Foldover-free maps in 50 lines of code

Abstract

Mapping a triangulated surface to 2D space (or a tetrahedral mesh to 3D space) is an important problem in geometry processing. In computational physics, untangling plays an important role in mesh generation: it takes a mesh as an input, and moves the vertices to get rid of foldovers. In fact, mesh untangling can be considered as a special case of mapping where the geometry of the object is to be defined in the map space and the geometric domain is not explicit, supposing that each element is regular. In this paper, we propose a mapping method inspired by the untangling problem and compare its performance to the state of the art. The main advantage of our method is that the untangling aims at producing locally injective maps, which is the major challenge of mapping. In practice, our method produces locally injective maps in very difficult settings, both in 2D and 3D. We demonstrate it on a large reference database as well as on more difficult stress tests. For a better reproducibility, we publish the code in Python for a basic evaluation, and in C++ for more advanced applications.

Practical construction of globally injective parameterizations with positional constraints

“…On the other hand, some methods take an inverted configuration as input and try to remove the flipped elements afterwards. These include a bounded distortion mapping method [9], projection-based methods [8,35], an assembly-based method [36], penalty-based methods [37,38], and area-based methods [39,40]. However, until now, there has been no theoretical guarantee that all flips will be totally removed.…”

“…Constructing flip-free parameterizations with positional constraints has attracted considerable research attention in recent years [7,14,16,36], but without the intersection-free constraint being considered. Many methods can be successfully applied to fixed-boundary mapping problems, thus leading to an intersection-free mapping as long as the initial boundary has no intersections [8,38,40]. However, existing methods provide no theoretical guarantee that they generate an initial parameterization satisfying the intersection-free condition, the flip-free condition, and arbitrary positional constraints.…”

Variational schemes and mixed finite elements for large strain isotropic elasticity in principal stretches: Closed‐form tangent eigensystems, convexity conditions, and stabilised elasticity

“…More specifically, we consider $$$ {\mu}_2=0 $$$ and $$$ \eta =1 $$$. Polyconvexity of the energy under such setting is well‐known 50,62 …”

“…Polyconvexity of the energy under such setting is well-known. 50,62 TA B L E 1 Comparison of displacements and Cauchy stress components at multiple points for planar Cooks membrane problem for P2 displacement-based and P2-P1 mixed principal stretch formulations. The variation of stresses is sparse for displacement-based FEM but in a tighter range for mixed principal stretch FEM.…”

Evocube: A Genetic Labelling Framework for Polycube‐Maps

“…Subsequently, we minimize the energy proposed by Garanzha et al. [GKK*21] under the same change of variable to restore inverted cells and further improve the quality of the volumetric polycube‐map.…”