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.

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