Order on Order Types

Given P and P\u27, equally sized planar point sets in general position, we call a bijection from P to P\u27 crossing-preserving if crossings of connecting segments in P are preserved in P\u27 (extra crossings may occur in P\u27). If such a mapping exists, we say that P\u27 crossing-dominates P, and if such a mapping exists in both directions, P and P\u27 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 crossing-dominated by points in convex position. Further, we give a full characterization of minimal and maximal abstract order types. Based on that, we provide a polynomial-time algorithm to check whether a point set crossing-dominates another. Moreover, we generate all maximal and minimal x-types for small numbers of points

Surface reconstruction between simple polygons via angle criteria

We consider the problem of connecting two simple polygons P and Q in parallel planes by a polyhedral surface. The goal is to find an optimality criterion which naturally satisfies the following conditions (i) if P and Q are convex, then the optimal surface is the convex hull of P and Q (without facets P and Q), and (ii) if P can be obtained from Q by scaling with a center c, then the optimal surface is the portion of the cone defined by P and apex c between the two planes. We provide a criterion (based on the sequences of angles of the edges of P and Q), which satisfies these conditions, and for which the optimal surface can be e ciently computed. Moreover, we supply a condition, so called angle consistency, which proved very helpful in preventing self intersections (for our and other criteria). The methods have been implemented and gave improved results in a number of examples

Convex Hulls of Random Order Types

We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3): (a) The number of extreme points in an n-point order type, chosen uniformly at random from all such order types, is on average 4+o(1). For labeled order types, this number has average 4-8/(n^2 - n +2) and variance at most 3. (b) The (labeled) order types read off a set of n points sampled independently from the uniform measure on a convex planar domain, smooth or polygonal, or from a Gaussian distribution are concentrated, i.e., such sampling typically encounters only a vanishingly small fraction of all order types of the given size. Result (a) generalizes to arbitrary dimension d for labeled order types with the average number of extreme points 2d+o(1) and constant variance. We also discuss to what extent our methods generalize to the abstract setting of uniform acyclic oriented matroids. Moreover, our methods allow to show the following relative of the Erd?s-Szekeres theorem: for any fixed k, as n ? ?, a proportion 1 - O(1/n) of the n-point simple order types contain a triangle enclosing a convex k-chain over an edge. For the unlabeled case in (a), we prove that for any antipodal, finite subset of the 2-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral or one of A?, S? or A? (and each case is possible). These are the finite subgroups of SO(3) and our proof follows the lines of their characterization by Felix Klein

Tail estimates for the space complexity of randomized incremental algorithms

We give tail estimates for the space complexity of randomized incremental algorithms for line segment intersection in the plane For n the number of segments m is the number of intersections and m n ln n ln n there is a constant c such that the probability that the total space cost exceeds c times the expected space cost is e mn ln