Fourier transform of Bernstein Bézier polynomials
|
|
- Avis Beasley
- 6 years ago
- Views:
Transcription
1 Illinois Wesleyan University From the SelectedWorks of Tian-Xiao He 27 Fourier transform of Bernstein Bézier polynomials Tian-Xiao He, Illinois Wesleyan University Charles K. Chui Qingtang Jiang Available at:
2 Fourier Transform of Bernstein-Bézier Polynomials Charles. K. Chui, Tian-Xiao He 2, and Qingtang Jiang 3 Department of Mathematics and Computer Science University of Missouri-St. Louis, St. Louis, MO 632, USA and Department of Statistics, Stanford University, Stanford, CA 9435, USA 2 Department of Mathematics and Computer Science Illinois Wesleyan University, Bloomington, IL , USA 3 Department of Mathematics and Computer Science University of Missouri-St. Louis, St. Louis, MO 632, USA Abstract Explicit formulae, in terms of Bernstein-Bézier coefficients, of the Fourier transform of bivariate polynomials on a triangle and univariate polynomials on an interval are derived in this paper. Examples are given and discussed to illustrate the general theory. Finally, this consideration is related to the study of refinement masks of spline function vectors. AMS Subject Classification: 42C4, 42C5, 4A5. Key Words and Phrases: Fourier transform, Bernstein-Bézier polynomials, minimal support, quasi-minimal support, refinement Introduction The objective of this paper is to present a compact formula of the Fourier transform of bivariate polynomials on a triangle and univariate polynomials on a bounded interval in terms of their Bernstein-Bézier BB coefficients see, for example, [3, p. 58. Of course the BB coefficients are formulated, as usual, in Research supported by ARO Grant #W9NF and DARPA/NGA Grant # HM The research of this author was partially supported by ASD Grant of the Illinois Wesleyan University.
3 2 Fourier transform of BB polynomials on triangles terms of the Barycentric coordinates, as opposed to the Cartesian coordinates x = x, x 2 for x R 2 or x R. We will focus on the bivariate setting and only consider the univariate formulae as simple consequences. In this regard, although our method of derivation can be extended to multivariate polynomials on simplexes, we have decided to present the detailed derivation only for bivariate polynomials, since the motivation of this research is the study of subdivision masks [ for spline curves and surfaces. Let P n x, x R 2, x = x, x 2, be a Bernstein-Bézier or Bézier polynomial see, for example, [3, p. 58 on a triangle A A 2 A 3 with vertices A i = a i, b i, where i =, 2, 3. We write P n x in terms of the Barycentric coordinates u, v, w of A A 2 A 3 as follows. P n x P n x, x 2 = j,k,l n, j+k+l=n a j,k,l n! j!k!l! uj v k w l,. where u, v, w is the Barycentric coordinate of x = x, x 2 A A 2 A 3 i.e., x, x 2 = a u + a 2 v + a 3 w, b u + b 2 v + b 3 w, u, v, w and u + v + w =, with Bernstein-Bézier BB coefficients a j,k,l. For m, s {, 2,, 3, 2, 3}, the forward-backward and backward-forward operators, m,s and m,s respectively, are defined on the sequences {a j,k,l } j,k,l with multi-indices j, k, l by: 2 a j,k,l := a j+,k,l a j,k,l, 2 a j,k,l := a j,k,l a j,k+,l, 3 a j,k,l := a j+,k,l a j,k,l, 3 a j,k,l := a j,k,l a j,k,l+,.2 23 a j,k,l := a j,k+,l a j,k,l, 23 a j,k,l := a j,k,l a j,k,l+ ; and in addition, we set k m,s := m,s k m,s and k m,s := m,s k m,s for k =, 2,. For a triangle T = A A 2 A 3, we use V T to denote its area, given by V T := 2 det a a 2 a 3 b b 2 b 3. The main result of this paper can be stated as follows. Theorem. The Fourier transform of P n x as in. over a triangle A A 2 A 3 has the explicit formulation:
4 C. K. Chui, T. X. He, and Q. T. Jiang 3 P n ξ := P n xe iξ x dx A A 2 A 3 n n l = 2V T k+l n! n k l! γ l+.3 l= k= { α k+ k 2 l 23a n l,l, e ia ξ +b ξ 2 k 2 l 23a,n, e ia 2ξ +b 2 ξ 2 β k+ k 3 l 23a n l,,l e ia ξ +b ξ 2 k 3 l 23a,,n e ia 3ξ +b 3 ξ 2 }, where ξ = ξ, ξ 2, and α := i a 2 a ξ + b 2 b ξ 2, β := i a3 a ξ + b 3 b ξ 2, γ := i a3 a 2 ξ + b 3 b 2 ξ 2. Of course the above formulation is valid for univariate polynomials, simply by setting w =, namely, p n x = j,k n,j+k=n b j,k n! j!k! uj v k,.4 with u = x b/a b, v = x a/b a, and x [a, b. Then, by using and to denote the forward-backward and backward-forward operators b j,k := b j+,k b j,k ; b j,k := b j,k b j,k+,.5 and k := k ; k := k, for k =, 2,, we have, as an immediate consequence of Theorem., the following formulation of the Fourier transform of univariate polynomials. Corollary.2 The Fourier transform of p n x in.4 over a bounded interval [a, b has the explicit formulation: ˆp n ξ := b a p n xe ixξ dx n = b ae ibξ k n! n k! ib aξ k+ k b n, k b,n..6 k= We will present the proof of Theorem. in the next section. In Section 3, we will compute the Fourier transforms of certain minimum-supported bivariate splines to illustrate the general theory and introduce the Fourier transform approach to computing subdivision or refinement masks.
