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

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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.