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

Abstract

We investigate the local geometry of generic smooth surfaces in $\mathbb R^4$ via the contact with $2$-planes and the associated apparent contour. We study the $\mathcal A$-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, at hyperbolic or elliptic (respectively, parabolic) points, there exist up to ten (respectively, seven) tangent directions determining planes along which the projection exhibits $\mathcal A$-singularities of type butterfly or worse. Moreover, we 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 of $M$. When the projection plane contains an asymptotic direction, we establish connections between the singularities of parallel projections, orthogonal projections to hyperplanes, and height functions. We further study the apparent contour associated with the parallel projection. When the projection is a fold, we prove a Koenderink-type theorem relating the Lipschitz--Killing curvature of the surface to the curvature of the apparent contour. Moreover, we show that elliptic or inflections points of the surface give rise to vertices of the apparent contour, whereas hyperbolic and parabolic points give rise to inflections. These phenomena are then characterized in terms of the singularities of the projection. In the non-fold case, we show that the apparent contour admits singularities of type $(t^k,t^\ell)$-cusp associated with the singularities of the projection.