Showing papers on "Minkowski addition published in 2005"

Reachability of uncertain linear systems using zonotopes

Abstract: We present a method for the computation of reachable sets of uncertain linear systems. The main innovation of the method consists in the use of zonotopes for reachable set representation. Zonotopes are special polytopes with several interesting properties : they can be encoded efficiently, they are closed under linear transformations and Minkowski sum. The resulting method has been used to treat several examples and has shown great performances for high dimensional systems. An extension of the method for the verification of piecewise linear hybrid systems is proposed.

Geometry of Log-concave Functions and Measures

Abstract: We present a view of log-concave measures, which enables one to build an isomorphic theory for high dimensional log-concave measures, analogous to the corresponding theory for convex bodies. Concepts such as duality and the Minkowski sum are described for log-concave functions. In this context, we interpret the Brunn–Minkowski and the Blaschke–Santalo inequalities and prove the two corresponding reverse inequalities. We also prove an analog of Milman’s quotient of subspace theorem, and present a functional version of the Urysohn inequality.

Tropical Discriminants

Abstract: Tropical geometry is used to develop a new approach to the theory of discriminants and resultants in the sense of Gel'fand, Kapranov and Zelevinsky. The tropical A-discriminant, which is the tropicalization of the dual variety of the projective toric variety given by an integer matrix A, is shown to coincide with the Minkowski sum of the row space of A and of the tropicalization of the kernel of A. This leads to an explicit positive formula for the extreme monomials of any A-discriminant, without any smoothness assumption.

Sums, Projections, and Sections of Lattice Sets, and the Discrete Covariogram

Abstract: Basic properties of finite subsets of the integer lattice ℤn are investigated from the point of view of geometric tomography. Results obtained concern the Minkowski addition of convex lattice sets and polyominoes, discrete X-rays and the discrete and continuous covariogram, the determination of symmetric convex lattice sets from the cardinality of their projections on hyperplanes, and a discrete version of Meyer’s inequality on sections of convex bodies by coordinate hyperplanes.

Toric surface codes and Minkowski sums

Abstract: Toric codes are evaluation codes obtained from an integral convex polytope $P \subset \R^n$ and finite field $\F_q$. They are, in a sense, a natural extension of Reed-Solomon codes, and have been studied recently by J. Hansen and D. Joyner. In this paper, we obtain upper and lower bounds on the minimum distance of a toric code constructed from a polygon $P \subset \R^2$ by examining Minkowski sum decompositions of subpolygons of $P$. Our results give a simple and unifying explanation of bounds of Hansen and empirical results of Joyner; they also apply to previously unknown cases.

Spacelike Normal Curves in Minkowski Space E^3_1

Abstract: In the Euclidean space E3, it is well known that normal curves, i.e., curves with position vector always lying in their normal plane, are spherical curves [3]. Necessary and sufficient conditions for a curve to be a spherical curve in Euclidean 3-space are given in [10] and [11]. In this paper, we give some characterizations of spacelike normals curves with spacelike, timelike or null principal normal in the Minkowski 3-space E31.

On f-vectors of Minkowski additions of convex polytopes

Abstract: The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of face lattice of the sum in terms of the face lattice of a given perfectly centered polytope. Exact face counting formulas are then obtained for perfectly centered simplices and hypercubes. The second type of results concerns tight upper bounds for the f-vectors of Minkowski sums of several polytopes.

Set-Valued Approximations with Minkowski Averages – Convergence and Convexification Rates

Abstract: Three approximation processes for set-valued functions (multifunctions) with compact images in ℝ n are investigated. Each process generates a sequence of approximants, obtained as finite Minkowski averages (convex combinations) of given data of compact sets in ℝ n . The limit of the sequence exists and and is equal to the limit of the same process, starting from the convex hulls of the given data. The common phenomenon of convexification of the approximating sequence is investigated and rates of convergence are obtained. The main quantitative tool in our analysis is the Pythagorean type estimate of Cassels for the “inner radius” measure of nonconvexity of a compact set. In particular we prove the convexity of the images of the limit multifunction of set-valued spline subdivision schemes and provide error estimates for the approximation of set-valued integrals by Riemann sums of sets and for Bernstein type approximation to a set-valued function.

