Maximal Demyanov difference

Minkowski sums of point sets defined by inequalities

Minkowski sums are a very simple geometrical operation, with applications in many different fields. In particular, Minkowski sums of polytopes have shown to be of interest to both industry and the academic world. This thesis presents a study of these sums, both on combinatorial properties and on computational aspects. In particular, we give an unexpected linear relation between the f-vectors of a Minkowski sum and that of its summands, provided these are relatively in general position. We further establish some bounds on the maximum number of faces of Minkowski sums with relation to the summands, depending on the dimension and the number of summands. We then study a particular family of Minkowski sums, which consists in summing polytopes we call perfectly centered with their own duals. We show that the face lattice of the result can be completely deduced from that of the summands. Finally, we present an algorithm for efficiently computing the vertices of a Minkowski sum of polytopes. We show that the time complexity is linear in terms of the output for fixed size of the input, and that the required memory size is independent of the size of the output. We also review various algorithms computing different faces of the sum, comparing their strong and weak points.

We investigate how the Minkowski sum of two polytopes affects their graph and, in particular, their diameter. We show that the diameter of the Minkowski sum is bounded below by the diameter of each summand and above by, roughly, the product between the diameter of one summand and the number of vertices of the other. We also prove that both bounds are sharp. In addition, we obtain a result on polytope decomposability. More precisely, given two polytopes $P$ and $Q$, we show that $P$ can be written as a Minkowski sum with a summand homothetic to $Q$ if and only if $P$ has the same number of vertices as its Minkowski sum with $Q$.

General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of [Formula: see text] arbitrary ellipsoids in [Formula: see text]-dimensional Euclidean space. Expressions for the principal curvatures of these Minkowski sums are also derived. These results are then used to obtain upper and lower volume bounds for the Minkowski sum of ellipsoids in terms of their defining matrices; the lower bounds are sharper than the Brunn–Minkowski inequality. A reverse isoperimetric inequality for convex bodies is also given.

An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in n -dimensional Euclidean space \mathbb R^n . It is proved that if n\ge 2 , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, \operatorname{GL}(n) covariant, and associative if and only if it is L_p addition for some 1\le p\le\infty . It is also demonstrated that if n\ge 2 , an operation * between compact convex sets is continuous in the Hausdorff metric, \operatorname{GL}(n) covariant, and has the identity property (i.e., K*\{o\}=K=\{o\}*K for all compact convex sets K , where o denotes the origin) if and only if it is Minkowski addition. Some analogous results for operations between star sets are obtained. Various characterizations are given of operations that are projection covariant, meaning that the operation can take place before or after projection onto subspaces, with the same effect. Several other new lines of investigation are followed. A relatively little-known but seminal operation called M -addition is generalized and systematically studied for the first time. Geometric-analytic formulas that characterize continuous and \operatorname{GL}(n) -covariant operations between compact convex sets in terms of M -addition are established. It is shown that if n\ge 2 , an o -symmetrization of compact convex sets (i.e., a map from the compact convex sets to the origin-symmetric compact convex sets) is continuous in the Hausdorff metric, \operatorname{GL}(n) covariant, and translation invariant if and only if it is of the form \lambda DK for some \lambda\ge 0 , where DK=K+(-K) is the difference body of K . The term “polynomial volume” is introduced for the property of operations * between compact convex or star sets that the volume of rK*sL , r,s\ge 0 , is a polynomial in the variables r and s . It is proved that if n\ge 2 , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, \operatorname{GL}(n) covariant, associative, and has polynomial volume if and only if it is Minkowski addition.

We describe a GPU-based computational platform for six-dimensional configuration mapping, which is the description of the configuration space of rigid motions in terms of collision and contact constraints. The platform supports a wide range of computations in design and manufacturing, including three and six dimensional configuration space obstacle computations, Minkowski sums and differences, packaging problems, and sweep computations. We demonstrate dramatic performance improvements in the special case of configuration space operations that determine interference-free or containment-preserving configurations between moving solids. Our approach treats such operations as convolutions in the six dimensional configuration space that are efficiently computed using the Fast Fourier Transform (FFT). The inherent parallelism of FFT algorithms facilitates a straightforward implementation of convolution on GPUs with existing and freely available libraries, making all such six dimensional configuration space computations practical, and often interactive.