5 4 Fourier transform of BB polynomials on triangles 2 Proof of Theorem Certain properties of the hypergeometric functions, along with the notion of forward-backward and backward-forward operators defined as in.5, will be used to prove Theorem.. More precisely, the integral of the Fourier transform P n of a bivariate polynomial P n restricted to the triangle is related to the hypergeometric function F, called a confluent hypergeometric function see, for example, [, defined by F α, β ; z := + α z β! + α α + β β + z 2 2! + α α + α + 2 β β + β + 2 where α, β C with β / {,, 2, }, so that z 3 +, z C, 3! F, β ; z =, F β, β ; z = e z. We need the following two properties of F α, β ; z that are valid for all nonnegative integers α and β : i For all integers k, m >, z m F k +, m + ; z = F k +, m; z F k, m; z, z C; 2. ii For all integers k, m, u, and ρ C, u ν k u ν m e ρν dν = k!m! k + m +! F k +, k + m + 2; uρ. 2.2 The interested reader is referred to [, p. 3 and [, p. 343 for more details. To prove Theorem., we need the following result. Lemma 2. For any integer s, real numbers b m,j, and ρ, z C, s m= = b m,s m s +! zs+ F m +, s + 2; ρz s l l s l l ρ l+ l b s, s l! eρz ρ l+ l b,s s l!. 2.3 l= l=
6 C. K. Chui, T. X. He, and Q. T. Jiang 5 Proof. By applying 2., we see that Left-hand side of 2.3 = = s m= b m,s m s m= b m,s m ρ s! ρ s! F m +, s + ; ρz = b s, ρ s! F s +, s + ; ρz + ρ s b,s ρ s! F, s + ; ρz = b s, ρ s! eρz + ρ b,s s m= s b m,s m m= s m= b m,s m F m +, s + ; ρz F m, s + ; ρz s m= b m,s m b m,s m s! F m +, s + ; ρz ρ s! F m, s + ; ρz s! F m +, s + ; ρz ρ s! F m, s + ; ρz ρ s! b m+,s m ρ s! F m +, s + ; ρz m= = ρ b s, s! eρz ρ b,s s! + ρ s m= b m,s m s! F m +, s + ; ρz. Now, repeating this process, we may conclude that the left-hand side of 2.3 is given by ρ b s, s! + ρ 2 b s, s! + + l l ρ l+ l b s l,l s l! + + s ρ s+ s b,s e ρz ρ b,s s! + ρ 2 b,s s! + + l l ρ l+ l b,s s l! + + s ρ s+ s b,s s l l s l l = ρ l+ l b s, s l! eρz ρ l+ l b,s s l!, l= l= where the last equality follows from the identity l b s l,l = l b s,. Hence, we obtain 2.3. We are now ready to prove the main result of the paper.
7 6 Fourier transform of BB polynomials on triangles Proof of Theorem.. By simple calculations, we have n n j P n ξ = e iξ x n! P n xdx = 2V T a j,k,n j k A A 2 A 3 j!k!n j k! j= k= u e iξ a 3 +a a 3 u+a 2 a 3 v e iξ 2 b 3 +b b 3 u+b 2 b 3 v u j v k u v n j k dvdu = 2V T e ia 3ξ +b 3 ξ 2 = 2V T e ia 3ξ +b 3 ξ 2 k= n n j j= k= n n j j= k= a j,k,n j k n! j!k!n j k! a j,k,n j k n! j!n j +! e βu u j u n j+ F k +, n j + 2; uγdu u e βu u j e γv v k u v n j k dvdu n = 2V T n!e ia 3ξ +b 3 ξ 2 βu uj e j= j! { n j } a j,k,n j k n j +! un j+ F k +, n j + 2; uγ du, where the third equality follows from 2.2. Applying Lemma 2. to the summation inside the curly brackets of the rightmost equality with z = u, ρ = γ, and s = n j, b m,l = a j,m,l so that b m,l = 23 a j,m,l, b m,l = 23 a j,m,l, we see that P n ξ = 2V T n!e ia 3ξ +b 3 ξ 2 { n j l= n j= βu uj e j! l u n j l n j γ l+ l 23a j,n j, n j l! e uγ l= } l u n j l γ l+ l 23a j,,n j du n j l!
8 C. K. Chui, T. X. He, and Q. T. Jiang 7 = 2V T n!e ia 2ξ +b 2 ξ 2 2V T n!e ia 3ξ +b 3 ξ 2 = 2V T n!e ia 2ξ +b 2 ξ 2 2V T n!e ia 3ξ +b 3 ξ 2 = 2V T n!e ia 2ξ +b 2 ξ 2 2V T n!e ia 3ξ +b 3 ξ 2 n n j j= l= n n j j= l= n n j j= l= n n j j= l= n l= n l= l γ l+ l γ l+ l 23a j,n j, j!n j l! l γ l+ l 23a j,,n j j!n j l! u j u n j l e αu du u j u n j l e βu du l γ l+ l 23a j,n j, n l +! F j +, n l + 2; α l γ l+ l γ l+ l 23a j,,n j n l +! F j +, n l + 2; β n l j= n l j= l 23a j,n j, n l +! F j +, n l + 2; α l 23a j,,n j n l +! F j +, n l + 2; β. Finally, applying Lemma 2. to the first and second terms in the above equation with s = n l, b m,k = l 23a m,k+l,, z =, ρ = α and s = n l, b m,k = l 23a m,,k+l, z =, ρ = β, respectively, we have P n ξ = 2V T n!e ia 2ξ +b 2 ξ 2 { n l k= k n l= l γ l+ α k+ k 2 l e α n l 23a n l,l, n l k! k= 2V T n!e ia 3ξ +b 3 ξ 2 { n l k= k n l= l γ l+ β k+ k 3 l e β n l 23a n l,,l n l k! k= } k α k+ k 2 l 23a,n, n l k! } k β k+ k 3 l 23a,,n n l k! n n l = 2V T k+l n! n k l! γ l+ l= k= { α k+ k 2 l 23a n l,l, e ia ξ +b ξ 2 k 2 l 23a,n, e ia 2ξ +b 2 ξ 2 β k+ k 3 l 23a n l,,l e ia ξ +b ξ 2 k 3 l 23a,,n e ia 3ξ +b 3 ξ 2 },
9 8 Fourier transform of BB polynomials on triangles as desired. Remark. By applying Lemma 2. and following the above procedure, the main result in this paper can be extended to higher dimensions. That is, explicit formulae of the Fourier transform of Bernstein-Bézier representations of multivariate polynomials on simplexes can be derived in a similar way. 3 Application to refinable bivariate splines on triangles Refinable spline functions see, for example, [ for a precise definition are instrumental to surface subdivisions. For example, the bi-cubic B-spline is used in the Catmull-Clark scheme [2 and the three-direction box-spline B 222 is used in the Loop scheme [. See, for example, [3, pp. 5-8 for the definition of box splines and their direction sets. The simple reasons are that firstly, the refinement masks of such spline functions immediately give the so-called local averaging rules for the subdivision schemes; and secondly, the spline representations are precisely the subdivision surfaces. In the Fourier domain, the refinement mask P of a refinable function, or function vector Φ, is defined by ˆΦ = P /2ˆΦ /2 provided that the Fourier transform ˆΦ exists. While the refinement masks of the bi-cubic B-spline and the box-spline B 222, being defined by convolutions of the characteristic function of the unit square along the appropriate directions, are readily computable, computation of the refinement masks for others, such as basis functions with minimum and quasi-minimum supports, is usually very