Computing All Faces of the Minkowski Sum of V-Polytopes

Abstract: We consider the problem of listing faces of the Minkowski sum of several V-polytopes in R^d. An algorithm for listing all faces of dimension up to j is presented, for any given 0<=j<=d-1. It runs in time polynomial in the sizes of input and output.

Computing the convolution and the Minkowski sum of surfaces

Abstract: In many applications, such as NC tool path generation and robot motion planning, it is required to compute the Minkowski sum of two objects. Generally the Minkowski sum of two rational surfaces cannot be expressed in rational form. In this paper we show that for LN spline surfaces (surfaces with a linear field of normal vectors) a closed form representation is available.

Random Closed Sets

Abstract: Concepts and results involving random sets appeared in probabilistic and statistical literature long time ago. The origin of the modern concept of a random set goes as far back as the seminal book by A.N. Kolmogorov [22] (first published in 1933) where he laid out the foundations of probability theory. He wrote [22, p. 46]

A Comprehensive and Robust Procedure for Obtaining the Nofit Polygon using Monkowski Sums

Abstract: The nofit polygon is a powerful and effective tool for handling the geometric requirements of solution approaches to irregular cutting and packing problems. Although the concept was first described in 1996, it was not until the early 90s that the general trend of research moved away from direct trigonometry to favour the nofit polygon. Since then, the ability to calculate the nofit polygon has practically become a pre-requisiste for researching irregular packing problems. However, realisation of this concept in the form of a robust algorithm is a highly challenging task with few instructive approaches published. In this paper, a procedure using the mathematical concept of Minkowski sums for the calculation of the nofit polygon is presented. The described procedure is more robust than other approaches using Minkowski Sum knowledge and includes details of the removal of internal edges to find holes, slits and lock and key positions. The procedure is tested on benchmark data sets and gives examples of complicated cases. In addition the paper includes a description of how the procedure is modified in order to realise the inner-fit polygon.

Toric resultants and applications to geometric modelling

Abstract: Toric (or sparse) elimination theory uses combinatorial and discrete geometry to exploit the structure of a given system of algebraic equations The basic objects are the Newton polytope of a polynomial, the Minkowski sum of a set of convex polytopes, and a mixed polyhedral subdivision of such a Minkowski sum Different matrices expressing the toric resultant shall be discussed, and effective methods for their construction will be described based on discrete geometric operations, namely the subdivision-based methods and the incremental algorithm The former allows us to produce Macaulay-type formulae of the toric resultant by determining a matrix minor that divides the determinant in order to yield the precise resultant Toric resultant matrices exhibit a quasi-Toeplitz structure, which may reduce complexity by almost one order of magnitude in terms of matrix dimension

On the theory of mainstay parallelohedra

Abstract: In 1998 the first author announced a theorem stating that every primitive -dimensional parallelohedron can be represented, up to an affine transformation, as a weighted Minkowski sum of parallelohedra belonging to a certain finite set of -dimensional mainstay parallelohedra situated in a special way. This paper contains a detailed proof of this theorem in a refined and definitive form.

Pianos are not flat: rigid motion planning in three dimensions

Abstract: Consider a robot R that is either a line segment or the Minkowski sum of a line segment and a 3-ball, and a set S of polyhedral obstacles with a total of n vertices in R3. We design near-optimal exact algorithms for planning the motion of R among S when R is allowed to translate and rotate. Specifically, we can preprocess S in time O(n4+e) for any e > 0 into a data structure that given two placements α and β of R, can decide in time O(log n) whether a collision-free rigid motion of R between α and β exists and if so, output such a motion in time asymptotically proportional to its complexity. Furthermore, we can find in time O(n4+e) for any e > 0 the largest placement of a similar (translated, rotated and scaled) copy of R that does not intersect S. A number of additional stronger results are provided. Our line segment motion planning algorithm improves the result of Ke and O'Rourke by two orders of magnitude and almost matches their lower bound, thus settling a classical motion planning problem first considered by Schwartz and Sharir in 1984. This implies a number of natural directions for future work concerning rigid motion planning in three dimensions.

