Order on Order Types

Abstract

Given P and P , equally sized planar point sets in general position, we call a bijection from P to P crossing-preserving if crossings of connecting segments in P are preserved in P (extra crossings may occur in P ). If such a mapping exists, we say that P crossing-dominates P, and if such a mapping exists in both directions, P and P are called crossing-equivalent. The relation is transitive, and we have a partial order on the obtained equivalence classes (called crossing types or x-types). Point sets of equal order type are clearly crossing-equivalent, but not vice versa. Thus, x-types are a coarser classification than order types. (We will see, though, that a collapse of different order types to one x-type occurs for sets with triangular convex hull only.) We argue that either the maximal or the minimal x-types are sufficient for answering many combinatorial (existential or extremal) questions on planar point sets. Motivated by this we consider basic properties of the relation. We characterize order types crossingdominated by points in convex position. Further, we give a full characterization of minimal and maximal abstract order types. Based on that, we provide a polynomialtime algorithm to check whether a point set crossing-dominates another. Moreover, we generate all maximal and minimal x-types for small numbers of points.Dedicated to Jacob E. Goodman and Richard Pollack on the occasion of their eightieth birthdays.

Packing plane spanning trees and paths in complete geometric graphs

“…Using the order type database for small point sets [2] it can be easily checked that GK 8 and GK 9 each contain 4 edge-disjoint plane spanning trees, and that GK 10 contains 5 edge-disjoint plane spanning trees. (The latter has been obtained by reducing the set of order types to so-called crossing-maximal ones, as characterized in [18].) Theorem 3.…”

An Optimal Algorithm for Reconstructing Point Set Order Types from Radial Orderings

“…Another variation on radial systems was studied by Tovar, Freda, and LaValle [21] in the context of a robot that can sense landmarks around it. Disser et al [6] and Chen and Wang [5] consider the polygon reconstruction problem from angles, where the objective is to reconstruct a polygon when given, for each vertex v, the angles with the other vertices of the polygon visible from v. Pilz and Welzl [18] describe a hierarchy on order types based on crossing edges; two order types are equivalent in their partial order if and only if they have the same radial system. We refer to the work by Aichholzer et al [1] for a more complete list of related work.…”

“…The following lemma reformulates the fact that an inner important triangle is not contained in a convex quadrilateral (see Figure 2). [18]). Let P be a generalized configuration of n points, and let χ be the abstract order type of P .…”

“…In this context, we mention that, if R is the radial system of some point set order type, then every abstract order type with radial system R can be realized as a point set (see Theorem 27 in Pilz and Welzl [18]). We do not deal with the realizability of abstract order types as point sets in this work.…”