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.


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