  Mapping a triangulated surface to 2D space (or a tetrahedral mesh to 3D
space) is the most fundamental 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, and with less distortion than the previous work, 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.
