ΒΙΒΛΙΟ: Differential Geometry, From Elastic Curves to Willmore Surfaces

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ΒΙΒΛΙΟ: Differential Geometry PDF)

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DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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Catalan's minimal surface

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

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.

The surface has the mathematical characteristics exemplified by the following parametric equation:

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

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