G 1 Interpolation of arbitrary meshes with Bézier patches

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1 Master s Degree i Applied Mathematics Supervised by Stefaie Hahma March 5th - July 5th 007 G Iterpolatio of arbitrary meshes with Bézier patches Jea Kutzma Lab. B.P Greoble Cedex 9

2 Cotets Ackowledgmets iii Itroductio. Cotext Defiitios Tesor-product surfaces Bézier patches The G Cotiuity Related work Our cotributios G Iterpolatio with 4-Splittig method 8. Global method Splittig method Why do we split? The 4-split i terms of Bézier Cotiuity coditios Notatios G coditios Methods The problematic The first iterpolatio scheme The secod iterpolatio scheme Theoretical formulatio 7 3. The first iterpolatio scheme Choice of scalar fuctios Φ i,ν i,µ i Costructig curve etwork Cross-boudary tagets The macro-patches i Bézier form A Optimizatio of the scheme What is the optimizatio? Geeral approach Optimizatio applied to boudary curve etwork The secod iterpolatio scheme assets Motivatios Choice of Φ i, µ i, ad ν i Boudary curves ad cross-boudary tagets i

3 CONTENTS 4 Results 4 4. First iterpolatio scheme Usig optimizatio Toward the secod iterpolatio scheme Coclusio Summig up Future works A Explicit Formulas 49 A. The first iterpolatio scheme A.. Computig the twist A.. Computig cross-boudary poits A..3 Computig first row ier cotrol poits A..4 Optimizig the secod derivatives A. The secod iterpolatio scheme A.. Computig fuctio H i A.. Computig fuctio W i ii

4 Ackowledgmets I would like to ackowledge greatly Stéfaie Hahma ad George-Pierre Boeau for havig give me this subject i Master s fial project. Eve if my itership s subject was based o their researchs, they let me work freely i orgaisatio ad directio. They were preset to help me whe I eeded their guidelie. Fially, I would like to thak my office s parters for the time we shared together. iii

5 Part Itroductio. Cotext I Computer Aided Geometric Desig (CAGD) history mathematical tools describig objects geometry were always fudametals. Maipulatio with computatioal methods of these free form objects implied a icreasig developmet i computer sciece ad applied mathematics ivestigatio. The works over the years were motivated by applicatios i CAD/CAM (Computer aided desig/ Computer aided maufacturig), i medical area with Magetic Resoace Imagig, i computer-aimated films, architecture,... At the same time, ew aesthetical costraits appeared i object modelig. For istace, i vehicle idustry, smoothess is a importat sellig criterio. For that matter, mathematical tools for geometry modelig were bored i vehicle idustry : Bézier for Reault, De Casteljau for Citroë, Coos for Ford,... A surface is commoly created from a data set which characterizes more or less precisely the surface shape. These data are ofte of oe of the followig forms : a etwork of curves with or without restrictios o its topology; or a set of poits without topological coectios. A restrictio o its topology meas it would be ay umber of curves meetig at the odes of the etwork. From those data sets, various models have bee developed for mathematically modelig the surface (Bézier patches, Coos patches,...). I my work, I use the Bézier patches. I iteractive CAGD applicatios, they are curretly oe of the most widely used models for free-form surface modelig. This is due to its umerous geometric properties ad its simplicity of maipulatio. However, objects, eve simple, ofte have complex shapes. Thus it is ofte difficult to represet them usig a global approach. Two differet research directios have bee pursued. Oe is based o subdivisio surfaces that recursively subdivide the cotrol-mesh util a global smooth surface. The other directio cosists of buildig a patchwork of smoothly joied parametric patches, with the same topology as the cotrol-polygo. My work deals with this last kid of surface. The most importat for modelig a surface usig piecewise represetatio, is to cotrol the desired cotiuity betwee adjacet patches for the resultig surface to be cotiuous. As we saw, i vehicle idustry smoothess is a very importat sellig criterio. Thus I will focuse my work o modelig surface with a visual cotiuity (ad ot a parametric cotiuity). I other words, I esure the

6 . Defiitios taget plae coituity betwee adjacet patches. It will be explaied more precisely below. I most of the modellig situatios, the Bézier patches are defied over a topologically rectagular mesh. Usig them to build a global surface will imply a global surface with a topological type equivalet as a square. Ufortuately, -maifold surfaces with arbitrary topological type are very commo i everyday life. For istace, we ca imagie a cup of coffee which has the topology of a torus. For this reaso, may researchs have bee devoted to the use of triagular Bézier patches for modelig that kid of surface. However, as we will see, iterpolatio with rectagular patches permits a lower degree of the Bézier patches tha i triagular iterpolatio. Moreover, i the idustry, the rectagular patches are widely used, like i CATIA, the famous Dassault Systems tool. Thus, to model ay surface with rectagular patches, the use of meshes of ay topology (i.e. with ay umber of patches meetig at the odes) becomes ecesary. To guaratee geometric cotiuity of the resultig surfaces, oe must solve the smooth coectio problem betwee adjacet patches, ad especially at a corer where N patches meet.. Defiitios.. Tesor-product surfaces Ituitively, a tesor-product surface ca be viewed as a curve guided by aother curve. Mathematically, the surface is the result of the tesor product betwee two parametric curves. The theoretical defiitio of a tesor product is ot useful for the work, therefore we will just give how we ca use it o surfaces. Simply, a tesor product betwee two tesors is the product of each compoet of the first tesor by each compoet of the secod (which dimesio would be differet). Here, a tesor meas a quatity which ca be represeted as a multidimesioal array, related to a space basis. For istace, matrices ad vectors are tesors ad scalar product is a particular tesor product. Now, we itroduce a algebraic otatio of the tesor product. Let X,Y be fiite fuctio spaces with the respective basis : X 0 (u),...,x (u) ad Y 0 (v),...,y m (v). We ca defie two parametric curves X(u),Y(v) by X(u) = p 0.X 0 (u) p.x (u) Y(v) = q 0.Y 0 (v) q m.y m (v) Let S(u, v) be the tesor-product surface costructed as the tesor-product of the two previous curves X,Y. It is the liear combiatio of each basis fuctio of X by each basis fuctio of Y. The coefficiets ca be easily calculated from X coefficiets ad Y coefficiets. S(u,v) = i=0 j=0 m ( ) pi.q j Xi (u).y j (v)

7 . Defiitios.. Bézier patches I the whole work, a Bézier patch ca be idistictly amed Bézier surface or Bézier patch. Defitios A Bézier Patch of degree (,m) is defied as the tesor-product of two Bézier curves. I additio, a Bézier curve of degree is defied by B(u) = b i Bi (u) where the b i are the cotrol poits for the Bézier curve, ad ( ) Bi (u) = ( u) i u i i i=0 are the Berstei polyomials of degree. They form a basis for the vector space Π of polyomials of degree. This leads to the parametric formula M(u,v) = i=0 j=0 m b i,j Bi (u)bj m (v) (.) M is the tesor-product Bézier surface of degree (,m), where u,v [0,] [0,]. The set of Bézier poits is refered to as the Bézier et. The coefficiets i. ca be either vector-valued or real-valued. Whe b i,j R 3,. leads to a surface i R 3 described by a fuctio defied over the uit square. Evaluatig. at v = 0 leads to a Bézier curve of degree : C (u) = i=0 b i,0bi (u); Likewise, evaluatig equatio. at u = 0 leads to a Bézier curve of degree m : C (v) = m j=0 b 0,jBj m(v). Operatios o Bézier patches Computig the partial derivatives at a poit o a Bézier surface is easy. Sice the cotrol poits are costats ad idepedet of the variables u, v, computig the partial derivatives reduces to the computatio of the derivatives: d r du r B i (u) ad d s B m j dv s (v) where i = 0,..., ad j = 0,...,m. The partial derivatives of M respect to u, resp. v, are give by the followig formulas: r r u r M(u,v) =! ( r)! s v s M(u,v) = m! (m s)! m i=0 j=0 m s i=0 j=0 r0 b i,j B r i (u)b m j (v) (.) 0s b i,j B i (u)b m s j (v) (.3) 3

8 . Defiitios where the forward differeces are give by r0 b i,j = r,0 b i+,j r,0 b i,j & 0s b i,j = 0,s b i,j+ 0,s b i,j Next, the mixed derivatives are give by r+s u r v s M(u,v) =! ( r)! with rs b i,j = r k=0 l=0 r m! (m s)! m s i=0 j=0 s ( )( ( ) k ( ) l r s k l rs b i,j B r i (u)b m s j (v) (.4) ) b i+r k,j+s l At last, the first derivative perpedicular to the boudary u = 0 is m u M(0,v) = (b,j b 0,j )Bj m (v) (.5) j=0 Like Bézier curves, it is possible to raise the degree of a Bézier surface. Let us recall this property: Lemma.. (Degree Raisig) Suppose X(t) is a Bézier curve of degree correspodig to Bézier poits b i. The, X ca be writte as a Bézier curve of degree + with ew Bézier poits b ew i = i + b i + ( i + ) b i, i = 0,..., + b = b + = 0 Raisig the degree of a Bézier surface, is raisig the degree i u-directio ad ext, i v-directio (or iversely). The, the Bézier et coverges to the surface (Weierstrass approximatio theorem). To evaluate a polyomial writte i terms of the Berstei basis at a give poit (u,v) = (u 0,v 0 ), we use the de Casteljau algorithm. First, the algorithm is applied at v = v 0 to compute the Bézier poits m b i = b i,j Bj m (v 0 ) i = 0... j=0 Ad the the de Casteljau algorithm is applied at u = u 0 with the previously computed Bézier poits. It gives the value of a Bézier surface at (u,v) = (u 0,v 0 ): m M(u 0,v 0 ) = b i,j Bj m (v 0 ) Bi (u 0 ) i=0 j=0 } {{ } b i Of course, the de Casteljau algorithm ca also be applied first with u = u 0, ad the i the secod step with v = v 0. Moreover, the ext-to-last elemets i the de Casteljau scheme give the directios of the partial derivatives M M u, resp. v. The de Casteljau algorithm implies a ice property for Bézier surface, preseted as a lemma Lemma.. A Bézier surface lies etirely i the covex hull of its Bézier poits. Geometrically, it meas that cotrol graph mimics the surface shape. It gives us a powerful tool to maipulate the surface. 4

