Geodesic-Preserving Polygon Simplification

Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon $\mathcal{P}$ by a polygon $\mathcal{P}'$ such that (1) $\mathcal{P}'$ contains $\mathcal{P}$, (2) $\mathcal{P}'$ has its reflex vertices at the same positions as $\mathcal{P}$, and (3) the number of vertices of $\mathcal{P}'$ is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both $\mathcal{P}$ and $\mathcal{P}'$, our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of $\mathcal{P}$

Threadable Curves

We define a plane curve to be threadable if it can rigidly pass through a point-hole in a line L without otherwise touching L. Threadable curves are in a sense generalizations of monotone curves. We have two main results. The first is a linear-time algorithm for deciding whether a polygonal curve is threadable---O(n) for a curve of n vertices---and if threadable, finding a sequence of rigid motions to thread it through a hole. We also sketch an argument that shows that the threadability of algebraic curves can be decided in time polynomial in the degree of the curve. The second main result is an O(n polylog n)-time algorithm for deciding whether a 3D polygonal curve can thread through hole in a plane in R^3, and if so, providing a description of the rigid motions that achieve the threading.Comment: 16 pages, 12 figures, 12 references. v2: Revised with brief addendum after Mikkel Abrahamsen pointed us to a relevant reference on "sweepable polygons." v3: Major revisio

The Voronoi Diagram of Rotating Rays With applications to Floodlight Illumination

We introduce the Voronoi Diagram of Rotating Rays, a Voronoi structure where the input sites are rays, and the distance function is the counterclockwise angular distance between a point and a ray-site. This novel Voronoi diagram is motivated by illumination and coverage problems, where a domain has to be covered by floodlights (wedges) of uniform angle, and the goal is to find the minimum angle necessary to cover the domain. We study the diagram in the plane, and we present structural properties, combinatorial complexity bounds, and a construction algorithm. If the rays are induced by a convex polygon, we show how to construct the ray Voronoi diagram within this polygon in linear time. Using this information, we can find in optimal linear time the Brocard angle, the minimum angle required to illuminate a convex polygon with floodlights of uniform angle. This last algorithm improves upon previous results, settling an interesting open problem

Flat Zipper-Unfolding Pairs for Platonic Solids

We show that four of the five Platonic solids' surfaces may be cut open with a Hamiltonian path along edges and unfolded to a polygonal net each of which can "zipper-refold" to a flat doubly covered parallelogram, forming a rather compact representation of the surface. Thus these regular polyhedra have particular flat "zipper pairs." No such zipper pair exists for a dodecahedron, whose Hamiltonian unfoldings are "zip-rigid." This report is primarily an inventory of the possibilities, and raises more questions than it answers.Comment: 15 pages, 14 figures, 8 references. v2: Added one new figure. v3: Replaced Fig. 13 to remove a duplicate unfolding, reducing from 21 to 20 the distinct unfoldings. v4: Replaced Fig. 13 again, 18 distinct unfolding

New Results on Abstract Voronoi Diagrams

Voronoi diagrams are a fundamental structure used in many areas of science. For a given set of objects, called sites, the Voronoi diagram separates the plane into regions, such that points belonging to the same region have got the same nearest site. This definition clearly depends on the type of given objects, they may be points, line segments, polygons, etc. and the distance measure used. To free oneself from these geometric notions, Klein introduced abstract Voronoi diagrams as a general construct covering many concrete Voronoi diagrams. Abstract Voronoi diagrams are based on a system of bisecting curves, one for each pair of abstract sites, separating the plane into two dominance regions, belonging to one site each. The intersection of all dominance regions belonging to one site p defines its Voronoi region. The system of bisecting curves is required to fulfill only some simple combinatorial properties, like Voronoi regions to be connected, the union of their closures cover the whole plane, and the bisecting curves are unbounded. These assumptions are enough to show that an abstract Voronoi diagram of n sites is a planar graph of complexity O(n) and can be computed in expected time O(n log n) by a randomized incremental construction. In this thesis we widen the notion of abstract Voronoi diagrams in several senses. One step is to allow disconnected Voronoi regions. We assume that in a diagram of a subset of three sites each Voronoi region may consist of at most s connected components, for a constant s, and show that the diagram can be constructed in expected time O(s2 n ∑3 ≤ j ≤ n mj / j), where mj is the expected number of connected components of a Voronoi region over all diagrams of a subset of j sites. The case that all Voronoi regions are connected is a subcase, where this algorithm performs in optimal O(n log n) time, because here s = mj =1. The next step is to additionally allow bisecting curves to be closed. We present an algorithm constructing such diagrams which runs in expected time O(s2 n log(max{s,n}) ∑2 ≤ j≤ n mj / j). This algorithm is slower by a log n-factor compared to the one for disconnected regions and unbounded bisectors. The extra time is necessary to be able to handle special phenomenons like islands, where a Voronoi region is completely surrounded by another region, something that can occur only when bisectors are closed. However, this algorithm solves many open problems and improves the running time of some existing algorithms, for example for the farthest Voronoi diagram of n simple polygons of constant complexity. Another challenge was to study higher order abstract Voronoi diagrams. In the concrete sense of an order-k Voronoi diagram points are collected in the same Voronoi region, if they have the same k nearest sites. By suitably intersecting the dominance regions this can be defined also for abstract Voronoi diagrams. The question arising is about the complexity of an order-k Voronoi diagram. There are many subsets of size k but fortunately many of them have an empty order-k region. For point sites it has already been shown that there can be at most O(k (n-k)) many regions and even though order-k regions may be disconnected when considering line segments, still the complexity of the order-k diagram remains O(k(n-k)). The proofs used to show this strongly depended on the geometry of the sites and the distance measure, and were thus not applicable for our abstract higher order Voronoi diagrams. The proofs used to show this strongly depended on the geometry of the sites and the distance measure, and were thus not applicable for our abstract higher order Voronoi diagrams. Nevertheless, we were able to come up with proofs of purely topological and combinatorial nature of Jordan curves and certain permutation sequences, and hence we could show that also the order-k abstract Voronoi diagram has complexity O(k (n-k)), assuming that bisectors are unbounded, and the order-1 regions are connected. Finally, we discuss Voronoi diagrams having the shape of a tree or forest. Aggarwal et. al. showed that if points are in convex position, then given their ordering along the convex hull, their Voronoi diagram, which is a tree, can be computed in linear time. Klein and Lingas have generalized this idea to Hamiltonian abstract Voronoi diagrams, where a curve is given, intersecting each Voronoi region with respect to any subset of sites exactly once. If the ordering of the regions along the curve is known in advance, all Voronoi regions are connected, and all bisectors are unbounded, then the abstract Voronoi diagram can be computed in linear time. This algorithm also applies to diagrams which are trees for all subsets of sites and the ordering of the unbounded regions around the diagram is known. In this thesis we go one step further and allow the diagram to be a forest for subsets of sites as long as the complete diagram is a tree. We show that also these diagrams can be computed in linear time