Algorithms for fat objects : decompositions and applications

Computational geometry is the branch of theoretical computer science that deals with algorithms and data structures for geometric objects. The most basic geometric objects include points, lines, polygons, and polyhedra. Computational geometry has applications in many areas of computer science, including computer graphics, robotics, and geographic information systems. In many computational-geometry problems, the theoretical worst case is achieved by input that is in some way "unrealistic". This causes situations where the theoretical running time is not a good predictor of the running time in practice. In addition, algorithms must also be designed with the worst-case examples in mind, which causes them to be needlessly complicated. In recent years, realistic input models have been proposed in an attempt to deal with this problem. The usual form such solutions take is to limit some geometric property of the input to a constant. We examine a specific realistic input model in this thesis: the model where objects are restricted to be fat. Intuitively, objects that are more like a ball are more fat, and objects that are more like a long pole are less fat. We look at fat objects in the context of five different problems—two related to decompositions of input objects and three problems suggested by computer graphics. Decompositions of geometric objects are important because they are often used as a preliminary step in other algorithms, since many algorithms can only handle geometric objects that are convex and preferably of low complexity. The two main issues in developing decomposition algorithms are to keep the number of pieces produced by the decomposition small and to compute the decomposition quickly. The main question we address is the following: is it possible to obtain better decompositions for fat objects than for general objects, and/or is it possible to obtain decompositions quickly? These questions are also interesting because most research into fat objects has concerned objects that are convex. We begin by triangulating fat polygons. The problem of triangulating polygons—that is, partitioning them into triangles without adding any vertices—has been solved already, but the only linear-time algorithm is so complicated that it has never been implemented. We propose two algorithms for triangulating fat polygons in linear time that are much simpler. They make use of the observation that a small set of guards placed at points inside a (certain type of) fat polygon is sufficient to see the boundary of such a polygon. We then look at decompositions of fat polyhedra in three dimensions. We show that polyhedra can be decomposed into a linear number of convex pieces if certain fatness restrictions aremet. We also show that if these restrictions are notmet, a quadratic number of pieces may be needed. We also show that if we wish the output to be fat and convex, the restrictions must be much tighter. We then study three computational-geometry problems inspired by computer graphics. First, we study ray-shooting amidst fat objects from two perspectives. This is the problem of preprocessing data into a data structure that can answer which object is first hit by a query ray in a given direction from a given point. We present a new data structure for answering vertical ray-shooting queries—that is, queries where the ray’s direction is fixed—as well as a data structure for answering ray-shooting queries for rays with arbitrary direction. Both structures improve the best known results on these problems. Another problem that is studied in the field of computer graphics is the depth-order problem. We study it in the context of computational geometry. This is the problem of finding an ordering of the objects in the scene from "top" to "bottom", where one object is above the other if they share a point in the projection to the xy-plane and the first object has a higher z-value at that point. We give an algorithm for finding the depth order of a group of fat objects and an algorithm for verifying if a depth order of a group of fat objects is correct. The latter algorithm is useful because the former can return an incorrect order if the objects do not have a depth order (this can happen if the above/below relationship has a cycle in it). The first algorithm improves on the results previously known for fat objects; the second is the first algorithm for verifying depth orders of fat objects. The final problem that we study is the hidden-surface removal problem. In this problem, we wish to find and report the visible portions of a scene from a given viewpoint—this is called the visibility map. The main difficulty in this problem is to find an algorithm whose running time depends in part on the complexity of the output. For example, if all but one of the objects in the input scene are hidden behind one large object, then our algorithm should have a faster running time than if all of the objects are visible and have borders that overlap. We give such an algorithm that improves on the running time of previous algorithms for fat objects. Furthermore, our algorithm is able to handle curved objects and situations where the objects do not have a depth order—two features missing from most other algorithms that perform hidden surface removal

Families of Invariant Divisors on Rational Complexity-One T-Varieties

We study invariant divisors on the total spaces of the homogeneous deformations of rational complexity-one T-varieties constructed by Ilten and Vollmert. In particular, we identify a natural subgroup of the Picard group for any general fiber of such a deformation, which is canonically isomorphic to the Picard group of the special fiber. This isomorphism preserves Euler characteristic, intersection numbers, and the canonical class.Comment: v4 minor changes: 17 pages and 3 figure

Discrete Geometry

[no abstract available

On Tetrahedralisations of Reduced Chazelle Polyhedra with Interior Steiner Points

