  We present Advancing Front Mapping (AFM), a provably robust algorithm for the
computation of surface mappings to simple base domains. Given an input mesh and
a convex or star-shaped target domain, AFM installs a (possibly refined)
version of the input connectivity into the target shape, generating a
piece-wise linear mapping between them. The algorithm is inspired by the
advancing front meshing paradigm, which is revisited to operate on two
embeddings at once, thus becoming a tool for compatible mesh generation. AFM
extends the capabilities of existing robust approaches, such as Tutte or
Progressive Embedding, by providing the same theoretical guarantees of
injectivity and at the same time introducing two key advantages: support for a
broader set of target domains (star-shaped polygons) and local mesh refinement,
which is used to automatically open the space of solutions in case a valid
mapping to the target domain does not exist. AFM relies solely on two
topological operators (split and flip), and on the computation of segment
intersections, thus permitting to compute provably injective mappings without
solving any numerical problem. This makes the algorithm predictable, easy to
implement, debug and deploy. We validated the capabilities of AFM extensively,
executing more than one billion advancing front moves on 36K mapping tasks,
proving that our theoretical guarantees nicely transition to a robust and
practical implementation.
