Subdividing Barycentric Coordinates

Transcription

1 Subdvdng Barycentrc Coordnates Dtry Ansov Chongyang Deng Ka Horann Abstract Barycentrc coordnates are coonly used to represent a pont nsde a polygon as an affne cobnaton of the polygon s vertces and to nterpolate data gven at these vertces. Whle unque for trangles, varous generalzatons to arbtrary sple polygons exst, each satsfyng a dfferent set of propertes. Soe of these generalzed barycentrc coordnates do not have a closed for and can only be approxated by pecewse lnear functons. In ths paper we show that subdvson can be used to refne these pecewse lnear functons wthout losng the key barycentrc propertes. For a wde range of subdvson schees, ths generates a sequence of pecewse lnear coordnates whch converges to non-negatve and C contnuous coordnates n the lt. The power of the descrbed approach coes fro the possblty of evaluatng the C lt coordnates and ther dervatves drectly. We support our theoretcal results wth several exaples, where we use Loop or Catull Clark subdvson to generate C coordnates, whch nhert the favourable shape propertes of haronc coordnates or the sall support of local barycentrc coordnates. Ctaton Info Journal Coputer Aded Geoetrc Desgn Volue, March 26 Pages 72 5 Note Proceedngs of GMP Introducton Suppose we are gven a planar n-sded sple polygon Ω 2 wth n vertces v,..., v n 2. For any p 2, the values b (p ),..., b n (p ) = b (p ) n are called barycentrc coordnates of p wth respect to Ω, f b (p ) = = and b (p )v = p. () = Non-negatvty s soetes entoned as an addtonal condton [], but snce ths precludes the exstence of barycentrc coordnates at ponts outsde the convex hull of the vertces v, we prefer to consder the condtons n () as the defnng propertes and regard non-negatvty as a desrable property only. It s well known [27] that the barycentrc coordnates of p are unque for n =, when Ω s a trangle, and they are non-negatve f and only f p Ω n ths case. Instead, for n > the condtons n () descrbe an (n )-densonal affne subspace of n fro whch b (p) can be chosen. For exaple, Waldron [9] suggests to consder barycentrc coordnates wth nal l 2 -nor and derves an explct forula for coputng the. He further shows that these affne barycentrc coordnates are non-negatve n a convex regon that contans the barycentre v = (v + + v n )/n of Ω. Another exaple are Floater s shape preservng coordnates [7] whch are well-defned and non-negatve for any p n the kernel of Ω and have been used successfully for esh paraeterzaton [7] and orphng [9]. Both applcatons rely on pontwse barycentrc coordnates, n the sense that b (p) wth the propertes n () ust be deterned for a sngle pont p n the kernel of soe polygon Ω. Instead, other applcatons, lke geoetrc odellng [22], colour nterpolaton [26], renderng [5], shape deforaton [7], and age warpng [], requre barycentrc coordnates for all p Ω and consder b (p) as a functon of p over Ω. In ths settng, the ndvdual barycentrc coordnate functons b : Ω ust satsfy the Lagrange property b (v j ) = δ j =, f = j,, otherwse,, j =,..., n (2) n addton to the defnng condtons n (), so that the functon f : Ω d wth f (p ) = b (p )f () =