We derive tight expressions for the maximum number of k-faces, k=0,...,d-1, of the Minkowski sum, P_1+...+P_r, of r convex d-polytopes P_1,...,P_r in R^d, where d >= 2 and r < d, as a (recursively defined) function on the number of vertices of the polytopes. Our results coincide with those recently proved by Adiprasito and Sanyal [1]. In contrast to Adiprasito and Sanyal's approach, which uses tools from Combinatorial Commutative Algebra, our approach is purely geometric and uses basic notions such as f- and h-vector calculus, stellar subdivisions and shellings, and generalizes the methodology used in [10] and [9] for proving upper bounds on the f-vector of the Minkowski sum of two and three convex polytopes, respectively. The key idea behind our approach is to express the Minkowski sum P_1+...+P_r as a section of the Cayley polytope C of the summands; bounding the k-faces of P_1+...+P_r reduces to bounding the subset of the (k+r-1)-faces of C that contain vertices from each of the r polytopes. We end our paper with a sketch of an explicit construction that establishes the tightness of the upper bounds.

We describe a graphics processing unit (GPU)-based computational platform for six-dimensional configuration mapping, which is the description of the configuration space of rigid motions in terms of collision and contact constraints. The platform supports a wide range of computations in design and manufacturing, including three- and six-dimensional configuration space obstacle computations, Minkowski sums and differences, packaging problems, and sweep computations. We demonstrate dramatic performance improvements in the special case of configuration space operations that determine interference-free or containment-preserving configurations between moving solids. Our approach treats such operations as convolutions in the six-dimensional configuration space that are efficiently computed using the fast Fourier transform (FFT). The inherent parallelism of FFT algorithms facilitates a straightforward implementation of convolution on GPUs with existing and freely available libraries, making all such configuration space computations practical, and often interactive.

Configuration products and quotients in geometric modeling

Let X be a finite dimensional vector space. The power space 2X, supplied with an order relation, a complementary operation, union, intersection and Minkowski sum, enjoys many algebraic structures.

A solution of polygon containment, spatial planning, and other related problems using Minkowski operations

The six-dimensional space SE(3) is traditionally associated with the space of configurations of a rigid solid (a subset of Euclidean three-dimensional space E3). But a solid can be also considered to be a set of configurations, and therefore a subset of SE(3). This observation removes the artificial distinction between shapes and their configurations, and allows formulation and solution of a large class of problems in mechanical design and manufacturing. In particular, the configuration product of two subsets of configuration space is the set of all configurations obtained when one of the sets is transformed by all configurations of the other. The usual definitions of various sweeps, Minkowski sum, and other motion related operations are then realized as projections of the configuration product into E3. Similarly, the dual operation of configuration quotient subsumes the more common operations of unsweep and Minkowski difference. We identify the formal properties of these operations that are instrumental in formulating and solving both direct and inverse problems in computer aided design and manufacturing. Finally, we show that all required computations may be implemented using a fast parallel sampling method on a GPU.

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.

The navigation function (NF) is widely used for motion planning of autonomous vehicles. Such a function is bounded, analytic, and guarantees convergence due to its Morse nature, while having a single minimum value at the target point. This results in a safe path to the target. Originally, the NF was developed for deterministic scenarios where the positions of the robot and the obstacles are known. Here we extend the concept of NF for static stochastic scenarios. We assume the robot, the obstacles and the workspace geometries are known, while their positions are random variables. We define a new Probability NF that we call PNF by introducing an additional permitted collision probability, which limits the risks (to a set value) during the robot’s motion. The Minkowski sum is generalized for the geometries of the robot and the obstacles with their respective Probability Density Functions (PDF), that represent their locations’ uncertainties. The probability for a collision is therefore the convolution of the robot’s geometry, the obstacles’ geometries and the PDFs of their locations The novelty of the proposed algorithm is in its ability to provide a converging trajectory in stochastic environment without inflating the ambient space dimension. We demonstrate our algorithm performances using a simulator, and compare its results with the conventional NF algorithm and with a version of the well known RRT* and Voronoi Uncertainty Fields methods for uncertain scenarios. Finally, we show simulation results of the PNF in disc-shaped world, as well as in star-shaped world.

Motzkin decomposition of closed convex sets