HW7suggestions - s u consisting of segments s u and s 1 u at u = 1 u = 2 u = 5 and u = 7 in order to draw s n and the respective Bezier polygons

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Jacobs University, Bremen School of Engineering and Science Prof. Dr. Lars Linsen, Orif Ibrogimov Spring Term 2010 Homework 7 120202: ESM4A - Numerical Methods Homework Problems 7.1. a) Use the extended Newton divided difference algorithm to determine a polynomial P with minimal degree that takes these values: P (1) = 1 P (2) = 13 P 0 (1) = 3 P 0 (2) = 26 P 00 (2) = 40 b) Suppose that f C 1 [ a, b ] and that x 0 , x 1 , . . . , x n in [ a, b ] are distinct. Prove that the unique polynomial of least degree agreeing with f and f 0 at x 0 , x 1 , . . . , x n is the polynomial of degree at at most 2 n + 1 given by H 2 n +1 ( x ) = n X j =0 f ( x j ) H n,j ( x ) + n X j =0 f 0 ( x j ) b H n,j ( x ) , where H n,j ( x ) = [1 - 2( x - x j ) L 0 n,j ( x j )] L 2 n,j ( x ) and b H n,j ( x ) = ( x - x j ) L 2 n,j ( x ) Here, L n,j ( x ) denotes the j -th Lagrange coefficient polynomial of degree n . ( 5+8=13 points ) 7.2. Given knots u 0 = 0 , u 1 = 3 , and u 2 = 9, points p 0 = 1 , p 1 = 2 , and p 2 = 1 , and derivatives d 0 = 1 , d 1 = 1 and d 2 = 0. (a) Determine the Beizier polygon of the Bezier curves s i ( u ), i = 0 , 1, that interpolate points p i , p i +1 and derivatives d i , d i +1 and knots u i , u i +1 with respect to the piecewise Hermite interpolation scheme. (b) Use the De Casteljau algorithm to evaluate the piecewise polynomial curve
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Unformatted text preview: s ( u ) consisting of segments s ( u ) and s 1 ( u ) at u = 1 , u = 2 , u = 5 , and u = 7 in order to draw s ( n ) and the respective Bezier polygons over the interval [0 , 9]. ( 10 points ) 7.3. For distinct points P , P 1 , . . . , P n , the Bezier curve of order n ( n = 0 , 1 , 2 , . . . ) can be recur-sively defined by    B ( t, P ) := P B n ( t, P , . . . , P n ) := (1-t ) · B n-1 ( t, P , . . . , P n-1 ) + t · B n-1 ( t, P 1 , . . . , P n ) ( n > 0) where 0 ≤ t ≤ 1. Show that for any non-negative integer n , the Bezier curve of order n is given by the equation B n ( t, P , . . . , P n ) = n X k =0 ± n k ² t k (1-t ) n-k · P k ( 7 points ) Due: 26.03.10, at 3 pm (in the mailbox labeled Linsen in the entrance hall of Res.I )...
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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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HW7suggestions - s u consisting of segments s u and s 1 u at u = 1 u = 2 u = 5 and u = 7 in order to draw s n and the respective Bezier polygons

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