the ease of freehand drawng wth the advantages of computer

14 CHAPTER. INTRODUCTION processng, to let desgner fully concentrate on the desgn process. The research ssues had manly focused on searchng for drawng and edtng algorthms that adequately support the user n creatng and modfyng 3D geometrc objects. Durng the earler sketch-based routne desgn, everythng about the desgn process must be known n advance. Ths requres long adaptaton tranng and strct conformty to fxed operaton order. A new paradgm to geometrc desgn, the so-called constrant-based desgn, has been proposed. The desgn work proceeds n an nteractve manner.e. there s no fxed order of desgn operaton. Instead, based on specfed constrants and the ntenton of the user, the bult-n logc of the modelng system derves the shape of the model. However the major dffculty of constrant-based approach s the constrant solvng, whch has to compute the values of degrees of freedom such that defned constrants are satsfed. Although sometmes t s suffcent for many applcatons, concentratng the effort on these cases hnders the extenson of ths paradgm to other areas, especally to fully free-form surface modelng. When a feature-based method s ntroduced nto the dgtal models, t mproves the modelng manpulaton to hgh-level semantcs whch support more complex shape modfcaton. The basc dea s to allow the desgner to apply a shape constrant to the parent surface. Features assocate specfc functonal meanng to groups of geometrc enttes. In ths way t s effcent to deform free-form surfaces, however so far feature-based representaton has been only assocated to the lmted geometrc enttes. Further the characterstc lnes, are decomposed nto low-level constrants, restrcts the scope of the deformaton process. The 2

15 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING feature taxonomy s also ncomplete and the mplementaton only based on the Boolean operaton. For ths t s stll of lmted use for full free-form modelng and manpulaton. All n all, the use of state-of-the-art modelers s far from beng ntutve and ntellgent for stylst and desgner whose man target s the aesthetc free form surfaces. Moreover, the majorty of these systems have been desgned as further extenson of classcal CAD tools; the lower geometrc dgtal shape representaton such as vector and pxel cannot be easly used when nformaton at a hgher, semantc level s to be decoded and manpulated. Moreover, the fully free form shape desgn process not only needs the ablty of ntellgent recognton and rapd 3D modelng, but t also needs ntutve, powerful modfcaton manpulaton. Therefore effcent and ntutve shape manpulaton technques are vtal to the refnement of geometrc models as well. Recently consderable achevements have been reached through the adopton of Free-Form Deformaton (FFD) and Extended Free-form Deformaton (EFFD). These embed the whole object nto a tensor product volume. The volume can be deformed by means of splne control ponts whle the embedded object s deformed accordngly. Unfortunately, manpulaton of splnes s not ntutve. Although other physcally-based manpulaton approaches mprove the natural operaton and a new Medal Axal Deformaton method (AxDf) s beng currently proposed to acheve better deformaton results, the degree of freedom avalable to control the shape s stll lmted My research tres to answer to the ncreasng demand for more ntutve methods for both creatng and modfyng free-form curves and surfaces. The work mproved the constrant-based modelng method and sketch-based nterface 3

16 CHAPTER. INTRODUCTION to develop a more ntellgent free form shape manpulaton system for conceptual stylng tasks. The paradgm proposed for advanced sketch modelng further brdges the gap between lower geometrc parameters manpulaton technque and hgher level features and semantc shape control. Durng the modelng phase, the system allows user to construct a B-splne surface along an arbtrary curve wth respect to the user s desgn ntenton. It addresses the ssue of tradtonal llustraton for depctng 3D subjects, where the creaton of 3D objects s usually preceded by a sequence of drawng steps usng few strokes. Meanwhle suffcent nterpretatons of user s freehand splne sketchng are provded for fully free shape creaton. Durng the mplementaton of the deformaton, t hghlghts the physcally-based deformaton technque and fnte element method. I further propose a novel method that s able to automatcally extract a seres of key ponts on a target splne and mposes adaptve external forces to relocate correspondng vertces on a parent surface. Furthermore, a seres of lnear nfluence functons are ntroduced to mprove the contnuty and the symmetry. Ths research work manly focuses on the correlaton between geometrc constrants and the surface propertes to offer the desgner drect NURBS parameter-ndependent surface control. Constrants restrct changes to the surface, whle propertes descrbe surface behavour under appled forces. There are fve constrant types: pont, dstance, tangency, curve and surface area, as well as two surface propertes: stretch and bend. Snce desgners are more nterested n how to obtan the aesthetc results nstead of carng about geometrc detals, my thess also hghlghts the mprovement from the lower geometrc dgtal shape representaton level to hgher 4

17 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING grammar-based 2D sketchng curve manpulaton level, I frst propose the symbolc shape descrpton by the ntrnsc curvature feature of the curve, then I dscuss the curve propertes based on the analyss of the curvature extrema. Fnally a set of semantc-based shape operators based on Leyton grammar process s proposed to assstant desgner to get aesthetc operaton of the curve..2 Structure of the Thess The thess has been organzed nto eght chapters: Chapter s an ntroducton of my research work about sketch-based surface stylng and deformaton. It hghlghts the natural and nteractve requrements for aesthetc desgnng. It further addresses the mprovement from hgh-level shape operators for fully free form shape control. Chapter 2 presents an overvew of the exstng sketchbased shape modelng methods and deformaton technques. Based on the systematc analyss of dfferent algorthms, t concludes wth the advantages and dsadvantages of current research works. Furthermore, t emphaszes the sgnfcance to provde mproved constrant-based surface modelng technques for user s creatve desgnng work. Especally t s necessary to provde more natural and ntellgent modfcaton by combnng the physcally-based and FEM method wth sketch modelng nteracton. Chapter 3 ponts out the paradgm of the sketch-based 3D shape modelng system. Ths chapter detals the archtecture 5

18 CHAPTER. INTRODUCTION of the sketchng system and the adaptve mult-layered constrants decson structure for shape manpulaton. Meanwhle, t not only gves the mplementaton of the basc 3D object modelng based on sketchng understandng, but t also further llustrates the mplementaton of complex shape modelng by 3D sketches and ts effcency for free form surface manpulaton. Chapter 4 addresses an optmzed curve recognton algorthm whch automatcally approxmates the arbtrary hand drawng curve nto cubc B-splne. The sequent approxmated geometrc curves from the user s sketchng wll be used for constraned surface constructon. Moreover, t descrbes an effectve local modfcaton method for surface trmmng through smply over-sketchng. Chapter 5 detals the sketch-based B-splne surface modelng, where a seres of free form curves s nterpolated. Meanwhle, t also presents the transformaton from 2D sketches to 3D descrpton and the flexble space control by usng a dragger n a 3D Vrtual envronment. Chapter 6 presents a splne-drven surface deformaton method. It ntroduces how the user-specfed constrants are used for ntutve shape control by combnng the bar network and mnmzaton process. It also descrbes the mplementaton of the global and local shape modfcaton by addng few curve sketches. Chapter 7 ntroduces the theory of Leyton shape grammar and t further smulates the mplementaton by ntegratng the group of geometrc constrants wth the B-splne deformaton operaton. Moreover, t llustrates the nteractve 6

19 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING shape manpulaton by addng the constrants to the characterstc ponts (curvature extrema ponts) whle the curve s represented as a symbolc descrpton through the analyss of the curvature dstrbuton. Fnally, t provdes quanttatve parameters for curve characterstc analyss, and a set of aesthetc operators are proposed. Chapter 8 concludes the dscusson and proposes some related future work. 7

20 Chapter 2 State of the Art 2. Sketch-based Geometrc Modelng Sketch-based nterfaces have been researched for long tme snce 70`s. As an ntutve tool t provdes perceptual and drect nteracton for desgner and, at the same tme, t also carres the ambguty of the sketched nput []. How to make sketch-based tools ntellgent enough for creatve work has been one of the man research topcs. Wthn ths research area we can dstngush three approaches: Gesture-based geometrc modelng: these systems use gestures as commands for generatng solds from 2D segments, where the user must conform to fx drawng sequence and style. Such a method s useful to create geometrc prmtves. Constrant-based geometrc modelng: ths approach stresses the ssue of free form modelng; t uses algorthms to reconstruct geometrc objects from sketches whle a seres of predefned logcs are used to match the user s dfferent drawng styles. Feature and semantc-based geometrc modelng: ths 8

21 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING approach mproves the two approaches prevously mentoned. The shape-based constrants are adopted to mplement hgher-level global and local shape control. 2.. Gesture-based Geometrc Modelng Earler systems allow sketchng 3D geometrc models by usng a set of pre-defned gestures for creatng and combnng geometrc prmtves whch convert sketches usng Gesture Recognton algorthms nto a seres of specfc commands [2] (see Fgure 2-). Fgure 2-: Gestures for prmtve constructon [2] The work n [3] uses a gesture nterface to allow users to create and manpulate geometres, and t transforms them nto a hgh level feature-based model. Ths system provdes mechansms to constran and parameterze 3D geometres. However ths system lacks n generalty and t also requres extendble gesture sets to hgh level 2D and 3D constructons. The work n [4] ntroduces a new type of so-called 9

22 CHAPTER 2. STATE OF THE ART suggestve nterfaces for 3D drawngs. It extends the gesture nterface through offerng multple canddates, whch are generated by a set of suggeston engnes. It s easy to use for novce users, but f the hnts gven are nadequate, the system never responds. Further complcated 3D scenes can make t dffcult to specfy hnts and to fnd the desred one. The system descrbed n [5] defnes geometrc features of objects through drawng of a set of auxlary lnes. It supports over-sketchng of real lnes over auxlary lnes and snappng and adjustment are preformed n real tme. Ths system dffers from prevous approaches n that speed of executon and fast feedback are more mportant than the ablty to produce models. However by usng constructon lnes ths method can not provde a sound bass for more sophstcated nput technques, especally curves and surfaces. In general a gesture-based geometrc modelng system must consder a seres of constrants such as parallel lnes, rght angles, symmetry and other concepts, whch have been defned before. These lower level topologcal constrants are dffcult to use for fully free-form surface modelng tools Free form Surface Modelng As a central topc of Computer Aded Geometrc Desgn, free-form curve and surface creaton attracted many researchers attenton. Constrants-based Free Form Modelng Parametrc curves and surfaces have been successfully appled to desgn of free form shapes. However the man lmt of them les n the fact that ther manpulaton requres sg- 0

23 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING nfcant mathematcal knowledge and sklls durng ther shapng. Ths makes the task tedous and cumbersome. Sometmes changng a small detal s dffcult and t can be as tme-consumng as recreatng the whole model. Furthermore, these models do not mantan suffcent dependency between the geometrc elements of an object. Recently a number of researchers have tred to mprove them. For example n [6] the authors analyze approxmaton of curves and reconstructon of surfaces. The usual approach uses a least squares formulaton that expresses the farness of the fnal result based on parametrc B-splne, however t s a dffcult to estmate the parameters. Further they propose that the actve curve or surface adapts to the model shape to be approxmated n an optmzaton algorthm through a quas-newton optmzaton procedure, whch completely avods the parameterzaton problem (see Fgure 2-2). Fgure 2-2: Approxmatng B-splne surface [6] The work n [7] presents a smple touch-and-replace technque to edt 2D and 3D curves. The authors ntroduce auxlary surfaces that allow for a relable nterpretaton of users pen-strokes n 3D and a new method for sketch and constrant-based surface sculptng (see Fgure 2-3). One lmtaton of ths approach s the lack of degrees of freedom,

24 CHAPTER 2. STATE OF THE ART whch are restrcted by the auxlary surfaces, partcularly when edtng complex scenes. The poor vsblty of the auxlary surface can also be a hndrance to users. Fgure 2-3: Constrant-based surface sculptng [7] Recently Teddy s [8] provdes us wth a more humanzed modelng technque whch s orented to free-form surface modelng by usng a very smple nterface. It creates smple objects of sphercal topology by usng a sequence of nflate, extrude and cut operatons. An obvous applcaton of Teddy s the rapd desgn of 3D approxmated models for character anmaton. Amng at Teddy s rough polygonal meshes, the work n [9] proposed an mproved approach to beautfy and refne polygonal meshes. However Teddy s cannot create multple objects; the models must have sphercal topology (equal to sphere) and no holes. It s mpossble to some edtng operatons (e.g. creatng a brdge between objects) and the modelng s carred out by repeatng cut and extruson operatons. Fnally there s no analogy to the process n way people sketch, create models. An alternatve approach to free-form modelng adopts Varatonal Implct Surfaces (VIS). These surfaces are defned by a set of constrants that specfy the ponts on the 2

25 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING surface boundary. Karpenko et al. [0] and Bruno et al [] descrbe how sketched slhouettes are converted nto a set of constrans whch are then used to create a blob by stroke nflaton. These blobs can be combned by usng the blob mergng operatons (see Fgure 2-4). Fgure 2-4: Surface mergng and edtng operaton [0] The advantage of the mplct surface representaton s that the natural modelng operatons nflaton, strokebased mergng s supported. However they cannot provde further nteror modfcatons. In [2] the approach allows the user to desgn more freely objects wth hgh topologcal complexty due to the nature of mplct surfaces. The approach also supports addtonal features: shapes wth holes, sem-sharp features, and free-form cross-secton shapes. However ths s not suffcent for fully free form shape modelng. Further as the number of constrants ncreases, the tme t takes to compute the coeffcents for the varatonal mplct surfaces grows as well. Fnally ts representaton cannot support sharp edges n a surface. 3

26 CHAPTER 2. STATE OF THE ART Feature-based Free Form Modelng Constrants-based modelng technques restrct the possbltes to deform a curve or surface. For ths reason t s dffcult to get the desred surface for aesthetc desgnng. In [3] [4] the authors propose a feature-based desgnng method where features are well known, n a mechancal envronment, as hgh semantc level enttes. These assocate functonal nformaton to pure geometry, they are sutable for faster modelng and t s possble to re-use the nformaton. The authors also suggest a possble classfcaton of free form features for recognzng and generatng the desred surface. P 0 P Fgure 2-5: Feature-based shape modelng by lmtng and target lnes constrants [5] Cheutet et al. [5] nsert the δ-f4 (fully free form deformaton features) n stylng desgn. The author deforms the geometry through hgher-level constrants, whch allows for a drect manpulaton of surfaces through a restrcted number of ntutve key lnes (see Fgure 2-5). The mportant thng s that they consder some modelng enttes as feature prmtves, and they establsh a lnk between geometrc level and semantc level n respect to stylst s purpose whch permts a fast access to the desred shape accordng to ts semantcs. However untl now, only the 4

27 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING classcal addton operator has been proposed to add deformaton features. Operators such as FFF extruson, Lmtng Lne fuson or FFF unon can be defned to extend the varety of accessble shapes nsde the proposed classfcaton, whch need more future works Free Form Surface Modelng n VR Currently vrtual realty technology for modelng s very popular for desgnng because of ts realstc vsualzaton. Earler work n [6] demonstrates an nterface to create cubes, smple prsms and pyramds constructed by lnes n vrtual envronment. However t s dffcult to draw any complcated objects and some curves. In [7] the authors present a smple method to create 3D curve, whch s determned by ts mage-space projecton and ts shadow. It s well sutable for fast approxmaton 3D curves; however t can be qute hard to judge what the shadow should look lke for complex 3D curves. In [8] the authors translate conceptual sketch strokes nto a sutable B-splne representaton wth a three-step method. Frst a data flter s used to elmnate redundancy and nose. Then a knowledge based algorthm tres to nterpret the user s ntenton accordng to drecton, speed and curvature nto two types of curves joned respectvely wth C 0 and C contnuty. Fnally an algorthm translates each segmented sketch stroke nto a cubc B-splne wth adaptve approxmaton (see Fgure 2-6). However ths system cannot support the post-sketchng edtng functon. A better tool s presented n [9] where the authors propose a two-handed 3D stylng system for free-form surfaces n a table-lke Vrtual Envronment. One hand holds 5

28 CHAPTER 2. STATE OF THE ART the model, the other carres out the edtng functon. In ths applcaton Cubc B-Splne curves are beng drawn by freehand, edted and automatcally ntegrated nto the exstng network. Functons lke smoothng, sharpenng and draggng are avalable. The system provdes the desgner wth an ntutve nterface relyng on famlar metaphors whch mask the complex mathematcal representaton of the Splne curves and surfaces (see Fgure 2-6). Fgure 2-6: 3D curve creaton [8] and 3D curve generaton from scratch [9] A VR system s a perfect mean to delver the percepton of 3D space, but t s dffcult to control for novce user and desgner, when they wear the stereo glasses to draw n the ar rather than usng the pen and paper. Furthermore most of the VR systems cannot easly support the fully and natural free form sketchng and manpulaton Concluson As a whole, current 3D desgn software n the CAD feld focus on provdng powerful shape control. Authors have developed advanced methods from constrants-based to fea- 6

29 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING ture-based; some of them are mplemented n perceptually surroundng - vrtual envronment. However, they are stll not enough ntellgent and complete for free form shape modelng. In fact the fully free form shape desgn process not only needs the ablty of ntellgent recognton and rapd 3D modelng, but t also needs ntutve, powerful modfcaton manpulaton and the knowledge reasonng technque for better understandng the sketched nput [20] Furthermore, wth the ncreasng of the freedom of free-form shape, and n partcular for aesthetc desgnng work, what the desgners really requre s the hgh-level shape handlng system, where only set of meanngful shape operators are provded to users, wth the encapsulaton of tedous geometry representaton and geometrc constrants. 2.2 Free Form Surface Deformaton Technques Snce the man characterstcs of conceptual stylng stage are naccuracy and nconsstence; the effcent and ntutve shape modfcaton technques are vtal to further refnement of geometrc models. Such refnement processes are not only the smple over-sketchng of the boundares but also the obtanng of predctable shape varatons by addng few sketches. The advanced deformable modelng technques have been ncorporated to the CAD/CAS the desgnng works. The mprovement of the deformaton methods further accelerates the effcency of modfcaton procedure. The exstng deformaton technques generally are classfed n three man categores of methods. 7

30 CHAPTER 2. STATE OF THE ART The geometrc approaches enable the modfcaton of geometrc models ether through drect geometrc parameters adjustment or by a set of dsplacement constrants or by usng volumes of nfluence. In earler research works, these geometrc models are represented as ether double-quadratc curves, B-splnes, ratonal B-splnes, or non-unform ratonal B-splnes (NURBS). The desgner adjusts the shape of the objects by movng control ponts to new postons, by addng or deletng control ponts, or by changng ther weghts or by removng and addng sequence knots. Ths parameter-based object representaton s computatonally effcent and t supports nteractve modfcaton. However, ths level of control s sometmes a dsadvantage: precse specfcaton or modfcaton of curves or surfaces can be laborous. Even a perceptually smple change may requre adjustment of many control ponts. The Free-form deformaton (FFD) proposed by Sederberg and Parry s a general method for deformng geometrc objects that provdes a hgher and more powerful level of control than adjustng ndvdual control ponts. These FFDs change the shape of an object by deformng the space n whch the object les. Ths technque can be appled to many dfferent graphcal representatons, ncludng: ponts, polygons, splnes, parametrc patches, and mplct surfaces. The basc FFD method has been extended by several others. Coqullart [2] provdes a toolkt of lattces wth dfferent szes, resolutons and geometres that can be postoned over the object for selectve control of sub-regons of the surface. Hsu [22] allows drect manpulaton of surface or curve ponts by convertng the desred movement of these ponts to equvalent grd pont movement. Ths under- 8

31 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING constraned problem s solved by choosng the grd pont movement wth the mnmum least-squares energy that produces the desred object manpulaton. MacCracken and Joy [23] perform FFD on lattces wth arbtrary topology, usng a subdvson algorthm to refne the lattce and to provde the desred shape. Specal lattce ponts along sharp edges and corners are handled separately. The results of these research works have also been exploted across several domans ncludng human body anmatons and dynamc flexble deformatons. However, whle these approaches ncrease a degree of flexblty n terms of control, on the other hand they are based on the soluton of complex nonlnear equatons through numercal methods. Another example of geometrc deformaton method s the new Medal Axal Deformaton method (AxDf) proposed by [24] ams at achevng better control over deformatons. The user creates a curvlnear path represented by a B-Splne curve and locates t wth respect to the object. By modfyng the shape of the path, the cylndrcal space surroundng ths path as well as the part of the object whch s mmersed n ths space s modfed. Ths method s further extended to generate a free form models by set of wres [25]. However, these geometrc approaches of modelng deformaton are stll lmted by the expertse and patence of the user, and the deformaton results are certanly nadequate by smply changng the axs and wres. The physcally-based approaches led to the modfcaton of geometrc models whle usng a physcal model enablng ether surface, or volume, or mxed deformaton. Here, the deformaton of geometrc models mmcs the deformaton of real objects,.e. objects havng mechancal propertes relatve to ther materal, ther shape and so on. 9

32 CHAPTER 2. STATE OF THE ART As desktop computng power and graphcs capabltes ncreased durng the 980's, the graphcs communty began explorng physcally based methods for anmaton and modelng. These methods use physcal prncples and computatonal power for realstc smulaton of complex physcal processes that would be dffcult or mpossble to model wth purely geometrc technques. Mass-sprng systems are one physcally based technque that has been used wdely and effectvely for modelng deformable objects. An object s modeled as a collecton of pont masses connected by sprngs n a lattce structure. The sprng forces are often lnear (Hookean), but nonlnear sprngs can be used to model tssues such as human skn that exhbt nelastc behavor. Terzopoulos and Waters were the frst to apply dynamc mass-sprng systems to facal modelng [26]. Furthermore, the technque has been used extensvely for anmaton. Chadwck et al. combned mass-sprng models wth free form deformatons to anmated muscles n human character anmaton [27]. Terzopoulos et al. descrbe a mass-sprng model for deformable bodes that experence state transtons from sold to lqud [28]. However such knd of method suffers of poor stablty and huge tme consumng. As an alternatve method, the Fnte Element methods (FEM) are proposed, where an approxmaton of a contnuous functon that satsfes some equlbrum expresson s used. The deformable object consders the equlbrum of a general body acted on by external forces. The object deformaton s a functon of these actng forces and the object's materal propertes. The object reaches equlbrum when ts potental energy s at a mnmum. In FEM, the contnuum, or object, s dvded nto elements joned at dscrete node ponts. A functon that solves 20

