An Integral Formula Related to Inner Isoptics

Abstract

An isoptic $C_{\alpha }$ of a strictly convex $C^2$-curve in the plane is the locus of all points from which $C$ is seen under the same fixed angle. 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 $C_{\alpha }$. We describe an integral formula for inner isoptics in terms of quantities that naturally occur in this geometric configuration.