Using general polar values as control points for polynomial curves

Abstract

Blossoming has proven to be a useful technique for understanding and generalizing polynomial curves through the use of the polar form. This paper shows that general polar values may be used to control polynomial curves when a related matrix is invertible. The inverse matrix provides a useful translation from these general blossom control points to well known ones such as those of Bézier. The special case in which the polar form can be evaluated through pairwise affine combinations is characterized, allowing the arguments of the blossom control points to be chosen in a manner akin to choosing the knot vector of a B-spline segment. The number of free parameters for specifying the blossom control points of polynomial curves is increased significantly over the B-spline case.