CAREER: Parameterization and Tessellation for Computer Graphics

随着廉价计算能力的广泛普及，数字时代已经改变了我们搜索信息、分发内容甚至导航城市的方式。计算也改变了我们设计形状的方式。在过去，工程师和艺术家经常用粘土雕刻物体，而今天大多数形状都是用电脑虚拟设计的。曲线、曲面和体积函数的计算机辅助设计在工业设计、娱乐工业甚至建筑设计等各个领域都有应用。然而，在所有这些应用程序中，设计人员受到所采用的表示的能力的限制。参数化曲面表示（如NURBS和细分曲面）的质量和几何特性强烈依赖于它们的参数化。然而，很少有设计师使用或理解这种自由度。此外，参数化与这些形状的几何性质之间的联系还没有得到很好的理解。我们目前只有基本的工具来控制高维形状（如表面或体积）的参数化。无法控制这种参数化导致形状质量差，设计师需要付出更多的努力来减轻这些工件。在这个项目中，PI将探索新的表面和体积表示，允许在创建底层形状时更多的几何自由。他将研究参数化与参数化曲线、表面和体积的表面形状/质量之间的基本联系。他将扩展曲面非均匀参数化的概念，以表明结间距（边长）不足以完全控制曲面或体积的参数化。他将设计新的表示，允许用户在设计过程中控制或自动适应这些形状的参数化，并结合目前不可能的非均匀参数化方法。作为这个过程的一部分，他将开发新的高阶重心坐标，专门用于解决这个问题。最后，他将研究参数化的效果和操作，用于这些参数化表面的镶嵌和渲染，并开发高质量的GPU镶嵌算法。更广泛的影响：项目成果不仅将显著提高计算机图形学和几何建模的技术水平，而且还将在应用数学和计算机科学的其他领域取得进展，在这些领域中，光滑自由形状的表示和精确控制发挥着关键作用。近似理论、建筑设计、娱乐业和工业制造业都将受益于本研究的结果。

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.