9 . Defiitios..3 The G Cotiuity We cosider ow the smooth coectio problem betwee two adjacet Bézier patches. Smooth coectio problem is refered to cotiuity costraits at the juctio. Traditioally, the costraits correspod to the equality of the parametric derivatives alog the commo boudary. For istace, two adjacet surfaces are said to be C 0 -cotious if they share a commo boudary of the same degree, ad, by extesio, commo cotrol poits for the boudary curve. They are said to be C -cotious if, i additio, the first parametric derivatives are idetical alog ad across their commo boudary. Thus, the parametric cotiuity depeds o the parameterizatio of the surfaces, ad so is less sigificat geometrically. Less strog ad geometrically more sigificat cotiuity must be idepedet of the parameterizatio of the surfaces. The cotiuity obtaied is called geometric cotiuity (or visual cotiuity). For istace, two adjacet patches are said to be G cotious if the taget plae is cotiuous alog the commo boudary. I this work, G cotiuity is sufficiet, ad the work highligths methods which esure such cotiuity for the global surface. Let P ad Q be two Bézier patches of degree (,m), resp. (m,p), such that: P(u,v) = Q(v,w) = i=0 j=0 m k=0 l=0 m b P i,jbi (u)bj m (v) p b Q k,l Bm k (v)bl (w) They share a commo boudary curve writte: Γ = m c i Bi m (v) i=0 See Fig... Mathematically, adjacet patches P ad Q joi at a commo boudary with G cotiuity if ad oly if there exist three scalar fuctios Φ,ν ad µ such that Φ(v) v P(0,v) = Φ(v) Q(v,0) = ν(v) Q(v,0) + µ(v) P(0,v) (.6) v w u where ν(v).µ(v) > 0 (preservatio of orietatio) ad P(0,v) Q(v,0) 0 (well defied ormal vectors). Usig derivative formulas see previously with. ad.3, equatio.6 ca be expressed i terms of Bézier cotrol poits 5

10 .3 Related work P v Q v P(0,v) u P(0,v) w Q(v,0) u w Figure.: Two adjacets patches which are joiig smoothly alog the commo boudary curve. v P(0,v) = m v Q(v,0) = m j=0 m = m i=0 m = m k=0 m w Q(v,0) = p = p i=0 m i=0 ( ) b P 0,j+ b P 0,j ( ) b Q i+,0 bq i,0 B m j (v) B m i (v) (c k+ c k ) B m k (v) ( b Q i, bq i,0 ) B m i (v) ( ) b Q i, c i Bi m (v) m u P(0,v) = ( ) b P,i b P 0,i Bi m (v) i=0 m ) = (b P,i c i Bi m (v) Thereby, P(u,v) ad Q(v,w) joi with G cotiuity if ad oly if there exist three scalar fuctios Φ,ν ad µ such that i=0 m Φ(v).m (c j+ c j ) B m j (v) = ν(v).p j=0 m j=0 +µ(v). ( ) b Q j, c j Bj m (v) (.7) m j=0 (b P,j c j ) B m j (v) To balace the degrees i coditio.7, the degree of Φ has to be greater by tha the degree of ν ad µ..3 Related work The problem of defiig a surface from a mesh of poits has received a great deal of attetio i the computer aided geometric desig literature. The geeral 6

11 .4 Our cotributios problem of costructig a surface from a irregular mesh i space has bee cosider by may, Piper [Pip87], Peters( [Pet90b], [Pet90a], [Pet9] ad [Pet93]), Loop [Loo94], Hahma & Boeau ( [HB00], [HBT00] ad [HB03]). A widely accepted ad popular way i defiig surfaces without ay limit of topologies is the use of smoothly joied triagular patches, where each patch is defied over the uit triagle. They have the advatage to provide a uiform descriptio for all possible topologies without ay restrictio o the umber of faces that meet at a vertex or the umber of edges of the faces. Previous works ca be divided ito differet groups depedig o how they solve the vertex cosistecy problem, which occurs whe joiig G -cotiuously a eve umber of patches aroud a vertex. This problem is defied i the review by Du ad Schmidt [DS90], ad i the icotrovertible [HL93] by Hoschek ad Lasser. Clough-Tocher-like domai splittig methods were developed, like i [Pip87]. Boudary curve schemes were developed by Peters i [Pet9] ad [Loo94] : oe first creates C cosistet boudary curves ad the fills i the patches polyomial. Loop uses patches of degree 6 to satisfy G costraits. The 4- splittig method itroduced i [HB00] permits to decrease the boudary curves degree, ad so to get 5-degree patches. I our method, we adapt this method for rectagular patches ad it allows bicubic patches. Previous works o rectagular patches or a mix betwee rectagular ad triagular patches were preseted by Peters i [Pet90a]. However the data here had to be cosistet with the secod fudametal form. This is a severe restrictio. Liu ad Zhou have preseted i [LZ05] coditios such that multisided Bézier patches G -cotiously fill ay -sided holes, formed by rectagular Bézier patches of ay degrees, so adapted for irregular meshes. Ufortuately, they used ratioal patches. Others tried to use optimizatio methods geeratig smooth surfaces, like Pottma i [Pot9] or Westgaard ad Nowacki i [WN0]. However, eve if the resultig surfaces are smooths, G -cotiuity costraits are o loger preserved..4 Our cotributios Oly a very few papers deal with quadrilateral patches. O this case, they are ofte of relatively high degree ad they make restrictios o the iput data. O the preset thesis, we itroduce a ew costructio of quadrilateral G surface of arbitrary topology with very low degree patches (bicubic i the first iterpolatio scheme). No restrictios are doe o iput data i cotrary to all existig previous works. We use the 4-split method preseted i [HB00] ad [HB03] ad show that smooth G surfaces ca be obtaied by iterpolatio o the vertices of ay arbitrary give quad mesh. We fially preset two differet methods. Oe cosists i a iterpolatig bicubic G rectagular splie surface which is based o [HB00] ad [Loo94]. Secod cosists i a likewise iterpolatio, avoidig malformatios eve whe iterpolatig irregular iput meshes. This implies usig biquitic G rectagular splie surface. 7

12 Part G Iterpolatio with 4-Splittig method. Global method Let M deote the iput surface mesh. It cosists of a list of vertices ad a list of edges. Together they describe a -maifold mesh i R 3 whose faces are quadrilaterals. The umber of faces / edges icidet i oe vertex is refered to as order of a vertex. The order of the vertices is assumed to be greater tha. I other words, oly closed surfaces are cosidered. We aim to costruct a piecewise rectagular surface S that iterpolates the give vertices V. The surface is composed of tesor-product patches M i which are i oe-to-oe correspodece with the mesh facets. They are all polyomial images of the uit rectagle i R joiig G cotiuously. The boudary curves of the patches M i are costructed i correspodece with a mesh edge. For each edge, we will have a Bézier curve, shared by two patches, which lik two vertices. Clearly, the boudary curves have the most importat role i the global surface shape. Moreover, we wat them of the lowest degree possible. All the patches will be of the same degree. Two patches will share the same cotrol poits alog the commo boudary. This esures G 0 cotiuity betwee patches. The G cotiuity implies costraits o the taget plae betwee two adjacet patches. Thereby, the cross-boudary tagets, alog the boudaries, have to belog to the same plae, amely the taget plae. Sice patches are Bézier surfaces, remaiig ier poits satisfy C cotiuity costraits. Therefore, the work is based o the global algorithm. Algorithm G Iterpolatio Algorithm Require: Quadrilateral mesh Esure: G surface iterpolatig the iput mesh. Step : Costruct boudary curves Step : Costruct cross-boudary tagets Step 3 : Fill-i the patches Ed The figure. shows the parameterizatio adopted. All the superscripts ad 8

13 . 4-Splittig method subscripts are take modulo where is the order of the mesh vertex. M i (ui,ui+) ui+ M i ui ui M i ui Figure.: Parameterizatio used i the work. u i ad u i+ belog to the real iterval [0, ]. 4-Splittig method.. Why do we split? The formula.7 is more descriptive tha costructive. Several authors have used coditio.7 to develop differet solutios for the G smooth coectio betwee two adjacet patches. These solutios ofte differ i the choice of Φ,µ,ν ad i the choice of the degree of the patches. Obviously, we would wat to fid patches of the lowest degree, ad keep at the same time, a local iterpolatio. I other words, whe oe modifies a vertex, the deformatio is effective just i the vertex eighbourhood. Loop, i [Loo94], chose sextic triagular patches, µ ad ν were costats, ad Φ quadratic. He had to take Φ quadratic i order to keep the local iterpolatio. That is, he does ot wat that the first derivative of Φ depeds o the opposite vertex iformatio, like its order. Thus Loop had to take it quadratic. However Hahma ad Boeau have foud a splittig method which permits keepig the local property ad havig Φ liear. I geeral, it allows to keep all fuctio degrees quite low. Their 4-splittig scheme cosists i dividig parameterizatio uit rectagle i four. It creates four sub-patches of the origial patch (see Fig..). A origial patch, correspodig to a mesh face, will be called macro-patch. The 4-split creates four sub-patches called micro-patches... The 4-split i terms of Bézier From the geeral formula of a Bézier patch (equatio.), a d-degree surface ca be deoted with the set of cotrol poits b k,l, where k,l = 0,...,d. Therefore, a d-degree patch S ca geerally be expressed as S(u,v) = d b k,l Bk(u)B d l d (v) k,l=0 The 4-split creates 4 differet parts of the macro-patch. These parts are cosidered as 4 differet d-degree patches. Obviously, they are ot etirely idepedet because of cotiuity costraits alog the boudaries. However, each 9

