Affine Geometry of Space Curves

TL;DR

This paper provides a complete classification of space curves under affine transformations, introduces invariants, and explores their applications in physics, computer vision, and image processing.

Contribution

It offers the first necessary and sufficient conditions for invariants of space curves and details the shapes of curves with constant curvatures.

Findings

Established invariants for affine classification of space curves

Derived necessary and sufficient conditions for these invariants

Explored applications in physics, computer vision, and image processing

Abstract

This paper is devoted to the complete classification of space curves under affine transformations in the view of Cartan's theorem. Spivak has introduced the method but has not found the invariants. Furthermore, for the first time, we propound a necessary and sufficient condition for the invariants. Then, we study the shapes of space curves with constant curvatures in detail and suggest their applications in physics, computer vision and image processing.