Computers and Mathematics with Applications. Discrete schemes for Gaussian curvature and their convergence
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1 Computers and Mathematcs wth Applcatons 57 (009) Contents lsts avalable at ScenceDrect Computers and Mathematcs wth Applcatons journal homepage: Dscrete schemes for Gaussan curvature and ther convergence Zhqang Xu, Guolang Xu LSEC, Insttute of Computatonal Math. and Sc. and Eng. Computng, Academy of Mathematcs and System Scences, Chnese Academy of Scences, Bejng, Chna a r t c l e n f o abstract Artcle hstory: Receved Aprl 008 Receved n revsed form 6 December 008 Accepted 9 January 009 Keywords: Dscrete surfaces Gaussan curvature Dscrete curvature Geometrc modellng Angular defect schemes The popular angular defect schemes for Gaussan curvature only converge at the regular vertex wth valence 6. In ths paper, we present a new dscrete scheme for Gaussan curvature, whch converges at the regular vertex wth valence greater than 4. We show that t s mpossble to buld a dscrete scheme for Gaussan curvature whch converges at the regular vertex wth valence 4 by a counterexample. We also study the convergence property of other dscrete schemes for Gaussan curvature and compare ther asymptotc errors by numercal experments. 009 Elsever Ltd. All rghts reserved.. Introducton Estmaton of ntrnsc geometrc nvarants s mportant n a number of applcatons such as n computer vson, computer graphcs, geometrc modellng and computer aded desgn. It s well known that Gaussan curvature s one of the most essental geometrc nvarants for surfaces. However, n the classcal dfferental geometry, ths nvarant s well defned only for C smooth surfaces. In the feld of modern-computer-related geometry, one often uses C 0 contnuous dscrete trangular meshes to represent smooth surfaces approxmately. Hence, estmaton of accurately Gaussan curvature for trangular meshes s demanded strongly. In the past years, a wealth of dfferent methods for estmatng Gaussan curvature have been proposed n the vast lterature of appled geometry. These methods can be dvded nto two classes. The frst class s for computng Gaussan curvature based on the local fttng or nterpolaton technque [ 5], whle the second class s for gvng dscretzaton formulas whch represent the nformaton about the Gaussan curvature [6 9]. In ths paper, our focus s on the methods n the second class and our man am s to present a new dscretzaton scheme for Gaussan curvature whch has better convergence property than the prevous dscretzaton schemes. Let M be a trangulaton of the smooth surface S n R 3. For a vertex p of M, suppose that {p } n = s the set of the one-rng neghbor vertexes of p. The set {p pp + }( =,..., n) of n Eucldean trangles forms a pecewse lnear approxmaton of S around p. Throughout the paper, we use the followng conventons p n+ = p and p 0 = p n. Let γ denote the angle p pp + and let the angular defect at p be π γ. A popular dscretzaton scheme for computng Gaussan curvature s n the form of (π γ )/E, where E s a geometrc quantty. In general, one takes E as A(p)/3 and obtans the followng approxmaton 3(π γ ) G () :=, A(p) () Zhqang Xu s supported by the NSFC grant Guolang Xu s supported by NSFC grant and Natonal Key Basc Research Project of Chna (004CB38000). Correspondng author. E-mal address: xuzq@lsec.cc.ac.cn (Z. Xu) /$ see front matter 009 Elsever Ltd. All rghts reserved. do:0.06/j.camwa
2 88 Z. Xu, G. Xu / Computers and Mathematcs wth Applcatons 57 (009) where A(p) s the sum of the areas of trangles p pp +. In [6], another scheme π G () := S p γ () s gven, where S p := [ η η + cos γ ] 4 sn γ (η + η ) + s called the module of the mesh at p. In [0,], the dscretzaton approxmaton G () s modfed as G (3) := π γ, (3) area(p pp + ) cot(γ 8 )d where d s the length of edge p p +. There are dfferent vewponts for explanng the reason why the angular defect closely relates to the Gaussan curvature, ncludng the vewponts of the Gaussan Bonnet theorem, Gaussan map and Legendre s formula (see the next secton n detals). Asymptotc analyss for the dscretzaton schemes have been gven n [4,6,9]. In [4], the authors show that the dscretzaton scheme G () s not always convergent to the true Gaussan curvature for the non-unform data. In [6], the authors prove that the angular defect s asymptotcally equvalent to a homogeneous polynomal of degree two n the prncpal curvatures and show that the scheme G () converges to the exact Gaussan curvature n a lnear rate provded p s a regular vertex wth valence sx. Moreover, n [6], the authors show that 4 s the only value of the valence such that the angular defect depends upon the prncpal drectons. In [9], Xu proves that the dscretzaton scheme G () has a quadratc convergence rate f the mesh satsfes the so-called parallelogram crteron, whch requres valence 6. Therefore, one hopes to construct a dscretzaton scheme whch converges over any dscrete mesh. But n [], Xu et al. show that t s mpossble