A comprehensive comparison of algorithms for evaluating rational Bézier curves

Bézier curves are very important tools in various fields and applications, such as computer graphics and computer-aided design. The de Casteljau algorithm is the first method introduced for evaluating polynomial Bézier curves, later also generalized to the rational case and surfaces. Although it presents an elegant definition through convex combinations and generally yields stable results, it has quadratic time complexity, which means that its computational cost can increase significantly with the number of control points. This represents a significant limitation, especially when dealing with high-degree curves and real-time applications. For this reason, numerous studies have been conducted in order to provide alternative approaches and more efficient algorithms. In this paper, we present a collection of the most commonly used algorithm in the state-of-the-art, also providing a comparison of their efficiency and their numerical stability.