David Sewell z. Sewell Development. Provo, Utah. Figure 1: Uniform and non-uniform Doo-Sabin surfaces.

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1 Non-Unform Recursve Subvson Surfaces Thomas W. Seerberg Janmn Zheng y Brgham Young Unversty Dav Sewell z Sewell Development Provo, Utah Malcolm Sabn x Numercal Geometry Lt. Cambrge, UK Abstract Doo-Sabn an Catmull-Clark subvson surfaces are base on the noton of repeate knot nserton of unform tensor prouct B-splne surfaces. Ths paper evelops rules for non-unform Doo- Sabn an Catmull-Clark surfaces that generalze non-unform tensor prouct B-splne surfaces to arbtrary topologes. Ths ae exblty allows, among other thngs, the natural ntroucton of features such ascusps, creases, an arts, whle else mantanng the same orer of contnuty astherunform counterparts. Categores an Subject Descrptors: I.3.5 [Computer Graphcs]: Computatonal Geometry an Object Moelng{ surfaces an object representatons. Atonal Key Wors an Phrases: B-splnes, Doo-Sabn surfaces, Catmull-Clark surfaces. INTRODUCTION Tensor prouct non-unform ratonal B-splne surfaces have become an nustry stanar n computer graphcs, as well as n CAD/CAM systems. Because surfaces of arbtrary topologcal genus cannot be represente usng a sngle B-splne surface, there has been conserable nterest n the generalzaton, base on knot nserton, calle `recursve subvson,' whch removes ths lmtaton. However, espte beng base on knot nserton, the recursve subvson technques publshe so far are the analogues of equal nterval, unform B-splnes rather than of nonunform B-splnes. Ths paper explores the possblty of achevng the extra exblty of unequal knot ntervals n a recursve subvson scheme nclung, for example, the ablty to express features such as creases an arts by smply settng some of the knot ntervals to zero. Schemes are presente for achevng nonunform Doo-Sabn an Catmull-Clark surfaces. We wll refer to these collectvely as Non-Unform Recursve Subvson Surfaces (NURSSes). Fgure (left) shows a Doo-Sabn surface, an Fgure (rght) shows an example of a non-unform Doo-Sabn surface n whch the knot spacngs along certan control eges have been set to zero (as labele), thereby creatng a G scontnuty along the oval ege on the left. Fgure shows two non-unform Catmull-Clark surfaces. The one on the left contans a art forme by settng two pars of control-ege knot spacngs to zero. The one on the rght shows shape mocaton nuce by changng the knot spacng along the top eges to an along the center horzontal eges to.. The control net use here has the topology of a B-splne control tom@byu.eu y zheng@cs.byu.eu (On leave from Zhejang Unversty) z ave@sewell.com x malcolm@geometry.emon.co.uk Fgure : Unform an non-unform Doo-Sabn surfaces. net, but these shapes cannot be obtane usng NURBS or unform Catmull-Clark surfaces. Fgure : Non-unform Catmull-Clark surfaces.. Backgroun The concept of mage space subvson as a graphcs technque ha been aroun for a long tme when recursve subvson appeare as an object enton technque. The rst relevant result n parametrc space subvson of sculpture surfaces was the e Casteljau algorthm, whch both evaluate a pont on a Bezer curve an prove the control ponts for the parts of the curve meetng there. Ths was generalze to B-splnes n the form of the Oslo algorthm [6] an Boehm subvson [3] an use as a bass for nterrogaton methos applyng parametrc space subvson. However, what sparke the magnaton of the graphcs an moelng communtes n 975 was a much more specc subvson of a quaratc B-splne, propose by Chakn as a curve renerng technque [5], an recognze for what t was by Forrest [] an by Resenfel [6]. It later turne out that the concept of a curve beng the lmt of a polygon uner the

