An integral formula related to inner isoptics (Q641035)

Let \(C\) be an oval, i.e., a simple closed \(C^2\)-curve in the plane with positive curvature. Let a coordinate system with the origin \(O\) in the interior of \(C\) be fixed. Denote by \(p(t)\), \(t\in {\mathbb R}\), the support function of \(C\) (a periodic function on \({\mathbb R}\) with the period \(2\pi\)). Let \(C_{\alpha}\) be the locus of apices of a fixed angle \(\pi - \alpha\), where \(\alpha\in (0, \pi)\), formed by two supporting lines of \(C\). Then \(C_{\alpha}\) is called \(\alpha\)-isoptic of \(C\). In this paper, the authors describe geometric properties of so-called inner isoptics which are derived from isoptics of the given curve \(C\) (the two supporting lines of \(C\) through such a point determine a secant of \(C\), and the envelope of all these secants is the inner isoptic of \(C\) and \(C_{\alpha}\)).