Collaborative RUI: Quadrilateral Surface Meshes with Provable Quality Guarantees

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The representation and reconstruction of complex three-dimensional objects is critical in a wide range of applications in computing today. Polygonal meshes have become the industry standard for the representation of surfaces with highly complex geometry and arbitrary genus in computer graphics and geometry processing applications. While triangle meshes are the most popular type of mesh representation for surfaces, quadrilateral meshes are better suited than triangle meshes in several applications such as character animation, texture mapping, spline-based surface modeling, mesh compression, and some specific finite element analysis applications. Provably good algorithms for generating triangle meshes from surfaces given by parametric or implicit functions or as point point clouds are widely available. However, algorithms to generate quadrilateral meshes with provable quality guarantees for such surfaces are not as prevalent, in part because the problem of generating a quadrilateral mesh from a given surface is intrinsically harder than its triangular counterpart. The goal of this project is to develop algorithms for quadrilateral meshes for various surface representations with provable guarantees on element quality as measured by commonly used metrics such as angle bounds or aspect ratio, and mesh quality as measured by mesh size or anisotropy. Direct and indirect methods (which generate a quad mesh from a triangle mesh), as well as parameterization guided methods will be utilized. Techniques from computational geometry and graph theory will play a central role in the design and development of algorithms. The automated generation of provably good quadrilateral meshes for surfaces is a fundamental problem that is of interest both in theory, as it raises several geometric, combinatorial, and graph-theoretic questions, as well as practice, where the computational pipeline from designing a model for the surface to the end stage of simulation or animation is frequently dominated by the meshing process. A formal understanding of these questions is critical not only for the theoretical underpinnings of automated mesh generation, but also for the sound practice of utilizing the meshes in a wide range of applications. As a collaborative effort between PIs at three undergraduate institutions, involvement of undergraduate students in research projects is an integral part of this project. An important and particular goal for this project is the creation of a larger peer group for female and minority undergraduate Computer Science majors by providing opportunities for collaboration and joint research projects between students across all three institutions. Through early and active involvement of undergraduates in the project, the PIs also seek to create a pipeline of female and minority students bound for graduate school in Computer Science.