Showing papers on "Minkowski addition published in 2001"

Decomposition of Polytopes and Polynomials

Abstract: Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral polygons is NP-complete then present a pseudo-polynomial-time algorithm for decomposing polygons. For higher-dimensional polytopes, we give a heuristic algorithm which is based upon projections and uses randomization. Applications of our algorithms include absolute irreducibility testing and factorization of polynomials via their Newton polytopes.

Minkowski Geometric Algebra of Complex Sets

Abstract: A geometric algebra of point sets in the complex plane is proposed, based on two fundamental operations: Minkowski sums and products. Although the (vector) Minkowski sum is widely known, the Minkowski product of two-dimensional sets (induced by the multiplication rule for complex numbers) has not previously attracted much attention. Many interesting applications, interpretations, and connections arise from the geometric algebra based on these operations. Minkowski products with lines and circles are intimately related to problems of wavefront reflection or refraction in geometrical optics. The Minkowski algebra is also the natural extension, to complex numbers, of interval-arithmetic methods for monitoring propagation of errors or uncertainties in real-number computations. The Minkowski sums and products offer basic 'shape operators' for applications such as computer-aided design and mathematical morphology, and may also prove useful in other contexts where complex variables play a fundamental role – Fourier analysis, conformal mapping, stability of control systems, etc.

Tight error bounds of geometric problems on convex objects with imprecise coordinates

Abstract: We study accuracy guaranteed solutions of geometric problems denned on convex region under an assumption that input points are known only up to a limited accuracy, that is, each input point is given by a convex region that represents the possible locations of the point. We show how to compute tight error bounds for basic problems such as convex hull, Minkowski sum of convex polygons, diameter of points, and so on. To compute tight error bound from imprecise coordinates, we represent a convex region by a set of half-planes whose intersection gives the region. Error bounds are computed by applying rotating caliper paradigm to this representation.

An Efficient Algorithm to Calculate the Minkowski Sum of Convex 3D Polyhedra

Abstract: A new method is presented to calculate the Minkowski sum of two convex polyhedra A and B in 3D. The method works as follows. The slope diagrams of A and B are considered as graphs. These graphs are given edge attributes. From these attributed graphs the attributed graph of the Minkowski sum is constructed. This graph is then transformed into the Minkowski sum of A and B. The running time of the algorithm is linear in the number of edges of the Minkowski sum.

Bour's Theorem in Minkowski Geometry

Abstract: In three dimensional Minkowski space, we show that a generalized helicoid is isometric to a rotation surface so that helices on the helicoid correspond to parallel circles on the rotation surface. Moreover, if these surfaces have the same Gauss map, we can determine them.

On the Busemann area in Minkowski spaces.

Abstract: Among the different notions of area in a Minkowski space, those due to Busemann and to Holmes and Thompson, respectively, have found particular attention. In recent papers it was shown that the Holmes-Thompson area is integral-geometric, in the sense that certain integral-geometric formulas of Croftontype, well known for the area in Euclidean space, can be carried over to Minkowski spaces and the Holmes-Thompson area. In the present paper, the Busemann area is investigated from this point of view. MSC 2000: 52A21 (primary); 46B20, 52A22, 53C65 (secondary)

Linear time computation of feasible regions for robust compensators

Abstract: We introduce an application of computational geometry, including figures of merit standard in the analysis of algorithms, to the design of robust control systems. With respect to system transfer function magnitude, we show how to compute feasible regions for compensators whose plant transfer function is the ratio of uncertain interval polynomials. Our solution sweeps the Minkowski quotient set of the corresponding Kharitonov rectangles. Enumerating the winding numbers of Minkowski sum convolution curves, we obtain optimal, linear time algorithms that eliminate three factors from the execution inefficiency of traditional gridding approaches. We illustrate with examples pertinent to quantitative feedback theory (QFT). Copyright © 2001 John Wiley & Sons, Ltd.