G 2 Surface Modelng Usng Mnmal Mean-Curvature-Varaton Flow Guolang Xu 1 Qn Zhang 2 1,2 LSEC, Insttute of Computatonal Mathematcs, Academy of Mathematcs and System Scences, Chnese Academy of Scences, Bejng 100080, Chna 2 Department of Basc Courses, Bejng Informaton Scence and Technology Unversty, Bejng 100085, Chna Abstract Physcal and geometrc based varatonal technques for surface constructon have been shown to be advanced methods for desgnng hgh qualty surfaces n the felds of CAD and CAGD. In ths paper, we derve an Euler-Lagrange equaton from a geometrc nvarant curvature ntegral functonal the ntegral about the mean curvature gradent. Usng ths Euler-Lagrange equaton, we construct a sxthorder geometrc flow, whch s solved numercally by a dvded-dfference-lke method. We apply our equaton to solvng several surface modelng problems, ncludng surface blendng, N-sded hole fllng and pont nterpolatng, wth G 2 contnuty. The llustratve examples provded show that ths sxthorder flow yelds hgh qualty surfaces. Keywords: Euler-Lagrange equaton; Mnmal mean-curvature-varaton flow; Energy functonal; Dscretzaton; Surface modelng 1 Introducton Varatonal surface desgn. Surface farng (see [12] for bascs on the subject), free-from surface desgn (see [25]), surface blendng (see [27] and Fg. 5.3) and N-sded hole fllng (see [31] and Fg. 1.1) have been mportant ssues n the areas of CAD and CAGD. These problems can be effcently solved by an energy-based varatonal approach (e. g. [2, 4, 5, 10, 11, 21, 22, 24]). Roughly speakng, the varatonal approach s to pursue a Project supported n part by NSFC grant 10371130 and Natonal Key Basc Research Project of Chna (2004CB318000). The second author s also supported n part by the NSFC grant 10571012 and the Bejng Educatonal Commttee Foundaton KM200411232006. curve or surface whch mnmzes certan type of energy smultaneously satsfyng prerequste boundary condtons. A problem one meets wthn ths approach s the choce of energy models. Energy models prevously used can be classfed nto the categores of physcal based and geometrc based. The class of physcal models encompasses membrane energy E 1 and stran energy E 2 of thn elastc plate (see [7, 24]): E 1 (f) := (fx 2 + fy 2 )dxdy, (1.1) E 2 (f) := (fxx 2 + 2fxy 2 + fyy)dxdy, 2 (1.2) where f(x, y) and are surface parametrzaton and ts doman, respectvely. These energes are generalzed as ( E 3 (M) := α11 r 2 u + 2α 12 r u r v + α 22 r 2 vrg ) dudv ( + β11 r 2 uu + 2β 12 r 2 uv + β 22 r 2 vv 2rg ) dudv by Terzopoulos et al. n [23] for a parametrc surface M := {r(u, v); (u, v) } whch can be regarded as a combnaton of E 1 (f) and E 2 (f), where α, β, g(u, v) are gven parameters and a vectorvalued functon. Recently, energy functonals based on geometrc nvarants begn to lead n ths feld. As s well-known, area functonal and total curvature functonal (see [13]) E 4 (M) := da, E 5 (M) := (k1 2 + k2)da 2 M are the most frequently used energes, where k 1 and k 2 are the prncpal curvatures. The mnmzng surfaces of E 4 (M) and E 5 (M) are mnmal surfaces and Wllmore surfaces, respectvely. The energy E 6 (M) := M [ (dk1 M de 1 ) 2 + ( dk2 de 2 ) 2 ] da 1
(a) (b) (c) Fg 1.1: (a) shows a head mesh wth several holes. (b) shows an ntal fller constructon. (c) s the smooth fllng surface, after 50 teratons, generated by usng equaton (3.8). proposed by Moreton et al. n [17] punshes the varaton of the prncpal curvatures, where e 1 and e 2 are prncpal drectons correspondng to the prncpal curvatures k 1 and k 2. The advantage of utlzng physcal based models s that the resultng equatons are lnear and therefore easy to solve. The dsadvantage s that the resultng equatons are parameter dependent. That means when a reparametrzaton s performed, dentcal surfaces may have dfferent energes. Energy models based on geometrc nvarants have no such a dsadvantage, whch are not affected by the choce of the parametrzaton. For certan specal cases, these two knds of functonal models are compatble when surfaces are sometrcally parameterzed. For nstance, E 4 (M) and E 5 (M) concde wth E 1 (f) and E 2 (f), respectvely. In fact, parametrzaton dependent functonals can be regarded as the lnear substtutes for geometrc nvarants. But n the general cases