Smooth Spline Surfaces over Irregular Meshes

Transcription

1 Smooth Splie Surfaces over Irregular Meshes Charles Loop Apple Computer, Ic. Abstract A algorithm for creatig smooth splie surfaces over irregular meshes is preseted. The algorithm is a geeralizatio of quadratic B-splies; that is, if a mesh is (locally) regular, the resultig surface is equivalet to a B-splie. Otherwise, the resultig surface has a degree 3 or 4 parametric polyomial represetatio. A costructio is give for represetig the surface as a collectio of taget plae cotiuous triagular Bézier patches. The algorithm is simple, efficiet, ad geerates aesthetically pleasig shapes. CR Categories ad Subject Descriptors: I.3.5 [Computer Graphics]: Computatioal Geometry ad Object Modelig - Curve, Surface, Solid, ad Object represetatios; J.6 [Computer-Aided Egieerig]: Computer-Aided Desig (CAD); G.1. [Approximatio]: Splie Approximatio. Additioal Key Words ad Phrases: Computer-aided geometric desig, B-splie surfaces, Triagular patches, Geometric cotiuity, Irregular meshes, Aribitrary topology. 1 Itroductio The B-splie paradigm for modelig smooth surfaces is limited by the requiremet that the cotrol poit mesh must be orgaized as a regular rectagular structure. Igorig this requiremet by collapsig cotrol mesh edges leads to surfaces with ambiguous surface ormals ad degeerated parameterizatios. A more geeral method is to costruct a surface from a mesh of poits without degeeracy. By costructig this surface usig piecewise polyomials, familiar algebraic tools ca be brought to bear for aalysis. This is the approach take i this paper. A ew type of splie surface is preseted for modelig surfaces of arbitrary topological type by smoothly approximatig a irregular cotrol mesh. The advatage of this techique over existig schemes is simplicity, efficiecy, ad piecewise polyomial form. The splie surface is simply costructed by computig a triagular Bézier represetatio of a etwork of surface patches. Beig of Author s address: Apple Computer, Ic., 1 Ifiite Loop, MS:301-3J, Cupertio, CA 95014; loop@apple.com fairly low polyomial degree (at most 4), these patches are efficiet to compute ad evaluate. The splie surface is smooth, sice the patches fit together with taget plae cotiuity. Aother advaage of this scheme is a close relatioship to quadratic B-splie surfaces. I regular regios of the mesh, the surface is equivalet to a B-splie represeted by bi-quadratic Bézier patches. This property ca represet a cosiderable savigs i time ad space, sice i practice cotrol meshes ofte have few irregularities. The splie algorithm takes a irregular cotrol mesh as iput. A ew refied mesh is created with more faces, vertices, ad edges tha the origial. The ew mesh has a simpler structure sice every vertex has exactly four edges icidet upo it. Next, a itermediate form called a quad-et is costructed correspodig to each vertex of the refied mesh. The quad-ets characterize local 4-sided regios of the surface i a uiform way. Fially, a group of four quartic triagular patches are costructed for each quadet as output. The uio of these patches costitutes a smooth splie surface. This paper is orgaized as follows: previous work is surveyed i. Relevat backgroud material, icludig Bézier forms ad B-splies are covered i 3. The splie algorithm is preseted as a sequece of pipelie stages i 4. A detailed developmet of the smoothess costraits used to costruct the surface is preseted i 5. Cocludig remarks are foud i 6. Special techiques for dealig with meshes with boudaries (i.e., meshes that are ot closed) are give i Appedix A. Previous Work The earliest attempts to overcome the topological limitatios of B-splie surfaces were based o the refiemet priciple[1, 4]. The idea is to refie, or subdivide, a irregular mesh by creatig a ew mesh, with more faces ad vertices, that approximates the old. By repeatig this process, a smooth surface is formed i the limit. Subdivisio algorithms are coceptually quite simple ad geerally geerate ice shapes. However, subdivisio surfaces do ot admit a aalytic form, complicatig their use i may practical applicatios. Despite this, algorithms based o subdivisio surfaces cotiue to appear[7]. Gregory patches have bee used to iterpolate the vertices of a irregular mesh[]. These patches have sigularities at corers ad are ot polyomial. Other o-polyomial surface patches used to defie B-splie-like surfaces over irregular meshes iclude the 3 ad 5-sided patches defied i[17], ad -sided S- patches[9, 1]. S-patch based schemes ca be iefficiet (i both time ad space) for >5 or 6. A geeralizatio of quartic triagular B-splies to strictly triagular meshes usig degree six polyomial patches appears i [11]. Other schemes use tesor

