On Self-Intersections of Cubic Bézier Curves

Ying-Ying Yu, Xin Li and Ye Ji ()
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Ying-Ying Yu: School of Mathematics, Liaoning Normal University, Dalian 116029, China
Xin Li: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
Ye Ji: Delft Institute of Applied Mathematics, Delft University of Technology, 2628 CD Delft, The Netherlands

Mathematics, 2024, vol. 12, issue 6, 1-11

Abstract: Cubic Bézier curves are widely used in computer graphics and geometric modeling, favored for their intuitive design and ease of implementation. However, self-intersections within these curves can pose significant challenges in both geometric modeling and analysis. This paper presents a comprehensive approach to detecting and computing self-intersections of cubic Bézier curves. We introduce an efficient algorithm that leverages both the geometric properties of Bézier curves and numerical methods to accurately identify intersection points. The self-intersection problem of cubic Bézier curves is firstly transformed into a quadratic problem by eliminating trivial solutions. Subsequently, this quadratic system is converted into a linear system that may be easily analyzed and solved. Finally, the parameter values corresponding to the self-intersection points are computed through the solution of the linear system. The proposed method is designed to be robust and computationally efficient, making it suitable for real-time applications.

Keywords: geometric modeling; Bézier curves; self-intersections (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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