G 2 Tensor Product Splines over Extraordinary Vertices

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1 Eurographics Symposium o Geometry rocessig 28 ierre Alliez ad Szymo Rusikiewicz Guest Editors Volume 27 28, Number 5 G 2 Tesor roduct Splies over Extraordiary Vertices Charles Loop 1 ad Scott Schaefer 2 1 Microsoft Research 2 Departmet of Computer Sciece, Texas A&M Uiversity Abstract We preset a secod order smooth fillig of a -valet Catmull-Clark splie rig with biseptic patches While a uderdetermied biseptic solutio to this problem has appeared previously, we make several advaces i this paper Most otably, we cast the problem as a costraied miimizatio ad itroduce a ovel quadratic eergy fuctioal whose absolute miimum of zero is achieved for bicubic polyomials This meas that for the regular 4-valet case, we reproduce the bicubic B-splies other cases, the resultig surfaces are aesthetically well behaved We exted our costraied miimizatio framework to hadle the case of iput mesh with boudary Categories ad Subject Descriptors accordig to ACM CCS: 35 [Computer Graphics]: Curve, surface, solid, ad object represetatios 1 troductio Catmull-Clark subdivisio surfaces have become a stadard modelig primitive i computer geerated films ad video games [CC78] The success of this algorithm is due to its ability to model surfaces of arbitrary geus, possibly with boudary [Nas87] The modelig paradigm is simple: a user specifies a coarse cotrol mesh cosistig of vertices, faces, ad edges that approximates a desired shape; the Catmull- Clark surface smoothly approximates the cotrol mesh i a ituitive fashio Artists easily grasp the behavior of these shapes relative to the cotrol mesh However, subdivisio surfaces cotai shape defects at extraordiary vertices where the umber of icidet edges is ot equal to 4 geeral the surface is oly C 1 at these isolated poits etertaimet scearios, the viewpoit is cotrolled or the presece of isolated shape defects is acceptable For modelig high quality shapes, subdivisio surfaces are iadequate Subdivisio surface behavior at extraordiary vertices has bee extesively studied ad their shape artifacts are by ow well uderstood [DS78, Rei95, R98, ra98, RS99] Taget plae cotiuity at extraordiary vertices was formally established i [Rei95] However, o modificatio to the subdivisio rules will result i curvature cotiuity at these poits [ra98] Modificatios that boud the otherwise ubouded curvature at extraordiary vertices have appeared [Sab91, ADS6, GU7] A weak form of curvature cotiuity has bee achieved by locally projectig the cotrol mesh to a flat spot with zero curvature [Rei98, U98] True curvature cotiuity has bee obtaied by bledig a disk shaped regio about the extraordiary vertex with a quadratic shape [Zor6, Lev6] All of these schemes are cocered with the limitig behavior of the subdivisio process, ad ot the removal of the uderlyig sigularities i the mappig from a maifold domai to a embeddig space Figure 1: a Each subdivisio step adds a ew splie rig to the iterior of the hole created by a extraordiary vertex b The work preseted i this paper fills the hole i a Catmull- Clark splie rig with biseptic patches,, 1 Examples of vertex types 1, 2, ad 3 are also show These sigularities, correspodig to the extraordiary vertices of the cotrol mesh, are a result of the iheretly Joural compilatio c 28 The Eurographics Associatio ad Blackwell ublishig Ltd ublished by Blackwell ublishig, 96 Garsigto Road, Oxford OX4 2DQ, UK ad 35 Mai Street, Malde, MA 2148, USA

2 C Loop & S Schaefer / G 2 Tesor roduct Splies over Extraordiary Vertices fuctioal splie machiery used by may subdivisio algorithms That is, Catmull-Clark surfaces are bicubic tesor product B-splies, albeit over procedurally defied cotrol meshes with ifiite structure Such a compoetwise fuctioal splie is a deformatio of a regular plaar lattice, so modelig a arbitrary geus surface with a more geeral tessellatio requires sigularities This fact is ivariable igored i the theoretical study of extraordiary vertex behavior uder subdivisio stead that study focuses o so-called splie rigs, a local collectio of surface patches that form a -sided hole about a extraordiary vertex, see Figure 1a As subdivisio proceeds, a ew splie rig is formed iside the hole such that the old ad ew splie rigs joi with the smoothess of the uderlyig B-splie the limit, the - sided hole becomes ifiitesimally small, but ever vaishes This ifiite set of polyomials ad the limitig behavior at extraordiary vertices complicates evaluatio ad processig of these surfaces [HKD93, Sta98] whereas surfaces composed of a fiite set of polyomials are substatially simpler 11 roblem Statemet ad Cotributios The problem we address ca be reduced to the followig: Fill the hole i a -valet Catmull-Clark splie rig with tesor product patches that joi each other ad the splie rig with secod order smoothess See Figure 1b for a illustratio Our solutio to this problem requires bidegree 7 patches This result was origially reported i [Loo4] While that work established the existece of a biseptic solutio space, ad-hoc meas were used to remove the extra degrees of freedom Here we make several improvemets ad cotributios; specifically 1 The derivatio of the uderdetermied biseptic solutio space is based etirely o properties of the correspodece maps betwee adjacet patches ad the ecessary cocycle coditio these maps must obey about vertices 2 Our surface is defied as a costraied miimizatio over a ovel eergy fuctioal that achieves a absolute miimum of zero for bicubic patches ad results i aesthetically pleasig shapes otherwise 3 We solve for data idepedet basis fuctios explicitly, as a off-lie preprocess Sice the basis fuctios are solved idepedet of the surface, we ca maipulate these surfaces i realtime 4 We defie basis fuctios to hadle meshes with boudary such that the surface iterpolates the cubic B-splie curve defied by the mesh boudary Fillig a splie rig with secod order smooth surfaces has practical applicatios i surface desig We use our results to costruct secod order smooth surfaces over refied quadrilateral cotrol meshes, where each quad has at most oe icidet extraordiary vertex Refiemet is eeded to isolate extraordiary vertices as is doe for Catmull-Clark evaluatio [Sta98] Ulike Catmull-Clark surfaces, our surfaces are secod order smooth everywhere ad cotai a fiite umber of polyomial patches 12 revious Work May papers addressig the problem of costructig first ad secod order smooth patch complexes have appeared over the last two decades We metio here oly those that explicitly joi tesor product polyomials with secod order smoothess at extraordiary vertices [ra97] a - valet Catmull-Clark splie rig is filled with bidegree 6 patches; however, 4 such patches are eeded Similarly, [GZ99] form a secod order smooth joi over extraordiary vertices with 4 bidegree 5 patches [et2] a combiatio of 2 bidegree 3 5 ad 2 bicubic patches would be eeded to fill a -sided hole surrouded by bicubics [K7], a collectio 16 patches of bidegree 4 4 ad 6 6 are used to form a smooth complex surrouded by bicubic patches While other works have achieved lower bidegree, with respect to total cotrol poit cout, bidegree 7 with patches is still the best result This makes the scheme attractive for GU implemetatio sice total data throughput is miimized this paper, we strive to improve shape quality, ad to make the results more practically applicable This paper is orgaized as follows Sectio 2 we preset aspects of geometric cotiuity ecessary to derive our results Sectio 3 we specify the correspodece maps betwee adjacet patches as required by the defiitio of geometric cotiuity Sectio 4 we use the correspodece maps to derive sets of costraits o the coefficiets of adjacet patches eeded for secod order smoothess Sectio 5 we preset a ovel quadratic eergy fuctioal, the show how this fuctioal is miimized subject to our costraits i Sectio 6 We solve for data idepedet basis fuctios, with support ad boudary costraits i Sectio 7 Fially, we preset results ad coclude with Sectio 8 2 Geometric Cotiuity Give a pair of surface patches i, i+1 : [,1] [,1] R m, we say i ad i+1 meet with k th order geometric cotiuity deoted G k [DeR85], if there exists a map θ such that i meets i+1 θ with parametric cotiuity C k, that is i G k = i+1 i C k = i+1 θ More formally, this coditio requires that the k th order derivatives of the two patches after reparameterizatio with respect to θ coicide We refer to the map θ : R 2 R 2 as the correspodece map betwee i ad i+1 Techically θ, alog with k of its trasversal derivatives, oly eeds to be defied o a lie correspodig to the commo patch boudary However we fid it coveiet to defie correspodece Joural compilatio c 28 The Eurographics Associatio ad Blackwell ublishig Ltd

