HARTMUT PRAUTZSCH and GEORG UMLAUF. Fakultat fur Informatik, Universitat Karlsruhe. [prau/umlauf

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1 A G AND A G SUBDIVISION SCHEME FOR TRIANGULAR NETS HARTMUT PRAUTZSCH ad GEORG UMLAUF Fakultat fur Iformatik, Uiversitat Karlsruhe D-768 Karlsruhe, Germay [prau/umlauf ]@ira.uka.de I this article we improve the buttery ad Loop's algorithm. As a result we obtai subdivisio algorithms for triagular ets which ca be used to geerate G - ad G - surfaces, respectively. Subdivisio, iterpolatory subdivisio, Loop's algorithm, buttery algo- Keywords: rithm.. Itroductio Subdivisio algorithms are popular i CAGD sice they provide simple, eciet tools to geerate arbitrary free form surfaces. For example, the algorithms by Catmull ad Clark [] ad Loop [] are geeralizatios of well-kow splie subdivisio schemes. Therefore the surfaces produced by these algorithms are piecewise polyomial ad at ordiary poits curvature cotiuous. At extraordiary poits however, the curvature is zero or iite. I geeral, sigularities at extraordiary poits is a iheret pheomeo of subdivisio, see [,, 5]. The smoothess of a subdivisio surface at its extraordiary poits depeds o the spectral properties of the associated subdivisio matrix. Doo ad Sabi [6] derived ecessary coditios o the eigevalues. Ball ad Storry [7, 8] made rst rigorous ivestigatios to prove the taget plae cotiuity for a class of Catmull/Clark type algorithms. The Reif [9] observed that taget plae cotiuous surfaces may have local self-itersectios ad itroduced the characteristic map deed by the subdomiat eigevectors. Moreover, for all statioary subdivisio schemes he derived ecessary ad suciet coditios which guaratee that the limitig surface is regular, i.e. taget plae cotiuous without local peetratios. Supported by IWRMM ad DFG grat # PR 565/-.

2 Fially, i [] Reif's characteristic map is used to parametrize the subdivisio surface. With this parametrizatio it is possible to extet Reif's result ad to obtai for all statioary subdivisio schemes ecessary ad suciet coditios which guaratee that the limitig surface is a regular G k -surface. Doo ad Sabi [6], Ball ad Storry [8] ad Loop [] used the smoothess criteria to d amog certai variatios of the Catmull/Clark ad Loop's algorithm the best. However, these best algorithms still produce curvature discotiuous surfaces, see e.g. [8]. I [] we took a dieret approach. Istead of varyig the subdivisio rules withi some bouds which are set heuristically, we chaged the spectrum of the subdivisio matrix so as to obtai the desired properties. Usig the G -characterizatio i [] we derived a G -subdivisio algorithm from the Catmull/Clark algorithm (which does ot produce iite curvatures), see []. Here we provide similar improvemets, a G - ad a G -algorithm based o the buttery ad Loop's algorithm.. Loop's algorithm Loop's algorithm geeralizes the subdivisio algorithm for surfaces expressed i terms of the symmetric quartic box splie over a regular triagulatio of IR. It geerates from ay triagular et N a ew et N, whose vertices are classied as E- ad V-vertices. Computig the weighted averages of the four vertices of ay two triagles i N sharig a commo edge with the weights show i Figure gives the E- vertices. Similarly computig the weighted averages of all vertices of all triagles i N aroud ay vertex with the weights show i Figure gives the V-vertices. For = 6 Loop chooses = 5=8 sice this correspods to box splie subdivisio. PSfrag replacemets =8?? =8 =8?? =8?? E-mask V-mask Fig.. The masks of the Loop algorithm { the V-mask is illustrated for = 6. The ew et N is obtaied by coectig for all triagles of N the associated three E-vertices ad for all edges of N the associated E-vertices with both associated V-vertices. By the same procedure a ext et N is obtaied from N ad so o.