ths equvalency s not correct any more. Another crtcal problem of the varatonal approach s how to fnd out those surfaces whch mnmze these energy functonals. Two approaches have been employed to solve ths problem. One method s usng the optmzaton approach (see [11, 17, 20, 24]), whch starts from a gven surface, and searches teratvely a next surface that has less energy. Usng local nterpolaton or fttng, or replacng dfferental operators wth dvded dfference operators, the optmzaton problems are dscretzed to arrve at fnte dmensonal lnear or nonlnear systems. Approxmate solutons are then obtaned by solvng the constructed systems. Another wdely accepted method s based on varatonal calculus. The frst step of ths method s to calculate the Euler-Lagrange equatons for the energy functonals, then solve these equatons for the ultmate surfaces. Ths method s superor to the optmzaton technque n general because optmzaton s lack of local shape control and computatonally expensve. Gradent descent flow method. Generally speakng, Euler-Lagrange equatons of the geometrc energy functonals are hghly nonlnear. Except for a very lmted number of smple cases do these equatons gve analytc and smple soluton, drectly solvng of the equatons s dffcult. Gradent descent flow method s therefore ntroduced to crcumvent ths problem. For nstance, from the Euler-Lagrange equaton H = 0 of E 4 (M), whch s also the defnton of mnmal surface nvestgated snce 250 years ago, we can construct a flow, called mean curvature flow, r t = Hn, here n s the normal vector feld of the surface, the auxlary varable t s denoted as a tme-marchng parameter. When the steady state of the flow s acheved, we obtan H = 0. Smlarly, Wllmore surfaces (see [26]), the soluton of the Euler-Lagrange equaton H + 2H(H 2 K) = 0 of the energy E 7 (M) := H 2 da, (1.3) M can be constructed by ths gradent descent flow method. Note that functonal (1.3) s equvalent to E 5 (M) after Gauss-Bonnet-Chern formula beng taken nto account. For the purpose of volumepreservng for closed surfaces, surface dffuson flow (see [16]) r t = Hn s sometmes employed, whch can be regarded as a smplfed verson of the Wllmore flow. Contnuty. It s well-known that the second-order flows, such as mean curvature flow or averaged mean curvature flow (see [8]), yeld G 0 contnuous surfaces at the boundares of the constructed surfaces. The fourth-order flows, such as surface dffuson flow and Wllmore flow ([14, 26]), result n G 1 contnuty. However, hgher order contnuty are sometmes requred n the ndustral and engneerng applcatons. For nstance, n the shape desgn of the 2
streamlned surfaces of arcraft, shps and cars, G 2 contnuous surfaces are crucal. Therefore, hgher order flows need to be consdered. On ths aspect, Xu et al. have utlzed a sxth-order flow n [31] to acheve G 2 contnuty and Zhang et al. have used another sxth-order PDE n [32, 33] to obtan C 2 contnuty. A sxth-order equaton s also proposed n [3] by Botsch and Kobbelt to conduct real-tme freeform medelng. But all these sxth-order flows and PDEs are nether physcal based nor geometrc based n the sense mentoned above. Our Contrbutons. In ths paper, a sxth-order geometrc based PDE s ntroduced. It s derved from the Euler-Lagrange equaton of the energy functonal F (M) := H 2 da, (1.4) M whch punshes the total varaton of mean curvature. Ths functonal s smlar to but dfferent from the functonal E 8 (M) := M (Hn) 2 da, whch s proposed by Grener n [11] for smplfyng energy E 6 (M) and s used to far splne surfaces by an optmzaton approach. A surface whch mnmzes functonal (1.4) s called Mnmal Mean-Curvature- Varaton Surface. We expect that G 2 contnuty can be acheved usng ths sxth-order PDE n solvng the surface modelng problems, such as surface blendng, N-sded hole fllng and scattered ponts nterpolaton. The expermental results show that hgh qualty surfaces are obtaned. The rest of ths paper s organzed as follows. In Secton 2, some used notatons and prelmnares are ntroduced. One sxth-order flow s derved n Secton 3. The numercal solvng technque of the flow s dscussed n Secton 4. The applcaton and examples are provded n Secton 5. Secton 6 concludes ths paper. 