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

Mesh based Scene Evaluation Metrics for LOD and Simplification

I present seven metrics to quantify attributes of different meshes in a scene. Each metricrepresents a different geometrical or topological aspect of the mesh. Theresulting ratingvalues serve to convey the underlying complex data to the user. These allow the user toswiftly compare several features of multiple meshes. The metricsmay thus guide usersand programs during the process of mesh modification, i.e. optimization, simplification orsmoothing, and scene modification as a whole.I evaluate each metric individually by applying them to a sample scene. To examine thecorrectness and expressiveness of the metrics I compare the automatically calculated ratingsto the raw base data. I find two of the metrics to be immediately useful and four of theratings promising, but in need of adjustments. The remaining last metric, however, requiressignificant rework to generate useful data on par with the other six metrics.This thesis first introduces the subject with a motivating example. It then presents importantconcepts and research on related topics. Afterwards it details the concept of the programand the mathematical considerations it is based on. It also lists my approach to solving thechallenges which emerged during the implementation. Subsequently, the thesis focusses onthe visualized output of the program and challenges said ouput. Finally, it contrasts theexpectations and goals of each metric with the respective actual result

Proceedings of the Second PHANToM Users Group Workshop : October 19-22, 1997 : Endicott House, Dedham, MA, Massachusetts Institute of Technology, Cambridge, MA

"December, 1997." Cover title.Includes bibliographical references.Sponsored by SensAble Technologies, Inc., Cambridge, MA."[edited by J. Kennedy Salisbury and Mandayam A. Srinivasan]

Applications of mirror symmetry to the classification of Fano varieties

In this dissertation we discuss two new constructions of Fano varieties, each directly inspired by ideas in Mirror Symmetry. The first recasts the Fanosearch program (Coates--Corti--Kasprzyk et al.) for surfaces in terms of a construction related to the SYZ conjecture. In particular we construct Q-Gorenstein smoothings of toric varieties via an application of the Gross-Siebert algorithm to certain affine manifolds. We recover the theory of combinatorial mutation, which plays a central role in the Fanosearch program, from these affine manifolds. Combining this construction and the work of Gross--Hacking--Keel on log Calabi--Yau surfaces we produce a cluster structure on the mirror to a log del Pezzo surface proposed by Coates--Corti--et al. We exploit the cluster structure, and the connection to toric degenerations, to prove two classification results for Fano polygons. This cluster variety is equipped with a superpotential defined on each chart by a so-called maximally mutable Laurent polynomial. We study an enumerative interpretation of this superpotential in terms of tropical disc counting in the example of the projective plane (with a general choice of boundary divisor). In the second part we develop a new construction of Fano toric complete intersections in higher dimensions. We first consider the problem of finding torus charts on the Hori--Vafa/Givental model, adapting the approach taken by Przyjalkowski. We exploit this to identify 527 new families of four-dimensional Fano manifolds. We then develop an inverse algorithm, Laurent Inversion, which decorates a Fano polytope P with additional information used to construct a candidate ambient space for a complete intersection model of the toric variety defined by P. Moving in the linear system defining this complete intersection allows us to construct new models of known Fano manifolds, and also to construct new examples of Fano manifolds from conjectured mirror Laurent polynomials. We use this algorithm to produce families simultaneously realising certain collections of 'commuting' mutations, extending the connection between polytope mutation and deformations of toric varieties.Open Acces

On realistic target coverage by autonomous drones

Low-cost mini-drones with advanced sensing and maneuverability enable a new class of intelligent sensing systems. To achieve the full potential of such drones, it is necessary to develop new enhanced formulations of both common and emerging sensing scenarios. Namely, several fundamental challenges in visual sensing are yet to be solved including (1) fitting sizable targets in camera frames; (2) positioning cameras at effective viewpoints matching target poses; and (3) accounting for occlusion by elements in the environment, including other targets. In this article, we introduce Argus, an autonomous system that utilizes drones to collect target information incrementally through a two-tier architecture. To tackle the stated challenges, Argus employs a novel geometric model that captures both target shapes and coverage constraints. Recognizing drones as the scarcest resource, Argus aims to minimize the number of drones required to cover a set of targets. We prove this problem is NP-hard, and even hard to approximate, before deriving a best-possible approximation algorithm along with a competitive sampling heuristic which runs up to 100× faster according to large-scale simulations. To test Argus in action, we demonstrate and analyze its performance on a prototype implementation. Finally, we present a number of extensions to accommodate more application requirements and highlight some open problems