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Algebraic Construction of Smooth Interpolants on Polygonal Domains

# Input Data

The interpolant can be constructed on any two-dimensional slice of any polytope, including polygons, polyhedra, or higher-dimensional objects. Accordingly, the dimension of the data must be defined. The examples are in two dimensions.

The list of vertices must also be defined; for example, here is a triangle with three nodes.

Another example is a triangle enclosing a quadrilateral and an interior point.

For a simple interpolant, the quantity that is going to be distributed smoothly over the domain must be defined at every nodal point. The nodal points not only determine the structure of the domain, they are also the points at which the value being distributed is given. The vertices of the polygon are a subset of the nodal points. Nodal points and interpolation points are one and the same.