Abstract: Let $\Delta$ be an arbitrary regular triangulation of a simply connected compact polygonal domain $\Omega\subset\RR^2$ and let $S_q^1(\Delta)$ denote the space of bivariate polynomial splines of degree $q$ and smoothness $1$ with respect to $\Delta$. We develop an algorithm for constructing point sets admissible for Lagrange interpolation by $S_q^1(\Delta)$ if $q\ge4$. In the case $q=4$ it may be necessary to slightly modify $\Delta$, but only if exeptional constellations of triangles occur. Hermite interpolation schemes are obtained as limits of the Lagrange interpolation sets.
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