Minkowski sum of HV-polytopes in Rn

TL;DR

This paper presents new methods for computing Minkowski sums of polytopes in n-dimensional space using both H- and V-representations, leveraging polyhedral cone intersections and primal cone approaches.

Contribution

It introduces two novel algorithms for Minkowski sums of polytopes that utilize normal fan intersections and primal cone techniques, applicable when both representations are available.

Findings

The normal fan-based method effectively computes Minkowski sums via dual cone intersections.

The primal cone approach offers an alternative computation method for Minkowski sums.

Both methods extend the applicability of Minkowski sum calculations in higher dimensions.

Abstract

Minkowski sums cover a wide range of applications in many different fields like algebra, morphing, robotics, mechanical CAD/CAM systems ... This paper deals with sums of polytopes in a n dimensional space provided that both H-representation and V-representation are available i.e. the polytopes are described by both their half-spaces and vertices. The first method uses the polytope normal fans and relies on the ability to intersect dual polyhedral cones. Then we introduce another way of considering Minkowski sums of polytopes based on the primal polyhedral cones attached to each vertex.