2 product polyomials, but require the coectivity of the cotrol mesh to be restricted[6, 18]. More recet approaches to the problem assume that irregular vertices (a vertex with other tha 4 edges icidet upo it) are isolated. That is, every irregular vertex is surrouded by oe or more layers of quadrilaterals ad regular vertices. G-splies[8, 16] take this approach. Severals schemes by Peters[13, 14, 15] isolate irregularities by applyig oe or more refiemet steps to a irregular mesh. The approach take i this paper is similar. The distiguishig features are that oly oe refiemet step is required, ad the mesh does ot have to be preprocessed to have excusively 3 or 4-sided faces. The trade-off for this simplificatio is fewer patches of higher degree. The patches computed here are at most polyomial degree 4, as opposed to degree 3 i [10, 13, 14]. 3 Backgroud This sectio gives a brief review of Bézier curves ad surfaces, ad B-splie surfaces. Cosult [5] for additioal details. 3.1 Bézier forms A degree d Bézier curve is defied B(t) = d b ibi d (t), i=0 where t [0, 1], the poits b i form the Bézier cotrol polygo, ad ( ) Bi d d (t) = (1 t) d i t i, i are degree d Berstei polyomials. As a coveiet otatio, a Bézier curve will be idetified by its cotrol polygo represeted by the vector [b 0, b 1,...,b d ]. A degree r by s tesor product Bézier patch is defied B(u, v) = r i=0 s b ijbi r (u)bj s (v), where u, v [0, 1], ad the b ij are a rectagular array of poits formig the Bézier cotrol et. A degree d Bézier triagle is defied B(u, v) = b ijk Bijk(u, d v), i+j+k=d where u, v, (1 u v) [0, 1], ij, ad k are o-egative itegers that sum to d, the b ijk form a triagular Bézier cotrol et, ad ( ) Bijk(u, d d v) = (1 u v) i u j v k, ijk are the degree d bi-variate Berstei polyomials where ( d ijk) is d! the triomial coefficiet. i!j!k! Bézier surfaces are a coveiet represetatio for idividual polyomial patches. Algorithms for rederig, raytracig, ad surface itersectio ofte utilize the Bézier form. Whe costructig smooth composite surfaces cosistig of several patches, satisfyig the ecessary smoothess costraits amog Bézier surfaces ca be quite complex. I this settig, it is preferable to use B-splie surfaces. 3. B-splies A tesor product B-splie surface is defied S(u, v) = d ijni r (u)nj s (v), i j where the d ij form a rectagular cotrol mesh ad the Nk d are order d (degree d 1) B-splie basis fuctios. Each basis fuctio is defied over a partitio of the real axis called a kot vector (see [5] for details). Two particular properties of B-splies are of iterest here. First, by itroducig a ew kot betwee each pair of existig kots, the cotrol mesh is refied, or subdivided, without chagig the shape of the surface. Secod, B-splies are piecewise polyomial, therefore it is possible to represet a B-splie surface as a collectio of idividual polyomial patches. The splie surface preseted i this paper is closely related to quadratic B-splie surfaces with uiform kots. A quadratic B-splie ca be represeted as a composite of biquadratic tesor product Bézier patches. A sigle such patch is costructed correspodig to each vertex d ij of the cotrol mesh as illustrated i Figure 1. The corer poits b 00, b 0, b 0, ad b are foud as the cetroids of the four faces surroudig d ij. The poits b 10, b 01, b 1, ad b 1 are foud as midpoits of the four edges icidet o d ij, ad the poit b 11 is equivalet to d ij. The refiemet algorithm for quadratic B-splies ivolves computig a ew vertex correspodig to each {vertex, face} pair of the origial mesh. The ew vertices are foud as weighted averages of the poits belogig to each face of the origial mesh. For the quadratic B-splie case, these weights (goig aroud a face) are { 9, 3, 1, 3 }. The ewly created vertices are the coected to form the faces of the refied cotrol mesh. b 00 b 01 b 0 b 10 d ij b 11 b 1 b 0 b 1 b Figure 1: The bi-quadratic Bézier patch correspodig to the B- splie cotrol mesh poit d ij. A B-splie surface is smooth because adjacet patches share positios ad first derivatives at all poits alog commo boudaries. This otio of matchig derivatives alog patch boudaries is sufficiet because the domai of each patch lies i a sigle uv plae. Therefore, a B-splie surface is a deformatio of this domai plae. For this reaso, B-splie surfaces ca oly model shapes that are topologically plaar. B-splie surfaces may also be defied over cyliders ad tori, as these domais ca tile the plae.

