CAREER: Parameterization and Tessellation for Computer Graphics

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The digital age with its widespread availability of cheap computing power has transformed the way we search for information, distribute content and even navigate cities. Computation has also transformed the way we design shapes. Whereas in the past engineers and artists often sculpted objects from clay, today most shapes are designed virtually with computers. Computer-aided design of curves, surfaces and volumetric functions has applications in diverse fields such as industrial design, the entertainment industry and even architectural design. However, in all of these applications the designer is limited by the capabilities of the representations employed. The quality and geometric properties of parametric surface representations such as NURBS and subdivision surfaces are strongly dependent on their parameterization. Yet this is a degree of freedom that few designers use or understand. Furthermore, the connection between parameterization and the geometric properties of these shapes is not well understood. We currently have only rudimentary tools for controlling the parameterization of higher dimensional shapes such as surfaces or volumes. The inability to control this parameterization leads to poor quality shapes and more effort on the part of designers to alleviate these artifacts. In this project, the PI will explore new representations of surfaces and volumes that allow for more geometric freedom in creating the underlying shapes. He will investigate the fundamental connection between parameterization and surface shape/quality for parametric curves, surfaces and volumes. He will expand upon the concept of non-uniform parameterization of surfaces to show that knot spacing (edge lengths) is not sufficient to completely control the parameterization of surfaces or volumes. And he will design new representations that allow the user to control or automatically adapt the parameterization of these shapes during the design process, and incorporate methods of non-uniform parameterization that are currently not possible. As part of this process, he will develop new higher order barycentric coordinates specifically adapted to this problem. Finally, he will investigate the effect and manipulation of parameterization for the purposes of tessellation and rendering of these parametric surfaces, and develop high quality GPU tessellation algorithms.Broader Impacts: Project outcomes will significantly advance the state of the art not only in computer graphics and geometric modeling, but also in other areas of applied mathematics and computer science where the representation and precise control of smooth freeform shapes play a key role. Approximation theory, architectural design, the entertainment industry and industrial manufacturing will all benefit from the results of this research.