2 Notatons and Prelmnares In ths secton, we ntroduce some notatons and several dfferental operators defned on surfaces. Let M be a regular parametrc surface represented as r(u, v) R 3, (u, v) R 2, whose unt normal vector s n = ru rv r u r v after sutable orentaton beng chosen, where the subscrpt of r denotes ts partal dervatve and x := x, x 1 2 := (x T x) 1 2 s the usual Eucldean norm. Superscrpt T stands for the transpose operaton. We assume at least r C 6 (, R 3 ). The coeffcents of the frst and the second fundamental forms are g 11 = r u, r u, g 12 = r u, r v, g 22 = r v, r v, b 11 = n, r uu, b 12 = n, r uv, b 22 = n, r vv. To smplfy the notaton we sometmes wrte w = (u, v) and u 1 = u, u 2 = v and [ g αβ ] = [ g αβ ] 1, g = det[ g αβ ], [ b αβ ] = [ b αβ ] 1, b = det[ b αβ ]. To ntroduce the mean curvature and Gaussan curvature, let us frst ntroduce the concept of Wengarten map. The Wengarten map or shape operator of surface M s a self-adjont lnear map on the tangent space T r M := span{r u, r v } defned by (see [9]) S : T r M T r M, S(v r ) = D v n, where v r s an arbtrary tangent vector of M at pont r and v s a tangent vector feld satsfyng v(r) = v r, and D v s the drectonal dervatve operator along drecton v. We can represent ths lnear map by a matrx as [ ] 1 b11 g S = 22 b 12 g 12 b 12 g 11 b 11 g 12 g 11 g 22 g12 2. b 12 g 22 b 22 g 12 b 22 g 11 b 12 g 12 The trace dvded by 2 and determnant of S, H = b 11g 22 + b 22 g 11 2b 12 g 12 2(g 11 g 22 g12 2 ) = k 1 + k 2, 2 K = b 11b 22 b 2 12 g 11 g 22 g12 2 = k 1 k 2 are the mean curvature and Gaussan curvature, respectvely. Now let us ntroduce some used dfferental operators defned on surface M. Tangental gradent operator. Let f be a smooth functon on M. Then the tangental gradent operator actng on f s gven by f = [r u, r v ][ g αβ ][f u, f v ] T R 3. (2.1) Second tangent operator. Let f be a smooth functon on M. Then the second tangent operator actng on f s gven by f = [r u, r v ][ K b αβ ][f u, f v ] T R 3. (2.2) These dfferental operators are all geometrc ntrnsc, though expressons (2.1) and (2.2) depend on a local surface parametrzaton. That s we have the followng lemma: 3
Lemma 2.1 Let r = r(u, v) and r = r(ū, v) be two dfferent parametrc representaton of surface M. Provded that the transformaton (u, v) (ū, v) preserves the orentaton of the surface, then f = f, f C 1 (M), (2.3) f = f, f C 1 (M), (2.4) where, and, are two groups of operators on M under dstnct coordnates. Proof. We only prove equaton (2.4). Smlar proof can be performed for (2.3). Assume that the change of parameters s σ : (ū, v) (u, v). then the transformaton of bass of the tangent space s [ ] [ u ] [ [ v rū = ū ū ru ru =: J r v u v v v r v ] r v ]. For the transformaton preservng the orentaton of the surface, n = n holds. Therefore the transformatons between the coeffcents of the frst and second fundamental forms are Hence [ ḡ αβ ] = J [ g αβ ] J T, [ bαβ ] = J [ b αβ ] J T. f = [ rū, r v ][ K b αβ ][ fū, f v ] T = [ r u, r v ]J T J T [ Kb αβ ]J 1 J[ f u, f v ] T = f. Dvergence operator. Let v be a C 1 smooth vector feld on M. Then the dvergence of v s defned by dv(v) = 1 [ g u, ] [ g [ g αβ ] [r u, r v ] T v ]. v Note that f v s a normal vector feld of M, dv(v) = 0. Laplace-Beltram operator. Let f C 2 (M). Then f s a smooth vector feld on M. The Laplace-Beltram operator (LBO) applyng to f s defned by f = dv( f) (see [6]). From the defnton of and dv, we can derve that f = 1 [ g u, ] [ g ] [ g αβ ] [f u, f v ] T. v It s easy to see that s a second-order dfferental operator whch relates closely to the mean curvature normal by the relaton r = 2H, where H := Hn s the mean curvature normal. It s well-known that these two dfferental operators are also geometrc ntrnsc though they are parametrcally defned. 3 A Sxth-Order Geometrc Flow In ths secton, we frst derve an Euler-Lagrange equaton for the functonal (1.4) and then construct a sxth-order geometrc flow. Theorem 3.1 Let F (M) be defned as (1.4). Then the Euler-Lagrange equaton of F (M) s 2 H+2(2H 2 K) H+2 H, H 2H H 2 = 0. (3.1) Proof. At frst, we can rewrte the functonal (1.4) as F (M) = H 2 gdu 1 du 2, (3.2) whch s parameter-nvarant. Consder now an extremal M of functonal (3.2) and a famly of normal varaton r(w, ε) of M defned by r(w, ε) = r(w) + εϕ(w)n(w), w, ε 1, where ϕ Cc () := {φ C (, R); suppφ }. Then we obtan 0 = d dε F (M(, ε)) ε=0 =: δf (M, ϕ), (3.3) where δf (M, ϕ) = δ( H 2 ) gdu 1 du 2 [ + H 2 (δ g)/ g ] gdu 1 du 2. (3.4) From δ(g αβ ) = 2ϕb αβ, δ(g) = 4gHϕ, δ( g) = 2Hϕ g, δ(h) = (2H 2 K)ϕ + 1 2 ϕ, we can deduce that δ( H 2 ) = δ( 1 g )(g 22H 2 u 1 + g 11H 2 u 2 2g 12H u 1H u 2) 4