3 Uder a less restrictive defiitio, a surface is cosidered smooth if at all poits it has a cotiuous, well-defied taget plae. This otio is kow as first order geometric cotiuity[3] ad deoted G 1. I the ext sectio, a splie surface is created by costructig a collectio of patches over idepedet domais such that the uio of this collectio is G 1 4 Costructig the Splie Costructig the splie surface begis with a user-defied cotrol mesh deoted M 0. A cotrol mesh is a collectio of vertices, edges, ad (ot ecessarily plaar) faces that ca ituitively be thought of as a polygoal surface that may, or may ot, be closed. The term valace is used to deote the umber of edges icidet o a vertex. The splie surface is costructed i the followig stages: Iput: irregular cotrol mesh 1. refie mesh. costruct quad-ets 3. costruct patches Output: collectio of triagular patches The mesh M 0 is passed to a refiemet procedure that creates a ew mesh M 1. The purpose of the refiemet procedure is to isolate irregularities. After the refiemet step, the mesh M 1 is used to costruct a set of quad-ets. The quad-ets characterize the surface locally, ad provide a uiform structure for the third ad fial step. From each quad-et, a collectio of four quartic triagular Bézier patches is costructed ad output. The details of each step are described i the ext three sectios, followed by some examples. 4.1 Mesh Refimet The first step takes a user-defied cotrol mesh M 0 ad creates a ew refied mesh M 1. The vertices of M 1 are costructed to correspoded to each {vertex, face} pair of M 0. Let F be a face of M 0 cosistig of vertices {P 0,P 1,...,P } with cetroid O (the average of the P i s). The poit P i of M 1 correspodig to {P i,f} is foud by P i = 1 O Pi Pi + 1 Pi+1, 8 where all subscripts are take modulo. The faces of M 1 are costructed correspodig to a vertex, face, or edge of M 0. Each k-valat vertex of M 0 will geerate a k-sided face belogig to M 1. Similarly, each -sided face of M 0 will geerate a -sided face belogig to M 1. Fially, each edge of M 0 will geerate a 4-sided face belogig to M 1. This costructio is illustrated i Figure. Note that all the vertices of M 1 are 4-valat, ad every o-4-sided face is surrouded by 4-sided faces. Special cosideratio for vertices ad edges that belog to the boudary of M 0 ca be foud i Appedix A. Remark : The refiemet rule give here is equivalet to quadratic B-splie refiemet for regular meshes. A more geeral costructio of the refied mesh poits due to Peters[13] associates a pair of scalar values u ad v with each poit P i such that P i =(1 u)(1 v)o + (1 u)v P i 1 + u+v P i + u(1 v) P i+1. The parameters u ad v are similar to kots of a B-splie i that they may be used to locally adjust the shape of the surface. More techically, a cotrol mesh is a tessellated, orieted -maifold (possibly with boudary). Figure : Mesh refiemet: The vertices of the refied mesh M 1 (thi lies) correspod to {vertex,face} pairs of the origial mesh M 0 (bold lies). 4. Quad-Nets I the secod step, 16 poits ad a pair of itegers collectively referred to as a quad-et are costructed correspodig to each vertex of M 1. Though quad-ets are i may ways like the cotrol ets of Bézier patches, their purpose here is oly as a itermediate stage betwee the refied mesh ad the fial triagular Bézier surface patches. A quad-et ad its labelig scheme are illustrated i Figure 3. V A 03 A 0 A 01 A 00 A 13 A 1 A 3 A V A 11 A 10 A 1 A 0 A 33 A 3 A 31 A 30 Figure 3: The quad-et correspodig to the vertex V of M 1. A quad-et locally characterizes a piece of the splie surface bouded by the four cubic Bézier curves [A 00,A 10,A 0,A 30], [A 30,A 31,A 3,A 33], [A 33,A 3,A 13,A 03], ad [A 30,A 0,A 01,A 00]. The corers A 00, A 30, A 03 ad A 33 lie at the cetroids of the four faces surroudig a vertex V. The iterior poits A 11, A 1, A 1,

