homework-06 - H OMEWORK 6 120202: ESM4A - N UMERICAL M...

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Unformatted text preview: H OMEWORK 6 120202: ESM4A - N UMERICAL M ETHODS Spring 2009 Prof. Dr. Lars Linsen Zymantas Darbenas School of Engineering and Science Jacobs University Due: Friday, March 27, 2009 at noon (in the mailbox labeled "Linsen" in the entrance hall of Research I). Problem 16: Generalized Hermite interpolation. Given knots u0 = 0, u1 = 3, and u2 = 9 and points p0 = 1, p1 = 2, and p2 = 1. (10 points) (a) Use the generalized Hermite interpolation scheme to develop the interpolation conditions for a curve p(u) of minimal degree that interpolates the given points at the respective knots and the derivatives p (u0 ) = 1 and p (u2 ) = 0 at the endpoints of the interval. (b) Compute the respective generalized Hermite polynomials and the interpolating curve p(u). Problem 17: Bernstein polynomials. (10 points) n (a) Show that the Bernstein polynomials Bi (u), i = 0, . . . , n, are linearly independent (u [0, 1]). n+1 (b) Show that for each Bernstein polynomial Bi (u) with i {0, . . . , n + 1} the following equation holds n+1 n n Bi (u) = u Bi-1 (u) + (1 - u) Bi (u) n n with B-1 (u) = Bn+1 (u) = 0 (u [0, 1]). Problem 18: Piecewise Hermite interpolation using Bezier curves. (10 points) Given knots u0 = 0, u1 = 3, and u2 = 9, points p0 = 1, p1 = 2, and p2 = 1, and derivatives d0 = 1, d1 = 1 and d2 = 0. (a) Determine the Bezier polygons of the Bezier curves si (u), i = 0, 1, that interpolate points pi , pi+1 and derivatives di , di+1 at knots ui , ui+1 with respect to the piecewise Hermite interpolation scheme. (b) Use the De Casteljau algorithm to evaluate the piecewise polynomial curve s(u) consisting of segments s0 (u) and s1 (u) at u = 1, u = 2, u = 5, and u = 7 in order to draw s(u) and the respective Bezier polygons over the interval [0, 9]. ...
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This note was uploaded on 05/18/2010 for the course MATHEMATIC 120102 taught by Professor Xxxxxxxxxyyyyyy during the Spring '10 term at Jacobs University Bremen.

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