{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

HW7suggestions - s u consisting of segments s u and s 1 u...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 (1) = 3 P (2) = 26 P (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 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 j =0 f ( x j ) H n,j ( x ) + n j =0 f ( x j ) H n,j ( x ) , where H n,j ( x ) = [1 - 2( x - x j ) L n,j ( x j )] L 2 n,j ( x ) and 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)
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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 )...
View Full Document

{[ snackBarMessage ]}