Oswin Aichholzer

A Novel Type of Skeleton for Polygons

A new internal structure for simple polygons, the straight skeleton, is introduced and discussed. It is composed of pieces of angular bisectores which partition the interior of a given n-gon P in a tree-like fashion into n monotone polygons. Its straight-line structure and its lower combinatorial complexity may make the straight skeleton preferable to the widely used medial axis of a polygon. As a seemingly unrelated application, the straight skeleton provides a canonical way of constructing a polygonal roof above a general layout of ground walls.

Medial axis computation for planar free-form shapes

We present a simple, efficient, and stable method for computing-with any desired precision-the medial axis of simply connected planar domains. The domain boundaries are assumed to be given as polynomial spline curves. Our approach combines known results from the field of geometric approximation theory with a new algorithm from the field of computational geometry. Challenging steps are (1) the approximation of the boundary spline such that the medial axis is geometrically stable, and (2) the efficient decomposition of the domain into base cases where the medial axis can be computed directly and exactly. We solve these problems via spiral biarc approximation and a randomized divide & conquer algorithm.

Quickest Paths, Straight Skeletons, and the City Voronoi Diagram

Abstract The city Voronoi diagram is induced by quickest paths in the L1plane, made faster by an isothetic transportation network. We investigate the rich geometric and algorithmic properties of city Voronoi diagrams, and report on their use in processing quickest-path queries. In doing so, we revisit the fact that not every Voronoi-type diagram has interpretations in both the distance model and the wavefront model. Especially, straight skeletons are a relevant example where an interpretation in the former model is lacking. We clarify the relationship between these models, and further draw a connection to the bisector-defined abstract Voronoi diagram model, with the particular goal of computing the city Voronoi diagram efficiently.

Matching Shapes with a Reference Point

For two given point sets, we present a very simple (almost trivial) algorithm to translate one set so that the Hausdorff distance between the two sets is not larger than a constant factor times the minimum Hausdorff distance which can be achieved in this way. The algorithm just matches the so-called Steiner points of the two sets. The focus of our paper is the general study of reference points (like the Steiner point) and their properties with respect to shape matching. For more general transformations than just translations, our method eliminates several degrees of freedom from the problem and thus yields good matchings with improved time bounds.

On the Crossing Number of Complete Graphs

Let (G) denote the rectilinear crossing number of a graph G. We determine (K11)=102 and (K12)=153. Despite the remarkable hunt for crossing numbers of the complete graph Kn – initiated by R. Guy in the 1960s – these quantities have been unknown forn>10 to date. Our solution mainly relies on a tailor-made method for enumerating all inequivalent sets of points (order types) of size 11.Based on these findings, we establish a new upper bound on (Kn) for general n. The bound stems from a novel construction of drawings of Kn with few crossings.

A lower bound on the number of triangulations of planar point sets

We show that the number of straight-edge triangulations exhibited by any set of n points in general position in the plane is bounded from below by Ω(2.33n).

Pseudotriangulations from Surfaces and a Novel Type of Edge Flip

We prove that planar pseudotriangulations have realizations as polyhedral surfaces in three-space. Two main implications are presented. The spatial embedding leads to a novel flip operation that allows for a drastic reduction of flip distances, especially between (full) triangulations. Moreover, several key results for triangulations, like flipping to optimality, (constrained) Delaunayhood, and a convex polytope representation, are extended to pseudotriangulations in a natural way.

The path of a triangulation

For a planar point set S let T be a triangulation of S and 1 a line properly intersecting T. We show that there always exists a unique path in T with certain properties with respect to 1. This path is then generalized to (non triangulated) point sets restricted to the interior of simple polygons. This so-called triangulation path enables us to treat several triangulation problems on planar point sets in a divide & conquer-like manner. For example, we give the first algorithm for counting triangulations of a planar point set which is observed to run in time sublinear in the number of triangulations. Moreover, the triangulation path proves to be useful for the computation of optimal triangulations.

Sequences of spanning trees and a fixed tree theorem

Let Ts be the set of all crossing-free spanning trees of a planar n-point set S. We prove that Ts contains, for each of its members T, a length-decreasing sequence of trees T0 ,..., Tk such that T0 = T, Tk = MST(S), Ti does not cross Ti-1 for i = 1,...,k, and k = O(logn). Here MST(S) denotes the Euclidean minimum spanning tree of the point set S.As an implication, the number of length-improving and planar edge moves needed to transform a tree T ∈ Ts into MST(S) is only O(n log n). Moreover, it is possible to transform any two trees in Ts into each other by means of a local and constant-size edge slide operation. Applications of these results to morphing of simple polygons are possible by using a crossing-free spanning tree as a skeleton description of a polygon.

New Lower Bounds for the Number of (≤ k)-Edges and the Rectilinear Crossing Number of K n

AbstractWe provide a new lower bound on the number of (≤ k)-edges of a set of n points in the plane in general position. We show that for