+ 1 g δ(g 22H 2 u 1 + g 11H 2 u 2 2g 12H u 1H u 2) = 4Hϕ H 2 + 2 [ (b 22 Hu 2 g + b 11H 2 1 u 2b 12H 2 u 1H u 2)ϕ +(g 22 H u 1 g 12 H u 2) [(2H2 K)ϕ + 1 2 ϕ] u 1 +(g 11 H u 2 g 12 H u 1) [(2H2 K)ϕ + 1 2 ϕ] ] u 2 = 4Hϕ H 2 2 H, H ϕ +2 H, [(2H 2 K)ϕ + 1 ϕ]. (3.5) 2 In the above dervaton, we have used the fact that the varaton operaton δ and dervatve operaton u are commutatve. Substtutng (3.5) nto (3.4), α we arrve at [ δf (M, ϕ) = 2H H 2 ϕ 2 H, H ϕ +2 H, [(2H 2 K)ϕ + 1 ] gdu 2 ϕ] 1 du 2 [ = 2H H 2 ϕ 2 H, H ϕ 2(2H 2 K) Hϕ H ϕ] gdu 1 du 2. Usng the Green s formula, f 2 h gdu 1 du 2 = f h gdu 1 du 2, for all f Cc (), we eventually wrte (3.3) as [ 2H H 2 2 H, H ] ϕ gdu 1 du 2 [ 2(2H 2 K) H + 2 H ] ϕ gdu 1 du 2 = 0, for any ϕ Cc (). In the end, the Euler-Lagrange equaton of functonal (1.4) s (3.1) and the theorem s proved. Obvously, equaton (3.1) s of sxth-order. It s easy to see that surfaces wth constant mean curvature, such as Delaunay surfaces (see [18], pp. 144-148) (nclude undulod and nodod), sphere, cylnder, and mnmal surfaces, are the solutons of the equaton. But tor and cone are not the soluton surfaces of the equaton. It s not dffcult to derve that Theorem 3.2 Equaton (3.1) s nvarant under the transforms of rotaton, translaton and scalng. Here the nvarant means that a soluton surface of (3.1) s stll a soluton under the three transforms mentoned. Now let us ntroduce the sxth-order flow used n ths paper. Let M 0 be a compact mmersed orentable surface n R 3. A curvature drven geometrc evoluton conssts of fndng a famly {M(t) : t 0} of smooth mmersed orentable surfaces n R 3 whch evolve accordng to the flow equaton r t = nu, M(0) = M 0. (3.6) Here r(t) s a surface pont on M(t), U denotes the normal velocty of M(t), whch depends on the curvatures H and K of M(t). Let M(t) be a closed surface wth outward normal. Then t has been shown that (see [15], Theorem 4) da(t) dt = 2 UHdA, M(t) dv (t) dt = M(t) UdA, (3.7) where A(t) denotes the area of surface M(t) and V (t) denotes the volume of the regon enclosed by M(t). If da(t) dt = 0, we say that the flow s areapreservng. Smlarly, the flow s volume-preservng dv (t) f dt = 0. Let M 0 be a compact orentable surface n R 3 wth boundary Γ. Then the sxth-order flow constructed from the Euler-Lagrange equaton (3.1) s r t = [ 2 H + 2(2H 2 K) H + 2 H, H 2H H 2] n, r M(t), M(0) = M 0, M(t) = Γ. (3.8) If M(t) s a closed constant mean curvature surface, (3.7) mples that da(t) dv (t) dt = 0 and dt = 0 for the flow (3.8). In general, ths area-preservng or volume-preservng propertes are not vald. In ths paper we name ths newly ntroduced flow as mnmal mean-curvature-varaton flow (abbrevated as MMCVF). Though the problems of the exstence and the unqueness of the solutons of the geometrc flow (3.8) are currently left open, the numercal solvng of the equaton could be conducted by ether the dvded-dfference-lke (generalzed dvded dfference) method or the fnte element approach. For smplcty, we solve t n ths paper by the dvded-dfference-lke method. 5
4 Numercal Solvng of the GPDE 4.1 Dscretzatons of curvatures and geometrc dfferental operators To solve the geometrc PDE (3.8) over a trangular surface mesh M wth vertex set {r } usng a dvdeddfference-lke method, dscrete approxmatons of the mean curvature, Gaussan curvature and varous dfferental operators are requred. In order to use a sem-mplct scheme, we requre the approxmatons of the dfferental operators mentoned above at r to have the followng form Θf(r ) = j N 1() w Θ jf(r j ), where Θ represents one of above mentoned dfferental operators and w Θ j R or wθ j R3, N k () s the ndex set of the k-rng neghbor vertces of r. Although there are several dscretzaton schemes of Laplace-Beltram operator and Gaussan curvature (see [28, 30] for a revew), the dscretzatons of Gaussan curvature are not n the requred