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

TL;DR

This paper extends the d'Ocagne formula to relate the local behavior of surface projections in 3D space with Gaussian curvature, using view-dependent invariants of projected plane curves.

Contribution

It introduces a view-parametric extension of the d'Ocagne formula connecting surface curvature with projection invariants in 3D.

Findings

Derived invariants of projected plane curves.

Extended d'Ocagne formula incorporating view parameters.

Link between projection behavior and Gaussian curvature.

Abstract

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.