Master of Science / Department of Mathematics / Todd Cochrane / Polygonal numbers are nonnegative integers constructed and represented by geometrical arrangements of equally spaced points that form regular polygons. These numbers were originally studied by Pythagoras, with their long history dating from 570 B.C, and are often referred to by the Greek mathematicians. During the ancient period, polygonal numbers were described by units which were expressed by dots or pebbles arranged to form geometrical polygons. In his "Introductio Arithmetica", Nicomachus of Gerasa (c. 100 A.D), thoroughly discussed polygonal numbers. Other Greek authors who did remarkable work on the numbers include Theon of Smyrna (c. 130 A.D), and Diophantus of Alexandria (c. 250 A.D). Polygonal numbers are widely applied and related to various mathematical concepts. The primary purpose of this report is to define and discuss polygonal numbers in application and relation to some of these concepts. For instance, among other topics, the report describes what triangle numbers are and provides many interesting properties and identities that they satisfy. Sums of squares, including Lagrange's Four Squares Theorem, and Legendre's Three Squares Theorem are included in the paper as well. Finally, the report introduces and proves its main theorems, Gauss' Eureka Theorem and Cauchy's Polygonal Number Theorem.

The principle concern of this work is the design of a lathe attachment for grinding non-circular 'polygonal' shaped workpieces suitable for use as torque transmitting machine elements. In the course of the work substantial attention is also given to the general theory and development of computer aided error analysis procedures for planar linkage mechanisms. A further smaller part of the work investigates the torsion of polygonal shafts. The non-circular shapes considered here may be loosely defined as polygonal profiles. Their application is in torque transmitting couplings for which they represent an alternative to keyed and splined couplings, although, in comparison to keys and splines, their application has been limited, mainly due to the specialised nature of their manufacture. The main objective of this work is to investigate suitable profiles and the means for their production using an attachment which can be mounted on a conventional machine tool, such as a lathe or grinding machine. The work progresses from initial consideration of shapes produced by various geometric generating methods and conception of an 'ideal' profile generating linkage mechanism through to detailed design of a precision, polygonal profile grinding, lathe attachment, and final assessment of its feasibility based on a profile precision criterion. In order to assess the precision of the attachment, computer-aided procedures are developed, after consideration of existing error analysis methods and their limitations for use in this case. These consider the various effects of tolerances, clearances and deflections upon mechanism output. As a coincidental investigation, the mechanical behaviour and strength of polygonal shaft-hub connections is reported. In particular, the torsion of a polygonal bar is theoretically analysed, using a stress function method, to determine maximum shear stresses.

No description available.

Recent innovations in design and construction of architectural buildings has led us to revisit the metrics for discretizing smooth freeform shapes in context with both aesthetics and fabrication. Inspired by the examples of the British Museum Court Roof in Britain and the Beijing Aquatic Centre in China, we propose solutions for generating aesthetic as well as economically viable solutions for tessellating smooth, freeform shapes. For the purpose of generating an aesthetic tessellation, we propose a simple linearized strain based metric to minimize dissimilarity amongst triangles in a local neighborhood. We do so by defining an error function that measures deformation required to map a pair of triangles onto each other. We minimize the error using a global non-linear optimization based framework. We also reduce the complexity associated with prefabricating triangulated panels for a given shape. To do so, we propose a global optimization based framework to approximate any given shape using significantly reduced numbers of unique triangles. By doing so, we leverage the economies of scale as well as simplify the process of physical placement of panels by manual labor.