14 . 4-Splittig method M i M i ui+ v ui M i ui Figure.: The 4-splittig method applied o rectagular Bézier patches micro-patch will be cosidered as a patch with its cotrol poits. The cotiuity costraits will be some coditios o the cotrol poits ear the boudaries. ui+ 0 ui Figure.3: The parameterizatio uit rectagle i R is splitted i four equal domais. The parameter domai is splittig too. It meas four differet domais for the couple (u i,u i+ ), see Fig..3 micro-patch : (u i,u i+ ) [ [ ] 0, ] 0, micro-patch : (u i,u i+ ) [,] [ ] 0, micro-patch 3: (u i,u i+ ) [,] [,] micro-patch 4: (u i,u i+ ) [ ] [ 0,,] Thereby, the 4 d-degree Bézier patches correspodig to the 4 micro-patches are writte: d micro-patch : S(u, v) = b k,l Bk(u)B d l d (v) (.) k,l=0 d micro-patch : S(u, v) = b k,l Bk(u d )Bl d (v) (.) k,l=0 d micro-patch 3: S(u, v) = b k,l Bk(u d )Bl d (v ) (.3) k,l=0 d micro-patch 4: S(u, v) = b k,l Bk(u)B d l d (v ) (.4) k,l=0 0

15 .3 Cotiuity coditios.3 Cotiuity coditios.3. Notatios The macro-patches aroud a vertex are deoted M i where all the superscripts i =,..., are take modulo, where is the order of the mesh vertex correspodig to M i (0,0). The parameter u i, drawig the boudary curve betwee M i ad M i, lies i the iterval [0,]. The tesor-product surface is M i (u i,u i+ ) with (u i,u i+ ) [0,]. The scalar fuctios used i the cotiuity formula are deoted Φ i,ν i,µ i i correspodece to idex i. Partial derivative of M i respect to a parameter u i is deoted M i u i. By extesio, the mixed derivatives of M i are M i u iu i+..3. G coditios G coditios betwee two adjacet patches Coditio.6 is a geeral coditio whe two adjacet patches are joiig G cotiously. Usig otatios itroduced, this coditio is re-writte, with u i [0,], i: Φ i (u i )M i u i (u i,0) = ν i (u i )M i u i+ (u i,0) + µ i (u i )M i u i (0,u i ) (.5) G coditios i a etwork of patches : N-order corer G smooth coectio betwee two adjacet patches is relatively simple : coditio.5 has to be satisfied. However, i a etwork of patches, cotrollig the G coditios at the vertices ca result difficult. Various surfaces with a arbitrary topological type have N-order vertices. Ad satisfyig coditio.5 for all edges ca preset serious difficulties. This problem is called vertex cosistecy problem. Let us itroduce it. Let M,...,M be patches satisfyig coditios.5 at the corer associated with u i = 0. Sice Bézier surfaces are i the cotiuity class C, the mixed partial derivatives (or twist) M i / u i u i+ ad M i / u i+ u i are idetical. By differetiatig previous coditio with respect to u i ad evaluatig it at u i = 0, the patches must satisfy the followig coditio : µ i (0)M i u i u i (0,0) + ν i (0)M i u iu i+ (0,0) = Φ i(0)m i u i (0,0) + Φ i (0)M i u iu i (0,0) ν i(0)m i u i+ (0,0) µ i(0)m i u i (0,0) (.6) The problem arises whe oly the boudary curves are kow. Ideed, the first ad secod derivatives ca be determied. If the boudary curves lie i the same taget plae at a commo corer, the Φ i (0),µ i (0) ad ν i (0) ca be deduced with coditio.5. Vertex cosistecy problem arises whe solvig.6 for the ukows Mu i iu i+ (0,0) ad Mu i iu i (0,0). I other words, solvig the followig system : µ i (0)M i u i u i (0,0) + ν i (0)M i u iu i+ (0,0) = r i

16 .4 Methods where r i = Φ i(0)m i u i (0,0) + Φ i (0)M i u iu i (0,0) ν i(0)m i u i+ (0,0) µ i(0)m i u i (0,0) Varyig i from to leads to a liear system: where. M u u (0,0) Lt = r ν (0) µ (0) µ (0) ν (0) L = µ (0) ν (0) µ (0) ν (0) Mu u (0,0) t =, ad r = Matrix L is a circulat matrix, ad so it is ot iversible whe is eve. Therefore, the system is i geeral ot solvable whe is eve ( [Dav79]). The vertex cosistecy problem has bee dealt with by domai splittig ( [Far8], [Je87] or [Pip87]), ratioal bledig of icosistet mixed partial terms ( [Gre74]) or by requirig that patches meet with G cotiuity at the corers ( [Pet9]). I this paper, both iterpolatio schemes will avoid the vertex cosistecy problem. However, both deal with differet poits of view ad avoid it distictly. G coditios of a etwork of patches : taget plae cotiuity Accordig to the algorithm, oce we have foud boudary curves satsifyig the G coditios ad the twist compatibility, the secod step is defie the cross boudary tagets : Mu i i+ (u i,0), ad Mu i i (0,u i ) for each boudary curve of the etwork. They must respect three coditios. First, they must satisfy the G coditios alog the boudary curves. The, they must satisfy the twist costrait at the ed poits. Ad, they must be cosistet to the curve etwork. With the curve etwork, the values of the cross-boudary tagets at the corers are already fixed..4 Methods Here, I will preset the two methods developed durig my itership. The first method uses a first 4-splittig approach itroduced i [HB00] by Hahma ad Boeau. The secod oe also uses the 4-split techique but shows that it is possible to get more degrees of freedom for better shape cotrol. These two approaches are based o the same geeral pla : costructig boudary curves; costructig cross boudary tagets; fillig-i the patches. Eve though, they basically differ i the first step of the algorithm. The first was implemeted with a optimizatio. Due to the lack of time, the secod method has bee developed theoretically but could ot be implemeted as well. It remais a future ad promisig work. r.. r

17 .4 Methods.4. The problematic Let us first itroduce the followig otatio for the first derivatives of the boudary curves at a mesh vertex: Γ i(0) = M i u i (0,0) With this otatio, equatio.5 evaluatig at u i = 0 gives : Φ i (0)Γ i(0) = ν i (0)Γ i+(0) + µ i (0)Γ i (0) (.7) Therefore it relates the values of the scalar fuctios aroud a vertex to the first derivatives of the boudary curves. These scalar fuctios play a very importat role i the shape surface. The simplest solutio is to choose ν i ad µ i costats. To balace the degree of the three terms i coditio.7, Φ i is ofte choose as a liear fuctio. I certai applicatios, such as the G cotiuous iterpolatio of a rectagular etwork of curves, it is geerally ecessary for the values of ν i ad µ i to be differet at the two eds of a boudary curve i order adequately to represet the surface. I this way, a solutio is to take them liears ad Φ i quadratic. Chagig the three scalar fuctios has a direct impact o the shape, ad o the degrees of freedom that we have to iterpolate the surface. First, if oe wats to get patches of degree as low as possible, oe has to keep the degree of these fuctios as low as possible. The, the possibility of cotrollig the variatio of Φ i,ν i ad µ i is paid for by the loss of degrees of freedom i the free positioig of the cotrol poits. To reveal this last property, two cases will be cosidered : ν i,µ i costats ad Φ i liear (used i the method ); ν i,µ i liears ad Φ i quadratic (used i the method ). We will ote the i-th patch by M i (u i,u i+ ) = k,l bi k,lb k (u i )B l (u i+ ), ad the cotrol poits o the boudary curve by b i j,0 = c j. Ad let the patches be of the same degree. Case : ν i,µ i costats ad Φ i liear Let ν i (u) = ν, µ i (u) = µ ad Φ i (u) = φ 0 B 0(u) + φ B (u). Simply, we have the followig formulas B0(u).B i i (u) = ( + ).B+ i (u) B(u).B i (u) = i + +.B+ i+ (u) Thus, coditio.7 ca be writte uder the followig set of costraits, for j = 0,...,: ( φ 0 j ) j (c j+ c j )+φ (c j c j ) = ν ( b i ) ( j, c j +µ b i,j c j) (.8) So the cotiuity costrait liks three cotrol poits of the boudary with oe of each side., see Fig..4. 3

18 .4 Methods b i j c j+ c j c j b i j Figure.4: The cotrol poits ivolved i the case ν i, µ i costats ad Φ i liear Case : ν i,µ i liears ad Φ i quadratic Let ν i (u) = ν 0 B 0(u) + ν B (u), µ i (u) = µ 0 B 0(u) + µ B (u) ad Φ i (u) = φ 0 B 0(u)+φ B (u)+φ B (u). The product betwee a -degree Berstei polyomial ad a -degree Berstei polyom implies the ew basis fuctios: B 0(u).B i B (u).b i B (u).b i (u) = ( + i)( + i) ( + )( + ).B + i (u) (u) = ( + i)(i + ) ( + )( + ).B+ i+ (u) (u) = (i + )(i + ) ( + )( + ).B+ i+ (u) As we wrote.8 from the G coditios, the ew scalar fuctios ivolve the followig set of costraits. Let A 0,A,A,B 0,B,C 0,C be ( ( + j)( j) A 0 = φ 0 B 0 = ν 0 j ) ( C 0 = µ 0 j ) ( + ) A = φ j( + j) ( + ) A = φ j(j ) ( + ) The, for j = 0,..., B = ν j C = µ j A 0 (c j+ c j ) + A (c j c j ) + A (c j c j ) = ( ) ( ) ( B 0 b i j, c j + B b i j, c j + C0 b i,j c ) ( j + C b i,j c ) j Here, the cotiuity costrait liks four cotrol poits of the boudary curve with two cotrol poits of each side, that is three cotrol poits additioally accordig to the previous case, see Fig..5. The cost to have more degrees of freedom o scalar fuctios is to lose free positioig cotrol poits. 4