2 nterpolates the data f,..., f n d at the vertces v,..., v n. Most applcatons further expect the barycentrc coordnate functons to be sooth, so that the barycentrc nterpolant f n () s C or even C 2 contnuous. And for soe applcatons t s crucal that the coordnates are non-negatve, because ths guarantees that the nterpolated values f (p ) are contaned n the convex hull of the data.. Related work Wachspress [] was the frst to descrbe a constructon of ratonal barycentrc coordnate functons for convex polygons n the context of generalzed fnte eleent ethods, but these Wachspress coordnates are not well-defned for arbtrary sple polygons. The sae holds for dscrete haronc coordnates, whch arse fro the classcal pecewse lnear fnte eleent approxaton to Laplace s equaton [6] and have been appled for coputng dscrete nal surfaces [29] and esh paraeterzaton [6]. Mean value coordnates [] overcoe ths drawback, as they are well-defned even for sets of nested sple polygons and for any p 2 []. However, ean value coordnates can be negatve nsde concave polygons, and the sae s true for etrc [7], ovng least squares [2], Posson [], and cubc ean value coordnates [9]. Postvty nsde arbtrary sple polygons s guaranteed by postve ean value [2] and postve Gordon Wxo coordnates [2], but both constructons delver only C contnuous coordnate functons. All the aforeentoned constructons provde closed-for coordnates, whch can be evaluated exactly for any p Ω n a fnte nuber of steps. At the sae te, nether of these coordnates are sooth and postve nsde non-convex polygons. So far the only barycentrc coordnates known to have both propertes are the haronc [6], axu entropy [], and local barycentrc coordnates [2], but they all are coputatonal coordnates n the sense that they lack a closed-for expresson and ust be treated nuercally. For exaple, haronc coordnates can be approxated by usng the coplex varable boundary eleent ethod [, Sec. 6.] or the ethod of fundaental solutons [25, Sec. 5]. The advantage of both approaches s that the resultng coordnates are sooth and haronc and can be wrtten n closed for after ntally solvng a rather sall but dense lnear syste, but they only approxate the pecewse lnear boundary condtons and thus do not satsfy the Lagrange property. Another coon strategy for coputng haronc coordnates [6, 6] s frst to create a dense trangulaton of Ω, then to fx the barycentrc coordnates of the boundary vertces accordng to the Lagrange property (2) and such that the coordnates are lnear along the edges of Ω, and fnally to deterne the coordnates at the nteror vertces usng the standard fnte eleent dscretzaton of the Laplace equaton wth Drchlet boundary condtons. Ths approach s qute effcent, because t only requres solvng a sparse lnear syste, but the resultng coordnate functons are erely pecewse lnear approxatons of the true haronc coordnates. Local barycentrc coordnates are approxated slarly, except that coputng the coordnates at the nteror vertces s ore nvolved as t leads to a convex optzaton proble wth a non-sooth target functon [2, Sec. ]. However, the advantage of the resultng coordnate functons s that ther support s saller than the support of haronc coordnate functons. In both cases a global proble s solved to deterne the barycentrc coordnates for all nteror vertces sultaneously. In contrast, axu entropy coordnates are coputed for any p Ω by solvng a local convex optzaton proble, whch n turn can be done very effcently wth Newton s ethod [, Sec. 5]..2 Contrbutons In ths paper we descrbe a novel way to construct non-negatve barycentrc coordnate functons. The an dea s to start wth a pecewse lnear approxaton of haronc or local barycentrc coordnates over a coarse trangulaton of Ω and then to use subdvson [] to refne the coordnate functons (Secton 2). Whle the coordnate functons rean pecewse lnear after any fnte nuber of subdvson steps, the refneent process gves C contnuous and non-negatve coordnates n the lt for any coon subdvson schees, and these lt coordnates can be evaluated lke axu entropy coordnates by solvng a local convex optzaton proble (Secton 2.). In partcular, we focus on Loop subdvson [2] and show that the resultng C lt coordnate functons cobne the favourable shape propertes of haronc coordnates or the sall support of local barycentrc coordnates wth the possblty of evaluatng the and ther dervatves effcently at any p Ω (Secton ). We further dscuss brefly how to obtan slar results wth Catull Clark subdvson [] (Secton ). Our an contrbutons are: We prove that subdvdng pecewse lnear barycentrc coordnates keeps all the desred propertes, even n the lt, as long as the subdvson rules respect certan condtons. 2

3 We show for Loop and Catull Clark subdvson how to avod fold-overs at concave corners of Ω whch would otherwse lead to not well-defned lt coordnate functons. We present several exaples that llustrate the propertes and advantages of the proposed approach. 2 Refnng pecewse lnear barycentrc coordnates Our an observaton, whch otvated us to nvestgate the dea of subdvdng barycentrc coordnates, s that affne cobnatons of ponts and barycentrc coordnates coute n the followng sense. Lea. Suppose we are gven ponts p,..., p 2 wth barycentrc coordnates b (p ),..., b (p ) and soe weghts α,...,α whch su to one, j = α j =. Let p = j = α j p j 2 be the pont gven by the affne cobnaton of the ponts p j wth the weghts α j and b (p) = j = α j b (p j ) be the affne cobnaton of the coordnates b (p j ) wth the sae weghts. Then b (p) are barycentrc coordnates of p. Moreover, f the coordnates b (p j ) and the weghts α j are non-negatve, then so are the coordnates b (p ). Proof. To prove the frst stateent, we refer to () and observe that b (p ) = α j b (p j ) = b (p j ) = α j = and = = = j = b (p )v = α j b (p j )v = = j = α j j = = α j j = = j = b (p j )v = α j p j = p. The second stateent follows, because convex cobnatons of non-negatve values are non-negatve. Another well known fact, whch turns out to be useful n ths context, s that affne cobnatons n general, and n partcular those of barycentrc coordnates, coute wth lnear functons. Lea 2. If the barycentrc coordnates b (p j ) of the ponts p j n Lea le on a coon lnear functon, that s, b (p j ) = Ap j + c for soe A n 2, c n, and j =,..., n, then so does ther affne cobnaton b (p ). Proof. The stateent holds because b (p ) = α j b (p j ) = α j (Ap j + c ) = A α j p j + c α j = Ap + c. j = j = Suppose now that T s a trangulaton of Ω and that we are gven for each vertex p of T soe ntal barycentrc coordnates b (p). We then consder the pecewse lnear functon b = [b,..., b n ]: T n whch nterpolates the gven barycentrc coordnates at the vertces of T. Corollary. If the ntal barycentrc coordnates at the vertces v of Ω are j = b (v ) = δ = [δ,,...,δ n, ], =,..., n, () then the coponents b of b are barycentrc coordnate functons. If all ntal barycentrc coordnates are non-negatve, then so are the functons b. Proof. For any p Ω, let T = [p, p 2, p ] be the trangle n T that contans p, so that p = j = α j p j, where α j are the unque barycentrc coordnates of p wth respect to T. By the defnton of b we have b (p) = j = α j b (p j ), and Lea assures not only that b (p) are vald barycentrc coordnates of p n the sense of (), but also the stateent about non-negatvty. Condton () further guarantees that b Lagrange property (2). j = j = satsfes the We then refne T successvely wth soe lnear subdvson schee S [] to generate the sequence of trangulatons T,T,... and apply the subdvson rules not only to the (x, y ) coordnates of the vertces, but also to the assocated barycentrc coordnates. That s, f the vertex p of T k+ s generated by the affne cobnaton p = j = α j p j of soe vertces p,..., p of T k, then we assocate wth p the values b (p ) = j = α j b (p j ), and t follows fro Lea that b (p ) are vald barycentrc coordnates of p. As above, we now consder at each level k the pecewse lnear functon b k = [b k,..., b k n ]: T k n whch nterpolates the generated barycentrc coordnates at the vertces of T k (see Fgure ).

