Jacob E. Goodman

Geometric Transversal Theory

Geometric transversal theory has its origins in Helly’s theorem: Theorem 1.1 (Helly’s Theorem) [49]. Suppose A is a family of at least d + 1 convex sets in IR d , and A is finite or each member of A is compact. Then if every d + 1 members of A have a common point, there is a point common to all the members of A.

On the combinatorial classification of nondegenerate configurations in the plane

Abstract We classify nondegenerate plane configurations by attaching, to each such configuration of n points, a periodic sequence of permutations of {1, 2, …, n } which satisfies some simple conditions; this classification turns out to be appropriate for questions involving convexity. In 1881 Perrin stated that every sequence satisfying these conditions was the image of some plane configuration. We show that this statement is incorrect by exhibiting a counterexample, for n = 5, and prove that for n ⩽ 5 every sequence essentially distinct from this one is realized geometrically by giving a complete classification of configurations in these cases; there is 1 combinatorial equivalence class for n = 3, 2 for n = 4, and 19 for n = 5. We develop some basic notions of the geometry of “allowable sequences” in the course of proving this classification theorem. Finally, we state some results and an open problem on the realizability question in the general case.

Upper bounds for configurations and polytopes inRd

We give a new upper bound onnd(d+1)n on the number of realizable order types of simple configurations ofn points inRd, and ofn2d2n on the number of realizable combinatorial types of simple configurations. It follows as a corollary of the first result that there are no more thannd(d+1)n combinatorially distinct labeled simplicial polytopes inRd withn vertices, which improves the best previous upper bound ofncnd/2.

Proof of Grünbaum's conjecture on the stretchability of certain arrangements of pseudolines

Abstract We prove Grunbaums conjecture that every arrangement of eight pseudolines in the projective plane is stretchable, i.e., determines a cell complex isomorphic to one determined by an arrangement of lines. The proof uses our previous results on ordered duality in the projective plane and on periodic sequences of permutations of [1,n] associated to arrangements of n lines in the euclidean plane.

On the Number of k-Subsets of a Set of n Points in the Plane

Abstract For a configuration S of n points in E 2 , H. Edelsbrunner (personal communication) has asked for bounds on the maximum number of subsets of size k cut off by a line. By generalizing to a combinatorial problem, we show that for 2 k n the number of such sets of size at most k is at most 2 nk − 2 k 2 − k . By duality, the same bound applies to the number of cells at distance at most k from a base cell in the cell complex determined by an arrangement of n lines in P 2 .

Helly-Type Theorems for Pseudoline Arrangements in P2

Abstract We prove duals of Radons theorem, Hellys theorem, Caratheodorys theorem, and Kirchbergers theorem for arrangements of pseudolines in the real projective plane, which generalize the original versions of those theorems for plane configurations of points. We also prove a topological generalization of the pseudoline-dual of Hellys theorem.

A theorem of ordered duality

We associate, to any ordered configuration of n points or ordered arrangement of n lines in the plane, a periodic sequence of permutations of [1, n] in a way which reflects the order and convexity properties of the configuration or arrangement, and prove that a sequence of permutations of [1, n] is associated to some configuration of points if and only if it is associated to some arrangement of lines. We show that this theorem generalizes the standard duality principle for the projective plane, and we use it to derive duals of several well-known theorems about arrangements, including a version of Hellys theorem for convex polygons.

Two computational geometry libraries: LEDA and CGAL

Over the past decades, two major software libraries that support a wide range of geometric computing have been developed: Leda, the Library of Efficient Data Types and Algorithms, and Cgal, the Computational Geometry Algorithms Library. We start with an introduction of common aspects of both libraries and major differences. We continue with sections that describe each library in detail. Both libraries are written in C++. Leda is based on the object-oriented paradigm and Cgal is based on the generic programming paradigm. They provide a collection of flexible, efficient, and correct software components for computational geometry. Users should be able to easily include existing functionality into their programs. Additionally, both libraries have been designed to serve as platforms for the implementation of new algorithms. Correctness is of crucial importance for a library, even more so in the case of geometric algorithms where correctness is harder to achieve than in other areas of software construction. Two well-known reasons are the exact arithmetic assumption and the nondegeneracy assumption that are often used in geometric algorithms. However, both assumptions usually do not hold: floating point arithmetic is not exact and inputs are frequently degenerate. See Chapter 45 for details.

Cell decomposition of polytopes by bending.

If a lineL crosses a polygonP, then bendingP up on both sides ofL yields a 3-polytope whose “upper” boundary projects back to a cell decomposition ofP transverse toL, and to a triangulation in the general case. We generalize this simple idea to polytopes of arbitrary dimension, and use it to answer several questions posed recently about possible decompositions of polytopes and of regions between polytopes.

The complexity of point configurations

Abstract There are several natural ways to extend the notion of the order of points on a line to higher dimensions. This article focuses on three of them—combinatorial type, order type, and isotopy class—and surveys work done in recent years on the efficient encoding of order types and on complexity questions relating to all three classifications.