2 operaton of cuttng o the corners ha been explore by e Rham n the 94s an 5s. Hs results were translate nto moern termnology by e Boor [7]. It was quckly apprecate that the curve eas coul gve surface technques just by applyng the concept of tensor prouct, but the mportant key concept, that subvson coul overcome the rg rectangular parttonng of the parametrc oman one of the major lmtatons of tensor proucts was reache more or less smultaneously by Catmull an Clark [4] an by Doo an Sabn [8]. Snce then there have been ve major rectons of evelopment:. The analyss of what happens near an extraornary pont, starte n the Doo-Sabn paper [8], was taken up by Ball an Storry. Ths le to an optmzaton of the coecents for the cubc case [, ] whch unfortunately msse one of the possble varatons, an the task was complete by Sabn [8], n a paper whch alsoente that a cubc constructon coul never gve fullg contnuty atthe sngular ponts, an that contnuty atsuch ponts was n fact a much morecomplcate queston than ha been assume. Further analyss was carre out by Ref [5]. The nature of the behavor aroun the extraornary ponts s now well unerstoo.. Constructons base on box-splnes, rather than on tensor proucts, were explore by Farn [] an by Loop [5], an a collecton of possble constructons was assemble by Sabn[7]. 3. Constructons that nterpolate the control ponts were explore by Dyn, Gregory an Levn [9], an an mprove scheme was erve by Zorn, Schroer an Swelens [3]. Kobbelt [4] propose an alternatve for quarlateral nets wth arbtrary topology. The smpler ones n ths category can be vewe as uals of quaratc B-splne constructons. 4. Nasr [7] stue the problems of ecent mplementaton an practcal ege-contons an extene ths to mocatons of the basc technque to acheve varous nterpolaton contons [8, 9, ]. Halstea, Kass, an DeRose showe that a farness norm coul be compute exactly for Catmull-Clark surfaces [], enablng the etermnaton of more far lmt surfaces. 5. The ea of usng just a small number of subvson steps, an then usng n-se combnatons of patches to ll n a conguraton mae more regular n some sense by those steps, was explore by Loop [6], Peters [,, 3] an Prautzsch [4]. Ball an Storry [9] took the opposte lne, of usng subvson to ene an n-se patch. What was not explore untl now was that the general topology subvson schemes were as rg as the equal nterval splnes from whch they were erve.. Overvew Secton revews knot-oublng for non-unform B-splne curves of egree two an three an ntrouces a smple approach for labelng the knot ntervals on the control polygon an ea that s crucal for the extenson to subvson surfaces. Secton 3 then gves the corresponng expresson for knot oublng of non-unform tensor prouct B-splne surfaces.secton 4 proposes subvson rules for non-unform Doo-Sabn an Catmull-Clark surfaces, whch reuce to nonunform B-splne surfaces when the control net s a rectangular gr an when all knot ntervals along every gven row an column are the same. A contnuty analyss s gven n secton 5, showng that non-unform Doo-Sabn surfaces are G an non-unform Catmull-Clark surfaces are generally G, but G at certan ponts. Secton 6 makes some observatons on NURSSes, an oers a concluson. CURVE KNOT DOUBLING For a quaratc peroc B-splne curve, each vertex of the control polygon correspons to a sngle quaratc curve segment. It s convenent then to express the knot vector by wrtng the knot nterval of each curve segment nexttots corresponng control vertex P. Ifanew knot s nserte at the mpont ofeach current knot nterval, the resultng control polygon has twce as many control ponts, an ther coornates Q k are: Q = ( + + )P + P + ( + + ) Q + = +P +( + + )P + ( + + ) as llustrate n Fgure 3. P Q - Q / - / Q+ + P + / - P Fgure 3: Non-unform quaratc B-splne curve. For cubc peroc B-splne curves, each ege of the control polygon correspons to a sngle cubc curve segment, an so we wrte the knot ntervals ajacent toeach ege of the control polygon. The equatons for the new control ponts Q k generate upon nsertng a knot mway through each knot nterval are: Q + = ( + + )P +( + ; )P + ( ; ) Q = Q ; +( ; + )P + ; Q + ( ; + ) as shown n Fgure 4. P - Q P Q Q + - / / P + + Fgure 4: Non-unform cubc B-splne curve. 