@article{Zhao2022, 
author = {Xuanyi Zhao and Jinggai Li and Ying Wang and Chungang Zhu},
title = {Improved algorithms for determining the injectivity of 2D and 3D rational Bézier curves},
year = {2022},
journal = {Electronic Research Archive},
volume = {30},
number = {5},
pages = {1799-1812},
keywords = {injectivity, Bézier curve, monotone chain, algorithm analysis},
url = {https://www.sciopen.com/article/10.3934/era.2022091},
doi = {10.3934/era.2022091},
abstract = {Bézier curves and surfaces are important to computer-aided design applications. This paper presents algorithms for checking the injectivity of 2D and 3D Bézier curves. An injective Bézier curve or surface is one that has no self-intersections. The proposed algorithms use recently proposed sufficient and necessary conditions under which Bézier curves are guaranteed to be non-self-intersecting. As well as a rigorous derivation of the proposed algorithms, we present a series of examples and derive the complexity and computation times of the proposed algorithms. We find that the complexity our algorithms is approximately  O(m), representing an improvement over previous injectivity-checking algorithms.}
}