Changing Views on Curves and Surfaces

TL;DR

This paper explores how the visual appearance of 3D curves and surfaces changes with viewpoint using algebraic geometry, deriving formulas and methods to compute visual event surfaces.

Contribution

It introduces algebraic techniques to analyze and compute visual event surfaces for curves and surfaces in 3D, linking geometric changes to singular loci.

Findings

Derived formulas for degrees of visual event surfaces.

Provided algebraic methods for exact computation of these surfaces.

Connected geometric changes to singular loci of coisotropic hypersurfaces.

Abstract

Visual events in computer vision are studied from the perspective of algebraic geometry. Given a sufficiently general curve or surface in 3-space, we consider the image or contour curve that arises by projecting from a viewpoint. Qualitative changes in that curve occur when the viewpoint crosses the visual event surface. We examine the components of this ruled surface, and observe that these coincide with the iterated singular loci of the coisotropic hypersurfaces associated with the original curve or surface. We derive formulas, due to Salmon and Petitjean, for the degrees of these surfaces, and show how to compute exact representations for all visual event surfaces using algebraic methods.