On the contact of surfaces in 4-space with 2-planes and their apparent contours

We investigate the local geometry of generic smooth surfaces in R⁴ R 4 through their contact with 2-planes and the associated apparent contour. We study the A A -singularities of parallel projections of such surfaces along planes to transverse planes. When the projection plane does not contain an asymptotic direction, we show that hyperbolic and elliptic points (respectively, parabolic points) admit up to ten (respectively, seven) tangent directions determining planes for which the projection has A A -singularities of type butterfly or worse. We also prove that the locus of points where the discriminant of the equation defining these directions vanishes generically forms regular curves in the hyperbolic and elliptic regions, and isolated points in the parabolic set. When the projection plane contains an asymptotic direction, we relate the singularities of parallel projections to those of orthogonal projections onto hyperplanes and height functions. We then investigate the associated apparent contour. When the projection is a fold, we establish a Mannheim–d’Ocagne–Koenderink-type theorem relating the Lipschitz–Killing curvature of the surface to the curvature of the apparent contour. We further show that elliptic and inflection points give rise to vertices of the apparent contour, whereas hyperbolic and parabolic points generate inflections. In the non-fold case, we prove that the apparent contour admits (tᵏ,t^ ) ( t k , t ℓ ) -cusp singularities associated with the singularities of the projection.