4 ad A help specify the taget plae alog each of the four boudary curves. I order to esure that the splie surface is G 1, some costraits must be satisfied betwee the poits of a pair of adjacet quad-ets. These costraits are as follows: (1 c)a 00 + ca 01 = 1 A Â10, (1) 1 A A0 = 1 A1 + 1 Â1, () A 03 = 1 A Â13, (3) where Â10, Â 1 ad Â13 are poits of a adjacet quad-et, ad c is a scalar to be determied. Similar costraits apply for the other three boudary curves i a symmetric maer. Justificatio for Costraits (1), (), ad (3) is provided i 5. Costrait (1) must hold betwee all pairs of adjacet quadets that share the poit A 00. This implies that all quad-et poits surroudig A 00 must be co-plaar. The followig theorem is the key to costructig quad-et poits that satisfy this requiremet: Theorem 4.1 Let P 0,...,P R 3 be a set of poits i geeral positio. The set of poits Q 0,...,Q foud by Q i = 1 satisfy where P j(1 + β(cos π(j i) + ta π si π(j i) )), (4) (1 cos π )O + cos π Qi = 1 Qi Qi+1, (5) O = 1 P j, ad are therefore co-plaar. Proof : See Appedix B. The factor β i Equatio (4) is a free parameter that may be set arbitrarily. Theorem 4.1 applies to the costructio at had by settig β = 3 (1 + cos π ), ad iterpretig the poits P 0,...,P as the vertices of a face belogig to mesh M 1, the poit O as A 00, ad the poits Q 0,..., Q as the quad-et poits surroudig A 00. Uder this iterpretatio it is immediately clear from (5) that Costrait (1) is satisfied with c = cos π. Costructig the poits A30, A03, ad A 33 ad the surroudig quad-et poits is similar. The observatio that every -sided face of M 1 ( 4) is surrouded by 4-sided faces, idicates that faces cotaiig A 30 ad A 03 are always 4-sided. Costrait (3) is satisfied sice cos π = 0 whe = 4. Applyig Theorem 4.1 to each of the four faces surroudig a vertex of M 1 will produce all of the quad-et poits except for the four iterior poits A 11, A 1, A 1, ad A. The costructio for the poit A 1 is as follows: let V be the vertex about which the quad-et is costructed, ad let ˆV be a edge sharig eighbor of V (see Figure 3). Compute ad by symmetry A 1 = 1 A A (V ˆV ), (6) Â 1 = 1 A A ( ˆV V ). (7) Averagig Equatios (6) ad (7) shows that Costrait () is satisfied. The costructio of the other three iterior quad-et poits is symmetric. The sixtee quad-ets poits do ot by themselves give eough iformatio to costruct surface patches that meet eighborig patches smoothly. The pair of itegers 0 ad 1 that correspod to the umber of sides belogig to the faces that cotai poits A 00 ad A 33 respectively are also eeded. These two itegers characterize the relatioship betwee a quad-et ad its eighbors whe cos π 0 or cos π 1 are substituted for c i Costrait (1). The quad-ets are ow passed to the ext step where patches are costructed. 4.3 Costructig Patches I the third ad fial step, parametric surface patches are costructed that iterpolate the iformatio ecoded by the quad-ets costructed i step. A sigle bi-cubic patch is ot sufficiet to iterpolate this data i geeral, sice the mixed partial or twist terms at the corers of a quad-et may ot be cosistet (i.e., u v v u, where u ad v correspod to boudary curve parameters). This difficulty ca be elimiated by usig four triagular patches that form a X with respect to the four quad-et boudary curves. Cubic triagular patches suffice to iterpolate the quad-et boudary curves, but do ot have eough degrees of freedom to satisfy smoothess costraits across quad-ets boudaries. By usig quartic patches, additioal degrees of freedom are itroduced that ca be used to esure smooth jois betwee adjacet triagular patches. The labelig scheme used for the Bézier cotrol ets of the four quartic patches is as follows: b 04 b 14 b 4 b 34 b 44 a 03 a 13 a 3 a 33 b 03 b 13 b 3 b 33 b 43 a 0 a 1 a a 3 b 0 b 1 b b 3 b 4 a 01 a 11 a 1 a 31 b 01 b 11 b 1 b 31 b 41 a 00 a 10 a 0 a 30 b 00 b 10 b 0 b 30 b 40 Formulas for the Bézier cotrol poits of oe of the triagular patches are ow give. Similar formulas for the other three patches ca be foud by symmetry. Iterpolatig the cubic boudary curves of a quad-et is achieved by degree raisig, resultig i b 00 = A 00, b 01 = 1 4 A A01, 4 b 0 = 1 A A0, b 03 = 3 4 A0 + 1 A03, 4 b 04 = A 03. Taget plae cotiuity is maitaied across quad-et boudaries by settig a 00 = 1 b b01, a 01 = c 3 3c A00 + A c 4 A A A1, a 0 = 3 c 8 A0 + c 8 A A1 + 1 A13, 8 a 03 = 1 b b14, where c = cos π 0 (ote that c = cos π 1 whe costructig a 31, a 3, a 13, ad a 3). These formulas are derived i 5.

