Metric-Driven 3D Frame Field Generation

Controlling the size and shear of elements is crucial in pure hex or hex-dominant meshing. To this end, non-orthonormal frame fields that are almost everywhere integrable (except for the singularities) can play a key role. However, it is often challenging or impossible to generate such a frame field under the tight control of a general Riemannian metric field. Therefore, we propose to solve a relatively weaker problem, i.e., generating such a frame field for a Riemannian metric field that is flat away from singularities. Such a metric field admits a local isometry to 3D Euclidean space. Applying Cartans first structural equation to the associated rotation field, i.e., the rotation part of the frame field, we show that the rotation field must have zero covariant derivatives under the 3D connection induced by the metric field. This observation leads to a metric-aware smoothness measure, equivalent to local integrability. The use of such a measure can be justified on meshes associated with locally flat metric fields. We also propose a method to generate smooth metric fields under a few intuitive constraints. On cuboid shapes, our method generates singularities aware of the metric fields, which makes the parameterization match the input metric fields better than the conventional methods. For generic shapes, while our method generates visually similar results to those using boundary frame fields to guide the metric field generation, the integrability and consistency of the metric fields are still improved, as reflected by the statistics.