19 .4 Methods b i j c j b i j c j+ c j c j b i j b i j Figure.5: The cotrol poits ivolved i the case ν i, µ i liears ad Φ i quadratic.4. The first iterpolatio scheme Let M deote the iput surface mesh. We will costruct Bézier patches i oe-to-oe correspodece betwee the mesh faces. The method is ispired by the Loop scheme, preseted i [Loo94]. The first step is to compute the scalar fuctios Φ i,ν i ad µ i. As we saw previously, the choice of the scalar fuctios is a importat step i the costructio. As we wat the lowest patch s degree, we will choose scalar fuctios with the lowest degree. Thereby, the fuctios ν i ad µ i are take costats ad Φ i liear. Coditio.5 evaluated at u i = 0 implies a liear system. This system ca be sigular or ot. Also, we compute Φ i i order to have a solutio for the system. With this choice, the first derivatives i terms of Bézier cotrol poits always form a affie trasformatio of a regular -go. I coditio.6, the right-had side oly cotais first ad secod derivatives of the patch boudary curves at the commo vertex. Whether or ot the liear system.6 ca be solved depeds therefore o the choice of the boudary curves. Boudary curves are called to be C -cosistet, if the right had side vectors lie i the image space of the system s matrix. The first method costructs the first ad secod derivatives i such a way that they lie i the colum space of the system s matrix. The solutio is guarateed. Next, cross-boudary tagets are computed i respect to.5. Fially, patches are fillig-i with C cotiuity..4.3 The secod iterpolatio scheme The motivatio was fidig a ew method with derivatives which ca represet better irregular meshes. The global structure does ot chage. We will also costruct Bézier patches i oe-to-oe correspodece betwee mesh faces. The method is ispired by the article [HB03] writte by Hahma ad Boeau. I this part, the scalar fuctios will ote defie the boudary curves. O the cotrary, the boudary curves will defie the scalar fuctios. I other words, we fid mesh adapted derivatives, followig Piper s method, which represet 5

20 .4 Methods well the boudary curves. The the scalar fuctio values at the corers are deduced from these derivatives. ν i ad µ i are take liears ad Φ i quadratic. This choice implies (5,5)-degree Bézier patches. So the scalar fuctio Φ i are arbitrarily chose ad they are ot take as parameters esurig a solutio to a system, like i the previous method. Cosequetly, the ukows will ot be the twists aymore. The twists will be choose, ad the we simply evaluate equatio.6 separately from each other i order to get values for the secod derivatives. Fidig arbitrarily the twists is ot a istictive method because they do t have as a importat geometrical role as the secod derivatives. So, they will be take as miimizatio parameters of a eergy fuctio. Nevertheless, chagig scalar fuctio degree ca ivolve ratioal patches. The method proposes a solutio to avoid that case : patches are computed as multiples of the scalar fuctios. The cross-boudary tagets are foud followig G coditios, ad the patch is fillig-i like i the first method. 6

21 Part 3 Theoretical formulatio 3. The first iterpolatio scheme 3.. Choice of scalar fuctios Φ i,ν i,µ i Oe of the most importat targets is to keep the degrees of the patches as low as possible. As we are workig with tesor-product Bézier patches, if M i (u i,u i+ ) is a rectagular surface of degree (,m), the Mu i i (u i,0) is of degree ad Mu i i+ (u i,0) ad Mu i i (0,u i ) are of degree. Degrees compatibility Whe joiig patches G cotiously, the coditios.5 ad.6 must be satisfied. It is importat to choose the scalar valued fuctios Φ i,ν i,µ i such that they do ot raise the degree of the fial patches. Moreover, we saw that Φ i will be of degree d + if d is the degree of ν i ad µ i. Ideally, this would mea to take Φ i liear ad ν i,µ i costats. I that way, the degree of the rectagular patches would ot be raised. For symmetry reaso we choose ν i = µ i = / ad as simplificatio we suppose that φ 0 = Φ i (0) ad φ = Φ i (0) for i =,...,. These asumptios imply that the G coditios ow state as follow: ad Φ i (u i )M i u i (u i,0) = Mi u i+ (u i,0) + Mi u i (0,u i ) (3.) ν i (0)M i u iu i+ (0,0) + µ i (0)M i u i u i (0,0) = φ im i u i (0,0) + φ 0 im i u iu i (0,0) (3.) Varyig 3. from to leads to the followig liear system of equatios where Tt = φ r + φ 0 r (3.3) T =

22 3. The first iterpolatio scheme Mu (0,0) Mu u (0,0) r =. r =. Mu (0,0) Mu u (0,0) Ad t is the vector of the twists. Importat remarks ca be doe about this system. First, the simplified matrix T has a circulat structure, that is to say, it would be sigular whe the system s size is eve. The secod remark cocers the right-had side i 3.3. It is reduced to the sum of scalar multiples of the colum vectors r ad r. This fact is the key to solvig the vertex cosistecy problem. By properly costructig a boudary curve etwork, the solutio t will always exist. The form of the Φ fuctios Before describig the costructio of the twist, ad so the costructio of the boudary curve etwork, a digressio ito the form of the scalar valued fuctio Φ i (u i ) that appears i 3. is i order. This coditio evaluated at u i = 0 leads to the homogeeous system : where P = Pr = 0 (3.4) φ φ φ φ 0 The matrix P must be sigular i order for the previous system to have a otrivial solutio r. The determiat has to be zero. Sice P is circulat, that meas det(p) = k=0 cos ( ) πk φ 0 = 0 So, it is clear that P is sigular if ad oly if φ 0 = cos ( ) πk for some iteger k. Settig k = isures that the ri spa a plae ad are ordered properly. Thus, we set: ( ) π φ 0 = cos Whe u i =, coditio 3. implies: ( cos π ) Mu ĩ i (,0) = i Mĩ u i+ (,0) + Mi ũ i (0,) where i is the umber of patches meetig at the corer associated with M i (,0), ad the ew vectors are defied by (see Fig. 3.): ũ i = u i, ũ i+ = u i+, ũ i = u i 8

23 3. The first iterpolatio scheme u i+ ũ i+ u i 0 ũ i u i ũ i Figure 3.: parameterizatio at u i = 0 ad equivalece at u i = Liearity of differetiatio implies: ( cos π ) Mu i i (,0) = i Mi u i+ (,0) + Mi u i (0,) Ad so, we deduce the value of Φ i at u i = : Φ i () = cos π i If oe takes the fuctios Φ i as liear fuctios, this would imply that φ = Φ i (0) i 3.3 depeds o the order i of the opposite vertex. This would make the algorithm global istead of local, which is ot acceptable. I our method, the 4-splittig of domai rectagles eables us to separate vertex iformatio by takig the fuctios Φ i piecewise liear, cotious, defied o [0,/] ad [/,], with Φ i (/) = 0 (see Fig. 3.). Thus, cos π ( u i) for u i [0,/] Φ i (u i ) = cos π i (u i ) for u i [/,] (3.5) Why do we choose Φ i (/) = 0? First, settig a value for Φ i at u i = / esures the local property, because if oe vertex is moved, the mid-edge value allows to fid the first derivative without opposite vertex iformatio. The, settig it at 0 is justified by the observatios that = i implies Φ i (0) = Φ i () ad therefore the Φ i is a sigle liear fuctio. Moreover, = i = 4 implies Φ i (u i ) = 0 for all u i [0,]. This particular case correspods to a tesorproduct cofiguratio, ad the C cotiuity is guaratied. This choice would ot have bee possible without 4-splittig the parameterizatio domai. 3.. Costructig curve etwork Here, we will see the costructio of a bicubic boudary curve etwork correspodig to the edges of a cotrol mesh M. This is the most importat step i the surface costructio method, because the shape of this curve etwork has great ifluece o the surface shape. The requiremets o the boudary curves are the followig: 9

24 3. The first iterpolatio scheme Φ i (0) 0 u i Φ i () Figure 3.: scalar fuctio Φ i. iterpolatig the vertices of M. satisfyig the G coditios at the ed poits 3. keepig the surface scheme local The locality requiremet imposes to costruct curves such that they satisfy coditios 3. ad 3. at oe vertex idepedetly from the opposite vertex. The costraits imply coditios o derivatives at the ed poits. The first ad secod derivatives at the curve s ed poits are ivolved i 3.. I Loop s scheme, a polyomial curve which separates these iformatio of both ed poits should be of degree greater tha 5. The advatage of the 4-split is that it allows to take piecewise C polyomial curves of degree 3. Therefore, each curve betwee two adjacet vertices cosists of two cubic pieces, which are costructed idepedetly from each other. The two cubic pieces meet at u i = / C - cotiously. Let deote the polyomial piece of the boudary curve betwee the eighborig vertex v of order ad the vertex p i of order i i Bézier form by the cotrol poits b i 0,b i,b i,b i 3. The three first cotrol poits b i 0,b i,b i ca be costructed idepedetly from the joiig curve piece, thaks to the previous coditios. The fourth poit b i 3 is costructed i order to joi the two cubic pieces with C cotiuity. This poit correspods to the parameter u i = /. From the opposite vertex p i, the cubic piece i Bézier form is defied by the cotrol poits b k 0,b k,b k,b k 3 where k is the idex of v from the eighbor poit p i (see Fig. 3.3). 0

25 3. The first iterpolatio scheme b i 3 b k 3 b k b k p i b k 0 b i b i b i 0 v Figure 3.3: The i-th boudary curve : two cubic curves cotrol poits. They joi at u i = / with cotiuity C For simplificatio, it is coveiet to adopt a matrix otatio: b 0 b 0 b 0 :=.. b := b := p := b 0 b b.. b where p is refered to the vertex eighborhood of v. Let v be the vector: v v v := Now the cotrol poits will be deduced to satisfy the coditios. Let us remid that the boudary curves have to be costructed i order to have oe-to-oe correspodece betwee the macro-patches ad the faces of M.. v b b.. b p p. p Fidig the b i 0 The oe-to-oe correspodece meas that boudary curve ed poits (i.e. patch corers) correspod to the vertices of M. Sice the problem is the iterpolatio of the itial mesh, these ed poits will correspod exactly to the vertices. b i 0 = v, i =,..., (3.6) This is a special case of the more geeral settig b i 0 = αv + ( α) j= p j, i =,...,

