Abstract
We introduce some basic constructions for sets of functions which solve the Lagrange interpolation problem from two or more sets of functions having the same property and sharing a common 'separating set'. We also investigate similar constructions for sets of functions which solve the Hermite interpolation problem. These constructions translate into constructions for geometries on surfaces which generalize and extend the fundamental cut and paste constructions for topological geometries on surfaces such as flat affine, projective, Mobius, Laguerre and Minkowski planes.
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