The Curvature of Plane Elastic Curves

In this paper plane elastic curves are revisited from a viewpoint that emphasizes curvature properties of these curves. The family of elastic curves is considered in dependence of a tension parameter Sigma and the squared global curvature maximum K2/m. It is shown that for any elastic curve K2/m is bigger than the tension parameter Sigma. A curvature analysis of the fundamental forms of the elastic curves is presented. A formula is established that gives the maximum turning angle of an elastica as a function depending on K2/m and Sigma. Finally, it is shown that an elastic curve can be represented as a linear combination of its curvature, arc length and energy function and that any curve with this property is an elastic.