## On isoptic curves.(English)Zbl 0725.52002

Let $$C$$ be a planar convex curve, and let $$\alpha >0$$. The $$\alpha$$-isoptic curve $$C_{\alpha}$$ of $$C$$ is the locus of points $$p$$ in the complement of $$C$$ such that the angle between the two lines through $$p$$ which are tangent to $$C$$ equals $$\alpha$$. Let $$q_1$$, $$q_2$$ be the two points of tangency.
The authors study the smallest $$\alpha$$ for which $$C_{\alpha}$$ is convex, in particular when $$C$$ is an ellipse. They show that $$C_{\alpha}$$ is tangent at $$p$$ to the circle through $$p$$, $$q_1$$, and $$q_2$$. They also reprove, using isoptic curves, a Crofton-type integral formula in the plane.

### MSC:

 52A10 Convex sets in $$2$$ dimensions (including convex curves) 53A04 Curves in Euclidean and related spaces 53C65 Integral geometry