Algorithmic and Combinatorial Results on Fence Patrolling, Polygon Cutting and Geometric Spanners

The purpose of this dissertation is to study problems that lie at the intersection of geometry and computer science. We have studied and obtained several results from three different areas, namely–geometric spanners, polygon cutting, and fence patrolling. Specifically, we have designed and analyzed algorithms along with various combinatorial results in these three areas. For geometric spanners, we have obtained combinatorial results regarding lower bounds on worst case dilation of plane spanners. We also have studied low degree plane lattice spanners, both square and hexagonal, of low dilation. Next, for polygon cutting, we have designed and analyzed algorithms for cutting out polygon collections drawn on a piece of planar material using the three geometric models of saw, namely, line, ray and segment cuts. For fence patrolling, we have designed several strategies for robots patrolling both open and closed fences

Single-Strip Triangulation of Manifolds with Arbitrary Topology

Triangle strips have been widely used for efficient rendering. It is NP-complete to test whether a given triangulated model can be represented as a single triangle strip, so many heuristics have been proposed to partition models into few long strips. In this paper, we present a new algorithm for creating a single triangle loop or strip from a triangulated model. Our method applies a dual graph matching algorithm to partition the mesh into cycles, and then merges pairs of cycles by splitting adjacent triangles when necessary. New vertices are introduced at midpoints of edges and the new triangles thus formed are coplanar with their parent triangles, hence the visual fidelity of the geometry is not changed. We prove that the increase in the number of triangles due to this splitting is 50% in the worst case, however for all models we tested the increase was less than 2%. We also prove tight bounds on the number of triangles needed for a single-strip representation of a model with holes on its boundary. Our strips can be used not only for efficient rendering, but also for other applications including the generation of space filling curves on a manifold of any arbitrary topology.Comment: 12 pages, 10 figures. To appear at Eurographics 200

Systems Issues in Solid Freeform Fabrication

This paper is concerned with the systems aspects of the Solid Freeform Fabrication (SFF) technology, i.e., the issues that deal with getting an external geometric CAD model to automatically control the physical layering fabrication process as directly as possible, regardless ofthe source of the model. The general systems issues are described, the state of systems research is given, and open research questions are posed.Mechanical Engineerin

The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings

Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain this phenomenon, we use Galois theory to show that many variants of these problems have solutions that cannot be expressed by nested radicals or nested roots of low-degree polynomials. Hence, such solutions cannot be computed exactly even in extended computational models that include such operations.Comment: Graph Drawing 201

3-D model construction using range and image data

This paper deals with the automated creation of geometric and photometric correct 3-D models of the world. Those models can be used for virtual reality, tele-presence, digital cinematography and urban planning applications. The combination of range (dense depth estimates) and image sensing (color information) provides data-sets which allow us to create geometrically correct, photorealistic models of high quality. The 3-D models are first built from range data using a volumetric set intersection method previously developed by us. Photometry can be mapped onto these models by registering features from both the 3-D and 2-D data sets. Range data segmentation algorithms have been developed to identify planar regions, determine linear features from planar intersections that can serve as features for registration with 2-D imagery lines, and reduce the overall complexity of the models. Results are shown for building models of large buildings on our campus using real data acquired from multiple sensors