5 The poits b 1, b 1, b 3, ad b 3 do ot affect taget plae behavior across quad-et boudaries, ad may be choose arbitrarily. Some care should be take i determiig the positio of these poits so that the resultig surface is free of uwated udulatios or other artifacts. A reasoable costructio is: b 1 = 7 8 A (A1 A11 A) (A10+A13) 1 16 (A00+A03). The remaiig Bézier cotrol poits are computed by b 11 = 1 a a01, a 11 = 1 b1 + 1 b1, b = 1 a1 + 1 a1. These costructios esure that the triples {b 01, a 00, b 10}, {a 01, b 11, a 10}, {b 1, a 11, b 1}, ad {a 1, b, a 1} are coliear ad share affie ratios. Therefore, the four triagular patches are C 1 alog the boudaries iteral to a quad-et. The collectio of quartic Bézier triagles costructed i this step are output as the fial step i the splie algorithm. Figures 5 ad 6 show several cotrol meshes ad the correspodig splie surfaces geerated by the algorithm. 4.4 Special Cases The costructio just preseted geerates a smooth splie surface over ay cotrol mesh that is topologically a -maifold. However, there are certai optimizatios that ca be implemeted to geerate patches of lower degree. These special cases arise whe 0 ad 1 equal 3 or 4. I each case, the boudary curves of a quad-et are quadratic rather tha cubic. If 0 = 1 = 4, a sigle bi-quadratic Bézier patch ca be used i place of the four quartic triagles. Otherwise, the four quartic Bézier triagles ca be replaced by cubics. A 0 A 01 A 1 A V A 00 A 11 A 10 A 1 A 0 Figure 4: The special case quad-et correspodig to the vertex V. To take advatage of these optimizatios, the special case quad-et show i Figure 4 correspodig to the V of M 1 is used. The poits A 00, A 0, A 0, ad A, are the cetroids of the four faces surroudig V. The poits A 10, A 01, A 1, ad A 1, are the midpoits of the four edges icidet o V, ad the poit A 11 is equivalet to V. This special case quad-et must also kow about the itegers 0 ad 1 (equal to the umber of sides belogig to the faces surroudig A 00 ad A respectively). The four cubic Bézier triagular patches costructed from the special case quad-et are labeled as: b 03 b 13 b 3 b 33 a 0 a 1 a b 0 b 1 b b 3 a 01 a 11 a 1 b 01 b 11 b 1 b 31 a 00 a 10 a 0 b 00 b 10 b 0 b 30 Formulas for the Bézier cotrol et of oe of these patches are give. The other three cotrol ets are foud by symmetry. The boudary curve is foud by b 00 = A 00, b 01 = 1 3 A00 + A01, 3 b 0 = 3 A A0, 3 b 03 = A 0. Taget plae cotiuity is maitaied across quad-et boudaries by settig a 00 = 1 b b01, a 01 = 1 c 1+c A00 + A A0 + 1 A11, 3 a 0 = 1 b0 + 1 b13, where c = cos π 0. A smooth joi across the iteral boudaries is esured by settig: b 11 = 1 a a01, a 11 = 1 b1 + 1 b1. If 0 = 1 = 4, the the special case quad-et is output as the cotrol et of a bi-quadratic tesor product Bézier patch. 