3 C Loop & S Schaefer / G 2 Tesor roduct Splies over Extraordiary Vertices maps i Bézier form over the etire uit square; this way trasversal derivatives o edges ad cosistet mixed partial derivatives at vertices are easily specified The derivatives of i+1 θ ca be foud via the chai rule ad yield a matrix equatio of the form D i = D i+1 θ Θ where [ D i is a vector of the ] partial derivatives of i ie D i = i i 2 i u v ad Θ is a matrix obtaied usig u 2 the chai rule that ecodes the partial derivatives of θ We refer to Θ as the chai rule matrix of θ f we make the simplifyig assumptio that θ is the idetity fuctio alog the commo boudary, the the above equatio reduces to D i = D i+1 Θ, 1 whe evaluated o the boudary This assumptio is reasoable as ay other choice of θ would lead to higher degree boudary curves with more smoothess costraits Equatio 1 tells us how to trasform the derivatives wrt the domai of i i terms of the derivatives of i+1 ad θ For a cyclic collectio of patches i icidet o a commo vertex with correspodece maps θ i betwee patches i ad i+1, i =,, 1 idices take modulo, satisfyig geometric cotiuity results i a cocycle coditio amog the patches f we evaluate equatio 1 at the commo vertex for all patches, we fid that D = D Θ 1 Θ 2 Θ 1 Θ, for patches icidet o that vertex Therefore, this relatioship results i the additioal requiremet that = Θ 1 Θ 2 Θ 1 Θ 2 whe evaluated at the commo vertex [Hah89] For G k cotiuity a correspodece map must ecode all k th order trasversal derivatives i the versal directio f we differetiate k times i this directio, will get mixed partials of order 2k These derivatives must agree at the commo vertex i order to get a polyomial parameterizatio Therefore, for G k cotiuity the chai rule matrices must ecode derivatives up to order 2k 3 Correspodece Maps We will costruct two types of correspodece maps o edges joiig three types of vertices: 1 a extraordiary vertex, 2 a edge adjacet eighbor of a type 1 vertex, 3 a face adjacet diagoal eighbor of a type 1 vertex, see Figure 1b Note that a type 1 vertex is -valet, vertex types 2 ad 3 are always 4-valet Over the edge betwee vertex types 1 ad 2 we defie iterior correspodece maps; over the edge betwee vertex types 2 ad 3 we defie exterior correspodece maps 31 terior Correspodece Maps We defie iterior correspodece maps i terms of the maps φ : u,v x,y defied by [ φ,x u,v = b 1 u T cos 2π 1 1 [ φ,y u,v = b 1 u T si 2π ta π ] b 1 v, ] b 1 v, where b d are degree d Berstei polyomials The geometry of φ is illustrated i Figure 21 Give that φ creates a agle of 2π aroud the extraordiary vertex, the correspodece map from patch i to i+1 is φ 1 r 1 φ where r is a couterclockwise rotatio of 2π about the origi We verify the cocycle coditio at a type 1 vertex usig the chai rule matrices Φ ad R for φ ad r respectively We form the compositio of the correspodece maps φ 1 r 1 φ from patches through 1 ad evaluate at,, correspodig to the extraordiary vertex to get = Φ 1 R 1 Φ = Φ 1 R 1 Φ Notice that R 1 = because r 1 is a rotatio of 2π The above expressio oly depeds o Φ beig locally ivertible at, The cocycle loop of correspodece maps icidet o a type 1 vertex is illustrated i Figure Exterior Correspodece Maps The secod type of correspodece map we eed is defied over a edge betwee vertex types 2 ad 3; this edge correspods to the boudary of the splie rig Ulike the iterior correspodece maps, we must carefully solve for the exterior correspodece maps previous work [Loo4] the same correspodece maps were derived by appealig to the embeddig space of the resultig patches Here we derive the correspodece maps strictly i terms of abstract adjacecy relatios of the uderlyig tessellatio We begi by defiig maps ψ : u,v x,y, where is the valece of the earby extraordiary vertex The exterior correspodece maps will be defied by ψ u,1 ad ψ 1,v, where u,v [,1] We require ψ to be the idetity o the edges u,,u, 1,, v, ad 1, v arameter values 1, ad, 1 correspod to type 2 vertices; 1, 1 correspods to the type 3 vertex Figure 22 illustrates the cocycle loop where four patches meet at a type 2 vertex Note that this vertex might share a edge with aother extraordiary vertex of valece m, which is possible give the miimum separatio of extraordiary vertices we require We ca factor the cocycle compositio ito two parts, correspodig to Joural compilatio c 28 The Eurographics Associatio ad Blackwell ublishig Ltd

