Projections of surfaces in ${\mathbb{R}}^4$ to ${\mathbb{R}}^3$ and the geometry of their singular images

Abstract

We study the geometry of germs of singular surfaces in ${\mathbb{R}}^3$ whose parametrisations have an $\mathcal{A}$-singularity of ${\mathcal{A}}_e$-codimension less than or equal to 3, via their contact with planes. These singular surfaces occur as projections of smooth surfaces in ${\mathbb{R}}^4$ to ${\mathbb{R}}^3$. We recover some aspects of the extrinsic geometry of these surfaces in ${\mathbb{R}}^4$ from those of the images of their projections.