to construct a dscrete scheme whch s convergent over any dscrete mesh. Hence, we have to be content wth the dscretzaton schemes whch converge under some condtons. Accordng to past experences [6,,3], we regard a dscretzaton scheme as desrable f t has the followng propertes: () It converges at regular vertexes, at least for suffcently large valence (the defnton of the regular vertex wll be gven n Secton ); () It converges at umblcal ponts,.e., the ponts satsfyng k m = k M where k m and k M are two prncpal curvatures. As stated before, the prevous dscretzaton schemes, ncludng G (), G () and G (3), only converge at the regular vertex wth valence 6. In [6], a method for computng the Gaussan curvature at the regular vertex wth valence unequal to 4 s descrbed. But the method requres two meshes wth valences n and n (n = 4, n = 4, n = n ). In ths paper, we wll construct a dscretzaton scheme whch converges at the regular vertex wth valence not less than 5, and also at umblcal ponts wth any valence. Moreover, the dscretzaton scheme requres only a sngle mesh. Hence, the new scheme s more desrable. Furthermore, we show that t s mpossble to construct a dscretzaton scheme whch s convergent at the regular vertex wth valence 4. Therefore, the convergence problem remans open for the regular vertexes wth valence 3. Here, t should be noted that the pontwse convergence dscussed n ths paper s dfferent from the convergence n norm as dscussed n [4,5]. The rest of the paper s organzed as follows. Secton descrbes some notatons and defntons and Secton 3 shows three vewponts for expressng the relaton between the angular defect and Gaussan curvature. In Secton 4, we study the convergence property of a modfed dscretzaton Gaussan curvature scheme. We present n Secton 5 a new dscretzaton scheme and prove that the scheme has a good convergence property, whch s the central result of the paper. In Secton 6, for the regular vertex wth valence 4, we show that t s mpossble to buld a dscretzaton scheme whch s convergent to the real Gaussan curvature. Some numercal results are gven n Secton 7.. Prelmnares In ths secton, we ntroduce some notatons and defntons used throughout the paper (see also Fg. ). Let S be a gven smooth surface and p be a pont over S. Suppose that the set {p pp + }, =,..., n, of n Eucldean trangles forms a pecewse lnear approxmaton of S around p. The vector from p to p s denoted as pp. The normal vector and tangent plane of S at the pont p are denoted by n and Π, respectvely. We denote the projecton of p onto Π as q, and defne the plane contanng n, p and p as Π. Then we let κ denote the curvature of the plane curve S Π at p. The dstances from p to p and q are denoted as η and l, respectvely. Let γ and β denote p pp + and q pq +. The two prncpal curvatures at p are denoted as k m and k M. Let η = max η. The followng results are presented n [6,9,3]: l η = + O(η), β = γ + O(η ), (4)
3 Z. Xu, G. Xu / Computers and Mathematcs wth Applcatons 57 (009) Fg.. Notatons. w pp = w κ η + O(η 3 ), (5) where w R. Now we gve the defnton of the regular vertex usng the notatons ntroduced above. Smlar defntons also appear n [6,3]. Defnton. Let p be a pont of a smooth surface S and let p, =,..., n be ts one-rng neghbor. The pont p s called a regular vertex f t satsfes the followng condtons: () the β = π n, () the η s all take the same value η. Remark. We can replace () n Defnton by requrng that the γ s all take the same value. Snce β = γ + O(η ), all the results n ths paper hold also for the alternatve defnton. 3. The angular defect and Gaussan curvature In ths secton, we summarze three dfferent vewponts for expressng the relaton between the angular defect and Gaussan curvature. These vewponts have been descrbed n the lterature [4,9]. We collect them here. 3.. Gaussan Bonnet theorem vewpont Let D be a regon of the surface S, whose boundary conssts of pecewse smooth curves Γ j s. Then the local Gaussan Bonnet theorem s as follows G(p)dA + k g (Γ j )ds + α j = π, D j Γ j j where G(p) s the Gaussan curvature at p, k g (Γ j ) s the geodesc curvature of the boundary curve Γ j and α j s the exteror angle at the jth corner pont p j of the boundary. If all the Γ j s are the geodesc curves, the above formula reduces to G(p)dA = π α j. (6) D j Let M be a trangulaton of the surface S. For the vertex p, each trangle p pp + can be parttoned nto three equal parts, one correspondng to each of ts vertexes. We let D be the unon of the part correspondng to p of trangles p pp +. Note that γ = j α j. Assumng G(p) s a constant on D, and usng (6), we can see that G(p) can be approxmated by G () (p), where G () (p) s the dscrete Gaussan curvature obtaned usng G () at p. 3.. Sphercal mage vewpont We now ntroduce another defnton of Gaussan curvature. Let D be a small patch of area A ncludng pont p on the surface S. There wll be a correspondng patch of area I on the Gaussan map. Gaussan curvature at p s the lmt of rato lm A 0 I A.