4 C Loop & S Schaefer / G 2 Tesor roduct Splies over Extraordiary Vertices Figure 2: The cocycle maps for the 3 vertex types : 1 extraordiary vertex, 2 edge sharig eighbor of a type 1 vertex 3 face sharig diagoal eighbor of a type 1 vertex the two eighborig extraordiary vertices to get {}}{ 1 = Q Ψ S R 1 Φ Φ Ψ 1 {}}{ 1 Q Ψ m S R 1 m Φ m Φm Ψ 1 m, 3 where Q is the chai rule matrix for qu,v = u, v the reflectio across the u axis, S is the chai rule matrix for su,v = v,u the reflectio across the diagoal u = v, R is the chai rule matrix for a 2π rotatio about the origi, ad Ψ is the chai rule matrix for ψ Note that parameter for evaluatio of these matrices correspods to the type 2 vertex Both factors of equatio 3 represet the idetity ad are the same up to valece The factor ivolvig will impose costraits of various partial derivatives o ψ at the type 2 vertex Additioal costraits o ψ come from the cocycle coditio at the type 3 vertex Figure 23 illustrates the cocycle loop where four patches meet at a type 3 vertex We assume that this vertex may be a diagoal eighbor of four extraordiary vertices By assumig the symmetry ψ u,v = ψ v,u, the cocycle compositio at a type 3 vertex ca be factored ito four parts, correspodig to the 4 arbitrary valece diagoal eighbors {}}{{}}{ = Ψ k S Ψ k S 1 Ψ l S Ψ l S 1 {}}{{}}{ Ψ m S Ψ m S 1 Ψ S Ψ S 1 4 This expressio holds sice S Ψ S = Ψ We use the costraits imposed o the derivatives of ψ by equatios 3 ad 4 to fid ψ,x u,v = 1 5c 3 3c 2 15c +18 c 2 +2c c 22c 3 12c 2 4 b 4 u T 1 c +3 c 2 +3c c 2 2 b 3 v, 3 4 c c where c = cos 2π By the symmetry coditio, we also have ψ,y u,v = ψ,x v,u The derivatio of this solutio is preseted i Appedix A Note that whe = 4, ψ 4 = 4 atch Smoothess Costraits We ow use the correspodece maps to determie secod order smoothess costraits o the coefficiets of the patches i,i =,, 1 41 Exteral Costraits A Catmull-Clark splie rig surface is completely characterized by a 3-rig of poits about a extraordiary vertex However, the secod order behavior of the splie rig boudary ca be described by a 2-rig We label the poits of this 2-rig i a cyclic fashio, as show i Figure 3 Note that we have seve poit sectios ad the cetral vertex is duplicated times; doig so will give us a circulat system of equatios i Sectio 43 We characterize the secod order behavior of the splie rig boudary with a set of bicubic patches The cotrol poits of these bicubic Bézier patches are foud by applyig the cubic kot isertio operator M = , 3 4 Joural compilatio c 28 The Eurographics Associatio ad Blackwell ublishig Ltd

5 C Loop & S Schaefer / G 2 Tesor roduct Splies over Extraordiary Vertices Figure 3: A 2-rig a with poit labelig o the 2-rig cotrol poits This will covert from B-splie to Berstei form resultig i bicubic Bézier patches H i u,v = b 3 u T M T a 1 i+2 a 11 i+1 a 21 i+1 a 1 i 1 a i a 1 i+1 a 2 i+1 a 11 i 1 a 1 i a 11 i a 12 i a 12 i 1 a 2 i a 21 i a 22 i Mb3 v, where i =,, 1 ad a jk i are cotrol mesh vertices from Figure 3 Note that, due to the udefied cotrol poit represeted by a above, the patches H i ca oly be evaluated alog the exteral boudary u,1 ad 1,v; however, all derivatives up to secod order are well defied o this boudary Furthermore, these derivatives will meet the surroudig splie rig with C k cotiuity Next, we use the maps ψ from Sectio 32 to reparameterize the bicubic patches H i to get costraits o the exteral edge of our patches; that is j u j i 1,t = j H u j i ψ 1,t, j =,1,2 Expadig this expressio usig the chai rule results i the followig exteral costraits o patch i i 1,t = H i 1,t, 5 u i1,t = x H i1,t u ψ,x1,t + y H i1,t u ψ,y1,t, 6 2 u 2 i 1,t = x H i1,t 2 ψ u 2,x 1,t + y H i1,t 2 ψ u 2,y 1,t H x 2 i 1,t u ψ,x1,t xy H i1,t u ψ,x1,t u ψ,y1,t H y 2 i 1,t u ψ,y 1,t 7 The costraits alog the boudary t, 1 are defied similarly All terms o the right had sides of Equatios 5, 6, ad 7 are polyomials of kow degree From this, we ca deduce by degree coutig, that the patches i must be bidegree 7 To determie the umber of costraits give by equatios 5, 6, ad 7, we ote that each of these 3 equatios is a degree 7 polyomial with 8 degrees of freedom resultig i 24 costraits Therefore we have 48 costraits for both edges of the exteral boudary However, at u = v = 1 correspodig to a type 3 vertex, the cocycle coditio