Approximation of Smooth Surfaces by Polyhedral Surfaces with Hidden Vertices

In R, one can construct polyhedral surfaces such that, for some point p not on the surface, none of the vertices of the surface is visible from p. For a given compact surface K and a point we study the relation between the set of partial-linear embeddings of K with hidden vertices into R and the set of embeddings of K whose image is a smooth manifold with at least one principal curvature pointing towards p at each point that is visible from p. In particular, we establish results that suggest that elements of any of these spaces can be C-approximated by elements of the other one. Considering that these approximations may be extended to work parametrically, we conjecture that the space of partial linear mappings with vertices hidden from a point p is weakly homotopy equivalent to the space of mappings whose image has at least one principal curvature pointing towards p at all visible points, endowed with the C-topology.