one needs to align more than just the tangents. Example If the fixed curve is a catenary and the rolling curve is a line, we have: The parameterization of the line is chosen so that Applying the formula above we obtain: If p = −i the expression has a constant imaginary part (namely −i) and the roulette is a horizontal line. An interesting application of this is that a square wheel could roll without bouncing on a road that is a matched series of catenary arcs. List of roulettes See also Rolling Gear Superposition principle Spirograph Tusi couple Rosetta (orbit) Notes References W. H. Besant (1890) Notes on Roulettes and Glissettes from Cornell University Historical Math Monographs, originally published by Deighton, Bell & Co. Further reading Roulette at 2dcurves.com Base, roulante et roulettes d'un mouvement plan sur plan Eine einheitliche Darstellung von ebenen, verallgemeinerten Rollbewegungen und deren Anwendungen