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Degree-Raising Formulas for Bernstein Polynomials

  1. Sep 24, 2012 #1
    Two part question...

    1. The problem statement, all variables and given/known data

    Question 1:
    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.)

    Question 2:
    Multiply the matrix in the previous part (on the left) by the conversion matrix from the standard basis of
    P3 to the Bernstein basis of P3. The resulting matrix should convert from degree 2 to degree 3 Bernstein
    representation (degree-raising). Test your matrix on the Bernstein polynomials of degree 2. Check by
    expanding everything back to the standard basis.

    2. Relevant equations
    the Bernstein basis of P2 is { (t-1)^2, -2t(t-1), t^2}
    the Bernstein basis of P3 is { -(t-1)^3, 3t(t-1)^2, -3t^2(t-1), t^3}


    3. The attempt at a solution

    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]


    For question 2, I am to multiple the the matrix above with the matrix that goes from P3 (standard) to P3 (BB)

    The matrix that goes from BB->S in p3 is [1,0,0,0; -3,3,0,0 ; 3,-6,3,0; -1,3,-3,1]^-1

    so I multiple both matrices together and that should be the answer to question 2, right?


    Thank you
     
  2. jcsd
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