Linear Precision for Parametric Patches
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1 Department of Mathematics Texas A&M University March 30, 2007 / Texas A&M University
2 Algebraic Geometry and Geometric modeling Geometric modeling uses polynomials to build computer models for industrial design and manufacture. Algebraic geometry investigates the algebraic and geometric properties of polynomials.
3 Bézier curves and surfaces Bézier curves and surfaces are the fundamental units for geometric modeling of curves and surfaces F (x) = P 0 (1 x) 3 +3P 1 x(1 x) 2 +3P 2 x 2 (1 x)+p 3 x 3, x [0, 1]
4 Bézier surfaces Bézier surfaces come in two basic shapes triangular Bézier patches and rectangular tensor product patches.
5 Bézier surfaces
6 Parametric representation of Bézier patches Bézier curves of degree m m i=0 Bm i (t)p m, Bi m (t) = ( ) m i t i (1 t) m i rectangular Bézier surfaces m n i=0 j=0 Bm i (u)b n j (v)p ij triangular Bézier surfaces i+j+k=n n! i!j!k! ui v j w k p ijk rational Bézier patches m n i=0 j=0 Bm i (u)b n j (v)w ij p ij m n i=0 j=0 Bm i (u)b n j (v)w ij
7 ... are a vast but well-controlled generalization of Bézier patches. are based on the geometry of toric varieties. depend on a polytope and some weights.
8 Let R 2 be a lattice polygon. Edges of define lines h i (t) = n i, t + a i = 0, with inward oriented normal primitive lattice vectors n i. Let ˆ = Z 2 be the set of lattice points of. Note h i (m) is a non-negative integer for all m ˆ. A toric patch associated to is a rational patch with domain and basis functions h h 1(m) 1 h h 2(m) 2 h hr (m) r.
9 Toric surface patches
10 Linear precision Bézier patches have linear precision. It underlies numerical stability of Bézier patches. Linear precision is the ability of a patch to replicate linear functions. Rimvydas Krasauskas Which toric Bézier patches have linear precision?
11 Parametric patch Let A be a finite set of points in R 2. A control point scheme for parametric patches A patch is a collection β = {β a a A} of non-negative functions, called blending functions. Partition of unity β a (x) = 1 a A The common domain of the blending functions is the convex hull of A.
12 Parametric representation of a patch Given a set {P a a A} R 3 of control points, define a smooth map F : R 3 by F (x) = a A β a (x)p a. F(x) = P 0 (1 x) 3 +3P 1 x(1 x) 2 +3P 2 x 2 (1 x)+p 3 x 3, x [0, 1]
13 Linear precision Parametric map F(x) = a A β a (x)p a. Tautological map τ(x) := a A β a (x)a. Definition A patch has linear precision if and only if its tautological map is the identity map on.
14 Bézier cubic in R 3 A = {0, 1 3, 2 3, 1} [0, 1]. Control points P 0, P 1, P 2, P 3 R 3 F (x) = P 0 (1 x) 3 + 3P 1 x(1 x) 2 + 3P 2 x 2 (1 x) + P 3 x 3 Linear precision τ(x) = x(1 x) 2 + 2x 2 (1 x) + x 3 = x
15 RP A πp β F (x) = a A β a(x)p a replacements F Map given by β = {β a a A} β : RP A, β : x [β a (x) a A] Linear projection given by P = {P a R 3 a A} π P : RP A π P RP 3, y = [y a a A] a A y a(1, P a )
16 Linear precision The parametric map is the composition β X β = β( ) RP A π P RP 3. F ( ) is the image of X β under the projection π P. The tautological map is the composition β X β RP A π A RP 2. Geometric criterion The patch has linear precision if this composition is the identity.
17 Main result Theorem If a patch β = {β a a A} has linear precision, then 1 Y β = X β is a rational variety, 2 Y β meets the center E A of the tautological projection in a maximally degenerate manner. In algebro-geometric terms, linear precision is a pathological situation This result gives a very strong and precise tool to study linear precision
18 Results Introduction Theorem Bézier simploids (higher-dimensional generalization of Bézier curves and surfaces) are the only toric patches based on a product of standard simplices which have linear precision. Theorem (Ranestad, Sottile) Triangular Bézier patches and rectangular tensor product patches are the unique toric surface patches having linear precision. There are no n-sided toric surface patches having linear precision for n > 4.
arxiv: v1 [math.ag] 14 Jun 2007
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