X-Splines : A Spline Model Designed for the End-User

Transcription

1 X-Splines : A Spline Model Designed for he End-User Carole Blanc Chrisophe Schlic LaBRI 1 cours de la libéraion, 40 alence (France) [blancjschlic]@labri.u-bordeaux.fr Absrac his paper presens a new model of spline curves and surfaces. he main characerisic of his model is ha i has been creaed from scrach by using a ind of mahemaical engineering process. In a firs sep, a lis of specificaions was esablished. his lis groups all he properies ha a spline model should conain in order o appear inuiive o a non-mahemaician end-user. In a second sep, a new family of blending funcions was derived, rying o fulfill as many iems as possible of he previous lis. Finally, he degrees of freedom offered by he model have been reduced o provide only shape parameers ha have a visual inerpreaion on he screen. he resuling model includes many classical properies such as affine and perspecive invariance, convex hull, variaion diminuion, local conroland C =G or C =G 0 coninuiy. Bu i also includesoriginal feaures such as a coninuum beween B-splines and Camull-Rom splines, or he abiliy o define approximaion zones and inerpolaion zones in he same curve or surface. 1 Inroducion Since he ground wor in CAD during he lae sixies, many differen models of splines have been inroduced. One specific characerisic of CAD is ha he mahemaical models developped by researchers are laer manipulaed by non-mahemaician end users (designers, archiecs, animaors). herefore, raher han is complee mahemaical properies, a major crierion for he evaluaion of a spline model may be he abiliy o undersand inuiively he degrees of freedom ha i provides. A full sudy of exising spline models on ha paricular poin lies no wihin he scope of his shor inroducion, bu le us jus ae one or wo examples. he popular NURBS model is a good example in which he user has o be familiar wih he mahemaical srucure o obain bes resuls. For insance, he manipulaion of he no vecor is really complex: firs he geomerical effecs generaed by hese manipulaions can hardly be prediced, second hese effecs are no robus because furher no manipulaions may move hem along he curve, and hird he effecs are propagaed along he whole isoparameric curves in he case of surfaces. Even he manipulaion of he weighs may someimes be confusing: for insance, he modificaions of wo adjacen weighs are muually cancelled [11]. he model ha accouns he mos for he ergonomics of he manipulaion is undoubedly he -spline model [1] which includes inuiive shape parameers (ension and bias). Ye, if he behaviour of he Laboraoire Bordelais de Recherche en Informaique (Universié Bordeaux I and Cenre Naional de la Recherche Scienifique). he presen wor is also graned by he Conseil Régional d Aquiaine. model is really naural when using global ension and bias, he exended model [] wih local parameers is less convincing, mainly because hese parameers are no direcly relaed o he conrol poins. Moreover, he C 0 =G coninuiy of he -splines is los by inerpolaion, his maes hem inadequae for many applicaions [9]. his paper proposes a new spline model ha has been designed o mae user manipulaions as inuiive as possible. Is formulaion is presened in four seps: Secion presens he lis of specificaions for he new model, Secion explains he principle and he basic formulaion, Secion 4 derives a more complee expression including an original shape parameer, finally, Secion deails he general formulaion. Bacground.1 Definiion In heir mos general definiion, splines can be considered as a mahemaical model ha associaes a coninuous represenaion (curve or surface) wih a discree se of poins of an affine space (usually IR or IR ). In he case of curves, his definiion can be expressed as follows: le IR wih ( = 0::n) be a se of poins called conrol poins, and le F : [0; 1]! IR (wih = 0::n) be a se of funcions called blending funcions, he spline curve generaed by he couples ( ; F ) is he curve C defined by he parameric equaion: 8 [0; 1] C() = nx =0 F () (1) According o he shape of he blending funcions, he resuling curve may eiher approximae he conrol poins or inerpolae hem. Figure 1 and Figure illusrae his disincion by showing wo classical examples of spline curves (cubic uniform B-splines [1] in Figure 1, cubic Camull-Rom [6] in Figure ). Each figure is divided in wo pars, he op shows he conrol laice and he curve, he boom shows he plos of he blending funcions. he same graphical framewor will be used hroughou he paper.. roperies he family of curves ha obeys Equaion 1 is exremely vas and hus many of is members are liely o be of lile ineres. In fac, he wor done over he years in he lieraure has exhibied many properies ha a spline model should include o become useful for geomeric modelling. In a recen survey, we have shown ha all hese properies can be obained by imposing specific consrains on he blending funcions [].

