published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy

An algebraic surface defined by an equation of the form z2 = (x+a1y) ... (x + any) (x - 1) is studied, from both an algebraic and geometric point of view. It is shown that the surface is rational and contains a singular point which is nonrational. The class group of Weil divisors is computed and the Brauer group of Azumaya algebras is studied. Viewing the surface as a cyclic cover of the affine plane, all of the terms in the cohomology sequence of Chase, Harrison and Roseberg are computed. / by Djordje Bulj. / Thesis (Ph.D.)--Florida Atlantic University, 2012. / Includes bibliography. / Mode of access: World Wide Web. / System requirements: Adobe Reader.

Chan, Pui Lam. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 72-75). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

Lee Keung Tat. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 81-86). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgments --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Motivation --- p.1 / Chapter 1.2 --- Contribution --- p.3 / Chapter 1.3 --- Previous Work --- p.5 / Chapter 1.4 --- Thesis Organization --- p.9 / Chapter 2 --- Overview --- p.10 / Chapter 3 --- Data Acquisition --- p.14 / Chapter 3.1 --- Acquiring the Surface Model --- p.14 / Chapter 3.2 --- Camera Calibration --- p.16 / Chapter 3.3 --- Capturing Image with Camera Pose --- p.18 / Chapter 3.3.1 --- Fastrack --- p.19 / Chapter 3.3.2 --- Tracking the Camera Pose --- p.21 / Chapter 3.3.3 --- Calibrating the Tracking System --- p.23 / Chapter 3.4 --- Summary --- p.32 / Chapter 4 --- Data Fusion --- p.33 / Chapter 4.1 --- Converting Surface Model to Point-Based Model --- p.33 / Chapter 4.2 --- Registering the Radiance Values onto the Point-Based Model --- p.36 / Chapter 4.3 --- Scattered Data Fitting --- p.40 / Chapter 4.3.1 --- Spherical Delaunay Triangulation --- p.41 / Chapter 4.3.2 --- Hierarchical Spherical Triangulation --- p.46 / Chapter 4.3.3 --- Interpolation --- p.49 / Chapter 4.4 --- Data Compression --- p.50 / Chapter 4.5 --- Summary --- p.52 / Chapter 5 --- Multiresolution Point-Based Representation and Rendering --- p.53 / Chapter 5.1 --- Multiresolution Point-Based Representation --- p.55 / Chapter 5.1.1 --- Construction --- p.57 / Chapter 5.2 --- Rendering --- p.62 / Chapter 5.2.1 --- Culling --- p.63 / Chapter 5.2.2 --- Drawing the Node --- p.66 / Chapter 5.3 --- Summary --- p.68 / Chapter 6 --- Experimental Results --- p.69 / Chapter 6.1 --- Tested Objects --- p.69 / Chapter 6.2 --- Evaluation --- p.70 / Chapter 6.3 --- Summary --- p.78 / Chapter 7 --- Conclusion --- p.79 / Chapter 7.1 --- Summary --- p.79 / Chapter 7.2 --- Future Direction --- p.80 / Bibliography --- p.81

A new approach is proposed which uses a combination of a Backprop paradigm neural network along with some perceptron processing elements performing logic operations to construct a numeric-to-symbolic converter. The design approach proposed herein is capable of implementing a decision region defined by a multi-dimensional, non-linear boundary surface. By defining a "two-valued" subspace of the boundary surface, a Backprop paradigm neural network is used to model the boundary surf ace. An input vector is tested by the neural network boundary model (along with perceptron logic gates) to determine whether the incoming vector point is within the decision region or not. Experiments with two qualitatively different kinds of nonlinear surface were carried out to test and demonstrate the design approach.

In Australia, in recent years there has been increasing demand for more streamlined lodgement of cadastral plans and for their later dissemination. There are a number of approaches to meeting this demand, one of which is developed in detail in this dissertation. The current status of digital lodgement and Digital Cadastral Databases (DCDB) throughout Australia and New Zealand is reviewed. Each of the states and territories in Australia and also New Zealand are examined, looking at the process involved in the lodgement of survey plans and the state of the DCDB in each jurisdiction. From this examination the key issues in digital lodgement and dissemination are extracted and a needs analysis for an Australia-wide generic system is carried out. This needs analysis is directed at technological change allied with sound cadastral principles. Extensible Markup Language (XML) is considered for the storage and transport of all the required data and to facilitate the dissemination of information over the Internet. The benefits of using XML are comprehensive, leading to its selection and the use of related technologies LandXML, Extensible Structured Query Language (XSQL) and Extensible Stylesheet Language (XSL). Vector graphics are introduced as the means to display plans and maps on the Internet. A number of vector standards and Web mapping solutions are compared to determine the most suitable for this project. A new standard developed by the World Wide Web Consortium (W3C), Scalable Vector Graphics (SVG), is chosen. A prototype Web interface and the underlying database and Web server were developed using Oracle as the database and Apache as the Web server. Each aspect of the development is described, starting with the installation and configuration of the database, the Web server and the XSQL servlet. Testing was undertaken using LandXML cadastral data and displaying plans using SVG. Both Internet Explorer and Mozilla were trialled as the Web browser, with Mozilla being chosen because of incompatibilities between Internet Explorer, LandXML and SVG. An operational pilot was created. At this stage it requires manual intervention to centre and maximise a plan in the display area. The result indicates that an automated system is feasible and this dissertation provides a basis for further development by Australian land administration organisations.