4 v v v T T T 2 b b b 2 Fgure : Man dea of refnng pecewse lnear barycentrc coordnates: A trangulaton T of the polygon Ω s refned by a lnear subdvson schee wth specal rules to keep the boundary fxed. In parallel, the sae subdvson rules are appled to the barycentrc coordnates assocated wth the vertces of the trangulaton, thus creatng a sequence of pecewse lnear barycentrc coordnate functons b k (shown for the red vertex) wth a C contnuous lt. The whte curves are the contour lnes at.,.2,...,.9. Theore. Let S be a subdvson schee that. s convergent, 2. generates C contnuous lts,. s equpped wth boundary rules for nterpolatng corner vertces and preservng straght boundary segents. Further assue that the trangulatons T k are regular n the sense that they do not contan any degenerate or flpped trangles, even n the lt. Then the coponents b k of b k converge to C contnuous barycentrc coordnate functons b : Ω as k. Moreover, f the ntal barycentrc coordnates at the vertces of T and the weghts of the subdvson rules are non-negatve, then so are the b Proof. Usng the approprate boundary rules along the edges of Ω ensures that T k s a trangulaton of Ω. Moreover, taggng the vertces of Ω as corners and applyng to the the nterpolatng subdvson rule guarantees that condton () s preserved at any level k. Wth the sae reasonng as n the proof of Corollary we then conclude that the b k are barycentrc coordnate functons. Note that we have to assue here that T k s regular, because otherwse t could happen that soe p Ω s contaned n ore than one trangle of T k and then b k would not be well-defned. To study the lt behavour of ths subdvson process, we recall that the natural paraeterzaton of a subdvson surface s the one wth respect to the dpont-subdvded control esh []. In our settng, ths eans that we use Ω as our doan, consder the sequence of trangulatons D,D,..., where D = T and D k+ s derved fro D k by dpont subdvson, and regard T k as the age of the pecewse lnear functon v k : Ω Ω that aps fro each trangle n D k to the correspondng trangle n T k (see Fgure 2). Under the gven condtons on S, ths sequence of functons converges to a C contnuous appng v : Ω Ω. Lkewse, subdvdng the ntal barycentrc coordnates gves a C contnuous appng b = [b,...,b n ]: Ω n n the lt. Puttng both together, we conclude that the barycentrc coordnate functons b k converge to the C contnuous functons b = b v. Note that we have to assue here the regularty of T k n the lt n order to ensure that v exsts and s C contnuous, accordng to the nverse functon theore. The functons b are barycentrc coordnate functons, because = b (p) = l b k k = (p ) = l k =. b k (p ) = l k = D T D T D 2 T 2 v v v 2 Fgure 2: Natural paraeterzaton v k of the subdvded trangulaton T k over the refned doan D k for k =,, 2.

