Optimal mapping is one of the longest-standing problems in computational mathematics. It is natural to measure the relative curve length error under map to assess its quality. The maximum of such error is called the quasi-isometry constant, and its minimization is a nontrivial max-norm optimization problem. We present a physics-based quasi-isometric stiffening (QIS) algorithm for the max-norm minimization of hyperelastic distortion. QIS perfectly equidistributes distortion over the entire domain for the ground-truth test (unit hemisphere flattening) and, when it is not possible, tends to create zones where all cells have the same distortion. Such zones correspond to fragments of elastic material that became rigid under stiffening, reaching the deformation limit. As such, maps built by QIS are related to the de Boor equidistribution principle, which asks for an integral of a certain error indicator function to be the same over each mesh cell. Under certain assumptions on the minimization toolbox, we prove that our method can build, in a finite number of steps, a deformation whose maximum distortion is arbitrarily close to the (unknown) minimum. We performed extensive testing: on more than 10,000 domains QIS was reliably better than the competing methods. In summary, we reliably build 2D and 3D mesh deformations with the smallest known distortion estimates for very stiff problems.

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.