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09/04/02 Dinesh Manocha, COMP258 Bezier Curves Interpolating curve Polynomial or rational parametrization using Bernstein basis functions Use of control points –Piecewise segments defining control polygon or characteristic polygon –Algebraically: used for linear combination of basis functions

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09/04/02 Dinesh Manocha, COMP258 Properties of Basis Functions Interpolate the first and last control points, P 0 and P n. The tangent at P 0 is given by P 1 - P 0 and at P n is given by P n - P n-1 Generalize to higher order derivatives: second derivate at P 0 is determined by P 0, P 1 and P 2 and the same for higher order derivatives The functions are symmetric w.r.t. u and (1-u). That is if we reverse the sequence of control points to P n P n-1 P n-2 … P 0, it defines the same curve.

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09/04/02 Dinesh Manocha, COMP258 Bezier Basis Function Use of Bernstein polynomials: P(u) = P i B i,n (u) u [0,1] Where B i,n (u) = u i (1-u) n-i

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09/04/02 Dinesh Manocha, COMP258 Cubic Bezier Curve: Matrix Representation Let B = [P 0 P 1 P 2 P 3 ] F = [B 1 (u) B 2 (u) B 3 (u) B 4 (u)] or F = [u 3 u 2 u 1] -1 3 -3 1 3 6 3 0 -3 3 0 0 1 0 0 0 This is the 4 X 4 Bezier basis transformation matrix. P(u) = U M B P, where U = [u 3 u 2 u 1]

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09/04/02 Dinesh Manocha, COMP258 Properties of Bezier Curves Invariance under affine transformation Convex hull property Variation diminishing De Casteljau Evaluation (Geometric computation) Symmetry Linear precision

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