Crofton formulas and convexity condition for secantoptics
The paper under review deals with some properties of secantoptics, defined in [\textit{M. Skrzypiec}, Beitr. Algebra Geom. 49, No.~1, 205--215 (2008; Zbl 1155.53003)]. An isoptic of a given curve \(C\) is a set of points from which this curve is seen under the fixed angle \(\alpha \in (0,\pi)\). A hedgehog is a curve which can be parametrized by the formula \(z(t)=\psi(t)e^{it}+\dot\psi(t)ie^{it}\) with \(\psi(t)=h(\cos t,\sin t), h\in C^2(\mathbb S^1,\mathbb R)\). A closed convex curve \(C\) of class \(C^2\) with nonvanishing curvature is said to be an oval. The evolutoid of angle \(\delta\) of a curve \(f(s)\) is the envelope of the lines making a fixed angle \(\delta\) with the normal vector at \(f(s)\). The authors show that any evolutoid of a given oval \(C\) is a hedgehog and that any secantoptic of an oval \(C\), is an isoptic of a pair of required evolutoids. They also prove some Crofton-type formulas for secantoptics and give a necessary and sufficient condition for a secantoptic to be convex.
scientific article; zbMATH DE number 5612289