Conversion Between Cubic Bezier Curves and Catmull-Rom Splines

TL;DR

This paper presents a simple, efficient method for converting cubic Bezier curves to Catmull-Rom splines and vice versa, enabling shape-preserving transformations in computer graphics and geometric design.

Contribution

It derives linear conversion equations for control points, allowing shape-approximate transformations between Bezier and Catmull-Rom curves with ease.

Findings

Conversion equations are simple and linear.

Method is validated with numerical examples.

Shape approximation is effective and efficient.

Abstract

Splines are one of the main methods of mathematically representing complicated shapes, which have become the primary technique in the fields of Computer Graphics (CG) and Computer-Aided Geometric Design (CAGD) for modeling complex surfaces. Among all, B\'ezier and Catmull-Rom splines are the most common in the sub-fields of engineering. In this paper, we focus on conversion between cubic B\'ezier and Catmull-Rom curve segments, rather than going through their properties. By deriving the conversion equations, we aim at converting the original set of the control points of either of the Catmull-Rom or B\'ezier cubic curves to a new set of control points, which corresponds to approximately the same shape as the original curve, when considered as the set of the control points of the other curve. Due to providing simple linear transformations of control points, the method is very simple, efficient, and easy to implement, which is further validated in this paper using some numerical and visual examples.