Showing papers on "Minkowski addition published in 1997"

Quantizations of Flag Manifolds and Conformal Space Time

Abstract: In this paper we work out the deformations of some flag manifolds and of complex Minkowski space viewed as an affine big cell inside G(2,4). All the deformations come in tandem with a coaction of the appropriate quantum group. In the case of the Minkowski space this allows us to define the quantum conformal group. We also give two involutions on the quantum complex Minkowski space, that respectively define the real Minkowski space and the real euclidean space. We also compute the quantum De Rham complex for both real (complex) Minkowski and euclidean space.

Convex Set Symmetry Measurement via Minkowski Addition

Abstract: We introduce and investigate measures of rotation and reflection symmetries for compact convex sets The appropriate symmetrization transformations are used to transform original sets into symmetrical ones Symmetry measures are defined as the ratio of volumes (Lebesgue measure) of the original set and the corresponding symmetrical set For the case of rotation symmetry we use as a symmetrization transformation a generalization of the Minkowski symmetric set (a difference body) for a cyclic group of rotations For the case of reflection symmetry we investigate Blaschke symmetrization Given a convex set and a hyperplane E in R^n we get a set symmetrical with respect to this hyperplane Analyzing all hyperplanes containing the coordinate center one gets the measure of reflection symmetry We discuss the lower bounds of this symmetry measure and also a derived symmetry measure In the two dimensional case a perimetric measure representation of convex sets is applied for convex sets symmetrization as well as for the symmetry measure calculation The perimetric measure allows also to perform a decomposition of a compact convex set into Minkowski sum of two sets The first one is rotationally symmetrical and the second one is completely asymmetrical in the sense that it does not allow such a decomposition We discuss a problem of the fast computation of symmetrization transformations Minkowski addition of two sets is reduced to the convolution of their characteristic functions Therefore, in the case of the discrete plane Z^2, Fast Fourier Transformation can be applied for the fast computation of symmetrization transformations

The Reflection Structures of Generalized Co-Minkowski Spaces Leading to K-Loops

Abstract: We extend the notion co- Minkowski plane to “unitary co- Minkowski space” (P, G, ∼) and discuss the problem whether there are subsets Q of P which can be turned into a geometric K- loop.

Podality-based time-optimal computations on enhanced meshes

Abstract: The main contribution of this paper is to present simple and elegant podality-based algorithms for a variety of computational tasks motivated by, and finding applications to, pattern recognition, computer graphics, computational morphology, image processing, robotics, computer vision, and VLSI design. The problems that we address involve computing the convex hull, the diameter, the width, and the smallest area enclosing rectangle of a set of points in the plane, as well as the problems of finding the maximum Euclidian distance between two planar sets of points, and of constructing the Minkowski sum of two convex polygons. Specifically, we show that once we fix a positive constant /spl epsiv/, all instances of size m, (n/sup 1/2 +/spl epsiv///spl les/m/spl les/n) of the problems above, stored in the first [m//spl radic/n] columns of a mesh with multiple broadcasting of size /spl radic/n/spl times//spl radic/n can be solved time-optimally in /spl Theta/(m//spl radic/n) time.

An algebra for slope-monotone closed curves

Abstract: The paper proposes a new operation for a class of slope-monotone closed curves. This operation is closely related to the Minkowski sum in that the Minkowski sum can be considered as the binary operation for figures bounded by those closed curves. In this sense, the proposed algebra is a generalization of the Minkowski sum. Conventionally, the inverse of the Minkowski sum can be defined naturally for convex figures, but the generalization of the inverse to nonconvex figures results in a complicated algebraic structure. This paper, on the other hand, gives a natural definition of the inverse for nonconvex figures, and thus constructs a simple algebraic structure.

Normals in a Minkowski Plane

Abstract: In this paper we will establish bounds on the average number of normals through a point in a convex body in a Minkowski plane for certain classes of convex bodies. Also, a related Euler relation is discussed.

Comparing Convex Shapes Using Minkowski Addition

Abstract: This report deals with similarity measures for convex shapes whose definition is based on Minkowski addition and the Brunn-Minkowski inequality. All measures considered here are invariant under translations. In addition, they may be invariant under rotations, multiplications, reflections, or affine transformations. Restricting oneselves to the class of convex polygons, it is possible to develop efficient algorithms for the computation of such similarity measures. These algorithms use a special representation of convex polygons known as the perimetric measure. Such representations are unique for convex sets and linear with respect to Minkowski addition. Although the paper deals exclusively with the 2-dimensional case, many of the results carry over almost immediately to higher-dimensional spaces.