Doctor of Philosophy

dissertationWhile boundary representations, such as nonuniform rational B-spline (NURBS) surfaces, have traditionally well served the needs of the modeling community, they have not seen widespread adoption among the wider engineering discipline. There is a common perception that NURBS are slow to evaluate and complex to implement. Whereas computer-aided design commonly deals with surfaces, the engineering community must deal with materials that have thickness. Traditional visualization techniques have avoided NURBS, and there has been little cross-talk between the rich spline approximation community and the larger engineering field. Recently there has been a strong desire to marry the modeling and analysis phases of the iterative design cycle, be it in car design, turbulent flow simulation around an airfoil, or lighting design. Research has demonstrated that employing a single representation throughout the cycle has key advantages. Furthermore, novel manufacturing techniques employing heterogeneous materials require the introduction of volumetric modeling representations. There is little question that fields such as scientific visualization and mechanical engineering could benefit from the powerful approximation properties of splines. In this dissertation, we remove several hurdles to the application of NURBS to problems in engineering and demonstrate how their unique properties can be leveraged to solve problems of interest

Vertex classification for non-uniform geometry reduction.

Complex models created from isosurface extraction or CAD and highly accurate 3D models produced from high-resolution scanners are useful, for example, for medical simulation, Virtual Reality and entertainment. Often models in general require some sort of manual editing before they can be incorporated in a walkthrough, simulation, computer game or movie. The visualization challenges of a 3D editing tool may be regarded as similar to that of those of other applications that include an element of visualization such as Virtual Reality. However the rendering interaction requirements of each of these applications varies according to their purpose. For rendering photo-realistic images in movies computer farms can render uninterrupted for weeks, a 3D editing tool requires fast access to a model's fine data. In Virtual Reality rendering acceleration techniques such as level of detail can temporarily render parts of a scene with alternative lower complexity versions in order to meet a frame rate tolerable for the user. These alternative versions can be dynamic increments of complexity or static models that were uniformly simplified across the model by minimizing some cost function. Scanners typically have a fixed sampling rate for the entire model being scanned, and therefore may generate large amounts of data in areas not of much interest or that contribute little to the application at hand. It is therefore desirable to simplify such models non-uniformly. Features such as very high curvature areas or borders can be detected automatically and simplified differently to other areas without any interaction or visualization. However a problem arises when one wishes to manually select features of interest in the original model to preserve and create stand alone, non-uniformly reduced versions of large models, for example for medical simulation. To inspect and view such models the memory requirements of LoD representations can be prohibitive and prevent storage of a model in main memory. Furthermore, although asynchronous rendering of a base simplified model ensures a frame rate tolerable to the user whilst detail is paged, no guarantees can be made that what the user is selecting is at the original resolution of the model or of an appropriate LoD owing to disk lag or the complexity of a particular view selected by the user. This thesis presents an interactive method in the con text of a 3D editing application for feature selection from any model that fits in main memory. We present a new compression/decompression of triangle normals and colour technique which does not require dedicated hardware that allows for 87.4% memory reduction and allows larger models to fit in main memory with at most 1.3/2.5 degrees of error on triangle normals and to be viewed interactively. To address scale and available hardware resources, we reference a hierarchy of volumes of different sizes. The distances of the volumes at each level of the hierarchy to the intersection point of the line of sight with the model are calculated and these distances sorted. At startup an appropriate level of the tree is automatically chosen by separating the time required for rendering from that required for sorting and constraining the latter according to the resources available. A clustered navigation skin and depth buffer strategy allows for the interactive visualisation of models of any size, ensuring that triangles from the closest volumes are rendered over the navigation skin even when the clustered skin may be closer to the viewer than the original model. We show results with scanned models, CAD, textured models and an isosurface. This thesis addresses numerical issues arising from the optimisation of cost functions in LoD algorithms and presents a semi-automatic solution for selection of the threshold on the condition number of the matrix to be inverted for optimal placement of the new vertex created by an edge collapse. We show that the units in which a model is expressed may inadvertently affect the condition of these matrices, hence affecting the evaluation of different LoD methods with different solvers. We use the same solver with an automatically calibrated threshold to evaluate different uniform geometry reduction techniques. We then present a framework for non-uniform reduction of regular scanned models that can be used in conjunction with a variety of LoD algorithms. The benefits of non-uniform reduction are presented in the context of an animation system. (Abstract shortened by UMI.)

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

Proceedings of the 2018 Canadian Society for Mechanical Engineering (CSME) International Congress

Published proceedings of the 2018 Canadian Society for Mechanical Engineering (CSME) International Congress, hosted by York University, 27-30 May 2018