4 90 Z. Xu, G. Xu / Computers and Mathematcs wth Applcatons 57 (009) Let us consder a dscrete verson of the defnton. The Gaussan map mage,.e. the sphercal mage, of the trangle p pp + (p p s the pont ) (p p + ). Jon these ponts by a great crcle formng a sphercal polygon on the unt sphere. The area of (p p ) (p p + ) ths sphercal polygon s π γ. Smlarly to the above, each trangle s parttoned nto three parts, one correspondng to each vertex. Then the Gaussan curvature can be approxmated by G () (p) Geodesc trangle vewpont Let T = ABC be a geodesc trangle on the surface S wth angles α, β, γ and geodesc edge lengths a, b, c. Let A B C be a correspondng Eucldean trangle wth angles α,β,γ and edge lengths a, b, c. Legendre presents the followng formula α α = area(t) G(A) + o(a + b + c ), 3 where area(t) s the area of the geodesc trangle ABC and G(A) s the Gaussan curvature at A. Usng Legendre s formula for each trangles wth p as a vertex, we arrve at the estmatng formula G () (p) agan. 4. Convergence of angular defect schemes In [9], Xu gves an analyss about the scheme G () and proves that the scheme converges at the vertexes satsfyng the so-called parallelogram crteron. In [6], the authors gve an elegant analyss about the angular defect and they show that the angular defct s asymptotcally equvalent to a homogeneous polynomal of degree two n the prncpal curvatures wth closed form coeffcents f the vertex p s regular. Moreover, they present another angular scheme G () := π γ. In fact, S p usng the law of cosne, we have area(p pp + ) cot(γ )d = [ 8 4 η η + sn γ ] cos γ (η + η 8 sn γ η + η + cos γ ) = [ η η + cos γ ] 4 sn γ (η + η ) + = S p. Ths shows that G () and G (3) are equvalent, whch means that these two schemes output the same value for the same trangular mesh. In [9], the author proves that the dscretzaton scheme G () has quadratc convergence rate under the parallelogram crteron. In the followng theorem, we shall show that the dscretzaton scheme G (3) has also quadratc convergence rate under the same crteron. Theorem. Let p be a vertex of M wth valence sx, and let p j, j =,..., 6 be ts neghbor vertexes. Suppose that p and p j, j =,..., 6 are on a suffcently smooth parametrc surface F(ξ,ξ ) R 3, and there exst u and u j R such that Then p = F(u), p j = F(u j ) and u j u = (u j u) + (u j+ u), j =,..., 6. π γ A(p, r) cot(γ 8 (r))d (r) = G(p) + O(r ), where, G(p) s the real Gaussan curvature of F(u) at p, A(p, r) := area[p (r)pp + (r)], p (r) := F(u (r)), and u (r) = u + r(u u), =,..., 6. Proof. Let A(p, r) = a 0 r + a r 3 + O(r 4 ) and A(p, r) cot(γ (r))d 8 (r) = b 0r + b r 3 + O(r 4 ) be the Taylor expansons wth respect to r. Accordng to Theorem 4. n [9], 3(π γ ) = G(p) + O(r ). A(p, r) (7)
