Affine geometry of equal-volume polygons in 3-space

TL;DR

This paper develops a theory of discrete affine invariants for equal-volume polygons in 3D space, derived from discretizations of smooth curves, with applications in estimating projective invariants of planar curves.

Contribution

It introduces a novel framework for discrete affine invariants of equal-volume polygons and links them to smooth curve invariants, expanding geometric analysis tools.

Findings

Discrete affine invariants for equal-volume polygons are established.

Equal-volume polygons can estimate projective invariants of planar curves.

The theory has potential applications in silhouette curve analysis.

Abstract

Equal-volume polygons are obtained from adequate discretizations of curves in 3-space, contained or not in surfaces. In this paper we explore the similarities of these polygons with the affine arc-length parameterized smooth curves to develop a theory of discrete affine invariants. Besides obtaining discrete affine invariants, equal-volume polygons can also be used to estimate projective invariants of a planar curve. This theory has many potential applications, among them evaluation of the quality and computation of affine invariants of silhouette curves.