Sum-of-squares geometry processing

Abstract

<jats:p> Geometry processing presents a variety of difficult numerical problems, each seeming to require its own tailored solution. This breadth is largely due to the expansive list of <jats:italic>geometric primitives</jats:italic> , e.g., splines, triangles, and hexahedra, joined with an ever-expanding variety of <jats:italic>objectives</jats:italic> one might want to achieve with them. With the recent increase in attention toward <jats:italic>higher-order surfaces</jats:italic> , we can expect a variety of challenges porting existing solutions that work on triangle meshes to work on these more complex geometry types. In this paper, we present a framework for solving many core geometry processing problems on higher-order surfaces. We achieve this goal through sum-of-squares optimization, which transforms nonlinear polynomial optimization problems into sequences of convex problems whose complexity is captured by a single <jats:italic>degree</jats:italic> parameter. This allows us to solve a suite of problems on higher-order surfaces, such as continuous collision detection and closest point queries on curved patches, with only minor changes between formulations and geometries. </jats:p>