26 3. The first iterpolatio scheme which should hold for approximatio. The shape parameter α is cotrollig the iterpolatio (approximatio). I our work, we set α = Fidig the b i The poits b i lie i the taget plae of the surface at b 0. The poits b i are related to the first derivatives take at (0,0) (cf. ad. to.4) r := Mu i i (0,0) = 6 ( b i b i ) 0 (3.7) Ad so, b i = b i r i (3.8) Additioally, r have to lie i the image space of T i 3.3. A solutio of that problem is to take r as the local averagig of the vertex eighborhood of v which is kow as first order discrete Fourier approximatio to p (see [Dav79]). r i = 6β ( ) π(j i) cos p j, i =,..., (3.9) j= where β is a shape parameter cotrollig the magitude of the taget vectors (ad so the amplitude of the taget plae at the curve s ed poits). To see that the r i costructed satisfies 3., substitute 3.9 i 3.4. The equatio holds (Pr ) i,j = 6β ( ( ) ( ) π(j i ) π(j i) cos φ 0 cos + ( )) π(j i + ) cos sice φ 0 = cos ( ) π = 0 Combiig equatios from 3.6 to 3.9, it leads to: b = v + B p (3.0) where Bi,j = β ( ) π(j i) cos, i,j =,..., Fidig the b i The poits b i are related to the secod derivatives at the boudary curves. Usig. ad. to.4: r i := Mu i i,u i (0,0) = 4 ( b i b i + b i ) 0 (3.) Moreover, r i have to lie i the image space of T. I fact, a sufficiet coditio for that, is to take b i as a barycetric combiatio of four poits (see [Loo94]): d i = 3 v + 3 p i + 6 p i+ + 6 p i (3.)

27 3. The first iterpolatio scheme Sice ay combiatios of poits, which lie i the image space of T, also does, let defie b i = γ 0 b i 0 + γ b i + γ d i where γ 0,γ,γ are shape parameters cotrollig the secod derivative. For istace, Loop used γ 0 = 0, γ = 0, γ = i [Loo94]. b = γ 0 v + γ ( v + B p ) + γ ( 3 v + Mp ) where Fially, where 3 6 M = 0... ( b = γ 0 + γ + γ ) v + B p (3.3) Bi,j = γ ( ) β π(j i) cos 6 if j = i,j = i + + γ 3 if j = i 0 otherwise Fidig the b i 3 The boudary curves have to be C -cotious at u i = / i order to get cross boudary tagets. Ideed, the cross boudary tagets at u i = ad u i = + have to agree. Thereby, two cubic curves joi themselves C cotiously if ad oly if { b i 3 = b k 3 C 0 cotiuity b i b i 3 = (b k b k 3) C cotiuity The secod coditio is the equality betwee the tagets. It ca be re-writte to fid the poit b i 3 like the followig: b i 3 = bi + b k (3.4) Solvig for the twist I this sectio, it will be show that the previously computed cotrol poits lie (as we wated) i the image space of the matrix T. This result will be preseted as lemmas (Lemmas 3.. to 3..3). Sice, r ad r are liear combiatios of b 0,b,b, it follows that r ad r must also lie i the colum space of T. The solutio for the twist will be formulated as a theorem (Theorem 3..). 3

28 3. The first iterpolatio scheme Before the lemmas, it would be useful to recall the matrix T: T = The, the algebraic defiitio of lyig i the image space of T ca be viewed geometrically. Let X be a vector which lies i the colum space of T, the there exists X such that: T X = X (3.5) It is clear from the structure of T that equatio 3.5 meas: ( X i + X i ) = X i I other words, the poits of X ca be costructed as cosecutive midpoits of a collectio X. Lemma 3.. Let b 0 be the first boudary curve s cotrol poit, the Tb 0 = b 0 Proof: Sice b i 0 = b i 0 = v for all i =,...,, it follows that (bi 0+b i 0 ) = b i 0. Thus, Tb 0 = b 0. Lemma 3.. simply states that cosecutive midpoits of a collectio of idetical poits is idetically the same poit. Lemma 3.. Let b be the first derivative cotrol poits. There exist a b such that T b = b Proof: Let b = v + B p where B i,j = β [ ( ) π(j i) ( ( )] π π(j i) cos + ta si, i,j =,..., ) This formula proposed permits, usig trigoometric idetities, to write: (T B ) i,j = ( B i,j + B i,j) = B i,j Ideed, ( B i,j + B i,j ) = β [ ( ) ( ) π(j i) π(j i + ) cos + cos ( ( ) π π(j i) ( ( )] π π(j i + ) +ta si + ta si ) ) 4

29 3. The first iterpolatio scheme The trigoometric idetities give: ( ) ( ) ( π(j i) π(j i + ) π(j i) cos + cos = cos + π ) ( ) ( ) ( π(j i) π(j i + ) π(j i) si + si = si + π Ad we have the result. Therefore T b = T( v + B p) = T v + T B p = v + B p = b ( π cos ) ( π ) ) cos Lemma 3.. is a simple fact of affie geometry that the midpoits of the legs of a affie -go form a iscribed affie -go. Lemma 3..3 Let b be the secod derivative cotrol poits. There exist a b such that T b = b Proof: Let b = (γ 0 + γ + γ 3 ) v + B p, where [ ( ) B i,j β π(j i) = γ cos + ta + γ { 3 if j = i,i otherwise It follows that Therefore ( π ) si i,j =,..., (T B ) i,j = ( B i,j + B i,j) = B i,j ( )] π(j i) T b = T( 3 v + B p) = 3 v + T B p = 3 v + B p = b Lemma 3..3 also has a simple geometric iterpretatio. The poits b are simply the cetroids of the triagles v, p i, p i+. Theorem 3.. There exist a t, such that Tt = φ r + φ 0 r Proof: Rewritig equatio 3.7 ad usig lemmas 3.. ad 3.. leads to r = 6(b b 0 ) = 6(T b Tb 0 ) = 6T( b b 0 ) Similary, combiig 3. ad lemmas shows: r = 4(b b + b 0 ) = 4(T b T b + Tb 0 ) = 4T( b b + b 0 ) 5

30 3. The first iterpolatio scheme It ow follows that Tt = φ r + φ 0 r = 6φ T( b b 0 ) + 4φ 0 T( b b + b 0 ) ] = T [4 b (48φ 0 6φ ) b + (4φ 0 6φ )b 0 Therefore t = 4φ 0 b (48φ 0 6φ ) b + (4φ 0 6φ )b 0 Theorem 3.. shows that the boudary curve etwork costructed is twist compatible. Therefore, polyomial surface patches ca be foud to iterpolate the boudary curve etwork. Moreover, a explicit expressio ca be foud, see Appedix A Cross-boudary tagets Oce the boudary curve etwork was costructed, the secod step is to defie the cross-boudary tagets Mu i i+ ad Mu i i for each boudary curve of the etwork. The first coditio for them is to satisfy the G coditios alog the boudary curves, see 3.. The, they have to be twist compatible at each corer. Fially, they have to be cosistet with the boudary curves, that is the first cross-boudary taget has to coicide with the first derivative of the ext boudary curve: Mu i i+ = Mu i+ i+ A coveiet way to defie cross-boudary tagets that esure G cotiuity is the followig: M i u i+ (u i,0) = Φ i (u i )M i u i (u i,0) + Ψ i (u i )V i (u i ) (3.6) M i u i (0,u i ) = Φ i (u i )M i u i (u i,0) Ψ i (u i )V i (u i ) I fact, as we said, the cross-boudary tagets have to satisfy 3.. However 3.6 is equivalet to 3.. To see that 3.6 implies 3., oe ca add the two equatios i 3.6. To see that 3. implies 3.6, choose It is clear that Ψ i (u i )V i (u i ) = Φ i (u i )M i u i (u i,0) M i u i (0,u i ) Ψ i (u i )V i (u i ) = Φ i (u i )M i u i (u i,0) + M i u i+ (u i,0) The advatage of formulatig the cross-boudary tagets as i the previous equatio is that both are writte i terms of a sigle fuctio V i (u i ) that, together with M i u i (u i,0), characterizes the taget plae behavior of the surface alog the boudary curve M i (u i,0). The degrees of Ψ i (u i ) ad V i (u i ) decide the degree of the surface. They will be costructed to be the miimum-degree polyomials such that the crossboudary tagets satisfy the taget plae ad twist costraits. Φ i (u i ) is 6

31 3. The first iterpolatio scheme liear, ad Mu i i (u i,0) is of degree. So Ψ i (u i )V i (u i ) must have a degree lower or equal tha 3. Moreover, Mu i i+ (u i,0) or Mu i i (0,u i ) are of degree 3. Hece, Ψ i (u i ) will be take liear ad V i must be take of degree. Thaks to the 4- split, it will be possible to costruct the fuctio V i (u i ) cotious ad piecewise quadratic. If the first equatio of 3.6 is evaluatig at u i = 0, it holds: M i u i+ (0,0) = Φ i (0)M i u i (0,0) + Ψ i (0)V i (0) r i+ = φ 0 r i + ψ 0 V i (0) From this equatio, it is possible to deduce the fuctio values ψ 0 ad V i (0). Rearragig the previous formula, leads to; where Sice r = B p, the ψ 0 V i (0) = r i+ φ 0 r i = (Nr ) i φ φ N = φ φ 0 ψ 0 V i (0) = (N.B p) i The (i,j)-th coefficiet of the matrix N.B is Thus, (N.B ) i,j = N i,i Bi,j + N i,i+bi+,j [ ( ) = 6β φ 0 cos π(j i) + cos [ = 6β cos ( ) ( ) π cos π(j i) + cos [ = 6β cos ( ) ( ) π cos π(j i) + cos ( ) +si π(j i) si ( ) ] π = 6β si ( ) ( ) π si π(j i) j= ( π(j i ) )] ( π(j i) π ( ) π(j i) )] cos ( π ψ 0 V i (0) = 6β ( ) π si ( ) π(j i) si p j (3.7) A appropriate choice of ψ 0 is therefore: ψ 0 = si ( π From the opposite ed( poit, the product ψ V i () ca be obtaied aalogously, which gives ψ π = si i ), where i is the order of the opposite vertex p i. Hece, the fuctio Ψ i ca be choose liear, which is miimal degree: ( ) ( ) π π Ψ i (u i ) = si ( u i ) + si u i (3.8) ) i ) 7