5 and = b (p )v = l b k k = and the stateent about the non-negatvty of b (p )v = l k = b k (p )v = l k p = p, follows edately fro the gven condtons. The condtons on S n Theore are not very restrctve and satsfed by any popular subdvson schees [, 2]. However, we recoend to use approxatng schees, because nterpolatng schees, lke the butterfly schee [5, 5] have subdvson rules wth negatve coeffcents, so that the non-negatvty of the lt coordnates b s not guaranteed. We further note that Corollary and Theore work for any ntal data, but n our exaples we anly focus on the settng where the ntal pecewse lnear barycentrc coordnates b are ether haronc or local barycentrc coordnates, coputed for soe trangulaton T of Ω. By constructon, the ntal coordnate functons b are lnear along the edges of Ω n these cases, and t and the lt follows fro Lea 2 and the thrd condton on S n Theore that the sae s true for b k coordnate functons b. 2. Evaluaton For the evaluaton of the lt coordnates b, there are three possble scenaros. Frst, there are any applcatons, where t s suffcent to have a pecewse lnear approxaton of the coordnates. In ths stuaton, we sply carry out a fnte nuber of, say k = 5 or k = 6 subdvson steps, and take b k as the desred pecewse lnear approxaton over T k. We can further use the lt rules of S to snap the vertces p of T k to ther lt postons p, thus gvng a new trangulaton T k. Concurrently we apply the sae lt rules to the correspondng coordnates b k (p) to copute b ( p). Overall ths results n pecewse lnear coordnates b k over T k, whch nterpolate the lt coordnates at the vertces p of T k nstead of only approxatng the. The other two scenaros requre the avalablty of a general routne for evaluatng the lt surfaces generated by S at arbtrary paraeter values, whch n our settng allows to copute v (p) and b (p) at any p Ω. For splne subdvson schees whch generate polynoal patches n regular regons, such a routne wth constant te coplexty can be desgned by followng the deas of Sta [, 5], and non-polynoal schees can be evaluated wth the approach of Schaefer and Warren [, 2]. On the one hand, we can then ap any p Ω to ts lt poston p = v (p ) Ω and copute the lt coordnates b ( p ) = b (p ) of p. Ths s suffcent, for exaple, for applcatons whch requre to evaluate b at a dense set of ponts, but where the exact postons of the ponts do not atter. On the other hand, we can also deterne the lt coordnates b (p ) of p Ω tself by frst fndng q = v (p ), whch n turn requres solvng the local convex optzaton proble n q Ω p v (q ) 2, (5) for exaple wth Newton s ethod [2]. Once q s found we copute the lt coordnates of p as b (p ) = b (q ). 2.2 Connecton to standard surface subdvson Before contnung wth soe concrete exaples, we would lke to pont out a dfferent perspectve on the subdvson process descrbed above. Suppose we attach the -th barycentrc coordnate b (p) as a z -coordnate to each vertex p of the trangulaton T k. Ths turns T k nto the D trangle esh M k, whch s nothng but the graph of the barycentrc coordnate functon b k, and generatng M k+ fro M k s just the standard surface subdvson process. Under the gven condtons on S n Theore, t s then clear that the sequence of eshes M,M,... converges to a C contnuous lt surface M, and the regularty of T k n the lt guarantees that M s the graph of a functon, naely the lt coordnate functon b. Loop coordnates In order to verfy the theoretcal results fro the prevous secton we decded to use Loop subdvson [2] wth the odfcaton proposed by Berann et al. []. That s, we ark vertces and edges of Ω as corners and creases to preserve the boundary of the polygon and use the subdvson rules n Fgure, where the paraeter α for an nteror vertex wth valency s α = 5 + cos 2π 2 /. The standard rules are used 5

6 α α α α α α (a) (b) (c) (d) (e) 2 2 β β θ (f) Fgure : Standard Loop subdvson rules for nteror vertces (a), nteror edges (b), boundary vertces (c), and boundary edges (d). Corner vertces are sply nterpolated. The odfed edge rule (e) depends on the nteror angle at the adjacent corner vertex (f). everywhere, except at nteror edges adjacent to exactly one corner. For these edges, the corner s weghted by the odfed coeffcent β = ( + cosθ )/, where θ s the nteror angle at the corner and s the nuber of adjacent trangles. As the subdvson rules have non-negatve weghts, they generate non-negatve coordnate functons b n the lt. These rules further guarantee that the coordnate functons are C 2 alost everywhere n the nteror and along the edges of Ω, and they are C at extraordnary nteror vertces wth valency other than 6 and at convex corners. The b are only C at concave corners, but ths s not surprsng, because non-negatve coordnate functons cannot be C at such corners (see Fgure ).. Evaluaton To evaluate these Loop coordnates b, we pleented the three strateges outlned n Secton 2. n C++ on a MacBook Pro wth 2. GHz Intel Core 7 processor and GB RAM. The frst opton s to subdvde the trangulaton T and the ntal barycentrc coordnates b untl T k has about one llon vertces and to snap both the vertces p of T k and the correspondng coordnates b k (p) to the lt usng the usual lt rules [2]. Ths gves a rather detaled pecewse lnear nterpolant of b. Our pleentaton takes about 2 seconds for subdvdng the (x, y ) coordnates of the vertces of the trangulaton and anagng the data structures, plus another.2n seconds for subdvdng the assocated barycentrc coordnates, where n s the nuber of vertces of Ω. By eans of the tangent vector rules, we can even deterne the gradents of the lt coordnate functons at the lt ponts at an addtonal cost of.2 +.5n seconds. Note that coputng haronc coordnates for a trangulaton wth the sae nuber of vertces costs about.2n seconds n our pleentaton, whch s based on Egen [2], plus 5 seconds for asseblng and factorzng the atrx. Hence, t s even a bt faster for sall n, but the resultng pecewse lnear coordnate functons do not nterpolate the true haronc coordnates at the vertces and gradents can only be approxated. The second opton s to evaluate for any p Ω the lt appngs v and b, so as to get the lt coordnates b ( p ) = b (p ) at p = v (p ). To ths end, we frst subdvde T and b twce n a preprocessng step, to separate extraordnary vertces, and then fnd the trangle T n T 2 that contans p. If T s not adjacent to the boundary of T 2, then we use Sta s algorth [5], otherwse we resort to the ethod of Zorn and Krstjansson [], whch s slghtly ore coplex but works for ponts near the boundary. Wth our pleentaton, whch uses a quadtree to store the trangles of T 2, evaluatng one llon ponts ths way takes about.2 seconds for fndng the trangle T and coputng p, plus.n seconds for evaluatng b ( p ). We can further use Sta s b approach to copute frst dervatves of b at p at roughly the sae cost and even second dervatves, except w v w v Fgure : Consder the barycentrc coordnate functon for the concave vertex v (red) and the cross secton along the lne defned by ths vertex and the neghbourng vertex w (dashed lne). If ths functon s C at v, then t ust be greater than one n the nteror of Ω, whch ples that another coordnate functon s negatve. 6

