Catalan's minimal surface

In differential geometry, Catalan's minimal surface is a minimal surface originally studied by Eugène Charles Catalan in 1855.^[1]

It has the special property of being the minimal surface that contains a cycloid as a geodesic. It is also swept out by a family of parabolae.^[2]

The surface has the mathematical characteristics exemplified by the following parametric equation:^[3]

${\displaystyle {\begin{aligned}x(u,v)&=u-\sin(u)\cosh(v)\\y(u,v)&=1-\cos(u)\cosh(v)\\z(u,v)&=4\sin(u/2)\sinh(v/2)\end{aligned}}}$