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Geometric hierarchies have proven useful for the problems of point location in planar subdivisions and 2- and 3-dimensional convex polytope separation on a sequential model of computation. In this thesis, we formulate a geometric hierarchy paradigm (following the work of Dobkin and Kirkpatrick) and apply this paradigm to solve a number of computational geometry problems on a shared memory (PRAM) parallel model of computation. For certain problems, we describe what we call cooperative algorithms, algorithms which exploit parallelism in searching geometric hierarchies to solve their respective problems. For convex polygons, the geometric hierarchies are implicit and can be exploited in cooperative algorithms to compute convex polygon separation and to construct convex polygon separating/common tangents. The paradigm is also applied to the problem of tree contraction which is, in turn, applied to a number of specialized point location applications including the parallel construction of 2-dimensional Voronoi Diagrams. For point location in planar subdivisions, we present parallel algorithms to construct a subdivision hierarchy representation. A related convex polyhedra hierarchy is constructed similarly and applied to the parallel construction of 3-dimensional convex hulls. The geometric hierarchy paradigm is applied further to the design of a data structure which supports cooperative point location in general planar subdivisions. Again, a related polyhedral hierarchy can be used to exploit parallelism for a cooperative separation algorithm for convex polyhedra. / Science, Faculty of / Computer Science, Department of / Graduate

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This thesis presents an analysis of several smoothness energies (also called smoothing energies) in geometry processing, and introduces new methods as well as a mathematical proof of correctness and convergence for a well-established method. Geometry processing deals with the acquisition, modification, and output (be it on a screen, in virtual reality, or via fabrication and manufacturing) of complex geometric objects and data. It is closely related to computer graphics, but is also used by many other fields that employ applied mathematics in the context of geometry. The popular Laplacian energy is a smoothness energy that quantifies smoothness and that is closely related to the biharmonic equation (which gives it desirable properties). Minimizers of the Laplacian energy solve the biharmonic equation. This thesis provides a proof of correctness and convergence for a very popular discretization method for the biharmonic equation with zero Dirichlet and Neumann boundary conditions, the piecewise linear Lagrangian mixed finite element method. The same approach also discretizes the Laplacian energy. Such a proof has existed for flat surfaces for a long time, but there exists no such proof for the curved surfaces that are needed to represent the complicated geometries used in geometry processing. This proof will improve the usefulness of this discretization for the Laplacian energy. In this thesis, the novel Hessian energy for curved surfaces is introduced, which also quantifies the smoothness of a functions, and whose minimizers solve the biharmonic equation. This Hessian energy has natural boundary conditions that allow the construction of functions that are not significantly biased by the geometry and presence of boundaries in the domain (unlike the Laplacian energy with zero Neumann boundary conditions), while still conforming to constraints informed by the application. This is useful in any situation where the boundary of the domain is not an integral part of the problem itself, but just an artifact of data representation---be it, because of artifacts created by an imprecise scan of the surface, because information is missing outside of a certain region, or because the application simply demands a result that should not depend on the geometry of the boundary. Novel discretizations of this energy are also introduced and analyzed. This thesis also presents the new developability energy, which quantifies a different kind of smoothness than the Laplacian and Hessian energies: how easy is it to unfold a surface so that it lies flat on the plane without any distortion (surfaces for which this is possible are called developable surfaces). Developable surfaces are interesting, as they can be easily constructed from cheap material such as paper and plywood, or manufactured with methods such as 5-axis CNC milling. A novel definition of developability for discrete triangle meshes, as well as a variety of discrete developability energies are also introduced and applied to problems such as approximation of a surface by a piecewise developable surface, and the design and fabrication of piecewise developable surfaces. This will enable users to more easily take advantages of these cheap and quick fabrication methods. The novel methods, algorithms and the mathematical proof introduced in this thesis will be useful in many applications and fields, including numerical analysis of elliptic partial differential equations, geometry processing of triangle meshes, character animation, data denoising, data smoothing, scattered data interpolation, fabrication from simple materials, computer-controlled fabrication, and more.