Convexity limit angles for isoptics

Abstract

Given an oval C in the plane, the alpha -isoptic C_alpha of C is the plane curve composed of the points from which C can be seen under the angle pi -alpha . We consider isoptics of ovals parametrized with the support function p(t)=a+cos n t, nin mathbb {N}, and present an example of an oval such that when alpha increases, the alpha -isoptics begin to be convex, then lose their convexity and finally are convex again along a curve intersecting the isoptics orthogonally. Next we give an example of a curve from the same family, for which the curvature of the isoptics changes its sign three times. These changes occur on the symmetry axes of the oval C and coincide with the orthogonal trajectories which start at the points with extremal curvature. Finally, we formulate the hypothesis concerning the general case where we expect n-1 convexity limit angles for the isoptics of an oval parametrized by p(t)=a+cos n t.