7 T T T 2 T Fgure 5: Exaple of an ntal trangulaton T for whch the subdvded trangulatons T k fold over at the concave corner. T T T 2 T Fgure 6: The foldovers n Fgure 5 can be avoded by our vertex adjustent strategy (see Fgure 7). when p s an extraordnary vertex of T 2 wthout well-defned second dervatves. To copute dervatves at ponts near the boundary, we subdvde the trangulaton around p locally untl the trangle that contans p s not adjacent to the boundary anyore before callng Sta s routne, because Zorn and Krstjansson do not dscuss how to copute dervatves wth ther ethod. However, the cost of these local subdvsons has a neglgble effect on the average runte. The thrd opton s to copute b (p ) for any p Ω, whch requres to solve the optzaton proble (5). We pleented a sple Newton ethod wth adaptve step sze, takng advantage of the fact that we can use Sta s ethod as explaned above to get the gradent and the Hessan of the objectve functon. At extraordnary vertces, where the Hessan s undefned, we resort to a fnte dfference approxaton of the Hessan. Our experents show that the optal pont q = v (p) s usually found n less than three teratons wth an accuracy of 7 at an average cost of 2 6 seconds per pont. Note that ths cost does not depend on n, snce t s a proble n 2. Once q s found, we proceed to copute b (p) = b (q ) as n the second opton above. Overall, our pleentaton takes about 5 +.n seconds for evaluatng one llon ponts ths way. Ths s roughly on par wth the runte of our pleentaton of axu entropy coordnates, whch takes about 2 +.5n seconds for the sae task. Whle the thrd opton s the least effcent, t s the only one that delvers the lt coordnates b (p ) at an arbtrary p Ω. Moreover, the addtonal cost wth respect to the second opton becoes argnal for large n, and n coparson to the frst opton t requres less eory, as t needs to store only T 2 nstead of T k..2 Concave corners In the reasonng above we tactly assued that the subdvson process gves regular trangulatons T k, even n the lt as k. However, as notced by Berann et al. [], foldovers ay occur at concave corners, not only n the lt, but already after a sall nuber of subdvson steps (see Fgure 5). Consequently, the lt coordnates wll not be well-defned n these regons. However, we can avod ths proble (see Fgure 6) by odfyng T before deternng the ntal barycentrc coordnates b. To ths end, we adjust the postons of the vertces p n the one-rng neghbourhood of a concave corner p as shown n Fgure 7. That s, we frst deterne the length r = n{r, r,..., r }, where r = p p, of the shortest edge adjacent to the concave corner. We then place all neghbours regularly spaced on a crcle wth radus r around p, p = p + (p p )r /r, p = p + R θ (p p ), =,...,, (6) 7

8 p p r θ p p p p p Fgure 7: Our vertex adjustent strategy relocates the vertces n the one-rng neghbourhood of a concave corner (left) so that all adjacent trangles have the sae shape and sze (rght). p where θ s defned as above and R γ denotes the rotaton atrx for counterclockwse rotaton by γ. If ths vertex adjustent strategy creates foldovers of T n the 2-rng neghbourhood of p, then we repeatedly halve r untl the foldovers dsappear. Note that ths strategy generally requres that the one-rng neghbourhoods of the concave corners do not contan coon vertces. Theore 5. If the neghbours of a concave corner p have been adjusted wth the strategy n (6), then the trangulatons T k are regular around p, even n the lt. Proof. We frst note that the adjusted neghbours of p satsfy Recallng that p ± = p + R ±θ (p p ), =,...,. R θ + R θ = 2 cosθ I = (β 2)I, where β s defned as above and I denotes the dentty atrx, we fnd that after one subdvson step wth the odfed edge rule n Fgure (e), the new nteror neghbours p of p for =,..., are just the edge dponts of the old nteror edges, p = [β p + (6 β)p + p + p + ]/ = [β p + (6 β)p + 2p + (R θ + R θ )(p p )]/ = [β p + (6 β)p + 2p + (β 2)(p p )]/ = [p + p ]/2. As the sae holds for the new boundary neghbours p and p, due to the boundary edge rule n Fgure (d), we conclude that each subdvson step sply scales the one-rng neghbourhood of p by a factor of /2. Ths clearly avods foldovers at p, even n the lt.. Internal foldovers Whle our vertex adjustent strategy takes care of foldovers at concave corners, nteror foldovers ay stll occur n the nteror of T k, even for convex ntal trangulatons T where all nteror vertces are regular wth valency 6 (see Fgure ). A foral analyss of ths proble s beyond the scope of ths paper, but we observed that ths proble does not appear f we construct the ntal trangulaton as shown n Fgure 9. Gven a polygon Ω and a target edge length h, we frst saple each edge of Ω wth unforly spaced vertces such that the spacng s as close as possble to h. We then use Trangle [] to copute a conforng constraned Delaunay trangulaton of Ω whch contans the saple vertces, does not create any further T T T 2 Fgure : For ths convex trangulaton wth regular nteror vertces, foldovers occur n the nteror after two subdvson steps.

