published_or_final_version / Mathematics / Master / Master of Philosophy

Geometry processing has witnessed tremendous development in the last few decades. Starting from acquiring 3D data of real life objects, people have developed practical methods for polishing the raw data usually in the format of point clouds, reconstructing surfaces from the point clouds, cleaning up the surfaces by denoising or fairing, texturing the object surfaces by parametrization to 2D domain, deforming the objects realistically and in real time, and many other advanced tasks. Along with the notable methods is the sophistication of knowledge for working with discrete geometric data, in particular points, triangles, quadrangles and polygons for object representation, with a large body summarized and principled in the field known as discrete differential geometry. Meanwhile, geometric modeling has come to a new era: unlike the previous industrial practice of spline-based modeling where people tune control points to search for aesthetic shapes, now people want novel ways of interaction. For example, find unknown shapes that are usually characterized to have variational and physical properties of interest. Also user-friendly modeling methods like sketching have gained remarkable attention and advances. We note that many of these surface modeling problems could be regarded as asking for surfaces with special differential geometric properties. To be specific, people find surfaces of minimal area for modeling soap films that are balanced under surface tension; surfaces that if fabricated could stand firmly and are therefore important in real life architectural structures, are described by having homogeneous relative mean curvatures; even for surfaces filling up sketched 3D curves, the significant property of a good filling surface is that the curves follow principal curvature directions of the surface. This thesis presents our results in developing effective algorithms for modeling the above mentioned surfaces, by adapting knowledge and techniques in geometry processing, especially from computational and discrete differential geometry. In particular, we extend surface remeshing techniques to model high quality Constant Mean Curvature (CMC) surfaces that are models of soap films and bubbles, use power diagrams and the dual regular triangulations to parametrize and process self-supporting surfaces, and apply direction field modeling and discrete curvature adaptation to surfacing sketch curve networks. / published_or_final_version / Computer Science / Doctoral / Doctor of Philosophy

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We introduce the naive notion of a relative open book decomposition for contact 3-manifolds with torus boundary. We then use this to construct nice, minimal genus open book decompositions compatible with all of the universally tight contact structures (as well as a few others) on torus-bundles over S¹, following Honda's classification. In an accurate sense, we find Stein fillings of 'half' of the torus bundles. In addition, these give the first examples of open books compatible with the universally tight contact structures on circle bundles over higher genus surfaces, as well, following a pattern introduced by a branched covering of B⁴. Some interesting examples of open books without positive monodromy are emphasized.

published_or_final_version / Economics / Master / Master of Social Sciences

No description available.

No description available.

No description available.

The Basal Unit of West Woodbourne Field in Barbados is a 250 m thick succession of finely-interbedded sandstones and mudstones deposited by Paleogene, fine-grained, deep-water systems off the northern South American margin and deformed as sediments were translated to the subduction zone of the Caribbean and Atlantic plates. Closely spaced gamma ray, neutron, density, spontaneous potential, formation microimager and dip meter logs, limited core, and published reports of local outcrops, were used to define three scales of vertical stratigraphic variation within this 1.5 km2 field: (1) decimeters to meters thick log facies; (2) meters to tens of meters thick log successions; and (3) tens to hundred meter thick intervals that are continuous laterally across the field. These variations record changes in sediment supply and depositional energy during progradation and abandonment events varying in scale from local shifts in distributary channels to regional changes in sediment transport along the basin. Well log correlations suggest the Basal Unit comprises a turbidite fan system (250 m thick) trending north to northeast, composed of six, vertically-stacked, distributary channel complexes. Three architectural elements are identified within each distributary channel complex: (1) Major amalgamated channels (30-40 m thick, 150-200 m wide and at least 900 m long) pass down depositional dip into proximal second-order channels that bifurcate basinward (15-20 m thick, symmetric successions); (2) Lobe deposits (20-50 m thick, 400 m wide, and at least 400 m long) are composed of upward-coarsening successions that contain distal second-order channels (1-10 m thick); and (3) Laterally extensive overbank deposits (5-10 m thick), which vertically separate distributary channel-lobe complexes. Reservoir heterogeneities within the Basal Unit are defined by the lateral extent and facies variations across a hierarchy of strata within channel-lobe complexes. Although laterally extensive muddy overbank deposits generally inhibit vertical communication between stacked channel-lobe complexes, in places where high-energy first-order channel sandstones incise underlying muddy overbank deposits, sandstones in subsequent intervals are partially connected. The Basal Unit is bounded on the southwest by a northwest-southeast trending fault that rises 30 degrees towards the northwest to define a structural trap on the northeast side of the field.

The intent of this work is to improve the machining speed of an existing 3 axis CNC wood working lathe. This lathe is unique in that it is a modi ed manual lathe that is capable of machining complex sculptured surfaces. The current machining is too slow for the lathe to be considered useful in an industrial setting. To improve the machining speed of the lathe, several modi cations are made to the mechanical, electrical and software aspects of the system. It was found that the x-axis of the system, the axis that controls the depth of cut of the tool, is the limiting axis. A servo motor is used to replace the existing stepper motor, providing the x-axis with more torque and faster response times, which should improve the performance of the system. To control the servo motor, a 1st-order linear transfer function model is selected and identi ed. Then, an adaptive sliding mode controller is applied to make the x-axis a robust and accurate positioning system. A new trajectory generator is implemented to create a smooth motion for all three axes of the lathe. This trajectory uses a 5th-order polynomial to describe the position curve of the feed pro le, giving the system continuous jerk motion. This type of pro le is much easier for motors to follow, as discontinuous motion will always result in errors. These modi cations to the lathe system are then evaluated experimentally using a test case. Three test pieces are designed to represent three of the common shapes that are typically machined on the wood turning lathe. These test cases indicated a minimum reduction in machining time of 52:91% over the previous lathe system. An algorithm is also developed that attempts to sacri ce work piece model geometry to achieve speed gains. The algorithm is used when a certain feedrate is desired for a model, but machining at that speed will cause toolpath following errors, leaving surface defects in the work piece. The algorithm will attempt to solve this problem by sacri cing model geometry. A simulation tool is used to detect where surface defects will occur during machining and a then the work piece model is modi ed in the corresponding area. This will create a smoother part, which allows each axis of the system to follow the new toolpath more easily, as the dynamic requirements are reduced. The potential of this algorithm is demonstrated in an experimental test case. A test piece is created that has features of varying di culty to machine. When the algorithm is run, Matlab/Simulink is used simulate the output of the lathe and locate the areas in the part geometry that will cause defects. Once located, the geometry features are smoothed in SolidWorks using the fi llet feature. The algorithm produces a work piece with smoothed geometry that can be machined at a feedrate approximately 42:8% faster than before. Although it is only the first implementation of the algorithm, the experimental results con rm the potential of the method. Machining speed gains are successfully achieved through the sacrifice of model geometry.