5 Smoothess Coditios The purpose of this sectio is to derive the costraits imposed o the quad-et costructio ( 4.), ad the formulas for Bézier cotrol poits that affect taget plae behavior alog quad-et boudaries ( 4.3 ad 4.4). This sectio is icluded for completeess; it is ot crucial to uderstadig the results of this paper. The purpose of the quad-ets is to characterize the curves ad taget plaes alog the boudaries of a quadrilateral piece of the splie surface. Oe such boudary curve is represeted by the cubic Bézier curve [A 00,A 01,A 0,A 03] costructed i 4.. Adjacet quad-ets sharig these poits will clearly lead to a cotiuous (but ot ecessarily smooth) surface. To see how adjacet quad-ets give rise to surfaces that are taget plae cotious, it must be demostrated how a quad-et ecodes a taget plae alog a boudary. The taget plae at a poit o a surface ca be represeted as the spa of a pair of vectors. Alog a quad-et boudary, oe of these vectors is the derivative of the boudary curve writte i Bézier form as R = 3[A 01 A 00,A 0 A 01,A 03 A 0].

6 The other vector poits iward alog the boudary ad is defied S = 3[A 10 (1 c)a 00 ca 01, A 1 A 01 A 0,A 13 A 03], where c = cos π 0. The taget plae ecoded by the quad-et alog the boudary is the spa of R ad S. Similar expressios hold for the other three edges of a quad-et. To see that a pair of adjacet quad-ets ecode the same taget plae alog a commo boudary, cosider the pair of quadets that share the boudary [A 00,A 01,A 0,A 03]. Let A 10, A 1, ad A 13 be the first row of poits belogig to the first quad-et adjacet to the commo boudary, ad let  10,  1, ad Â13 be the first row of the secod quad-et. Clearly, both quad-ets will share the taget vector R sice they share a commo boudary curve. By defiitio Ŝ = 3[Â10 (1 c)a00 ca01, Â1 A01 A0, Â13 A03], is the iward poitig taget vector of the secod quad-et. Adjacet quad-ets will ecode the same taget plae alog the commo boudary if R, S, ad Ŝ are liearly depedet. This follows by costructio, sice it is easily verified that Costraits (1-3) are equivalet to the coditio S = Ŝ. Therefore, the quad-ets costructed i 4. ecode idetical taget plaes alog commo boudaries. Next, it is show that the triagular patches costructed i 4.3 iterpolate the taget plaes ecoded alog quad-et boudaries. Let P be the quartic triagular patch costructed to iterpolate a quad-et boudary curve. The taget plae of P alog this boudary is the spa of the partial derivatives ad P u = 4[b 01 b 00,b 0 b 01,b 03 b 0,b 04 b 03], P v = 4[a 00 b 00,a 01 b 01,a 0 b 0,a 03 b 03]. (8) The taget plae of P will iterpolate the quad-et taget plae if R, S, P u, ad P v are liearly depedet. By costructio P u = R, ad P v = φr + ψs, (9) where φ =[ 1+c, 1 ] ad ψ =[1], are scalar valued fuctios i Bézier form. Therefore P will iterpolate the taget plae alog the boudary of the quad-et. Expadig the right had side of (9) ad equatig this result to the right had side of (8) gives the formulas used to costruct poits a 00, a 01, a 0, ad a Smoothess i Special Cases The special case outlied i 4.4 is similar except the taget plae ecoded by the special case quad-et is the spa of the vectors R = [A 01 A 00,A 0 A 01], ad S = [A 10 (1 c)a 00 ca 01,A 11 A 01,A 1 A 0]. Expadig the right had side of (9) with these defiitios of R ad S, ad equatig this result to P v = 3[a 00 b 00,a 01 b 01,a 0 b 0], gives the formulas used to costruct poits a 00, a 01, ad a 0. It must also be demostrated that a special case quad-et of 4.4 ecodes the same boudary curve ad taget plae as a ormal quad-et. Let A be a ormal quad-et ad à be a special case quad-et defied over the same vertex. By costructio A 00 = Ã00, ad A03 = Ã0. The weights from Theorem 4.1 for the cases = 3 ad = 4 are { 4, 4, 1 } ad { 5, 5, 1, 1 } respectively. Sice these weights are used to costruct A 10, A 01, A 0, ad A 13, it is straightforward to show that i either case A 10 = 1 3 Ã00 + Ã10, 3 A01 = 1 3 Ã00 + Ã01, 3 A 0 = 1 3 Ã0 + Ã01, 3 A13 = 1 3 Ã0 + Ã1, 3 ad A 1 = 1 6 à à Ã0 + 1 Ã11. 