A Ray Intersection Algorithm for Fast Growth Distance Computation Between Convex Sets

2026-04-11

AI Summary

This work addresses the efficient computation of the minimum scaling factor—known as the growth distance—required for two compact convex sets to first intersect under uniform scaling about prescribed centers. By reformulating the problem as a ray-intersection query on the Minkowski difference, the authors propose a novel algorithm that simultaneously ensures primal-dual feasibility and monotonic convergence, providing a unified treatment for both intersecting and separating configurations. The method represents convex sets via their support functions and iteratively refines inner and outer polyhedral approximations to converge to the ray intersection point, substantially improving computational efficiency. An open-source implementation demonstrates superior performance across diverse convex geometries and has been successfully applied to robotic motion planning and rigid-body simulation.

Technology Category

• Search and Optimization: Non-convex Optimization
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

In this paper, we discuss an efficient algorithm for computing the growth distance between two compact convex sets with representable support functions. The growth distance between two sets is the minimum scaling factor such that the sets intersect when scaled about some center points. Unlike the minimum distance between sets, the growth distance provides a unified measure for set intersection and separation. We first reduce the growth distance problem to an equivalent ray intersection problem on the Minkowski difference set. Then, we propose an algorithm to solve the ray intersection problem by iteratively constructing inner and outer polyhedral approximations of the Minkowski difference set. We show that our algorithm satisfies several key properties, such as primal and dual feasibility and monotone convergence. We provide extensive benchmark results for our algorithm and show that our open-source implementation achieves state-of-the-art performance across a wide variety of convex sets. Finally, we demonstrate robotics applications of our algorithm in motion planning and rigid-body simulation.

Problem

• growth distance
• convex sets
• ray intersection
• Minkowski difference
• set intersection

Innovation

• growth distance
• ray intersection
• Minkowski difference
• polyhedral approximation
• convex sets