Jacobians and Hessians of mean value coordinates for closed triangular meshes

Mean value coordinates provide an efficient mechanism for the interpolation of scalar functions defined on orientable domains with a nonconvex boundary. They present several interesting features, including the simplicity and speed that yield from their closed-form expression. In several applications though, it is desirable to enforce additional constraints involving the partial derivatives of the interpolated function, as done in the case of the Green coordinates approximation scheme (Ben-Chen, Weber, Gotsman, ACM Trans. Graph.:1–11, 2009) for interactive 3D model deformation. In this paper, we introduce the analytic expressions of the Jacobian and the Hessian of functions interpolated through mean value coordinates. We provide these expressions both for the 2D and 3D case. We also provide a thorough analysis of their degenerate configurations along with accurate approximations of the partial derivatives in these configurations. Extensive numerical experiments show the accuracy of our derivation. In particular, we illustrate the improvements of our formulae over a variety of finite differences schemes in terms of precision and usability. We demonstrate the utility of this derivation in several applications, including cage-based implicit 3D model deformations (i.e., variational MVC deformations). This technique allows for easy and interactive model deformations with sparse positional, rotational, and smoothness constraints. Moreover, the cages produced by the algorithm can be directly reused for further manipulations, which makes our framework directly compatible with existing software supporting mean value coordinates based deformations.