Visual Contours and Planar Sections of Affine Immersions

TL;DR

This paper investigates the geometric properties of affine immersions, revealing that the third order Taylor expansion of visual contours and planar sections relates directly to the cubic form, and characterizes affine spheres with special coordinate systems.

Contribution

It establishes a precise link between visual contours, planar sections, and the cubic form in affine immersions, and characterizes affine spheres with parallel-meridian coordinates.

Findings

Third order Taylor expansion equals the cubic form.

Affine spheres with parallel-meridian coordinates are quadrics.

Characterization of special coordinate systems on affine immersions.

Abstract

In this paper we consider planar sections and visual contours of co-dimension one affine immersions. The main theorem says that the third order Taylor expansion of the difference between the visual contour and planar section functions is exactly the cubic form. We also consider parameterizations on two dimensional affine immersions whose coordinate lines are geodesics, in one direction, and both planar sections and visual contours in the other direction. We call such coordinates parallel-meridian. Among other results, we show that any two dimensional affine sphere that admits parallel-meridian coordinates with a pole must be a quadric.