## Parametric Cubic

Given four points A, B, C, D, one can define the parametric cubic p(t) = A + Bt + Ct^2 + Dt^3. Except A, the other points lie not, in general, on the cubic. They control however the shape of the curve. The present cubic, for example, is controlled by the vertices of a parallelogram. In this case the cubic passes always through the origin (0,0). The shape and its dynamic dependence on the four points are constructed in EucliDraw by a [user-tool]. The corresponding script is in the file [EUC_Scripts \ EUC_User_Tools \ ParamCubicTool].

It would be more interesting to construct a cubic passing through four points. But which is the distinguished way to do something like that? If in the previous parametric representation p(t), we require that p(t1) = p1, p(t2) = p2, etc., then a linear system results, expressing p1, p2, ... through A, B, C and D. The coefficients of this system are powers of t1, t2, ... and the corresponding matrix is of Vandermonde type. Such a cubic, drawn again through a [user-tool], is contained in the file CubicFitting4.html . There the cubic passes through the 4 points, for corresponding parameter-values: 0, 1, 2, 3 and the script uses the inverse of the corresponding Vandermonde matrix.