2 F1 F 1 0 Figure 1: Uniform B-spline curve 4 6 F1 F Figure : Camull-Rom spline curve Using ha resul, we are going o lis now all he properies (as well as he corresponding consrains on he blending funcions) ha we have found vial or simply desirable o include in our user-oriened spline model: Affine invariance: he affine ransformaion of a spline should be obained by applying he ransformaion o is conrol poins. his is provided by he normaliy consrain: 8 [0; 1] nx =0 F () = 1 () Convex hull: he spline should be enirely conained in he convex hull of is conrol laice. his is provided by he normaliy consrain combined wih he posiiviy consrain: 8 = 0::n 8 [0; 1] F () 0 () Variaion diminuion: he number of inersecions beween he spline and a plane (or a line, for D splines) should be a mos equal o he number of inersecions beween he plane and he conrol laice, which means ha he spline should have less oscillaions han is conrol laice. his propery is provided by combining he normaliy, he posiiviy wih he regulariy consrain: 8 = 0::n 9 [0; 1] = (4) 8 < () 0 and 8 0 = +1::n F 0() F () 8 > () 0 and 8 0 = 0::?1 F 0() F () his consrain may appear complex a a firs glance, bu i simply says ha he blending funcions are bell-shaped and ha wo funcions canno cross each oher in he zone where hey are simulaneously increasing or decreasing. Local conrol: Each conrol poin should only influence he shape of he spline in a resriced zone. his propery is provided by he localiy consrain: 8 = 0::n 9(? ; + ) [0; 1] = () 8 <? F () = 0 and 8 > + F () = 0 A spline may offer more or less local conrol according o he exen of he influence of a given conrol poin. o quanify his aspec, he noion of L p localiy [] can be used: a spline curve (resp. surface) has go L p localiy when each conrol poin influences p segmens (resp. paches) a mos. Smooh shapes/sharp shapes: he spline model should allow boh smooh shapes and sharp shapes and more precisely mixing smooh zones and sharp ones in he same curve. I is well nown ha parameric coninuiy does no provided any informaion on he shape of he curve; herefore one has o use geomeric coninuiy: smooh shapes are G a leas, sharp shapes are G 0 a mos. On he oher hand, parameric coninuiy is needed o provide smooh moion in animaion; herefore he model should also provide C coninuiy. Inuiive shape parameers: In addiion o he conrol poins, he spline should also provide oher degrees of freedom, usually called shape parameers. Bu o be usable by a non-mahemaician end user, he role of hese parameers should be as inuiive as possible. Among all he shape parameers (nos, weighs, ension, bias, curvaure) ha we can find in exising spline models, only he local ension effec (which allows he user o pull he curve locally oward one or several conrol poins) appears oally inuiive. Exisence of refinemen algorihms: he spline model should allow he use of refinemen or subdivision echniques which are powerful ools ha increase he number of degrees of freedom for a spline (conrol poins or shape parameers) wihou modifying is shape. Represenaion of conics: he spline model should be able o represen conic secions, and consequenly a large se of curves and surfaces (circles, ellipses, spheres, cylinders, surfaces of revoluion, ec) ha are inensively used in CAD. he exac represenaion of conics is one reason for he populariy of he NURBS model []. Neverheless, having only a close approximaion (up o he resoluion of he display, for insance) is sufficien for mos applicaions. Approximaion/Inerpolaion: For some applicaions or some users, approximaion splines are preferable, whereas for ohers, inerpolaion splines are imperaive. For ha reason, he model should provide approximaion splines and inerpolaion splines in a unified formulaion. Among he exising models, only he general Camull- Rom model [6] includes such a feaure; bu we would lie o ge a sep furher by allowing he creaion of approximaion zones and inerpolaion zones in he same curve. In he following secions, we describe a new spline model which was designed o fulfill as many iems as possible of he previous lis. A he curren sage in his developmen, all iems bu one (he exisence of refinemen echniques) are fulfilled by he model. he possibiliy of including he las iem will be discussed in he conclusion. Basic X-Splines.1 rinciple Building a new spline model from scrach implies defining a new family of blending funcions. Among he consrains ha have been lised