Crystal Growth Simulations: a new Mathematical Model based on the Minkowski Sum of Sets

Abstract: Realistic crystal growth simulators can give information on what would be the surface structure of a crystal grown under specific physical-chemical conditions, avoiding the real growth in a laboratory. By suitable upscaling, simulations can therefore be useful for industrial purposes to foresee and control the final product. We initially present Monte-Carlo micro-scale simulations based on spatial stochastic processes; we followed the Hartman and Perdok method for the classification of facets, using Growth Units with cubic shape. For a better description of a self-similar growth of a faceted surface, we propose a new mathematical micro-scale model based on the Minkowski sum of sets and we present some results obtained by the numerical implementations.

Accurate sampling-based algorithms for surface extraction and motion planning

Abstract: Boolean operations, Minkowski sum evaluation, configuration space computation, and motion planning are fundamental problems in solid modeling and robotics. Their applications include computer-aided design, numerically-controlled machining, tolerance verification, packing, assembly planning, and dynamic simulation: Prior algorithms for solving these problems can be classified into exact and approximate approaches. The exact approaches are difficult to implement and are prone to robustness problems. Current approximate approaches may not solve these problems accurately. Our work aims to bridge this gap between exact and approximate approaches. We present a sampling-based approach to solve these geometric problems. Our approach relies on computing a volumetric grid in space using a sampling condition. If the grid satisfies the sampling condition, our algorithm can provide geometric and topological guarantees on the output. We classify the geometric problems into two classes. The first class includes surface extraction problems such as Boolean operations, Minkowski sum evaluation, and configuration space computation. We compute an approximate boundary of the final solid defined using these geometric operations. Our algorithm computes an approximation that is guaranteed to be topologically equivalent to the exact surface and bounds the approximation error using two-sided Hausdorff error. We demonstrate the performance of our approach for the following applications: Boolean operations on complex polyhedral models and low degree algebraic primitives, model simplification and remeshing of polygonal models, Minkowski sums and offsets of complex polyhedral models, and configuration spare computation for low degrees of freedom objects. The second class of problems is motion planning of rigid or articulated robots translating or rotating among stationary obstacles. We present an algorithm for complete motion planning, i.e., finding a path if one exists and reporting a failure otherwise. Our algorithm performs deterministic sampling to compute a roadmap that captures the connectivity of free space. We demonstrate the performance of our algorithm on challenging environments with narrow passages and no collision-free paths.

Boundary evaluation algorithms for Minkowski combinations of complex sets using topological analysis of implicit curves

Abstract: Minkowski geometric algebra is concerned with sets in the complex plane that are generated by algebraic combinations of complex values varying independently over given sets in ℂ. This algebra provides an extension of real interval arithmetic to sets of complex numbers, and has applications in computer graphics and image analysis, geometrical optics, and dynamical stability analysis. Algorithms to compute the boundaries of Minkowski sets usually invoke redundant segmentations of the operand-set boundaries, guided by a “matching” criterion. This generates a superset of the true Minkowski set boundary, which must be extracted by the laborious process of identifying and culling interior edges, and properly organizing the remaining edges. We propose a new approach, whereby the matching condition is regarded as an implicit curve in the space ℝn whose coordinates are boundary parameters for the n given sets. Analysis of the topological configuration of this curve facilitates the identification of sets of segments on the operand boundaries that generate boundary segments of the Minkowski set, and rejection of certain sets that satisfy the matching criterion but yield only interior edges. Geometrical relations between the operand set boundaries and the implicit curve in ℝn are derived, and the use of the method in the context of Minkowski sums, products, planar swept volumes, and Horner terms is described.

Separability of H-Convex Sets

Abstract: We consider the problem of H-separability for two H-convex sets A,B ⊂ R. There are two types of H-separability. The first one, called “strict H-separability", is the separation (in the usual sense) of the sets A and B by an H-convex hyperplane. The second one (“weak H-separability") means to look for an H-convex half-space P such that A is situated in P , whereas B has no point in common with the interior of P . We give necessary and sufficient conditions for both these types of H-separability; the results are connected with H-convexity of the Minkowski sum of H-convex sets, see [7]. Some examples illustrate the obtained results.

