The generation of quad meshes based on surface parametrization techniques has proven to be a versatile approach. These techniques quantize an initial seamless parametrization so as to obtain an integer grid map implying a pure quad mesh. State-of-the-art methods following this approach have to assume that the surface to be meshed either has no boundary, or has a boundary which the resulting mesh is supposed to be aligned to. In a variety of applications this is not desirable and non-boundary-aligned meshes or grid-parametrizations are preferred. We thus present a technique to robustly generate integer grid maps which are either boundary-aligned, non-boundary-aligned, or partially boundary-aligned, just as required by different applications. We thereby generalize previous work to this broader setting. This enables the reliable generation of trimmed quad meshes with partial elements along the boundary, preferable in various scenarios, from tiled texturing over design and modeling to fabrication and architecture, due to fewer constraints and hence higher overall mesh quality and other benefits in terms of aesthetics and flexibility.
Field-guided parametrization methods have proven effective for quad meshing of surfaces; these methods compute smooth cross fields to guide the meshing process and then integrate the fields to construct a discrete mesh. A key challenge in extending these methods to three dimensions, however, is representation of field values. Whereas cross fields can be represented by tangent vector fields that form a linear space, the 3D analog---an octahedral frame field---takes values in a nonlinear manifold. In this work, we describe the space of octahedral frames in the language of differential and algebraic geometry. With this understanding, we develop geometry-aware tools for optimization of octahedral fields, namely geodesic stepping and exact projection via semidefinite relaxation. Our algebraic approach not only provides an elegant and mathematically-sound description of the space of octahedral frames but also suggests a generalization to frames whose three axes scale independently, better capturing the singular behavior we expect to see in volumetric frame fields. These new odeco frames, so-called as they are represented by orthogonally decomposable tensors, also admit a semidefinite program--based projection operator. Our description of the spaces of octahedral and odeco frames suggests computing frame fields via manifold-based optimization algorithms; we show that these algorithms efficiently produce high-quality fields while maintaining stability and smoothness.