Question

Consider the second degree polynomial p(u) = c0 + c1u + c2u2 = uTc = [1 u u2][c0 c1 c2]T and the control points p = [p0 p1 p2] T. Derive the matrix A so that the coefficients c = A−1p make p(u) interpolate all control points (for u∈[0, 1]).

Answer 1

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The polynomial is

The points are .

The polynomial must satisfy the above points in order to interpolate the given polynomial.

Thus we have

and

We write the above information in the form of a matrix as:

------------------------(1)

and

--------------------------(2)

Thus the required matrix is obtained.

Also

--------------------------(3)

Thus the required matrices so that the co-efficients make interpolate all control points are given by the equations