3 SURFACE KNOT DOUBLING The knot-oublng formulae for B-splne curves exten easly to surfaces. Non-unform B-splne surfaces are ene n terms of a control net that s topologcally a rectangular gr, for whch all horzontal knot vectors are scales of each other, an all vertcal knot vectors are scales of each other. 3. Quaratc Case The formulae for the new control ponts F A can be wrtten n Doo-Sabn form, whch s sgncant because n ths form () () (3)

3 the new control ponts are seen as beng n groups, creatng a new face n each olface, an the vertces of each such new face are n : corresponence wth the vertces of the ol, as uner the tensor prouct form we merely see all the new vertces as formng a new regular array (see Fgure 5). F A = V + A V = + ac(b + C ; A ; D) 4(a + ac + bc + b) (4) ba + ab + bcc + acd : (5) b + a + bc + ac a C c/ a/ F A c a/ A c/ a Fgure 5: Knot oublng, quaratc B-splne. 3. Cubc Case For non-unform cubc B-splne surfaces, the renement rules can be wrtten as follows (see Fgure 6). Frst, each face s replace wth a new vertex F.For example, F =[(e 3 +e 4 )( + )P +(e 3 +e 4 )( + 3 )P +(e 3 +e )( + 3 )P 5 +(e 3 +e )( + )P ] =[4(e + e 3 + e 4 )( )]: (6) Then, each ege s splt wth an ege vertex E, e.g. E = e F + e 3 F 4 +(e + e 3 )M (7) (e + e 3 ) M = ( + )P +( + 3 )P : (8) ( ) Fnally, each orgnal control pont s replace wth a vertex pont V V = P 4 + 3e F + e F + e 3 F e 3 F 4 + 4( + 3 )(e + e 3 ) [ 3 (e + e 3 )M + e ( + 3 )M + (e + e 3 )M 3 +e 3 ( + 3 )M 4 ]=[4( + 3 )(e + e 3 )]: (9) b D B b c e 4 e P 4 e P 4 5 P e M 3 F e 3 F e 3 E E E3 P M P M 3 P 3 V 3 4 e F M 4 e 4 F 3 e E 4 P 8 P 4 P 7 e 3 e e 4 Fgure 6: Face, ege an vertex ponts. 4 NURSS REFINEMENT In unform B-splne surfaces, all knot spacngs are the same. Doo-Sabn an Catmull-Clark propose generalzatons of unform B-splne surface schemes that allow for vertces of the control mesh to have valence other than four, an the faces of the control mesh to have other than four ses. Ther subvson rules were esgne such that when the control mesh happens to be a rectangular gr, the subvson rules are equvalent toknot oublng of unform B-splne surfaces. The subvson surfaces are then ene as the lmt of the control meshes when these subvson rules are apple an nnte number oftmes. We here ene generalzatons of non-unform B-splne surfaces. As n the cubc curve case, eachegenthecontrol polyheron of a non-unform Catmull-Clark surface s assgne a knot spacng. For a non-unform Doo-Sabn surface, each vertex s assgne a knot spacng (possbly erent) for each ege raatng from t. Our objectve stoevse a set of re- nement rules for NURSSes such that f all knot ntervals are equal, the quaratc NURSS reuces to Doo-Sabn an the cubc NURSS reuces to Catmull-Clark. There are actually two stnct rules to be evse. Frst, we neetorevse the Doo-Sabn an Catmull-Clark rules for the new pont coornates, takng the knot spacngs nto account. Secon, we nee rules for etermnng the new knot spacngs. Note that \NURSS" coul just as well stan for \Non- Unform Ratonal Subvson Surfaces," because t s a smple matter to rst project ratonal control ponts to 4-D, then apply our rules, an nally to project back to3-d. In ths secton, bol captal letters stan for ponts, an non-bol typeface for knot spacngs. The nces for knot spacng k j ncate that the spacng pertans to an ege wth P as one enpont. Referrng to Fgure 8, the notaton j ncates the knot