3 in Secion, he mos difficul o fulfill is he normaliy consrain. Indeed, finding a family of funcions F () ha sum o one whaever he value of is a ricy as. For ha reason, we have chosen o build our blending funcions independenly of he normaliy consrain, and hen o apply in a final sep, a normalizaion process which replaces F () by F (): () 8 [0; 1] F () = nf F() (6) =0 hus, he acual blending funcions F () will be normalized raional polynomials which, as a side-effec, adds he projecive invariance propery o he resuling curves. By combining he differen properies recalled in Secion, we can esablish ha for a normal, posiive, regular and local spline, each blending funcion F () is bell-shaped, sars o grow a a given value?, reaches is unique maximum a a second value and drops o zero a a hird value + (see Figure ). In classical spline models, F () is defined by a piecewise polynomial or a raional piecewise polynomial, composed of as many segmens as consecuive inervals beween? and + (e.g. four segmens wih cubic B-splines) ae he case of a uniform no vecor: 8 = 1::n??1 = If we apply he following reparamerizaion o he curve, u() =???? =?? we are assured ha u = 0 a no? where F () sars o grow and u = 1 a no where F () reaches is maximum. herefore, we have o find a polynomial f(u) defined on he range [0; 1] which can be lined o he lef par of F () by: F? () = f?? Because we wan a C coninuous curve, he following consrains for he funcion f(u) can be immediaely derived: (7) (8) f(0) = 0 f 0 (0) = 0 f 00 (0) = 0 (9) As he maximum of he blending funcion is reached a u = 1, is firs derivaive is necessarily null. Moreover, we can se f(1) = 1 because he normalizaion sep will reduce he maximum o is exac value anyway. Finally, he second derivaive a u = 1 can be se o a given consan(we call his consan?p o simplify he formulaion): f(1) = 1 f 0 (1) = 0 f 00 (1) =?p (10) Figure : Configuraion of he blending funcions he driving idea of he new model ha we propose here is he following: he non-null par of he blending funcion should be composed of only wo segmens 1. he firs, called F? (), is defined beween? and ; he second, called F + (), is defined beween and +. In order o mae his idea clearer, le us ae he case of a spline in which each conrol poin influences four segmens of he curve (i.e. L 4 localiy). his is a usual case (shared by every classical model of cubic splines, for insance) and is ofen considered [] as he bes rade-off beween low degree splines on one hand (which are closely relaed o he conrol laice and hus can hardly provide very smooh shapes) and high degree splines on he oher hand (which can hardly provide very sharp shapes). By definiion, for an L 4 spline, each blending funcion is non-null over four consecuive inervals of he no vecor: F () becomes non-null a no?, is maximal a no and becomes null again a no + (he nos are shown on he op of Figure ). As F () is composed only of wo segmens, i depends only on?, and +. hus here is a ind of alernaion in he way he nos are aen ino accoun (even poins use even nos and odd poins use odd nos). Moreover, as we will see, he blending funcions F? and F + cross each oher a no and all he derivaion of he model is based on his crossing. For ha reason, we have called his new model, cross-splines or X-splines, for shor.. Formulaion In fac, once he general principle has been esablished, he basic formulaion of he new model can be derived quie naurally. Le us firs hus we have derived a sysem of six consrains. As we search for a polynomial soluion, i will necessarily be quinic, in order o ge six degrees of freedom. By maching he consrains and he coefficiens of he polynomial, we obain: f p(u) = u?10? p + (p? 1) u + (6? p) u (11) Moreover, he propery of regulariy requires an increasing funcion on he range [0; 1] and hus a posiive derivaive. herefore here is an addiional condiion on p: 0 p 10 he funcion f p(u) (see Figure 4) provides he lef par of F () according o Equaion 8. By reversing he direcion and he origin of he reparamerizaion, he righ par of F () is obained similarly: F + () = fp +? (1) he wo funcions F? and F + join a no wih C coninuiy ( 00 () = 0 and F () =?p= ) which means ha he global blending funcion F (), and herefore he whole curve C(), are C coninuous. 1 In fac, we have also ried he case where he non-null par is composed of only one segmen. Bu his maes he modelmuch moreexpensive (degree8 raionalpolynomials) wih no addiional feaures.

