Closed-Form Minkowski Sums of Convex Bodies with Smooth Positively Curved Boundaries
This work provides new mathematical tools for researchers and practitioners working with geometric operations on convex bodies, particularly in fields like motion planning and collision detection, where accurate and efficient computation of Minkowski sums is crucial. It offers an incremental improvement in the mathematical formulation of these sums.
This paper presents closed-form parametric formulas for Minkowski sums of convex bodies in d-dimensional Euclidean space, specifically for bodies with smooth, positively curved boundaries. The authors provide two theorems: one using unit normal vectors and a more computationally practical one using unnormalized gradients. The derived expressions are validated through numerical comparisons of superquadric bodies and shown to be consistent with prior work on ellipsoids.
This article derives closed-form parametric formulas for the Minkowski sums of convex bodies in d-dimensional Euclidean space with boundaries that are smooth and have all positive sectional curvatures at every point. Under these conditions, there is a unique relationship between the position of each boundary point and the surface normal. The main results are presented as two theorems. The first theorem directly parameterizes the Minkowski sums using the unit normal vector at each surface point. Although simple to express mathematically, such a parameterization is not always practical to obtain computationally. Therefore, the second theorem derives a more useful parametric closed-form expression using the gradient that is not normalized. In the special case of two ellipsoids, the proposed expressions are identical to those derived previously using geometric interpretations. In order to examine the results, numerical validations and comparisons of the Minkowski sums between two superquadric bodies are conducted. Applications to generate configuration space obstacles in motion planning problems and to improve optimization-based collision detection algorithms are introduced and demonstrated.