5 Z. Xu, G. Xu / Computers and Mathematcs wth Applcatons 57 (009) Hence, to prove the theorem, we need to show that b 0 = a 0 /3 and b = a /3. Accordng to [9], we have a = 0, whch mples that we only need to prove b 0 = a 0 /3 and b = 0. Note that the u and u j, j =,... 6, satsfy the parallelogram crteron. Wthout loss of generalty, we may assume u =[0, 0] T and u =[, 0] T. Then there exst a constant a > 0 and an angle θ such that u =[a cos θ, a sn θ] T. Hence, u 3 =[a cos θ, a sn θ] T and u j+3 = u j, j =,, 3. Let u j = s j d j = s j [g j, l j ] T, j =,..., 6, where s j = u j and d j =. Then, we have s =, s = a, s 3 = a ac +, s 4 = s, s 5 = s, s 6 = s 3, g =, g = c, g 3 = (ac )/s 3, g 4 = g, g 5 = g, g 6 = g 3, l = 0, l = t, l 3 = at/s 3, l 4 = l, l 5 = l, l 6 = l 3, where (c, t) := (cos θ, sn θ). Note that A(p, r) = cot(γ j (r)) = 6 j= p j (r) p p j+ (r) p p j (r) p, p j+ (r) p, (8) p j (r) p, p j+ (r) p p j (r) p p j+ (r) p p j (r) p, p j+ (r) p, (9) and d (r) j = p j(r) p + p j+ (r) p p j (r) p, p j+ (r) p. (0) Let F k d j denote the kth order drectonal dervatve of F n the drecton d j. Then usng the Taylor expanson wth respect to r, we have that p j (r) p j = s j r F dj, F dj + s 3 j r 3 F dj, F d j + 4 s4 j r 4 F d j, F d j and + 3 s4 j r 4 F dj, F 3 d j + 6 s5 j r 5 F d j, F 3 d j + s5 j r 5 F dj, F 4 d j + O(r 6 ), () p j (r) p, p j+ (r) p = s j s j+ r F dj, F dj+ + s js j+ r 3 F dj, F d j+ + s j s j+r 3 F dj+, F d j + 4 s j s j+ r 4 F d j+, F d j + 6 s js 3 j+ r 4 F dj, F 3 d j+ + 6 s3 j s j+r 4 F dj+, F dj+ + s j s3 j+ r 5 F d j, F 3 d j+ + s j+ s3 j r 5 F d j, F 3 d j+ + 4 s4 j+ s jr 5 F dj, F 4 d j+ + 4 s j+s 4 j r 5 F 4 d j, F dj+ + O(r 6 ). () To compute all the nner products n the two equatons above, we let t = F(ξ,ξ ) ξ, t j = F(ξ,ξ ) ξ ξ j, t jk = 3 F ξ ξ j ξ k, t jkl = for, j, k, l =,, and let 4 F ξ ξ j ξ k ξ l g j = t T t j, g jk = t T t jk, e jkl = t T t jkl, e jklm = t T t jklm, f jklm = t T j t klm. Snce F k d j can be wrtten as a lnear combnaton of t, t j, t jk and t jkl, all the nner products n () and () can be expressed as a lnear combnaton of g j, g jk, g jkl, e jkl, e jklm and f jklm. Substtutng () and () nto (8) (0), and then substtutng (8) (0) nto the expresson A(p, r) cot(γ 8 (r))d (r), and usng Maple to conduct all the symbolc calculatons, we have b 0 = a 0 /3 = a t (g g g ), b = 0. The theorem follows.
