On the topology of walkable environments

Motivated by motion planning applications, we study 2-dimensional surfaces embedded in 3dimensional space with the property that their vertical projection is an immersion. We provide bounds on the complexity of a triangulation of such a surface, given that the projection of the boundary is a polygon with m segments. We then show how these bounds lead to efficient algorithm to compute such a triangulation. Finally, we relate our result to concrete motion planning setting and review related open questions.