Equipartition algorithm for a flat parametric curve based on the intersection between it and a moving circle

Abstract

The problem of partitioning a curve in a parametric vector form on the Euclidean plane into segments equal in chord length, having the “classical” formulation, was considered. A method of partitioning a flat parametric curve into equal-chord segments by crossing a circle of constant radius with the subsequent movement of the circle's center to the intersection point is proposed. The problem of the multivalued solution of the intersection equation was considered, which complicates the application of this method. This circumstance limits the use of circular partitioning by the lower limit of the values of the number of segments. The proposed algorithm was presented in pseudocode and described. It consists of the following procedures: the procedure for the initial initialization of the radius of a circle based on a partition with a uniform distribution by a parameter, procedures for partitioning the curve by a circle for different directions of the circle`s move (direct, reverse, two-way); the procedure for obtaining an equal-chord partition with a specified tolerance of determining the chord length. For the real curve`s example, experiments were conducted on its equipartition by this algorithm, implemented in the Julia programming language. It was established that with an increase in the degree of discretization of the value of the curve, the number of iterations required to achieve the specified accuracy stabilizes. This leads to a linear dependence of the partition execution time with an increase in the number of segments. It was found that when the accuracy of the partition is increased, the number of iterations increases slightly compared to the increase in accuracy.