Mapping a given surface onto a piece of the plane is an usual problem, and conformal mappings with border free approaches are classical as non-distorting methods. However, algorithms of this category are only able to produce a mapping from a surface topo-logically equivalent to a disc. In the general case, a topological cutting is thus required. The approach we are reporting here is a method that does not only take into account the topologi-cal properties by cutting the original surface into a disc [3], but also the the geometrical properties by globally selecting the local extrema of original arm-like surfaces (denoted organic surfaces in the following).