Explicit parameterization of the mirror curve of a reflecting sundial

We obtain the explicit parameterization of a convex planar curve that, when used as mirror of a reflecting sundial, its caustic curves have cusps along a straight line segment. In a recent paper the intrinsic equation of such a curve was obtained and integrated numerically. In the first part of this note we discuss in detail such intrinsic equation and we show how it can be reduced, integrated, and expressed implicitly, and why explicit integration in terms of elementary functions is not possible. In the second part we show that, despite the impossibility of the explicit integration, the implicit solution can be used to obtain an explicit parameterization of the mirror curve defined by such a function as its curvature. Finally, a comparison with Lowne’s curve shows how close it was to the solution.