Injectivity of 2D Toric B\'{e}zier Patches

Rational B\'{e}zier functions are widely used as mapping functions in surface reparameterization, finite element analysis, image warping and morphing. The injectivity (one-to-one property) of a mapping function is typically necessary for these applications. Toric B\'{e}zier patches are generalizations of classical patches (triangular, tensor product) which are defined on the convex hull of a set of integer lattice points. We give a geometric condition on the control points that we show is equivalent to the injectivity of every 2D toric B\'{e}zier patch with those control points for all possible choices of weights. This condition refines that of Craciun, et al., which only implied injectivity on the interior of a patch.