On canonical representation of convex polyhedra

Abstract: Convex polyhedra are important objects in various areas of mathematics and other disciplines. A fundamental result, known as Minkowski-Weyl theorem, states that every polyhedron admits two types of representations, either as the solution set to a finite system of linear inequalities or the Minkowski sum of a finite set of points and half-rays. These are usually referred to as its H-representations and its V-representations, respectively. Neither H-representations nor V-representations are unique. Hence, deciding whether two H-representations, or two V-representations describe the same polyhedron is a nontrivial problem. We identify particular representations which are easy to determine, are compact, and reflect the geometrical properties of the underlying polyhedron. Key ingredients to this discussion are affine transformations of polyhedra, duality and complexity of polyhedral decision problems. In this dissertation, we discuss in detail the problem of refining the definitions of Hand V- representations such as to guarantee a one to one correspondence between polyhedra and their representations. As a convenient formalism, we introduce the notion of the canonical representation rule, which is any function assigning to each polyhedron P a particular H-representation, and a particular V-representation. The orthogonal representation rule is a well-known example of such a function. We define the lexico-smallest representation rule, an improved canonical representation rule that produce representations which are nonredundant, make the underlying geometry transparent and, furthermore, are sparse. These rules also exhibit nice polarity properties that can be exploited for computation. The key computational tool is linear programing. Furthermore, we show that the lexico-smallest H-representation of a perfect bipartite matching polyhedron is easy to determine using the combinatorial properties of the underlying graph. Finally, we discuss the complexity of various polyhedral verification problems related to representation of convex polyhedra. The standard technique to identify the redundancies within an H- or a V-representation is to use linear programming. We discuss the complexity of the converse reduction. More precisely, we show that deciding the feasibility of a system of linear inequalities can be reduced to deciding redundancy in an H- or in a V- representation in linear time. Also, we will study the equivalence of these problems with those of identifying the implicit linearities within an H- or a V- representation, and the boundedness of a polyhedron.

Computing faces up to k dimensions of a Minkowski sum of polytopes

Abstract: We consider the problem of listing faces of the Minkowski sum of several V-polytopes in R^d. An algorithm for listing all faces of dimension up to j is presented, for any given 0<=j<=d-1. It runs in time polynomial in the sizes of input and output.

Decomposability and Indecomposability of Binary Shapes: A Theory of Shape Arithmetics

Abstract: We observe that the resemblance between the integer number system with multiplication & division and the system of convex objects with Minkowski addition & decomposition is really striking. We present an idea of the shape decomposition into prime shapes, which are analogue of the prime numbers and indecomposable ones. Here, we concentrate the discussion on binary images, and present some propositions on the indecomposability problem.

Successive linear programs for computing all integral points in a minkowski sum

Abstract: The computation of all integral points in Minkowski (or vector) sums of convex lattice polytopes of arbitrary dimension appears as a subproblem in algebraic variable elimination, parallel compiler code optimization, polyhedral combinatorics and multivariate polynomial multiplication. We use an existing approach that avoids the costly construction of the Minkowski sum by an incremental process of solving Linear Programming (LP) problems. Our main contribution is to exploit the similarities between LP problems in the tree of LP instances, using duality theory and the two-phase simplex algorithm. Our public domain implementation improves substantially upon the performance of the above mentioned approach and is faster than porta on certain input families; besides, the latter requires a description of the Minkowski sum which has high complexity. Memory consumption limits commercial or free software packages implementing multivariate polynomial multiplication, whereas ours can solve all examined data, namely of dimension up to 9, using less than 2.7 MB (before actually outputting the points) for instances yielding more than 3 million points.

A robust procedure for obtaining the nofit polygon using Minkowski sums

Abstract: The nofit polygon is a powerful tool for handling the geometry of nesting problems. A procedure using the mathematical concept of Minkowski sums for calculating the nofit polygon is presented. It is more efficient and reliable than other Minkowski Sum approaches. Computational experience shows that it is general and accessible.

Regular Metric: Definition and Characterization in the Discrete Plane

Abstract: We say that a metric space is regular if a straight-line (in the metric space sense) passing through the center of a sphere has at least two diametrically opposite points. The normed vector spaces have this property. Nevertheless, this property might not be satisfied in some metric spaces. In this work, we give a characterization of an integer-valued translation-invariant regular metric defined on the discrete plane, in terms of a symmetric subset B that induces through a recursive Minkowski sum, a chain of subsets that are morphologically closed with respect to B.