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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. 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