6 9 Z. Xu, G. Xu / Computers and Mathematcs wth Applcatons 57 (009) Remark. The calculaton of b 0 and b nvolves a huge number of terms. It s almost mpossble to fnsh the dervaton by hand. Maple completes all the computaton n 6 s on a PC equpped wth a 3.0 GHZ Intel(R) CPU. The Maple code that conducts all dervatons of the theorem s avalable at The nterested readers are encouraged to perform the computaton. Remark 3. It should be ponted out that there s another dscretzaton scheme π γ G (4) :=, A M (p) where A M (p) s the area of the Vorono regon. Snce area(p pp +) could be approxmated by 3A M (p) under some condtons, for example the condtons of Theorem, G (4) s easly derved from G (). 5. A new dscretzaton scheme of the Gaussan curvature and ts convergence In ths secton, we ntroduce a new dscretzaton scheme for Gaussan curvature whch converges at umblcal ponts and regular vertexes wth valence greater than 4. Ths s the man result of the paper. Before ntroducng the new dscretzaton, we dscuss some propertes about the dscrete mean curvature. Settng α = p p p and δ = p p + p, we let (cot α + cot δ ) pp H () :=, (3) (cot α + cot δ )η whch s a popular dscrete scheme for the mean curvature at vertex p (c.f. [6]). Moreover, the real mean curvature and the real Gaussan curvature at p are denoted as H and G respectvely. Then, we have: Lemma. Suppose that p s a regular vertex or a umblcal pont. The dscrete scheme H () converges lnearly to the mean curvature H as η = η 0. Proof. Frstly, let us consder the regular vertex. Snce p s a regular vertex, one has cot α +cot δ cot α j +cot δ j = + O(η ), for the dfferent and j. By (5), we have (cot α + cot δ ) pp = (cot α + cot δ )η k + O(η 3 ). Hence, one has H () = (cot α + cot δ )η κ (cot α j + cot δ j )η + O(η) = κ + O(η) = H + O(η). (4) n j j Secondly, we consder the umblcal ponts. Accordng to the defnton of umblcal ponts, one has k = k j = H for any and j. Hence, (cot α + cot δ ) pp H () := (cot α + cot δ )η (cot α + cot δ )η k + O(η 3 ) = = H + O(η). (5) (cot α + cot δ )η Combnng (4) and (5), the theorem follows. Now, we turn to a new dscrete scheme for Gaussan curvature. Set ϕ := γ j= j and π γ (S p A)(H () ) G (5) :=, A S p where A := ( η η + 4 sn γ S p := [ 4 sn(γ ) ( cos ϕ cos ϕ + ) cos γ ) 4 (η sn ϕ + η + sn ϕ + ), η η + cos(γ ] ) (η + η + ).
7 Z. Xu, G. Xu / Computers and Mathematcs wth Applcatons 57 (009) Then, we have: Theorem. Suppose that p s a regular vertex wth valence greater than 4 or a umblcal pont. The dscretzaton scheme G (5) converges towards the Gaussan curvature G as η 0 at p. Proof. We frstly consder the case where p s a regular vertex. Set θ(n) := π. Snce p s a regular vertex, we have n γ = θ(n) + O(η ) for any accordng to (4). After a bref calculaton, we have A = A + O(η 4 ) and S p = S + p O(η4 ), where A = 6 sn(θ(n)) [n n cos(θ(n)) n cos(θ(n))]η, S = n p 4 sn(θ(n)) [ cos(θ(n))]η. Hence, we have ( π γ (S p A)(H () ) ) /(A S p ) = ( π γ (S p A )(H () ) ) /(A S p ) + O(η ). Note that η max η mn = + O(η). Accordng to Theorem 3 n [6], we have π γ = A G + B (k M + k m ) + o(η ), where, B = Note that 6 sn(θ(n)) [n + n cos(θ(n)) 3n cos(θ(n))]η. A G + B (k M + k m ) = A G + B [(k M + k m ) k M k m ] Hence, we have = A G + 4B H B G = (A B )G + 4B H. π γ = (A B )G + 4B H + o(η ). (6) Note that A = O(η ), B = O(η ) and A B = 0 provded n = 3. Combnng (6), Lemma and S p = A + B, one has G = = π γ 4B H + o() A B π γ (S p A )(H () ) A S + o() = G (5) + o() P provded n 5. Therefore, G (5) converges to the Gaussan curvature. Now, let us turn to the umblcal pont. For umblcal ponts, each drecton s the prncpal drecton. Accordng to Lemma 4 n [6], we have π γ = (AG + (S p A)k m ) + o(η ) at umblcal ponts. Snce k m = H = G, we have π γ = (AG + (S p A)k m ) + o(η ) = (AG + (S p A)H (S p A)G) + o(η ). From the equaton above, we arrve at π γ (S p A)(H () ) G = + o() = G (5) + o(). A S p The theorem follows.
