Linear-time geometric algorithm for evaluating B\'ezier curves

TL;DR

This paper introduces a linear-time geometric algorithm for evaluating Bézier curves that uses convex combinations of control points, offering efficient computation with minimal memory usage.

Contribution

It presents a novel geometric algorithm for Bézier curve evaluation with linear complexity and constant memory, improving efficiency over existing methods.

Findings

Computational complexity is linear in the number of control points.

Memory complexity is constant, O(1).

Applicable to polynomial and rational Bézier curves.

Abstract

A new algorithm for computing a point on a polynomial or rational curve in B\'{e}zier form is proposed. The method has a geometric interpretation and uses only convex combinations of control points. The new algorithm's computational complexity is linear with respect to the number of control points and its memory complexity is $O(1)$. Some remarks on similar methods for surfaces in rectangular and triangular B\'{e}zier form are also given.