**Abstract:** We describe an algorithm for constructing point sets
which admit unique Lagrange and Hermite interpolation by the space $S^1_3(
\Delta)$ of splines of degree 3 defined on a general class of triangulations
$\Delta$. The triangulations $\Delta$ consist of nested polygons whose
vertices are connected by line segments. In particular, we have to determine
the dimension of $S^1_3 (\Delta)$ which is not known for arbitrary triangulations
$\Delta$. Numerical examples are given.

**Preprint version available:** pdf

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