This is a classic result in differential geometry and is worth mentioning in these posts on minimal surfaces. Before we can talk about the deformation we need a definition.

Definition. A minimal surface described by the Weierstrass-Enneper data (f,g) or F(\tau) has an associated family of minimal surfaces given by, respectively, (e^{it}f,g) or e^{it}F(\tau).

The catenoid has Weierstrass-Enneper representation (f,g)=(-\frac{e^{-z}}{2},-e^z). Thus, the associated family of surfaces of the catenoid has Weierstrass-Enneper representation (f,g)=(-\frac{e^{-z}}{2}e^{it},-e^z), which corresponds to the following standard parametrization.

\textbf{x}(u,v)=(x^1(u,v),x^2(u,v),x^3(u,v)), for any fixed t, where

x^1(u, v) = \cos(t)\cos(v)\cosh(u) + \sin(t)\sin(v)\sinh(u)

x^2(u, v) = \cos(t)\cosh(u)\sin(v) - \cos(v)\sin(t)\sinh(u)

x^3(u, v) = u\cos(t) + v\sin(t)

A very beautiful result in minimal surface theory. The catenoid can be continuously deformed into the helicoid by the transformation given above, where t=0 represents the catenoid and t=\frac{\pi}{2} represents the helicoid. It should be pointed out that the parametrization above represents a minimal surface for any value of t. That is, any surface in the associated family of a minimal surface is also minimal.

The surfaces below, plotted for different values of t, represent the associated family of minimal surfaces of the catenoid.

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