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 or has an *associated family* of minimal surfaces given by, respectively, or

The catenoid has Weierstrass-Enneper representation . Thus, the associated family of surfaces of the catenoid has Weierstrass-Enneper representation , which corresponds to the following standard parametrization.

, for any fixed , where

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

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

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