tedious. Examples of the recent development in this direction are the refinable bivariate C 2 -cubic, C 2 -quartic, and C 2 -quintic spline functions in [7, 8, 9, introduced for matrix-valued surface subdivisions to gain such desirable properties as surface geometric shape control parameters, smaller subdivision template size to better address the often unavoidable extraordinary vertices, and interpolation of the position components of the initial control vertices. Computation of the refinement masks of these bivariate spline functions requires formulating and solving large linear systems in terms of Bernstein-Bézier coefficients, see [8, 9. For this reason, the original motivation for this research was to extend the standard Fourier approach to computing the scalar-valued refinement masks of refinable spline functions to computing the matrix-valued refinement masks of refinable spline function vectors. As an application, let us consider the Fourier transform of the minimumsupported ms and quasi-minimum-supported qms bivariate spline functions in Smn n i, i = and 2, where and 2 denote the 3-direction mesh and 4-direction mesh, respectively, in R 2 with integer grid points, obtained by partitioning R 2 with the two sets of grid lines x = j, y = j, x y = j and x = j, y = j, x y = j, x + y = j, respectively, where j =...,,,.... Here, the bivariate spline spaces S n mn i, i = and 2, are the spaces of functions in C n
10 C. K. Chui, T. X. He, and Q. T. Jiang 9 whose restrictions on the triangular cells of i are polynomials of total degree mn, where mn denotes the smallest nonnegative integer for which S n mn i contains at least one nontrivial locally supported function. It is well-known that for, we have m2n = 3n and m2n = 3n + ; and for 2, we have m3n = 4n, m3n = 4n +, and m3n + = 4n + 2 see [3, 4 for details. In the following, B jkl and B jklm denote the box splines in the spaces S n mn and S n mn 2 for mn = j +k +l 2 or mn = j +k +l +m 2, respectively. Here, j, k, l and m denote the numbers of repetition of the vectors,,,,,, and,, respectively, in the direction sets that determine i. See, for example, [3, pp. 7-8 for details. Bivariate splines with local supports that are ms and qms, as studied in [3, 4, 5, are defined as follows. The support of a locally supported function f in a spline space is the closure of the set on which f does not vanish and is denoted by suppf. A set S is called a minimal support of a spline space if there is some f, called an ms spline, in the space with suppf = S, but there does not exist a nontrivial g in the space with suppg properly contained in S. A function f in a spline space is called a qms spline if i f cannot be written as a finite linear combination of ms splines in the space, and ii for any h in the space properly contained in suppf, h is some finite linear combination of ms spines in the space. For example, for the 3-direction mesh, while the spline space S has only one ms spline g, which is the box spline B with direction set {,,,,, }, the space S has two ms splines g2 and g 3 which are not box-splines, but are characteristic functions of the sets χ A and χ B, respectively, where A is the triangle bounded by the 3 lines: x =, y =, and y = x, and B the triangle bounded by the 3 lines: x =, y = and y = x. On the other hand, for the 4-direction mesh 2, the space S2 2 has only one ms spline f = B, the box spline with direction set {,,,,,,, }; but there are two ms splines f2 and f 3 in the space S 2, where f2 is the Courant hat function with support given by the diamond having vertices at,,,,,, and,, and with the value of at the center, of the support; and f3 is the other Courant hat function supported on the unit square [, 2 with the value at its center /2, /2. Furthermore, there are two ms splines f4 and f 5 and one qms spline f 6 in S2 4 2 see [3, 4, 5. In the following, let us consider the k-fold 2-dimensional convolutions: g k = g g g and f k }{{} = f f f of g }{{} and f, respectively. Now, the k+ k+ spline function vectors of interest are: [ G k g k 2 g k 3 [ := g k g 2 g3 [, F k f k 2 f3 k [ := f k f 2 f3, 3.
11 Fourier transform of BB polynomials on triangles [ g where k and f h := [ f g f h. We remark that among the functions gi k, f i k, i =, 2, 3, only gk and f k are box splines, while all of them are the unique ms and qms splines in the corresponding spline spaces where the notion of uniqueness is according to the statement of Theorem 3.2 in [3. The Fourier transforms of the initial ms splines gi, f i, i = 2, 3, can be evaluated by using the formula provided in this paper, and the Fourier transforms of the other splines are given by the corresponding products with those of g k or f k. Finally, the refinement masks of G k, F k can be easily computed by making use of 3. from the refinements of the initial G and F. Example. The Fourier transform of G k is given by Ĝk ξ, ξ 2 = [ĝk 2, ĝk 3 T ξ, ξ 2, where ĝk 2 = ĝ 2 ĝ k and ĝk 3 = ĝ 3 ĝ k. Here, we have ĝ ξ, ξ 2 = B ξ, ξ 2 = e iξ e iξ2 iξ iξ 2 e iξ+ξ2 iξ + ξ 2, and ĝ 2 ξ, ξ 2 = e iξ +ξ 2 ξ ξ + ξ 2 ĝ 3 ξ, ξ 2 = e iξ +ξ 2 ξ 2 ξ + ξ 2 e iξ 2 ξ ξ 2, 3.2 e iξ ξ ξ Next, let us compute the Fourier transform of F k. For this purpose, recall that φa k ξ = deta e iξt A k φ A T ξ, 3.4 for any invertible matrix A of dimension s. Example 2. Let f2, f 3 be the two Courant hat functions in S 2 as introduced previously. To compute the Fourier transform of f2, we use the x-y axes to partition its support into four triangles:, 2, 3, and 4 in the the first, second, third, and fourth quadrants, respectively. Then f2 can be written as the sum of four functions: φ, φ 2, φ 3, and φ 4, with supports given by, 2, 3, and 4, respectively. By the formula.3, the Fourier transform of φ is given by φ ξ, ξ 2 = ξ ξ 2 + i e iξ ξ 2 ξ ξ 2 + i e iξ2 ξ 2 2 ξ 2 ξ. 3.5