form and may be not consstent n the followng sense. Defnton 4.1 A set of approxmatons of dfferental geometrc operators s sad consstent f there exsts a smooth surface S, such that the approxmate operators concde wth the exact counterparts of S. Here we use a bquadratc fttng of the surface data and functon data to calculate the approxmate dfferental operators. The algorthm we adopted s from [29]. Let r be a vertex of M wth valence n, r j be ts neghbor vertces for j N 1 (). Then approxmatons of the used dfferental operators are represented as (see [29] for detal) f(r ) j N 1() w j f(r j), w j R3, f(r ) j N 1() w j f(r j), w j R3, f(r ) j N 1() w j f(r j), w j R, K(r ) j N 1() (wk j )T r j, w K j R3. Usng the relaton r = 2H = 2Hn, we have H(r ) 1 2 H(r ) 1 2 j N 1() j N 1() w j r j, w j n(r ) T r j. Remark 4.1. Now we explan why we use the dscretzed dfferental operators based on the parametrc fttng rather than other methods. The frst reason s that ths fttng scheme yelds a convergent approxmaton as the mesh sze (the maxmal edge length) h 0. The second reason s that the computaton of these operators s consstent. The thrd reason s that the fttng scheme yelds the requred form expressons, whch are ready for use n the sem-mplct dscretzaton of the PDE. The last reason s that all the dfferental operators used n ths paper nvolve the frst and second order dervatves of the surface or functons on the surface. Hence, quadratc functon s enough to provde these partal dervatve data. 4.2 Sem-mplct dscretzaton of the GPDE Let us now consder the dscretzaton of (3.8). An explct scheme for solvng the equaton (3.8) n general s unstable, therefore requres a small tme stepsze. To make the evoluton process more effcent, an mplct scheme s more desrable. However, snce the used PDE s hghly nonlnear, a complete mplct scheme s hard to solve. In the followng we present a sem-mplct scheme, whch leads to a lnear system of equatons. The basc dea for formng the lnear equatons s to decompose each of the terms of (3.1) as a product of a lnear term and a remanng term. The lnear term s dscretzed usng the dscretzed dfferental operator. The remanng term s computed from prevous approxmaton of the surface. Specfcally, the terms of the equaton (3.8) are approxmated as follows: r t r(k+1) r (k), τ 2 H ( H (k+1) ), 2(2H 2 K) H (2H (k+1) H (k) +[2(H (k) ) 2 K (k) ] H (k+1), 2 H, H H (k+1), H (k) + H (k), H (k+1), 2H H, H H (k+1) +H (k) H (k+1), H (k), K (k+1) H (k), H (k) ) H (k) where the subscrpt denotes the correspondng quantty s evaluated at the vertex r, the superscrpt (k) denotes the quantty s at the tme kτ, the superscrpt (k + 1) denotes the quantty s at the tme (k + 1)τ. The quanttes at (k + 1)τ are 6
unknowns. Usng these approxmatons, we can dscretze the equaton (3.8) recursvely, and derve a lnear system wth r (k+1) as unknowns. For nstance, n ( H (k+1) ) n 1 2 Smlarly, j N 1() w j H (k+1) j wj j N 1() l N 1(j) wj j N 1() l N 1(j) j N 1() w jl(n n T l )H (k+1) l w jl(n n T l ) m N 1(l) n H (k+1), H (k) wj, H (k) n H (k+1) j 1 2 j N 1() j N 1() where n := n(r (k) w j, H (k) (n n T j )H (k+1) j wj, H (k) (n n T j ) l N 1(j) w lmr (k+1) m. w jlr (k+1) l, ) s the surface normal at r (k). Note that the dscretzed equaton at the vertex r nvolves three-rng neghbor vertces. 4.3 Boundary condton Suppose we are gven a trangular surface mesh M wth certan vertces are tagged as nteror. The nteror vertces are subject to change. The remanng vertces are fxed. Usng the above mentoned approxmaton of dfferental operators, we can dscretze recursvely the GPDE for each nteror vertex r and fnally derve a lnear equaton. Ths equaton s a lnear combnaton of the three-rng neghbor vertces of r. r (k+1) + τ j N 3() w j r (k+1) j = r (k), w j R 3 3. If an nvolved vertex r (k+1) j s not an nteror one, r (k+1) j = r j s fxed and the term τw j r (k+1) j s moved to the rght hand sde of the equaton. Such a treatment of the boundary condton leads to a system of n equatons wth n unknowns. Here n s the number of nteror vertces. The dea of ths boundary treatment s adopted from [31]. 