33 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING the equlbrum equaton s found for each element. The soluton s subject to constrants at the node ponts and the element boundares so that contnuty between the elements s acheved. Unlke mass-sprng methods, where the equlbrum equaton s dscretzed and solved at fnte mass ponts, n FEM, the system s dscretzed by representng the desred functon wthn each element as a fnte sum of element-specfc nterpolaton, or shape, functons. Although fnte element methods provde a more physcally realstc smulaton than mass-sprng methods wth fewer node ponts and t requres the soluton of a smaller lnear system, the appled forces must be converted to ther equvalent force vectors, whch can requre numercally ntegratng dstrbuted forces over the volume at each tme step. The mxed approaches use both a geometrc approach and a physcally-based approach of the deformaton. The objectve s to get closer to physcal behavor whle usng an energetc functonal or a mechancal model coupled to the vertces defnng the geometrc model (nodes of the control polyhedron/polygon of a parametrc surface/curve, vertces of a mesh) Geometrc Approaches From a rather general pont of vew, geometrc approaches deform a curve or a surface through drect or ndrect methods. The frst method s to adjust the shape of the objects by drectly changng geometrc detals, such as control ponts, weghts, knots sequence and meshes so on. And the latter s that the desgners control the objects through ndrectly user-defned constrants changng the mmersng space or 2

34 CHAPTER 2. STATE OF THE ART volume or auxlary axs. Deformaton by Drect Vector Control Many curve researches have been developed n the past. Subjectng to some requrement for modfcaton, the curve s normally mposed by some constrants; and then the correspondng adjustment responds approprately. For splne curves, such adjustments may nvolve changng knots, control vertces, and/or weghts (f the curve s ratonal). [29] were the frst to descrbe constrant modfcaton of mathematc curves. The propertes to modfy the curve at a desgnated pont nclude poston, tangency and curvature whle constranng one ore more of the curve s dervatves at one or more selected parametrc ponts, whch leaves the burden to understand the relaton of geometrc parameters to the users. The work n [30] proposed easy edton of B-splne curves by cuttng and sketchng control polygons. In [3] they use the knot removal technque to adjust curve shape, [32] presents an effcent technque for approxmatng a planar parametrc curve by a small set of ellptc arcs; they also ntroduce a smple approach to smooth replaced sectons of desgn curves wth sectons of the French curve. Ths system s of great value n smplfyng and neatenng curve, but t would nvolve n a number of computaton, especally t s not sutable for the users who wthout the professonal experence. 22

35 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Fgure 2-7: A cubc B-splne curve. The shape of the curve s edted by movng control ponts [30] Deformaton wth Parametrc Volume (FFD, EFFD) [33] have proposed a method called Free Form Deformaton (FFD). The analogy often used to explan the prncple of ths approach s to consder that the objects to deform are mmersed nsde an elastc block havng a parallelepped shape. The manpulatons of the block produce the deformaton of the objects. Therefore, two steps can be dstngushed: the deformaton of the block whch can requre physcs-based approaches to smulate the behavor of a real object, and the deformaton of the mmersed object once the new shape of the block obtaned. Ths secton smply gathers together the works related to the second step. : Parametrc Volume of Deformaton (FFD) The bass of the deformaton model s an elastc block, called FFD block, represented by a Bezer volume defned 23

36 CHAPTER 2. STATE OF THE ART by a 3D network of control ponts S jk whch determnes the porton of space to be deformed. Thus, a pont P s located by ts (u, v, w) parametrc coordnates defnng ts 3D poston nsde the volume (see Fgure 2-8): l m n P( u, v, w) = Bl ( u) B jm ( v) Bkn ( w) Sjk (2-) = 0 j= 0 k = 0 Fgure 2-8: Free Form Deformaton by a parallelepped [33] Once the poston of the control ponts modfed, the new poston of the ponts defnng the surrounded object can be computed. When deformng a parametrc surface, the equaton (2-) s appled to the control ponts of the surface to be deformed, whereas t s drectly appled to the vertces n case of a polyhedral model. In the ntal formulaton, the block s necessarly parallelepped. The soluton s unque. Moreover, usng a Bezer volume, no dscontnutes can be nserted and the contnuty of the model s preserved. 2: Extended Parametrc Volume of Deformaton (EFFD) In order to overcome the lmt nherent to the ndrect ma- 24

37 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING npulatons, [22] have proposed a method whch enables the specfcaton of constrant ponts drectly on the geometry to deform. Startng from ths set of user-specfed ponts, the algorthm computes the new postons of the S jk. As a result, the dsplacements generated do not match properly the prescrbed dsplacements snce a mnmzaton s used. Fgure 2-9: The deformed lattce and the deformed surface [2] Several extensons have also been proposed to enable the use of more complex blocks [2, 23]. Recently, a new method called t-ffd has been proposed for the manpulaton of large scale polygonal meshes [35]. In ths approach, an orgnal shape of mesh or pont-cloud s deformed by usng a control mesh, whch formed by a set of trangles wth arbtrary topology and geometry, ncludng the confguratons of dsconnecton or self-ntersecton. For modelng purposes, a desgner can handle the shape drectly or ndrectly and also locally or globally. Ths method uses a smple mappng mechansm. Frst, each pont of the orgnal shape s parameterzed by the local coordnate system on each trangle of the control mesh (Fgure 2-0.a). After modfyng the control mesh (Fgure 2-0.b), the pont s mapped accordng to each modfed trangle (Fgure 2-0.c). Fnally, the mapped locatons are blended as a new poston of the orgnal pont, and then a smoothly deformed shape s 25

38 CHAPTER 2. STATE OF THE ART produced. Fgure 2-0: Deformaton steps of t-ffd [35] Although FFD and EFFD based methods can acheve a very varety of deformatons, the user s forced to defne some control ponts around the space to be deformed and then move these control ponts. Ths ndrect nterface may be unnatural for some applcatons. Hsu et al addressed ths problem and proposed a drect nterface that nvolves solvng a complex equaton system [2], but ts cost s expensve Deformaton by User-specfed Constrants Ths concept of surface deformaton by user-appled pont constrants has been ntroduced by [36] n the case of n- dmensonal objects. Snce ths approach has been mproved afterwards, t seems mportant to menton ts prncple. In order to smply the explanatons, the space wll be supposed of dmenson 3. Let P be a pont of the 3D-space n whch 26

39 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING the object to deform s mmersed. The dsplacement d (P) of P s gven by the equaton: n c d( P) = M f ( P) (2-2) = Where n c represents the number of constrants, each constrant beng characterzed by ts drecton M and ntensty functon f. ; When D j s the user-specfed dsplacement vector of the constraned pont C j, t comes that: n c d( C j ) = D j = M f ( C j ), = j {... nc} (2-3) The ntensty functons f beng specfed for each constrant ponts, the M drectons have to be computed by solvng the (3.nc 3.nc) system formed by the equatons (2-3). Thus, the deformaton of the object can be obtaned through the equaton (2-2) for each pont of the geometrc model. In ther approach, Borrel and Bechmann apply ths prncple to the deformaton of trangular polyhedral models. The deformaton functons f affect the whole space surroundng the object to deform and the constrant pont C correspond to some vertces of the geometrc models. Dependng on the nature of ths functon (B-Splne, polynomal and so on), dfferent local or global behavors can be obtaned. Heterogeneous behavors could also be obtaned whle specfyng dfferent types of functons. Unfortunately, ther approach requres a structured polyhedral model and the toolbox s restrcted to pont constrants. The contnuty of the model s preserved. 27

40 CHAPTER 2. STATE OF THE ART Fgure 2-: B-splne bass functon (a); Computaton of the dstance between C and P ether n 3D (b) or nsde the 2D parametrc space(c) of a surface Stll n the case of polyhedral models, Borrel [37] has proposed to mprove the prevous scheme whle addng the concept of radus of nfluence R of each constrant use the followng functons: f P) = B ( U ( P)) wth U ( P) = P C R ( (2-4) where B s a B-Splne bass functon centered at 0 and varyng between 0 and. Ths functon s equal to 0 for the pont C and s null for pont located outsde the radus of nfluence. Borrel has also extended the concept of radus of nfluence to the three drectons of the space. The extended radus of nfluence of the th constrant becomes R = t[r x,r y,r z ] and the correspondng functon can be wrtten 2 Cx ) 2 x 2 Cy ) 2 y 2 Cz ) 2 z ( P ( x Px ( Px f ( P) = B ( + + (2-5) R R R [38] have also extended the noton of radus of nfluence whle proposng the concept of super-quadrc nfluence hull. More recently, they have extended ther constrants toolbox whle enablng the specfcaton of curvlnear dsplacement 28

41 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING constrants [39]. Usng ths approach, the dsplacement constrant affected to C s defned by a user-specfed Bezer curve whch drves the shape deformaton. All these concepts have also been appled to the doman of parametrc surfaces. [40] defne a surface S (u, v) resultng from the deformaton of an ntal surface S 0 (u, v) as follows: S( u, v) = D D LP ( u, v) = LP n c ( u, v) + S = M 0 ( u, v) f ( u, v, u, v ). wth (2-6) As already stated, M represents the control vector of the th constrant. The prescrbed dsplacement D s appled at C located by ts (u, v ) parametrc coordnates. The dsplacement functon D LP (u, v) represents how much the ntal surface s deformed. Usng an analogy wth the Cao En surface model where a closed parametrc surface s defned by a mappng from a sphere, or hyper sphere, to a correspondng closed parametrc surface; the functon of nfluence f s defned by the equaton 2-7. The nfluence radus R s defned n the parametrc space of the surface, whch strongly dffers from the prevous technques where ths concept s assocated to a 3D dstance. As a consequence, the specfcaton of ths value must be dffcult for a neophyte user. F ( u, v, u, v ) f ( u, v, u, v ) = n c F (,,, ) j = j u v u v Where the blendng functon F j are defned by the followng rule 29

42 CHAPTER 2. STATE OF THE ART 2 3 ( ( d / r ) ) ; f F ( u, v, u, v ) = 0; otherwse. d R 2 2 Where d = ( u u ) + ( v v ) (2-7) Moreover, when the parameterzaton of the surface s not homogeneous, the part of the surface correspondng to ths crcle can have a shape very dfferent from the one of a crcle. Smlarly to the prevous approaches usng the concept of nfluence functons, the M vectors are obtaned through the resoluton of the (3.nc 3.nc) system comng from the constrants specfcaton. In ther approach, the constrant ponts can come from the dscretzaton of a target surface. A smlar approach has been developed by [4]. It dffers n the sense that the dsplacements are performed accordng to the normal to the surface. The constrant toolbox has been enhanced wth curve constrants defned by specfc blendng functons enablng the treatment of trmmed surfaces. Fgure 2-2: Lne segment constrant and polylne constrant [4] Meanwhle, another effectve work n [42] further mproved by to the extent whch, not only t conducts to deformaton of pont constrants, but also lnes and surfaces constrants (Fgure 2-2). They proposed a new local deformaton model based on generalzed meta-balls. The user specfes a seres of constrants, whch can be made up of 30

43 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING ponts, lnes, surfaces and volumes, ther effectve rad and maxmum dsplacements; the deformaton model creates a generalzed meta-ball for each constrant. Axal Deformaton (AxDf, SuIf) In ths category of approaches, the object s deformed accordng to specfed drectons. These drectons can be defned ether through the use of B-Splne curves that defne the ampltude of the deformatons (AxDf), or accordng to surfaces of nfluence (SuIf). Axal deformatons (AxDf) [24] have generalzed the dea of 3D axal deformaton. To modfy an object, the user creates a curvlnear path represented by a B-Splne curve and locates t wth respect to the object. By modfyng the shape of the path, the cylndrcal space surroundng ths path as well as the part of the object whch s mmersed n ths space s modfed (see Fgure 2-3). Ths concept of curves used to deform an object has been extended by [25] who have defned the noton of wres. Ther approach s nspred by armatures used by sculptors. It ams at brngng geometrc and deformaton modellng closer together by usng a collecton of wres both as a coarse-scale representaton of the object surface, and a drectly manpulated prmtve that hghlghts and tracks the salent deformable features of the object. There are two stages n the wre deformaton process. In the frst one, whch s typcally computed once, an object s bound to a set of wres. In the second one, any manpulaton of a wre trggers a deformaton of the object. More precsely, an mplct functon smlar to the one of equaton s assocated to each wre. It s montored by an nfluence radus that defnes whether the shape surroundng the object has to be modfed or not and f modfed nto whch extent t has to be mod- 3

44 CHAPTER 2. STATE OF THE ART fed. In order to further descrbe the deformaton doman, the concept of doman curves s also ntroduced. Fgure 2-3: Example of axal deformaton [24] Fgure 2-4: The wres deformaton technque [25] One lmt of the approach ntroduced by Lazarus les n the fact that the object cannot be twsted by manpulatng the axal curve. To overcome ths lmt, [43] have defned the concept of axal curve-pars. The object s not anymore located accordng to the Frenet local reference frames of a sngle curve, but accordng to the local reference frames defned by a par of curves: the prmary curve and the orentaton curve. The orentaton curve s an approxmate offset of 32

45 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING the prmary curve. Fgure (2-5.a) shows the constructon of a leaf structure and the defnton of ts axal curve-pars. The man stem can be ether smply deformed (Fgure 2-5.b) by modfyng the prmary curve or twsted (Fgure 2-5.c) by modfyng the orentaton curve. Each leaf can also be deformed (Fgure 2-5.d). Hu s approach seems well suted to the manpulaton of objects defned by axs, e.g. a leaf structure or a rbbon. Consequently, the use of ths technque for the manpulaton of any 3D object can be nadequate and one would prefer the use of Wres. Fgure 2-5: Leave pattern deformaton [43] Even f all these technques, that use curves as nput parameters, are ndependent of the type of underlyng geometrc model, they do not consder the manpulaton of hybrd models. The contnuty of the model s preserved and the proposed constrant toolbox s stll restrcted. Skeleton-based shape deformaton [44] method propose that the drecton from a mesh vertex to ts correspondng skeleton vertex s a good approxmaton for the normal drecton at the mesh vertex; and the Skeleton s a generalzaton of the Vorono dagram for a contnuous curve. Ths method s well used for global and local deformatons. 33

46 CHAPTER 2. STATE OF THE ART Fgure 2-6: The skeleton extracton and the deformaton, the fnger deformaton by usng skeleton control. [44] Surfaces of Influence (SuIf) [45] have extended the axal deformaton concepts to the manpulaton of objects through the use of two nfluence surfaces. [46] have used a smlar approach for the manpulaton of objects wth a sensor glove. As shown on fgure (2-7), the nfluence surface H(u, v) s a b-cubc B-Splne surface nterpolatng or approxmatng key data ponts Q,j of a sensor glove,.e. fnger jonts and palm center of the user s hand. By settng up a correspondng mappng between the vrtual object to deform and the nfluence surface, the object can be deformed wth the control of the sensor glove. Each pont P of the object surface s located accordng to ts projecton P onto the nfluence surface H 0 (u, v) correspondng to the glove n a poston ntally planar. Each pont P s then transformed to a new poston Q by the followng mappng: Q = H(u p, v p ) + d p.n(u p, v p ) (2-8) 34

47 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING where N s the unt normal of H(u, v) at (u p, v p ) and d p s the drectonal dstance of P to the ntal surface H 0. As the user bends hs/her fngers, the object changes ts shape accordngly. Snce ths formulaton s senstve to the normal varaton of the nfluence surface, especally f the glove s far away form the object surface, the mappng of equaton 2-7 s modfed as follows: Q = H(u p, v p ) + d p. [( α).n(u p, v p ) + α.n H 0 ] (2-9) where N s the normal to the ntal nfluence surface H 0. By selectng dfferent values of α, the user can acheve dfferent effects. Such a control can be ether local or global. For local deformaton, they have ntroduced a regon flter functon whch localzes the applyng/deformaton. These approaches enable the manpulaton of an object surface ndependently of ts underlyng geometrc representaton. The contnuty of the model s preserved. Unfortunately, the manpulatons are not curve-orented. Fgure 2-7: Surfaces deformaton usng an nfluence surface defned by a sensor glove [46]. 35

48 CHAPTER 2. STATE OF THE ART Physcally-based Approaches By usng those geometrc approaches for shape deformaton, the desgn process s lmted by the expertse and patence of the user. Deformatons must be explctly specfed and the system has no knowledge about the nature of the objects beng manpulated, therefore, the ndrect shape refnement remans ad hoc and ambguous. To amelorate the ndrect desgn process, the physcsbased deformaton technques have been further proposed, where the deformable models, governed by physcal laws, dynamcally respond to appled smulated forces n a natural and predctable way. Mass-sprng systems are one physcally based technque that has been used wdely and effectvely for modelng deformable objects. An object s modeled as a collecton of pont masses connected by sprngs n a lattce structure (Fgure 2-8). The sprng forces are often lnear (Hookean), but nonlnear sprngs can be used to model tssues such as human skn that exhbt nelastc behavor. In a dynamc system, Newton's Second Law governs the moton of a sngle mass pont n the lattce: m & x γ x& + g + f (2-0) = j where, m s the mass of the pont, x s ts poston, and the terms on the rght-hand sde are forces actng on the mass pont. The frst rght-hand term s a velocty-dependent dampng force, g j s the force exerted on mass by the sprng between masses and j, and f s the sum of other external forces, (e.g. gravty or user appled forces), actng on mass. j 36

49 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Fgure 2-8: A porton of a mass-sprng model. Sprngs connectng pont masses exert forces on neghborng ponts when a mass s dsplaced from ts rest postons Terzopoulos and Waters were the frst to apply dynamc mass-sprng systems to facal modelng [26]. Waters and Terzopoulos later developed a technque to generate facal models for partcular ndvduals from a radal laserscanned mage data. And the seres of mprovement n [27][28] are used n dfferent applcaton areas. Snce Mass-strng models start wth a dscrete object model, such a dscrete model s an approxmaton of the true physcs that occurs n a contnuous body, but the lattce whch turned through ts sprng constants are not always easy to derve from measured materal propertes. In addton, the certan constrants are not naturally expressed n the model. In fact, more accurate physcal models treat deformable objects as a contnuum: sold bodes wth mass and energes dstrbuted throughout. The ncorporaton of physcally-based method wth contnum models n the geometrc modelng context orgnates from the works of [47] whch have been developed later by Barr, Flesher, Metaxas, Pentland, Platt, Qn, 37

50 CHAPTER 2. STATE OF THE ART Zvelsk, Terzopoulos, Tonnesen and Wllams n the context of modelng wth NURBS. The man dea of ths type of approach les n the fact that the deformaton of the geometrc model has to mmc the deformaton of a real object havng specfc mechancal propertes. One can dstngush approaches usng a surface mechancal model (SuDf), those usng a volumc mechancal model (FFD-D) and those mxng the two (MEF). Lke the geometrc approaches, ths secton gathers together a bref revew of some of the methods proposed before 999, thus enablng a better ntroducton to the new mprovements and/or applcatons that have been proposed durng the last years. For complementary nformaton, the reader should refer to Deformable B-Splne and NURBS: The ntroducton of the deformable B-Splne by [48] has ntalzed ths concept. [49] have then taken nto account lnear constrants whereas [50] have extended ths concept to herarchcal trmmed surfaces. A deformable B-Splne surface s an extenson of the B-Splne model whch ncorporates explctly the tme: m n P( u, v) = N ( u) N ( v) S ( t) = J ( u, v) S( t) (2-) = 0 j= 0 m jn where the J matrx gathers together the products of the bass functons N m and N jn, whereas the S(t) vector gathers together the coordnates of the control ponts S j (t). The concept of deformable B-Splne has then been generalzed to NURBS by [5] who have also extended ther so-called D-NURBS model to the manpulaton of trangular patches [52], subdvson surfaces [53] and mplct surfaces. Qn has also tested hs approach n the context of haptc sculptng [54]. More recently, Zhang and Qn have 38 j

51 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING proposed the concept of herarchcal DNURBS surfaces [55]. Herarchcal D-NURBS can be seen as a collecton of standard D-NURBS fnte elements, organzed herarchcally n a tree structure and subjected to contnuty constrants across the shared boundares of adjacent D-NURBS elements at dfferent levels. Even f the proposed constrants toolbox seems promsng (normal and curvature constrants as well as curve and surface constrants, the resultng of deformed surface suffers from qualty problems (undesred undulatons appear around the deformaton area). Moreover, ths method s not adapted to the general case of trmmed patches. However, the possblty of applyng external forces onto the geometrc model can be consdered as a frst step n the defnton of hgher level operators conveyng semantc nformaton, e.g. push and pull operators. Formulaton of the problem: by applyng the Lagrangan formulaton to the deformable B-Splne surfaces, a second order non-lnear equaton s obtaned: M S& + D S& + K S = F p + G p & (2-2) where F p are the forces obtaned by the prncple of vrtual works gven by a dstrbuton of forces F(u, v, t) on the surface, and G p are the nerta forces that can eventually be assocated. The mechancal model often corresponds to the mnmzaton of the energy. The varous matrces are then defned by: M = D = K = t µ ( u, v) J J dudv t γ ( u, v) J J dudv t t t [ α J u J v + α 22 J u J v + β J uu J uu t t + β2 J uv J uv + β 22 J vv J vv ] dudv (2-3) 39

52 CHAPTER 2. STATE OF THE ART where the functons µ(u, v), γ(u, v), α j (u, v) and β j (u, v), generally contnuous, are the control parameters of the physcal model. They can produce several solutons satsfyng the same set of constrants. The specfcaton of these parameters can be dffcult and some smplfcatons can be requred, e.g. the control functons µ, γ, α and β are constant. Ths type of mnmzaton, or a smplfed verson, has also been used n [56, 57]. In ther herarchcal formulaton of the B-Splne deformaton, Xe and Farn use the mnmzaton of the thn plate energy to solve the often underconstraned system of equatons. Hu et al. have proposed a method whch uses the mnmzaton of the thn plate energy to modfy the poston of the control ponts together wth ther as- assocated weghts. Unfortunately, ther applcatons are restrcted to the deformaton of a sngle untrmmed patch, whch does not ft the engneerng desgn requrements. The work of Zhang et al. s also nterestng snce t enables stretchng of panels whle preservng the shape of some parts of the surface, and partcularly the shape of the holes as well as the contnuty between the varous connected patches (Fgure 2-9). (a) (b) Fgure 2-9: Metamorphoss between two planar polygonal curves usng D-NURBS nterpolatng surface 40