guaratees that the mixed partial derivatives will agree up to secod order, meaig that 9 of these costraits will be depedet; so there are oly 39 exteral costraits per patch, or 39 for all patches sharig the type 1 vertex 42 teral Costraits We ow derive costraits alog iteral patch edges We combie the iteral correspodece map φ 1 r 1 φ betwee the pair of surface patches i,t ad i+1 t,, with the defiitio of secod order geometric cotiuity to get the relatio j+k u j v k i,t = j+k u j v k i+1 φ 1 r 1 φ t, where j+k =,1,2 Expadig this gives us G, G 1 ad G 2 costraits i,t = i+1 t,, 8 c 1 t u i+1 t,+ v i,t = v i+1 t,+ u i,t, 9 2c 2 1 t v i+1 t, u i,t 2c 1 t1+c 1 t = 1+c 1 t + 2 uv i+1t, 2 2 v 2 i+1 t, 2 u 2 i,t uv i,t 1 We cout the umber of costraits determied by these equatios as follows Sice patches i are biseptic, we ca deduce by degree coutig the umber of costraits i equatios 8, 9, ad 1, are 8, 8, ad 9 respectively; therefore we have 27 costraits per iteral edge However, the exteral costraits derived i the previous sectio specify the 9 secod order mixed partial derivatives correspodig to the type 2 vertex at i+1 1, ad i,1 By the cocycle coditio, these derivatives will automatically satisfy 8, 9, ad 1 resultig i 9 depedet costraits Therefore the geometric cotiuity coditios itroduce 16 iteral costraits 43 Costrait System The 16 iteral costraits combied with the 39 exteral costraits results i a system of 55 equatios Each biseptic patch i has 64 coefficiets, so we have a system of 55 Joural compilatio c 28 The Eurographics Associatio ad Blackwell ublishig Ltd

6 C Loop & S Schaefer / G 2 Tesor roduct Splies over Extraordiary Vertices equatios i 64 ukows All of our costraits are polyomial equatios that ca writte i terms of biseptic Bézier cotrol poits p jk i ad two-rig cotrol mesh vertices a lm i We ca write the costraits as a block circulat system that expads to Cp = Wa, 11 c c 1 p p 1 c = p 1 c 1 c w w 1 w 1 w 1 w w1 w 1 w 1 w a a 1 a 1 where p i are the ukow Bézier cotrol poits of patch i, ad a i the seve kow vertices of the i th two-rig sectio as labeled i Figure 3 Sice this system is uderdetermied, we itroduce a eergy fuctioal ad fid the solutio that miimizes this fuctioal with respect to the costraits 5 Bicubic Eergy Existig eergy fuctioals, eg thi plate or biharmoics, have the disadvatage that they are ot zero for bicubic surfaces Our costrait system will be cosistet with tesor product B-splies whe = 4 order to geeralize this case, we must defie a ew eergy fuctioal We therefore itroduce a eergy fuctioal, that to our kowledge has ot previously appeared i the CAGD literature, whose absolute miimum of zero is achieved for a bicubic tesor product patch, ad all higher degree parameterizatios of such a patch Let Fu, v be a tesor product patch Our bicubic eergy fuctioal is defied as the itegral 1 1 eergy = 4 2 Fu,v u Fu,v 4 v dudv 4 our case, F = i is a biseptic patch Sice the coefficiets of i are at most squared, our eergy fuctioal ca be writte as a quadratic form p T i E p i, where E is a symmetric matrix Because 4 / u 4 ad 4 / v 4 are idetically zero for all bicubic patches, ay vector p i cosistet with a bicubic patch or a degree elevated form will be i the kerel of E, implyig p T i E = E p i = We fid the bicubic eergy of the collectio of patches i by the product p T Ep, where E is a block diagoal matrix whose blocks are the sigle patch bicubic eergy matrix E, 6 Costraied Miimizatio We ca miimize quadratic eergy p T Ep subject to costraits Cp = Wa usig Lagrage multipliers Differetiatig with respect to p yields p p T Ep = p 2Ep = C T Λ p T C T a T W T Λ, These equatios ca be represeted by a sigle block matrix [ ][ ] [ ] E C T p = a C Λ W This system ca be put ito block circulat form by permutig its rows ad colums to get E c T c c T 1 c 1 = c 1 E c T c c T 1 c 1 c T 1 w w 1 w 1 w w 1 w 1 w 1 w E c T c w 1 a a 1 a 1 p λ p 1 λ 1 p 1 λ 1 Rather tha solvig this system directly, we trasform the problem from the spatial to frequecy domai by applyig the Discrete Fourier Trasform to the blocks The Discrete Fourier Trasform ad verse Discrete Fourier Trasform are give by 1 ˆx j = i= e 2π 1i j x i ad x i = 1 1 j= e 2π 1 i j ˆx j We use the DFT to put the above system ito block diagoal form, where the solutio is foud by solvig the idividual blocks [ E ĉ H j ĉ j ][ ] [ ˆp j = ˆλ j ŵ j ] â j, for j =,, 1, where ĉ H j deotes the cojugate traspose of ĉ j The patch coefficiets p i are foud by takig the DFT of the solutios ˆp j 7 Basis Fuctios To this poit, we have assumed that the two-rig data a i R m comes from a cotrol mesh f we let a i take the form [,,,1,,,] the we are istead solvig for the basis fuctio associated with some a jk i, correspodig to the o-zero elemet Sice our costrait systems do ot ivolve ay actual data, oly valece, we ca solve for the, Joural compilatio c 28 The Eurographics Associatio ad Blackwell ublishig Ltd