3 Substitutig these equatios (for A 00,...,A 1) ito the defiitio of the taget vector S for the ormal case yields the defiitio of S for the special case. Therefore, both types of quad-ets ecode the same taget plaes ad may be used iterchagably whe 0 ad 1 are equal to 3 or 4. 6 Coclusios A algorithm has bee preseted for costructig a taget plae smooth splie surface that approximates a irregular cotrol mesh. The splie surface is i geeral a composite of quartic triagular Bézier patches. I certai special cases, cubic triagular patches may be used i place of the quartics. Over regular regios of the mesh, a bi-quadratic Bézier patch may be used i place of four quartic triagular patches. I fact, the four quartic triagular patches costructed over a regular regio represets exactly the same polyomial map as the sigle bi-quadratic Bézier patch. Although this has ot bee proved, its plausability is evidet sice the total degree of a bi-quadratic surface is 4. The splie algorithm as preseted was factored ito 3 steps. Each of these steps was a geometric costructio that ivolved takig weighted averages (affie combiatios) of poits. Therefore, the splie surface is affie ivariat (i.e., idepedet of ay affie trasformatio applied to the cotrol mesh). It is ot clear that the cocateatio of the geometric costructios leads to covex combiatios i all cases (although the special case costructios of 4.4 are covex). Over regular regios of a mesh, the refiemet step ( 4.1) is ot eeded ad will result i more patches beig costructed tha are actually required. It should be possible to avoid this uecessary splittig of patches as a optimizatio. Appedix A Treatmet of Boudaries A method of dealig with mesh boudaries i a reasoable way is ow preseted. The problem is that quad-ets are ot defied over boudary vertices of M 1. As a result, the boudary of the splie surface does ot approximate the boudary of the M 0 very well. A solutio is to modify step 1 ( 4.1) so that ew faces are added to M 1 that correspod to vertices ad edges belogig to the boudary of M 0. The followig costructio has the property that the boudary of the resultig splie surface will be the quadratic B-splie curve correspodig to the boudary vertices of M 0. There are two cases to cosider, faces of M 1 correspodig to boudary edges of M 0, ad faces of M 1 correspodig to boudary vertices of M 0.

7 Figure 5: A pair of irregular cotrol meshes ad resultig splie surfaces. The patch structure of the splie surfaces are idicated by color: blue ad yellow patches are quartic, red ad gree patches are cubic, ad gray patches are bi-quadratic. Figure 6: More examples: the color codig of patches is the same as above. The boudaries of meshes are hadled by the scheme outlied i Appedix A. This approach may be used to create creases o a surface as illustrated by the two shapes i the lower right had corer. Disjoit meshes that share boudary geometry will result i a crease.

