Convexity limit angles for isoptics

Given an oval C in the plane, the $$\alpha $$ α -isoptic $$C_\alpha $$ C α of C is the plane curve composed of the points from which C can be seen under the angle $$\pi -\alpha $$ π - α . We consider isoptics of ovals parametrized with the support function $$p(t)=a+\cos n t$$ p ( t ) = a + cos n t , $$n\in \mathbb {N}$$ n ∈ N , and present an example of an oval such that when $$\alpha $$ α increases, the $$\alpha $$ α -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 $$n-1$$ n - 1 convexity limit angles for the isoptics of an oval parametrized by $$p(t)=a+\cos n t$$ p ( t ) = a + cos n t .