On Self-Intersections of Cubic Bézier Curves

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.

On Intersections of B-Spline Curves

“…The flexibility to adjust the shape of B-spline curves through manipulation of knot vectors and control points has made them popular in computer graphics for curve and surface modeling, animation, interpolation and shape editing, and font design. With their different characteristics and advantages, Bézier curves and B-spline curves become important tools for representing and manipulating curves in computer graphics, and many scholars have conducted research on the intersection problems of Bézier [13,14] and B-spline curves/surfaces [15][16][17] .…”