Maximal intersection of spherical polygons by an arc with applications to 4-axis machining

Abstract

Many geometric optimization problems in CAD/CAM can be reduced to a maximal intersection problem on the sphere: given a set of N simple spherical polygons on the unit sphere and a real number constant L≤2π, find an arc of length L on the unit sphere that intersects as many spherical polygons as possible. Past results can only solve this maximization problem for two very restricted special cases: the arc must be either a great circle or a semi-great circle. In this paper, a simple and deterministic algorithm based on domain partitioning is presented for solving this maximal arc intersection problem in the general case when the number L is arbitrary. The algorithm is made possible by reducing the domain of the arcs to a continuous sub-space in R 2 and then establishing a quotient space partitioning in this sub-space based on a congruence relation. The number of the constituting congruent sub-regions in this quotient space partitioning is shown to have an upper-bound O( E 3), where E is the total number of edges on the polygons. The proposed algorithm has a worst-case upper bound O( ME) on its running time, where M is an output-sensitive number and is bounded by O( E 3). Examples including two realistic tests for 4-axis NC machining are presented.