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.