12 C. K. Chui, T. X. He, and Q. T. Jiang Since φ 2 x, y = φ x, y, it follows from 3.5 and 3.4 that φ 2 ξ, ξ 2 = φ ξ, ξ 2 = e iξ i ξ ξ 2 ξ 2ξ + ξ 2 + i e iξ2 ξ2 2ξ + ξ 2. Similarly, since φ 3 x, y = φ x, y and φ 4 x, y = φ x, y, we have Consequently, we arrive at f 2 ξ, ξ 2 = φ 3 ξ, ξ 2 = e iξ + i ξ ξ 2 ξ 2ξ 2 ξ + i e iξ2 ξ2 2ξ ξ 2 ; φ 4 ξ, ξ 2 = + i e iξ ξ ξ 2 ξ 2ξ + ξ 2 i e iξ2 ξ2 2ξ + ξ 2. 4 e φ iξ 2 e iξ 2 j ξ, ξ 2 = 2i ξ 2 ξ + ξ 2 ξ 2 ξ + 2i e iξ e iξ ξ ξ + ξ 2 ξ ξ 2. j= [ x Next, observe that the linear transformation B y B = [, [, with 3.6 maps the suppf3 to suppf 2, with the vertices,,,,,, and, of suppf3 corresponding to the vertices,,,,,, and, of suppf2, respectively. Hence, we may write [ [ f3 x, y = f2 x, B y and apply 3.4 to obtain f 3 ξ, ξ 2 = 2 e iξ 2 B[,T f 2 2 Bξ = 2 e i ξ +ξ 2 = 2i e iξ 2 e iξ ξ ξ 2 ξ 2 ξ + e iξ+ξ2 ξ + ξ 2 2 f 2 ξ + ξ 2 2, ξ ξ Finally, to compute the refinement masks of G k and F k in 3., we observe that the convolution of two finitely refinable functions i.e., refinable functions whose refinement masks consist of finitely many terms remains to be finitely refinable, and that G k is the convolution of G with the refinable box spline g k, and F k the convolution of F with the refinable box spline f k. Hence, we only
13 2 Fourier transform of BB polynomials on triangles need to compute the refinement masks of the initial ms splines. Precisely, from ĝ2 and ĝ 3 given by 3.2 and 3.3, we have Ĝ 2ξ, 2ξ 2 = Q ξ, ξ 2 Ĝ ξ, ξ 2 where [ + z2 + z Q ξ, ξ 2 = z 2 z 2 z + z + z z 2, z = e iξ, z 2 = e iξ 2 ; and from f 2 and f 3 given by 3.6 and 3.7, we have F 2ξ, 2ξ 2 = R ξ, ξ 2 F ξ, ξ 2 where R ξ, ξ 2 = 4 [ + 2 z + z 2 z + z z + z 2 z z z + z 2, z = e iξ, z 2 = e iξ 2. The interested reader is referred to [6 for computing the refinement mask for F x by calculating the BB coefficients of F B x directly. Therefore, we conclude, from the definitions in 3., that with Ĝ k 2ξ, 2ξ 2 = Q k ξ, ξ 2 Ĝk ξ, ξ 2, F k 2ξ, 2ξ 2 = R k ξ, ξ 2 F k ξ, ξ 2 Q k ξ, ξ 2 = qξ, ξ 2 k Q ξ, ξ 2, R k ξ, ξ 2 = rξ, ξ 2 k R ξ, ξ where qξ, ξ 2 = 8 + e iξ + e iξ 2 + e iξ +ξ 2 is the mask or two-scale symbol of g, and rξ, ξ 2 = 6 +e iξ +e iξ 2 +e iξ +ξ 2 +e iξ ξ 2 the mask or two-scale symbol of f. That is, we have the following result. Theorem 3. The vector-valued functions G k and F k are finitely refinable with refinement masks given by 3.8. Similarly, f k is also finitely refinable, with refinement mask given by rξ, ξ 2 k S ξ, ξ 2, f 4 where S is the refinement mask of f5. Computation of S as well as the f 6 refinement masks of other initial ms and qms bivariate splines in general is usually nontrivial and requires further investigation. f 4 f 5 f 6
14 C. K. Chui, T. X. He, and Q. T. Jiang 3 References [ A. S. Cavaretta, W. Dahmen, and C. A. Micchelli, Stationary Subdivision, Mem. Amer. Math. Soc. # 453, Amer. Math. Soc., Providence, RI, 99. [2 E. Catmull and J. Clark, Recursively generated B-splines surfaces on arbitrary topological meshes, Comput. Aided Design 978, [3 C. K. Chui, Multivariate Splines, NSF-CBMS Series #54, SIAM Publ., Philadelphia, 988. [4 C. K. Chui and T. X. He, Computation of minimally and quasi-minimally supported bivariate splines, J. Comput. Math. 8 99, 8 7. [5 C. K. Chui and T. X. He, On minimal and quasi-minimal supported bivariate splines, J. Approx. Theory , [6 C. K. Chui and Q. T. Jiang, Multivariate balanced vector-valued refinable functions, In: Modern Development in Multivariate Approximation, ISNM 45, V.W. Haussmann, K. Jetter, M. Reimer, and J. Stöckler eds., Birhhäuser Verlag, Basel, 23, pp [7 C. K. Chui and Q. T. Jiang, Surface subdivision schemes generated by refinable bivariate spline function vectors, Appl. Comput. Harmon. Anal. 5 23, [8 C. K. Chui and Q. T. Jiang, Refinable bivariate C 2 -splines for multi-level data representation and surface display, Math. Comp , [9 C. K. Chui and Q. T. Jiang, Refinable bivariate quartic and quintic C 2 - splines for quadrilateral subdivisions, J. Comp. Appl. Math., to appear. Available: jiang [ I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th edition, Academic Press, New York, 2. [ C. Loop, Smooth subdivision surfaces based on triangles, Master s Thesis, University of Utah, Dept. of Math., Salt Lake City, 987.
Fourier Transform of Bernstein-Bézier Polynomials
Fourier Transform of Bernstein-Bézier Polynomials Charles. K. Chui, Tian-Xiao He 2, and Qingtang Jiang 3 Department of Mathematics and Computer Science University of Missouri-St. Louis, St. Louis, MO 632,
More informationRefinable bivariate quartic and quintic C 2 -splines for quadrilateral subdivisions
Refinable bivariate quartic and quintic C 2 -splines for quadrilateral subdivisions Charles K. Chui, Qingtang Jiang Department of Mathematics and Computer Science University of Missouri St. Louis St. Louis,
More informationMatrix-valued 4-point Spline and 3-point Non-spline Interpolatory Curve Subdivision Schemes
Matrix-valued 4-point Spline and -point Non-spline Interpolatory Curve Subdivision Schemes Charles K. Chui, Qingtang Jiang Department of Mathematics and Computer Science University of Missouri St. Louis
More informationEdge Detections Using Box Spline Tight Frames
Edge Detections Using Box Spline Tight Frames Ming-Jun Lai 1) Abstract. We present a piece of numerical evidence that box spline tight frames are very useful for edge detection of images. Comparsion with
More informationA C 2 Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extensions
A C 2 Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extensions Nira Dyn Michael S. Floater Kai Hormann Abstract. We present a new four-point subdivision scheme that generates C 2 curves.
