09/25/02 Dinesh Manocha, COMP258 Triangular Bezier Patches Natural generalization to Bezier curves Triangles are a simplex: Any polygon can be decomposed.

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Presentation transcript:

09/25/02 Dinesh Manocha, COMP258 Triangular Bezier Patches Natural generalization to Bezier curves Triangles are a simplex: Any polygon can be decomposed into triangles Formulation based on Barycentric coordinates and linear interpolation

09/25/02 Dinesh Manocha, COMP258 Barycentric Coordinates Given a triangle with vertices A, B, C and a fourth point P, P can be expressed as a barycentric combination of A, B, and C: P = u A + v B + w C, and u + v + w = 1 The coefficients (u,v,w) are called barycentric coordinates of P with respect to A, B, C Given A,B,C and P, the barycentric coordinates can be computed as:

09/25/02 Dinesh Manocha, COMP258 Barycentric Coordinates Barycentric coordinates are affinely invariant, i.e. an affine map or tranformations preserves the barycentric coordinates If a point is outside the triangle one of the Barycentric coordinate may be negative For all points inside the triangle, the Barycentric coordinates are non-negative

09/25/02 Dinesh Manocha, COMP258 de Casteljau Algorithm for Triangular Patches Given a triangular patch of degree n with control points ( b i = b ijk ), where i = ijk and |i| = i + j + k; e1 = (1,0,0); e2 = (0,1,0); e3 = (0,0,1) The de Casteljau evaluation algorithm is: where r = 1,…..,n and |i| = n – r, and u = (u,v,w) are the barycentric coordinates of a point, where the function is evaluated. and is the point with parameter value u on the triangular Bezier patch.

09/25/02 Dinesh Manocha, COMP258 Properties of Triangular Patches Affine invariance Convex hull property Invariance under affine parameter transformation Boundary curves are Bezier curves of degree n

09/25/02 Dinesh Manocha, COMP258 Bernstein polynomials The Bernstein polynomials are defined as:, where |i| = n and a triangular patch can be written in terms of Bernstein polynomials as: