Invariance of the area of ovaloids (Q6614566)

Consider a two dimensional convex body \(\mathcal A\) in the Euclidean plane \(\mathbb E^2\) with smooth boundary \(\partial A\) lying tangent to a smooth, oriented base curve in \(\mathbb E^2\) at a point \(O\in \partial A\). A second convex body comes by reflecting \(\mathcal A\) at the (common) tangent of the base curve and \(\partial A\) in \(O\). When the two convex bodies are rolling simultaneously without slipping on the base curve, the point \(O\) traces out two curves bounding an \textit{ovaloidal region}. \N\NThe authors show that the area of the ovaloidal region is independent of the shape of the base curve if some curvature conditions on the base curve and the boundary curve \(\partial A\) are fulfilled. The assertion keeps true when replacing \(\mathbb E^2\) by \(\mathbb H^2\) or \(\mathbb S^2\). The present paper generalizes results of [\textit{T. M. Apostol} and \textit{M. A. Mnatsakanian}, New horizons in geometry. Washington, DC: Mathematical Association of America (MAA) (2012; Zbl 1268.51001); \textit{H. Choi}, Am. Math. Mon. 127, No. 6, 537--544 (2020; Zbl 1439.51025)].