Progressive Convex Hull Simplification

2026-04-15

AI Summary

This work addresses the high computational complexity arising from the linear growth of convex hulls with input size by proposing an efficient, conservative simplification method in dual space based on a greedy strategy. The approach approximates a convex hull using a prescribed number of halfspaces while minimizing the increase in volume or surface area. The authors introduce, for the first time, a dual-space greedy algorithm with $O(n \log n)$ time complexity that simultaneously achieves geometric conservativeness, simplification efficiency, and tightness—overcoming the longstanding trade-off among these three criteria in existing methods. Experimental results demonstrate that the proposed technique consistently outperforms state-of-the-art approaches across diverse input geometries and downstream applications, including collision detection and ray intersection.

Technology Category

• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• 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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Applications of cryptography

Abstract

Convex hulls are useful as tight bounding proxies for a variety of tasks including collision detection, ray intersection, and distance computation. Unfortunately, the complexity of polyhedral convex hulls grows linearly with their input. We consider the problem of conservatively simplifying a convex hull to a specified number of half-spaces while minimizing added volume or surface area. By working in the dual representation, we propose an efficient $O(n \log n)$ greedy optimization. In comparisons, we show that existing methods either exhibit poor efficiency, tightness or safety. We demonstrate the success of our method on a variety of input shapes and downstream application domains.

Problem

• convex hull
• simplification
• half-spaces
• volume minimization
• surface area

Innovation

• convex hull simplification
• dual representation
• greedy optimization
• conservative approximation
• computational geometry