EVOLUTES OF FRONTS IN THE EUCLIDEAN PLANE

The evolute of a regular curve in the Euclidean plane is given by not only the caustics of the regular curve, envelope of normal lines of the regular curve, but also the locus of singular points of parallel curves. In general, the evolute of a regular curve has singularities, since such points correspond to vertices of the regular curve and there are at least four vertices for simple closed curves. If we repeat an evolute, we cannot define the evolute at a singular point. In this paper, we define an evolute of a front and give properties of such an evolute by using a moving frame along a front and the curvature of the Legendre immersion. As applications, repeated evolutes are useful to recognize the shape of curves.