A reflecting sundial with cusps of the caustics on a straight line segment

Given a convex mirror curve and a sheaf of parallel incident rays, a caustic curve with at least one cusp appers. This is the principle on which the construction of reflecting sundials is founded. When the sheaf of parallel incident rays changes along the day, the cusp of the caustic curve traces out a new curve. The initial convex mirror curve for which its caustic curves have cusps along a straight line segment is studied. An intrinsic equation is obtained for it and, after numerical integration, the resulting curve is compared with the solution given by C. M. Lowne.