More informationMatrix-valued Subdivision Schemes for Generating Surfaces with Extraordinary Vertices
Matrix-valued Subdivision Schemes for Generating Surfaces with Extraordinary Vertices Charles K. Chui, Qingtang Jiang Department of Mathematics & Computer Science University of Missouri St. Louis St. Louis,
More informationA C 2 Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extensions
A C 2 Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extensions Nira Dyn School of Mathematical Sciences Tel Aviv University Michael S. Floater Department of Informatics University of
More informationC 1 Quintic Spline Interpolation Over Tetrahedral Partitions
C 1 Quintic Spline Interpolation Over Tetrahedral Partitions Gerard Awanou and Ming-Jun Lai Abstract. We discuss the implementation of a C 1 quintic superspline method for interpolating scattered data
More informationNormals of subdivision surfaces and their control polyhedra
Computer Aided Geometric Design 24 (27 112 116 www.elsevier.com/locate/cagd Normals of subdivision surfaces and their control polyhedra I. Ginkel a,j.peters b,,g.umlauf a a University of Kaiserslautern,
More informationBernstein-Bezier Splines on the Unit Sphere. Victoria Baramidze. Department of Mathematics. Western Illinois University
Bernstein-Bezier Splines on the Unit Sphere Victoria Baramidze Department of Mathematics Western Illinois University ABSTRACT I will introduce scattered data fitting problems on the sphere and discuss
More informationDepartment of Mathematics and Computer Science University of Missouri St. Louis St. Louis, MO 63121
Triangular 7 and Quadrilateral 5 Subdivision Schemes: Regular Case Charles K. Chui 1, Qingtang Jiang 2 3, Rokhaya N. Ndao Department of Mathematics and Computer Science University of Missouri St. Louis
More informationFrom Extension of Loop s Approximation Scheme to Interpolatory Subdivisions
From Extension of Loop s Approximation Scheme to Interpolatory Subdivisions Charles K. Chui, Qingtang Jiang 2 Department of Mathematics and Computer Science University of Missouri St. Louis St. Louis,
More informationOn the Dimension of the Bivariate Spline Space S 1 3( )
On the Dimension of the Bivariate Spline Space S 1 3( ) Gašper Jaklič Institute of Mathematics, Physics and Mechanics University of Ljubljana Jadranska 19, 1000 Ljubljana, Slovenia Gasper.Jaklic@fmf.uni-lj.si
More informationA NEW PROOF OF THE SMOOTHNESS OF 4-POINT DESLAURIERS-DUBUC SCHEME
J. Appl. Math. & Computing Vol. 18(2005), No. 1-2, pp. 553-562 A NEW PROOF OF THE SMOOTHNESS OF 4-POINT DESLAURIERS-DUBUC SCHEME YOUCHUN TANG, KWAN PYO KO* AND BYUNG-GOOK LEE Abstract. It is well-known
More informationTangents and curvatures of matrix-valued subdivision curves and their applications to curve design
Tangents and curvatures of matrix-valued subdivision curves and their applications to curve design Qingtang Jiang, James J. Smith Department of Mathematics and Computer Science University of Missouri St.
More informationNormals of subdivision surfaces and their control polyhedra
Normals of subdivision surfaces and their control polyhedra I. Ginkel, a, J. Peters b, and G. Umlauf a, a University of Kaiserslautern, Germany b University of Florida, Gainesville, FL, USA Abstract For
More informationOptimal Quasi-Interpolation by Quadratic C 1 -Splines on Type-2 Triangulations
Optimal Quasi-Interpolation by Quadratic C 1 -Splines on Type-2 Triangulations Tatyana Sorokina and Frank Zeilfelder Abstract. We describe a new scheme based on quadratic C 1 -splines on type-2 triangulations
More informationA new 8-node quadrilateral spline finite element
Journal of Computational and Applied Mathematics 195 (2006) 54 65 www.elsevier.com/locate/cam A new 8-node quadrilateral spline finite element Chong-Jun Li, Ren-Hong Wang Institute of Mathematical Sciences,
More informationThe goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a
The goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a coordinate system and then the measuring of the point with
More informationOn the graphical display of Powell-Sabin splines: a comparison of three piecewise linear approximations
On the graphical display of Powell-Sabin splines: a comparison of three piecewise linear approximations Hendrik Speleers Paul Dierckx Stefan Vandewalle Report TW515, January 008 Ò Katholieke Universiteit
More informationNon-Uniform Recursive Doo-Sabin Surfaces (NURDSes)
Non-Uniform Recursive Doo-Sabin Surfaces Zhangjin Huang 1 Guoping Wang 2 1 University of Science and Technology of China 2 Peking University, China SIAM Conference on Geometric and Physical Modeling Doo-Sabin
More informationDesigning Subdivision Surfaces through Control Mesh Refinement
Designing Subdivision Surfaces through Control Mesh Refinement by Haroon S. Sheikh A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of
More informationQuad/triangle Subdivision, Nonhomogeneous Refinement Equation and Polynomial Reproduction
Quad/triangle Subdivision, Nonhomogeneous Refinement Equation and Polynomial Reproduction Qingtang Jiang a,, Baobin Li b, a Department of Mathematics and Computer Science, University of Missouri St. Louis,
More informationGeneralizing the C 4 Four-directional Box Spline to Surfaces of Arbitrary Topology Luiz Velho Abstract. In this paper we introduce a new scheme that g
Generalizing the C 4 Four-directional Box Spline to Surfaces of Arbitrary Topology Luiz Velho Abstract. In this paper we introduce a new scheme that generalizes the four-directional box spline of class
More informationComputation of interpolatory splines via triadic subdivision
Computation of interpolatory splines via triadic subdivision Valery A. Zheludev and Amir Z. Averbuch Abstract We present an algorithm for computation of interpolatory splines of arbitrary order at triadic
More informationInterpolation by Spline Functions
Interpolation by Spline Functions Com S 477/577 Sep 0 007 High-degree polynomials tend to have large oscillations which are not the characteristics of the original data. To yield smooth interpolating curves