4 Figure 6: Similariy of he cubic uniform B-splines and he basic X-splines blending funcions (for p = 8) Figure 4: Funcion f p(u) for p = 0; ; 4; 6; 8; 10 Finally, we ge he formulaion for a segmen of he curve C() on he parameer range [ +1; +], defined by he four conrol poins ; +1; +; +: A0() + A1() +1 + A() + + A() + C() = A 0() + A 1() + A () + A () (1) A 0() = f p +? A () = f p? A 1() = f p +? A () = f p?+1 he process defined above has provided a quinic raional approximaion spline model ha includes he properies of normaliy, posiiviy, regulariy, localiy and C coninuiy. Moreover, he curves conain a degree of freedom p [0; 10] which allows a (sligh) modificaion of heir shapes. I should be noiced ha a very ineresing case is obained for p = 8. Indeed, afer he normalizaion sep, he blending funcions are very close o he cubic uniform B-splines basis funcions (see Figure 6). I means ha he resuling curves call hem basic X-splines are almos idenical o he uniform cubic B-splines (compare Figure and Figure 1) F1 F Figure : Basic X-spline curve 4 Exended X-Splines 4.1 Formulaion he degree of freedom p in Equaion 11 does no offer enough variey in he shapes of he blending funcions (see Figure 4) o provide ineresing effecs on he resuling curves. herefore, i appears somewha useless in he formulaion of he new model. In fac, he exisence of his degree of freedom will be hidden o he end user. As we will see below, his parameer p is needed o manage anoher parameer s, ha we are going o inroduce now and which is he acual degree of freedom accessible by he end user. Among he iems of our lis of specificaions, ension and angular shapes (G 0 coninuiy) can be included in our model by he same derivaion. Indeed, he basic idea which has led o he concep of ension in he spline lieraure is o be able o srain he curve (or he surface) in order o pull i oward he conrol laice. A is limi, his process forces he curve o inerpolae one or several conrol poins, and due o he convex hull propery, his inerpolaion will creae sharp edges. o bring he curve closer o a given par of he conrol laice, one has o increase he influence of he corresponding conrol poins. A sraighforward idea o realize his process is o add a specific weighing coefficien o each conrol poins. Bu, as we have recalled in Secion 1, his soluion (which is used in every classical raional spline) does no wor in a saisfying way, because he influences of neighbouring weighs are muually cancelled. herefore, we propose here an original soluion o include he concep of ension, which does no conain he drawbac of he exising models. o illusrae his new soluion, le us ae he blending funcions F ; and F 4 in Figure. We now ha reaches is maximum a. Bu, as F and F 4 are no null a, he normalizaion process has se he acual maximum o =(F + + F 4). herefore, a way o increase his maximum, in order o bring he curve closer o he conrol poin, is o decrease F ( ) and F 4( ). We now ha in he area of ineres, F (respecively F 4) decreases (respecively increases) monoonically in he range [ ; 4]. hus, o obain smaller values for hese funcions a, one has o speed up he decrease of he former and o slow down he increase of he laer. o realize hese wo operaions symmerically, we acually push he crossing poin of F and F 4 down oward he horizonal axis. For ha, we inroduce a new degree of freedom s [0; 1] a poin. his parameer will be used, firs o compue he value + (where F becomes null) by inerpolaion beween 4 and : + = + s ( 4? ) = + s and second, o compue he value? 4 by inerpolaion beween and :? 4 = + s (? ) =? s (where F4 becomes non null) In oher words, i means ha F (respecively F 4) is null all over he range [ + ; 4] (respecively [;? 4 ]). he same operaion can be

5 done for each. he resuling values (? ; + ) have o be replaced in he reparamerizaion equaions (Equaion 8 and Equaion 1) as follows: F? () = fp???? F + () = fp? +? + (14) he wo pars of F () sill join a, heir firs derivaives are sill null bu heir second derivaives are differen: 0 (? ) =?p (?? ) 0 ( + ) =?p (? + ) (1) Here is he poin where our parameer p will finally be used. Indeed, in order o equal he lef and righ expressions, he only hing o do is o use a specific value for p (noed p?1) in F? and anoher one (noed p +1) in F +. aing p?1 =? (? ) and p +1 =? ( + ) (16) and herefore he curve is usually (when?1; and +1 are no aligned) only G 0 a. In oher words, i means ha, even if i is always C, he model enables he creaion of angular poins or sharp edges. 4. Examples his secion demonsraes he role of he parameer s by showing is influence on he resuling shapes. he basic formulaion defined in Secion is a paricular case of he exendedone, where all parameers s are seo one. As we have seen,basic X-splines are almosidenical o uniform cubic B-splines. A firs varian consiss in seing s 0 and s n o zero in order o inerpolae he end poins of he conrol laice and hus o enable beer conrol of he curve boundaries. he resuling curves call hem exremal X-splines (see Figure 7) are very close o he classical exremal cubic B-splines (also called non-periodic cubic B-splines). 