53 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Resoluton of the problem: n general, the equatons (2-3) cannot be solved analytcally and Fnte Element Methods (FEM) has to be used. Each patch can be seen as a specfc fnte element where the postons of the control ponts correspond to the degree of freedom of the element. For a surface composed of several trmmed patches, the dfferent matrces have to be assembled. In case of NURBS surfaces, the computaton of these matrces has to be performed at each step of the resoluton process. In FEM, object, s dvded nto elements joned at dscrete node ponts. A functon that solves the equlbrum equaton s found for each element. The soluton s subject to constrants at the node ponts and the element boundares so that contnuty between the elements s acheved. Unlke mass-sprng methods, where the equlbrum equaton s dscretzed and solved at fnte ponts, n FEM, the system s dscretzed by representng the desred functon wthn each element as a fnte sum of element-specfc nterpolaton, or shape, functons. In the case when the desred functon s a scalar functon Φ ( x, y, z), the value of Φ at the pont (x, y, z) s approxmated by: Φ( x, y, z) h ( x, y, z) Φ (2-4) where the h are the nterpolaton functons for the element contanng (x; y; z), and the Φ are the values of Φ ( x, y, z) at the element's node ponts. Solvng the equlbrum equaton becomes a matter of determnng the fnte set of node values Φ that mnmze the total potental energy n the body. 4

54 CHAPTER 2. STATE OF THE ART Concluson As prevously stated, the geometrc approaches often requre a polyhedral model as nput geometry. In the case of parametrc surfaces, some dscretzaton and reconstructon steps would be necessary. The man drawback of the physcs-based approaches comes from the computaton tme that can ncrease sgnfcantly when manpulatng surfaces composed of lots of patches. Moreover, the mechancal parameters can be dffcult to set up. The approach usng the FDM seems to be the best compromse between the preservaton of the geometry, the speed of the deformaton process and the easness of the manpulatons I further propose a new deformaton method whch mproves the FDM by extended constrants (geometrc and parametrc constrants n 3D) Based on the FDM and applcable to every type of geometrc model defned by control vertces (polylnes and meshes, parametrc curves and surfaces). The new formulaton enables the smultaneous deformaton by user s sculptng operatons. The defnton of hgher level operators has been consdered n the thrd part. They should enable shape-orented manpulatons of the geometrc models. 42

55 Chapter 3 Sketch-based Modelng Paradgm Durng the last two decades, sketch based free-drawng nterfaces, combnng the ease of freehand drawng wth the advantages of computer processng, has been wdely adopted for creatve desgnng work. Moreover, the adopton of Splne-based free-form surfaces has ntroduced hgher complexty n the process of creaton and manpulaton of shapes whlst provdng greater freedom for desgners. Therefore, the trade-off between the degree of the freedom for hand sketchng and the accuracy of the understandng of user s orgnal ntenton s stll a crtcal topc for computer-aded product stylng. Recent years experence suggests that a great mprovement to desgn such tools can be ntroduced by mprovng ther level of ntellgence ; t should be able to understand the desgner s behavor. That s, they should be able to comprehend the users actons, the way they dentfy a shape, the way they nteract wth the drawng tools durng the desgn process. Fnally such a system should present the nformaton consequently provded to the user n a flexble, effcent and supportve manner. The aforementoned requrements have fuelled recent years research on sketch- 43

56 CHAPTER 3. SKETCH-BASED MODELING PARADIGM based modelng systems. These allow the user to quckly create 3D models by smple freehand strokes rather than by typng n parameters. The systems developed range from those pursung a constrant-based or feature-based approach. However, the majorty of these systems have been desgned as further extenson of classcal CAD tools. They are stll lmted n tedous geometrc parametrc presentaton and manpulaton. These cannot be used when nformaton at a hgher, semantc level s to be decoded and manpulated. Furthermore these systems propose nteracton metaphors far from the desgner s tradtonal approach, typcally featurng a top down and stepwse refnement process. We try to brdge ths gap by proposng a formal theory whch models the process of comprehenson of the user s sketches. Such methodology, appled to conceptual desgnng, s able to support the stepwse process of sketchng by contnuously nterpretng the flow of data and constrants generated by the desgner s actons. Further the approach developed s capable to dynamcally adapt to the dfferent users styles by constantly modelng ther personal behavours. Especally we further proposed free form surface manpulaton structures, where we descrbe how we mplement hgher lever shape control by sequent sketched B-splnes, whlst the semantc operators, grouped by seres of basc geometrc constrants, provde desgners more ntutve shape manpulaton concepts. Ths chapter s organzed as follows: we frst present our general archtecture for sketch-based modelng systems (SMBS) that ntegrates the nterface module, the user adaptaton module and the renderng module. Then we wll detal the developed nteracton algorthm. Ths s based on sketchng and features stroke recognton, dynamc shape modelng and on mult-user adaptaton. Fnally we detal free form splne-based surface manpulaton archtec- 44

57 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING ture, where the sequent sketched splnes combned wth optmsed deformaton algorthm mplement the ntutve surface control [34]. 3. Sketch-based 3D Modelng Paradgm We consder the process of sketchng as the nformaton flow from/to the desgner s bran. For ths reason the sketch-based system developed had to be able to show the evoluton of the correspondng desgnng behavour by provdng ntellgent reasonng of the user s nput. As llustrated n Fgure 3-, the proposed archtecture for such sketch-based modelng system conssts of an nterface module, a user adaptaton module and renderng module. Fgure 3-: The proposed archtecture for a sketch-based modelng system. 45

58 CHAPTER 3. SKETCH-BASED MODELING PARADIGM The nterface module provdes the nteractve behavour and t supports stroke recognton. Snce usually the raw stroke nput nformaton s unclear, the module plays a key role for the effectve extracton of geometrc features and constrants. At ths stage we analyze the nput data, represented as pont and tme stamp nformaton, and we obtan the speed, length, vertex sets, other object attrbutes so on. Then, based on the evaluaton of these features and correspondng fuzzy sets, we assess the shape and save t wthn a shapelst. As llustrated n the class dagram of Fgure 3-2 whch shows the man components features, as a result the system s capable to generate a number of geometrc prmtves as t s shown n Fgure 3-3, whch are sent, together wth other data such as topologcal nformaton, tme sequences etc., to the user adaptaton module for further processng. One stroke Recognzer Gesture Input Pont() Tme() Speed () Length () BoundngBox() LargestTrangle() Convex Hull() -EnclosedRec() Evaluate() Feature() Fuzzyset() ShapeLst() Lne -Lengt -Vertex Getname();Gettype();Pregesture() Ellpse -Center -Vertex Trangle -Length -Vertex Fgure 3-2: Class dagram for the sngle stroke recognton The user adaptaton module accepts such basc nformaton (e.g. geometrc and topologcal constrants, basc geometrc enttes and tme sequences) and t processes t through a stepwse refnement process. To do so the module 46

59 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING constantly extracts attrbutes from the latest sketches arrvng from the nterface module through a seres of real-tme operatons. Ths data s then transformed accordng to the constrant solvng model adopted. Ths s based on the characterstc features and behavours typcal of the conceptual stage. The user adaptaton module, whch represents the centre of the whole system, combnes adaptve reasonng wth dynamc user modelng n order to deal wth the uncertanty typcal of hand drawn sketches. As a result, the module, whch takes nto account the dfferences between users drawng styles, decodes the correspondng desgn behavour and t nserts ts dfferent possble nterpretatons nto a ranked probablty lst. Parallel Intersect Vertcal Dsjont Fgure 3-3: Sngle stroke recognton. The black strokes are user s free drawng, and our sngle stroke recognton system 47

60 CHAPTER 3. SKETCH-BASED MODELING PARADIGM automatcally generates basc 2D objects such as lne, crcle, ellpse, B-splne trangle and square so on (red shapes) together wth the topologcal nformaton between the sequent strokes. Fnally, the renderng model performs the postprocessng and vsualzaton process allowng the use of talored representatons such as non-photorealstc renderng or alke. 3.. Automatc Sketches Identfcaton In order to deal wth the unclarty typcal of the sketched nput n the ntal stages of the desgnng, we have modeled the data flow whch characterzes the early desgn process. The process of recognton, whch s performed at the nterface module level, s made of a stroke and shape recognton sub-processes. When each stroke s processed at ths frst stage, the system automatcally performs the constrant reasonng and matchng process accordng to the dfferent characterstcs typcal of each user s drawng style. The adaptve constrant reasonng process takes places at two levels: ether by the stoke recognton process (for sngle-stroke dentfcaton) or by the combnaton of stoke recognton process and user adaptaton module (for more complex shape understandng). The latter s responsble to compare the nformaton contaned n the relevant model of the user wth the nformaton contaned n a database of geometrc templates organzed by parameters and constrants. Durng ths process the stroke represents the base unt of the response mechansm. When the pen s lfted the system automatcally reacts by extractng the relevant features and, f requred, by performng the adaptve reasonng. As a result of ths frst process a graphc object s produced. 48

61 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING The detals of ths process are llustrated n Table 3- where the nformaton on each raw stroke nput s represented by a 4-tuple G (L, S, P, V ) where L s the length of the stroke, S s the speed, P s the pont set extracted, V s the vertex set extracted. We also assume that T represents a graphc object whle C s a constrant. From Table t s possble to see how, durng STEP 2, the basc geometrc object T whch satsfes a seres of specfc constrants s retreved through the stroke recognton process. If the user s sketch s not a sngle stroke (see STEP 4), the constrant solvng module s called to analyze the hstory of the prevous strokes and to perform the constrant matchng. As a result the T m, whch represents the most sutable graphc object, s selected and used for hgher level analyss. Table 3-: the algorthm based on adaptve reasonng STEP: Raw stroke nformaton G (L, S, P, V ), STEP2: Stroke recognton: G (L, S, P, V ) T (C j ) STEP3: If (Sngle-stroke) then GOTO END. Else GOTO STEP4 STEP4: Composte constrant reasonng: C m (T ΛT - T ) T m (T j (C k ), T n ( C l ), C m ) STEP5: Check T m whether t exsts n the pre-defned model database: If (!exsted ) then INSERT (T m ) Else f (Contnue) then GOTO STEP Else GOTO END. END: Results confrmated by user s feedback and vsualzaton. 49

62 CHAPTER 3. SKETCH-BASED MODELING PARADIGM (a) The prsm s generated when a trangle s ntersected wth a lne wthn the predefned crcle constrant (b) The cylnder s formed when a lne stoke s vertcal and outsde a crcle pre-stroke (c) The cone s constructed only f the lne stroke s nsde and vertcal to a crcle stroke. 50

63 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING (d) Once the sequent three strokes satsfy the predefned spatal topologcal and sequence constrants, a polygon volume s shown. (e) The B-splne s constructed by recognzng the actons Extruson and Sknnng based on analyzng the spatal relatonshp between two splne strokes Fgure 3-4: Stroke recognton stage; the geometrc feature s extracted after each stroke. The spatal nformaton and topologcal nformaton s obtaned after each par of strokes; based on the matchng between the basc geometrc features and topologcal nformaton, basc 3d models are constructed. 5

64 CHAPTER 3. SKETCH-BASED MODELING PARADIGM 3..2 Intellgent Stroke Interpretaton System It s known that desgners tend to draw shapes through several prmtve sub-shapes. Lkewse they tend to defne a prmtve shape ether by a sngle stroke or by several consecutve strokes. In our recognton algorthm, we frst dscover latent prmtve shapes among user strokes. Then we recognze and regularze them and, fnally, we show the regularzed drawng on the screen. After beng recognzed and regularzed, the prmtve shapes whch belong to the same graphc object are grouped together. Eventually they are segmented and combned to form an object skeleton. In our system the geometrc enttes are postoned usng parameters and constrants rather than usng a specfc coordnates system. Each tme we extract the features parameters from the free-form drawng and then we match them on the bass of a probablty rank. The presence of a hgh number of shapes ncreases the dffculty of sketch understandng snce t can be dffcult for the system even to assess correctly whether or not a shape s completely drawn. Lkewse t s essental to make the system capable of automatcally re-organzng the relevant features when a synchronous edtng operaton takes place. The answer to these ssues s provded by the adaptve decson-makng system whch s descrbed n the followng secton. 52

65 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING 3..3 Adaptve Decson-makng for Shape Understandng The process of shape understandng s grounded upon an ad hoc nternal model whch adopts a decson tree to arrange and organze parameters and constrants. As llustrated n Fgure 3-5 the tree contans basc constrants (geometry prmtves, geometrc relatonshps, algebrac relatonshps) as well as hgh-level features whch are composed of basc constrants used for descrbng the operaton behavour. All the features and parameters are arranged startng from the low-level constrant layer to a hgh-level feature layer. Sandwched between these two levels, the constrants are cross-checked and shared, untl eventually the semantc descrptons are attached. Ths s a reversed tree structure where the top level contans the leaf nodes whlst the bottom level holds the root. Leaf nodes represent a seres of basc geometrc prmtves and constrants. Furthermore, n order to better capture the user s sketchng commands, we have adopted an adaptve method whch adapts the decson tree accordng to dynamcally upgraded user models. Specfcally, we set the attrbute of beng a possble shape to each branch node whlst the root represents the fnal geometrc object whch s reached only when the constrant branches are satsfed. The user tree s created to match constrants wth the pre-defned decson tree of database. As a result f we fnd that the model s not present n the pre-defned decson tree wthn the database, we place the object nto the relevant tree as a new node. Then the decson tree wthn the database s synchronously updated. 53

66 CHAPTER 3. SKETCH-BASED MODELING PARADIGM Fgure 3-5: The structure of the user adaptaton module. An example of ths process, llustrated n Fgure 3-6, wll help better llustrate the process. In the llustraton the ndexes from to refer to dfferent constrants. Specfcally we suppose that constrans C 7, C 8, C 9, C 0 are used to obtan general geometrcal relatonshps between graphcal objects, whlst C, C 2, C 3, C 4, C 5, C 6 are used to recognze the geometrc prmtves. Let T be a geometrc object set, whch ncludes all the nodes n the graph. The raw stroke nput nformaton s represented as a 4-tuple G (L, S, P, V ), where L s the length, S s the speed, P the pont set and V the vertex set. For each sngle stroke a number of specfc features are extracted and the relevant constrants are analyzed. As a result of ths process a number of regularzed geometrc prmtves are produced whose relatve logc representatons are lsted below: 54

67 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING C 0 (L 0, S 0, P 0, V 0 ) cubc B-splne; C (L, S, P, V ) crcle; C 2 (L 2, S 2, P 2, V 2 ) trangle; C 3 (L 3, S 3, P 3, V 3 ) lne; C 4 (L 4, S 4, P 4, V 4 ) square; C 5 (L 5, S 5, P 5, V 5 ) rectangle; C 6 (L 6, S 6, P 6, V 6 ) damond; C 7 (X, Y) =(X s overlap to Y) (X s precede Y) (X T; Y T); C 8 (X, Y) =(X s parallel to Y) (X s precede Y) (X T; Y T); C 9 (X, Y) =(X s vertcal to Y) (X s precede Y) (X T; Y T); C 0 (X, Y) =(X s touched to Y) (X s precede Y) (X T; Y T); C (C 8, C 9 ) =( C 8 T m ) ( C 9 T n ) ( T m s nsde T n ); We now assume that a decson tree (see Fgure 3-6a) s already defned n the database. Each node represents an object and n partcular each leaf node (top-level) s a geometrc prmtve, whlst each branch node s a possble shape (where the ndex refers to the relevant constrant). For nstance n the scene T 9 (Fgure 3-6a, second level on the left), where a trangle s adjacent to a crcle, the constrants such as C, C 2, C 0 must be satsfed. Ths condton s represented wth the notaton T 9 (T 2 (C 2 ), T (C ), C 0 ). Ths way each node can be represented as a mult-tuple T (T p (C j ), T q (C k ), C n, and, p, q, j k, n, nteger). When the new object T 3 s added, a dynamc user model s created. Frst, based on sketched nput, we extract a seres of features n tme sequence and then, accordng to 55

68 CHAPTER 3. SKETCH-BASED MODELING PARADIGM the pre-defned constrants, we create the dynamc tree (see Fgure 3-6b) and we produce new object nformaton T 3, whch can be represented as T 3 (T 9, T, C ). Eventually, the new object features are used to update the orgnal tree structure (see Fgure 3-6c). (a) Before the new object s nserted (pre-defned decson-tree structure) 56

69 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING (b) Dynamc user tree for new object T 3 (e) The new object T 3 s nserted n database Fgure 3-6: The process of adaptve reasonng In order to optmze the whole reasonng process, we defne the nodes n a pre-defned database as chan structures (see Table 3-2). Each node s represented wth two lnk felds (n Table 3-2 referred to as Pre-Node and Curr- Node) where the frst ponts to the prevous stroke node whle the second ponts to the current node. Snce the value of any lnk refers to a node s ndex, t s possble to fnd the nformaton of a certan node by followng the correspondng ndex. Fnally the Constrant feld n Table 3-2 defnes the new node s condton. 57

70 CHAPTER 3. SKETCH-BASED MODELING PARADIGM Table 3-2: the structure of the predefned decsontree storage structure. Index Name Pre-Node Curr- Node Constrant T NULL NULL C 2 T 2 NULL NULL C 2 3 T 3 NULL NULL C 3 4 T 4 NULL NULL C 4 5 T 5 NULL NULL C 5 6 T 6 NULL NULL C 6 7 T C 8 8 T C 7 9 T 9 2 C 0 0 T 0 9 C 9 T 2 5 C 8 2 T C 7 The specfc user modelng and updatng process can be descrbed accordng to the followng pseudo-code excerpt: PROCEDURE: USER MODELING Begn P=NULL, /*Intalze a node ponter*/ Stop=0, FndFlag=0 Whle (! Stop) Q= head /* head of pre-defned lnk table*/ Accept (T) /* T: the current node ponter*/ Constrant (P, T) C n If (P Pre-Node == NULL) then P Pre-Node = T, P Constrant= C n Else If (P Curr-Node == NULL) then P Curr- Node = T Whle (Q! =NULL) 58

71 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING If (Q= = P) then P Pre-Node = Q /*set node to prev node*/ P Curr-Node = NULL FndFlag= End whle If (!FndFlag) then GOTO INSERT(P)/*update table*/ If (Stop) then END End whle END PROCEDURE PROCEDURE: INSERT (P) Begn Q=End /* the end of the pre-defned lnk table*/ Q=ALLOC (new node); Q=P End END PROCEDURE If we assume that the number of nodes n a pre-defned database s equal to n, then the modelng algorthm and nserton procedure are bound by a complexty of O (n). From Table 3-2 t s also possble to easly assess the partal structural smlartes between users drawng. Ths s done by trackng all the pre-node lnks. If two nodes have the same value then the correspondng nodes wll be canddates. For nstance, assumng that T 2 s the frst stroke, f the system can not extract from the second stroke the exact features then, by tracng the Table 2, t s possble to fnd that T 8, T 9, T have the same pre-node lnk value 2. As a result of ths, T 8, T 9, T would be offered to the user for selecton. The whole process would be stll bound to O (n). In general, ths adaptve reasonng method effectvely solves the stepwse refnement desgnng process. In fact the approach proposed s truly ncremental snce each stroke lnks to the nformaton about the prevous one and t depends to the constrants defned up to that pont. Further- 59

72 CHAPTER 3. SKETCH-BASED MODELING PARADIGM more the database can be extended through the nserton of new constrants and new nodes. The dynamc user tree model, whch tracks the nformaton on the strokes, focuses on the extracton of features and matchng of constrants, and then t effcently hdes the dversty between dfferent users nput style. When an object s fnshed the tree model s dynamcally stored and the tree s replaced by the one correspondng to next free-form drawng. In the followng secton, we further propose the free form surface manpulaton mechansm, where we mprove the parametrc geometrc control to hgher level shape manpulaton. Sets of geometrc constrants are grouped to mplement functonal operatons, n order to provde the user a more natural and nteractve surface creaton and modfcaton process. 3.2 Free Form Surface Manpulatons Desgners sketchng s an actvty usually drven by sgnfcant curves and t s performed n 2D. However, the geometrc nformaton extracted from a 2D sketch forms the nput for a 3D sketch, where the desgner ought to fnd tools to carry on the adjustment of the 3D surface generated to hs/her ntent as t s n hs/her mnd. Our proposal s to ncorporate and structure the splne constrants chosen by the desgners and to develop the functon necessary to let them adjust the 3D shape by relaxng some shape constrants for a user-frendly nteracton way. In ths way, dgtal surfaces can be drectly controlled by sketched splnes, makng creaton and manpulaton 60

73 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Hgh 2 3 Shape Geometrc Constrants Mathematcal Model Free Form Surface Manpulaton Low Hgh Level Manpulaton Tools Basc Manpulaton Tools Drect Manpulaton Tools Fgure 3-7:Shape descrpton levels and manpulaton tools In fact, effectve surface manpulaton tools based on the specfc curves would certanly help desgners by provdng functonal and aesthetcal detals.therefore, the surface behavors whch are mportant to assocate to the splne are not only contnuty and tangency condtons, but also related to the shape tself. The brdge between the basc geometrc surface manpulaton technque and hgher level feature and semantc surface manpulaton s descrbed n Fgure 3-8. The frst level drectly corresponds to the mathematcal model of the free form surface defned by many geometrc parameters (control ponts, weghts, knot sequences, contnuty condtons and so on) whose nstantaton requres lots of sklls and tme. At ths level, the noton of shape s closely assocated to the noton of control ponts whch does not ft well wth the way stylsts and desgners thnk. Snce the surface mathematcal models are qute stable now, nothng new has been proposed at ths level. The two next levels defne the 6

74 CHAPTER 3. SKETCH-BASED MODELING PARADIGM framework of the approach proposed n ths document. At the second level, a shape s defned through a restrcted set of geometrc constrants (ponts, curves and so on) the surface has to nterpolate. In the proposed approach, the use of an adapted deformaton technque reduces the tme of requred for manpulaton but the constrants are handled separately wthout knowng that they concur to the defnton of the same shape. The thrd level descrbes drectly a free form shape as a hgh level deformaton entty defned by a restrcted set of ntutve control parameters D Splne-based Surface Sketchng Our proposed method supports desgner free hand splnes sketchng. The free form splnes are automatcally recognzed and reconstructed as cubc B-splnes. A B-splne surface s then constructed by consderng the user s drawng style, where the constrants among the sequence of B-splne curves are defned wth respect to the user s ntenton. Furthermore, n order to provde the user s stepwse refnement operaton, the pont and curve constrants can be freely attached to the parent surface whlst keepng the specfc tangency, contnuty and curvature constrants. A semantc operaton lnks the features to the geometrc representaton of the curve and surface. A deformaton engne based on a curve-based deformaton method has been mplemented, tryng to be as flexble as possble, and t wll be detaled n chapter 6. Obvously, the shaded representaton of a shape n splne sketches s not defnng explctly the correspondng 3D surface. Hence, the deformaton mechansm should pro- 62