The non-convex polyhedron constructed by Chazelle, known as the Chazelle polyhedron [4], establishes a quadratic lower bound on the minimum number of convex pieces for the 3d polyhedron partitioning problem. In this paper, we study the problem of tetrahedralising the Chazelle polyhedron without modifying its exterior boundary. It is motivated by a crucial step in tetrahedral mesh generation in which a set of arbitrary constraints (edges or faces) need to be entirely preserved. The goal of this study is to gain more knowledge about the family of 3d indecomposable polyhedra which needs additional points, so-called Steiner points, to be tetrahedralised. The requirement of only using interior Steiner points for the Chazelle polyhedron is extremely challenging. We first “cut off” the volume of the Chazelle polyhedron by removing the regions that are tetrahedralisable. This leads to a 3d non-convex polyhedron whose vertices are all in the two slightly shifted saddle surfaces which are used to construct the Chazelle polyhedron. We call it the reduced Chazelle polyhedron. It is an indecomposable polyhedron. We then give a set of (N + 1)2 interior Steiner points that ensures the existence of a tetrahedralisation of the reduced Chazelle polyhedron with 4(N + 1) vertices. The proof is done by transforming a 3d tetrahedralisation problem into a 2d edge flip problem. In particular, we design an edge splitting and flipping algorithm and prove that it gives to a tetrahedralisation of the reduced Chazelle polyhedron

On indecomposable polyhedra and the number of interior Steiner points

The existence of 3d {\it indecomposable polyhedra}, that is, the interior of every such polyhedron cannot be decomposed into a set of tetrahedra whose vertices are all of the given polyhedron, is well-known. While the geometry and combinatorial structure of such polyhedra are much less studied. In this article, we first investigate the geometry of some well-known examples, the so-called {\it Sch\"on\-hardt polyhedron}~\cite{Schonhardt1928} and the Bagemihl's generalization of it~\cite{Bagemihl48-decomp-polyhedra}, which will be called {\it Bagemihl polyhedra}. We provide a construction of an interior point, so-called {\it Steiner point}, which can be used to tetrahedralize the Sch\"on\-hardt and the Bagemihl polyhedra. We then provide a construction of a larger class of three-dimensional indecomposable polyhedra which often appear in grid generation problems. We show that such polyhedra have the same combinatorial structure as the Sch\"onhardt and Bagemihl polyhedra, but they may need more than one interior Steiner point to be tetrahedralized. Given such a polyhedron with $n \ge 6$ vertices, we show that it can be tetrahedralized by adding at most $\left\lceil \frac{n - 5}{2}\right\rceil$ interior Steiner points. %, is sufficient to decompose it. We also show that this number is optimal in the worst case

On tetrahedralisations of reduced Chazelle polyhedra with interior Steiner points

The polyhedron constructed by Chazelle, known as Chazelle polyhedron [4], is an important example in many partitioning problems. In this paper, we study the problem of tetrahedralising a Chazelle polyhedron without modifying its exterior boundary. It is motivated by a crucial step in 3d finite element mesh generation in which a set of arbitrary boundary constraints (edges or faces) need to be entirely preserved. We first reduce the volume of a Chazelle polyhedron by removing the regions that are tetrahedralisable. This leads to a 3d polyhedron which may not be tetrahedralisable unless extra points, so-called Steiner points, are added. We call it a reduced Chazelle polyhedron. We define a set of interior Steiner points that ensures the existence of a tetrahedralisation of the reduced Chazelle polyhedron. Our proof uses a natural correspondence that any sequence of edge flips converting one triangulation of a convex polygon into another gives a tetrahedralization of a 3d polyhedron which have the two triangulations as its boundary. Finally, we exhibit a larger family of reduced Chazelle polyhedra which includes the same combinatorial structure of the Schönhardt polyhedron. Our placement of interior Steiner points also applies to tetrahedralise polyhedra in this family

An Introduction to Geometric Topology

This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces, and the third to three-manifolds. It contains complete proofs of Mostow's rigidity, the thick-thin decomposition, Thurston's classification of the diffeomorphisms of surfaces (via Bonahon's geodesic currents), the prime and JSJ decomposition, the topological and geometric classification of Seifert manifolds, and Thurston's hyperbolic Dehn filling Theorem

Discrete Geometry

The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants

Decomposing and packing polygons / Dania el-Khechen.

In this thesis, we study three different problems in the field of computational geometry: the partitioning of a simple polygon into two congruent components, the partitioning of squares and rectangles into equal area components while minimizing the perimeter of the cuts, and the packing of the maximum number of squares in an orthogonal polygon. To solve the first problem, we present three polynomial time algorithms which given a simple polygon P partitions it, if possible, into two congruent and possibly nonsimple components P 1 and P 2 : an O ( n 2 log n ) time algorithm for properly congruent components and an O ( n 3 ) time algorithm for mirror congruent components. In our analysis of the second problem, we experimentally find new bounds on the optimal partitions of squares and rectangles into equal area components. The visualization of the best determined solutions allows us to conjecture some characteristics of a class of optimal solutions. Finally, for the third problem, we present three linear time algorithms for packing the maximum number of unit squares in three subclasses of orthogonal polygons: the staircase polygons, the pyramids and Manhattan skyline polygons. We also study a special case of the problem where the given orthogonal polygon has vertices with integer coordinates and the squares to pack are (2 {604} 2) squares. We model the latter problem with a binary integer program and we develop a system that produces and visualizes optimal solutions. The observation of such solutions aided us in proving some characteristics of a class of optimal solutions