7 C Loop & S Schaefer / G 2 Tesor roduct Splies over Extraordiary Vertices basis fuctios a priori, store the resultig basis patches ad multiply them by local mesh data at rutime Sice a tworig cotais sectios of seve poits, we oly eed to fid seve basis fuctios for each valece We label these seve basis fuctios A, A 1, A 2, A 11, A 21, A 12, ad A 22, where the idices correspod to the idices of a jk i to which they are associated A obvious questio to ask about these basis fuctio is: are they o-egative? By ispectig the Berstei coefficiets of the basis patches, we fid the aswer to be o, but just barely There are a few egative coefficiets o the order of 1 3, so this prevets us from claimig the covex hull property We should poit out that these egative values do ot come from the exteral costraits; so a differet yet to be determied eergy fuctioal could lead to the covex hull property Alteratively, we could fid the covex miimum to our costraied miimizatio problem; but this is complicated by the fact that we solve our system i the frequecy domai, ad covexity must be satisfied i the spatial domai We leave this optio for future work Aother issue cocerig the basis fuctio is their support; that is, the collectio of faces where a basis fuctio o-zero resetly, every basis fuctio supported over a extraordiary vertex, will have support over all faces icidet o that vertex deally, the support of a basis fuctio should ot exted beyod the two-rig of its correspodig vertex Otherwise, a extraordiary patch will deped o vertices outside the 1-rig of a quadrilateral face ie, vertices ot coected by a edge to oe of the quad face vertices This exteded support occurs for basis fuctios A 2, A 21, A 12, ad A 22 Note that the patches outside the two-rig of the basis fuctio are early zero, but ot absolutely We illustrate the poit for the basis fuctio A 21 i Figure 4 To deal with this issue, we eforce additioal support costraits o patches associated with basis fuctios A 2, A 21, A 12, ad A Support Costraits To eforce ideal support we solve modified costrait systems The cases of A 2, ad A 21 are similar; we treat A 12 as a istace of A 21 sice they are the same up to diagoal reflectio these cases, we wat the basis fuctios to have support over two adjacet patches i u,v ad i+1 u,v icidet o the extraordiary vertex We impose exteral costraits 5, 6, ad 7 o these two patches, as well as the iteral geometric cotiuity costraits 8, 9, ad 1 o the shared boudary i,t = i+1 t, Next, we add costraits to force the basis fuctio support boudary i t,, ad i+1,t to go to zero, alog with all derivatives up to secod order k v k i t, =, k u k i+1,t =, for k =,1,2 All patches other tha i ad i+1 are set to zero We the solve for miimum bicubic eergy as before Sice we oly eed to solve for two patches, we do so i the spatial domai Eforcig two-rig support o the basis fuctio A 22 is similar, except we oly eed to solve for a sigle patch We use the exteral boudary costraits, alog with costraits that force all derivatives up to secod order alog boudaries i,t ad i t, to zero rior to eforcig the support costraits, the basis fuctio were guarateed to sum to 1 sice the colums of the matrix W i equatio 11 sum to 1 Addig support costraits to select basis fuctios will cause this property to be violated We ca restore a partitio of uity by simply defiig A = 1 A i jk i jk 72 Boudary Basis Fuctios Figure 4: Cotours from 1/2 to 1/2 by 1/2 Red are positive cotours, gree zero, ad blue egative This basis fuctio reaches a maximum height 4/9 at the solid dot, so the bad of cotours show represets 225% of the height of the basis fuctio The basis fuctio A 21 is show a without support costraits, ad b with support costraits Not all cotrol meshes are closed Frequetly, artists model cotrol meshes with boudary For Catmull-Clark subdivisio surfaces, the subdivisio rules are