Distance, Normals and Double Normals for Real Plane Curves with Singularrities

For real algebraic curves in the plane with singularities we investigate the relation between normals and double normals and the critical points of the squared distance function (up to topological equivalence). For the distance to a given point we show that the topological discriminant consists of the (traditional) evolute, together with some distinguished normal lines at algebraic singular points. We pay special attention to the behavior at points, where the curve is smooth, but only $C^1$ embedded. We discus the differences and similarities with the ED-discriminant in the complex theory. We give counting formula's for normals and double normals, relating them with maxima and minima of the distance function.