9 (a) (b) (c) (d) Fgure 9: To create the ntal trangulaton T for a gven polygon Ω (a), we frst specfy unforly spaced vertces on the edges of Ω (b), then copute a constraned Delaunay trangulaton (c), and fnally odfy the one-rng neghbourhood of each concave corner wth our vertex adjustent strategy (d). boundary vertces, and has trangles wth areas less than A = h 2 /, the area of the equlateral trangle wth target edge length h. Ths usually generates a trangulaton wth a axu edge length h close to h and not too any extraordnary vertces. In a fnal step we apply the vertex adjustent strategy fro Secton.2 to create the ntal trangulaton T of Ω. To test our conjecture that the lt appng v s regular so that v exsts and the lt coordnates are well-defned, we generated rando sple polygons and trangulated the for dfferent values of h. We then subdvded each ntal trangulaton untl the nuber of trangles was above one llon and appled three tests. For each trangle wth three regular vertces (usually ore than 99.9% of all trangles), we checked the condton n [, Lea ] to verfy that the correspondng lt patch s regular. We further coputed the lt tangents at all extraordnary vertces and checked that the lt appng does not fold at these vertces. Both tests restrct the potental occurrence of foldovers to the trangles adjacent to extraordnary vertces and we evaluated the lt tangents at rando ponts nsde each of these trangles as explaned n Secton.. All ntal test trangulatons passed these tests, whch akes us confdent that our conjecture s true.. Exaples Fgure shows soe exaples of Loop coordnate functons for haronc ntal coordnates. Despte the low resoluton of the trangulaton T, the functons are sooth and no vsual artefacts are recognzable at the extraordnary nteror vertces. Not too surprsngly, they actually look very slar to haronc coordnates, and Fgures and 2 further llustrate ths behavour. In both exaples, we frst coputed haronc coordnates over a esh wth two llon trangles and took ths approxaton as referental true haronc coordnates b H. As expected, the log-log plots show that the pecewse lnear haronc coordnate functons b over T converge to b H as the axu edge length h of T tends to, and that the sae holds for the Drchlet energy D (f ) = 2 Ω f 2, f : Ω, whch s of course nal for b H. The plots also show that the Loop coordnate functons b wth b as ntal coordnates and ther Drchlet energes converge at the sae rate and are consstently closer to b H. The behavour s confred by the error vsualzatons whch llustrate that Loop subdvson effectvely soothes out the error between approxate and true haronc coordnates. T b b.. b 5 b Fgure : Exaple of Loop coordnate functons and the nor of ther gradents (shown for the red vertces) usng pecewse lnear haronc coordnates over T as ntal coordnates b. 9

10 2 Ω h? =.5 h? =.29 h? =.5 h? = h h Fgure : Coparson of the errors bh b (centre) and bh b (botto) between the true haronc coordnate functon bh, the pecewse lnear approxaton b, and the Loop coordnate functon b for dfferent resolutons of the trangulaton T wth axu edge lengths h?. The top log-log plot shows the axu errors kbh b k (blue) and kbh b k (red) over h?, and the botto log-log plot shows the dfferences of the Drchlet energes D (b ) D (bh ) (blue) and D (b ) D (bh ) (red). Ω h? =.5 h? =.25 h? =.26 h? = h h Fgure 2: Coparson of the errors bh b (centre) and bh b (botto) between the true haronc coordnate functon bh, the pecewse lnear approxaton b, and the Loop coordnate functon b for dfferent resolutons of the trangulaton T wth axu edge lengths h?. The top log-log plot shows the axu errors kbh b k (blue) and kbh b k (red) over h?, and the botto log-log plot shows the dfferences of the Drchlet energes D (b ) D (bh ) (blue) and D (b ) D (bh ) (red). For the exaple n Fgure, the authors of [2] provded us wth local barycentrc coordnates for dfferent resolutons of T. Although the theory suggests that these coordnate functons are locally supported, the nuercal solver used n [2] generates sall functon values even outsde the probable support and Zheng et al. suggest to consder all values below as nuercally zero. We odfed ther data n the followng way. For each vertex p of T wth one or ore coordnates b (p ) < we set b (p ) to exact zero and perturbed the other coordnates n a least squares sense to restore the barycentrc propertes n (). We then used these odfed coordnates as b. The plots show that the correspondng Loop coordnates are truly locally supported and that the support s slghtly larger, but also soother than the nuercal support of the orgnal local coordnates. We further observe that the shape of the Loop coordnates for the ntal trangulatons wth 7 and 6979 vertces s vsually the sae, whch suggests that, gven the exponental

