SKRZYPIEC MagdalenaOn orthogonal trajectories and zeroes of curvature of isopticsFor a plane curve $C$ the locus of the points in which the support lines to $C$ intersect under the fixed angle $pi-alpha$ can be considered. The curve $C_alpha$, where $alphain(0,pi)$, is called the $alpha$-isoptic, or simply the isoptic. What are the orthogonal trajectories of the isoptics of a given convex curve? We will present a Cauchy problem, the solution of which produces the parametric form of the orthogonal trajectories we seek out. We consider the existence and uniqueness of the solution of this problem for isoptics of ovals. The convexity of curves is an important and frequently discussed topic. Using the example of isoptics of conic sections, for which the implicit equations are known, we present here a method for determining and describing the curves formed by the points where convexity of isoptics is lost. |