Elastica and Computer Vision

Abstract

I want to discuss the problem from differential geometry of describing those plane curves C which minimize the integral

$$\int\limits_C {(\alpha k^2 + \beta )ds.}$$

Here α and β are constants, kis the curvature of C, ds the arc length and, to make the fewest boundary conditions, we mean minimizing for infinitesimal variations of C on a compact set not containing the endpoints of C. Alternately, one may minimize

$$\int\limits_C {k^2 ds}$$

over variations of C which preserve the total length.