Abstract: We present an algorithm for mapping a triangle mesh, which is homeomorphic to a disk, to a planar domain with arbitrary fixed boundaries. The algorithm is guaranteed to produce a globally bijective map when the boundary is fixed to a shape that does not self-intersect. Obtaining a one-to-one map is of paramount importance for many graphics applications such as texture mapping. However, for other applications, such as quadrangulation, remeshing, and planar deformations, global bijectively may be unnecessarily constraining and requires significant increase on map distortion. For that reason, our algorithm allows the fixed boundary to intersect itself, and is guaranteed to produce a map that is injective locally (if such a map exists). We also extend the basic ideas of the algorithm to support the computation of discrete approximation for extremal quasiconformal maps. The algorithm is conceptually simple and fast. We demonstrate the superior robustness of our algorithm in various settings and configurations in which state-of-the-art algorithms fail to produce injective maps.