8 94 Z. Xu, G. Xu / Computers and Mathematcs wth Applcatons 57 (009) Fg.. A sequence of the regular vertexes wth valence n = 4 for the functon f (x, y) = x + xy + y. At these regular vertexes, t s mpossble to construct a dscrete Gaussan curvature scheme whch converges to the correct value. Remark 4. Theorem shows that the new scheme G (5) converges at regular vertexes wth valence greater than 4. And hence the new scheme has better convergence propertes than the avalable scheme. Remark 5. In [3], the authors also prove that the dscrete scheme H () converges to the real mean curvature at regular vertexes. However, the defnton of the regular vertex n [3] s dfferent from our defnton. Remark 6. Accordng to the conclusons above, the Gaussan curvature and the mean curvature can be approxmated at regular vertexes wth valence greater than 4. Hence, usng the formulas k m = H H G and k M = H + H G, one can approxmate the prncpal curvatures at regular vertexes wth valence greater than A counterexample for the regular vertex wth valence 4 In [], we have constructed a trangular mesh and shown that t s mpossble to construct a dscrete Gaussan curvature scheme whch converges for the mesh. But the vertex n the mesh s not regular. In ths secton, we shall show that t s also mpossble to buld a dscrete Gaussan curvature scheme whch converges at regular vertexes wth valence 4. Suppose that the xy plane s trangulated around (0, 0) by choosng 4 ponts q = (r, 0), q = (0, r ), q 3 = ( r, 0) and q 4 = (0, r ). For a bvarate functon f (x, y), the graph of f (x, y),.e. F(x, y) =[x, y, f (x, y)] T, can be regarded as a parametrc surface. Let p 0 = F(0, 0) and p = F(q ), =,, 3, 4. The set of trangles p p 0 p + forms a trangular mesh approxmaton of F at p 0 and we denote the trangular mesh as M f. When f (x, y) s n the form of x + cxy + y where c R, t s easy to prove that p 0 := (0, 0, 0) T s a regular vertex wth valence 4. Moreover, we can see that p = (r, 0, r )T, p = (0, r, r )T, p 3 = ( r, 0, r )T and p 4 = (0, r, r )T. Now we show that t s mpossble to construct a dscretzaton scheme for Gaussan curvature whch converges over the vertex p 0 (see Fg. ). Suppose for the purpose of contradcton that there s a dscrete scheme for Gaussan curvature, whch s denoted as G(M f, p 0 ; p, p, p 3, p 4 ) and s convergent at the regular vertex p wth valence 4. It s easy to calculate that the Gaussan curvature of F(x, y, z) at p 0 s 4 c. Accordng to the assumpton, we have lm r 0 G(M f, p 0 ; p, p, p 3, p 4 ) = 4 c. Note that the trangular mesh M f s ndependent of c,.e. for any functon f (x, y) whch s n the form of x + cxy + y, the trangular mesh M f s the same. Hence, lm r 0 G(M f, p 0 ; p, p, p 3, p 4 ) s ndependent of c. A contradcton occurs. The assumpton of G(M f, p 0 ; p, p, p 3, p 4 ) beng convergent at the regular vertex p wth valence 4 does not hold. Remark 7. The counterexample n ths secton justfes the concluson n [6], whch says that 4 s the only value of valence such that π γ depends upon the prncpal drectons. Remark 8. An open problem s to fnd a dscretzaton scheme for Gaussan curvature whch converges at the regular vertex wth valence Numercal experments The am of ths secton s to exhbt the numercal behavors of the dscrete schemes mentoned above. For a vector a = (a 0, a, a 0 ) R 3, we defne a bvarate functon f a (x, y) := a 0 x + a xy + a 0 y, and regard the graph of the functon f a (x, y) as a parametrc surface F a (x, y) =[x, y, f a (x, y)] T R 3. The Gaussan curvature of F a (x, y) at the orgn s 4a 0 a 0 a. The doman around (0, 0) s trangulated locally by choosng n ponts: q k = l k (cos θ k, sn θ k ), θ k = (k )π/n, k =,..., n. Let p k = F a (q k ) and p 0 = (0, 0, 0) T. Hence, the set of trangles {p k p 0 p k+ } forms a pecewse lnear approxmaton of F a around p 0. We set e k := f a (cos θ k, sn θ k ) and select + 4e k (l + k l4 k e ) k l k =, k (7) e k so that p 0 s a regular vertex.
