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.

Given a complex three-dimensional domain delimited by a closed and non-degenerate input triangle mesh without any self-intersection, a common geometry processing task consists in cutting up the domain into cells through a set of planar cuts, creating a "cut-cell mesh", i.e., a volumetric decomposition of the domain amenable to visualization (e.g., exploded views), animation (e.g., virtual surgery), or simulation (finite volume computations). A large number of methods have proposed either efficient or robust solutions, sometimes restricting the cuts to form a regular or adaptive grid for simplicity; yet, none can guarantee both properties, severely limiting their usefulness in practice. At the core of the difficulty is the determination of topological relationships among large numbers of vertices, edges, faces and cells in order to assemble a proper cut-cell mesh: while exact geometric computations provide a robust solution to this issue, their high computational cost has prompted a number of faster solutions based on, e.g., local floating-point angle sorting to significantly accelerate the process --- but losing robustness in doing so. In this paper, we introduce a new approach to planar cutting of 3D domains that substitutes topological inference for numerical ordering through a novel mesh data structure, and revert to exact numerical evaluations only in the few rare cases where it is strictly necessary. We show that our novel concept of topological cuts exploits the inherent structure of cut-cell mesh generation to save computational time while still guaranteeing exactness for, and robustness to, arbitrary cuts and surface geometry. We demonstrate the superiority of our approach over state-of-the-art methods on almost 10,000 meshes with a wide range of geometric and topological complexity. We also provide an open source implementation.

We introduce an approach to quadrilateral meshing of arbitrary triangulated surfaces that combines the theoretical guarantees of Morse-based approaches with the practical advantages of parameterization methods. We first construct, through an eigensolver followed by a few Gauss-Newton iterations, a periodic four-dimensional vector field that aligns with a user-provided frame field and/or a set of features over the input mesh. A field-aligned parameterization is then greedily computed along a spanning tree based on the Dirichlet energy of the optimal periodic vector field, from which quad elements are efficiently extracted over most of the surface. The few regions not yet covered by elements are then upsampled and the first component of the periodic vector field is used as a Morse function to extract the remaining quadrangles. This hybrid parameterization- and Morse-based quad meshing method is not only fast (the parameterization is greedily constructed, and the Morse function only needs to be upsampled in the few uncovered patches), but is guaranteed to provide a feature-aligned quad mesh with non-degenerate cells that closely matches the input frame field over an arbitrary surface. We show that our approach is much faster than Morse-based techniques since it does not require a densely tessellated input mesh, and is significantly more robust than parameterization-based techniques on models with complex features.

Abstract Representing rotational symmetry vector as a set of vectors is not suitable for design due to lacking of a consistent ordering for measurement. In this paper we introduce a spectral method to find rotation invariant harmonic functions for symmetry vector field design. This method is developed for 3D vector fields, but it is applicable in 2D. Given the finite symmetry group G of a symmetry vector fieldv(x) on a 3D domain Ω, we formulate the harmonic function h(s) as a stationary point of group G. Using the real spherical harmonic (SH) bases, we showed the coefficients of the harmonic functions are an eigenvector of the SH rotation matrices corresponding to group G. Instead of solving eigen problems to obtain the eigenvector, we developed a forward constructive method based on orthogonal group theory. The harmonic function found by our method is not only invariant under G, but also expressive and can distinguish different rotations with respect to G. At last, we demonstrate some vector field design results with tetrahedron‐symmetry, cube‐symmetry and dodecahedron‐symmetry groups.

The polycube-based hexahedralization methods are robust to generate all-hex meshes without internal singularities. They avoid the difficulty to control the global singularity structure for a valid hexahedralization in frame-field based methods. To thoroughly utilize this advantage, we propose to use a frame field without internal singularities to guide the polycube construction. Theoretically, our method extends the vector fields associated with the polycube from exact forms to closed forms, which are curl free everywhere but may be not globally integrable. The closed forms give additional degrees of freedom to deal with the topological structure of high-genus models, and also provide better initial axis alignment for subsequent polycube generation. We demonstrate the advantages of our method on various models, ranging from genus-zero models to high-genus ones, and from single-boundary models to multiple-boundary ones.

This paper presents a new technique for frame field generation. As generic frame fields (with arbitrary anisotropy, orientation, and sizing) can be regarded as cross fields in a specific Riemannian metric, we tackle frame field design by first computing a discrete metric on the input surface that is compatible with a sparse or dense set of input constraints. The final frame field is then found by computing an optimal cross field in this customized metric. We propose frame field design constraints on alignment, size, and skewness at arbitrary locations on the mesh as well as along feature curves, offering much improved flexibility over previous approaches. We demonstrate the advantages of our frame field generation through the automatic quadrangulation of man-made and organic shapes with controllable anisotropy, robust handling of narrow surface strips, and precise feature alignment. We also extend our technique to the design of n -vector fields.