There is a linear system expressing the coefficients q1, q2, ... as functions of A, B, C and D. The coefficients of the matrix of the system are powers of the integers 0, 1, 2, 3, the matrix being a special case of the well-known Vandermonde type. The script uses the inverse of the special corresponding Vandermonde matrix.

The script and the figure could be generalized to parametric curves of n-degree of the form:

The determination of the Ai from the n+1 interpolation points would require the inversion of a (n+1)x(n+1) matrix of Vandermonde type.

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