The Signature of a Plane Curve

The signature of a plane curve $\Gamma $ associated with every point p of $\Gamma $ the length of $\Gamma $ to the left of or on the line tangent to $\Gamma $ at p. The signature has properties that make it a useful tool for pattern recognition: it discards the location, orientation, and scale, and “slant” in special cases, but preserves symmetries. Its integral is a measure of convexity. This paper explores the theoretical properties of this concept. It is shown that in the special case of closed rectilinear curves, the signature retains enough information to permit exact reconstruction of the curve. Computing the signature and reconstructing curves from their signatures are interesting computational problems; time complexity bounds on these problems are presented. Several challenging open questions are posed.