differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, Catalan's minimal surface is a

minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...

originally studied by

Eugène Charles Catalan Eugène Charles Catalan (; 30 May 1814 – 14 February 1894) was a French and Belgian mathematician who worked on continued fractions, descriptive geometry, number theory and combinatorics. His notable contributions included discovering a periodi ...

in 1855. It has the special property of being the minimal surface that contains a

cycloid In geometry, a cycloid is the curve traced by a point on a circle as it Rolling, rolls along a Line (geometry), straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette (curve), roulette, a curve g ...

as a

geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...

. It is also ''swept out'' by a family of parabolae. The surface has the mathematical characteristics exemplified by the following

parametric equation In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters. In the case ...

:Gray, A. "Catalan's Minimal Surface." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, Florida: CRC Press, pp. 692–693, 1997 :

\begin
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

External links

* Weisstein, Eric W. "Catalan's Surface." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/CatalansSurface.html * Weiqing Gu, The Library of Surfaces. https://web.archive.org/web/20130317011222/http://www.math.hmc.edu/~gu/curves_and_surfaces/surfaces/catalan.html

References

{{DEFAULTSORT:Catalans minimal surface Minimal surfaces Differential geometry