Welcome

Eurographics 2021

We consider the problem of injectively embedding a given graph connectivity (a layout) into a target surface. Starting from prescribed positions of layout vertices, the task is to embed all layout edges as intersection-free paths on the surface. Besides merely geometric choices (the shape of paths) this problem is especially challenging due to its topological degrees of freedom (how to route paths around layout vertices). The problem is typically addressed through a sequence of shortest path insertions, ordered by a greedy heuristic. Such insertion sequences are not guaranteed to be optimal: Early path insertions can potentially force later paths into unexpected homotopy classes. We show how common greedy methods can easily produce embeddings of dramatically bad quality, rendering such methods unsuitable for automatic processing pipelines. Instead, we strive to find the optimal order of insertions, i.e. the one that minimizes the total path length of the embedding. We demonstrate that, despite the vast combinatorial solution space, this problem can be effectively solved on simply-connected domains via a custom-tailored branch-and-bound strategy. This enables directly using the resulting embeddings in downstream applications which cannot recover from initializations in a wrong homotopy class. We demonstrate the robustness of our method on a shape dataset by embedding a common template layout per category, and show applications in quad meshing and inter-surface mapping.

Eurographics 2021

We present a robust and fast method for the creation of conforming quad layouts on surfaces. Our algorithm is based on the quantization of a T-mesh, i.e. an assignment of integer lengths to the sides of a non-conforming rectangular partition of the surface. This representation has the benefit of being able to encode an infinite number of layout connectivity options in a finite manner, which guarantees that a valid layout can always be found. We carefully construct the T-mesh from a given seamless parametrization such that the algorithm can provide guarantees on the results' quality. In particular, the user can specify a bound on the angular deviation of layout edges from prescribed directions. We solve an integer linear program (ILP) to find a coarse quad layout adhering to that maximal deviation. Our algorithm is guaranteed to yield a conforming quad layout free of T-junctions together with bounded angle distortion. Our results show that the presented method is fast, reliable, and achieves high quality layouts.

Computer-Aided Design

We present octree-embedded BSPs, a volumetric mesh data structure suited for performing a sequence of Boolean operations (iterated CSG) efficiently. At its core, our data structure leverages a plane-based geometry representation and integer arithmetics to guarantee unconditionally robust operations. These typically present considerable performance challenges which we overcome by using custom-tailored fixed-precision operations and an efficient algorithm for cutting a convex mesh against a plane. Consequently, BSP Booleans and mesh extraction are formulated in terms of mesh cutting. The octree is used as a global acceleration structure to keep modifications local and bound the BSP complexity. With our optimizations, we can perform up to 2.5 million mesh-plane cuts per second on a single core, which creates roughly 40-50 million output BSP nodes for CSG. We demonstrate our system in two iterated CSG settings: sweep volumes and a milling simulation.