Circle-intersection-based equal-chord partitioning algorithm for a planar parametric curve

Abstract

This paper addresses the problem of equal-chord partitioning of planar parametric curves, which consists of determining a sequence of points on a curve such that the Euclidean distances (chords) between consecutive points are equal. Equal-chord discretization is important in computer-aided design, CNC machining, robotics, and computer graphics, as it provides stable geometric and technological properties compared to parameter-uniform or arc-length-based sampling. This work aims to develop efficient algorithms for equal-chord partitioning of planar parametric curves in the classical setting, i.e., with fixed endpoints and a prescribed number of segments, while explicitly accounting for the multivalued nature of solutions arising from curve–circle intersections. The proposed approach is based on successive intersections of the curve with a moving circle of fixed radius. Starting from one or both endpoints, the circle center is placed at the current partition point, and subsequent points are obtained as intersections between the curve and the circle in a specified direction. To handle non-convex curves and curves with inflection points, all possible intersection solutions are organized into a tree structure, where each branch represents an alternative partitioning trajectory. The optimal partition is selected by minimizing the difference between the residual segment length and the circle radius. An experimental evaluation of planar Bézier curves of various orders demonstrates stable convergence of the proposed algorithms across a wide range of segment counts. As the number of segments increases, the multiplicity of curve–circle intersections decreases, and the computational complexity approaches linear behavior in practical scenarios. The proposed method avoids numerical arc-length integration and auxiliary grid-based approximations and can be naturally extended to spatial curves.