Delaunay's roulette

Delaunay's roulette is the track of a focus, when a conic section rolls over a line:

• ellipse: elliptic catenary
• hyperbola: hyperbolic catenary
• parabola: parabolic catenary
  The latter curve is in fact the (ordinary) catenary.

The first to distinguish the roulette was Charles-Eugène Delaunay (1816-1872), in 1841. Lindelöf studied the curve in 1861.

The elliptic catenary

The hyperbolic catenary

notes

1)