More informationElement Quality Metrics for Higher-Order Bernstein Bézier Elements
Element Quality Metrics for Higher-Order Bernstein Bézier Elements Luke Engvall and John A. Evans Abstract In this note, we review the interpolation theory for curvilinear finite elements originally derived
More informationHighly Symmetric Bi-frames for Triangle Surface Multiresolution Processing
Highly Symmetric Bi-frames for Triangle Surface Multiresolution Processing Qingtang Jiang and Dale K. Pounds Abstract In this paper we investigate the construction of dyadic affine (wavelet) bi-frames
More informationEvaluation of Loop Subdivision Surfaces
Evaluation of Loop Subdivision Surfaces Jos Stam Alias wavefront, Inc. 8 Third Ave, 8th Floor, Seattle, WA 980, U.S.A. jstam@aw.sgi.com Abstract This paper describes a technique to evaluate Loop subdivision
More informationInterpolation and Splines
Interpolation and Splines Anna Gryboś October 23, 27 1 Problem setting Many of physical phenomenona are described by the functions that we don t know exactly. Often we can calculate or measure the values
More informationSubdivision of Curves and Surfaces: An Overview
Subdivision of Curves and Surfaces: An Overview Ben Herbst, Karin M Hunter, Emile Rossouw Applied Mathematics, Department of Mathematical Sciences, University of Stellenbosch, Private Bag X1, Matieland,
More informationGeneralised Mean Averaging Interpolation by Discrete Cubic Splines
Publ. RIMS, Kyoto Univ. 30 (1994), 89-95 Generalised Mean Averaging Interpolation by Discrete Cubic Splines By Manjulata SHRIVASTAVA* Abstract The aim of this work is to introduce for a discrete function,
More informationLecture 15: The subspace topology, Closed sets
Lecture 15: The subspace topology, Closed sets 1 The Subspace Topology Definition 1.1. Let (X, T) be a topological space with topology T. subset of X, the collection If Y is a T Y = {Y U U T} is a topology
More informationDeficient Quartic Spline Interpolation
International Journal of Computational Science and Mathematics. ISSN 0974-3189 Volume 3, Number 2 (2011), pp. 227-236 International Research Publication House http://www.irphouse.com Deficient Quartic
More informationOn Smooth Bicubic Surfaces from Quad Meshes
On Smooth Bicubic Surfaces from Quad Meshes Jianhua Fan and Jörg Peters Dept CISE, University of Florida Abstract. Determining the least m such that one m m bi-cubic macropatch per quadrilateral offers
More informationSung-Eui Yoon ( 윤성의 )
CS480: Computer Graphics Curves and Surfaces Sung-Eui Yoon ( 윤성의 ) Course URL: http://jupiter.kaist.ac.kr/~sungeui/cg Today s Topics Surface representations Smooth curves Subdivision 2 Smooth Curves and
More informationConvergence of C 2 Deficient Quartic Spline Interpolation
Advances in Computational Sciences and Technology ISSN 0973-6107 Volume 10, Number 4 (2017) pp. 519-527 Research India Publications http://www.ripublication.com Convergence of C 2 Deficient Quartic Spline
More informationAn interpolating 4-point C 2 ternary stationary subdivision scheme
Computer Aided Geometric Design 9 (2002) 8 www.elsevier.com/locate/comaid An interpolating 4-point C 2 ternary stationary subdivision scheme M.F Hassan a,, I.P. Ivrissimitzis a, N.A. Dodgson a,m.a.sabin
More informationG 2 Interpolation for Polar Surfaces
1 G 2 Interpolation for Polar Surfaces Jianzhong Wang 1, Fuhua Cheng 2,3 1 University of Kentucky, jwangf@uky.edu 2 University of Kentucky, cheng@cs.uky.edu 3 National Tsinhua University ABSTRACT In this
More informationFinite Element Methods
Chapter 5 Finite Element Methods 5.1 Finite Element Spaces Remark 5.1 Mesh cells, faces, edges, vertices. A mesh cell is a compact polyhedron in R d, d {2,3}, whose interior is not empty. The boundary
More informationExample: Loop Scheme. Example: Loop Scheme. What makes a good scheme? recursive application leads to a smooth surface.
Example: Loop Scheme What makes a good scheme? recursive application leads to a smooth surface 200, Denis Zorin Example: Loop Scheme Refinement rule 200, Denis Zorin Example: Loop Scheme Two geometric
More informationKNOTTED SYMMETRIC GRAPHS
proceedings of the american mathematical society Volume 123, Number 3, March 1995 KNOTTED SYMMETRIC GRAPHS CHARLES LIVINGSTON (Communicated by Ronald Stern) Abstract. For a knotted graph in S* we define
More informationSPERNER S LEMMA MOOR XU
SPERNER S LEMMA MOOR XU Abstract. Is it possible to dissect a square into an odd number of triangles of equal area? This question was first answered by Paul Monsky in 970, and the solution requires elements
More informationarxiv: v1 [math.na] 20 Sep 2016
arxiv:1609.06236v1 [math.na] 20 Sep 2016 A Local Mesh Modification Strategy for Interface Problems with Application to Shape and Topology Optimization P. Gangl 1,2 and U. Langer 3 1 Doctoral Program Comp.
More informationUsing Semi-Regular 4 8 Meshes for Subdivision Surfaces
Using Semi-Regular 8 Meshes for Subdivision Surfaces Luiz Velho IMPA Instituto de Matemática Pura e Aplicada Abstract. Semi-regular 8 meshes are refinable triangulated quadrangulations. They provide a
More informationAdaptive and Smooth Surface Construction by Triangular A-Patches
Adaptive and Smooth Surface Construction by Triangular A-Patches Guoliang Xu Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, China Abstract
More informationLacunary Interpolation Using Quartic B-Spline
General Letters in Mathematic, Vol. 2, No. 3, June 2017, pp. 129-137 e-issn 2519-9277, p-issn 2519-9269 Available online at http:\\ www.refaad.com Lacunary Interpolation Using Quartic B-Spline 1 Karwan
More informationIsoparametric Curve of Quadratic F-Bézier Curve
J. of the Chosun Natural Science Vol. 6, No. 1 (2013) pp. 46 52 Isoparametric Curve of Quadratic F-Bézier Curve Hae Yeon Park 1 and Young Joon Ahn 2, Abstract In this thesis, we consider isoparametric
More informationFall CSCI 420: Computer Graphics. 4.2 Splines. Hao Li.
