Curvature extrema of planar parametric polynomial cubic curves

Abstract

Parametric polynomial cubic curve segments are widely used in computer-aided design and computer-aided geometric design applications because their flexibility makes them suitable for use in the interactive design of curves and surfaces. Shape properties of these segments have been studied and results on the occurrence of cusps, and whether the segment is S- or C-shaped, are available in the literature. Their critical points, however, are not as well known. This paper presents results on the number and location of curvature extrema of these segments. All possible numbers of curvature extrema are found and it is demonstrated that all possible curvature extrema can be obtained numerically for any planar parametric cubic curve. These results are useful in the study of the fairness of curves designed with parametric polynomial cubic curve segments.