Inflection points and singularities on planar rational cubic curve segments

Abstract

We obtain the distribution of inflection points and singularities on a parametric rational cubic curve segment with aid of Mathematica (A System of for Doing Mathematics by Computer). The reciprocal numbers of the magnitudes of the end slopes determine the occurrence of inflection points and singularities on the segment. Its use enables us to check whether the segment has inflection points or a singularity (a loop or a cusp) and to get an idea how to place control vertices and how to choose weights for the rational Bézier cubic curve segment to preserve the fair shape.