Approximation of Convex Bodies by Polytopes (original) (raw)

Hausdorff approximation of 3D convex polytopes

Information Processing Letters, 2008

Let P be a convex polytope in R d , d = 3 or 2, with n vertices. We present linear time algorithms for approximating P by simpler polytopes. For instance, one such algorithm selects k < n vertices of P whose convex hull is the approximating polytope. The rate of approximation, in the Hausdorff distance sense, is best possible in the worst case. An analogous algorithm, where the role of vertices is taken by facets, is presented.

Constructing a polytope to approximate a convex body

Geometriae Dedicata, 1995

We develop an algorithm to construct a convex poIytope P with n vertices, contained in an arbitrary convex body K in R a, so that the ratio of the volumes IK\P[/[K[ is dominated by c. d/n 2/(a-1).

Linear time approximation of 3D convex polytopes

Computational Geometry, 2002

We develop algorithms for the approximation of a convex polytope in R 3 by polytopes that are either contained in it or containing it, and that have fewer vertices or facets, respectively. The approximating polytopes achieve the best possible general order of precision in the sense of volume-difference. The running time is linear in the number of vertices or facets. (M.A. Lopez), reisner@math.haifa.ac.il (S. Reisner). 0925-7721/02/$ -see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0925-7721(02 )0 0 10 0-1

Hausdorff approximation of convex polygons

Computational Geometry, 2005

We develop algorithms for the approximation of convex polygons with n vertices by convex polygons with fewer (k) vertices. The approximating polygons either contain or are contained in the approximated ones. The distance function between convex bodies which we use to measure the quality of the approximation is the Hausdorff metric. We consider two types of problems: min -#, where the goal is to minimize the number of vertices of the output polygon, for a given distance ε, and min -ε, where the goal is to minimize the error, for a given maximum number of vertices. For min -# problems, our algorithms are guaranteed to be within one vertex of the optimal, and run in O(n log n) and O(n) time, for inner and outer approximations, respectively. For min -ε problems, the error achieved is within an arbitrary factor α > 1 from the best possible one, and our inner and outer approximation algorithms run in O(f (α, P ) · n log n) and O(f (α, P ) · n) time, respectively. Where the factor f (α, P ) has reciprocal logarithmic growth as α decreases to 1, this factor depends on the shape of the approximated polygon P .  (M.A. Lopez), reisner@math.haifa.ac.il (S. Reisner).

Inner and outer approximations of polytopes using boxes

Computational Geometry, 2004

This paper deals with the problem of approximating a convex polytope in any finite dimension by a collection of (hyper)boxes. More exactly, given a polytope P by a system of linear inequalities, we look for two collections I and E of boxes with non-overlapping interiors such that the union of all boxes in I is contained in P and the union of all boxes in E contains P. We propose and test several techniques to construct I and E aimed at getting a good balance between two contrasting objectives: minimize the volume error and minimize the total number of generated boxes. We suggest how to modify the proposed techniques in order to approximate the projection of P onto a given subspace without computing the projection explicitly.

New analysis of the sphere covering problems and optimal polytope approximation of convex bodies

Journal of Approximation Theory, 2005

In this paper, we show that both sphere covering problems and optimal polytope approximation of convex bodies are related to optimal Delaunay triangulations, which are the triangulations minimizing the interpolation error between function x 2 and its linear interpolant based on the underline triangulations. We then develop a new analysis based on the estimate of the interpolation error to get the Coxeter-Few-Rogers lower bound for the thickness in the sphere covering problem and a new estimate of the constant del n appeared in the optimal polytope approximation of convex bodies.