1 gives 0 (? ) = 0 ( + ) =? which provides C coninuiy bu assures also ha he parameers p are in he range [0; 8] as needed o ge he propery of regulariy and o obain he cubic B-splines as a limi case. herefore we can derive a new formulaion for he segmen of he curve C() on he range [ +1; +] defined by he four conrol poins ; +1; +; +: 4 F1 F A0() + A1() +1 + A() + + A() + C() = A 0() + A 1() + A () + A () (17) A 0() = > +? 0 : f p?1? +? + A 1() = > + +1? 0 : f p? ? + +1 A () = <? +? 0 : f p+1?? + +?? + A () = <? +? 0 : f p+?? + +?? + Figure 7: Exremal X-spline curve s 0 = 0; s 1 = 1; s = 1; s = 1; s 4 = 1; s = 1; s 6 = 0 Le us now decrease he value of one parameer s (say s ). By comparing Figure 7 and Figure 8, one can see ha he crossing poin of F and F 4 a no has been pushed down. 1 p?1 = (? + ) p = (+1? + +1 ) p +1 = (+?? + ) p + = (+?? + ) he expression of C() seems complex bu in fac i can be implemened very compacly and efficienly (1 lines of source code in C language). So for he end user, an exended X-spline is oally defined by a se of quadruples (x ; y ; z ; s ) wih = 0:::n. All hese degrees of freedom have a very simple inerpreaion. he parameers (x ; y ; z ) IR are he coordinaes of he conrol poins. he parameer s [0; 1] symbolizes he disance beween he curve and he conrol laice: when s = 1, he curve passes relaively far away from poin ; when s decreases, he curve comes closer and closer o ; finally when s = 0, he curve passes hrough. I should be noiced ha he curve is always C (due o he consrucion process ha has been used), even when i inerpolaes a conrol poin. Bu in ha case, he firs and second derivaives drop o zero a We use here he (es? a : b) operaor borrowed from he C programming language which allows one o wrie muliple expressions in a compac way F1 F Figure 8: Augmenaion of he influence of poin s 0 = 0; s 1 = 1; s = 1; s = 0:; s 4 = 1; s = 1; s 6 = 0 herefore, afer he normalizaion sep, he maximum of has been increased and he curve has been pulled oward. Moreover, neiher

6 he maximum of F nor he maximum of F 4 has been modified. his means ha he curve has no changed near or 4: all he modificaions are localized in he neighbourhood of poin. More precisely, one can show ha a shape parameer s influences only wo segmens of he curves which is half he exen of he oher hree coordinaes (x ; y ; z ) of poin (i.e. L localiy raher han L 4 ). While s decreases, he maximum of increases. Finally, for s = 0, his maximum is se o one, which provides a sharp (G 0 coninuous) inerpolaion of poin F1 F F1 F Figure 11: Sharp inerpolaion of, e 4 s 0 = 0; s 1 = 1; s = 0; s = 0; s 4 = 0; s = 1; s 6 = Figure 9: Sharp inerpolaion of poin s 0 = 0; s 1 = 1; s = 1; s = 0; s 4 = 1; s = 1; s 6 = 0 he L localiy of he influence of he parameers s allows he same ind of acion on several adjacen conrol poins. For insance, if we decrease s ; s and s 4, he curve is pulled simulaneously oward ; and 4, 4 F1 F F 1 F F 4 F Figure 10: Augmenaion of he influence of, e 4 s 0 = 0; s 1 = 1; s = 0:; s = 0:; s 4 = 0:; s = 1; s 6 = 0 and if we se he hree parameers o zero, we obain a sharp inerpolaion of ; and 4 (see Figure 11). Finally, for he limi case where all he parameers s are se o zero, he curve merges wih he conrol laice (see Figure 1). Bu noice ha he curve is no a linear spline because he paramerizaion is C here, whereas i is only C 0 for linear splines. F 6 Figure 1: Sharp inerpolaion of every conrol poin s 0 = 0; s 1 = 0; s = 0; s = 0; s 4 = 0; s = 0; s 6 = 0 his abiliy o mix smooh curves and sharp edges in an unresriced way maes he exended X-spline model a candidae of choice for many applicaions. In vecorial fon design, for insance, one swiches frequeny beween smoohness and sharpness. herefore, he use of X-splines enables he design of characers wih one single spline for he ouline (plus evenually one spline for each hole) defined by a small number of conrol poins (see lef par of Figure 1) o conclude his secion, noe ha a very useful case is obained when he conrol laice forms a regular polygon and all he s are se o one: he resuling curve is a circle (see righ par of Figure 1. In fac, his circle is only an approximaed one bu his approximaion is so close (for 8 conrol poins, he ampliude of he oscillaions of he curve around he rue circle represens less han a facor 10? of he radius, and for 1 conrol poins, his variaion is less han 10?6 ) ha i is sufficien for mos of he applicaions. Saring from ha ernel case, oher conics can be approximaed as well wih a similar accuracy []. A similar resul is obained wih B-splines [4], herefore i is no surprizing ha i holds also for X-splines which approximaeb-splines in ha paricular configuraion.

