CubicFitting4

Cubic interpolation

Given four points A, B, C, D, there is a parametric cubic curve p(t) = q0 + q1*t + q2*t^2 + q3*t^3, passing through the points A=p(0), B=p(1), C=p(2), D=p(3). Its shape is controlled by the quadrilateral with vertices at A, B, C, D. The figure and its dynamic dependence is constructed in EucliDraw through a [user-tool]. The corresponding script is contained in the file [EUC_Scripts \ EUC_User_Tools \ CubicFitting4]. The figure is related to that of the file ParametricCubic.html .

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.

http://users.math.uoc.gr/~pamfilos/