modified so that the surface iterpolates the cubic B-splie curve defied by the boudary edges [Nas87, BLZ] We ca adapt this behavior to our surfaces by imposig mesh boudary costraits that force the surface to iterpolate the boudary cubic B- splie curve We assume a boudary extraordiary vertex is surrouded by a half two-rig of mesh cotrol poits We deote the valece of a boudary extraordiary vertex by m, the umber of icidet faces We use the same correspodece maps as before, with the caveat that = 2m First we eed to modify the exteral boudary costraits to take ito accout the presece of a mesh boudary We Joural compilatio c 28 The Eurographics Associatio ad Blackwell ublishig Ltd

8 C Loop & S Schaefer / G 2 Tesor roduct Splies over Extraordiary Vertices defie two bicubic patches H u,v = b 3 u T M T ad a 1 2 a 11 1 a 21 1 a a 1 1 a 2 1 a 1 a 11 a 12 a 2 a 21 a 22 Mb 3 v, H m 1 u,v = where b 3 u T M T a 1 m 2 a m a 1 m a 2 m a 11 m 2 a 1 m 1 a 11 m 1 a 12 m 1 a 12 m 2 a 2 m 1 a 21 m 1 a 22 m 1 M = We defie two ew mesh boudary costraits as k u k 1,t = k H u k ψ 1,t, k v k m 1 t,1 = k H v k m 1 ψ t,1, Mb 3 v, for k =,1,2 We use the exteral patch boudary costraits 5, 6 ad 7 for boudaries t,1 ad m 1 1,t, ad for both exteral boudaries of patches i,i = 1,,m 2 We iclude the iteral geometric cotiuity costraits 8, 9, ad 1 betwee pairs of patches i, i+1,i =,,m 2 Fially, we iclude the boudary cubic B-splie costraits t, = [ a 1 m a a 1 a 2 m 1,t = [ a 1 a m a 1 m a 2 m ] M b 3 t, ] M b 3 t We the solve to miimize bicubic eergy as before Figure 6: From left to right: the cotrol mesh, the patch structure of a Catmull-Clark surface with bicubic patches show i gray top ad our surface with biseptic patches show i blue bottom, reflectio lies ad a zoom i close to a extraordiary boudary vertex For the Catmull-Clark surface, we use the boudary rules of [BLZ] Figure 7: Compariso of Gaussia curvature for a Catmull- Clark surface left, our techique middle ad [Loo4] right 8 Results We show several examples of our surfaces i figures 5 to 11 The quality of our shapes is quite good i geeral compared with Catmull-Clark subdivisio, especially ear boudary ad high valece extraordiary vertices Oe issue that we have oticed are curvature hotspots, high positive Gauss curvature, for valece = 3; similar to the behavior of Catmull-Clark surfaces We have ot treated this as a special case i ayway; though doig so may result i better shapes The lack of a covex hull property warrats additioal work o this scheme Refereces Figure 5: The cotrol mesh, patch structures of the Catmull- Clark top ad our surface bottom ad the reflectio lies Notice that the valece 8 vertex from the Catmull-Clark surface causes a oticeable kik i the reflectio lies that is abset with our G 2 surface [ADS6] AUGSDÖRFER U, DODGSON N, SABN M: Tuig subdivisio by miimisig gaussia curvature variatio ear extraordiary vertices Computer Graphics Forum 25, 3 26, [BLZ] BERMANN H, LEVN A, ZORN D: iecewise smooth subdivisio surfaces with ormal cotrol SGGRAH : roceedigs of the 27th aual coferece o Computer graphics ad iteractive techiques 2, pp [CC78] CATMULL E, CLARK J: Recursively geerated B-splie surfaces o arbitrary topological meshes Computer Aided Desig 1, , [DeR85] DEROSE A: Geometric Cotiuity: A arametrizatio depedet Measure of Cotiuity for Computer Aided Geoc 28 The Authors Joural compilatio c 28 The Eurographics Associatio ad Blackwell ublishig Ltd

9 C Loop & S Schaefer / G 2 Tesor roduct Splies over Extraordiary Vertices Figure 8: Compariso of Gaussia curvature for a Catmull-Clark surface left, our techique middle ad [Loo4] right Figure 9: lots of the basis fuctios we compute for a valece 6 vertex The Bezier ets top row of the basis fuctios bottom row are composed of bicubic polyomials i the two-rig of a extraordiary vertex ad biseptic patches i the oe-rig Figure 1: Gaussia curvature plots of a boudary vertex with four quads for a Catmull-Clark surface left ad our techique right metric Desig hd thesis, Berkeley, 1985 also tech report UCB/CSD 86/255 2 [DS78] DOO D, SABN M: Behaviour of recursive divisio surfaces ear extraordiary poits Computer Aided Desig 1, , [GU7] GNKEL, UMLAUF G: Tuig subdivisio algorithms usig costraied eergy miimizatio Mathematics of Surfaces X 27, Marti R, Sabi M, Wikler J, Eds, Spriger-Verlag, pp [GZ99] GREGORY J, ZHOU J: rregular C 2 surface