Three-pencil lattices on triangulations

Abstract

In this paper, three-pencil lattices on triangulations are studied. The explicit representation of a lattice, based upon barycentric coordinates, enables us to construct lattice points in a simple and numerically stable way. Further, this representation carries over to triangulations in a natural way. The construction is based upon group action of S 3 on triangle vertices, and it is shown that the number of degrees of freedom is equal to the number of vertices of the triangulation.

Barycentric coordinates for Lagrange interpolation over lattices on a simplex

“…Combining these with the results in [4], where three-pencil lattices on triangulations have been studied, a continuous piecewise polynomial interpolant over the triangulation can be obtained (Fig. 3).…”

“…This approach heavily depends on homogeneous coordinates, and a nice illustrative explanation can be found in [7], where the cases perhaps most often met in practice, i.e., d = 2 and d = 3, are outlined. Here our goal is an explicit representation in barycentric coordinates using a novel approach, since this enables a natural extension from a simplex to a simplicial partition (see [4] for the case d = 2).…”

“…. , ξ d ) as in (4). Then the lattice is determined by parameters α and β β β β β β β β β that satisfy…”

“…Note that, in contrast to parameters β β β β β β β β β, the introduced parameters ξ ξ ξ ξ ξ ξ ξ ξ ξ have a clear geometric meaning, and can be used as shape parameters of the lattice (see [4] and Fig. 3, e.g.…”

“…. On the other hand, we provide fundamental tools for the construction of continuous splines over simplicial partitions, the problem which has already been observed in [4] but only for triangulations.…”