spacng for ege P {P j.rotatng counterclockwse about P, j enotes the knot spacng for the rst ege encountere, j ncates that of the secon ege, etc. For the cubc case, each ege has a sngle knot spacng, so j = j. 4. Quaratc Case In the quaratc case, renementprocees n a manner entcal to Doo-Sabn subvson: A polyheron spawns a rene polyheron for whch new faces (of type F, type E an type V respectvely) are create for each face, ege, an vertex of the prevous polyheron. Durng the subvson step, each face s replace by a new face connecte across the ol eges an across the ol vertces by other new faces. In such renement schemes, the extraornary ponts are at the \center"of n-se faces wth n 6= 4. After one teraton, every vertex of the new polyheron wll have valence four, an the numberof faces wth other than four ses wll reman constant. 3

4 Refer to Fgure 7 for labels. The new vertex P s compute: P = V + P P n ;np + j= +( ; ; ;3 ;) + cos j;jj n 8 P n k= k; k k+ k P n k= V = k; k P k+ k k P n : k= k; k k+ k P j () = 3 = 43 = - 4 = 3 34 P 3 P = 4 F 43 P 4 = P 4 4 = 4 Fgure 8: Face pont. P = 4 -,-,+ P _ -, -,- _,+ +, _,-,+ P P + - P -,- +,+ j = j + ; j ( + j ; + j ; + ) (4) j j f j + ; + j ; + j j 6=an j =otherwse. M = ( j + j + ;)P j +( j + j + ;)P j j + j + j ; + + j j + j ; j (5) Fgure 7: Quaratc renement rules. 4.. New knot spacngs New knot spacngs k j can be spece n numerous ways. Here are two straghtforwar optons: or + = ; ; = += ; = + = ;= + = += ; ; =( + + ; ; )=4 ; = ;= + =( ; + +)=4 The former allows the renement matrx to reman constant after a few teratons. The latter seems to prouce more satsfactory shapes. 4. Cubc Case Our evelopment parallels that for Catmull-Clark surfaces. As shown n Fgure 8, the face pont for a face wth n ses s compute as F = P n; = w P P n; = w () w =( ; + + ; ; + ; ;+ ; ; ; ) ( ; + ; + ; ; ; + + )() The ege pont s compute (see Fgure 9a): E =(; j ; j )M + j F j + j F j (3) - j j - j j F j - j j M j P P E P j j V F j Fgure 9: a) Ege pont. j - j j M b) Vertex pont. P M + F,+ P + The vertex pont for a pont ofvalence n s expresse (see Fgure 9b): V = cp + 3P n P = (m M + f + F + ) n n = (m (6) + f + ) M are ene as (5), F + as (), an m =( + ; )( + ; )= (7) f j = ; j (8) c = n ; 3 n otherwse, c =. 4.. New knot spacngs f P n = (m + f + ) 6= (9) Each n-se face s splt nto n four se faces, whose knot spacngs are etermne as shown n Fgure. 5 CONTINUITY ANALYSIS For each constructon, we conser the behavor of the lmt surface for all the knot spacngs beng postve. 4

5 P / 3 / (3 + 4 )/ / 34 / ( + 34 )/4 F ( + 3 )/4 P 4 / / 4 4 / 4 / ( 34 + )/4 ( 4 + )/4 P P / / P Theorem For orers 3 to 8, fallknot spacngs j >, then the renement matrx S n s not efectve an ts egenvalues are => = 3 = > j 4j j 5 j j 4n j: By an argument smlar to one use n [] we can conclue that, prove that all the knot spacngs are greater than zero, the lmt surface generate by the non-unform Doo- Sabn scheme s G contnuous both at all ornary ponts an at extraornary ponts of valence less than Quaratc case Fgure : New knot spacngs. 5.. Lmt surface structure After one teraton, all type V an type E faces are four-se, wth the property that the ntervals crossng the orgnal eges are equal as seen from the two ses. Thus each such type V face reuces to a mesh that s equvalent to a unform bquaratc B-splne, whle each type E face reuces to a mesh that s equvalent to a non-unform bquaratc B-splne. Ths leaves only the regresson n the type F faces to analyze for contnuty. The lmt surface for a face conssts of patches n the structure as shown n Fgure a. The face s four-se n ths example. 