Barycentric coordinates for Lagrange interpolation over lattices on a simplex

In this paper, a (d?+?1)-pencil lattice on a simplex in ${\mathbb{R}}^d$ is studied. The lattice points are explicitly given in barycentric coordinates. This enables the construction and the efficient evaluation of the Lagrange interpolating polynomial over a lattice on a simplex. Also, the barycentric representation, based on shape parameters, turns out to be appropriate for the lattice extension from a simplex to a simplicial partition.