On the curvature extrema of special cubic B\'ezier curves

TL;DR

This paper proves that special cubic Bézier curves, derived from quadratic curves with a scalar parameter, have at most one local curvature extremum within the parameter interval (0,1).

Contribution

It establishes a theoretical limit on the number of curvature extrema for a specific class of cubic Bézier curves, enhancing understanding of their geometric properties.

Findings

At most one local curvature extremum exists in (0,1) for these curves.

The result applies to curves generated from quadratic curves using a scalar parameter.

Provides a mathematical proof of the curvature extremum limitation.

Abstract

It is proved that special cubic B\'ezier curves, generated from quadratic curves by the use of a scalar parameter, have at most one local curvature extremum in the $(0,1)$ parameter interval.