8 A.1 Boudary Edges Let the vertex pair {V 0,V 1} be a boudary edge of M 0 belogig to face F. Let P 0 ad P 1 be the vertices of M 1 costructed i step 1 correspodig to the vertex-face pairs {V 0,F} ad {V 1,F} respectively. Two ew vertices Q 0 = 3 V0 + 1 V1 P0, Q 1 = 1 V0 + 3 V1 P1, ad oe ew face {P 0,P 1,Q 1,Q 0} are added to M 1. A. Boudary Vertices Let V be a vertex o the boudary of M 0. Let k be the umber of faces icidet o V. Let P 1,...,P k be the vertices of M 1 correspodig to V costructed i step 1, ad let P 0 ad P k+1 be the vertices foud by the boudary edge costructio give i Appedix A.1. Whe k = 1, V is a corer of M 0. By treatig this vertex as a discotiuity i the boudary B-splie curve, the splie surface boudary will have a corer. A ew face {P 0,P 1,P,P 3} is added to M 1 where P 3 = 4V P 0 P 1 P. Whe k > 1 a ew = k-sided face {P 0,...,P } is added to M 1 where P i = (uq 0 +(1 u)q 1) P i+1, i = k +,..., 1, with Q 0 = 1 P0 + 1 P1, Q1 = 1 P k + 1 P k+1, ad u = 1 πi (1 + cos + ta π πi si ). These costructios, offered without proof, are icluded because they are of practical value. The boudaries of the splie surfaces illustrated i Figures 5 ad 6 were dealt with usig this techique. B Proof of Theorem 4.1 Let M k = β(cos πk +ta π of Equatio (5) as follows: 1 Qi Qi+1 = 1 = 1 = 1 = 1 πk si ). Expad the right had side P j(1 + M j (i 1) )+ 1 P j( + M j i+1 + M j i 1), P j( + cos π Mj i), P j(1 + M j (i+1) ), P j(1 cos π )+ 1 P j cos π (1 + Mj i), = (1 cos π )O + cos π Qi. The key step of combiig M j (i 1) +M j (i+1) to get cos π Mj i comes about usig the well kow trigoometric idetities: cos θ + cos φ = cos 1 (θ + φ) cos 1 (θ φ), ad si θ + si φ = si 1 (θ + φ) cos 1 (θ φ). Clearly the poits Q i are co-plaar sice from (5) ay Q i ca be foud as a liear combiatio of O, Q 0, ad Q 1, ad must therefore lie i the plae spaed by these three poits. Refereces [1] E. Catmull ad J. Clark. Recursively geerated B-splie surfaces o arbitrary topological meshes. Computer Aided Desig, 10(6): , [] H. Chiyokura ad F. Kimura. Desig of solids with free-form surfaces. I Proceedigs of SIGGRAPH 83, pages [3] T. DeRose. Geometric Cotiuity: A Parametrizatio Idepedet Measure of Cotiuity for Computer Aided Geometric Desig. PhD thesis, Berkeley, [4] D. Doo. A subdivisio algorithm for smoothig dow irregularly shaped polyhedros. I Proceedigs o Iteractive Techiques i Computer Aided Desig, pages Bologa, [5] G. Fari. Curves ad Surfaces for Computer Aided Geometric Desig. Academic Press, third editio, [6] T. N. T. Goodma. Closed biquadratic surfaces. Costructive Approximatio, 7(): , [7] M. Halstead, M. Kass, ad T. DeRose. Efficiet, fair iterpolatio usig Catmull-Clark surfaces. I Proceedigs of SIGGRAPH 93, pages [8] K. Höllig ad Harald Mögerle. G-splies. Computer Aided Geometric Desig, 7:197 07, [9] C. Loop. Geeralized B-splie Surfaces of Arbitrary Topological Type. PhD thesis, Uiversity of Washigto, 199. [10] C. Loop. Smooth low degree polyomial splie surfaces over irregular meshes. Techical Report 48, Apple Computer Ic., Cupertio, CA, Jauary [11] C. Loop. A G 1 triagular splie surface of arbitrary topological type. Computer Aided Geometric Desig, to appear. [1] C. Loop ad T. DeRose. Geeralized B-splie surfaces of arbitrary topology. I Proceedigs of SIGGRAPH 90, pages [13] J. Peters. C 1 free-form surface splies. Techical Report CSD-TR , Dept. of Comp. Sci., Purdue Uiversity, W-Lafayette, IN, March [14] J. Peters. Smooth free-form surfaces over irregular meshes geeralizig quadratic splies. Computer Aided Geometric Desig, 10: , [15] J. Peters. Costructig C 1 surfaces of arbitrary topology usig biquadratic ad bicubic splies. I N. Sapidis, editor, Desigig Fair Curves ad Surfaces to appear. [16] U. Reif. Biquadratic G-splie surfaces. Techical report, Mathematisches Istitut A, Uiversität Stuttgart, Pfaffewaldrig 57, D-7000 Stuttgart 80, Germay, [17] M. Sabi. No-rectagular surface patches suitable for iclusio i a B-splie surface. I P. te Hage, editor, Proceedigs of Eurographics 83, pages North-Hollad, [18] J. va Wijk. Bicubic patches for approximatig orectagular cotrol-poit meshes. Computer Aided Geometric Desig, 3(1):1 13, 1986.