costructio usig b-polyomial rectagular patches Computer Aided Geometric Desig 16, , [Hah89] HAHN J: Geometric cotiuous patch complexes Computer Aided Geometric Desig 6, , [HKD93] HALSTEAD M, KASS M, DEROSE T: Efficiet, fair iterpolatio usig catmull-clark surfaces Computer Graphics 27, Aual Coferece Series 1993, [K7] KARCAUSKAS K, ETERS J: Guided C 2 splie surfaces with v-shaped tessellatio MA Coferece o the Mathematics of Surfaces 27, pp [Lev6] LEVN A: Modified subdivisio surfaces with cotiuous curvature Computer Graphics 25, Aual Coferece Series 26, [Loo4] LOO C: Secod order smoothess over extraordiary vertices roceedigs of the Symposium o Geometry rocessig 24, pp , 3, 8 [Nas87] NASR A H: olyhedral subdivisio methods for freeform surfaces ACM Tras Graph 6, , , 7 [et2] ETERS J: C 2 free-form surfaces of degree 3,5 Computer Aided Geometric Desig 19, 2 22, [R98] ETERS J, REF U: Aalysis of geeralized B-splie subdivisio algorithms SAM Joural o Numerical Aalysis 35, 2 Apr 1998, [ra97] RAUTZSCH H: Freeform splies Computer Aided Geometric Desig 14, , Joural compilatio c 28 The Eurographics Associatio ad Blackwell ublishig Ltd

10 C Loop & S Schaefer / G 2 Tesor roduct Splies over Extraordiary Vertices symposium o Geometry processig 26, Eurographics Associatio, pp Appedix A: Solvig for ψ We ow use the costraits implied by the cocycle coditios o vertices to solve for a mappig ψ that will miimize the bidegree of our G 2 surface splie the G 2 case, the cocycle coditio must hold for all secod order mixed partial derivatives geeral, the k th order derivatives will trasform via Equatio 1 i terms of all k th order derivatives of the correspodece maps However at type 2 ad 3 vertices this is ot the case sice at these 4-valet vertices, various mixed partial derives of the mappigs φ ad ψ vaish Therefore, the matrices i Equatios 3 ad 4 are 8 8 ad cotai the derivatives: Specifically, let Ψ = u, v, 2, 2 u 2 u v, 2, 3 v 2 u 2 v, 3, 4 u v 2 u 2 v 2 1 x 11 x 21 x 12 x 22 1 y 11 y 21 y 21 y x 11 2x x y 11 2x 11 2x x 11 y y y 11 2y y x y 11 1, Figure 11: Compariso of a valece 17 vertex for Catmull- Clark surface left, ad our techique right Below cotrol mesh: top row patch structure, middle row Gaussia curvature, ad bottom row reflectio lies [ra98] RAUTZSCH H: Smoothess of subdivisio surfaces at extraordiary poits Advaces i Computatioal Mathematics , [U98] RAUTZSCH H, UMLAUF G: A G 2 -subdivisio algorithm Geometric Modellig, volume 13 of Computig suppl 1998, G Fari H Bieri G B, DeRose T, Eds, Spriger- Verlag 1 [Rei95] REF U: A uified approach to subdivisio algorithms ear extraordiary vertices Computer Aided Geometric Desig , [Rei98] REF U: Turbs topologically urestricted ratioal b- splies Costructive Approximatio 14, , [RS99] REF U, SCHRÖDER : Curvature smoothess of subdivisio surfaces Tech Rep TR--3, Caltech, [Sab91] SABN M: Cubic recursive divisio with bouded curvature Curves ad Surfaces 1991, Lauret, Méhauté A L,, Schumaker L, Eds, Academic ress, pp Bosto 1 [Sta98] STAM J: Exact evaluatio of Catmull-Clark subdivisio surfaces at arbitrary parameter values Computer Graphics 32, Aual Coferece Series 1998, [Zor6] ZORN D: Costructig curvature-cotiuous surfaces by bledig SG 6: roceedigs of the fourth Eurographics where x i j = i+ j ψ u i u j,x 1, While y i j ca be defied similarly, we ote that sice by assumptio the x ad y compoets of ψ are symmetric, that is ψ,x v,u = ψ,y u,v, we eed oly solve for ψ,x Therefore y i j = i+ j u j u ψ i,x,1 Multiplyig together the matrices i Equatio 3 leads to the followig four costraits o ψ,x x 11 = c, x 21 =, y 12 = 2c y 11 + c, y 22 = 2c 2y 21 y 12 t is possible to satisfy these costraits with a bidegree 3 mappig; with this solutio, the derivative u ψ,x1,v will be degree 3 This term gets squared uder the chai rule leadig to higher degree surfaces tha ecessary stead we ca satisfy the above costraits with a bidegree 4 3 polyomial, leavig additioal freedom to lower the overall surface degree We ca write this polyomial ψ,x u,v = b 4 u T 1 cc 2 7c+6 18c+2α+27 c 2 +3c 18α c 32c 3 12c c α c β b3 v We ow solve for the 2 degrees of freedom α ad β so that u ψ,x1,v is degree 2 ad v ψ,xu,1 is degree 3 These costraits result i α = c2 +3c 9 9c 2, β = 9+c 12 As show i a Sectio 41, the bidegree of the resultig surfaces are miimized Joural compilatio c 28 The Eurographics Associatio ad Blackwell ublishig Ltd