3 A G ad a G subdivisio scheme for triagular ets A vertex of ay et N i ; i ; is called extraordiary, if it is a iterior vertex with valece 6= 6. A extraordiary vertex of N i is a V-vertex associated with a extraordiary vertex of N i?. Thus the umber of extraordiary vertices is costat for all ets N i ; i ; ad these vertices are separated by more ad more ordiary vertices as i grows. I particular if N is a regular triagular et, i.e. without extraordiary vertices, Loop's algorithm coicides with the subdivisio algorithm for quartic box splie surfaces. Thus also for a arbitrary et N the sequece N i coverges to a piecewise quartic surface with oe extraordiary poit for each extraordiary vertex of N. The limitig surface is a C -surface everywhere except at its extraordiary poits. Loop's aalysis shows that the limitig surface has a cotiuous taget plae at its extraordiary poits for a certai rage of 's, see [].. The buttery algorithm The buttery algorithm of Dy et al. [] geerates a sequece of triagular ets N i ; i ; similar to Loop's algorithm. Oly the masks used to compute the E- ad V-vertices are dieret. They are give i Figure. PSfrag replacemets?! =! =?!?!!?! E-mask V-mask Fig.. The masks of the buttery algorithm. A sequece of ets N i obtaied by the buttery algorithm with small positive! coverges to a surface that is dieretiable everywhere except at its extraordiary poits of valece [, ] ad 8. At extraordiary poits of valece 8 the surface is taget plae cotiuous but it has self-itersectios ad therefore is ot regular. We checked this for several!. However, i the sequel we always work with! = =. Variatios of the buttery algorithm have bee proposed by Zori et al. []. Recetly these variatios were proved to geerate regular G -surfaces [5].. A smoothess coditio I Sectios 5 ad 6 we preset modicatios of Loop's ad the buttery algo-

4 rithm givig G - or G -surfaces i the limit. The method used to derive these modicatios is based o the G k -aalysis of subdivisio schemes give i [] ad ca also be used for subdivisio schemes for quadrilateral ets []. For more details we eed to recall a result from []. We preset it i the theorem below for ay subdivisio scheme S that is idetical with the buttery or Loop's algorithm except that E- ad V-masks may be dieret. We assume that the limitig surface associated with ay iitial triagular et N obtaied by the subdivisio scheme S has C k -parametrizatios aroud all its ordiary poits. Extraordiary poits are isolated as observed i Sectio. Therefore, to aalyze the smoothess of the limitig surface at extraordiary poits it suces to cosider a subet M of N cosistig of oe extraordiary vertex surrouded by say r rigs of ordiary vertices as illustrated i Figure for r =. Fig.. A et with oe extraordiary vertex of valece 5 (marked by ) surrouded by r = rigs of ordiary vertices. Further let M be the largest subet of N whose vertices deped oly o M. This et M also has oly oe extraordiary vertex surrouded by say r rigs of ordiary vertices ad i case of the buttery algorithm by a further icomplete rig of V-vertices. To make M of a similar form as M we delete such a icomplete rig which modies the deitio of M. Note that r is roughly twice as lage as r. For example i Loop's algorithm r = 5 if r = ad i the buttery algorithm r = 6 if r =. Let r be so large that r? r. The discardig the r? r outer rigs of M gives a et K with the same size ad coectedess as M. Let m ; : : : ; m m ad k ; : : : ; k m deote the vertices of M ad K, respectively. Sice the vertices k i are ae combiatios of the m j, there is a m m

5 A G ad a G subdivisio scheme for triagular ets 5 matrix A such that [k : : : k m ] t = A[m : : : m m ] t : Let s deote the limitig surface associated with M uder the subdivisio scheme S. Applyig S to M gives the same limitig surface s, but the surface s associated with the subet K is smaller ad oly a part of s. Takig s away from s gives the here so-called rst surface rig associated with M. Now we are able to preset the followig theorem which is prove i a more geeral form i [] : Theorem Let A have the m (possibly complex) eigevalues ; ; ; ; : : :;, where > jj jj jj ad assume two eigevectors c ad d associated with the double for simplicity real eigevalue. If the rst surface rig of the et give by [c : : :c m ] t = [c d] is regular without self-itersectios ad jj k > jj; k ; () the the limitig surface is a G k -surface for almost all iitial ets M. Remark More precisely, if Theorem is satised, the limitig surface is a G k -surface for all iitial ets M whose expasio by the eigevectors of A ivolves c i oe ad d i a secod coordiate. The eigevalue coditio () goes back to Doo ad Sabi [6]. The rst surface rig associated with the eigevectors c ad d is called the characteristic map of A by Reif who used it to prove this Theorem for k = [9]. A example for the characteristic map of Loop's algorithm is show i Figure. Fig.. = 7. The characteristic map of Loop's algorithm at a extraordiary vertex of valece If the limitig surface i Theorem is a C k -maifold, k, the the extraordiary poit is a at poit. This fact is also true for more geeral subdivisio schemes, see [, 5].

