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We study the flat geometry of the least degenerate singularity of a singularsurface in $\mathbb R^4$, the $I_{1}$ singularity parametrised by$(x,y)\mapsto(x,xy,y^{2},y^{3})$. This singularity appears generically whenprojecting a regular surface in $\mathbb R^5$ orthogonally to $\mathbb R^4$along a tangent direction. We obtain a generic normal form for $I_1$ invariantunder diffeomorphisms in the source and isometries in the target. We thenconsider the contact with hyperplanes by classifying submersions which preservethe image of $I_1$. The main tool is the study of the singularities of theheight function.

The flat geometry of the $I_{1}$ singularity: $(x,y)\mapsto(x,xy,y^{2},y^{3})$