11 6.6 s 5 s 9 s 5972 s 66 vertces 7 vertces 6979 vertces 2765 vertces Fgure : Coparson of local barycentrc coordnates (top) and correspondng Loop coordnates (botto) for dfferent resolutons of the ntal trangulaton T. For local barycentrc coordnates, the contour lne at s shown n green, and the orange lne arks the support of the Loop coordnates. The tngs for coputng local barycentrc coordnates are gven at the top. Ω T LC HC MEC MVC Fgure : Coparson of dfferent barycentrc coordnate functons for the two red vertces. Loop coordnates were coputed for the ntal trangulaton T n the botto left. The nsets show the cross sectons along the dashed lne. cost of coputng local barycentrc coordnates, t s better to sooth the wth Loop subdvson nstead of further ncreasng the resoluton of the ntal trangulaton. The fact that the coordnate functons for the trangulaton wth 2765 vertces look apparently dfferent fro the others s probably due to the fact that the solver had not fully converged, even after the ndcated 5972 seconds. For the exaple n Fgure, we coputed Loop coordnates for a trangulaton T wthout nteror vertces, usng only the barycentrc coordnates n () as ntal coordnates at the vertces v of Ω. Because of the lack of extraordnary nteror vertces, the resultng coordnate functons are C 2 n the nteror of Ω. The coparson to haronc (HC), axu entropy (MEC), and ean value coordnates (MVC) shows that Loop coordnates are ore local at convex and less steep at concave corners. One potental drawback of our approach s that the Loop coordnates b depend on the ntal trangulaton T. An exaple of ths effect s gven n Fgure 5, whch shows two coordnate functons for two dfferent ntal trangulatons as well as the dfference between the. For convex corners, ths dfference s usually less than.5% and less than 2% for concave corners, but the contour and gradent plots confr that the global shapes of the coordnate functons are very slar.

12 b T k b k 2 b 9 k b k 2 6 T..5 b k b k.5 b b b. k b k. b b Fgure 5: Coparson of barycentrc coordnate functons for the two red vertces and dfferent ntal trangulatons T and T usng pecewse lnear haronc coordnates as ntal coordnates. Catull Clark coordnates The refneent process descrbed n Secton 2 works analogously wth lnear subdvson schees for quadrlateral eshes. We start fro an ntal quadrangulaton Q of Ω wth gven barycentrc coordnates at the vertces p of Q, and use the schee to generate the sequence of quadrangulatons Q, Q,... as well as to copute barycentrc coordnates at the vertces of each Qk. Under the sae condtons as n Theore, ths gves C contnuous barycentrc coordnate functons n the lt. The an dfference s that we need to be careful wth the defnton of the functons b k = [bk,..., bnk ]: Qk Rn, whch nterpolate the ntal or coputed barycentrc coordnates at the vertces of Qk, so as to guarantee the equvalent of Corollary. One possblty s to splt each quadrlateral of Qk nto two regular trangles and let b k be pecewse lnear over the trangles obtaned ths way. Another choce s to let b k be sooth over each quadrlateral by utlzng ean k value coordnates n the followng way. For any p Ω, let Q = [p, p2, p, p ] be the quadrlateral P n Q that contans p, let α,..., α be the ean value coordnates of p wth respect to Q, that s, p = j = α j p j, and P set b k (p ) = j = α j b k (p j ). Lea then guarantees that b k (p ) are vald barycentrc coordnates of p, and snce the weghts α j are non-negatve, even for a concave quadrlateral Q [5], the non-negatvty stateent n Corollary carres over to the quadrlateral settng. As a case study for ths quadrlateral settng we decded to use Catull Clark subdvson [] wth the odfcatons proposed by Berann et al. []. As n Secton, we ark vertces and edges of Ω as corners and creases to preserve the boundary of the polygon and use the subdvson rules n Fgure 6, where the paraeters for an nteror vertex wth valency are α = 2 and α2 = and the coeffcent for the odfed edge rules s β = ( + 2 cos θ )/ wth θ as n Secton. Lke Loop coordnates, these Catull Clark coordnates are C 2 alost everywhere, except at extraordnary nteror vertces and convex corners, where they are only C and at concave corners, where they are C. To evaluate the, we pleented the sae three optons as descrbed n Secton. wth slar runtes, usng Sta s algorth [] for the evaluaton of v and b n the nteror and the ethod of Zorn and Krstjansson [] near the boundary. The vertex α2 α α2 α α α2 α α α α β 6 6 β α2 (a) (b) (c) (d) (e) (f) Fgure 6: Standard Catull Clark subdvson rules for faces (a), nteror vertces (b), nteror edges (c), boundary vertces (d), boundary edges (e), and odfed edge rule (f ). Corner vertces are sply nterpolated. 2

