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Texture mapping with positional constraints is an important and challenging problem in computer graphics. In this paper, we first present a theoretically robust, foldover-free 2D mesh warping algorithm. Then, we apply this warping algorithm to handle mapping texture onto 3D meshes with hard constraints. The proposed algorithm is experimentally evaluated and compared with the state-of-the-art method for examples with more challenging constraints. These challenging constraints may lead to large distortions and foldovers. Experimental results show that the proposed scheme can generate more pleasing results and add fewer Steiner vertices on the 3D mesh embedding.