75 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING vde flexable solutons to the users. Takng nto account the uncertanty of a sketch around the extremtes of lnes, Secton 5.2 presented the tools as ntutve as possble to let users cope wth the uncertantes through free control n 3D space Hgher Level Surface Manpulaton The basc geometrc parameter elements are curves, whch specfy the extent of the deformaton and help defnng the shape of the feature. For each type of constrant splne, the curve s ntally contnuous and then dscretzed to reduce the number of constrants on the surface to a fnte value. Gven the number of ponts and a dstrbuton law, the postons of the sampled ponts are then defned. These are used to specfy the behavor of the deformed surface accordng. It has to be consdered that several control ponts nfluence both the area nsde and outsde the specfed lne. Thus, the localzed modfcaton of the surroundng surface s obtaned by several splne sketches. Ths process s a way to smplfy the task of the desgner rather than requrng long and tedous actons to produce a very accurate freeform surface. Therefore, these sketches, decoded as geometrc constrants, are used to defne the surface behavor. The goal s to produce a soluton close to the desgner s needs as fast as possble. In the followng chapters we wll detal how we construct the B-splne surface by smply splne sketchng and how these splnes are used to drve the shape varaton n a predctable way. 63

76 Chapter 4 Curve Sketchng and Modfcaton In CAD/CAS feld, the stylsts stll prefer to llustrate the mpulsve and creatve mental deas by usng 2D sketched curves. Even f the frst dgtal model s acqured drectly n 3D, curves also have a leadng mportance n the consequent refnng process. Therefore, nowadays more and more researches have focused on the technque for hgh degree freeform curve creaton and ntellgent manpulaton n order to accelerate the desgnng process. So far authors have already developed advanced algorthms rangng from manually geometrc parameters control (knot removal, control pont adjustment and control polygon cuttng) to sophstcated tangent, normal and curvature constrants control. However, they ether requre cumbersome understandng of the math or nvolve n huge tme consumng for the numercal computaton. Although some works have tred tackled sketchng-based curve manpulaton to mprove the user s nteracton durng the desgnng process, they are stll far from beng enough ntellgent and complete for free form curve control. In fact the hgh degree curve control process not only needs the ablty of rapdly 3D modelng, but t also needs ntutve, powerful modfcaton and manpulaton. 64

77 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING 4. 3D Curve Sketchng and Approxmaton Earler works on curves [29, 59] tend to modfy parameters, such as splne control ponts or knot values and sophstcated curvature, tangent constrants, and consequently they affect the curve s shape. Although the representaton of curves matchng the constrants of the objects provdes perfect shape control and useful geometrc propertes, e.g. C2 contnuty where needed and easy computaton of boundng surface or volume, the users usually are requred to have suffcent mathematcal background enablng them to master easly the peculartes of the nteracton technque. Sgnfcant achevements n user nteracton for free curve drawng were made by [60]. The approach s based on the fact that graphc desgners are usually very good at sketchng. Usng graphcs programs, however, much of the desgner s concerns are placed on curve representaton by control ponts, refnements, etc., nstead of creatve work. Baudel proposed a method by whch desgners can freely modfy ther drawngs by pen strokes. One restrcton n ths approach s that t uses off-lne splne curves, whch means that the edted representatons are based on pecewse lnear curves. Only after an explct user request (e.g. by pushng a button) the program generates tangent G contnuous Bezer curves. Therefore ths approach could not be drectly extended to B-splne surface sketchng. Pror to Baudel, Fowler and Bartels [29] found a new way of drectly manpulatng B-splne curves, by usng constraned mnmzaton technques whch solve underdetermned systems of equatons, mnmzng 2-norms of the modfcaton. Ths method s nherently dependent on the structure of knot vectors. 65

78 CHAPTER 4. CURVE SKETCHING AND MODIFICATION Banks and Cohen [6] proposed easy edtng by B- splne curves, cuttng and sketchng control polygons. It stll means addtonal work to edt control ponts, rather than curves. However, ther examples ndcate that edtng the control polygon by strokes s qute ntutve, although t s not as drect as one would lke. Zheng and Chan proposed deformng a curve, locally matchng t wth another curve [62]. The authors used knot removal technques [24] whch ncrease computaton tme, and do not guarantee smooth shape changes n the transton between the orgnal curve and a local modfcaton by the target curve. [7, 63] proposed methods by whch desgners can freely modfy ther drawngs by smple touch-and-replace or projecton technques. They ntroduce auxlary surfaces sculptng that allows for a relable nterpretaton of users pen-strokes n 3D. However the edtng operaton s mplemented by nterpolatng, and t cannot guarantee the smoothness of surfaces. In addton, t lacks of degree of freedom snce t s restrcted by the auxlary surface and t shows the poor vsualzaton for sculptng operatons. In [32] the author further proposed a soluton to the smoothness of transton ntervals for curve local modfcaton. However t s not robust enough to guarantee the effectvty n the case of loop curves. Indeed when the user s freehand sketchng, the result s not a basc B-splne or Bezer curve, nstead t s arbtrary splne, and t becomes more mportant to ft the fuzzy data nto geometrc curve. In [65] they propose real-tme nteracton capable to reduce the data ncrementally by usng knot removal method. In [66] an ad-hoc curve splttng method s adopted to approxmate B-splne wthn a VR envronment, but t nvolves n large computaton and the complex pa- 66

79 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING rameters are not sutable for supportng the ntutve postprocessng n conceptual desgnng stage. We present an nnovatve method, whch s capable of automatcally approxmate mprecse free-drawng curves to adaptve cubc B splnes base on a dynamc-threshold samplng mechansm. We further propose a stroke-replacement local modfcaton algorthm whch supports fully free form curve modfcaton. Furthermore we mprove the smoothness of transton ntervals through the defnton dynamc scalng factors and the constraned tangent features. Our approach s dvded n two stages: mprecse freeform 3D curve approxmaton and post-refnement. The adopton of an optmal samplng mechansm effectvely mproves the approxmaton result. Post-refnement wll be mplemented by ntutve stoke replacement process whch s so-called oversketchng. We present a novel soluton to smooth n transton ntervals by usng the dynamc scale factor and the constraned length and tangent features Our system does not put any partcular constrant on the way the use s drawng sketches. Users can use any pen-lke nput devce or even a mouse to freely draw n a vrtual 3D envronment. Each stroke s recoded as a seres of coordnates generated by the pen between a sngle par of mousedown and mouse-up events. The stroke s assocated wth a tme stamp to get correspondng speed feature whch s mportant for extractng geometrc features. Once the pen stroke s fnshed, t s automatcally analyzed and adapted nto B-splne. Three steps are thus performed as followng: 3D sketchng nput: n a predefned perspectve vew, the user s freehand sketchng s automatcally translated nto a seres of 3D vector descrptons. 67

80 CHAPTER 4. CURVE SKETCHING AND MODIFICATION Samplng: the corner detecton algorthm based on the analyss of speed, curvature and drecton features splts the curve nto several parts. We then adopt a dynamc threshold whch combnes arc length and speed features to obtan optmal samplng ponts. Approxmaton: the algorthm of approxmaton translates all the samplng data nto an adaptve B- splne. 4.. Dynamc Threshold-based Samplng Method Durng the drawng of freehand 3D curve, the corner ponts are very mportant nformaton for keepng the curve shape. Here we adopt two parameters to detect corners: drecton and curvature. When sketchng free-form curve, the drecton and curvature for each pont are automatcally computed (see formula 4-): where d n and C n represent the drecton and curvature of the n-th stroke pont respectvely, k s a small nteger defnng the neghbourhood sze around the n-th pont, and D (n-k, n+k) stands for the path length between the (n-k)-th and (n+k)-th stroke ponts. We set k to 2 emprcally as a trade-off between the suppresson of nose and the senstvty of vertex detecton. The ϕ shfts ts angle parameter from π to π see Fgure 4-. yn+ yn d n = arctan( ) x x n+ n 68

81 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING C n = n+ k = n k ϕ( d + d ) D( n k, n + k) (4-) Drecton graph 2,5 0,5 0-0,5 - -, Curvature graph 0,45 0,4 0,35 0,3 0,25 0,2 0,5 0, 0, Fgure 4-: An example curve wth ts drecton graph and curvature graph 69

82 CHAPTER 4. CURVE SKETCHING AND MODIFICATION In Fgure 4- we can fnd the drecton graph where the π drecton angle s shfted from to π. However n the 2 curvature graph we can easly judge the corner pont regon that are necessary to preserve the shape. In fact the freehand sketch s an unsteady movement (see Fgure 4-2-upper). When the speed s low, t s easy to obtan much more samplng ponts and to understand the user s ntenson. However, ths wll certanly produce redundant ponts, whch thus ncrease the computaton effort for the next-step B-splne approxmaton. On the contrary, when the speed s hgh t s good to get smooth shapes but the samplng ponts are few and t would be dffcult for post-processng too. Therefore, we combne two cases to adaptvely obtan samplng ponts. In a smlar manner to the approach proposed n [59], we dscrmnate them by usng dfferent samplng threshold. Amng at the usual low speed n corner regon and at hgh speed n other parts, we combne the arc-length and speed features to get optmal samplng ponts, as defned by: Savg m = round( ) S (4-2) 2 Savg m = 7 S (4-3) where m s the adaptve length threshold for samplng, and m s the dstance from (-)-th pont to -th pont. S avg s the average sketchng speed whch s defned by the arclength and tme stamp, and S s the speed at -th pont (the purpose of m s to avod pckng too close ponts). Consderng the fast speed n non-corner regons we employ formula 4-3 to control the dstance between ponts; on the other hand 70

83 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING we use formula 4-2 to get samplng ponts n corner regons where the speed s slow. Fgure 4-2: (upper) Dfferent denstes of samples only correspond to the speed feature. (Lower) The approxmated B-splne curve wthout usng our optmsed samplng mechansm (the red lne s the freeform sketchng. The yellow curve s an approxmated B-splne, and the yellow balls denote the control ponts). It s evdent that there are more samplng ponts n the corner regons when we only consder the drawng speed feature. Ths thus produces redundant constrants of control ponts (yellow balls) durng the approxmaton process (see Fgure 4-2-lower), whch consequently nvolves large com- 7

84 CHAPTER 4. CURVE SKETCHING AND MODIFICATION putatons to generate approxmated B-splne curves. In fact the deal method should acheve better contnuty wth low tme consumng. Therefore, we further balance the dstrbuton of all the samplng ponts through our dynamc threshold mechansm. The results are compared n Fgure 4-3, they demonstrate that the resultng B-splne curve not only keeps C 2 contnuty but also produces optmzed number of control ponts wth the use of our optmsed samplng mechansm control. In Fgure4-3 (a) the frst sketch s approxmated nto a B-splne by more than 46 control ponts (DOF). In Fgure4-3 (b) the smlar frst B-splne s reconstructed by only 8 control ponts wth the use of our dynamcal threshold method control. (a) Wthout usng our optmzed samplng mechansm. 72

85 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING (b) After usng our dynamc threshold samplng mechansm to optmze the samplng ponts. Fgure 4-3: Approxmated B-splne based on dfferent samplng mechansm. The whte lnes trace the free hand sketchng, and the red lne denotes the approxmated cubc B-splne, red balls represent the seres of control ponts generatng by Carl de Boor's least squares approxmaton method. (a) The samplng ponts are obtaned by only consderng the speed feature of hand sketchng, snce there are more samplng ponts on the corner regons, we get more control ponts too (red balls). (b) By usng our dynamc threshold samplng method, we can get optmzed samplng ponts and keep curve hgh contnuty wth sutable number control ponts Adaptve B-splne Approxmaton The fnal result of the complete algorthm s to provde a set of mathematcal representatons of the curves keepng them smple for further modfcaton. Ths s usually done by n- 73

86 CHAPTER 4. CURVE SKETCHING AND MODIFICATION terpolatng all the data ponts or by approxmatng the data to get desrable curves. It s well-known that hand-sketchng s susceptble to unsteady and mprecse hand movements of the desgners. Interpolatng all the data generally suffers from excessve undulatons, especally when the number of data ponts ncreases or mantans hgh contnuty, ths results n sharp wggles. Therefore, we adopt the Carl de Boor's least square approxmaton method to re-parameterze sampled ponts, where the B-splne s specfed by ts non-decreasng knots sequence and sampled control ponts sequence. (See formula 4-4, 4-5, and 4-6). n C (u) = N =0 74, p u) P ( (4-4) ; f t pupt+ ; and t pt + N,0 ( u) = 0 ; otherwse u t t+ p+ u N, p( u) = N, ( u) N, p ( u) t t p p t+ p+ t+ (4-5) where P represents the control ponts, n s the number of control ponts. Here we assume the degree of B-splne p as 3, and N,p are cubc B-splne basc functons over a knot vector t, ts value ranges from 0.0 to.0. Suppose we are gven m+ data ponts D 0, D 2,..., D m, and we wsh to fnd a B-splne curve that can follow the shape of the data polygon wthout actually contanng the data ponts. To do so, we need two more nput: the number of control ponts (.e., n+) and a degree (p), where m > n

87 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING 75 >= p >= must hold. Thus, approxmaton s more flexble than nterpolaton, because we not only select a degree but also the number of control ponts. Because parameter t k corresponds to data pont D k, the dstance between D k and the correspondng pont of t k on the curve s D k - C(t k ). Snce ths dstance conssts of a square root whch s not easy to handle, we have chosen to use the squared dstance D k - C(t k ) 2. Hence, the sum of all squared error dstances s: = = 2 ) ( ),..., ( m k k n t C D P P f (4-6) Our goal s, of course, to fnd those control ponts P,..., P n- such that the functon f() s mnmzed. Thus, the approxmaton s done n the sense of least square. Frst, let us rewrte D k - C(t k ) nto a dfferent form: [ ] = = = + + =,, 0 0,,, 0 0, ) ) ( ( ) ) ( ) ( ( ) ( ) ) ( ( ) ( ) ( n k p m k p n k p k n m k p n k p k p k k k P t N D t N D t N D D t N P t N D t N D C t D (4-7) In the above, D 0, D k and D m are gven, and N 0,p (t k ) and N n,p (t k ) can be obtaned by evaluatng N 0,p (u) and N n,p (u) at t k. For convenence, we defne a new vector Q k as: m k p n k p k k D t N D t N D Q ) ( ) (, 0, 0 = (4-8) Then, the sum-of-square functon f ( ) can be wrtten as follows:

88 CHAPTER 4. CURVE SKETCHING AND MODIFICATION 76 = = = 2, ) ) ( ( ),..., ( m k n k p k n P t N Q P P f (4-9) Next, we can fnd out what the squared error dstance looks lke. Recall the dentty x. x = x 2. Ths means the nner product of vector x wth tself gves the squared length of x. Thus, the error square term can be rewrtten as: ) ) ( ( ) ) ( ( ) ) ( 2( )) ) ( ( ( )) ) ( ( ( ) ) ( (,,,,, 2, = = = = = = + = = n k p n k p k n k p k k n k p k n k p k n k p k P t N P t N Q P t N Q Q P t N Q P t N Q P t N Q (4-0) Then, functon f ( ) becomes + = = = = =,,, ) ) ( ( ) ) ( ( ) ) ( 2( ),..., ( n k p n k p k m k n k p k k n P t N P t N Q P t N Q Q P P f (4-) How do we mnmze ths functon? Functon f ( ) s actually an ellptc parabolc n varables P,..., P n-. Therefore, we can dfferentate f () wth respect to each p g and fnd the common zeros of these partal dervatves. These zeros are the values at whch functon f () reaches ts mnmum. In

89 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING computng the dervatve wth respect to P g, note that all Q k and N,p (t k ) are constants (.e., no P k nvolved) and ther partal dervatves wth respect any P g must be zero. Therefore, we have: ( Qk Qk ) = 0 P (4-2) g Consder the second term n the summaton, whch s the sum of N,p (t k )P. Q k. The dervatve of each sub-term s computed as follows: Pg P ( N p ( tk ) P Qk ) = N, p ( tk ) Qk Pg, (4-3) The partal dervatve of P wth respect to P g s nonzero only f = g. Therefore, the partal dervatve of the second term s the followng: Pg n ( N, p ( tk ) P Qk ) = N g, p ( tk ) Qk = (4-4) The dervatve of the thrd term s more complcated, but t s stll smple. The followng uses the multplcaton rule (f. g)' = f'. g + f. g'. 77

90 CHAPTER 4. CURVE SKETCHING AND MODIFICATION 78 ) ) ( ( ) ) ( 2( ) ) ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) ( (,,,,,,,, = = = = = = = = = + = n g k p n k p n k p n g k p n g k p n k p n k p n k p g P P t N P t N P t N P P t N P P t N P t N P t N P t N P (4-5) Snce the partal dervatve of P wth respect to P g s zero f s not equal to g, the partal dervatve of the thrd term n the summaton wth respect to p g s: ) ) ( )( ( 2 ) ) ( ( ) ) ( (,,,, = = = = n k p k p g n k p n k p g P t N t N P t N P t N P (4-6) Combnng these results, the partal dervatve of f () wth respect to p g s: = + =,,, ) ( ) ( 2 ) ( 2 n k p k p g k k p g g P t N t N Q t N P f (4-7) By settng t to zero, we have the followng: k m m k k p g n k p k p g Q t N P t N t N = = = =,,, ) ( ) ( ) ( (4-8)

91 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING 79 Snce we have n- varables, g runs from to n- and there are n- such equatons. Note that these equatons are lnear n the unknown P 's. What s the coeffcent of P? Before we go on, let us defne three matrces P,Q, N: Here, the k-th row of P s the vector P k, the k-th row of Q s the rght-hand sde of the k-th equaton above, and the k-th row of N contans the values of evaluatng N,p (u), N 2,p (u),..., N n-,p (u) at t k. Therefore, f the nput data ponts are s-dmensonal vectors, P, N and Q are (n-) s, (m- ) (n-) and (n-) s matrces, respectvely. = P n P P P K = = = = k k p n m k k k p m k k k p m k Q t N Q t N Q t N Q ) ( ) ( ) (, 2,, K = ) ( )... (... ) ( ) ( )... (... ) ( ) ( )... ( ) (, 2,, 2, 2 2, 2,, 2,, m p n m p m p p n p p p n p p t N t N t N t N t N t N t N t N t N N K (4-9) Now, let s rewrte the g-th lnear equaton nto a dfferent form so that the coeffcent of p can easly be read off: = = = =,,, ) ( )) ( ) ( ( m k k k p g n m k k p k p g Q t N P t N t N (4-20)

92 CHAPTER 4. CURVE SKETCHING AND MODIFICATION Therefore, the coeffcent of P s m k = N g, p ( tk ) N, p ( tk ) (4-2) From matrx N we can see that N g,p (t ), N g,p (t 2 ),..., N g,p (t m- ) s the g-th columns of N, and N,p (t ), N,p (t 2 ),..., N,p (t m- ) s the -th columns of N. Note that the g-th column of N s the g-th row of N's transpose matrx N T, and the coeffcent of P s the "nner" product of the g-th row of N T and the -th column of N. Wth ths observaton, the system of lnear equatons can be rewrtten as: ( N T N ) P = Q (4-22) Snce N and Q are known, solvng ths system of lnear equatons for P gves us the desred control ponts. Then these control ponts P are appled to get cubc B-splne curve. 4.2 Modfcaton based on Oversketchng Due to the stepwse refnement characterstc n conceptual desgnng stage, ntutve and natural modfcaton operaton s the most mportant feature for the fast llustratons of desgners deas. We have consdered the problem of nteractve creaton and refnement. An over-sketchng operator allows users to nteractvely redefne 3D curves through free sculptng. The so called over-sketchng s appled to replace the prevous pen stroke by a new sketched curve (see Fgure 4-80

93 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING 4). For ths, there are generally two modfcaton methods, one s the replacement of control ponts, and another one s the replacement of all samplng ponts. Due to ths we already get optmal control ponts durng the approxmaton procedure, and n order to mprove the smooth and flexble characterstcs of curves, we wll detal the local modfcaton process based on the replacement of control ponts. Durng the modfcaton, the mportant task s to fnd the replacement part n the orgnal curve. We adopt the projecton method to fnd the nearest projected ponts from the begnnng pont and the endng pont on the target curve to the orgnal one (see Fgure 4-4). Where C (u) represents the orgnal curve, and C 2 (u) s the new stroke that would be approxmated as target curve. P s and P e are the projectons of the begnnng pont P s and the endng pont P e, f we drectly replace the part of the curve P s ' Pe ' wth C 2 (u), t would surly produce sharp breaks (see Fgure 4-5). Transton P s C 2(u) P s C (u) P e P e Transton 2 Fgure 4-4: Oversketchng: the yellow curve C 2 (u) wll locally replace the orgnal curve C (u) 8

94 CHAPTER 4. CURVE SKETCHING AND MODIFICATION To avod that, the curve s smoothenng the transton ntervals as descrbed n the next part. Then the new revsed curve wll nclude three parts, C (u)=c (a, a 2, a m ) + C 2 (b,b 2,,b n )+C 2 ( c,c 2, c r ), where C and C 2 are the two parts of the orgnal curve, C 2 s the target curve, a, b, c represent seres of control ponts on the curves respectvely. (a) (b) (c) Fgure 4-5: (a) Before modfcaton. The red curve s the target curve and the blue one s the prevous splne stoke. (b) The curve s generated by only replacng the closest control ponts, and the curve leads to undesred undulant effect (c). Applyng our length and tangent constrants, the resultng curve keeps well smooth feature. 82

