Extensions of Koenderink’s formula

R , where U ⊂ R is an open subset. We denote the singular set of πξ◦f by S(πξ◦f). The set πξ ◦ f(S(πξ ◦ f)) is the (apparent) contour or profile of f as viewed from the direction ξ, and f(S(πξ ◦ f)) is the rim or contour generator of f as viewed from ξ. Koenderink showed the following theorem. Theorem 1.1 ([12, Appendix], [13, page 433]). Suppose p ∈ S(πξ ◦ f), namely, ξ is in Tpf(U), and πξ ◦ f(S(πξ ◦ f)) is a regular curve near p. Let κc be the curvature of the plane curve πξ ◦ f(S(πξ ◦ f)) ⊂ ξ, and κs the curvature of the normal section of M at p by the plane that contains ξ. Then