4.4 Solvng the lnear system The result system s hghly sparse. An teratve approach for solvng the system s desrable. We employ Saad s teratve method (see [19]), named GM- RES, to solve the system. The experment shows that ths teratve method works very well. 5 Illustratve Examples 5.1 Recover property to some surfaces We have mentoned that constant mean curvature surfaces are the solutons of equaton (3.1). Fg. 5.1 and 5.2 are used to llustrate that constant mean curvature surfaces can be recovered from ther perturbed counterparts by the sxth-order geometrc flow. The test s performed as follows. We frst replace certan parts of a gven constant mean curvature surface wth another surface, and then we use our geometrc flow to evolve the surface. The frst row of Fg. 5.1 shows that a cylnder s recovered, where (a) s a cylnder wth certan parts mssng. Fgure (b) shows the mnmal surface fllng of the mssng parts. Ths mnmal surface acts as an ntal surface M 0 for the geometrc flow. (c) shows the evoluton result. It can be seen that the cylnder s correctly recovered. The second row of Fg. 5.1 shows that a sphere s recovered, where (d) s a wre-frame of a sphere wth eght openngs. These openngs are flled wth mnmal surfaces as shown n (e). These mnmal surfaces act as ntal surfaces M 0 of the geometrc flow. (f) shows the evoluton result. It s easy to see that the sphere s perfectly recovered. Fg. 5.2 shows a more complex shaped surface example, where fgure (b) shows a Delaunay surface patch wth a mssng part, whch s flled wth a surface patch as shown n (c). Ths Delaunay surface s generated by rotatng the parametrc curve { x(u) = 1 4 10 + 6 sn(4u), y(u) = u 1+3 sn(4t) dt, 0 10+6 sn(4t) as shown n (a). (d) shows evoluton result of the fllng surface. (e) shows the nput surface wth the fllng surface. It s not dffcult to observe that the Delaunay surface s recovered. It should be ponted out that the recovery property s vald only f the provded perturbed surface s nearby the soluton surface. 7
(a) (b) (c) (d) (e) (f) Fg 5.1: (a) s a cylnder wth certan parts are removed. (b) shows the mnmal surface fllng of the removed parts. (c) shows the evoluton results. (d) s a wre-frame of a sphere wth eght openngs. These openngs are flled wth mnmal surfaces as shown n (e). (f) shows the evoluton result. 1.25 1.0 0.75 0.5 0.25 0.0 0.6 0.8 1.0 (a) (b) (c) (d) (e) Fg 5.2: (b) shows a Delaunay surface wth a mssng part, whch s flled wth a surface patch as shown n (c). The Delaunay surface s the rotaton result of the curve as shown n (a). (d) shows the evoluton result of the fllng surface. (e) shows the nput surface wth the fllng surface. 5.2 Smooth blendng of surfaces Gven a collecton of surface meshes wth boundares, we construct a far surface to blend smoothly the meshes at the boundares. Fg 5.3 shows the case, where surfaces to be blended are gven (fgure (a), (d) and (g)) wth ntal mnmal surface constructons (fgure (b), (e) and (h)) usng [1] and then mean curvature flow. The surfaces (c), (f) and () are the blendng meshes generated usng our sxthorder flow. These fgures show the results after 60 teratons wth tme step szes 0.01. 5.3 N-sded hole fllng Gven a surface mesh wth holes, we construct a far surface to fll smoothly the holes wth G 2 contnuty at the boundary. Fg 1.1 shows such an example, where a head mesh wth several holes n the nose, face and jaw subregons s gven as the nput (fgure (a)). An ntal G 0 fller of the holes are shown n (b) usng [1] and then evolved wth the mean curvature flow. The blendng surface (c) s generated usng flow (3.8). 5.4 Pont nterpolaton For the pont nterpolaton problem, we are gven some ponts as the nput data, and we wsh to construct a far surface mesh to nterpolate ths multdmensonal data. Fg. 5.4 shows ths surface constructon approach, where a dodecahedron s served as nput as shown n fgure (a). The constructed surface s requred to nterpolate the vertces of the nput polygon. Each face of the nput polygon s trangulated by subdvdng the 5-sded face nto three trangles. Then each trangle s subdvded nto 64 sub-trangles. The GPDE s appled to the trangulated polygon wth the nput vertces fxed. (b) s 8