7 g(0) = 0 g 0 (0) = q g 00 (0) = 4q g(1) = 1 g 0 (1) = 0 g 00 (1) =?p h(0) = 0 h 0 (0) = q h 00 (0) = 4q (19) h(?1) = 0 h 0 (?1) = 0 h 00 (?1) = 0 where q is a degree of freedom ha conrols he value of he firs derivaive a u = 0 (he same degree of freedom has been used by Duff in his ensed inerpolaion spline model [8]. All hese consrains can be fulfilled by wo quinic polynomials: Figure 1: Fon design and represenaion of he circle g(u) = q u + q u + (10?1q?p) u + (p+14q?1) u 4 + (6?q?p) u (0) General X-Splines.1 Formulaion As hey have been formulaed above, exended X-splines fulfill many of he properies lised in Secion. Neverheless, even if hey allow inerpolaing one or several conrol poins, exended X-splines are sill approximaion splines, because only sharp inerpolaions are provided. he las feaure of our lis was he abiliy o manipulae he same model eiher as an approximaion spline or as an inerpolaion spline. he goal of his secion is o show how his characerisic can be included in he X-spline model. Bu, as recalled in Secion, using inerpolaion splines implies forsaing he posiiviy of he blending funcions and herefore he convex hull propery. For some applicaions (and for some users), his is inconceivable. For ha reason, we have purposely separaed his exension from he previous secion. So, he reader may choose beween he formulaion ha fulfills he convex hull propery and he formulaion ha provides he approximaion/inerpolaion dualiy. In Secion 4, we saw ha when he value of he parameer s is decreased, he blending funcion F? (respecively +1 F + ) becomes?1 null beween?1 and? (respecively +1 +?1 and +1). A he limi case, when s = 0, F? (respecively +1 F +?1 ) is null over he whole range [?1; ] (respecively [ ; +1]). Saring from ha configuraion of sharp inerpolaion, o ge a smooh (G coninuiy) inerpolaion of poin, we mus allow F? and +1 F +?1 o become negaive over hese ranges. Moreover, in he same manner as we have sough o approximae cubic B-splines wih he basic formulaion, we will ry o approximae cubic Camull-Rom splines wih his general formulaion. If we apply he following reparamerizaion o he curve, u() =? +1? =? (18) we are assured ha u =?1 a no?1 where F? +1 ges negaive, u = 0 a no where F? +1 ges posiive, and u = 1 a no +1 where F? +1 reaches is maximum. herefore, we have o find wo polynomials: g(u) defined on [0; 1] which represens he posiive par of F? +1 and h(u) defined on [?1; 0] which represens is negaive par. hese wo funcions mus join up a u = 0 wih C coninuiy. As in Secion, we can derive a sysem of consrains bu his ime here are wo funcions, which means 1 consrains: h(u) = q u + q u? q u 4 + q u Saring from hese equaions, he same consrucion process deailed in Secion provides a raional quinic inerpolaion spline model ha includes he properies of normaliy, localiy and C coninuiy. Moreover, he curves conain a degree of freedom q which allows modificaion of heir shapes. Figure 14: Similariy of he cubic Camull-Rom splines and he general X-splines blending funcions (for q = 1=) wo imporan remars should be made abou his model. Firs, as in every inerpolaion spline model, he regulariy propery is los, hus he curve may have unwaned oscillaions. We have observed experimenally ha hese oscillaions can usually be avoided by limiing q o he range [0; 1=]. Second, an ineresing case is obained for q = 1= because he blending funcions are very close o he Camull-Rom funcions (see Figure 14). Bu i should be noiced ha he new funcions are C coninuous insead of C 1. he final sep of he consrucion of our new spline model will be o merge he parameer s of he approximaion model and he parameer q of he inerpolaion one. Here again, he goal is o simplify he degrees of freedom manipulaed by he end user. racically, only one shape parameer s per conrol poin will be used. his is done wih he following convenion: When he user ses all s in he range [0; 1], i means ha he wans o manipulae approximaion splines. In ha case, s is he curve/laice disance parameer defined in Secion 4 (in paricular, uniform cubic B-splines are approximaed for s = 1). When he user ses all s in he range [?1; 0], i means ha he wans o manipulae inerpolaion splines. In ha case, q is obained from s by q =?s = (so, s =?1 provides q = 1= which approximaes cubic Camull-Rom splines). he posiive/negaive disincion for s indicaes clearly ha here is a breaing poin: for posiive s, he convex hull propery is fulfilled, for negaive s, i is no he case anymore. On he oher hand he