95 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING The whole modfcaton process can be llustrated as followng: Calculatng the projectons: as llustrated n Fgure 4-4 we can obtan the projecton ponts P s and P e from P s and P e. Furthermore, we locate the nearest control ponts CT s and CT e on C (u). Replacng: the part curve P s ' Pe ' n orgnal curve C (u) wll be replaced by target curve C 2 (u). Defnng the transton ntervals: we ntroduce a dynamc scale factor and tangent features to adjust the poston of CT s and CT e n order to obtan the optmal transton ntervals. Re-Parameterzng: based on correspondng control ponts we re-parameterze the knots sequence and get a new B-splne curve The Resoluton to Transton Interval To avod the sharp beaks n the curve, we adopt two methods to mprove the smoothness: Dynamc scale factors The constraned length d and tangent angel φ for the defnton of transton ntervals In [42] the author proposed a smlar scale feature to resolve the smoothness of transton parts. However the adopton of an nterpolaton method results n the whole curve sharp undulaton. It s neffectve n the case of loop curves or more complex curves. We further mprove ths flexblty through an approxmaton method, whch s based on a se- 83

96 CHAPTER 4. CURVE SKETCHING AND MODIFICATION res of optmzaton control ponts that we already obtaned durng the recognton process. In ths method the scale s used for weghtng the degree of approxmaton to the target curve, and t s defned as float type where the value ranges from 0 to. When the scale s 0, the orgnal curve wll not be modfed, and f t s set to the new curve wll follow exactly all the control ponts n the target curve. Ths way the revsed curve wll pass through the target curve (see Fgure 4-6). All the projecton dstances from the control ponts n C 2 (u) to C (u) are computed (see Fgure 4-6). We suppose y s the one control pont n C 2 (u), the projecton dstance s d, then we can produce the new control pont y through a scale (see formula 4-23): y = y (-scale) * d (4-23) (a) The target control pont y t s scaled by consderng the projecton dstance d t and t wll thus produce the new control pont y t 84

97 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING (b) Before local modfcaton (c) scale = (d) Scale s vared from.0, 0.5 to 0.2 Fgure 4-6: (a) Scalng the replacement of control ponts. (b) (c) (d) The examples wth dfferent scalng. (c) The dot square ndcates the sharp feature of the transton part. (d) It shows three modfed B-splne curves by varyng the scale factor from.0, 0.5 to 0.2. Even f the scale factor adjusts the degree of approxmaton to the target curve, however when the dstance from P s to CT s or from P e to CT e, s too close, the hard break would stll exst (Fgure 4-6 (d)). So we further ntroduce the constrants of length d and tangent angle φ to obtan sut- 85

98 CHAPTER 4. CURVE SKETCHING AND MODIFICATION able control pont CT s and CT e n C (u). Consequently we can decde the optmal transton regons. M = n = S round(2 S n avg + 0.5) (4-24) M d.5 M (4-25) The length d s the dstance of CT s P s or P e CT e s emprcally defned as a trade-off that ts value should be satsfed wth between M and.5 M (see formula 4-24). M s the mean dstance between dfferent neghbour control ponts n orgnal curve C (u), where S s the speed of -th pont and the S avg s the average speed of curve C (u). We further refne the transton nterval by adoptng the constraned feature φ. φ s used for evaluatng the dfference of tangent angle between vector P s CT s and CT or 86 sct s P e CT e and CT ect e. For nstance, we suppose P s (x 0, y 0 ) s the startng pont of target curve C 2 (u), the correspondng nearest control pont s CT s (x s, y s ) n C (u) and CT s- (x s-, y s- ) s the prevous control pont, as descrbed n formula We can obtan the tangent angle dfference φ : y0 ys y d = arctan( ) s ys x x ; d 2 = arctan( ) ; x x 0 s s s. It 0 < φ = d -d 2 <π /2 (4-26) Durng the whole process of defnng of transton ntervals, we frst use the length threshold d to roughly calculate transton regon. We then adopt the least deference of tangent angle to determne the exact transton ntervals.

99 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING The combnaton between scale factor and the constrants of length and tangent angle provdes the effectve manner to mprove the smoothness feature n the transton ntervals. Furthermore ths method s sutable for fully free form 3d curve sketchng (see Fgure 4-7). (a) (b) (c) (d) Fgure 4-7: The length constraned d (0.8 d.2) s adopted to determne the transtons. (a) The loop curve s used for the target curves for the sequent local modfcaton. (b) The new curve s constructed, where the scale s equal to.0 and t keeps well contnuty n the transton ntervals (n dot squares). (c)(d) They show how the resultng curve s changng wth the 87

100 CHAPTER 4. CURVE SKETCHING AND MODIFICATION dfferent scale (the upper curve s generated by settng scale to, the lower one sets the scale to 0.8) Ths local modfcaton method can be further appled to free-form surfaces where the surface boundary s revsed. The process follows four steps: STEP: free form 3D curves sketchng and automatcally approxmatng nto B-splne curves based on our algorthm. STEP2: usng GeomFll functon to create surfaces where the B-splne curves work as boundary. STEP3: adoptng our local modfcaton approach to revse surface s boundary. STEP4: creatng new surface based on the revsed boundary. A large number of experments has verfed the effectvty of ths ntutve operaton. Consequently, ths further extends the feld of surface modelng by usng the approxmaton free form 3D curve sketchng (see Fgure 4-8). (a) Before modfcaton b) Scale=0.3 88

101 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING (c) Scale=0.8 (d) Scale= Fgure 4-8: Modfyng the boundary of a surface based on our oversketchng method (.2 d.8) 89

102 Chapter 5 Sketch-based Surface Modelng The desgner mght start surface modelng by usng the standard parametrc desgn technque, a so-called ndrect control. The desgners are requred strong knowledge about the underlyng curve and surface representaton, so that they can manpulate seres geometrc parameters, such as the control ponts, knots sequence, dervatves and a sere of curve-surface constrants. On the other hand, the desgners mght use freely curve-lne drawng to llustrate and edt the shapes through what t s so-called drect control. The modelng tools wll then automatcally reconstruct the shape respectng the desgner s ntenton. It s evdent the latter one s preferred by most desgners durng the conceptual desgnng stage. Consequently the sketch-based ntellgent shape modelng systems have been becomng prevalng n CAD/CAS felds. So far there are dfferent research works carred on about sketch-based systems. However, the current free-form shape modelng tools are stll far away from generatng sculpted free-form surface wth desred propertes by few pen-strokes or mouse-clcks. They stll can t effectvely accelerate the desgner s creatve work process to some extent. The creaton of complex freeform surface models s stll a tme-consumng and cumbersome procedure. Here we un- 90

103 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING derlne how we accurately nterpret desgner s ntenton and how we provde themn wth nteractve control. 5. Constrant-based Surface Sketchng As llustrated n Fgure 5-, ths system allows users to drectly sketch subsequent pen-strokes n a 3D envronment (mplemented n Open Inventor). These strokes wll be automatcally approxmated nto cubc B-splnes as t s descrbed n the last chapter. The converson of these curves to B-splne surface depends on a so-called surface nterpreter. We already mplemented robust surface nterpreters to match a varety of user s drawng styles. Especally we present a dynamcal modelng mechansm that provdes desgner nteractve vsual feedback durng both modelng and edtng process by usng the so-called sensor and dragger metaphor. In ths sketchng system, the splne can be adjusted by ether usng 3D dragger or by an over-sketchng operator. Whenever the splne s changed, the sensor wll be trggered, and the correspondng surface then s dynamcally updated. In ths way a seres of mddle resultng models are produced, the desgner thus has more possbltes to obtan desred models. In the next sectons, I wll further descrbe how the pen stroke s nterpreted nto a 3D descrpton, and how these constrant-based B-splne surfaces are effectvely constructed wth respect to the user s ntenton. 9

104 CHAPTER 5. SKETCH-BASED SURFACE MODELING Approxmaton Intal Pen Stroke Cubc B-splne representaton YES Modfcaton Over-Sketchng Addtonal Stroke NO Sequent B-splne curves Surface Interpreters Boundary-based Rotaton Surface Ral-based Rotaton Surface Splne Sculptng Surface YES Updatng surface Sensor NO END Fgure 5-: The flowchart of our 3D splnes sketchng system 5.. 3D Splne Sketchng Ths system supports freehand splnes sketchng. It takes as nput strokes, sketched by usng a common 2D mouse or a tablet. Each stroke here s drawn on a predefned plane, whch mples a paper for the desgner to draw on. Such a plane can be added whenever the user wants to 92

105 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING sketch more new splnes. Meanwhle, each plane can be freely controlled n 3D space by attachng a dragger metaphor (see Fgure 5-2 b) that mplements grabbng, draggng and rotatng operatons. Each pen-stroke results n a set of ordered ponts on the current plane (see Fgure 5-2 c), whch are automatcally translated nto 3D vector descrptons n the predefned perspectve vew. Once the stroke s fnshed, a cubc B-splne curve s approxmated. (a) (b) (c) Fgure 5-2: (a) Sketchng (red curve) on a default plane (shown wth a grd); (b) (c) the sequent splnes are nterpreted nto B- splne surface, each plane where the splne les, s controlled n 3D space by a dragger Mappng 2D Sketchng nto 3D Snce the perspectve vew provdes user a more userfrendly renderng effect, we here defned a perspectve vewer to map 2D coordnates to 3D (Fgure 5-3). We assume a so-called vewport as the actve vew regon n a 93

106 CHAPTER 5. SKETCH-BASED SURFACE MODELING dsplay wndow. The vewport n our system s defned as the full wndow. It can be set ether n terms of screen-space pxels or as normalzed coordnates, where (0, 0) s the lower-left corner of the wndow and (, ) s the upper-rght corner. The defnton of the vew volume structure s based on a perspectve camera. In Open Inventor there s a parameter AspectRato. We can automatcally modfy the vew volume based on the aspect rato of the vewport. It s easer to mplement n OpenGL too. In order to mprove the nteractvty and effectveness of modelng process, we further adopted a projecton plane for the desgner to draw on. Such a plane can be freely adjusted by a dagger (see Fgure 5-3) n world space. Usng ths plane we can convert from wndow space (usually based on the mouse locaton) nto 3D world space. Ths s done by projectng the wndow coordnate as a 3D vector onto a geometrc functon n 3-space, and computng the ntersecton pont (see Fgure 5-5). Most projectors actually compute ncremental changes and produce ncremental rotatons and translaton as needed. We manly use t to wrte 3D nteractve manpulators and vewers. Our plane projector projects the mouse onto a plane. Then the gven mouse event (2D coordnates) returns the pont vector n three dmensons, and these ponts are normalzed from 0 to, wth (0, 0) at the lower-left. 94

107 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Frustum Vew Volume Back Plane Front Plane wndow Projected Plane Fgure 5-3: The perspectve vew, where the green plane s defned as projected plane n the world space. It can be freely controlled n 3D by a so-called dragger. Projected plane wndow Fgure 5-4: Sketched 2D splne n a screen wndow s projected onto a default plane n the world space. The 2D freehand sketch then s transformed nto 3D vector descrptons by usng the normalzed 3D transformaton equatons (see formula 5-) 95

108 CHAPTER 5. SKETCH-BASED SURFACE MODELING X Ps(x,y,z) P (x,y,z ) P(xp,yp) α C d screen z Z Fgure 5-5: Mappng 2D pont P(xp,yp) onto a predefned plane (lght blue) to get the correspondng 3D pont P s (x,y,z); when the projected plane s rotated along X axs, then the P s s updated by pont P As t s shown n Fgure 5-5, C denotes the camera poston; the 2D stroke pont P(xp, yp) s projected onto a default plane (lght blue) whch s parallel to the screen wndow n a predefned perspectve vewer. The mapped 3D pont P s (x, y, z) s calculated by formula 5-. As mentoned before, each projected plane can be freely controlled n our system. Once the projected plane s rotated or translated by usng the dragger metaphor, the 3D mappng pont wll be changed as t s wrtten as followng: 96

109 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING = ) ( ) ( z d z d yp d z d xp z y x (5-) = ) cos( ) sn( ) sn( ) cos( 0 0 ' ' ' z y x dz dy dx z y x α α α α (5-2) In the formula 5-2, we exemplfy the case where the projected plane s rotated around X axs, and the translaton s S ( dx, dy, dz ); the P (x,y,z ) then can be calculated. As for other cases of 3D transformaton we can smply refer to the formalzed transformaton equatons n 3D graphcs. Each pen stroke thus s automatcally transformed nto 3D vector descrptons and they then are approxmated nto B-splne curve as we descrbed n last chapter. These sequent B-splnes can be used for constructng the correspondng B-splne surface. In the next secton we wll detal how we construct the constrant-based surface wth respectng the desgner s ntenson.

110 CHAPTER 5. SKETCH-BASED SURFACE MODELING 5.2 Splne-based Surface Constructon The surface nterpreters that we proposed support surface creaton wth respect to the dversty of desgner s drawng styles. They nclude the usual modes such as sknnng, extruson, revolvng and sweepng modes; besdes these we also provde our characterstc ways to generate large varetes of 3D shapes by few sketched splnes.. Geom-fllng mode: the desgner s allowed to sketch two or more 3D curves whch serve as the constraned boundary of the surface (see Fgure 5-6(a)). 2. Revolvng mode: the desgner can generate a surface by revolvng a curve around one axs. Frst, the desgner draws a splne by free hand sketchng. Then the surface s shown as soon as the desgner fnshes defnng the axs. (see Fgure 5-6(b)) 3. Sknnng mode: the desgner sketches a surface by usng the well-known concept of extruson. He/she frst draws a free-form 3D curve, then the curve s attached to the ponter and when the ponter s movng, the process of surface generaton starts and the shape s mmedately shown. (see Fgure 5-6(c)) 4. Sweepng mode: one nput splne s nterpreted as a profle; the second splne s so-called path, sculpted by further pen strokes nsde of an orthogonal plane whch s perpendcular to the profle curve. In ths way, the surface can be obtaned by sweepng the profle along the path. (see Fgure 5-6(d)) 98

111 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING (a) (b) (c) (d) Fgure 5-6: (a) Two sketched curves (shown by set of whte balls) serve as surface boundares by usng Geom-Fllng mode.(b) The surface s generated by revolvng one sketched splne around an axs (green one). (c) Smlarly to the Geom- Fllng mode, t s used for dynamcally generatng a surface whle the second splne s drawng. (d) The surface s constructed by sweepng one profle splne around a path. By usng these drawng modes, the desgner can create many smple models. (See Fgure 5-7 a, b) 99

112 CHAPTER 5. SKETCH-BASED SURFACE MODELING (a) (b) Fgure 5-7: (a) The bottle body and ts plug are constructed by only two splne strokes usng revolvng mode; the handle s formed by two splnes usng sweepng mode, the bland (red part) s constructed by usng sknnng mode. (b) The boat model s generated by 8 strokes usng sknnng model. However, n tradtonal llustratons, the depcton of 3D forms s usually acheved by a seres of drawng steps by usng few strokes. The desgner ntally draws the outlne of the subject to depct ts basc masses and boundares. Ths ntal outlne, known as constructve curves, usually results n very smple geometrc forms. Our proposed methods are nspred by the tradtonal llustraton methods. We have developed new algorthms to facltate the rapd modelng of a varety of free-form 3D objects, constructed and edted from just a few sketched 3D splnes. And these dfferent surface constructon modes can be swtched by smple hot keys. 00

113 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING 5. Boundary-based rotaton mode: a rotaton surface s generated by two constructve 3D splnes, where these splnes serve as the outlne form. 6. Ral-based rotaton mode: a surface s generated by three sketched splnes; two closed splnes, connected by the thrd curve, serve as a ral; the thrd one then s movng along the ral generatng a surface. 7. Splne-based sculptng mode: The user draws one splne as a target curve to affect the curves ncdence on a selected surface n order to obtan desred shape. As a result, these B-splne surfaces are constructed and represented by a mult-patch whch s composed of a compatble network of parametrc curves as t s shown n formula 5-3, where numrow and numcol represent the number of parametrc curves n U and V drectons. C (u) s standard B-splne curve, {P k } are the control ponts and {N k,p } are the normalzed B-splne bass functons of p degree. numrow C ( u) = numcol C ( v) j j= n S ( ; ); C( u) = P N ( u ) k k, p k (5-3) k= 0 In the followng secton, we wll further descrbe how the constrant-based resultng surface s constructed. 5.2.The Boundary-based Rotaton Surface Ths approach combnes the surface of revoluton and the ruled surface to fnd the parametrc descrpton of a boundary-based rotaton surface. The two sketched splnes are used for the outlne of the surface. 0

114 CHAPTER 5. SKETCH-BASED SURFACE MODELING Let C l (u) and C r (u) be the 3D B-splne curves (strokes) defned by the user. We would lke to use C l (u) and C r (u) as the constructve curves. Then the surface S (u,v) s generated by a seres of crcles between these constructve curves (Fgure 5-8 a). Let A denote the plane where the curve Q (u) exsts, the curve Q (u) s formed by the mdpont of C l (u) and C r (u) at each u. Assume that O (u), for fxed u, parameterzes the crcle perpendcular to A wth the centre Q (u) and passng through C l (u) and C r (u) as follows: O (0) = C l (u); O ( π ) = C r (u) (5-4) Q (u) = Cl ( u) + Cr ( u) 2 2 (5-5) X ( u j ) X ( u j ) Y ( u j ) Y ( u j ) p j = M = T( ω) R( α) Z( u j ) Z( u j ) 0 u j X ( θ) R Cos( θ) + Pj ( X ) + Y ( θ) R Sn( θ) Pj ( Y) O( θ) = = Z ( θ) Pj ( Z) 0 θ 2π ; R = Q( u j ) p j 02 (5-6) (5-7) Here M s a transformaton matrx whch represents the poston of the -th plane n 3D space, t s formulated by orentaton α and translaton ω constrans. Then the space poston of the ponts on the splne (C l (u) or C r (u)) based on each u j can be calculated by ths matrx as shown

115 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING n formula 4. Thus for gven u j, we can obtan the crcle O (v), and the pont P (X, Y, Z ) n the crcle O (v) s represented as shown n formula 5-6. It also ndcates that the pont P s obtaned by rotatng the pont (X, Y, Z) along the centre Q (u j ). In ths way, we can sample a seres of ponts n the crcle by ncrementally ncreasng the angleθ. We then get all the samplng ponts through varyng u j from 0 to (see Fgure 5-8 b). C l (u) p j R C r (u) Q (u) (a) (b) Fgure 5-8: (a) The constructve curves and crcles. (b) The surface s generated from nterpolatng all the samplng ponts n the crcles. As mentoned before, each plane can be freely adjusted by a 3D dragger, whch provdes the orentaton and translaton control by user s draggng and rotatng. When the sensor attached to each plane detects any change, the correspondng splne exstng n the plane wll be recalculated (see formula 5-7), and the ncdent surface s then updated accordngly. 03

116 CHAPTER 5. SKETCH-BASED SURFACE MODELING Based on the parameterzed crcles O (v) we nterpolate a seres of 3D ponts sampled from these crcles to obtan adaptve B-splne surface. As t s shown n formula 6, each NURBS curve nterpolates (m+) data ponts {Q k }, we assume that the knot vector U s defned by U= {0,0,...,0, u p+,, u m,,,,}. No, p ( u )... Nm p ( u ) 0, 0 N u N u o, p ( )... m p ( ),... No, p ( u )... Nm p ( um m, ) Po Qo P Q = P m Q m (5-8) Then, once a common knot vector s selected for each of the u and v drectons, nterpolatng of a set of (m+) (n+) array data ponts {Q k,l },(k=0,,m; l=0, n} by a B- splne surface can be accomplshed by applyng the technque of B-splne curve nterpolaton (formula 5-8) twce. The correspondng least squares approxmaton method has been descrbed n the last chapter. Ths boundary-based rotatonal surface approach can create a varety of models (see Fgure 5-9) due to ts smlarty wth user s tradtonal drawng style, and these surfaces absolutely respect the user s ntenton. Furthermore, when the constructve curves have corner ponts or sharp features, the fnal surface wll also well keep sharp features and rotatonal creases. 04

117 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Fgure 5-9: The apple and candle are formed by four and sx splnes respectvely by usng boundary-based rotaton method Ral based Rotaton Surface The surface s formed by the moton of a curve along a ral. Here the user draws two closed curves C (v) and C 2 (v) as a ral, and the thrd curve C 3 (u), connectng wth these two closed curves, s movng along ths trajectory to construct a rotatonal surface (see Fgure 5-0). Comparng wth the boundary-based rotatonal surface approach, ths method further respects the desgner s conventonal drawng style by usng a sequent outlnes to construct free form 3D objects. It then provdes more possbltes for shape control through usng smple over-sketchng operator (see Fgure 5-0). p p M = pp 0 0 ; C 3( u) M = C ( u) (5-9) We defne an axs OO by the center of the boundng boxes of these closed curves. For any gven angle θ, we can obtan the correspondng ponts P (v) and P (v) n curve C (v) and C 2 (v). The Matrx M then mplements the transfor- 05

118 CHAPTER 5. SKETCH-BASED SURFACE MODELING maton from lne P 0 P 0 to lne PP. As a result we can obtan the new curve C (u) by applyng ths transformaton matrx to curve C 3 (u) (see Formula 5-9). The surface S (u,v) s fnally constructed by nterpolatng seres of transformed curves C (u). Ths approach can be used for creatng large number of cylndrcal objects (see Fgure 5-0). And t s more flexble than sngle rotatonal surface generaton, snce these sketched outlnes can precsely lead to the user s expected shape. Moreover, t s much easer to support the restylng process by smply usng our over-sketchng operator. C 3 (u) P P 0 θ P O O C (v) C 2 (v) C (u) P (a) (b) (c) (d) (e) 06

119 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Fgure 5-0: (a) (b) Ral-based rotatonal surface (c) (d) the skrt model s created by user's splne strokes; (e) one splne s locally modfed by smple oversketchng, and then the surface s changed as well Splne-based Surface Sculptng In order to effcently support desgner s restylng process, we present an approach to drve the shape changng by smply sculptng few splnes on the surface. The method adopted to manpulate a surface usually makes use of ncdent curves. The parameterzaton of a B- splne surface S and the curve G n the doman of S can be descrbed as H =S (G), where H denotes matrces of 3D control ponts of the curve G. The 3D sketched curve H (whch serves as a shape parameter) s mapped to ths surface. Then the control ponts of ths surface, whch satsfes H = S (G), can be determned as a soluton of a system of lnear equatons. However t nvolves huge computatons to determne the rank of the system matrx. In our system we nstead propose an adaptve dscretzaton approach where the contnuous curve-surface ncdence problem s dscretzed by consderng pont-surface constrants ordered along gven 3D curve. Frstly the sketched target splne H s nterpreted n an auxlary plane and then t s transformed nto key ponts m set ( k ) ( k H ), where m s the number of the key = ponts. They wll mpose external forces to the surface S (see Fgure 5- left) along the normal vector N t ; meanwhle 07