6 6 5. Modicatios of Loop's algorithm The subdivisio matrix A of Loop's algorithm associated with a extraordiary vertex of valece has a sigle domiat eigevalue ad satises the G -coditios of Theorem [, 6], but ot the G -coditio [7]. To obtai a subdivisio matrix A that represets a modicatio of Loop's algorithm satisfyig the G -coditio we diagoalize the matrix A, A = V V? ; chage the modal matrix to where = diag(; ; ; ; : : :; ); = + diag(; ; ; ; : : : ; ); where j + j; : : :; j + j < ; ad compute the ew subdivisio matrix as A = V V? : () Lemma The matrices A ad A have the same characteristic maps. Proof The eigevectors associated with are the same for A ad A. They dee a plaar cotrol et N. Subdividig N by Loop's algorithm ad also by the modicatio results both times i the same sequece of ets N i. The extraordiary vertex ad its three surroudig rigs of cotrol poits i N i are scaled versios of N. The other cotrol poits of N i are computed by the subdivisio rules for regular ets. Thus Loop's algorithm ad its modicatio applied to N produce the same surface i the limit. The symmetry of Loop's scheme meas that the subdivisio matrix A is block-circulat. Therefore a discrete Fourier trasformatio ca be used to aalyze the spectral properties of A. If =, the matrix A has the subdomiat eigevalue = = ad exactly six eigevalues with modulus i the half-ope iterval [jj ; jj). These are the two triple eigevalues =8 ad =6. Chagig just these triple eigevalues to the triple eigevalues =8 + " ad =6 + ", respectively, such that j=8 + " j ad j=6 + " j are less tha jj, results i a matrix A, which represets the same masks as the origial matrix A except for the E- ad V-masks show i Figure 5, where = 8 + " (? ) ; = 6 + "?? " 6? 7 ; = 8? (6? 7)" 6(? ) ; = 5 8? (6? 7)"? (6? )" (? ) (6? 7) ; = 8 + " ; = 6 + " ; = (? )" ; 6(? = ) 6 + "? " ; = 6 + (? )" (? ) + (? )" (6? 7) :

7 PSfrag replacemets A G ad a G subdivisio scheme for triagular ets 7 E-mask V-mask Fig. 5. The E- ad V-masks of the modied Loop algorithm ear a vertex of valece =. If, the matrix A has k := b(?)=c? double eigevalues besides. We deote these eigevalues by ; : : : ; k ad assume j j j k j. Furthermore, ay eigevalue of A with modulus i the half-ope iterval [jj ; jj) is oe of these double eigevalues i but ot vice versa. Chagig just these double eigevalues i to the double eigevalues i + i results i a matrix A, which represets the same masks as the origial matrix except for the E-mask illustrated i Figure 6, where i = f i + kx i(j + ) j cos ; i = ; : : :; b=c j= ad f i = 8 < : =8 =8 if i = i = i : Note that Loop's masks, see Figure, are obtaied if all 's ad "'s are zero. Figure 7 shows a example. The left surface is geerated usig Loop's algorithm while the right oe is produced with the above modied masks, where = :755 ad = = k =. The surfaces are show with the visualizatio of their Gaussia curvature. To compute the curvature we iterated the algorithm util the hole became smaller tha oe pixel ad the used the piecewise quartic parametrizatio of the surface ad ot a discrete approximatio based o the subdivided cotrol et. The commo cotrol et of both surfaces is give i Figure 8.