32 3. The first iterpolatio scheme The fuctio V i ca ot be take liear, because its derivatives deped o the twists. It will be take quadratic. That is it ca be writte usig the Berstei polyomials, basis of polyomials space of degree. V i (u i ) = v 0 B 0(u i ) + v B (u i ) + v B (u i ) (3.9) Whe differetiatig the first equatio of 3.6 with respect to u i ad evaluatig at u i = 0, it gives M i u iu i+ (0,0) = Φ i(0)m i u i (0,0) + Φ i (0)M i u iu i (0,0) + Ψ i(0)v i (0) + Ψ i (0)V i (0) It is clear that the derivative V i (0) appears i relatio to the twist, ad so V i could ot be liear. Usig the same otatio as usual, V i (0) ca be expressed as: V i (0) = t i φ r i φ0 r i ψ i V i(0) ψ 0 (3.0) ) (π where ψi = Ψ i (0) = si i si ( ) π depeds o i. Thereby, Vi (0),V i (0) are kow, ad V i (),V i () are kow from the opposite vertex. Oe ca fid a cubic curve usig the Hermite iterpolatio of these four values, but it will raise the degree. The 4-split allows to take a C 0 piecewise quadratic fuctio requirig that V i ( + ) = Vi ( ). Usig 3.7, we deduce: v i 0 = j= 6β si ( π(j i) ) p j (3.) Usig 3.0, we deduce the explicit formula of v i. The detailled calculus ca be see i Appedix A... Fially, v i ca be take freely to have V i ( + ) = Vi ( ) The macro-patches i Bézier form Here, the macro-patches are cosidered idividually. The parameterizatio will be adapted. I this sectio, the previous results are combied to exhibit explicit formulas used to compute the Bézier cotrol poits for a macro-patch. Whe these cotrol poits have bee foud, a de Casteljau algorithm allows to draw the patch. The border ad the first ier row of cotrol poits ca be foud from the previous sectios : the boudary curves ad the cross-boudary tagets. It remais the ier poits. They will be choose i order to joi the four micropatches C -cotiously. Fidig them ivolves lettig free four cotrol poits per macro-patch ad deducig the remaiig poits from them. Notatios The four rectagular Bézier patches of degree 3 wich compose the macro-patch M (see figure 3.4) are deoted by S,S,S 3,S 4. Therefore, the Bézier cotrol poits of M are respectively deoted s i,j,s i,j,s3 i,j,s4 i,j where i,j = 0,...,3. Moreover, the ew parameterizatio of the patch M cosists i u k = a k+ a k, where 8

33 3. The first iterpolatio scheme a k, k =,...,4 are the four rectagle s corers. The u k are vectors betwee domai vertices ad defie the directioal derivatives D uk M. I the rectagular case, the directioal derivatives are simple partial derivatives. Furtheremore, let E k (u) = ( u).a k + u.a k+, for u [0,], defie a edge fuctio ad let k be the order of the mesh vertex which is iterpolated by M(E k (0)). Fially, the superscripts L ad R will be used. They refer to the left ad right side of the boudary splittig. It will be useful to express the scalar fuctios Φ,ν,µ,Ψ or the cross-derivative fuctio V whe the computatio is focused o the boudary curve or the cross-boudary tagets. M a k+3 = (0,) u k+ a k+ = (,) S 4 S 3 uk+3 u k+ u k a k = (0,0) a k+ = (,0) S S Figure 3.4: parameterizatio of the macro-patch M, labellig of micro-patches ad derivatives directios Fidig boudary ad first derivative cotrol poits of M The boudary cotrol poits are computed : explicit expressios of b 0, b, b, b 3 are kow from the previous sectio. The first row ier poits, which correspod to the cross-boudary tagets, will be computed from the boudary cotrol poits. The formula is recallig here: M i u i+ (u i,0) = Φ i (u i )M i u i (u i,0) + Ψ i (u i )V i (u i ) The secod formula chages i the sig before Ψ i (u i )V i (u i ), ad so the calculus will be the same. This formula ca be writte usig the ew parametrizatio: ( Duk+3 M) (E k (u)) = Φ k (u)(d uk M) (E k (u)) + Ψ k (u)v k (u) v M(E k(u)) = Φ k (u) u M(E k(u)) + Ψ k (u)v k (u) Usig the formulas,. to.4, it is clear that 3 s i0bi 3 (u) u [0,/] M(E k (u)) = i=0 3 s i0bi 3 (u ) u [/,] i=0, k =,,3,4 (3.) 9

34 3. The first iterpolatio scheme where the s i0 ad s i0 are the cotrol poits which were called respectively bi j ad b k j with j = 0,...,3. The, the derivatives are kow sice. ad.3: (D uk M) (E k (u)) = 6 (s i+,0 s i0)bi (u) u [0,/] u M(E i=0 k(u)) = 6 (s i+,0 s i0)bi (u ) u [/,] Where k =,,3,4. The, the cross-derivative fuctio V k ca be writte v L i Bi (u) u [0,/] i=0 V k (u) =, k =,,3,4 v R i Bi (u ) u [/,] i=0 It remais the scalar fuctios Φ k ad Ψ k. They are recalled with the followig: φ L 0 B0(u) = cos π ( u) u [0,/] Φ k (u) = k φ R B(u π, k =,,3,4 ) = cos (u ) u [/,] k+ Ψ k (u) = i=0 i=0 ψ L i B i (u) = si π k ( u) + si π k+ (u), u [0,],k =,...,4 Let us ow cosider the boudary of M correspodig to u which is commo to the patches S ad S. The cotrol poits are labeled as i figure 3.5. S S s 0 s s s 0 s 3 s s s 3 s 00 s 00 s 30 s 0 s 0 s 0 s 0 s 30 Figure 3.5: Boudary ad first derivative cotrol poits of M correspodig to boudary u First the boudary curve cotrol poits are kow sice the sectio 3... The first row of ier poits s j,k ad s j,k ca be foud usig the equatio 3.. The left had side, ad the right had side are polyomials of the same degree. Fidig the first row of ier poits is makig correspod the left-had side polyomial s coefficiets with the right-had side polyomial s coefficiets. 30

35 3. The first iterpolatio scheme The variable k is settig to. The cross boudary derivatives respect to u 4 are: 3 6 (s i s i0)bi 3 (u) u [0,/] ( D u4 M) (E (u)) = i=0 3 6 (s i s i0)bi 3 (u ) u [/,] i=0 Thus, the i-th ier poit s i is obtaied by: s i = s i0 + [( i ) ] (6φ0 (s i+,0 s ) i 6 3 i0) + ψ 0 v i + 3 ψ v i (3.3) It gives all the first row ier poits we wated (explicit formulas are writte i Appedix A..3). Fillig-i the macro-patches by piecewise Bézier rectagles To fill-i the macro-patches, several cotrol poits are ivolved. Oe questio is posed : how do we fill the macro-patch to joi the 4 micro-patches with C - cotiuity? I case of C cotiuity, the ier poits must adopt a special cofiguratio. Actually, the derivatives, o each side of the boudary, must have their orm equal ad their directio opposed. I other words, the cotrol poits satisfy a simple equatio. S 4 S 3 S S Figure 3.6: Fillig-i the macro-patches : it remais 9 cotrol poits to fid i order to build etirely the piecewise Bézier rectagles The ier poits are called A to A 4 for the corers, B to B 4 for the poits o boudaries ad C the uique cotrol poit where the four micro-patches are meetig (see figure 3.7). All the subscripts are take modulo 4 i this case. The equatios will be: C = B i + B i+ B i = A i + A i+ = A + A + A 3 + A 4 4 3

36 3. A Optimizatio of the scheme A 4 B 3 A 3 C B 4 B A B A Figure 3.7: Notatios for the ier poits Nie cotrol poits have to be foud, the previous equatios give us five coditios. Hece, it remais four degrees of freedom. These degrees are the positio of the cotrol poits A,A,A 3 ad A 4. These poits ca be used to chage the surface shape. For istace, it is clear that if oe takes them very far from the macro-patch boudaries ad the first row cotrol poits, the surface o this macro-patch will be swelled. I the work A,A,A 3 ad A 4 are take as a liear combiatio of their eighborhourig (see Fig. 3.8). S s A s s Figure 3.8: The A i are computed as a liear combiatio of the three poits i its eighborhood 3. A Optimizatio of the scheme 3.. What is the optimizatio? I this sectio, the secod derivative will be optimize i order to get a smooth surface with the lowest eergy. The previous results are quite good, but it ca take very strage shape whe oe modifies the shape parameters. I fact, the algorithm is very sesitive accordig to parameter modificatios. So, to improve the global shape, the most importat is to improve the boudary curves. I other words, the four cotrol poits of each cubic piece. The first poit ca ot be moved : the iterpolatio imposes b 0 = v. The first derivatives ca 3