Fall 2014 CSCI 420: Computer Graphics 4.2 Splines Hao Li http://cs420.hao-li.com 1 Roller coaster Next programming assignment involves creating a 3D roller coaster animation We must model the 3D curve
More informationDiscrete Cubic Interpolatory Splines
Publ RIMS, Kyoto Univ. 28 (1992), 825-832 Discrete Cubic Interpolatory Splines By Manjulata SHRIVASTAVA* Abstract In the present paper, existence, uniqueness and convergence properties of a discrete cubic
More informationNatural Quartic Spline
Natural Quartic Spline Rafael E Banchs INTRODUCTION This report describes the natural quartic spline algorithm developed for the enhanced solution of the Time Harmonic Field Electric Logging problem As
More information2D Spline Curves. CS 4620 Lecture 18
2D Spline Curves CS 4620 Lecture 18 2014 Steve Marschner 1 Motivation: smoothness In many applications we need smooth shapes that is, without discontinuities So far we can make things with corners (lines,
More informationSurfaces for CAGD. FSP Tutorial. FSP-Seminar, Graz, November
Surfaces for CAGD FSP Tutorial FSP-Seminar, Graz, November 2005 1 Tensor Product Surfaces Given: two curve schemes (Bézier curves or B splines): I: x(u) = m i=0 F i(u)b i, u [a, b], II: x(v) = n j=0 G
More informationHarmonic Spline Series Representation of Scaling Functions
Harmonic Spline Series Representation of Scaling Functions Thierry Blu and Michael Unser Biomedical Imaging Group, STI/BIO-E, BM 4.34 Swiss Federal Institute of Technology, Lausanne CH-5 Lausanne-EPFL,
More informationOn the packing chromatic number of some lattices
On the packing chromatic number of some lattices Arthur S. Finbow Department of Mathematics and Computing Science Saint Mary s University Halifax, Canada BH C art.finbow@stmarys.ca Douglas F. Rall Department
More informationQuadratic and cubic b-splines by generalizing higher-order voronoi diagrams
Quadratic and cubic b-splines by generalizing higher-order voronoi diagrams Yuanxin Liu and Jack Snoeyink Joshua Levine April 18, 2007 Computer Science and Engineering, The Ohio State University 1 / 24
More informationBS-Patch: Constrained Bezier Parametric Patch
BS-Patch: Constrained Bezier Parametric Patch VACLAV SKALA, VIT ONDRACKA Department of Computer Science and Engineering University of West Bohemia Univerzitni 8, CZ 06 14 Plzen CZECH REPUBLIC skala@kiv.zcu.cz
More informationSubdivision based Interpolation with Shape Control
Subdivision based Interpolation with Shape Control Fengtao Fan University of Kentucky Deparment of Computer Science Lexington, KY 40506, USA ffan2@uky.edu Fuhua (Frank) Cheng University of Kentucky Deparment
More information751 Problem Set I JWR. Due Sep 28, 2004
751 Problem Set I JWR Due Sep 28, 2004 Exercise 1. For any space X define an equivalence relation by x y iff here is a path γ : I X with γ(0) = x and γ(1) = y. The equivalence classes are called the path
More informationOn the deviation of a parametric cubic spline interpolant from its data polygon
Computer Aided Geometric Design 25 (2008) 148 156 wwwelseviercom/locate/cagd On the deviation of a parametric cubic spline interpolant from its data polygon Michael S Floater Department of Computer Science,
More informationAPPROXIMATION BY FUZZY B-SPLINE SERIES
J. Appl. Math. & Computing Vol. 20(2006), No. 1-2, pp. 157-169 Website: http://jamc.net APPROXIMATION BY FUZZY B-SPLINE SERIES PETRU BLAGA AND BARNABÁS BEDE Abstract. We study properties concerning approximation
More informationu 0+u 2 new boundary vertex
Combined Subdivision Schemes for the design of surfaces satisfying boundary conditions Adi Levin School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel. Email:fadilev@math.tau.ac.ilg
More informationLecture 25: Bezier Subdivision. And he took unto him all these, and divided them in the midst, and laid each piece one against another: Genesis 15:10
Lecture 25: Bezier Subdivision And he took unto him all these, and divided them in the midst, and laid each piece one against another: Genesis 15:10 1. Divide and Conquer If we are going to build useful
More informationVertex Colorings without Rainbow or Monochromatic Subgraphs. 1 Introduction
Vertex Colorings without Rainbow or Monochromatic Subgraphs Wayne Goddard and Honghai Xu Dept of Mathematical Sciences, Clemson University Clemson SC 29634 {goddard,honghax}@clemson.edu Abstract. This
More informationspline structure and become polynomials on cells without collinear edges. Results of this kind follow from the intrinsic supersmoothness of bivariate
Supersmoothness of bivariate splines and geometry of the underlying partition. T. Sorokina ) Abstract. We show that many spaces of bivariate splines possess additional smoothness (supersmoothness) that
More informationOn the Number of Tilings of a Square by Rectangles
University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange University of Tennessee Honors Thesis Projects University of Tennessee Honors Program 5-2012 On the Number of Tilings
More informationDimensions of Spline Spaces over 3D Hierarchical T-Meshes
Journal of Information & Computational Science 3: 3 (2006) 487 501 Available at http://www.joics.com Dimensions of Spline Spaces over 3D Hierarchical T-Meshes Xin Li, Jiansong Deng, Falai Chen Department
More informationCS354 Computer Graphics Surface Representation IV. Qixing Huang March 7th 2018
CS354 Computer Graphics Surface Representation IV Qixing Huang March 7th 2018 Today s Topic Subdivision surfaces Implicit surface representation Subdivision Surfaces Building complex models We can extend
More informationKeywords: Non-stationary Subdivision, Radial Basis Function, Gaussian, Asymptotical Equivalence, 1. Introduction
ANALYSIS OF UNIVARIATE NON-STATIONARY SUBDIVISION SCHEMES WITH APPLICATION TO GAUSSIAN-BASED INTERPOLATORY SCHEMES NIRA DYN, DAVID LEVIN, AND JUNGHO YOON Abstract: This paper is concerned with non-stationary
More informationTopic: Orientation, Surfaces, and Euler characteristic
Topic: Orientation, Surfaces, and Euler characteristic The material in these notes is motivated by Chapter 2 of Cromwell. A source I used for smooth manifolds is do Carmo s Riemannian Geometry. Ideas of
More information08 - Designing Approximating Curves
08 - Designing Approximating Curves Acknowledgement: Olga Sorkine-Hornung, Alexander Sorkine-Hornung, Ilya Baran Last time Interpolating curves Monomials Lagrange Hermite Different control types Polynomials
More informationNeed for Parametric Equations
Curves and Surfaces Curves and Surfaces Need for Parametric Equations Affine Combinations Bernstein Polynomials Bezier Curves and Surfaces Continuity when joining curves B Spline Curves and Surfaces Need
More informationParameterization of triangular meshes
Parameterization of triangular meshes Michael S. Floater November 10, 2009 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to
More informationPS Geometric Modeling Homework Assignment Sheet I (Due 20-Oct-2017)
Homework Assignment Sheet I (Due 20-Oct-2017) Assignment 1 Let n N and A be a finite set of cardinality n = A. By definition, a permutation of A is a bijective function from A to A. Prove that there exist
