Reliable Computation of the Singularities of the Projection in $$\mathbb R^3$$ of a Generic Surface of $$\mathbb R^4$$

Computing efficiently the singularities of surfaces embedded in \(\mathbb R^3\) 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 \(C^\infty \) functions from \(\mathbb R^4\) to \(\mathbb R\) and \(\mathcal M=\{(x,y,z,t) \in \mathbb R^4 \, | \, F(x,y,z,t)=G(x,y,z,t)=0\}\) be the surface they define. Generically, the surface \(\mathcal M\) is smooth and its projection \(\Omega \) in \(\mathbb R^3\) is singular. After describing the types of singularities that appear generically in \(\Omega \), 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 \(\Omega \) 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.