37 3. A Optimizatio of the scheme be modified, but this chage is more fudametal ad becomes the subject of the secod method. The fourth poit is fixed accordig to the C cotiuity. Thereby, the idea is to modify the third poit, which represets the secod derivative of the boudary curves. Before optimizig the secod derivatives, we have to precise what optimize meas. I fact, optimizatio ofte ivolves a eergy fuctio which will be miimized. I our case, the eergy fuctio is take like the followig: E = X (u) du (3.4) where the orm used is the usual euclidea orm, ad the fuctio X represets the curve. Miimizig this eergy meas miimizig the curvature. A effective miimizatio meas fidig a curvature equal to 0 : a straight lie from oe vertex to oe eighbour vertex. Sice first derivatives are fixed, the curvature will ever be zero. Nevertheless, the boudary curve will be flatteed 3.. Geeral approach The otatio will temporarily chage to ot overweight the mathematical formulatio with too may subscripts or superscripts. Let deote the curve, of ay degree, by: C(u) = b i B i (u) i The secod derivative of C respect to u will be : C (u) = i b ib i (u). If U is a vector i the polyomial space, let deote the scalar product from the orm by U = U,U Thereby, the eergy ca be writte: E = X (u) du = X (u),x (u) du = b i B i (u), b j B j (u) du i j = b i,b j B i (u)b j (u)du i,j A coveiet way to formulate a eergy problem is usig the matrix otatio. Let deote by B the cotrol poit matrix ad E the itegral matrix: B i = b i E ij = B i (u)b j (u)du Thus, the eergy ca be expressed uder matrix otatio as: E = Trace (t BEB ) (3.5) I fact, the eergy must act o the cotrol poits, ad aboveall o the b which represets the secod derivative. Let deote the followig fuctio : (B) = t BEB 33

38 3. A Optimizatio of the scheme E is a symmetric, positive ad defiite ad so, it caracterizes a quadratic form which is deoted by. The miimizatio will operate o this quadratic form. To extract the right ukow from the matrix B, we write : Ad so the miimizatio problem is B = A.X + B (X) = t (A.X + B)E(A.X + B) mi The miimizatio problem for a quadratic form is equivalet to ullify the s gradiet. That is to say, it is equivalet to resolve a liear system which ca be writte: (X) = 0 ( t AEA)X = ( t AE B) 3..3 Optimizatio applied to boudary curve etwork Now, we would like to apply the optimizatio scheme o the boudary curves, precisely o the secod derivative poits : s i 0, where i =,...,4. Let us cosider a boudary curve which belogs to the boudary curve etwork. It has bee show previously a coveiet otatio to represet its cotrol poits. Left cotrol poits are deoted by b L i, i = 0,...,3; likewise b R i, i = 0,...,3 cocer the right cotrol poits. Let C be the boudary curve: C(u) = i b i B i (u) = i b L i B i (u) + i b R i B i (u ) = C L (u) + C R (u) The eergy is writte: E = 0 C (u) du = = i,j b L i,b L j Adoptig matrix otatio: A ij = C L(u) du + C 0 R(u) du B i (u)b j (u)du + b R i,b R j i,j B i (u)b j (u)du = B L = b L 0 b L b L b L 3 B R = B i (u )B j (u )du It leads to this ew matrix formulatio, amely the previous fuctio: b R 0 b R b R b R 3 (B L,B R ) = t B L AB L + t B R AB R = t (B L + B R )A(B L + B R ) B i (u )B j (u )du Now, the eergy fuctio has to be expressed as b L ad b R fuctio. I that way, the followig system will be resolved : = 0 b L b R = 0 34

39 3.3 The secod iterpolatio scheme assets The cotrol poits b L ad b R ca be extracted from B L ad B R followig: B L = B R = b L 0 b L b L b L 3 b R 0 b R b R b R 3 = = ( b L b R ( b L b R ) + ) + b L 0 b L b R b R 3 = M LX + B L = M RX + B R Therefore, let us re-defie the fuctio by: (X) = t( (M l + M R )X + B L + B R ) A ( (Ml + M R )X + B L + B R ) (3.6) Sice miimizig is equivalet to ullifyig the s gradiet, the followig liear system has to be resolved: ( t (M L + M R )A(M L + M R ) ) X = t ( B L + B R )A(M L + M R ) (3.7) The explicit formula of matrix A ca be foud i appedix, see A The secod iterpolatio scheme assets 3.3. Motivatios The first scheme applied to irregular mesh does t give good results. Its first derivatives form a affie trasformatio of a regular -go. It may happe whe the mesh has irregularities like adjacet edges with very differet sizes, or very differet agles betwee successives edges (see Fig. 3.9). Figure 3.9: The first iterpolatio scheme is restricted to havig first derivatives that form a affie trasformatio of a regular -go. 35

40 3.3 The secod iterpolatio scheme assets As we saw previously, boudary curve etwork cotrols the surface shape. Playig with its curves ca ivolve a importat improvemet. We saw a optimizatio o secod derivatives. I this sectio, the first derivatives are the shape parameters. So, we wat to create a ew iterpolat which allows free choice of all first derivatives at each iput vertex, alog each iput edge Choice of Φ i, µ i, ad ν i G coditios remai valid i this sectio ad we will still use them. For simplificatio, let us first itroduce the followig otatio for the first derivatives of the boudary curves at a mesh vertex : d i = M i u i (0,0) Let us re-write coditio.5 with the previous otatios: Φ i (u i )d i ν i (u i )d i+ µ i (u i )d i = 0 (3.8) As we said at the begiig, the secod iterpolatio scheme does t defie the boudary curves from the scalar fuctios, but o the cotrary, the scalar fuctios are defied from the boudary curves. Let us explai how to determie the quatities which are related by 3.8. If oe multiplies, usig cross product for each idex i the equatio 3.8 by the vectors d i,d i+,d i, respectively, oe gets three vector valued equatios: ν i (u i )d i+ d i µ i (u i )d i d i = 0 Φ i (u i )d i d i+ µ i (u i )d i d i+ = 0 Φ i (u i )d i d i ν i (u i )d i+ d i = 0 The, these equatios are multiplied, usig dot product, by (ormal vector to the d i), oe gets the followig: d µ i (0) = i,d i+, d i,d i+, Φ i(0) d ν i (0) = i,d i,,i =,..., (3.9) d i,d i+, Φ i(0) Fidig the scalar values µ i (0),ν i (0) is takig arbitrarily the tagets of the boudary curve, as log as they belog to the same taget plae. With them, the scalar values are fixed up to a scalar, amely Φ i (0). I practice, we choose the ormalizatio factor Φ i (0) such that µ i (0).ν i (0) = 4. This choice is motivated by a eed of ormalizatio that geeralizes the regular case where µ i (0) = ν i (0) =. Previous { equatios have a geometrical iterpretatio. For each triple of tagets d i,d i,d } i+, the value of µi (0) is proportioal to the area of the triagle (v,v + d i,v + d i+). I [Loo94] it was show that, i the case µ i = ν i = cst, the first derivatives i terms of Bézier cotrol poits always form a affie trasformatio of a regular 36

41 3.3 The secod iterpolatio scheme assets plaar -go. The, these scalar fuctios are take piecewise liears. The Φ i fuctio ca t be take liear too because of degree compatibility i.5. Ad so, it is take piecewise quadratic. We ca remark that choosig abritrarily the first derivatives implies a degreeraisig of the scalar fuctios ad, de facto, of the patch. Remark: The special case, amely tesor-product surface, ivolves d i,d i+, = 0. therefore, µ i(0) ad µ i(0) are ot defied. However, such cofiguratio implies Φ i = 0 as we see. So the ew G coditio is: ν i(u i)d i+ + µ i(u i)d i = 0 Sice ν i.µ i =, these scalar fuctios are etirely foud. A special implemetatio case has to be doe, but it will ot cotradict the method. The, coditio.6 has to be satisfied too. It ivolves the twists. Previously, first ad secod derivative vectors had to lie i the image space of a system matrix to allow a solutio for the twists. Here, as we choose arbitrarily the first derivatives, they will ot lie i the image space. Let us show how to determie the twist. First we recall the coditios with the ew otatio: Φ(0)d i = µ i(u i )d i + ν i(u i )d i+ Φ i(u i )d i ν i (u i )t i µ i (u i )t i where d i = Mu i iu i ad t i = Mu i iu i+. The equatio was writte to show the ukow : d i. Actually, the twist are computed miimizig a eergy fuctio, ad the secod derivatives are deduced directly from the equatio. I coclusio, at each vertex, we ow have fixed the data which is ecessary to satisfy G cotiuity there: the positio (iterpolatio), the tagets, the secod derivatives of the patch boudary curves ad the twists Boudary curves ad cross-boudary tagets The previous choice does ot allow to deduce the boudary curve etwork simply. I fact, if oe chooses quitic Hermite curves, oe gets ratioal patches. Actually, if we re-call the cross-tagets equatios esurig G cotiuity betwee adjacet patches, it leads to: ν i (u i )M i u i+ (u i,0) = Φ i (u i )M i u i (u i,0) + V i (u i ) µ i (u i )M i u i (0,u i ) = Φ i (u i )M i u i (u i,0) V i (u i ) (3.30) Addig up the two previous equatios, leads to coditio.5. Ad choosig V i (u i ) = ν i (u i )Mu i i+ (u i,0) Φ i (u i )Mu i i (u i,0) gives V i (u i ) = µ i (u i ) Mu i i (0,u i ) + Φ i (u i )Mu i i (u i,0). The formulatios are clearly equivalet. I the first iterpolatio scheme, the scalar fuctios µ i ad ν i were costats. Here, they are liears, so if oe tries to simplify by µ i or ν i i oe of the equatios 3.30, oe gets ratioal patches. The oly possibility that yields a polyomial solutio of equatios i 3.30 is to esure that polyomials µ i ad ν i divide Mu i i (u i,0). It meas, there exists a polyomial H i such as: M i u i (u i,0) = µ i (u i ).ν i (u i ).H i (u i ), for i =,..., (3.3) 37