8 inuiive noion of curve/laice disance is preserved even for negaive s. Indeed, as we will see below, he more s depars from zero, he more he curve depars from he conrol laice.. Examples We already now ha a sharp (G 0 coninuiy) inerpolaion of he conrol laice can be obained by seing all s o zero (see Figure 1). If we wan o realize a smooh (G coninuiy) inerpolaion, he only hing o do is o se hese parameers o negaive values. For insance, by seing all s o?1, an inerpolaion spline almos idenical o he Camull-Rom spline is obained (compare Figure 1 and Figure ). As expeced, he blending funcions become parly negaive, and hus he convex hull propery is los. And finally, wha is perhaps he mos ineresing feaure of he X- spline model, one can combine wihou any resricion, posiive and negaive shape parameers s in order o creae approximaion zones and inerpolaion ones in he same curve (see Figure 17) F1 F 4 F1 F 0 Figure 1: Smooh inerpolaion of every conrol poin s 0 = s 1 = s = s = s 4 = s = s 6 =? F1 F Figure 17: Approximaion zones and inerpolaion zones s 0 = 0; s 1 = s = s =?1; s 4 = s = 1; s 6 = 0 6 Surfaces he exension of he new model from curves o surfaces is sraighforward. he only hing o do is o compue he ensor produc of wo non-normalized X-spline curves and hen o apply he normalizaion sep 4. he characerisic of he X-splines o creae all possible geomeric effecs by using only uniform no vecors is vial here because, as we have recalled in Secion 1, effecs due o no manipulaions (e.g. sharp edges for B-splines) are propagaed along he whole isoparameric curves. On he conrary, he shape parameers of he X-spline model are direcly relaed o he conrol poins and hus can be localized precisely on a given zone of he surface. Because of he ensor produc, wo shape parameers r and s are provided for each conrol poin where r acs in he u direcion of he surface and s acs in he v direcion. A nice consequence is ha non-isoropic manipulaions are allowed (for insance, creaing sharpness in one direcion and smoohness in he oher one). As a counerpar, he behaviour of hese parameers is a bi more suble han previously: r > 0, s > 0 : is a C =G approximaion poin r = 0, s = 0 : is a C =G 0 inerpolaion poin r < 0, s < 0 : is a C =G inerpolaion poin r = 0, s > 0 : is an approximaion poin providing C =G 0 coninuiy in u and C =G coninuiy in v Figure 16: Modificaion of he inerpolaion s 0 = 0; s 1 = s = s =?1; s 4 = s =?0:; s 6 = 0 By providing differen values for he parameer s, he shape of he inerpolaion curve can be conrolled precisely. For insance, one can enable very slac inerpolaion for a specific zone of he laice and a much igher inerpolaion for anoher zone (see Figure 16). r = 0, s < 0 : is an inerpolaion poin providing C =G 0 coninuiy in u and C =G coninuiy in v Figure 19 and Figure 18 shows some examples of X-spline surfaces. You should noice he abiliy o creae inerpolaion of adjacen conrol poins, localized sharp edges as well as sof ransiions beween sharp and smooh zones; hree feaures ha are impossible (or a bes, only possible in specific cases) wih any exising spline model. 4 his process is someimes called generalized ensor produc [11]

