Showing papers on "Minkowski addition published in 1995"

Computing minkowski sums of plane curves

Abstract: The Minkowski sum of two plane curves can be regarded as the area generated by sweeping one curve along the other. The boundary of the Minkowski sum consists of translated portions of the given curves and/or portions of a more complicated curve, the “envelope” of translates of the swept curve. We show that the Minkowski-sum boundary is describable as an algebraic curve (or subset thereof) when the given curves are algebraic, and illustrate the computation of its implicit equation. However, such equations are typically of high degree and do not offer a practical basis for tracing the boundary. For the case of polynomial parametric curves, we formulate a simple numerical procedure to address the latter problem, based on constructing the Gauss maps of the given curves and using them to identifying “corresponding” curve segments that are to be summed. This yields a set of discretely-sampled arcs that constitutes a superset of the Minkowski-sum boundary, and can be regarded as a planar graph. To extract the true boundary, we present a method for identifying and “trimming” away extraneous arcs by systematically traversing this graph.

Existence of solutions with singularities for the maximal surface equation in minkowski space

Abstract: Let be a domain in , and a finite tuple of points in . The problem is considered of the existence of a solution for the maximal surface equation in , where Dirichlet boundary data are given on , and the flows of the time gradient on the graph of the solution in the Minkowski space are given at the points .Bibliography: 8 titles.

The complexity of a single face of a minkowski sum.

Abstract: This note considers the complexity of a free region in the con guration space of a polygonal robot translating amidst polygonal obstacles in the plane Speci cally given polygonal sets P and Q with k and n vertices respectively k n the number of edges and vertices bounding a single face of the complement of the Minkowski sum P Q is nk k in the worst case The lower bound comes from a construction based on lower envelopes of line segments the upper bound comes from a combinatorial bound on Davenport Schinzel sequences that satisfy two alternation conditions Introduction and Background Let A and B be two sets in IR The Minkowski sum or vector sum of A and B denoted A B is the set fa b j a A b Bg The Minkowski sum is a useful concept in robot motion planning and related areas For example consider an obstacle A and a robot B that moves by translation We can choose a reference point r rigidly attached to B and suppose that B is placed such that the reference point coincides with the origin If we let B denote a copy of B rotated by then A B is the locus of placements of the reference point where A B This sum is often called a con guration space obstacle or C obstacle because B collides with A under rigid motion along a path exactly when the reference point r moved along intersects A B We con ne ourselves to the Minkowski sum of polygonal sets which is a polygonal set Let P and Q be two polygonal sets not necessarily connected with k and n vertices respectively The boundary of P Q comes from an arrangement of O nk line segments which has complexity bounded by O n k and this bound is tight in the worst case In applications such as motion planning and assembly planning however we only need to know the face complexity the number of segments that bound a single face of the complement of the Minkowski sum P Q in the worst case Figure depicts the outer face of a sum P Q Davenport Schinzel sequence analysis which is described in section shows that the face complexity is O nk nk where is the functional inverse of Ackermann s function School of Mathematical Sciences Tel Aviv University Tel Aviv Israel Department of Computer Science University of British Columbia Vancouver Canada Supported in part by an NSERC Postgraduate fellowship Department of Computer Science Polytechnic University New York USA Supported by NSF Grant CCR Robotics Lab Dept of Comp Sci Stanford University California USA Supported by NSF ARPA Grant IRI and by a grant from the Stanford Integrated Manufacturing Association SIMA Department of Computer Science University of British Columbia Vancouver Canada Supported in part by an NSERC Research Grant and a fellowship from the B C Advanced Systems Institute P Q P ⊕ Q outer boundary of P ⊕ Q v1

The Cauchy Problem for Harmonic Maps on Minkowski Space

Abstract: In this article we shall be reporting on recent progress in the study of harmonic maps from Minkowski space (M, ŋ) into a Riemannian manifold (N, g). These maps (also called wave maps or sigma models) are solutions to the wave equation with partial derivatives replaced by covariant derivatives. These equations are naturally nonlinear because the image lives on a manifold instead of a vector space, as is the case for the linear wave equation. A useful way to describe the problem would be when the target manifold N is a hypersurface in ℝk+1.

New euclidean theorems by the use of Laguerre transformations — Some geometry of Minkowski (2+1)-space

Abstract: Examples of the use of Laguerre transformations to discover theorems in the Euclidean and Minkowski planes.

Minkowski operations and vector spaces

Abstract: Mathematical morphology started as a set of tools for analysing images by the use of transformations based on set-theoretical operations which are the Minkowski sum and subtraction. It was first developed for the analysis of binary images. Its extension to grey-level images was a later development with the extension of the Minkowski operations to real-valued functions in terms of sup-convolution and inf-convolution. The purpose of this paper is to define a type of convolution between set-valued maps, to study its properties, and to establish some associated differential relations. This set-convolution map allows us to extend the Minkowski sum and substraction to multivalued functions and to functions with vectorial values.

An elementary renewal theorem for random compact convex sets

Abstract: A set-valued analog of the elementary renewal theorem for Minkowski sums of random closed sets is considered. The corresponding renewal function is defined as H(K)= P{S, c K}, n=O where S, = A1 D - A are Minkowski (element-wise) sums of i.i.d. random compact convex sets. In this paper we determine the limit of H(tK)/t as t tends to infinity. For K containing the origin as an interior point,