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Math Help - Degree-Raising Matrix for Bernstein Polynomials

  1. #1
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    Degree-Raising Matrix for Bernstein Polynomials

    Question :
    Let v = (a, b, c)T be a column vector which represents a coordinate vector of a polynomial in P2 with
    respect to the Bernstein basis. Find the 4 3 matrix which transforms v to the standard basis of P3.
    (Hint: First transform v to the standard basis of P2, then transform to the standard basis of P3 by just
    adding an extra row of zeros.)

    So, the Bernstein basis of P2 is { (t-1)^2, -2t(1-t), t^2}

    so the change of basis matrix, from BB to standard is given by [1,0,0 ; -2,2,0 ; 0, 0, 1]

    So, the question asks for a 4x3 matrix that will take a P2 Bernstein polynomial and transform it to p3 Bernstein polynomial.

    So, the answer should be [1,0,0 ; -2,2,0 ; 0, 0, 1 ; 0, 0 ,0]

    is this right?

    Thank you
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  2. #2
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    Re: Degree-Raising Matrix for Bernstein Polynomials

    in the the standard basis {1,t,t2} (note: the order matters here), the coordinate form of (t-1)2 is (1,-2,1), and the coordinate form of (-2t)(1-t) = 2t(t-1) = 2t2-2t is (0,-2,2).

    EDIT: according to wikipedia, your second basis vector should be 2t(1-t), which would be (0,2,-2).

    Bernstein polynomial - Wikipedia, the free encyclopedia

    what i get for the matrix is:

    P = \begin{bmatrix} 1&0&0\\-2&2&0\\1&-2&1\\0&0&0 \end{bmatrix}
    Last edited by Deveno; September 24th 2012 at 12:53 PM.
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