120 CHAPTER 5. SKETCH-BASED SURFACE MODELING the determned projecton pont set ( Q j ) ( Q j S) on the surface S serve as the senstve sprngs whch wll be relocated by respondng to these forces. The resultng surface s fnally obtaned by nterpolatng all the revsed vertces (see Fgure5-2). m j= f ( k, p) Deformed Curve P Orgnal Curve Fgure 5-: (Left) the. k s the key pont whch mposes the external force f to the curve. (Rght) An example shows the patch s sculpted by a target curve, where the target curve exactly les on the surface. (a) (b) 08

121 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING (c) (d) (e) (f) Fgure 5-2: Splne-based surface sculptng (a) The user sculpts a target splne on a selected surface. (b) The surface responds to the target splne. (c)(when the target curve s over sketched by user,(d) the surface s changed as well. (e)(f) One more example wth dfferent target splne. In the next chapter, we wll detal how we effectvely dstrbute the forces among the surface, how we manage geometc constrants to mplement ntutve surface deformaton. 09

122 Chapter 6 Physcally-based Surface Control Geometrcal methods, where ndvdual or groups of control ponts or shape parameters are manually adjusted for shape edtng and desgn, manly rely on the mathematcal skll of desgner. A perceptually smple change mght requre adjustment of large of geometrc parameters. Thus they are lmted by the expertse and the patence of the user. Researchers have nstead explored ways to add physcal behavor to the tradtonal geometrc modelng prmtves, partcularly parametrc surface patches. Those methods use physcal prncples and computatonal power for realstc smulaton of complex physcal processes and ntutve shape manpulaton that would be dffcult or mpossble to model wth purely geometrc technques. The deformaton process starts wth varous geometrc models connected together wth parametrc constrants n 3D space. These constrants, encoded as user-appled sculptng forces that modfy the surface n predctable ways, mpose the physcal effects on the deformable models. In our system, for each geometrc model, a bar network s bult from ts control vertces (see Fgure 6-). The set of external forces on the bar network deforms the model accordng to the geometrc and parametrc pont s constrants. 0

123 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING The mnmzaton of the varaton of the external forces has been also used; as a result the least possble adjustment to the control vertces s nvolved. The resultng object s fnally obtaned by such an adjustment of the control pont network. Fgure 6-: The control pont s network of curve and surface, where the external forces are produced by user-appled constrants. Our deformable model s comng from the method for sketchng B-splne surface whch s specfed n the last chapter, n order to mprove the deformaton process, we represent our surface as a mult-patch structure, where the surface s composed of a compatble network of parametrc curves as t s shown n formula 5-3, Then each external force produced by user-appled parametrc constrants only mposes the energy along one curve whch s decded by combnng least squares dstance method (SDM) and our curve-determnaton approach as they wll be detaled n the followng sectons. The correspondng parameter vectors (control vertex) on ths curve wll be adjusted, and the methods for constraned optmzaton are used to fnd a state vector that mnmzes the energy whle satsfyng the constrants. Those updated control ver-

124 CHAPTER 6. PHYSICALLY-BASED SURFACE CONTROL texes consequently produce new constrants to the nvolved curves. In ths way the surface manpulaton s converted to the correspondng curves control. Therefore, we frst specfy how the constrant-based mnmzaton method works for the drect curve control [29]. 6. Drect Curve Manpulaton Drect curve manpulaton s an attempt to make curve control more user-frendly. The user s allowed to select any ponts on the curve and to apply parametrc constrants, such as the sngle poston constrant, dstance constrant and tangency constrant. The program then automatcally transfers them nto external-force constrants to adjust the control vertex postons. Snce a typcal curve depends upon m + control vertces, P 0 ; P ;... ; P m, once the constrants are mposed, there are somethng lke m degrees of freedom left unspecfed, provdng no unque soluton to the problem. In such a case, we shall resolve the ambguty by requrng the least possble adjustments to the curve's control vertces n some metrc. To be more specfc, let: m = 0 P n + d N ( u ) (6-) represent the curve after modfcaton. Modfcatons are specfed by requrng a target functonal, φ; e.g. to acheve a mnmum [curve-constrants manpulaton] m n φ [ P + ] ( ) d N u (6-2) = 0 Constrants are specfed by constrant functonals f j ; j = 0;... ; c and constrant values γ j. 2

125 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING f j m = 0 P + d N n ( u ) = γ j or f j m = 0 P + d N n ( u ) γ j (6-3) A least adjustment wll be gven by a metrc µ and the requrement that µ(d 0,..., d m ) = mn. We shall restrct the generalty to a further degree by requrng that φ s lnear attanng a specfed value, that only lnear equalty constrants are present, that µ be m µ ( do,..., dm) = ( d 2 ) = mn (6-4) = 0 where represents the ordnary Eucldan norm, and that there are no more functonals than control vertces. Wth these restrctons, and wth the assumpton that the functonals are all lnearly ndependent, a unque set of d exsts. In ths settng, φ need not be dstngushed n any way from the f j, snce all are lnear and each s requred to attan a specfed value. Thus, we shall consder fndng d 0,..., d m. f j m = 0 P + d N n ( u ) = γ The j th equaton of (6-5) can be rewrtten as: j (6-5) m F j d = = 0 n β j where Fj = f j ( N ( u)) and β j = γ j 3 m = 0 n P f j ( N ( u)) or, more compactly n matrx-vector notaton: Fd =b (6-6) Snce we have no more functonals than control vertces, F wll be square, or t wll have fewer rows than columns (the usual case). The lnear ndependence of the f j mples

126 CHAPTER 6. PHYSICALLY-BASED SURFACE CONTROL that the rows of F wll be lnearly ndependent. Usually some of the f j take on the value γ j for the orgnal, unmodfed curve, and correspondngly β = 0. To fnd a soluton to (6-6), we wrte d n terms of mutually orthogonal components, one n the row space of F and the other n the null space of F [5]: d = F T v + z (6-7) b= Fd = F(F T v + z)=ff T v +Fz = FF T v v = (FF T ) - b (6-8) The components of z consttute parameters rrelevant to the system, yet they contrbute to the length of d, so we obtan the mnmum length soluton to (6-7) when settng z = 0. Substtutng (6-8) nto (6-7) wth ths settng t leads to: d = F T (FF T ) - b where the matrx FF T s nonsngular under our assumpton that the rows of F are ndependent. Snce the nverse of Fˆ = FF T s gven by adj( Fˆ )/det( Fˆ ), where adj( Fˆ ) s the adjont matrx of Fˆ, ths soluton can be re-expressed as d = F T adj( Fˆ )b/ fˆ (6-9) Where fˆ = det ( Fˆ ). By lettng F and F j denote the th and j th rows of F, the, j th element of Fˆ s ˆ = F F j. The cofactor consttutng the, j th element of adj( Fˆ ) wll be denoted by α,j, these cofactors may be expressed compactly n terms of the elements of Fˆ. If these constrants are to be appled durng nteractve manpulaton, we dvde the soluton nto a preprocessng stage and an teratve stage. In the preprocessng stage we compute C = F T adj ( Fˆ )/ fˆ. The teratve stage conssts merely of multplyng a sequence of 4 j f, j

127 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING b's by C to obtan a sequence of d's. Only those columns of C correspondng to nonzero components of b need be computed. The preprocessng stage s performed at the begnnng of an nteracton, whle the teratve stage s appled throughout the nteracton. A system composed of a sngle constrant s gven by: F 0 d = β 0 the soluton for ths system s: d = F T 0 β 0 / ˆf 0, 0 (6-0) At a gven pont u on the curve, (6-0) can be used to constran poston, tangency, or a hgher dervatve by settng the th component of F 0 equal to the value or approprate dervatve of N n (u) evaluated at u. However, the poston of the curve at u wll vary f any dervatve s constraned. Fgure 6-2 llustrates one sngle poston constrant s appled on curves, whle hgher dervatves are free to vary. whose soluton s: F0 0 d = F β T T ˆ ˆ 0, 0,0 β 2 fˆ 0,0 fˆ ˆ, f 0, d= F [ f f ] / f d= F [ α α ] / f (6-) ˆ (6-2) fˆ = (6-3) Lkewse, for the three constrants: F0 0 F d = 0 F 2 β 2 (6-4) Where T α ˆ (6-5) T 0,2,2 2,2 β2 5

128 CHAPTER 6. PHYSICALLY-BASED SURFACE CONTROL α α α 0,2,2 2,2 fˆ = = fˆ = fˆ fˆ 0,2 0, 0, = fˆ 0,0 α fˆ fˆ 0,2,2 0,2 fˆ, fˆ fˆ fˆ + fˆ,2, 0,0 2 0, α fˆ,2 0,2 fˆ,2 + fˆ 2,2 α 2,2 (6-6) Target curve E p Intal curve Modfed control ponts network E p (free) (a) Blocked (b) Blocked Pont poston constrant (c) (d) Fgure 6-2: One poston constrant s appled to the curve whle hgher dervatves are free to vary (a) a postonal constrant s appled to the free pont E p by user s draggng process. (b) As a consequence, ths entty can adapt ts poston to further mnmze the correspondng adjustment of the control vertces (dot polylne). These control vertces can be defned ether as free or blocked states to mplement local deformatons. In ths case, the begnnng and endng control ponts (blue balls) are blocked, and others are free to vary. (c) and (d) show the curve s passng 6

129 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING at the new pont and the control vertces are changng to the least possble extent. In Fgure 6-3, we llustrate how the user-appled mult-constrants are teratvely performed for the complex curve manpulaton, where the curve s appled an orthogonal dstance constrant, the tangency and poston are varyng whle preservng the orthogonal angle between two ponts. E p Dstance Constrant d 0 X.2* d 0 E p2 (a) Tangency Constrant (b) (c) (d) Fgure 6-3: (a) It shows the applcaton of a trple constrant at free ponts E p and E p2. The tangent drecton and the poston are vared whle varyng the dstance d 0 and preservng certan or- 7

130 CHAPTER 6. PHYSICALLY-BASED SURFACE CONTROL thogonal angle between E p and E p2. (b) The curve s deformed by applyng orthogonal dstance constrant.2d 0 between two free ponts. (c)(d) They llustrate the adjustment of the correspondng bar network of control vertces. 6.2 User-appled Constrants for surface Control The user may freely attach a varety of features, such as ponts and flexble curves, whch then serve as handlers for drect manpulaton of the surface. The constrants appled for surface modfcaton can be classfed as followng: Pont poston constrant: t constrans any pont on a surface to nterpolate any pont n space (see Fgure 6-4 a). Pont normal constrant: t constrans the drecton of the normal at any pont on a surface to a pont n any drecton (see Fgure 6-4 b). Curve poston constrant: t constrans any curve wthn the surface to nterpolate a curve n space (see Fgure 6-4 c). Curve tangent constrant: t constrans the surface tangent across any curve n the surface to pont n a gven drecton. A combnaton of the curve poston and curve tangent constrants fxes the surface normal along the length of the curve constrant (see Fgure 6-4 d). 8

131 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Dstance constrants: the two surface ponts are constraned by a certan dstance constant. These constrants are further specfed n Table 6-, where P c and P s denote the general poston vectors on the curve and surface. Lkewse, the t c and t s are correspondng tangent vectors, n represents the unt of normal. The thrd column shows f the constrant s lnear or not. t 0 p 0 n 0 b 0 p c n c p s t sv n s t su p s n s t sv t su (a) (b) C m n c p c t c n s t su C s C s p s t sv (c) (d) 9

132 CHAPTER 6. PHYSICALLY-BASED SURFACE CONTROL Fgure 6-4: (a) Pont poston constrant: the target space pont P 0 mposes force constrant to the pont P s on the surface. (b) Pont normal constrant: where the normal n s of the pont P s on the surface s constraned by the target pont P c and the normal n c by satsfyng parallel or orthogonal or specfc angle relatons. (See Table 6.) (c) Curve poston constrant causes a curve wthn a deformable surface to approxmate the shape of a curve n space. The target curve C m and the correspondng curve C s on the surface are dscretzed nto a set of ponts, then the curve on the surface s gong to nterpolate all theses target ponts. In ths way the curve constrant s converted nto set of lnear pont poston constrants and tangent constrants. (d) The curve C s on the surface s constraned by the target pont P c and the tangency t c. As a result the resultng curve C s keeps the tangency t s t c = β, whereβ s a user-defned constant. It thus causes the ncdent surface deformed. Table 6- the defntons of varous constrants Constrants Equaton Lnear Pont poston Pc ( u ) Ps ( u2, v2) 0 Yes Tangency. Parallel 2. Arbtrary value β 3. Orthogonal =. t u ) Λt ( u, v ) 0 c ( s 2 2 = 2. t c u ) β t( u, v ) 0 ( 2 2 = 3. t u ) n ( u, v ) 0 c ( s 2 2 =. No 2. Yes 3. No Normal Drecton. Concdent 2. Angular α. u ) Λn ( u, v ) 0 2. n( s 2 2 = n ( u ) n s ( u2, v2 ) = cos( α ). No 2. No 20

133 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Dstance 2 Pc ( u ) Ps ( u2, v2 ) d Yes Wth deformable modelng, users can add any combnaton of the above constrants to a deformable model at run tme. Constrants can be appled to ndvdual ponts wthn the surface or may be appled to curves wthn the surface Pont Constrant Any number of ponts n a deformable model may be smultaneously constraned. A pont constrants locaton n a deformable model s specfed by a parametrc locaton, e.g. a uv pont. For deformable surfaces, the current set of shape propertes that can be constraned by a pont constrant ncludes poston and surface normal drecton. Pont constrants have the ablty to track. For example, the mage space poston of a poston pont constrant can be moved and the deformable model s automatcally deformed to track the moton of the pont (see Fgure6-5 c-d). f (k t p) p n k t Q,j (a) (b) 2

134 CHAPTER 6. PHYSICALLY-BASED SURFACE CONTROL (c) (d) Fgure 6-5: (a, b) One pont poston constrant f (k t,p) produced by pont k t. In ths case we blocked all the other control vertces except for the closest vertex Q,j. (c, d). One dragger s appled to provde the trackng of the pont s movement whch produces a sequence of deformaton results. As t s shown n Fgure 6-5, the decded projecton pont P s the one whose normal s concdent wth the target n k Λn p = 0 normal n k ( ). Then the closest control vertex Q s obtaned by usng the least squares dstance method. Snce we defned our surface model by U, V so-parametrc curves, the force energy s consequently dstrbuted along these curves. Assume that the pont constrant s mposng the forces on the selected curve along U drecton frst (see Fgure 6-7 a). Then the updated control vertces of Q su, sv and Q su+,sv (orange balls) serve as new dstance constrants to produce the forces to the correspondng curve along V drecton respectvely. Therefore, the k step pont constrant s optmzaton can be descrbed as: [k] k T d = F ] [ k] ] β / ˆ k f (6-7) [ 0 0 [ 0, 0 22

135 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING [0] In the preprocessng stage we compute T d = F 00 [ ] β / [0 ˆf ], [0] where the F s equal to the value or approprate dervatve of 0 numcol N u) ΦV = sv ( evaluated at u, and the numcol s the number of control vertex on selected parametrc curve Φ V =sv (see Fgure 6-6 a). The teratve stage ( k numrow) s preceded due to the [0] updated control vertces (Q,sv = Q,sv + d ; Q +,sv= Q +,sv [0] + d + N ;.). [ 0 k ] [ k] β = Q k,sv ; k F = N u) Φ U 0 numrow [0] 0 0,0 ( (6-8) k Φ U s the k th curve along the V drecton, whle numrow k U ( u) Φ denotes the value of (u) N numcol Φ. on the curve k U In ths way we can compute all the optmal control-pont postons of the deformable surface. The pont constraned surface can be fnally reconstructed by numcol tmes curve optmzaton process. Impose one pont constrant along ths curve (U drecton) Impose two dstance constrants on the curves along V drecton V Q su,sv V U U (a) (b) 23

136 CHAPTER 6. PHYSICALLY-BASED SURFACE CONTROL Fgure 6-6: Pont poston constrants are frst appled to the curve along U drecton (orange polylne); (b) the updated new control vertces wll produce new dstance constrants to the curves along V drecton respectvely, whle the boundary control vertces are blocked. Lkewse, we can also handle the pont constrant along V drecton frst, then the optmzaton process wll be recursvely proceed numrow tmes. Our method supports pont poston, pont tangent and curvature constrants on the deformable curves. Ths means that the curve can be made to deform to exactly nterpolate a pont poston or a pont tangent. Ths combnaton can be seen n equatons (6-2, 6-3, and 6-4), whch s able to mpose a set of constrants smultaneously whle stll deformng a curve to reman far. Fgure 6-7 llustrates the local pont constraned surface and global constraned surface. Pont Constrant P(0,)) P(,) P(0,0)) P(,0) (a) (b) (c) Fgure 6-7: (a, b) Sngle pont replacement for local deformaton where only one control vertex s free, others are blocked. (c) When all the control vertex nodes are free to vary, t mplements global deformaton. 24

137 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Under these constrants the behavor of physcal objects provde the user wth a famlar metaphor for modfyng shape wth forces n an ntutve manner: surfaces can be pushed, pulled, and nflated to get desred shapes Curve Constraned Surface Deformaton We further adopt curve constrant to mpose contnuous forces on the surface; such a curve-drven method provdes desgners wth more ntutve shape control just as sculptng n the deformable models (see Fgure 6-8). I propose an adaptve dscretzaton approach where the contnuous curve-surface ncdence problem s dscretzed by consderng many pont-surface constrants ordered along a gven 3D curve thus mprovng the accuracy and valdty of surface generaton. Durng the mplementaton we further present a farness measure n order to obtan the effectve key ponts from the sketched target splne whch serve as set of force constrants. Fnally we demonstrate the dstrbuton of adaptve forces among a parent surface. Departng from what descrbed n [32], our algorthm supports more nteractve shape control through userappled sketchng operatons. Especally we have optmzed the splne-drven deformaton process by usng adaptve force dstrbuton nstead of evaluatng every vertex n the parent surface. We optmze the forces only to the correspondng curves n U or V drecton by predctng the moton tendency of the target splne. Those renewed control vertces further serve as new pont constrants to symmetrcally dstrbute the energes along correspondng curves. 25

138 CHAPTER 6. PHYSICALLY-BASED SURFACE CONTROL (a) (b) (c) (d) (e) (f) Fgure 6-8: (a,b,c) The cap model s deformed by one curve. The dscrete ponts mpose the full forces on the surface, where the stff (force densty) s set to, so the surface exactly nterpolates all the target ponts. (d,e) Wthout a predefned local regon the key ponts n the target curve (blue balls) produce force sprngs n the cup surface and only mpose stran to these senstve sprngs, where the stffness s set to 0.8. (f) Wth the predefned local regon, the force s nfluence can be well localzed whle settng the force densty to Force f (k t,p) The target curve ) (u ψ s frst dscretzed nto set of k t as key ponts (see formula 6-9), where m s the number of key 26

139 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING ponts whch s adaptvely determned n our system as t s shown n the followng pages. Snce our system supports nteractve curve and surface manpulaton as aforementoned n last chapter. So we ntroduce a dynamc factor A to detect the state of current target splne whch s formulated as a transformaton matrx through the orentaton constrant R ( α ) and the translaton constrant T ( ω). Whenever the target splne or a surface s adjusted n 3D space by a dragger, t wll be used to dynamcally update the forces f and then to redstrbute the effect of these forces on the resultng surface. Ths way a number of shape varatons are avalable to users. The D (k t ) s the dstance between the projecton pont p(u,v) on the surface and k t. Such a pont and dstance constrants mpose the force f (k t, p) to the correspondng parameter curve. λ s a user-defned constant, that represents the force stff (densty) whch evaluates the degree that the surface approxmates to the target curve. m ψ ( u) = ( kt x( u A t ), y( ut ), z( u ) t ) (6-9) t= 0 A = ( T( ω), R( α)) ft ( kt, p ) = λ D( kt ) D( kt ) = kt ( ut ) p( u, v) 0 λ ; (6-20) num F t ( C( u)) = ft + Q, j d N, p ( u) (6-2) = 0 We fnally suppose F t ( C( u)) as the force s nfluence on a curve C (u) along U drecton, where m represents the number of control vertex n ths curve C (u). The optmzed functon s then adopted to dstrbute the energes among these vertces (see formula 6-2) whch refer to the curve manpulaton procedure as aforementoned. 27

140 CHAPTER 6. PHYSICALLY-BASED SURFACE CONTROL Bound Curve In order to precsely localze the nfluence of the external forces onto the orgnal surface, we proposed a so-called bound curve supportng nteractve defnton of the deformaton regon by user. The bound curve φ (u, v) s a sketched closed curve n 3D space. If the surface vertex Q,j les nsde the boundng curve, E (Q,j ) s equal to and t can be nfluenced by force constrants. Otherwse E (Q,j ) s set to 0 and t wll thus keep a statc status. We ntalze the default value E (Q,j ) =, whch mples that the system wll automatcally proceed to global deformaton operatons when the boundng curve s not sketched. Then the equaton 6-2 can be rewrtten as equaton ϕ ( u j, v j ) 0 E ( Q, j ) = 0 ϕ ( u j, v j ) > 0 (6-22) num F t ( C( u)) = ft + Q, j d E( Q, j ) N, p ( u) (6-23) = 0 Fgure 6-9 : The bound curve on the surface and the E (Q) dstrbuton of the bar network. 28

141 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Fgure 6-0: Target Curve mposes the global and local forces on the surface. The green polylne llustrates how the control vertces are adjusted. In the followng sectons we wll detal how to effectvely obtan the key ponts on the sketched target curve and how to classfy three constrant confguratons (overconstraned, under- constraned and well-constraned); n the end we wll further mprove boundary features of resultant surface The Determnaton of the Force Constrants Snce the desgner s sketchng actvty produces only an approxmaton of the desred shape, t s mportant that the resultant surface captures the shape features of the target curve. However, n the free-form doman, the number of constrants s usually unknown. Most current approaches provde only a soluton that s the result of a pre-determned crteron. We nstead propose a method whch adaptvely 29