5. Cubc Case 5.. Lmt surface structure After one subvson, every face s four-se, an after two more, the pattern of knot ntervals over the group of 6 subfaces replacng each orgnal face s as shown n Fgure a. The h are n arthmetc sequence as are the v. h h h h v5 v4 v3 v v h h h h v5 v4 v3 v v h3 h3 h3 h3 v5 v4 v3 v v h4 h4 h4 h4 v5 v4 v3 v v h5 h5 h5 h5 E (j-), C j E j a. b. P j P j- P j+ E j Fgure : a) Knot ntervals after three subvsons b) Sequence of cubc polynomal peces. a. b. Fgure : a) Sequence of quaratc polynomal peces for a four-se face b) Conguraton surrounng type F face. 5.. Contnuty atface-centers It s convenent to conser four-se faces alongse the more general n-se faces. After at most two subvsons, the con- guraton surrounng a type F face may be represente as n Fgure b. In the center les the face (of type F) P P...P n. Each egeofths face (for example P j P j+ )sajacent toa four-se face of type E wth vertces P j P j+ E j E j. The neghborhoo of each vertex P j s complete by afour-se face of type V wth vertces P j E j C j E (j;). Let the conguraton aroun ths type F face be represente by the vector of ponts M =[P... P n E E... E n E n C... C n] T an M be the corresponng conguraton after subvson. Then M = S nm, S n s a4n4n matrx calle the re- nementmatrx. Here we only conser the rst opton for the new knot spacngs n whch case S n remans constantthrough all the subsequent subvson steps. Thus we canusethe egenstructure of S n to analyze contnuty. We carre out an algebrac egenanalyss for orers 3 to 8 base on the screte Fourer transform technque n an exercse reporte n [3]. Ths leas us to When these values are substtute nto (), (3) an (6), the postons of the new vertces n, or on the bounary of, the nnermost four sub-faces are exactly the same as f all the horzontal ntervals ha been equal an all the vertcal ntervals equal lkewse. The nnermost four subfaces therefore converge towars unform bcubc B-splnes, gvng a pattern of bcubc peces as shown n Fgure b, the largest square represents the face of a non-unform Catmull-Clark net wth erent knot ntervals along each ege. The pattern of smaller squares shows schematcally the nnte progresson of Bezer patches that make up the lmt surface. The nteror of the lmt surface of every such face s therefore G.We nee only concern ourselves wth the contnuty at the eges an at the vertces, there s a regresson. 5.. Contnuty atvertces In ths secton we conser contnuty atvertex ponts of valence 3 (.e., exceptonal ponts, as well as vertex ponts of valence four). Unfortunately, snce the renement matrx changes at each teraton, t s cult to perform an egenanalyss to etermne f non-unform Catmull-Clark surfaces are G at vertex ponts, except for smple numercal cases. One of the few cases that yel a constant renement matrx s the valence three vertex n Fgure 3 (rght). In ths case, the secon an the thr egenvalues are generally erent. To noutwhat s gong on n the neghborhoo of ths pont, we performe a numercal stuy, chosng a cube as a control polyheron wth varous knot spacngs. Fgure 3 shows the neghborhoo of a vertex pont V (that began as a corner of the cube) after 5 teratons. The gure on the left shows the unform knot spacng everythng s symmetrc as expecte. The gure on the rght camefromthesame vertex on the same cube, only wth knot ntervals of value 5

6 E E 3 V θ Unform E E V E E 3 Non-unform θ spacng of, except for the four eges labele wth a. Two steps of non-unform Catmull-Clark subvson result n the meshes shown (mnus a few outer layers of quarlaterals). Ths conguraton of knot spacngs causes the lmt surface to nterpolate the center pont wth G contnuty. Fgure 3: Neghborhoo of valence 3 vertex pont after 5 teratons. assgne to one set of four parallel eges on the cube of value assgne to another set of four parallel eges, an of assgne to the remanng four eges, agan after =5 teratons. Notce that the angles are no longer equal. In fact, t turns out