The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull Algorithm with the general-dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and Delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it uses less memory. Computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floating-point arithmetic, this assumption can lead to serious errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick ” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.

The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.-Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...

polymake is a software tool designed for the algorithmic treatment of polytopes and polyhedra. We give an overview of the functionality as well as of the structure. This paper can be seen as a first approximation to a polymake handbook. The tutorial starts with the very basics and ends up with a few polymake applications to research problems. Then we present the main features of the system including the interfaces to other software products. polymake is free software; it is available on the Internet at http://www.math.tu-berlin.de/diskregeom/polymake/.

. The double description method is a simple and useful algorithm for enumerating all extreme rays of a general polyhedral cone in IR d , despite the fact that we can hardly state any interesting theorems on its time and space complexities. In this paper, we reinvestigate this method, introduce some new ideas for efficient implementations, and show some empirical results indicating its practicality in solving highly degenerate problems. 1 Introduction A pair (A; R) of real matrices A and R is said to be a double description pair or simply a DD pair if the relationship Ax 0 if and only if x = R for some 0 holds. Clearly, for a pair (A; R) to be a DD pair, it is necessary that the column size of A is equal to the row size of R, say d. The term "double description" was introduced by Motzkin et al. [MRTT53], and it is quite natural in the sense that such a pair contains two different descriptions of the same object. Namely, the set P (A) represented by A as P (A) = fx 2 IR d : Ax...

This paper describes an improved implementation of the reverse search vertex enumeration/convex hull algorithm for d-dimensional convex polyhedra. The implementation uses a lexicographic ratio test to resolve degeneracy, works on bounded or unbounded polyhedra and uses exact arithmetic with all integer pivoting. It can also be used to compute the volume of the convex hull of a set of points. For a polyhedron with m inequalities in d variables and known extreme point, it finds all bases in time O(md²) per basis. This implementation can handle problems of quite large size, especially for simple polyhedra (where each basis corresponds to a vertex and the complexity reduces to O(md) per vertex). Computational experience is included in the paper, including a comparison with an earlier implementation.

Algorithms specified for parametrically sized problems are more general purpose and more reusable than algorithms for fixed sized problems. For this reason, there is a need for representing and symbolically analyzing linearly parameterized algorithms. An important class of parallel algorithms can be described as systems of parameterized affine recurrence equations (PARE). In this representation, linearly parameterized polyhedra are used to describe the domains of variables. This paper describes an algorithm which computes the set of parameterized vertices of a polyhedron, given its representation as a system of parameterized inequalities. This provides an important tool for the symbolic analysis of the parameterized domains used to define variables and computation domains in PARE's. A library of operations on parameterized polyhedra based on the Polyhedral Library has been written in C and is freely distributed. 1 Introduction In order to improve the performance of scientific programs...

Every convex polytope can be represented as the intersection of a finite set of halfspaces and as the convex hull of its vertices. Transforming from the halfspace (respectively vertex) to the vertex (respectively halfspace) representation is called vertex enumeration (respectively facet enumeration). An open question is whether there is an algorithm for these two problems (equivalent by geometric duality) that is polynomial in the input size and the output size. In this paper, we extend the known polynomially solvable classes of polytopes by looking at the dual problems. The dual problem of a vertex (facet, respectively) enumeration problem is the facet (vertex) enumeration problem for the same polytope where the input and output are simply interchanged. For a particular class of polytopes and a fixed algorithm, one transformation may be much easier than its dual. In this paper, we propose a new class of algorithms that take advantage of this phenomenon. Loosely speaking, primal--dual ...

We show that in the worst case, Ω(n dd=2e\Gamma1 +n log n) sidedness queries are required to determine whether the convex hull of n points in R^d is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follows from a straightforward adversary argument. A key step in the proof is the construction of a quasi-simplicial n-vertex polytope with Ω(n dd=2e\Gamma1 ) degenerate facets. While it has been known for several years that d-dimensional convex hulls can have Ω(n bd=2c ) facets, the previously best lower bound for these problems is only Ω(n log n). Using similar techniques, we also obtain simple and correct proofs of Erickson and Seidel's lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane. As a related result, we show that detecting simplicial convex hulls in R^d is ⌈d/2⌉-hard, in the in the sense of Gajentaan and Overmars.

Abstract. Computing reachable sets is an essential step in most analysis and synthesis techniques for hybrid systems. The representation of these sets has a deciding impact on the computational complexity and thus the applicability of these techniques. This paper presents a new approach for approximating reachable sets using oriented rectangular hulls (ORHs), the orientations of which are determined by singular value decompositions of sample covariance matrices for sets of reachable states. The orientations keep the over-approximation of the reachable sets small in most cases with a complexity of low polynomial order with respect to the dimension of the continuous state space. We show how the use of ORHs can improve the efficiency of reachable set computation significantly for hybrid systems with nonlinear continuous dynamics.

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