Assume we are given the set $S$ of $n$ points on the real plane and want to draw a parametrized cubic curve (actually a segment of Bezier spline) with fixed startpoint in such a way, that the resulting curve is closest to $S$. $S$ may be assumed to contain at least three different points. By closest I mean such a curve, that the sum of distances between it and every point of $S$ is minimal (however I think the sum of squares of distances will be ok either). Drawing a cubic curve with fixed startpoint amounts to specifying 6 coefficients in the parametric presentation and the problem is to find way of expressing these coefficients through the given coordinates of points in $S$. Is there any solution to the problem?