9 Z. Xu, G. Xu / Computers and Mathematcs wth Applcatons 57 (009) Table The asymptotc maxmal error ε () (n). n ε () (n) ε () (n) ε (4) (n) ε (5) (n) e e e e e e e e + 0η 6.6e + 0η.903e + 0η.903e + 0η.488e + 0η e e e e 0η e e e e + 0η Table The asymptotc error ε () over a sphere. N η ε () ε () ε (4) ε (5) e 0.905e 0.905e 0.6e e e e 0.9e e 0.80e 0.80e 0.730e e e e e e e e e 03 We let G () (p 0 : F a ) denote the approxmated Gaussan curvatures of F a at p 0, whch are obtaned by usng the dscretzaton scheme G (). Suppose that A s a set consstng of M randomly chosen vectors a. Then, we let ε () (n) = G () (p 0 : F a ) (4a 0 a 0 a ) /M. a A Snce p s a regular vertex, each edge has the same length η. Table shows the asymptotc maxmal error ε () (n) when M = 0 4. The convergence property and the convergence rate are checked by takng l = /8, /6, /3,... (when k, l k can be obtaned by (7)). From Table, we can see that all methods work well on valence 6 but only new method works well for valence 5. We compute the Gaussan curvature over a randomly trangulated unt sphere by the dscretzaton schemes to test ther convergent property at umblcal ponts. The vertexes of the random trangulaton are unform dstrbuton on the sphere. Denote the vertexes n the random trangulaton as p, =,..., N where N s the number of the vertexes n the random trangulaton. We let G (j) (p ) denote the approxmate Gaussan curvature at the vertex p whch s obtaned by the dscrete scheme G (j). Smlarly to the above, we use ε (j) = N = (G(j) (p ) ) /N to measure the error of dscretzaton scheme G (j) and use η to denote the average length of the edges. Table lsts ε (j) for dfferent N. From these numercal results, we can draw the followng conclusons: For the regular vertexes wth the valence greater than 4, or the umblcal ponts, the new dscretzaton scheme G (5) converges to the real Gaussan curvature, whch agrees wth the theoretcal result. References [] F. Cazals, M. Pouget, Estmatng dfferental quanttes usng polynomal fttng of osculatng jets, Computer Aded Geometrc Desgn (005) [] I. Douros, B.F. Buxton, Three-dmensonal surface curvature estmaton usng quadrc surface patches, n: Scannng 00 Proceedngs, Pars, 00. [3] R. Martn, Estmaton of prncpal curvatures from range data, Internatonal Journal of Shape Modelng 4 (998) 99. [4] D.S. Meek, D.J. Walton, On surface normal and Gaussan curvature approxmatons gven data sampled from a smooth surface, Computer Aded Geometrc Desgn 7 (000) [5] Guolang Xu, Dscrete Laplace Betram operators and ther convergence, Computer Aded Geometrc Desgn (004) [6] V. Borrell, F. Cazals, J.M. Morvan, On the angular defect of trangulatons and the pontwse approxmaton of curvatures, Computer Aded Geometrc Desgn 0 (003) [7] C.R. Calladne, Gaussan curvature and shell structures, n: J.A. Gregory (Ed.), The Mathematcs of Surfaces, Clarendon Press, Oxford, 986. [8] M. Meyer, M. Desbrun, P. Schröder, A.H. Barr, Dscrete dfferental-geometry operator for trangulated -manfolds, n: Proc. VsMath 0, Berln, Germany, 00. [9] Guolang Xu, Convergence analyss of a dscretzaton scheme for Gaussan curvature over trangular surfaces, Computer Aded Geometrc Desgn 3 (006) [0] E. Box, Approxmaton lnéare des surfaces de R 3 et applcatons, Ph.D. Thess, Ecole polytechnque, 995. [] V. Borrell, Courbures dscrètes, Master Thess, Claude Bernard Unversty, Lyon, 993. [] Zhqang Xu, Guolang Xu, Jaguang Sun, Convergence analyss of dscrete dfferental geometry operators over surfaces, IMA Conference on the Mathematcs of Surfaces, 005, pp [3] T. Langer, A.G. Belyaev, H.P. Sedel, Analyss and desgn of dscrete normals and curvatures, Techncal Report, Max-Planck-Insttut für Informatk, 005. [4] D. Cohen-Stener, J.M. Morvan, Restrcted Delaunay trangulatons and normal cycle, n: Proceedngs of the nneteenth annual symposum on Computatonal geometry, 003. [5] K. Hldebrandt, K. Polther, M. Wardetzky, On the convergence of metrc and geometrc propertes of polyhedral surfaces, Geometrae Dedcata 3 (006) 89. [6] U. Pnkall, K. Polther, Computng dscrete mnmal surfaces and ther conjugates, Expermental Mathematcs (993) 5 36.
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