8 8 PSfrag replacemets b=c =8 E-mask Fig. 6. The E-masks of the modied Loop algorithm ear the vertices of valece illustrated for = Fig. 7. Visualizatio of the Gaussia curvature of the surface geerated from the et show i Figure 8 by Loop's algorithm (left) ad our modicatio (right). Fig. 8. Topview of the cotrol et used for Figure 7. It lies o a parabolic cylider.

9 A G ad a G subdivisio scheme for triagular ets 9 Remark I some cases better lookig surfaces are obtaied if Loop's algorithm is gradually modi ed after each subdivisio iteratio. For example, startig from the et N show i Figure the sequece of ets Ni ; i = ; : : :; 6; leadig to the surface show i Figurep9 (bottom left) has bee obtaied by Loop's algorithm modi ed with " = " = i=8 whe applied to the et Ni. I further iteratios we would chose " ad " costat as i step 6. Note that the modi ed subdivisio matrix satis es the coditios of Theorem for i. The adaptive liear combiatio of Loop's ad our scheme produces a surface with a more eve curvature distributio ad without i ite curvature Fig. 9. Visualizatio of the Gaussia curvature of the surface geerated from the et show i Figure by Loop's algorithm (top left), our modi ed scheme (top right), a adaptive liear combiatio of Loop's ad our scheme (bottom left) ad our modi ed scheme usig the thi plates eergy to determie the weights of the masks (bottom right). Remark 5 The eigevalues of A with modulus less tha j j eed ot be chaged. However, the masks of the modi ed algorithms deped liearly o ; : : :;, see (). Therefore quadratic eergy fuctioals ca be used to determie the optimal

10 Fig.. Topview of the cotrol et used for Figure 9. It lies o a hyperbolic paraboloid. values for ; : : : ;. A example surface is show i Figure 9 (bottom right). Startig from the iitial et of Figure the surface is computed by the modied Loop algorithm usig the thi plates eergy to provide the weights for the masks. 6. Modicatios of the buttery algorithm A limitig surface obtaied by the buttery algorithm is ot dieretiable at extraordiary poits, i geeral. This ca be see from the associated subdivisio matrix A which is block-circulat A = 6 A A A? A? A A?..... A A? A 7 5 : Let b A = diag( b A ; : : : ; b A? ) be the discrete Fourier trasform of A. The the blocks b A i ; i = ; : : :;? ; are give by ba i = 6 i; b L i R bi O 7 5 ;

11 A G ad a G subdivisio scheme for triagular ets with the Kroecker symbol i;, the zero-matrix O ad bl i = br i = 6 6 (ci? c i )!?( + a?i )! +a i? (a?i + a i )!?( + a i )!!?i?!?a!?!?! 7 5 ad a = exp( p?=); c i = Re(a i ). The eigevalues of A are the eigevalues of the blocks b A i ; i = ; : : :;?, ad are as follows: a simple eigevalue, a (6? )-fold eigevalue, a -fold eigevalue?! ad the eigevalues of b L i ; i = ; : : : ;?. For a extraordiary vertex of valece, the largest eigevalue of b L ; b L ad bl is ( + p? 6!)=. Therefore the subdomiat eigevalue is the triple eigevalue = ( + p? 6!)= whose associated eigevectors are liearly idepedet. So the limitig surface is ot dieretiable at a extraordiary poit for valece. However, the leadig eigevalues of b L ad b L are associated with eigevectors formig a regular ad ijective characteristic map. As i Sectio 5 we write b L as bl = b V b V? ; where = diag(; ;!); ad (see [8]) bv =!! 5 ( +!? ) ; =! where f; ;!g. Chagig the leadig eigevalue to + ", such that j + "j is less tha results i a ew modal matrix ad a modied block b L. The iverse, discrete Fourier trasform gives the modied matrix A. It diers from the iitial matrix A at those blocks that deped o b L. These are the secod diagoal blocks, L i, of the blocks A i. Their modicatios are i+ i+ i+ i+ i+ i+ i+ i+ i+ 7 5 ; 5 = L i + b V diag("; ; ) b V? ; i = ; ; : From this we read of the masks of the modied buttery algorithm which are show i Figure, where we used 5 = 9X 5? j= j j j 5 :

