ODF Using a Quartic Bézier

Abstract

A quartic Bézier is fit to a hypoTrochoid shape. The quartic Bézier is initialized using a cubic Bézier. The ODF results are much more complex than either the cubic Bézier or B-spline results. However, the solution set can be separated into two disjoint sets of solutions, one of which is topologically identical to the cubic Bézier, while the other is of little practical interest due to its’ high rms error. There is evidence of temporary branches which are double saddle points, in addition to the more common single saddle points. A pair of rules is developed to test the internal consistency of the solution set, based on the number of symmetric and anti-symmetric solutions, and on the number and types of saddle points versus local minima. A tentative link with Euler’s characteristic is proposed. Reasons for abnormal termination of solutions are discussed. The minimum rms error is almost uniformly the best of all the splines studied here, but the spline is not supported by standard graphics rendering packages.