13 Q b b 2 b b Fgure 7: Exaple of Catull Clark coordnate functons and ther gradents (shown for the red vertces) wth haronc coordnates over Q as ntal coordnates b. T Q Fgure : Coparson of Loop (left) and Catull Clark (rght) coordnate functons (shown for the red vertces) wth haronc coordnates over T and Q, respectvely, as ntal coordnates b. adjustent around a concave corner p s done as n (6) for the adjacent neghbours p,..., p of p, and the opposte corners q,..., q of the adjacent quadrlaterals are oved to q = p + p + p, =,...,, so that all adjacent quadrlaterals becoe congruent parallelogras. Wth the sae arguents as n Theore 5, one can then show that ths confguraton scales by a factor of /2 wth each subdvson step, thus avodng foldovers at p, even n the lt. We dd not further nvestgate the ssue of nternal foldovers, but dd not experence any probles n our nuercal exaples.. Exaples Fgure 7 s the analogue to Fgure and shows soe exaples of Catull Clark coordnate functons for haronc ntal coordnates. Snce the ntal quadrangulaton has no extraordnary nteror vertces, these functons are C 2 n the nteror of Ω, and the plots confr that they are also vsually sooth. In Fgure we coputed haronc coordnates over a trangulaton T and a quadrangulaton Q of the sae polygon and used the as ntal coordnates for Loop and Catull Clark coordnates, respectvely. The exaple shows that both subdvson schees have a very slar soothng effect and that the coordnate functons are alost dentcal. Snce quadrangulatng a gven polygon s uch harder than trangulatng t, ths suggests that Loop coordnates are probably the ethod of choce n ost cases. Ω Q CCC HC MEC LBC Fgure 9: Coparson of dfferent barycentrc coordnate functons for the two red vertces. Catull Clark coordnates were coputed for the ntal quadrangulaton Q shown n the botto left. The nset shows the cross secton along the dashed lne. The contour lne at s shown n green, and the orange lne arks the support.

14 However, for certan polygons lke the one n Fgure 9 t s ore natural to use Catull Clark coordnates. Slarly to the exaple n Fgure, ths fgure shows Catull Clark coordnates for a quadrangulaton Q wthout nteror vertces, usng only the barycentrc coordnates n () as ntal coordnates at the vertces of Ω. Consequently, the resultng coordnate functons are C 2 n the nteror of Ω. The coparson to haronc (HC) and axu entropy coordnates (MEC) shows that Catull Clark coordnates (CCC) are ore local at convex and less steep at concave corners, and they are soother, but less local than local barycentrc coordnates (LBC). 5 Conclusons Mesh subdvson s wdely known n coputer graphcs as a technque for creatng sooth surfaces wth arbtrary topology by repeatedly refnng an ntal base esh wth sple local rules. In ths paper we show that subdvson can also be used to construct barycentrc coordnates wth favourable propertes. Whle the theory developed n Secton 2 s general and works for a large class of subdvson schees, we beleve that Loop subdvson s the ethod of choce, for two reasons. On the one hand, t s sple and coes wth well-understood boundary rules and exact evaluaton routnes. On the other hand, our exaples confr that the an shape of the lt coordnate functons b s dctated by the ntal functons b, and we do not expect other subdvson schees to yeld qualtatvely better results. However, t stll reans future work to develop a strategy for constructng ntal trangulatons T, for whch t can be forally proven that the refned trangulatons T k are regular n the nteror, even n the lt. Note that ths proble s not restrcted to the constructon of well-defned Loop coordnates, as t addresses the general queston under whch condtons the two-densonal Loop appng v : Ω Ω s bjectve. Another drecton for future work s the extenson of our approach to D by usng voluetrc subdvson schees [, ]. Acknowledgeents We thank the anonyous revewers for ther valuable coents and suggestons, whch helped to prove ths paper. We further thank Teseo Schneder for useful dscussons and help wth the pleentaton, and Zshun Lu for provdng the local barycentrc coordnates data. References [] H. Berann, A. Levn, and D. Zorn. Pecewse sooth subdvson surfaces wth noral control. In Proceedngs of SIGGRAPH 2, Annual Conference Seres, pages 2, New Orleans, LA, July 2. [2] T. J. Cashan. Beyond Catull Clark? A survey of advances n subdvson surface ethods. Coputer Graphcs Foru, ():2 6, Feb. 22. [] E. Catull and J. Clark. Recursvely generated B-splne surfaces on arbtrary topologcal eshes. Coputer-Aded Desgn, (6):5 55, Nov. 97. [] Y.-S. Chang, K. T. McDonnell, and H. Qn. A new sold subdvson schee based on box splnes. In K. Lee and N. M. Patrkalaks, edtors, Proceedngs of the Seventh ACM Syposu on Sold Modelng an Applcatons, pages 226 2, Saarbrücken, Gerany, June 22. [5] N. Dyn, D. Levn, and J. A. Gregory. A butterfly subdvson schee for surface nterpolaton wth tenson control. ACM Transactons on Graphcs, 9(2):6 69, Apr. 99. [6] M. Eck, T. DeRose, T. Duchap, H. Hoppe, M. Lounsbery, and W. Stuetzle. Multresoluton analyss of arbtrary eshes. In Proceedngs of SIGGRAPH, pages 7 2, Los Angeles, Aug [7] M. S. Floater. Paraeterzaton and sooth approxaton of surface trangulatons. Coputer Aded Geoetrc Desgn, ():2 25, Apr [] M. S. Floater. Mean value coordnates. Coputer Aded Geoetrc Desgn, 2():9 27, Mar. 2. [9] M. S. Floater and C. Gotsan. How to orph tlngs njectvely. Journal of Coputatonal and Appled Matheatcs, ( 2):7 29, Jan [] M. S. Floater, K. Horann, and G. Kós. A general constructon of barycentrc coordnates over convex polygons. Advances n Coputatonal Matheatcs, 2( ):, Jan. 26. [] I. Gnkel, J. Peters, and G. Ulauf. Norals of subdvson surfaces and ther control polyhedron. Coputer Aded Geoetrc Desgn, 2(2):2 6, Feb. 27.

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