Reliable Computation of the Singularities of the Projection in R3 of a Generic Surface of R4

Computing efficiently the singularities of surfaces embedded in R^3 is a difficult problem, and most state-of-the-art approaches only handle the case of surfaces defined by polynomial equations. Let F and G be C∞ functions from R 4 to R and M = {(x, y, z, t) ∈ R 4 | F (x, y, z, t) = G(x, y, z, t) = 0} be the surface they define. Generically, the surface M is smooth and its projection Ω in R 3 is singular. After describing the types of singularities that appear generically in Ω, we design a numerically well-posed system that encodes them. This can be used to return a set of boxes that enclose the singularities of Ω as tightly as required. As opposed to state-of-the art approaches, our approach is not restricted to polynomial mapping, and can handle trigonometric or exponential functions for example.