42 3.3 The secod iterpolatio scheme assets Compute boudary curves This last coditio gives us the tools to fid a explicit formula of the boudary curves. First, µ i (u i ) ad ν i (u i ) are of degree. If we wat to keep the degree of the patch as low as possible, H i have to be chose liear. However, the C coditio at the mid-edge poit ca t be sastified i the geeral case. Actually, there are ot eough degrees of freedom. The miimum degree is. Thus, M i u i becomes of degree 4. Ad so, the patches are of degree (5,5). To be sure, let H i be of degree. It ca be writte H i (u i ) = h 0 B 0(u i ) + h B (u i ). The product µ i (u i ).ν i (u i ) is of degree, ad ca be expressed as µ i (u i ).ν i (u i ) = a 0 B 0(u i ) + a B (u i ) + a B (u i ). The B k j (u i) are the Berstei polyomials. Thus, M i is of degree 4, ad the left-had side i 3.3 is: M i u i (u i,0) = 4 3 (b j+,0 b j,0 )Bj 3 (u i ) = 4 j=0 Fially, equatio 3.3 gives: 4(b b 0 ) = a 0.h 0 3 (b j+ b j )Bj 3 (u i ) j=0 4(b b ) = 3 a.h a 0.h 4(b 3 b ) = 3 a.h a.h 4(b 4 b 3 ) = a.h Due to 4-split, M i is piecewise quartic. The cotrol poits of each both quartic piece will be oted b L i ad b R i, i = 0,...,4, respectively the left ad right side. Similarly, a i, ad h i coefficiets will be superscripted. The C cotiuity coditios are : C 0 cotiuity ad the same taget with opposed directios. It meas: { b L 4 = b R 4 C 0 coditio b L 3 b L 4 = b R b R 0 Same taget coditio The secod equatio leads to: a L.h L = a R 0.h R 0 (3.3) Sice µ i (u i ) ad ν i (u i ) are piecewise cotious, µ i (u i ).ν i (u i ) is cotious too ad a L = a R 0. So h L = h R 0. Cotrol poits we kow are b L 0,b L,b L ad b R 4,b R 3,b R. Moreover, a L i ad a R i, for i = 0,...,, are kow. From this, we ca easily compute h L 0 ad h R. Hece, coditio 3.3 is expressed i terms of kow coefficiets: 4 a L 0 ( ) ( ) b L b L 6aL (a L b L 0 ) b L 0 = 8 a R 0 ( ) b R 0 b R (3.33) Above coditio is ot esured to be verified. There are ot ay free parameters to make it satisfied. Addig free parameters meas raisig the degree of H i to : H i (u i ) = h 0 B 0(u i ) + h B (u i ) + h B (u i ). From b L 0,b L,b L ad b R 4,b R 3,b R, h L 0,h L ad h R,h R are deduced. C coditio gives us h L ad h R 0 (see Appedix A..) : 38

43 3.3 The secod iterpolatio scheme assets where, h L = βr + β L α R α L = hr 0 (3.34) α L = ( 6a L a L + a L ) 0 β L = ( 3a L a L ) h L + al h L 0 + b L α R = ( 6a R a R + a R ) β L = ( 3a R a R ) h R + ar 0 h R b R 3 From h L we ca easily compute b L 3,b L 4,b L 5. Likewise, it ca be doe o the right side. It gives the complete boudary curve. Compute cross-boudary tagets As we have doe for the boudary curves, the cross-boudary tagets ca be ratioals. Similarly, µ i ad ν i must divide V i. For that matter, there exists a polyomial W i such that V i is defied as: V i (u i ) = µ i (u i ).ν i (u i ).W i (u i ), for i =,..., (3.35) I equatios 3.30, the cross-boudary tagets Mu i i+ (u i,0) ad Mu i i (0,u i ) are of degree 5. The, the right-had side are of degree 6. Thus, the degree of W i have to be lower or equal tha 4. Let W i be: 4 W i (u i ) = w j Bj 4 (u i ) j=0 Simply, w 0 ad w ca be computed from W i (0) ad W i (0) ad equatios 3.30 (see Appedix A.. for more details). However, the two first coefficiets of a -degree polyomial P are sufficiet to compute the two first coefficiets due to a degree reductio o P. Hece, from the 4-degree polyomial s coefficiets w 0 ad w we will compute w () 0 ad w () correspodig to the same curve but degree decreased to : w () 0 = w 0 w () = w w 0 With these cotrol poits o each side, it is easy to compute the third poit of each piecewise quadratic curves. It is free subject to W i ( ) = Wi ( + ). Oce all the cotrol poits have bee foud, a degree-raisig gives us the complete polyomial W i ad the V i. Thus, first row ier poits ca be computed from equatios Ideed, if we write: 5 Φ(u i )µ i (u i )H i (u i ) = e j Bj 5 (u i ) µ i (u i )W i (u i ) = j=0 5 f j Bj 5 (u i ) j=0 39

44 3.3 The secod iterpolatio scheme assets The, first equatio i 3.30 leads to explicit formulas of ier cotrol poits: b j = 0 (e j + f j ) + b j0, j = 0,...,5 40

45 Part 4 Results 4. First iterpolatio scheme The first example shows how a simple cube ca be iterpolated. We used for this the followig parameters : β = 0.60,γ 0 = 0.6,γ =.05,γ = 0.9. The figure 4. shows the itial mesh : a cube. The cube is a perfect simple example, because its faces are rectagulars. All the vertices are of order 3, so we are ot i a clasical tesor-product case. The iterpolatio, figure 4., shows that each vertex has a large taget plae, otably the figure o the left. It makes the surface swelled o each face. Moreover, the secod derivatives emphazise this effect. Although, the surface is perfectly smooth, ad theoretically G. Figure 4.: O the left : the iitial mesh which is a simple cube. Each vertex has a order of 3. O the right : The iitial mesh iterpolated, usig the parameters β = 0.60, γ 0 = 0.6, γ =.05, γ = 0.9 The surface shape ca chage deeply i case where parameters chaged. A little variatio i the parameters ca make a surface very swelled or tagled. Let see that with a example. Let us set β to 0,55;0,50;0,45;0,40 successively (see Fig. 4

46 4. Usig optimizatio Figure 4.: o the left : A zoom i the taget plae at a vertex. It ca be see the affi trasformatio of a 3-go. O the right : The iterpolatio surface with the boudary curves ad the taget plaes at each vertex. 4.3). This parameter defies taget plaes amplitude. Previous value was 0.60, so the taget plae is reduced. Actually, whe taget plae is reduced, we ca remark itersectios i the cotrol poit etwork. Obviously, such itersectios imply very bad shapes. Figure 4.3: From left to the right, shape parameter β chages successively β = 0, 55; β = 0, 50; β = 0, 45; β = 0, 40. The taget plae is reduced. Whe β = 0, 40, there are itersectios wicked for the surface shape. Figure 4.4 shows the resultig shape with β = 0, Usig optimizatio The optimizatio implemeted acts o the secod derivatives. It fids the optimal third cotrol poits of the cubic boudary curve which miimizes the secod derivative of this curve. Thus, the eergy fuctio is directly related to the curve s curvature. Fially, the boudary curve is flatteig. Cosequetly, it avoids the itersectios see previously. Figure 4.5 shows the results after optimizatio. The cotrol polygo does t cotai ay itersectios ad so the micro-patches are well costructed. The global shape ca be see with figure 4.6. I it, the shape parameter is still β = 0,40 which gives a global surface closer to the mesh tha previously. It is globally smooth, ad eergetically optimal. 4

47 4. Usig optimizatio Figure 4.4: Shape parameter β is settig at 0, 40. O the left, the figure shows the surface with boudary curve etwork ad each taget plae. At its right side, the same surface with the cotrol polygo which presets itersectios Figure 4.5: From left to the right, shape parameter β chages successively β = 0, 55; β = 0, 50; β = 0, 45; β = 0, 40. The taget plae is reduced. The optimizatio o secod derivatives avoids itersectios. The cotrol polygo remais well costructed. Figure 4.6: Iterpolatio of the cube, with β = 0.40 ad secod derivatives optimizatio. O the left, the global smooth surface ca be see. O the right, the cotrol polygo ad each taget plae are prited. Optimized first iterpolatio scheme ca be applied o iput meshes with a differet topology. For istace figures 4.7 ad 4.8 have the same topological 43

48 4.3 Toward the secod iterpolatio scheme type as that of a torus. The resultig iterpolat is satisfactory. Eve if the iput mesh is ot covex, iterpolatig surface is really smooth, ad it remais of degree (3,3). Figure 4.7: Iterpolatio of the torus usig the optimized first method. O the left the iput mesh with the boudary curve etwork, ad o the right the resultig iterpolatio. Figure 4.8: Iterpolatio of a figure with the same topology as a torus usig the optimized first method. It shows the iput mesh with the boudary curve etwork (o the left) ad the resultig iterpolatio (o the right) The, applyig optimized first iterpolatio scheme o a irregular mesh, amely a vase, gives figures 4.9 ad 4.0. It is clear that regular first derivatives do t give a satsfyig iterpolatio. A zoom o the hadle eighborhood (Fig. 4.0) shows malformatios due to o-adapted taget plaes. 4.3 Toward the secod iterpolatio scheme The aim for havig worked o this secod iterpolatio scheme is the case of irregular meshes, as figure 4.9. Irregular meas that iital mesh cotais very differet faces, or edges. I this sectio, a cube elogated (see Fig. 4.) will be iterpolated with the two methods developed. Actually, the secod method 44

49 4.3 Toward the secod iterpolatio scheme Figure 4.9: Iterpolatio of the vase usig the optimized first method. O the left the iput mesh with the boudary curve etwork, ad o the right the resultig iterpolatio. Figure 4.0: Iterpolatio of a figure with the same topology as two torus usig the optimized first method. A zoom o the hadle eighborhood shows the malformatios due to o-adapted first derivatives. is ot completed, so we chose to show the differece i the first derivatives ad the taget plae. Figure 4.: The iitial mesh to compare two iterpolatio methods is a elogated cube. It ivolves differet size of faces ad edges 45

50 4.3 Toward the secod iterpolatio scheme Figure 4. shows the resultig taget plaes with both methods. O the left, the first iterpolatio scheme implies wide taget plaes. Sice first derivatives are costructed to form the affie of a regular -go whose cetroid is the correspodig vertex, taget plae is too wide ad the iterpolat will ot be satisfactory. I the secod scheme, first derivatives are computed followig Piper s method ad are closed to mesh geometry. Figure 4.: Zoom o the first derivatives eighborhood. O the left, the first method ivolves first derivatives ot adapted to the mesh. O the right the first derivatives are clearly better : their directio ad value are adapted to the irregular mesh. This secod iterpolatio scheme, theoretically developed, seems to be a promisig method ad ca be a great cotiuatio of this project. 46