A numerical approach to compute the topology of the Apparent Contour of a smooth mapping from R2 to R2

A rigorous algorithm for computing the topology of the Apparent Contour of a generic smooth map is designed and studied in this paper. The source set is assumed to be a simply connected compact subset of the plane and the target space is the plane. Whitney proved that, generically, critical points of a smooth map are folds or cusps [9]. The Apparent Contour is the set of critical values, that is, the image of the critical points. Generically speaking, the Apparent Contour does not have triple points and double points are normal crossings (i.e. crossing without tangency). Each of those particular cases, cusp and normal crossing, is described in order to be rigorously handled by an interval analysis based scheme. The first step of the presented method provides an enclosure of those particular points. The second part of the designed method is a computation of a graph which is homeomorphic to the Apparent Contour. Edges of this graph are computed by testing connectivity of those particular points in the source space. This paper also defines a concept called portrait. Relations between this notion and the more classical notion of Apparent Contour are discussed.