Simple quad domains for field aligned mesh parametrization

We present a method for the global parametrization of meshes that preserves alignment to a cross field in input while obtaining a parametric domain made of few coarse axis-aligned rectangular patches, which form an abstract base complex without T-junctions. The method is based on the topological simplification of the cross field in input, followed by global smoothing

Frame Fields for CAD models

International audienceGiven a triangulated surface, a unit length tangent vector field can be used to orient entities located on the surface, such as glyphs or strokes. When these entities are invariant under a π/2 rotation (squares, or curvature hatching), the orientation can be represented by a frame field i.e. four orthogonal tangent unit vectors at each point of the surface. The generation of such fields is a key component of recent quad meshing algorithms based on global parameterization, as it defines the orientation of the final facets. State-of-the-art methods are able to generate smooth frame fields subject to some hard constraints (direction and topology) or smooth constraints (matching the curvature direction). When we have a surface triangular mesh, and a vector defined on each facet, we can't directly know if all the vectors are colinear. We first have to define the (so called) parallel transport of every edge to compare the vectors on a common plan. When dealing with CAD models, the field must be aligned with feature edges. A problem occurs when there is a low angle corner formed by two colliding feature edges. Our solution not only defines the parallel transport to obtain smoothed frame fields on a surface triangular mesh, it also redefines the parallel transport wherever there is a low angle corner, to smooth a frame field as if these corners' angles were π/2

Génération de maillage quadrangulaire d'un domaine du plan via les équations de Ginzburg-Landau

National audienceGénérer un maillage d'une surface est un pré-requis souvent indispensable à de nombreuses applications. Certaines (la subdivision de surfaces, la simulation de couches limites) nécessitent l'utilisation de maillage quadrangulaire. L'état de l'art procède en trois étapes. Il s'agit d'abord de calculer un champ de croix, puis de l'intégrer pour obtenir une paramétrisation et enfin d'extraire un maillage quadrangulaire à partir de la paramétrisation. Nous montrerons que les deux premières étapes réfèrent aux mêmes équations et peuvent donc être traitées de la même manière. Cette approche permet de résoudre des problèmes (imprécision loin des bords, mauvaise localisation des singularités) qui se posaient jusqu'alors

Subdivision Directional Fields

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure-preserving: it reproduces curl-free fields precisely, and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover's distance computation, for efficient and robust computation

Quad Meshing

Triangle meshes have been nearly ubiquitous in computer graphics, and a large body of data structures and geometry processing algorithms based on them has been developed in the literature. At the same time, quadrilateral meshes, especially semi-regular ones, have advantages for many applications, and significant progress was made in quadrilateral mesh generation and processing during the last several years. In this State of the Art Report, we discuss the advantages and problems of techniques operating on quadrilateral meshes, including surface analysis and mesh quality, simplification, adaptive refinement, alignment with features, parametrization, and remeshing