Dielectric waveguides are important passive devices in optical communication systems. Circular-core fibers with slight ellipticity may lead to polarization-mode dispersion. A clear understanding of the propagation characteristics of the elliptical fibers thus becomes important for theoretical as well as practical purposes. Although mesh-dependent methods such as the finite-element method or finite-difference method, can be used to study such a complex structure, its computational task is very high. Strictly speaking, mesh-based solution does satisfy the Helmholtz equation and the solution only provided four to five significant digits. On the other hand, the highly accurate solution based on solving the Helmholtz equation of the elliptical coordinate system spend most its computational resources on computing the functional value and the zeros of the modified Mathieu functions of the first kind. Our method is based on linear combination of the exact mode-field solutions of the dielectric optical fiber. We apply the analytical continuity principle to obtain the simultaneous equation of the expansion coefficient vector. Since each basis solution satisfies the Helmholtz equation exactly, the overall solutions are very accurate and provide more than six significant digits for fibers with small elliptical eccentricity. In addition, only the Bessel functions are needed in our computation. Using cylindrical coordinate and symmetry, together with ACM principle, we simplify the problem of modal analysis of dielectric elliptical waveguides. This method also can be applied to some regular polygonal dielectric waveguides such as the large area VCESL.

No description available.

There has been a disparity between the conventional method of describing topographic surfaces (i.e. contour line definition) and a format of surface description often used in continuous-line computer graphics (i.e. panel definition). The two differ enough that conversion from contours to panels is not a trivial problem. A computer program that performs such a conversion would greatly facilitate continuous tone display of topographical surfaces, or any other surface which is defined by contour lines. This problem has been addressed by Keppel and alluded to by Fuchs. Keppel's is a highly systematic approach in which he uses graph theory to find the panel arrangement which maximizes the volume enclosed by concave surfaces. Fuchs mentions an approach to the problem as part of an algorithm to reconstruct a surface from data retrieved from a laser scan sensor. This thesis elaborates on a general conversion system. Following a brief overview of computer graphics, a simple algorithm is described which extracts a panel definition from a pair of adjacent contour loops subject to the restriction that the two loops are similarly sized and shaped, and are mutually centered. Next, a mapping procedure is described which greatly relaxes the above restrictions. It is also shown that the conversion from contours to panels is inherently ambiguous (to various degrees) and that occasionally the ambiguity is great enough to require user interaction to guide the conversion algorithm. An important complication addressed in this thesis is the problem of handling cases where one contour loop branches into two or more (or vice versa). Attention turns next to a contour line definition of the human brain, and special problems encountered in preparing those data for continuous tone display, The final chapters explain the fortran implementation, present an example-problem, and show sample pictures of the brain parts.

Wind energy is one of the most efficient renewable energies. The most used wind towers are tubularand lattice wind towers. Parts of lattice are easier to transfer, especially in the inland areas. Also, it is easier to build higher lattice tower in order to have more efficient energy conversion in inland areas.However, most of the cross sections for lattice tower are tubular cross sections.This thesis represents the parametric study of polygonal cross section of lattice tower. It consists ofthe numerical analysis based on finite element method (ABAQUS) and analysis based on EN 1993-1-3. The objective of this thesis is to find regular patterns of parametric influences on polygonal crosssection, and to compare them against calculation based on EN 1993-1-3. Also, to find regular patternsof parametric influences on the stiffness of the bolts on the lips.

<p>3D Models of botanical trees are very important in video games, simulation, virtual reality, digital city modeling and other fields of computer graphics. However, since the early days of computer graphics, the modeling of trees has been challenging, because of the huge dynamical range between its smallest and largest structures and their geometrical complexity. Trees are also ubiquitous which makes it even hard to model them in a realistic way, Current techniques are limited in that they model a tree either in a rule-based way or in an approximated way. These methods emphasize appearance while sacrificing its real structure. Recent development in range scanners are making 3D aquisition feasible for large and complex objects. This report presents the semi-automatic technique developed for modeling laser-scanned trees. First, the user draws a few strokes on the depth image plane generated from the dataset. Branches are then extracted through the 2D Curve detection algorithm originally developed. Afterwards, those short branches are connected together to generate the skeleton of the tree by forming a Minimum Spanning Tree (MST). Finally, the geometry of the tree skeleton is produced using allometric rules for branch thickness and branching angles.</p>

none