Fast Linking Numbers for Topology Verification of Loopy Structures

Abstract: It is increasingly common to model, simulate, and process complex materials based on loopy structures, such as in yarn-level cloth garments, which possess topological constraints between inter-looping curves. While the input model may satisfy specific topological linkages between pairs of closed loops, subsequent processing may violate those topological conditions. In this paper, we explore a family of methods for efficiently computing and verifying linking numbers between closed curves, and apply these to applications in geometry processing, animation, and simulation, so as to verify that topological invariants are preserved during and after processing of the input models. Our method has three stages: (1) we identify potentially interacting loop-loop pairs, then (2) carefully discretize each loop's spline curves into line segments so as to enable (3) efficient linking number evaluation using accelerated kernels based on either counting projected segment-segment crossings, or by evaluating the Gauss linking integral using direct or fast summation methods (Barnes-Hut or fast multipole methods). We evaluate CPU and GPU implementations of these methods on a suite of test problems, including yarn-level cloth and chainmail, that involve significant processing: physics-based relaxation and animation, user-modeled deformations, curve compression and reparameterization. We show that topology errors can be efficiently identified to enable more robust processing of loopy structures.