Implicit Real Vector Automata

peer reviewedThis paper addresses the symbolic representation of non-convex real polyhedra, i.e., sets of real vectors satisfying arbitrary Boolean combinations of linear constraints. We develop an original data structure for representing such sets, based on an implicit and concise encoding of a known structure, the Real Vector Automaton. The resulting formalism provides a canonical representation of polyhedra, is closed under Boolean operators, and admits an efficient decision procedure for testing the membership of a vector

Efficient Generation of Correctness Certificates for the Abstract Domain of Polyhedra

Polyhedra form an established abstract domain for inferring runtime properties of programs using abstract interpretation. Computations on them need to be certified for the whole static analysis results to be trusted. In this work, we look at how far we can get down the road of a posteriori verification to lower the overhead of certification of the abstract domain of polyhedra. We demonstrate methods for making the cost of inclusion certificate generation negligible. From a performance point of view, our single-representation, constraints-based implementation compares with state-of-the-art implementations

A Survey of Methods for Converting Unstructured Data to CSG Models

The goal of this document is to survey existing methods for recovering CSG representations from unstructured data such as 3D point-clouds or polygon meshes. We review and discuss related topics such as the segmentation and fitting of the input data. We cover techniques from solid modeling and CAD for polyhedron to CSG and B-rep to CSG conversion. We look at approaches coming from program synthesis, evolutionary techniques (such as genetic programming or genetic algorithm), and deep learning methods. Finally, we conclude with a discussion of techniques for the generation of computer programs representing solids (not just CSG models) and higher-level representations (such as, for example, the ones based on sketch and extrusion or feature based operations).Comment: 29 page

Combinatorial optimization in geometry

AbstractIn this paper we extend and unify the results of [Rivin, Ann. of Math. 143 (1996)] and [Rivin, Ann. of Math. 139 (1994)]. As a consequence, the results of [Rivin, Ann. of Math. 143 (1996)] are generalized from the framework of ideal polyhedra in H3 to that of singular Euclidean structures on surfaces, possibly with an infinite number of singularities (by contrast, the results of [Rivin, Ann. of Math. 143 (1996)] can be viewed as applying to the case of non-singular structures on the disk, with a finite number of distinguished points). This leads to a fairly complete understanding of the moduli space of such Euclidean structures and thus, by the results of [Penner, Comm. Math. Phys. 113 (1987) 299–339; Epstein, Penner, J. Differential Geom. 27 (1988) 67–80; Näätänen, Penner, Bull. London Math. Soc. 6 (1991) 568–574] the author [Rivin, Ann. of Math. 139 (1994); Rivin, in: Lecture Notes in Pure and Appl. Math., Vol. 156, 1994], and others, further insights into the geometry and topology of the Riemann moduli space.The basic objects studied are the canonical Delaunay triangulations associated to the aforementioned Euclidean structures.The basic tools, in addition to the results of [Rivin, Ann. of Math. 139 (1994)] and combinatorial geometry are methods of combinatorial optimization—linear programming and network flow analysis; hence the results mentioned above are not only effective but also efficient. Some applications of these methods to three-dimensional topology are also given (to prove a result of Casson's)

The three-dimensional art gallery problem and its solutions

This thesis addressed the three-dimensional Art Gallery Problem (3D-AGP), a version of the art gallery problem, which aims to determine the number of guards required to cover the interior of a pseudo-polyhedron as well as the placement of these guards. This study exclusively focused on the version of the 3D-AGP in which the art gallery is modelled by an orthogonal pseudo-polyhedron, instead of a pseudo-polyhedron. An orthogonal pseudopolyhedron provides a simple yet effective model for an art gallery because of the fact that most real-life buildings and art galleries are largely orthogonal in shape. Thus far, the existing solutions to the 3D-AGP employ mobile guards, in which each mobile guard is allowed to roam over an entire interior face or edge of a simple orthogonal polyhedron. In many realword applications including the monitoring an art gallery, mobile guards are not always adequate. For instance, surveillance cameras are usually installed at fixed locations. The guard placement method proposed in this thesis addresses such limitations. It uses fixedpoint guards inside an orthogonal pseudo-polyhedron. This formulation of the art gallery problem is closer to that of the classical art gallery problem. The use of fixed-point guards also makes our method applicable to wider application areas. Furthermore, unlike the existing solutions which are only applicable to simple orthogonal polyhedra, our solution applies to orthogonal pseudo-polyhedra, which is a super-class of simple orthogonal polyhedron. In this thesis, a general solution to the guard placement problem for 3D-AGP on any orthogonal pseudo-polyhedron has been presented. This method is the first solution known so far to fixed-point guard placement for orthogonal pseudo-polyhedron. Furthermore, it has been shown that the upper bound for the number of fixed-point guards required for covering any orthogonal polyhedron having n vertices is (n3/2), which is the lowest upper bound known so far for the number of fixed-point guards for any orthogonal polyhedron. This thesis also provides a new way to characterise the type of a vertex in any orthogonal pseudo-polyhedron and has conjectured a quantitative relationship between the numbers of vertices with different vertex configurations in any orthogonal pseudo-polyhedron. This conjecture, if proved to be true, will be useful for gaining insight into the structure of any orthogonal pseudo-polyhedron involved in many 3-dimensional computational geometrical problems. Finally the thesis has also described a new method for splitting orthogonal polygon iv using a polyline and a new method for splitting an orthogonal polyhedron using a polyplane. These algorithms are useful in applications such as metal fabrication

Euclidean distance geometry and applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.Comment: 64 pages, 21 figure

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