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On Self-Intersections of Cubic Bézier Curves

Author

Listed:
  • 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)

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.

Suggested Citation

  • Ying-Ying Yu & Xin Li & Ye Ji, 2024. "On Self-Intersections of Cubic Bézier Curves," Mathematics, MDPI, vol. 12(6), pages 1-11, March.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:6:p:882-:d:1358584
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