Global parametrization based on Ginzburg-Landau functional

International audienceQuad meshing is a fundamental preprocessing task for many applications (subdivision surfaces, boundary layer simulation). State-of-the-art quad mesh generators proceed in three steps: first a guiding cross field is computed, then a parametrization representing the quads is generated, and finally a mesh is extracted from the parameterization. In this paper we show that in the case of a periodic global parameterization two first steps answer to the same equation and inherently face the same challenges. This new insight allows us to use recent cross field generation algorithms based on Ginzburg-Landau equations to accurately solve the parametrization step. We provide practical evidence that this formulation enables us to overcome common shortcomings in parametrization computation (inaccuracy away from the boundary, singular dipole placement)

On spherical harmonics possessing octahedral symmetry

In this paper, we present the implicit representation of one special class of real-valued spherical harmonics with octahedral symmetry. Based on this representation we construct the rotationally invariant measure of deviation from the specified symmetry. The spherical harmonics we consider have some applications in the area of directional fields design due to their ability to represent mutually orthogonal axes in 3D space not relatively to their order and orientation

Dev2PQ: Planar Quadrilateral Strip Remeshing of Developable Surfaces

We introduce an algorithm to remesh triangle meshes representing developable surfaces to planar quad dominant meshes. The output of our algorithm consists of planar quadrilateral (PQ) strips that are aligned to principal curvature directions and closely approximate the curved parts of the input developable, and planar polygons representing the flat parts of the input. Developable PQ-strip meshes are useful in many areas of shape modeling, thanks to the simplicity of fabrication from flat sheet material. Unfortunately, they are difficult to model due to their restrictive combinatorics and locking issues. Other representations of developable surfaces, such as arbitrary triangle or quad meshes, are more suitable for interactive freeform modeling, but generally have non-planar faces or are not aligned to principal curvatures. Our method leverages the modeling flexibility of non-ruling based representations of developable surfaces, while still obtaining developable, curvature aligned PQ-strip meshes. Our algorithm optimizes for a scalar function on the input mesh, such that its level sets are extrinsically straight and align well to the locally estimated ruling directions. The condition that guarantees straight level sets is nonlinear of high order and numerically difficult to enforce in a straightforward manner. We devise an alternating optimization method that makes our problem tractable and practical to compute. Our method works automatically on any developable input, including multiple patches and curved folds, without explicit domain decomposition. We demonstrate the effectiveness of our approach on a variety of developable surfaces and show how our remeshing can be used alongside handle based interactive freeform modeling of developable shapes