12 6 5? 6 5?? 6? E-mask 5?? V-mask E-mask Fig.. The E- ad V-masks of the modied buttery algorithm ear a vertex of valece = 8. For extraordiary vertices of valece = ; : : : ; 7 the limitig surface is a regular C -surface, see [9]. For a extraordiary vertex of valece 8 the subdomiat eigevalue comes from L b i with i 6= ;?. This meas that the characteristic map of the subdivisio matrix A overlaps itself, cf. [5, ]. However, the largest eigevalue of L b is associated with two eigevectors represetig the cotrol et of a regular ijective surface rig. So let i deote the largest eigevalue of b L i ; i = ; : : : ;? ;. The we chage the eigevalues i ; i = ; : : : ;? ; with modulus i [j j; ) to j i + i j < as i Sectio 5 so that becomes the subdomiat eigevalue. The eigevectors of b L i ; i = ; : : :;? ; form the matrices b Vi give by (see [8]) bv i = i e i i i ei i 5 ; = ( +!(ci? c i!c i= )? ) for f i ;e i ; i g = spec(b L i ). This yields agai the masks of Figure with the weights i+ i+ i+ i+ i+ i+ 5 = L i +? X a ij V b j diag( j ; ; ) b? V j i+ i+ i+ j= for i = ; : : :;? ad = =; = ; =!. Note that the weights are always real if i =?i for i = ; : : : ; b=c. a i=

13 A G ad a G subdivisio scheme for triagular ets Figure shows a example with a extraordiary vertex of valece =. The top row shows the surfaces geerated usig the buttery scheme (left) ad the above modied masks (right) with the parameters! = = ad i =? i? : for i ; i = ; : : :;?. The bottom row shows a selective elargemet of some viciity of the extraordiary poit of the two surfaces, respectively. Note that the left surface has self-itersectios while the right surface as well as the commo cotrol et of both surfaces, see Figure, have o self-itersectios. Fig.. The surface geerated from the et show i Figure by the buttery scheme (top left) ad our modicatio (top right). The bottom row shows a elargemet of some viciity aroud the extraordiary poit of the two surfaces from a dieret perspective as the top row with their respective boudary curves. Remark 6 The surface obtaied by the modied buttery algorithm does ot iterpolate all vertices of the iitial cotrol et. However, if we use the buttery

14 Fig.. The cotrol et used for Figure. algorithm i the rst iteratio ad the modicatio i all further iteratios, all vertices of the iitial et are iterpolated. Ackowledgemets We wish to thak Kei Berardi, Matthias Joh ad Uwe Klotz who helped us to geerate Figures 7, 9 ad. Refereces. E. Catmull ad J. Clark. Recursive geerated b-splie surfaces o arbitrary topological meshes. CAD, (6):5{55, C.T. Loop. Smooth Subdivisio Surfaces Based o Triagles. Master's thesis, Departmet of Mathematics, Uiversity of Utah, August M.A. Sabi. Cubic recursive divisio with bouded curvature. I P.J. Lauret, A. Le Mehaute, ad L.L. Schumaker, editors, Curves ad Surfaces, pages {. Academic Press, Bosto, 99.. U. Reif. A degree estimate for subdivisio surfaces of higher regularity. Proceedigs of the America Mathematical Society, (7):67{7, H. Prautzsch ad U. Reif. Degree estimates for C k -piecewise polyomial subdivisio surfaces. Computatioal Mathematics, :9{7, D.W.H. Doo ad M. Sabi. Behaviour of recursive divisio surfaces ear extraordiary poits. CAD, (6):56{6, A.A. Ball ad D.J.T. Storry. A matrix approach to the aalysis of recursively geerated b-splie surfaces. CAD, 8(8):7{, A.A. Ball ad D.J.T. Storry. Coditios for taget plae cotiuity over recursiveley geerated b-splie surfaces. ACM Trasactios o Graphics, 7():8{,

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