Affine Curvature - Characterization Up To Affine Congruence

Characterization Up To Affine Congruence

The special affine curvature of an immersed curve is the only (local) invariant of the curve in the following sense:

• If two curves have the same special affine curvature at every point, then one curve is obtained from the other by means of a special affine transformation.

In fact, a slightly stronger statement holds:

• Given any continuous function k : → R, there exists a curve β whose first and second derivatives are linearly independent, such that the special affine curvature of β relative to the special affine parameterization is equal to the given function k. The curve β is uniquely determined up to a special affine transformation.

This is analogous to the fundamental theorem of curves in the classical Euclidean differential geometry of curves, in which the complete classification of plane curves up to Euclidean motion depends on a single function κ, the curvature of the curve. It follows essentially by applying the Picard–Lindelöf theorem to the system

where C_β = . An alternative approach, rooted in the theory of moving frames, is to apply the existence of a primitive for the Darboux derivative.