The Catenoid stands at the beginning of the theory of minimal surfaces. Leonhard Euler, in 1744, showed that among surfaces of revolution, it has minimal area.

Note that the catenoidal ends grow exponentially in radius while linearly in height, so appear from far away more like planes than funnels. The conjugate surface of the catenoid is the helicoid.

Weierstrass data:

G(z)=z

dh=\frac1z\, dz

The catenoid also provides solutions to the Plateau problem to find a minimal surface that has two horizontal circles with radius r and centers at (0,0,\pm 1). For r larger than the solution of r tanh(r)=1, which is approximately 1.2, there are two catenoidal solutions. One of them, the stable catenoid, you get via soap films experiments, the other you’ll probably never see in nature.

Resources:

Leonhard Euler:  Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, 1744,  chapter 5, § 47.

PoVRay source for unstable catenoids.

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