(a) (b) (c) (d) (e) (f) (g) (h) () Fg 5.3: Fgure (a), (d) and (g) show surfaces to be blended wth ntal mnmal surface constructons (fgure (b), (e) and (h)). The surfaces (c), (f) and () are the blendng meshes generated usng the sxth-order flow. the evoluton result usng the sxth-order flow. (c) shows an ntermedate result of the evoluton. 5.5 Comparson wth fourth-order flows Now we compare the used sxth-order flow MM- CVF wth three well-known lower order flows (see [31]): mean curvature flow (MCF), surface dffuson flow (SDF) and Wllmore flow (WF). From the defnton of MMCVF, we know that the man dfference of MMCVF from the lower order flows s that the former yelds G 2 and mean curvature unformly dstrbuted surfaces. Fg. 5.5 shows the evoluton results of the sxth- and fourth-order flows for the nput (d) of Fg. 5.3, where (a), (b) and (c) show the mean curvature plots of the evoluton results usng the MMCVF, SDF and WF, respectvely. From these fgures, we can observe that the surface produced by MMCVF s mean-curvature contnuous at the blendng boundares, whle the surfaces produced by SDF and WF are not. Fg. 5.6 and 5.7 show the dfferent effects of MM- CVF from the MCF and WF. Fg. 5.6 (a), (b) 9
(a) (b) (c) Fg 5.4: (a) shows the nput dodecahedron. (b) s the evoluton result of the sxth-order flow. The surface s requred to nterpolate the vertces of the nput polygon. (c) shows an ntermedate result of the evoluton process. (a) (b) (c) Fg 5.5: (a), (b) and (c) are the mean curvature plots of the evoluton results of MMCVF, SDF and WF, respectvely. (a) (b) (c) Fg 5.6: (a), (b) and (c) are the evoluton results of MCF, WF and MMCVF of an nput rng wth two fxed crcles on the rng. and (c) show the evoluton results (mean curvature plots) usng the MCF, WF and MMCVF from an nput rng wth two fxed crcles on t. It s easy to see that MMCVF produces more far surface (the rng s recovered). In Fg. 5.7, the surface to be evolved s defned as a graph of a functon g: x(u, v) = [u, v, g(u, v)] T, g(u, v) = e(u, v) + e(u + 1, v) + e(u, v + 1) + e(u + 1, v + 1), wth (u, v) := [ 1, 1] 2 and e(u, v) = exp[ 81 16 (u 0.5)2 + (v 0.5) 2 ]. Ths surface s unformly trangulated usng a 60 60 grd over the doman. We evolve a part of the surface, where g > 1.5. Fgures (a), (b) and (c) show the results of MCF, WF and MMCVF, respectvely. Fgures (d), (e) and (f) are the mean curvature plots of (a), (b) and (c), respectvely. It s easy to see that the second and the fourth-order flows are not curvature contnuous at the boundares of the evolved surface patch. 10
(a) (b) (c) (d) (e) (f) Fg 5.7: (a), (b) and (c) are the evoluton results of the MCF, WF and MMCVF. (d), (e) and (f) are the mean curvature plots of (a), (b) and (c), respectvely. 5.6 Runnng Tmes We summarze n Table 5.1 the computaton tme needed by some of our examples. The algorthm was mplemented n C++ runnng on a Dell PC wth a 3.0GHz Intel CPU. All the examples presented n ths secton are the approxmate steady soluton (t ). Hence, the total tme costs depend greatly on how far we go n the tme drecton, whch n turn depend on how far the ntal surface away from the fnal soluton. In Table 5.1, we lst the costs for a sngle teraton. The second column n Table 5.1 s the number of unknowns. These numbers are counted as 3n 0 (each vertex has x, y, z varables). Here n 0 s the number of nteror vertces. The thrd column s the used tme step-sze. The fourth column n the table s the tme (n seconds) for formng the coeffcent matrx. The ffth column s the number of evoluton steps. The last column s the tme for solvng the lnear systems for one tme step. 6 Conclusons We have derved a sxth-order nonlnear geometrc flow from the functonal F = M H 2 da. Ths flow can be used to solve several surface modelng problems, such as surface denosng, surface blendng, N-sded hole fllng and free-form surface desgn, when G 2 contnuty at the boundary s requred. The expermental results show that the dvded-dfference-lke method for solvng the nonlnear equaton s effcent and the flow yelds hgh qualty and hgh order contnuty surfaces. References [1] C. Bajaj and I. Ihm. Algebrac surface desgn wth Hermte nterpolaton. ACM Transactons on Graphcs, 11(1):61 91, 1992. [2] G. P. Bonneau, H. Hagen, and St. Hahmann. Varatonal surface desgn and surface nterrogaton. Computer Graphcs Forum, 12(3):447 459, 1993. [3] M. Botsch and L. Kobbelt. An ntutve framework for real-tme freeform modelng. ACM