that angle = 6 E ; V ; E 3 tens to zero as!.however, the three faces become coplanar at a much faster rate, as shown n the followng table. Here, N refers to the maxmum angle between the planes E ; V ; E, E ; V ; E 3, an E ; V ; E 3. Unform Non-unform N N raans raans raans raans 5 ; :5 5 ; :5 5 5 ;3 :6665 ; : ; : ;8 :5 5 ; : ;5 :43 75 ;3 : ; :3 6 ;4 : ;8 :4 The normals are becomng parallel at a rate that s roughly 7 tmes faster than the rate at whch s approachng zero. After teratons, the conguraton s as close to G as anyone coul possbly have nee for. The facets are 5 orers of magntue smaller than they nee to be for any practcal use (ve teratons are plenty for most graphcs applcatons). We also a smlar stuy on valence four, usng wely varyng knot spacngs, an agan observe a fast convergence of normal vectors, but no tenency of any face angles to ten to zero. Hence, we are conent that valence four ponts are G. Prelmnary experments wth n>4ncate very smlar behavor Contnuty across eges Across the nteror of an ege, the stuaton may be regare as a stanar non-unform B-splne wth a perturbaton ue to the orgnal varaton between the knot ntervals. Ths perturbaton tens to zero wth a convergence rate of O( ; ). Note that the non-unform B-splne s C an the surrounng vertces converge to a plane conguraton wth a rate of O(4 ; ). Therefore, n the lmt, the non-unform Catmull-Clark surface s G across every ege. 6 DISCUSSION Fgure 4 shows the eect that knot spacng can have ona surface. In the gr at the left, all eges are assgne knot Fgure 4: Eect of non-unform knot spacng. In the absence of non-unform subvson surfaces, Hoppe et. al. propose a scheme for mposng features such as creases, corners, an arts on an otherwse G subvson surface that uses specal-case \masks" [3]. NURSSes can prove for such features wthout the nee for specal masks. For example, Fgure shows a crease mpose on a Doo-Sabn surface by settng three knot spacngs to zero. Fgure (left) shows a art on a non-unform Catmull-Clark surface, create by settng four knot ntervals to zero. Fgures 6{9 show avarety ofshapes that can be attane usng NURSSes, but not usng unform subvson surfaces. The ntal control polyhera are shown n wreframe. Sharp features can be mpose by settng to zero the knot spacng of approprate eges. Another use for knot spacng s n shape mocaton. Fgure (rght) shows the eect of alterng the knot spacng on several control polygon eges of a torus-shape Catmull-Clark surface. Fgure 7 shows a unform Doo-Sabn surface (on the left) an two non-unform counterparts, forme by choosng erent knot spacngs as shown. The sphere n Fgure 8 cannot be expresse exactly usng unform subvson surfaces, ratonal or otherwse. Fgure 9c,e,h are other examples of non-unform Catmull-Clark surfaces. In summary, ths metho extens the known general topology methos by permttng unequal knot ntervals, thus allowng a sngle surface escrpton the strengths of both the stanar non-unform tensor proucts an the unform recursve subvson surfaces n one representaton. Even n the stuaton there are no extraornary ponts, ths theory extens current capablty by gvng a G surface when the knot ntervals are chosen nvually for every ege n the control polygon, not constrane to support a tensor prouct structure. Ths scheme proves a lot of freeom to ajust the shape of the surface. In partcular, t can moel sharp features by properly settng certan knot spacngs to zero. Future work wll esgn a convenentmoelng nterface for the nteractve purpose an etermne how to use knot spacng to best avantage. ACKNOWLEDGEMENTS The rst two authors receve partal nancal support for ths project through an NSF grant. Krs Klmaszewsk mae several helpful suggestons, as the referees. 6

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