More informationIntro to Curves Week 1, Lecture 2
CS 536 Computer Graphics Intro to Curves Week 1, Lecture 2 David Breen, William Regli and Maxim Peysakhov Department of Computer Science Drexel University Outline Math review Introduction to 2D curves
More informationCOMPACTLY SUPPORTED TIGHT AFFINE SPLINE FRAMES IN L 2 (R d ) AMOS RON AND ZUOWEI SHEN
MATHEMATICS OF COMPUTATION Volume 67, Number 1, January 1998, Pages 191 07 S 005-5718(98)00898-9 COMPACTLY SUPPORTED TIGHT AFFINE SPLINE FRAMES IN L (R d ) AMOS RON AND ZUOWEI SHEN Abstract. The theory
More informationApplications of Wavelets and Framelets
Applications of Wavelets and Framelets Bin Han Department of Mathematical and Statistical Sciences University of Alberta, Edmonton, Canada Present at 2017 International Undergraduate Summer Enrichment
More information= f (a, b) + (hf x + kf y ) (a,b) +
Chapter 14 Multiple Integrals 1 Double Integrals, Iterated Integrals, Cross-sections 2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals
More informationCurves and Surfaces for Computer-Aided Geometric Design
Curves and Surfaces for Computer-Aided Geometric Design A Practical Guide Fourth Edition Gerald Farin Department of Computer Science Arizona State University Tempe, Arizona /ACADEMIC PRESS I San Diego
More informationThe B-spline Wavelet Recurrence Relation and B- spline Wavelet Interpolation
Illinois Wesleyan University Digital Commons @ IWU Honors Projects Mathematics 1996 The B-spline Wavelet Recurrence Relation and B- spline Wavelet Interpolation Patrick J. Crowley '96 Illinois Wesleyan
More informationSubdivision Curves and Surfaces: An Introduction
Subdivision Curves and Surfaces: An Introduction Corner Cutting De Casteljau s and de Boor s algorithms all use corner-cutting procedures. Corner cutting can be local or non-local. A cut is local if it
More informationCS 536 Computer Graphics Intro to Curves Week 1, Lecture 2
CS 536 Computer Graphics Intro to Curves Week 1, Lecture 2 David Breen, William Regli and Maxim Peysakhov Department of Computer Science Drexel University 1 Outline Math review Introduction to 2D curves
More information2 Solution of Homework
Math 3181 Name: Dr. Franz Rothe February 6, 2014 All3181\3181_spr14h2.tex Homework has to be turned in this handout. The homework can be done in groups up to three due February 11/12 2 Solution of Homework
More informationOn Planar Intersection Graphs with Forbidden Subgraphs
On Planar Intersection Graphs with Forbidden Subgraphs János Pach Micha Sharir June 13, 2006 Abstract Let C be a family of n compact connected sets in the plane, whose intersection graph G(C) has no complete
More informationCommutative filters for LES on unstructured meshes
Center for Turbulence Research Annual Research Briefs 1999 389 Commutative filters for LES on unstructured meshes By Alison L. Marsden AND Oleg V. Vasilyev 1 Motivation and objectives Application of large
More informationAdaptive osculatory rational interpolation for image processing
Journal of Computational and Applied Mathematics 195 (2006) 46 53 www.elsevier.com/locate/cam Adaptive osculatory rational interpolation for image processing Min Hu a, Jieqing Tan b, a College of Computer
More informationFreeform Curves on Spheres of Arbitrary Dimension
Freeform Curves on Spheres of Arbitrary Dimension Scott Schaefer and Ron Goldman Rice University 6100 Main St. Houston, TX 77005 sschaefe@rice.edu and rng@rice.edu Abstract Recursive evaluation procedures
More informationInterpolatory 3-Subdivision
EUROGRAPHICS 2000 / M. Gross and F.R.A. Hopgood (Guest Editors) Volume 19 (2000), Number 3 Interpolatory 3-Subdivision U. Labsik G. Greiner Computer Graphics Group University of Erlangen-Nuremberg Am Weichselgarten
More informationModeling. Simulating the Everyday World
Modeling Simulating the Everyday World Three broad areas: Modeling (Geometric) = Shape Animation = Motion/Behavior Rendering = Appearance Page 1 Geometric Modeling 1. How to represent 3d shapes Polygonal
More informationSubdivision on Arbitrary Meshes: Algorithms and Theory
Subdivision on Arbitrary Meshes: Algorithms and Theory Denis Zorin New York University 719 Broadway, 12th floor, New York, USA E-mail: dzorin@mrl.nyu.edu Subdivision surfaces have become a standard geometric
More informationParallel and perspective projections such as used in representing 3d images.
Chapter 5 Rotations and projections In this chapter we discuss Rotations Parallel and perspective projections such as used in representing 3d images. Using coordinates and matrices, parallel projections
More informationDiscrete Coons patches
Computer Aided Geometric Design 16 (1999) 691 700 Discrete Coons patches Gerald Farin a,, Dianne Hansford b,1 a Computer Science and Engineering, Arizona State University, Tempe, AZ 85287-5406, USA b NURBS
More informationLecture notes for Topology MMA100
Lecture notes for Topology MMA100 J A S, S-11 1 Simplicial Complexes 1.1 Affine independence A collection of points v 0, v 1,..., v n in some Euclidean space R N are affinely independent if the (affine
More informationG 2 Bezier Crust on Quad Subdivision Surfaces
Pacific Graphics (2013) B. Levy, X. Tong, and K. Yin (Editors) Short Papers G 2 Bezier Crust on Quad Subdivision Surfaces paper 1348 Figure 1: Two examples of Bezier crust applied on Catmull-Clark subdivision
More informationTopological Invariance under Line Graph Transformations
Symmetry 2012, 4, 329-335; doi:103390/sym4020329 Article OPEN ACCESS symmetry ISSN 2073-8994 wwwmdpicom/journal/symmetry Topological Invariance under Line Graph Transformations Allen D Parks Electromagnetic
More informationOn the Linear Independence of Truncated Hierarchical Generating Systems
On the Linear Independence of Truncated Hierarchical Generating Systems Urška Zore a, Bert Jüttler a,, Jiří Kosinka b a Institute of Applied Geometry, Johannes Kepler University, Linz, Austria b Johann
More informationApproximation of 3D-Parametric Functions by Bicubic B-spline Functions
International Journal of Mathematical Modelling & Computations Vol. 02, No. 03, 2012, 211-220 Approximation of 3D-Parametric Functions by Bicubic B-spline Functions M. Amirfakhrian a, a Department of Mathematics,
More informationarxiv: v1 [math.gt] 28 Feb 2009
Coverings and Minimal Triangulations of 3 Manifolds William Jaco, Hyam Rubinstein and Stephan Tillmann arxiv:0903.0112v1 [math.gt] 28 Feb 2009 Abstract This paper uses results on the classification of
More informationarxiv: v2 [math.na] 26 Jan 2015
Smooth Bézier Surfaces over Unstructured Quadrilateral Meshes arxiv:1412.1125v2 [math.na] 26 Jan 2015 Michel Bercovier 1 and Tanya Matskewich 2 1 The Rachel and Selim Benin School of Computer Science and
More informationOptimization Problems Under One-sided (max, min)-linear Equality Constraints
WDS'12 Proceedings of Contributed Papers, Part I, 13 19, 2012. ISBN 978-80-7378-224-5 MATFYZPRESS Optimization Problems Under One-sided (max, min)-linear Equality Constraints M. Gad Charles University,
More information