A view-parametric extension of the d'Ocagne formula for a surface in ${\mathbb{R}}^3$

In this paper, we consider the orthogonal projection of a surface in ${\mathbb{R}}^3$ for a given view direction. We then introduce and investigate several invariants of the families of the plane curves that locally configure the projection image of the surface. Using the invariants, we also show an extension of the d’Ocagne formula that associates a local behavior of the projection image of a surface with Gaussian curvature of the surface.