Some remarks on robust features of surfaces in 4-space

We study robust features of smooth surfaces in 4-space associated with contact with lines and hyperplanes. We investigate the local and multi-local singularities of orthogonal projections and height functions that describe these contacts and their corresponding geometric loci on the surface. At an A₄-singularity of a height function, we show that the A₃-curve and the A₁A₂-curve are generically tangent with contact of order $2$. We further establish criteria for the versality of the families of height functions and orthogonal projections in terms of the regularity of the B₂-curve and S₂-curve, respectively. Finally, we obtain the [A₂]- and (A₀S₀)₂-curves arising from multi-local singularities of orthogonal projections and describe their tangency properties at the B₃- and H₃-singularities.