9 Figure 19: Smooh exrusion from a sharp objec Figure 18: Sharp exrusion from a smooh objec Noe ha he sar-shaped fla face on he op of he objec is composed of wo sides wih sraigh edges (lef and boom) and wo sides wih rounded edges (op and righ). Sraigh sides creae sharp edges ha are propagaed all along he exrusion whereas he sharp edges smoohly vanish when hey come near he rounded sides of he op face. 7 Conclusion In his paper, we have presened a new model of spline curves and surfaces. his model includes many classical properies such as affine and perpecive invariance, convex hull, variaion diminuion, local conrol and C =G or C =G 0 coninuiy, as well as some original feaures such as a coninum beween (an approximaion of) B-splines and (an approximaion of) Camull-Rom splines, or he abiliy o define approximaion zones and inerpolaion zones in he same curve or surface. hese properies have been obained by defining a new family of blending funcions ha are quinic raional polynomials and inroducing an original shape parameerha provides, for each conrol poin, a smooh ransiion beween approximaion, sharp inerpolaion and smooh inerpolaion. his paper is only inended as an iniial presenaion of X-splines. For space limiaions, several opics could no be included here. We propose some addiional resuls in [] which should be considered as he companion paper of his one. More precisely, he following opics are discussed in i: Some precisions on efficien implemenaion of X-splines: For insance, one can show ha even if hey are quinic, raional and provide more geomerical effecs, uniform X-splines are less expensive o compue han non-uniform cubic B-splines). Lower order and higher order X-splines: Quinic polynomials have been chosen here because we sough for C =G coninuiy, bu in fac a similar consrucion process can be used for any polynomial of degree +1 providing splines wih C =G coninuiy. Exension o non-uniform no vecors: Geomerical effecs generaed by non-uniformiy in classical splines can be creaed by he shape parameers, so his exension is no ha vial. Neverheless, non-uniform nos may be useful for ey-frame animaion or daa-fiing.

10 Refinemen algorihms: his is clearly a much harder as. For he momen, we propose only some preliminary resuls on a ind of De Caseljau subdivision algorihm. Acnowledgemens We wish o han all he anonymous reviewers for many helpful commens and suggesions. We also han Brian Smih (from Lawrence Bereley Laboraory), he mainainer of he xfig pacage, who gave us he permission o include he X-splines model in his sofware and o use i for public demonsraion. Finally, special hans o C. Feuille, S. Grobois, L. Mazière and L. Miniho who modified xfig for us and discovered he L (raher han L 4 ) localiy of he shape parameers. 8 References [1] B. Barsy, he Bea-Spline: a Local Represenaion based on Shape arameers and Fundamenal Geomeric Measures, hd hesis, Universiy of Uah, [] R. Barels, J. Beay, B. Barsy, An Inroducion o Splines for Compuer Graphics and Geomeric Modeling, Morgan Kaufmann, [] C. Blanc, echniques de Modélisaion e de Déformaion de Surfaces pour la Synhèse d Images, hd hesis, Universié Bordeaux I, 1994 (in french). [4] C. Blanc, C. Schlic, More Accurae Represenaion of Conics by NURBS, echnical Repor, LaBRI, 199 (submied for publicaion). [] C. Blanc, C. Schlic, X-Splines: Some Addiional Resuls, echnical Repor, LaBRI, 199 (available by H a [6] E. Camull, R. Rom, A Class of Inerpolaing Splines, in Compuer Aided Geomeric Design, p17-6, Academic ress [7] E. Cohen,. Lyche, R. Riesenfeld, Discree B-Splines and Subdivision echniques, Compuer Graphics & Image rocessing, v14, p87-111, [8]. Duff, Splines in Animaion and Modelling, SIGGRAH Course Noes, [9] G. Farin, Curves and Surfaces for Compuer Aided Geomeric Design, Academic ress, [10] D. Forsey, R. Barels, Hierarchical B-Spline Refinemen, Compuer Graphics, v, n4, p0-1, [11] L. iegl, On NURBS: a Survey, Compuer Graphics & Applicaions, v11, n1, p-71, [1] R. Riesenfeld, Applicaions of B-Spline Approximaion o Geomeric roblems of Compuer Aided Design, hd hesis, Universiy Syracuse, 197.