Morphing Contact Representations of Graphs

We consider the problem of morphing between contact representations of a plane graph. In a contact representation of a plane graph, vertices are realized by internally disjoint elements from a family of connected geometric objects. Two such elements touch if and only if their corresponding vertices are adjacent. These touchings also induce the same embedding as in the graph. In a morph between two contact representations we insist that at each time step (continuously throughout the morph) we have a contact representation of the same type. We focus on the case when the geometric objects are triangles that are the lower-right half of axis-parallel rectangles. Such RT-representations exist for every plane graph and right triangles are one of the simplest families of shapes supporting this property. Thus, they provide a natural case to study regarding morphs of contact representations of plane graphs. We study piecewise linear morphs, where each step is a linear morph moving the endpoints of each triangle at constant speed along straight-line trajectories. We provide a polynomial-time algorithm that decides whether there is a piecewise linear morph between two RT-representations of a plane triangulation, and, if so, computes a morph with a quadratic number of linear morphs. As a direct consequence, we obtain that for 4-connected plane triangulations there is a morph between every pair of RT-representations where the "top-most" triangle in both representations corresponds to the same vertex. This shows that the realization space of such RT-representations of any 4-connected plane triangulation forms a connected set

Q-Polynomial Association schemes with Irrational Eigenvalues

We work towards classifying the feasible parameter sets of irrational Q-polynomial association schemes with three classes. We aimed to provide a synopsis of the subject as well as provide some theorems and conjectures to understand these combinatorial objects

Cometric Association Schemes

The combinatorial objects known as association schemes arise in group theory, extremal graph theory, coding theory, the design of experiments, and even quantum information theory. One may think of a d-class association scheme as a (d + 1)-dimensional matrix algebra over R closed under entrywise products. In this context, an imprimitive scheme is one which admits a subalgebra of block matrices, also closed under the entrywise product. Such systems of imprimitivity provide us with quotient schemes, smaller association schemes which are often easier to understand, providing useful information about the structure of the larger scheme. One important property of any association scheme is that we may find a basis of d + 1 idempotent matrices for our algebra. A cometric scheme is one whose idempotent basis may be ordered E0, E1, . . . , Ed so that there exists polynomials f0, f1, . . . , fd with fi ◦ (E1) = Ei and deg(fi) = i for each i. Imprimitive cometric schemes relate closely to t-distance sets, sets of unit vectors with only t distinct angles, such as equiangular lines and mutually unbiased bases. Throughout this thesis we are primarily interested in three distinct goals: building new examples of cometric association schemes, drawing connections between cometric association schemes and other objects either combinatorial or geometric, and finding new realizability conditions on feasible parameter sets — using these conditions to rule out open parameter sets when possible. After introducing association schemes with relevant terminology and definitions, this thesis focuses on a few recent results regarding cometric schemes with small d. We begin by examining the matrix algebra of any such scheme, first looking for low rank positive semidefinite matrices with few distinct entries and later establishing new conditions on realizable parameter sets. We then focus on certain imprimitive examples of both 3- and 4-class cometric association schemes, generating new examples of the former while building realizability conditions for both. In each case, we examine the related t-distance sets, giving conditions which work towards equivalence; in the case of 3-class Q-antipodal schemes, an equivalence is established. We conclude by partially extending a result of Brouwer and Koolen concerning the connectivity of graphs arising from metric association schemes

Automatic mesh generation

The objective of this thesis project is a study of Pre-Processors and development of an Automatic Mesh Generator for Finite Element Analysis. The Mesh Generator developed in this thesis project can create triangular finite elements from the geometric database of Macintosh Applications. The user is required to give the density parameter to the program for mesh generation. The research is limited to Mesh Generators of planar surfaces. Delauny Triangulation method maximizes the minimum angles of a triangle. Watson\u27s Delauny Triangulation method can mesh only the \u27convex hull\u27 of a set of nodes. This algorithm has been modified to create triangular elements in convex and non-convex surfaces. The surfaces can have holes also. A node generation algorithm to place nodes on and inside a geometry has been developed in this thesis project. The mesh generation is very efficient and flexible. Geometric modeling methods have been studied to understand and integrate the Geometric Modeler with the Finite Element Mesh Generator. Expert Systems can be integrated with Finite Element Analysis. This will make Finite Element Method fully automatic. In this thesis project, Expert Systems in Finite Element Analysis are reviewed. Proposals are made for future approach for the integration of the two fields

Reconstructing triangulated surfaces from unorganized points through local skeletal stars

Surface reconstruction from unorganized points arises in a variety of practical situations such as range scanning an object from multiple view points, recovery of biological shapes from twodimensional slices, and interactive surface sketching. [...]Reconstrução da superfície de pontos desorganizados surge em uma variedade de situações práticas, tais como rastreamento de um objeto a partir de vários pontos de vista, a recuperação de formas biológicas de fatias bi-dimensionais, e esboçar superfícies interativas. [...