Oriented combinatorial maps and extensions – Specification , prototyping , applications

We propose a formal specification based on a functional prototype of the oriented combinatorial maps embedded in the plane and revisit classical problems in computational geometry. We first recall the basic properties of the maps and their modelling power for closed orientable surfaces. Then, we mainly focus on logical, topological and geometric aspects of algorithms rather than on efficiency and round-off error questions. All the descriptions are written into the Ocaml language and tested. Progressively enriched by topological and geometric features and operations, the prototype is a basis of discussion before doing a real efficient implementation in a computational geometry library. The paper is illustrated by some high-level operations, particularly a Delaunay triangulation. Finally, we present and discuss some extensions in higher dimensions, particularly the generalized maps and their features. We show how, at dimension 2, they model the topology of general surfaces, closed or open, orientable or not.