Mannheim-d’Ocagne-Koenderink type formulas for asymptotic directions

We consider a surface embedded in the Euclidean 3-space and fix a tangential vector $\textbf{v}$ at a given point $p$ on the surface. We first review the history of the formula obtained by Mannheim, d’Ocagne and Koenderink, which asserts that the Gaussian curvature of the surface at $p$ can be obtained if one knows “the normal curvature at $p$ with respect to $\textbf{v}$” and “the curvature at $p$ of the contour line $\Gamma$ of the surface” with respect to the orthogonal projection induced by $\textbf{v}$. However, this formula breaks down when $\textbf{v}$ is an asymptotic direction. In such cases, we present analogues of the formula that involve an invariant associated with cusp singularities of $\Gamma$.