142 CHAPTER 6. PHYSICALLY-BASED SURFACE CONTROL provdes such crtera through the predcton of the moton of the target curve (see Fgure6-). For ths we adopt the partal dervatves θ and θ 2 (see equaton 6-24). As shown n Fgure, we can easly get the ponts P s (u s,v s ) and P e (u e,v e ) by projectng K s and K e onto the parent surface S, where the H s the lne connected wth these two projecton ponts. Then we extract the span of the patches where Cs and Ce defne respectvely the curve poston n the V drecton, whle Rs and Re descrbe the curve poston n the U drecton. H H θ = ; θ 2 = ; u v Ce Cs ; θ θ2 m = Re Rs ; θ p θ2 Re, Rs, Cs, Ce Integer (6-24) When θ s not less than θ 2 (θ θ 2 ), the target curve s leadng towards the V drecton. Therefore the number of key ponts (constrants number) on the target curve m s determned by the dfference between Ce and Cs. Vce versa, when θ < θ 2, m s calculated by the dfference between Re and Rs. In ths way the key ponts on the target curve wll produce set of pont constrants and they then mpose the force s sprng to the curves. 30

143 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING K e K s Ce Pe Ps Cs Re Rs U Fgure 6-: The target curve (n orange) and ts projected lne H= Ps P e onto the parent surface. The yellow crcles represent the key ponts whch are adaptvely produced by consderng the orentaton of H. 6.3 Splne-based Local Deformaton The shape manpulaton technque we have mplemented s able to deal wth geometrc detals n an easer manner. Based on the llustrated sculptng technque, we have further mproved t to solve specfc deformatons that satsfy our requrements. 3

144 CHAPTER 6. PHYSICALLY-BASED SURFACE CONTROL The algorthm for the mplementaton of local deformaton s llustrated through the help of the followng pseudo code. Intalzaton: E (Q,j ) =; λ ( ( ω) = 0, Rot( α) = 0) Step: The user creates a free form surface S n the preferred mode (e.g. by boundary-based rotaton mode, sculptng mode or revolvng mode and so on). Step2: The user sketches the bound curve ϕ and the target splne H. Step3: The system consequently calculates local senstve regon E (Q j ).. The system symmetrcally dstrbutes lnear force energy f (k, P) to surface S. 2. The system automatcally reparameterzes the resultng surface. Step4: The system renders the resultng surface IF (Sensor -checkng s true) 32 Tran ; λ( Tran ω α ); Update (Dynamc factor ( ), Rot( )) Update External Forces f (k, P); goto Step3 Else goto Step5 Step5: End Durng the process of local deformaton, we have excluded the opton of havng all the vertces outsde the boundng curve fxed and havng to operate only on those nsde. However ths choce could stll result n an naccurate and nsuffcent deformed shape around the boundng curve. Furthermore, the leadng target curve may result over-constraned or just show unacceptable undulatons. To

145 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING avod these ssues, we propose two ways of mprovng the qualty of the deformaton. Frst, we classfy the constrants nto three cases: Over-constraned: f the target curve completely les outsde the boundng curve. Under-constraned: f the target curve partly les nsde the boundng curve. Well-constraned: f the target curve les well nsde the boundng curve. When the confguraton s over-constraned the parent surface s not affected. Conversely when the confguraton s well-constraned, we use the aforementoned Formula 6-25 to get the adaptve constrants. In the case of underconstraned, we adopt and four extremes (see Fgure 6-2), t s easer to obtan the ntersecton part between the target curve and boundng curve. In ths way the effectve span of the target curve can be calculated. A so-called Relaxaton Interval s used to provde the transton parts from the two endng ponts on the target curve to the parent surface. We defne the transton parts by computng the mnmum boundng box of boundng curve. Then we calculate the followng four extremes: MnRow, MaxRow, MnCol and MaxCol, the relaxaton ntervals are calculated accordng to the patches span Re MaxRow and MnRow Rs. The force f ( Ks, P) and ( K, P) f e wll gradually decrease to reach zero wthn these two parts as t s shown n Formula 6-25 and Formula

146 CHAPTER 6. PHYSICALLY-BASED SURFACE CONTROL f(k e, P) f(k s, P) MaxCol MaxRow MnCol MnRow Fgure 6-2: The relaxon nterval s determned by cosderng the boundng box (lght blue) on the surface and the endng ponts on the target lne. MnRow Np MnRow s { δ F( C ( u)) } (6-25) r r= Rs = r= Rs = Np 34 f ( K, P)( r MnRow) Rs MnRow MaxRow MaxRow f ( Ke, P)( MaxRow r) { δ F ( Cr ( u)) } = r Re (6-26) = Re r= Re = MaxRow Snce the target curve s used to drve the surface s deformaton process ths mght be characterzed by a sharp lne behavour. For ths we propose a smoothng functon whch mproves the symmetry of the deformed surface. Ths provdes strong vsual mpact n terms of qualty of the surface. Wthout the need for any new patches nserton, we mantan the same topology by symmetrcally dstrbutng the external force nfluence to the correspondng curve. Np Tolerance = (6-27) 2 C( u)

147 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING δf ( C( u)) = Np t= F( Q ) = t r t= λe( Q ) D( K ) t t Tolerance + Np t= r λe( Qt ) D( K) t Tolerance Q t C(u) t, r Np (6-28) The detals are shown n formula 6-27 where the value C(u) s the length of curve C, whle Np s the number of vertces on each curve. The Tolerance factor s used to determne the dstrbuton step along the correspondng curve. From formula 6-28, we can acheve symmetrc deformaton by symmetrcally and gradually dstrbutng f ( K, P) to the dfferent vertex Q t on the parent surface. Fgure 6-3: (Left) One key pont that s able to be controlled by dragger ) mposes the force nfluence to a localzed regon (yellow balls represent the sketched bound curve). (Rght two) Two target splnes controlled by two 3D draggers only mpose strans to predefned local regons. These draggers movements drve the shape varaton at nteractve rate. 35

148 CHAPTER 6. PHYSICALLY-BASED SURFACE CONTROL Fgure 6-4: (Top) the model of pengun s created by sequent splnes sketchng where the yellow closed curve s sketched as a bound curve and the orange one serves as target curve used to produce the mouth part. (Bottom) the body of the fsh s created by usng the boundary based rotaton mode wth 4 strokes, and the fns are generated by usng a target splne to locally deform the model. 6.4 Dynamc 3D Sensor In order to mprove the flexblty of the splne-drven deformaton, we adopt a 3D dragger whch provdes dynamc orentaton and translaton control. As descrbed before, all the planes are freely controlled by a 3D dragger. Whenever the dragger s moved n 3D 36

149 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING space, the sensor attached to ths plane wll start to record the poston of the dragger and t wll then relocate the key ponts. Sequently the dsplacements of the forces wll be used to update the resultng surface. At any moment durng the manpulaton phase, our system computes a space warp that takes as nput the orgnal postons of the plane constrants C (defned by a local coordnate system U, V, W) to the current poston of 3D dragger C (defned by U,V,W ), assumng that W = U V and W = U V. We wanted a smooth space warp that takes the startng poston to the endng poston, ths condton results n three translaton constrants and three rotaton constrants (see Fgure 6-5). f ( kt, P) α U UU U W ω α W WW P V VV V Fgure 6-5: (Left) the green curve s formed by mposng force energy f (k t, P) to the orgnal curve (green dot lne). The blue splne s obtaned by rotatng and dsplacng the key pont k t whch s actvated by a 3D dragger. (Rght) Computaton of orentaton ( α ) and translaton ( ω ) constrants. Snce the dragger controls the whole plane, we can use unfed scale to update all the key ponts. We assume the set 37

150 CHAPTER 6. PHYSICALLY-BASED SURFACE CONTROL of orgnal constranng ponts [k, k 2 k n ] on target splne and the warped postons A (T ( ω ), R( α )). When α s NULL, the deformaton s pure translaton and the dstance D can be calculated by the formula Then all the target constranng ponts [k, k 2, k n ] can be descrbed as formula D=UU VV +VV WW +WW UU ; ω = D/ D ; (6-29) f (k,p) = f(k,p) + ω (6-30) UU = 2sn ( ) 2 D U α (6-3) k (U, V, W ) =k (U, V, W) tan( α ) (6-32) It s obvous that translaton constrant ω determnes the dsplacement of the pont and t thus affects the ntensty of the force energy appled to the surface. On the other hand, the orentaton constrant α wll adjust the senstve regon on the orgnal surface as descrbed n followng formula Here k s the fnal revsed poston of key ponts. In ths way the senstve curves n the surface defned by these key ponts wll be updated. Therefore, once the sensor detects the dragger s movement, the system automatcally updates the dynamc factor λ, and then the resultng surface s recalculated, n ths way we can get the sequence of the deformatons (see Fgure 6-6). 38

151 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Fgure 6-6: Dynamcal shape control by splne-drven deformaton: (left) the boat model s created by 7 sequent splnes. (Mddle) The target curve (blue splne) mposes the force energes onto the red sal. (Rght) When the target curve s movng away the nfluenced regon and the correspondng force energy are recomputed. In order to preserve the smoothness of the deformed surface we allow the settng of a sensor to each node. However, as descrbed before, whenever the sensor s actvated the space-warp operator s startng to recalculate the resultng surface. Snce ths behavour would requre expensve numercal computaton we have attached a tme sensor whch trggers the resultng calculatons only at specfc ntervals. The nterval was expermentally set to 0.02 sec to acheve good vsual feedback wth approprate numercal computaton. The performance of our splne-based deformaton method has been shown n Table 6-2, and t s obvous that the requred tme s proportonal to the number of key ponts and DOF n the parent surface (see Fgure 6-7). Moreover, we apply two methods to test the nfluence of the forces to the surface. Frst the forces produced by the target curve mpose the global nfluence onto the parent model. Ths way 39

152 CHAPTER 6. PHYSICALLY-BASED SURFACE CONTROL a seres of senstve curves are gong to respond to the force strans from the target curve. In the second method, we drectly defne a local regon by sketchng a boundng curve as aforementoned. Table 6-2: The smulaton tme of our splne-based deformaton method Control Vertces (DOF: U V) Key Ponts Polygon Mesh Update tme (s) Control Vertces Kye Ponts Tme (s) 40

153 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING *0 20*20 26*26 28*0 35*8 24*8 Kye Ponts Tme (s) Fgure 6-7: It shows that the consumng tme s proportonal to the number of control vertces wthn the surface and the effectve key ponts on the target splne. In our system, the functon s centered at the nteractve surface creaton and the ntutve splne-drven deformaton. Furthermore the dynamc vsual feedback actvated by 3D sensor leads to the creaton of the resultng objects n a natural and predctve manner. Compared wth other methods, ths approach has the followng advantages of ntuton, localty and smplfcaton. In the next chapter we further dscuss about the hgh level aesthetc curve operaors, where we propose a method to brge the gap betweent the geometc constrants wth semantc shape operators. 4

154 Chapter 7 Semantc Shape Operators Ths chapter nvestgates the use of a shape-grammar n a new set of deformaton operators for nteractve manpulaton of 2D B-splne curves, whch focuses on buldng the brdge between low level geometrc constrants wth hgher semantc shape control. The work s based on the Leyton grammar whch proposed a theory for the classfcaton and manpulaton of 2D curves. We further depct each of these processes (L-operators) by transformng them nto group of geormetc constrants. Moreover, we dscuss quanttatve parameters whch are used to analyze the curve propertes. As a concequence, sets of aesthetc curve operators are llustrated to mplement hgher-level planar curve control. 7. Leyton s Shape Process Grammar To descrbe the dfferent shapes that curves can reach, Leyton bases on analyzng the evoluton of an ntrnsc quantty: 42

155 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING the curvature. He s nterested n the curvature extrema and he demonstrates that they are at the bass of the defnton and manpulaton of shapes. Leyton goes deeper nto hs reasonng by statng that each curvature extrema mples a process whose trace s the unque symmetry axs assocated wth, and termnatng at, that extrema. Processes are understood as creatng larger curvature varatons. Therefore, a curve havng a complex shape,.e. lots of curvature varatons, can be seen as the result of a successon of processes appled along the symmetry axes assocated wth the curvature extrema of a smpler curve (the smplest one beng a crcle because of ts constant curvature). Fgure 7-: (a, b) The PISA analyss enables the defnton of symmetry axes. (c) An example corresponds to the traces of the processes that have generated these extrema. (d) The curvature analyss hghlghts the curvature extrema and nflecton ponts. 43

156 CHAPTER 7. SEMANTIC SHAPE OPERATORS The examples of Fgure (7-) show how the curvature analyss of a curve (Fgure 7-.d) enables the dentfcaton of a set of extrema (Fgure 7-.c), and complementarly the PISA analyss (Fgure 7-.b) enables the dentfcaton of the symmetry axes (Fgure 7-.a). Desgnatng M and m respectvely as a local maxmum and local mnmum, followed by + or dependng on ther postve or negatve value, four types of extrema can be dstngushed: M +, m +, M and m. It s mportant to notce that these extrema may be assocated to a semantc meanng accordng to the process orgnatng them: protruson m- ndentaton m+ squashng M- nternal resstance M + (7-) In a relatvely ntutve way, arrows can be assocated to the symmetry axs, to ndcate the drecton of the possble mechancal actons mantanng the shape (Fgure 7-.c). Thus, the curve of Fgure (7-) wll be classfed and represented by the strng M + 0m M m 0M + m + M m +, where the zeros correspond to the nflecton ponts,.e. to the ponts havng a null curvature. Usng such a classfcaton, t s then possble to defne the set of curves havng a gven number of extrema. As an example, exactly twenty one dfferent shapes, havng a maxmum of eght extrema, can be defned and named [84]. Snce the curves are assumed to be closed, one can notce that the proposed classfcaton s nvarant under crcular permutaton and can be read n both drectons,.e. from left to rght or rght to left. 44

157 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Fgure 7-2: Contnuaton and bfurcaton processes as a mean to modfy the curvature evoluton. Gven ths shape descrpton vocabulary, Leyton proposes a set of processes to transform a shape nto another one. Two categores of processes enable a curvature varaton modfcaton: the contnuaton processes, and the bfurcaton processes, whch are both appled at the characterstc ponts of the curve,.e. ts curvature extrema. The behavor of such processes can easly be deduced from the curvature evoluton analyss. What happens when the process havng generated a M + contnues? Effectvely, Fgure (7-.c) shows that whether the value of a M + contnues to ncrease, the general curvature evoluton wll not produce new extrema. Smlarly, whether the value of a m contnues to decrease, the general curvature evoluton wll not be modfed. Therefore, the contnuaton at a M + and the contnuaton at a m (resp. CM + and Cm ) are not consdered as process snce they do not modfy/complcate the curvature evoluton. On the contrary, f one contnues to decrease the value of a m +, t can become negatve and therefore be 45

158 CHAPTER 7. SEMANTIC SHAPE OPERATORS transformed nto a sequence of 0m 0 (Fgure 7-2-upper). Analogously, what happens f the process havng produced a M + bfurcates? In ths case, the M + transforms nto a sequence of M + m + M + (Fgure 7-2-lower). Through the analyss of all the possble confguratons havng an mpact on the curvature evoluton, Leyton has defned a restrcted set of sx NURBS curves deformaton operators processes formng a process-grammar llustrated n the next secton: Cm + : m + 0m 0 CM : M 0M + 0 Bm : m m M m BM + : M + M + m + M + Bm + : m + m + M + m + BM : M M m M Contnuaton Bfuraton (7-2) In Fgure 7-3 t s the overvew of our shape operator. It llustrates the two categores of Leyton grammar,.e. contnuaton and bfurcaton. Ths grammar s then appled to shape adjustment based on the curvature extrema modfcaton. The term grammar s used to emphasze the fact that ths set of rules enables the defnton of sentences (complex shapes n terms of curvature evoluton) from a set of words (smpler shapes). In ths way the tedous lower geometrc dgtal shape representaton level s shfted to grammar based hgher level where the geometrc concepts such as pxel and vector are advanced to characterstc key curves. These curves, represented as curvature descrpton, can be controlled by smply extrema adjustment. All those adjustments, mplemented by addng set of geometrc constrants on the characterstc ponts (curvature extrema), are capsu- 46

159 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING lated as specfc semantc operators. Shape operator Shape form Grammatcal form Shape adjustment operator Acts on Morphologcal operator Dsplacement of extrema Acts on Curvature modfcaton Dscontnuous operator Contnuous operator Contnuaton Contnuaton Bfurcaton Extrema value modfcaton Modfcaton of curvature behavour CM- Cm+ BM+ Bm- Bm+ BM- Fgure 7-3: The Leyton Grammar structure In the followng secton we propose to mplement target curvature-based curve manpulaton by dynamcally detectng the curvature varaton. Furthermore, we detal how we defne the grammar-based operators and mplement aesthetc curve manpulaton. 47

160 CHAPTER 7. SEMANTIC SHAPE OPERATORS 7.2 Deformaton operators based on L-grammar To apply the Leyton grammar to B-splne curves defned by specfc control parameters (control ponts, knot vectors), [92] has expressed a subset of rules. For each assocated process, a correspondng deformaton operator enables to apply automatcally the set of geometrc and parametrc constrants. The deformaton tself s produced by modfyng the statc equlbrum of a bar network coupled to the control polygon of the curve (see chapter.5 for more). In the followng, some examples of these deformaton operators are expressed. To create these operators, a set of the curve constrants are used (see Fgure 7-4): where the constrants are composed by geometrc constrants and parametrc constrants (the pont on a parametrc curve). Those can be represented as tangency, dstance and poston constrants. Based on the fxed control ponts, these constrants can be mplemented as local deformatons. Poston constrants between a curve parametrc pont P(u) and a geometrc pont P 0 n 2D space: P(u) = P 0 ; thus enablng the specfcaton of dsplacement constrants: P(u) = δ; or the specfcaton of blocked ponts: P(u) = 0. Poston and tangency constrants between the curve(s) ponts that must stay connected and smooth durng the manpulaton: P (u ) = P 2 (u 2 ) and P / u (u ) ^ P 2 / u (u 2 ) = 0. 48

161 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Dstance constrants between two parametrc ponts on the curve: P 2 (u 2 ) - P (u ) = d. A dsplacement constrant such that P(u ) = λ. b(u ) where u s the parameter of the characterstc pont m +, b(u ) s the second vector n the Frenet local reference frame n m + and λ corresponds to the norm of the dsplacement. Three blocked ponts: P(u M+ ) = P(u M2+ ) = P(u M3+ ) = 0. Constrants Control ponts fxaton Local constrant Global constrant Poston Tangency Parametrc constrant Constant area Constant length Dstance Blocked Poston Geometrc Constrant 2 Curve parametrc pont Constrant entty Dsplacement Tangency Dstance Acts on Geometrc pont Fgure 7-4: The constrants structure 49

162 CHAPTER 7. SEMANTIC SHAPE OPERATORS All examples use the mnmsaton of the shape varaton n order to access one among the possble shapes. Fgure 7-5 shows how the dsplacement constrant s appled. From a user pont of vew, only the parameter λ has to be specfed, e.g. through a draggng mechansm. The frst example treats the case of the deformaton process assocated to the contnuous rule Cm+. The m+ contnuaton operator (Fgure 7-5), appled at the m -extremum, adds automatcally the followng constrants: ntal shape deformed shape external forces Bar network Fgure 7-5: Quanttatve contnuaton n m+ A dsplacement constrant such that P(u ) = +λ. b(u ), where u s the parameter of the characterstc pont m +, b(u ) s the second vector n the Frenet local reference frame n m + and λ corresponds to the norm of the dsplacement. two blocked ponts such that P(u M+ ) = P(u M2+ ) = 0. 50

163 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING all the NURBS control ponts on the M +M 2 +M 3 + area of the curve are automatcally blocked. Smlarly to the prevous cases, the M+ bfurcaton operator adds automatcally the followng set of constrants (Fgure 7-6): prescrbed dsplacement fxed ponts Fgure 7-6: Bfurcaton n M+ A dsplacement constrant P(u ) = -λ. b(u ) where u s the parameter of the characterstc pont M +, b(u ) s the second vector n the Frenet local reference frame n M + and λ s the norm of the dsplacement. Two fxaton constrants such that P(u O2 ) = P(u O3 ) = 0. A dstance constrant such that P(u O ) - P(u O4 ) = d. 5

164 CHAPTER 7. SEMANTIC SHAPE OPERATORS Fxaton of the curve control ponts havng an nfluence over the curve segment O 2 O 3. Some problems appear n the development of such operators: No control of the operators have been dentfed, to determne a acceptable range of values for the user parameter: for nstance, n the case of the contnuaton rule Cm+, a value too large for λ has for consequence a self-ntersecton of the curve. A hypothess has been posed n ths work: the number of control ponts s large enough to descrbe the resultng curve. Based on these ssues, we further mproved our deformaton operators through nteractve user control and dynamcal curvature varaton detecton, so that we can precsely manpulate curve shape. Meanwhle, a set of semantc curve operators are well defned by combnng basc deformaton procedure and Leyton grammar (L-operators). Those operators provde desgners wth more ntutve and nteractve curve control to capture ther desgnng ntenton n conceptual stylng phase. 52

165 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING 7.3 Dynamc Curvature-based B-splne Manpulaton In free-form shape desgn, far curve segments are requred to be free from undesrable wggles and can be smoothly manpulated by desgner. The optmal aesthetc curve manpulaton for free form shape desgnng should be easer and effectve to capture desgner s ntenton. It thus needs to provde the user wth meanngful enttes for the creaton, manpulaton and analyss the shapes n an ntutve way. Addressng these ssues, we propose hgh level shape operators, encapsulated by a group of basc L-operators, to release the desgners from tedous mathematc understandng of curves. Each operator only apples geometrc and parametrc constrants on few characterstc extrema ponts. We frst present a dynamcal curvature-extrema detecton mechansm, where the characterstc ponts are obtaned through real-tme montorng of the curvature extrema dstrbuton. As a concequence, the curve features can be obtaned by analyzng quanttatve parameters, and then we mplement the semantc curve control by only applyng a group of constrants to these characterstc ponts. In ths way the desgner can ntutvely and nteractvely manpulate the curve to express the mental deas for the aesthetc desgnng. Frst of all let us overvew the propertes decded by curve curvature. In fact, t s easy to determne curve features such as curve orentaton by consderng the curvature dstrbuton. In [93] the work provdes the theory to judge the ncreasng orentaton or decreasng orentaton through evaluatng the 53