Transacton on Graphcs, 23(3):630 634, 2004. Proceedngs of the 2004 SIGGRAPH Conference. [4] G. Brunnett, H. Hagen, and P. Santarell. Varatonal desgn of curves and surfaces. Surv. Math. Industry, 3:1 27, 1993. [5] G. Celnker and D. Gossard. Deformable curve and surface fnte-elements for free-form shape desgn. ACM Computer Graphcs, 25(4):257 266, 1991. [6] M. P. do Carmo. Remannan Geometry. Boston, 1992. [7] H. Du and H. Qn. Dynamc PDE-based surface desgn usng geometrc and physcal constrant. Graphcal Models, 67(1):43 71, 2005. [8] J. Escher and G. Smonett. The volume preservng mean curvature flow. Proceedngs of the Amercan Mathematcal Socety, 126(9):2789 2796, 1998. 11
Table 5.1: Runnng Tmes Examples # unknowns Tme step-sze Form matrx # steps Solvng Tme Fg 5.1(c) 3480 1.0 0.12 100 2.11 Fg 5.1(f) 11160 1.0 0.42 350 6.72 Fg 5.3(c) 3222 1.0 0.12 5 1.62 Fg 5.3(f) 4410 1.0 0.17 50 2.75 Fg 5.3() 2400 0.01 0.09 10 1.43 Fg 1.1(c) 1296 0.5 0.05 10 1.26 Fg 5.7(c) 2520 2.0 0.09 5 1.50 [9] A. Gray. Modern Dfferental Geometry of Curves and Surfaces wth Mathematca. CRC Press, second edton, 1998. [10] G. Grener. Surface constructon based on varatonal prncples. In P. J. Laurent, A. LeMéhauté, and L. L. Schumaker, edtors, Wavelets, Images, and Surface Fttng, pages 277 286. Wellesley MA, 1994. AKPeters. [11] G. Grener. Varatonal desgn and farng of splne surface. Computer Graphcs Forum, 13:143 154, 1994. [12] J. Hoschek and D. Lasser. Grundlagen der geometrschen Datenverarbetung. B. G. Teubner, Stuttgart, 1992. [13] M. Kallay. Constraned optmzaton n surface desgn. In B. Falcdeno and T. L. Kun, edtors, Modelng n Computer Graphcs, pages 85 93. Sprnger-Verlag, Berln, 1993. [14] E. Kuwert and R. Schätzle. The Wllmore flow wth small ntal energy. J. Dfferental Geom., 57(3):409 441, 2001. [15] H. B. Lawson. Lectures on Mnmal Submanfolds. Publsh or Persh, Berkeley, CA, 1980. [16] U. F. Mayer. Numercal solutons for the surface dffuson flow n three space dmensons. Comput. Appl. Math., 20(3):361 379, 2001. [17] H. P. Moreton and C. H. Séqun. Functonal optmzaton for far surface desgn. SIGGRAPH 92 Conference Proceedngs, pages 167 176, 1992. [18] J. Oprea. Dfferental Geometry and Its Applcatons. Pearson Educaton, Inc., second edton, 2004. [19] Y. Saad. Iteratve Methods for Sparse Lnear Systems. Second Edton wth correctons, 2000. [20] N. Sapds. Desgnng Far Curves and Surfaces: Shape Qualty n Geometrc Modelng and Computer-Aded Desgn. Socety for Industral and Appled Mathematcs, Phladelpha, PA, USA, 1994. [21] R. Schneder and L. Kobbelt. Geometrc far meshes wth G 1 boundary condtons. In Geometrc Modelng and Processng, pages 251 261, 2000. Hongkong, Chna. [22] R. Schneder and L. Kobbelt. Geometrc farng of rregular meshes for free-form surface desgn. Computer Aded Geometrc Desgn, 18(4):359 379, 2001. [23] D. Terzopoulos, J. Platt, and A. Barr. Elastcally deformable models. ACM Computer Graphcs, 24(4):205 214, 1987. [24] W. Welch and A. Wtkn. Varatonal surface modelng. Computer Graphcs, 26:157 166, 1992. [25] W. Welch and A. Wtkn. Free-from shape desgn usng trangluated surfaces. In SIGGRAPH 94 Proceedngs, volume 28, pages 247 256, 1994. [26] T. J. Wllmore. Remannan Geometry. Clarendon Press, Oxford, England, 1993. [27] L. R. Woodwark. Blends n geometrc modellng. In R. R. Martn, edtor, The Mathematcs of Surface II, pages 255 297. Oxford Unversty Press, 1987. [28] G. Xu. Dscrete Laplace-Beltram operators and ther convergence. Computer Aded Geometrc Desgn, 21(8):767 784, 2004. [29] G. Xu. Consstent approxmaton of some geometrc dfferental operators. Research Report No. ICM-06-01, Insttute of Computatonal Mathematcs, Chnese Academy of Scences, 2006. [30] G. Xu. Convergence analyss of a dscretzaton scheme for Gaussan curvature over trangular surfaces. Computer Aded Geometrc Desgn, 23(2):193 207, 2006. [31] G. Xu, Q. Pan, and C. L. Bajaj. Dscrete surface modellng usng partal dfferental equatons. Computer Aded Geometrc Desgn, 23(2):125 145, 2006. [32] L. H. You, P. Comnnos, and J. J. Zhang. PDE blendng surfaces wth C 2 contnuty. Computers and Graphcs, 28(6):895 906, 2004. [33] J. J. Zhang and L. H. You. Fast surface modellng usng a 6th order PDE. Comput. Graph. Forum, 23(3):311 320, 2004. 12