Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.

💡 Research Summary

The paper presents two exact, output‑sensitive algorithms for computing the Minkowski sum of two convex polyhedra in three‑dimensional space, together with a robust software implementation built on the CGAL library. Unlike many previous approaches, the authors do not assume general position; their methods handle degenerate inputs (e.g., coincident faces or overlapping edges) without failure and rely solely on exact rational arithmetic. The core of the implementation is an arrangement data structure that represents arcs of great circles (geodesic arcs) embedded on two‑dimensional parametric surfaces, most notably the unit sphere. This structure, provided by CGAL’s Arrangement_on_surface_2 package, supports insertion, intersection, subdivision, and geometric queries (area, volume, adjacency) in a fully exact manner.

The algorithms are output‑sensitive: if the two polyhedra have f₁ and f₂ facets, the worst‑case combinatorial complexity of their Minkowski sum is shown to be Θ(f₁·f₂), a tight bound that improves on earlier looser estimates. The runtime of the proposed methods scales proportionally to the actual size of the output, making them efficient in practice even for highly degenerate cases. Extensive experiments confirm that the exact rational implementation produces mathematically correct results where floating‑point based methods would either crash or discard valid configurations.

Beyond Minkowski sums, the paper demonstrates additional applications that exploit spherical arrangements of geodesic arcs. One notable example is an exact algorithm for partitioning an assembly of polyhedra using two “hands” that translate infinitely in space. By mapping each hand’s translation direction to a point on the sphere, the arrangement’s cells correspond to feasible motion pairs. Exact computation guarantees that no admissible partitioning motion is erroneously eliminated—a risk that would be severe in imprecise arithmetic.

All software components are released as part of CGAL, allowing users to instantiate the arrangement data structure with any number type, though the presented algorithms require only exact rational numbers. The work thus delivers a powerful, reusable toolkit for exact geometric computation on curved surfaces, and it establishes a clear pathway for applying these tools to robotics, CAD, and computational geometry problems where correctness cannot be compromised.