Exact and Efficient Construction of Minkowski Sums of Convex Polyhedra with Applications

Abstract

We present an exact implementation of an efficient algorithm that computes Minkowski sums of convex polyhedra in R 3 . Our implementation is complete in the sense that it does not assume general position. Namely, it can handle degenerate input, and it produces exact results. We also present applications of the Minkowski-sum computation to answer collision and proximity queries about the relative placement of two convex polyhedra in R 3 . The algorithms use a dual representation of convex polyhedra, and their implementation is mainly based on the Arrangement package of Cgal, the Computational Geometry Algorithm Library. We compare our Minkowski-sum construction with the only three other methods that produce exact results we are aware of. One is a simple approach that computes the convex hull of the pairwise sums of vertices of two convex polyhedra. The second is based on Nef polyhedra embedded on the sphere, and the third is an output sensitive approach based on linear programming. Our method is significantly faster. The results of experimentation with a broad family of convex polyhedra are reported. The relevant programs, source code, data sets, and documentation are available at http://www.cs.tau.ac.il/~efif/CD, and a short movie [16] that describes some of the concepts portrayed in this paper can be downloaded from

Robust polyhedral Minkowski sums with GPU implementation

“…The first approach decomposes the polyhedra into convex components, computes the Minkowski sums of the components with a specialized algorithm [6], and returns their union. This approach is inefficient because a polyhedron with r reflex edges can have Ω(r 2 ) convex pieces, so an input with n edges can entail a union of Ω(n 4 ) component Minkowski sums.…”

“…Fogel and Halperin [6] compute the Minkowski sum of two convex polyhedra. The kinetic convolution is the Minkowski sum boundary, so arrangement is trivial.…”

Contributing vertices-based Minkowski sum computation of convex polyhedra

“…Halperin [35]. The authors represented the convex polyhedra in a dual space they called the Cubical Gaussian Map (CGM) and implemented their algorithm on the base of the arrangement package of CGAL.…”

“…The first one is based on Nef polyhedra embedded on the sphere and implemented by Hachenberger [24], the second one is a method based on linear programming implemented by Weibel [38] (following the work of Fukuda [39]), and the third one is the Cubical Gaussian Map-based (CGM-based) algorithm of Fogel and Halperin [35].…”

Dynamic Minkowski sums under scaling

“…The Minkowski sum of two shapes, P and Q is defined as P ⊕Q = {p+q | p ∈ P, q ∈ Q}. Though its study dates back to the early 70s (see the cited surveys for more details [15,29,14]), recent work has also emerged studying the idea of dynamic Minkowski sums, rapid methods of updating the Minkowski sum under various transformations such as rotation [6,7,26]. However, as we have seen in our earlier discussion, rotation is not the only common transformation under which the Minkowski sum may be recomputed.…”