166 CHAPTER 7. SEMANTIC SHAPE OPERATORS sgn of the second dervatves d 2 dx y 2 of curve y=f(x). If postve, the slope of the tangents to the curve ncrease as x ncreases (see Fgure7-7 a). We say that a curve s concave up on an nterval I when: Lkewse, f d 2 dx 2 d y 2 dx y 2 d dy ( ) dx dx = >0 for all x n I. s negatve, the slope of the tangents to the curve decrease as x ncreases (see Fgure 7-7 b). We say a curve s concave down on an nterval I when: 2 d y 2 dx d dy ( ) dx dx = <0 for all x n I And a pont of nflecton occurs at a pont where 2 y 2 dx d 2 dx y 2 d = 0, then there s a change n concavty of the curve at that pont. s (a) Concave up (b) concave down. Fgure 7-7: Curve concavty Lkewse, the curvature extrema M+, m+ are the concave down ponts, and m-, M- are the concave up ponts. These features wll be adopted for the followng shape operators. 54

167 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING 7.3. Cubc Planar B-splne Curvatures Normally, a m th -order B-splne curve can be defned by: n = 0 [ t t ] C( u) = N ( u) P u (7-3), m, m, n+ Where {P } are control ponts, {N,m (u)} are the m th -order B- splne bass functons defned on knot vector : T= {t 0 = =t m-,t m,, t n, t n+ = = t n+m }. Our applcaton s concentrated on planner cubc B-splnes,.e. m= 4. For a planar curve C (u) = (x (u), y (u)), the curvature s computed by xy &&& &&& xy ( u) = ( x& + y& ) 2 ( κ (7-4) In the case of B-splne curve, the κ t ) and the frst and second dervatves at t can be obtaned by equatons 7-5 and 7-6: dc du s abbrevated toκ, n + ( u) = m ( ) N+, m ( u), u m, m+ = u+ m+ u+ P 2 d c m( m ) ( u) = 2 du ( u u + m+ P + 2 ) n 2 P + 2 P + P + P (( ) ( )) N+ 2, m 2 ( u) = 0 um+ + 2 u+ 2 um+ + u+ [ t t ] (7-5) (7-6) An orented B-splne curve s a curve such that at every pont a unt normal vector n s defned, provded n(u) s contnuous along the curve. The curvature depends on the 55

168 CHAPTER 7. SEMANTIC SHAPE OPERATORS orentaton, when the orentaton s changed, the sgn of curvature changes accordngly. The curvature vector κ and normal n do not depend on the orentaton. We denote ϕ as the angle between the tangent pont and the postve drecton of the x-axs (see Fgure 7-8). Then the unt of normal vector can be obtaned by equaton 7-7: dt = [ sn ϕ,cosϕ] = n (7-7) dϕ Y n(u) t(u) P (x(u), y(u)) φ X Fgure 7-8: Defnton of tangency, normal and curvature. Lkewse, p s called a convex pont f curvature at p s postve (the curve segment s concave up) and a concave pont f t s negatve (the curve segment s concave down). Therefore, we can obtan the symbolc descrpton of any B-splne curve based on the analyss of curvature dstrbuton. Fgure 7-9 shows an example of the B-splne curve; t s represented as (left to rght) m-0m+m+m+0m- 0M+m+M+. Such symbolc descrpton of a curve thus not only provdes an easy way to precsely classfy the curves but also t further mproves the lower geometrc representaton level to a hgher level. In the followng we further dscuss the curve propertes based on the ntrnsc curvature analyss. 56

169 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING (a) The B-splne curve s represented as m-0m+m+m+0m- 0M+m+M+ wthout consderng the begnnng and endng ponts (orange balls). The blue balls represent three nflexon ponts, and the other pnk balls then denote the curvature extrema. m+ M+ m+ 0 0 M- 0 M+ m+ M+ m- (b) The curvature plot and the extrema dstrbuton. Fgure 7-9: The symbolc descrpton of B-splne and the curvature plot. As t s shown n the curvature plot (Fgure7-9), t s easy to determne the concave pont and convex pont by detectng the sgn of the curvature extrema. Then ths B-splne can be decomposed nto four curve segments based on three nflexon ponts, whlst between each curve segment t 57

170 CHAPTER 7. SEMANTIC SHAPE OPERATORS keeps C 2 contnuty. For each curve segment we further propose a set of quanttatve parameters to evaluate the curve propertes. Fnally we apply deformaton operators to match desgner s aesthetc desgn requrements based on the analyss of specfc curve propertes whch wll be detaled n the followng sectons Quanttatve Parameters for Curve Analyss Leyton s grammar allows a dscrete classfcaton of shapes nto classes of equvalence. However, desgners typcally need hgh capactes for the specfcaton of aesthetc curves. To take nto account ths problem, we have extended the descrptve capactes by addng some quanttatve (and contnuous) characterstcs, allowng dstngushng curves of a same class, and some quanttatve operators, allowng manpulatng a curve wthout modfyng ts name. These quanttatve characterstcs are always attached to the characterstc ponts whch are descrbed as followng (Fgure 7- ): Curvature varaton k X wth respect to a characterstc pont CP, where X desgnate ether the left L or the rght R evaluaton drecton. The left (resp. rght) value s determned by the dfference between the curvature value of CP and the characterstc pont on the left (resp. rght) accordng to the curve parametersaton. 58

171 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Dstance D X denotes the dstance between a characterstc pont and the one on the left (resp on the rght), computed on the curve. Range of nfluence R X of a characterstc pont CP. To determnate the left (resp. rght) range, we take the average of the left (resp. rght) curvature varaton and we search for the pont P X on the left (resp. rght) of the characterstc pont whch has curvature value equal to (k (CP) ± k X /2). The curve porton between P X and CP s the left (resp. rght) range of the characterstc pont. It can be noted that all the ranges of characterstc ponts cover the entre curve. Curvature a) M+ M+ Curvature b) M+ M+ k R ½ k R 0 0 m+ s 0 0 m+ s m- (a) D L Quanttatve characterstcs m- (b) R R 59

172 CHAPTER 7. SEMANTIC SHAPE OPERATORS Curvature Small vsblty Curvature Sharp corner Redundant extrema s (c) (d) Fgure 7-0: The quanttatve parameters and the property analyss. (c)small vsblty (d) Sharp corner and redundant extrema We further present some propertes of curve based on the quanttatve parameter analyss: nvsblty, sharp corner and redundant extrema propertes. Those are descrbed as followng: Small vsblty: Ths characterstc s used to classfy the detals, consderng that a characterstc pont s a detal f t has both a small dstance and a small curvature varaton. The codon s usually 0m-0 or 0M-0. It s calculated as the product of the dstance and the curvature varaton at that CP. IF (D L < γ) OR (D R < γ) AND IF ( k L < β)or ( k R < β) As the user-defned parameters for the evaluaton, the γ and β denote the thresholds of dstance varaton and curvature changng. Sharp corner: t s used to detect f there s dramatc curvature varaton between two curva- 60

173 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING ture extrema (see Fgure 7- a). IF (D LR < γ) AND ( k L >µ) AND ( k R > µ) Symmetrc property: when the characterstc pont CP has equvalent curvature varaton between left and rght curvature extremum, whlst they have the same codon (C X ) such as M+m+M+, m-m-m- (see Fgure 7- b). IF ( k L = k R ) AND (D L = D R ) AND (C L = C R ) Redundant extrema: when the curve segment has frequent curvature varaton wthn a small dstance, t certanly causes unnecessary undulatons. The codon of such curve segments mght be M+m+M+, m+m+m+, m-m-m-, and M-m-M-. Then we can optmze them as M+. m+, m- and M- to obtan smoother effects (see Fgure 7- c). C L = M + C R = M + 2 m + D L D R (a) Ths curvature plot shows two undesred features on the curve,.e. sharp corner and redundant extrema 2. The red balls represent the correspondng curvature extrema. The dot lne represents the prevous state; the bold one s the current optmza- 6

174 CHAPTER 7. SEMANTIC SHAPE OPERATORS ton. (b) The curve segment M+m+M+ s symmetrc. Redundant extrema M+m+M+ M+ Sharp (c) (d) Fgure 7-: Handle the undesred propertes on the curve. (c) The undesred features on the curve. (d) The correspondng manpulaton Aesthetc Propertes of Curve As seen n the prevous sectons, the quanttatve parameters can detect undesred features of curves. Furthermore consderng the concepts of the aesthetc desgnng, we propose a set of hgh-level aesthetc operators. It has long been recognzed that to have aesthetcally good shapes for curve stylng, the number of extrema n the curvature dstrbuton s manly consdered. Normally the number of separate segments wth monotone curvature should be the mnmum requred to meet the aesthetc ntent of the desgner. Although dfferent types of aesthetc propertes exst n the doman of desgn, aesthetc propertes need to be structured, at least partally, and desgnated to enable desgners 62

175 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING to communcate among themselves and wth other actors durng the PDP (Product Development Process). For ths reason, ths complementary study concentrates on the propertes dentfed durng the FIORES-II European project [88] connected to the terms used by stylsts for expressng the desred shape modfcatons and to help them communcatng ther deas. To ths end, the study has been performed through desgners ntervews to dentfy ther specfc terms and ther correspondng meanng. The defntons and relatons between words and geometrc quanttes have been fnalzed. In partcular, the concepts of acceleraton, softness/sharpness, tenson, convexty/ concavty, flatness, crown, have been specfed together wth ther measures. It s mportant to notce that several geometrc characterstcs/varables contrbute to a sngle property, thus requrng a further level of nterpretaton to gve a formal descrpton both of the property and of ts measure. Also, t must be noted that whle t s n general mpossble to generate a curve wth a gven specfc property quantfcaton, t s much more meanngful to modfy an exstng one by ncreasng/decreasng the parameters quantfyng such a property. Here, the defntons are summarzed and ther measurement parameters attached are lsted [89]. Consderng a whole open curve, ts acceleraton s related to how much the varaton of the tangent to the curve s dstrbuted along t. Fast or slow acceleraton means that the curvature ncreases fast/slowly at proxmty of one end pont of the curve (see Fgure 7-2). If a curve changes curvature slowly t may show no acceleraton at all. One could defne a measure of acceleraton by the rato of curvature dfference k and arc length l where the dfference happens: 63

176 CHAPTER 7. SEMANTIC SHAPE OPERATORS acceleraton = k l Fgure 7-2: Curves and ther curvature plot, wth acceleraton ncreasng top-down The term softness/sharpness s used to descrbe the propertes of transtons between two adjacent curves or surfaces. In the stylng actvty, the term radus s generally used to ndcate the curvature dstrbuton n the neghborhood of the transton blend between these two curves or surfaces. Ths property s therefore local to an area around the common pont/curve shared by the two adjacent enttes. Generally, a small radus desgnates a sharp area, and a large one characterzes a soft transton (see Fgure 7-3). Then, makng a radus sharper (softer) means to decrease (ncrease) the radus of the blend. The meanng of bg and small depends on the szes and proportons of the curves to be connected. When we gve measures we wll concentrate on the mnmum radus of a gven blendng curve and say that a sharp radus s a small radus, whle a soft radus s a large one: softness = radusmn = sharpness 64

177 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Fgure 7-3:Example of sharpenng property (sharper from the left to the rght). Flatness s smply related to how much the curve gets closer to a straght lne. Curves ncorporatng some nflexon ponts are referred to wet curves or S-shaped. Flatness could be measured by the rato of maxmum and mnmum curve elongaton, whch s the wdth and the heght of the curve s mnmum-area encasng rectangle (see Fgure 7-4). In order to make the flatness range from 0 to we use: d Flatness = mn d max Fgure 7-4: Examples of straght lnes (straghter from the top to the bottom). When desgners make an open curve more convex (or 65

178 CHAPTER 7. SEMANTIC SHAPE OPERATORS concave, n the opposte drecton), they are movng towards ts enclosng sem-crcle (Fgure 7-5). Thus, the deal convex curve s an arc of crcle, assumng t s compatble wth the contnuty constrants at the endponts. Otherwse, t s the curve presentng the lowest varaton n curvature that satsfes the gven contnuty constrants,.e. a spral. Convex can be measured by, where the sgned area under the curve s lmted by the lne between the two curve end ponts, wth postve values standng for convex and negatve ones for concave: d Convexty = sgnedarea * mn d max Fgure 7-5: Example of convex property (more convex from the bottom to the top). Accordng to a user s feelng, Straght lnes have ether no tenson or an nfnte one. Tenson has been defned as the nternal energy of an open curve subjected to contnuty constrants at ts boundares, provded t s not a straght lne. Ths can be geometrcally translated nto an evoluton of curvature along the curve, whch means that ncreasng the tenson of the curve leads to a larger area hav- 66

179 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING ng a small curvature (Fgure 7-6). A frst attempt for measurng tenson would be the rato of curvature extrema, but n order to be more global one could set the curvature dfference n relaton to the average curvature: Tenson = k max kmn k avg Fgure 7-6: Curves and ther curvature plot, wth tenson ncreasng top-down. Crown means lftng or rasng a certan part of the curve n a gven drecton, wthout changng the end ponts, and elmnatng the nflexon ponts, f any, whle creatng a convex part. As already mentoned, desgners are used to adopt a lmted set of curves and to express ther own aesthetc percepton of the product. Some characterzatons are nterpreted n a standard way by desgners; therefore the APs (Aesthetc operators) are concepts ntrnsc to the stylst s envronment and are not referrng to any geometrc model. These concepts defne some of the ntrnsc parameters needed to model and manpulate semantcally a shape n the context of a 2D sketch. The next chapters wll present these 67

180 CHAPTER 7. SEMANTIC SHAPE OPERATORS 2D semantc aesthetc operators and how they are connected by set of geometrc constrants based on basc Leyton shape grammar. 7.4 A Mappng between Aesthetc Operators (ASP) and Process Grammar Once havng dentfed how the APs (Aesthetc operators) are lnked to the curve characterstcs, the second step s to dentfy how to modfy these quanttes through the use of the L-system curve modfcaton operators. Straghtness Operator In aesthetc desgn, such an operator does not create a straght lne but t deforms a curve to tend t towards flatness. In our understandng, ths operator can be decomposed nto two sub-operators dependng on the ntal shape. Frst, f the ntal lne s nosy,.e. wth some undulatons along t, the operator frst elmnates such undulatons. The am of the process s to manly make the codons havng the smallest vsblty dsappear. Furthermore, the operator decreases the complexty of the name of the lne f t has the redundant extrema feature by usng the nverses of grammatcal bfurcaton and contnuaton. As an example, n Fgure7-7, the black curve has name M+0m-0M+, and the codon 0m-0 s the one wth 68

181 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING the smallest vsblty or undulaton property. The nverse of the process assocated to the rule Cm+ s performed, to obtan the green curve (see Fgure7-7-c green curve), wth name M+m+M+. The process can be descrbed as: Frst, the nverse of the quanttatve contnuaton C*m- s appled to the characterstc pont m-, untl ts curvature value s equal to zero. At ths pont, the nverse of the Cm+ rule s appled, to transform the codon 0m-0 nto the characterstc pont m+. Fnally, the nverse of the quanttatve contnuaton C*m+ s appled. The green curve s stll a nosy one, snce ts name s complex accordng our redundant extrema analyss, and the nverse of the process assocated to the rule BM+ s appled on the curve to obtan the red one n the followng step: The nverse of the quanttatve contnuaton C*m+ (appled n the prevous operaton) s stll appled on the m+, untl ts curvature value s equal to the curvature value of one of the two M+ surroundng the consdered pont. The nverse of the BM+ rule s appled, to transform the mddle codon M+m+M+ nto the characterstc pont M+. The fnal result s a curve wth only one curvature extrema M+, as t s for the red curve n Fgure7-7-b. 69

182 CHAPTER 7. SEMANTIC SHAPE OPERATORS M + M + m - M + (a) Before manpulaton, the curve and the curvature llustraton (red part). (b) After manpulaton. (c) the black curve s orgnal curve, and the green one s the result after the frst nverse Cm- operator, the name s M+m+M+ ; the red curve s fnal one by usng nverse BM+ operator, the representaton of red curve s M+ M+ M+ M+ M+ 0 0 M+ m- m+ (d)curvature plots of the black curve, the green one and the red one Fgure 7-7: The orgnal B-splne curve (black one) s manpulated by our straghtness operator. The each result s llustrated by correspondng curvature plot Sharpness/softness Operator Ths operator acts on a curve segment connectng two regons of small curvature, and n practce, t acts on the char- 70

183 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING acterstc pont of M+ or m-. Makng ths blendng sharper means ncreasng the promnence of a corner between the two curves. At the same tme, the process cannot generate unwshed undulatons,.e. cannot nsert new characterstc ponts n the curve. As a consequence, the sharpness/softness operator wll be drectly translated by the operator C*M+ (or C*m-, dependng on the ntal stage for the curve): the curvature value of the characterstc pont wll ncrease or decrease (Fgure7-8). We here ntroduce a known mplementaton by smultaneously addng set of poston constrants and tangency constrants on the characterstc ponts. Addng dynamc poston constrant on the mddle-codon curvature extremum m- or M+ by user s nteractve draggng operaton n the follwong way: P (u m-) = d; In order to avod producng new characterstc ponts of ths curve segment, we smultaneously fx the tangences of the left extrema pont P l and the rght extrema pont P r. P l Pl u( u ) u( u l l P r Pr 0 ; 0 ) u( u ) u( u ) r r 7

184 CHAPTER 7. SEMANTIC SHAPE OPERATORS m m m (a) (b) (c) Fgure 7-8: Sharpness/softness operators. (a) Before manpulaton. (b) After softness operaton. (c) After sharpness operaton Convex/concave Operator The am of ths operator s to transform a curve so that t becomes closer to the enclosng half-crcle. The frst step of ths operator s to delete the small vsualty (nflexons) n order to obtan a sequence of type M+m+M+ or m-m-m- : the nverse of the processes assocated to the protruson rules are used on each characterstc pont that has a small vsblty. The second step s to apply on the mddle-codon curvature extremum the dsplacement operator n the drecton of the perpendcular bsector of the segment connectng the curve endponts n order to equlbrate the left and rght dstance (from the lower curve to the mddle one n Fgure7-9): n ths way, we obtan a more symmetrc curve. The mplementaton by smultaneously addng geometrc constrants on the characterstc ponts are shown as followng: 72

185 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING Addng dynamc poston constrant on the mddle-codon curvature extremum m+ or M- by user s nteractve draggng operaton: P (u m+) = d; or P (u M-) =d; Meanwhle the changng of the mddle-codon curvature extremum wll produce the adjustment of two nflexon ponts along ther normal vector drecton n (u), the scales of the changng of these two nflexons are determned by the dstance between the mddle-codon curvature extremum and ths nflexon ponts. We thus can symmetrcally control the curve varaton,as n: P(u 0L) =λ L n(u 0L ) d ; P(u 0R)= λ R n(u 0R ) d; λ L = D L ; λ R = D R ; When there s no nflexon on ths curve, the λ L and λ R are automatcally set as 0. Then the poston constrant s only appled on the mddlecodon curvature extremum to adjust the promnence of the concave or convex. 73

186 CHAPTER 7. SEMANTIC SHAPE OPERATORS Fgure 7-9: Curve change by usng concave operator and the curvature plots. Tenson Operator Tenson can be perceved when one curvature mnmum wth a small curvature value n-between two curvature maxma wth hgh curvature values. As for the prevous operators, the frst step of the tenson operator s to suppress the undulatons n order to obtan a name of type M+m+M+ or m-m-m-. After that, more tenson s obtaned by applyng the quanttatve contnuaton operator C*m+ (C*M- dependng of the ntal stage) to decrease the curvature value of the curvature extremum to tend to 0 (Fgure7-20). Ths operator can be also appled wth tangency contnuty condtons at the endponts of the curve. In ths case, 74

187 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING the curvature value of the two other extrema wll ncrease feasbly. m- M- m- M- m- m- Fgure 7-20: Tenson operator and two examples. Acceleraton Operator A curve s sad accelerated when the varaton of the tangent s bgger around one end pont when movng towards that pont, a straght lne or a true radus have no acceleraton at all. The acceleraton operator s meanngful only f appled to curves whch have already the acceleraton property. Acceleratng the extremum on the rght means ncreasng the curvature value of the extremum wthout modfyng the poston of ths pont. To obtan ths result, two quanttatve operators have to be appled smultaneously: the quanttatve contnuaton operator C*M+ (resp. C*m-) to ncrease the curvature value of the curvature extremum, and the curvature extremum dsplacement DM+ (resp. Dm-) to correct the dsplacement of the extremum generated by the contnuaton operator (Fgure7-2). As a consequence the left dstance for ths extremum and the rght range of nflu- 75

188 CHAPTER 7. SEMANTIC SHAPE OPERATORS ence of the next extremum (or nflexon pont) wll ncrease. Ths means: The tangency constrant s added to one endng pont to ncrease the value; and t keeps another endng pont statc: P / u (u ) = β And P 2 / u (u 2 ) = 0 Fgure 7-2: Acceleraton operator: one endng pont s dynamcally ncreasng the tangency value by user s draggng operaton 7.5 The Management of Geometrc Constrants One of the man problems, when desgnng a deformatonbased envronment, s to provde wth an n adequacy the number of constrants, mposed by the system to obtan a specfc shape, and the number of degrees of freedom of the shape, essentally the number of control ponts n case of B- splne curves. The symbolc descrpton can allow the system to have some hnts to know where and how many degrees of freedom to add n the geometrc model. 76

189 INTERACTIVE SHAPE MODELING AND DYNAMIC DEFORMATION BASED ON SPLINE SCULPTING m- 0 0 M+ M+ m- 0 0 M+ M+ (a) (b) Fgure 7-22: Planer curves wth 0 DOF and wth 8 DOF For nstance, when we mplement the softness/sharpness operaton, we add poston constrants on two endng ponts to fx the B-splne segment (blue balls n Fgure7-22), and smultaneously we keep the tangency and poston of the left characterstc pont and rght one (grey balls). We then nteractvely drag the selected extrema pont m- (pnk ball); n ths way we mpose 4 constrants to ths B-splne segment. For the case of (a), t s obvous there s no enough freedom to obtan soluton for ths operaton, on the contrary, when we ncrease the DOF of the B- splne curve (see Fgure7-22 (b)); we can thus have more possbltes to get adequate deformaton results. Another example s durng the creaton of nflexon ponts nto a subset of a curve. A theorem n case of Bézer curves expresses the fact that the number of nflexon ponts on a curve s an nferor of the number of ntersecton ponts between the curve and the control polyhedron [90] (Fgure7-23a). The value s only an nferor, so t means that an ntersecton pont between the curve and the control polyhedron 77