Abstract: We develop an algorithm for constructing Lagrange and Hermite interpolation sets for spaces of cubic $C^1$-splines on general classes of triangulations built up of nested polygons whose vertices are connected by line segments. Additional assumptions on the triangulation are significantly reduced compared to the special class given in [4]. Simultaneously, we have to determine the dimension of these spaces, which is not known in general. We also discuss the numerical aspects of the method.

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