The helicoid-catenoid family of surfaces are also isometric. (and they are all minimal...Helicoid and Catenoid are two surfaces having the same local geometry....If b=0, the surface is a helicoid as pictured...
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The helicoid is the conjugate minimal surface of the catenoid....The mathematical background here is that the catenoid and the helicoid are conjugate minimal surfaces ....
One can bend a catenoid into the shape of a helicoid without stretching....Animation showing the deformation of a helicoid into a catenoid....can make a continuous and isometric deformation of a catenoid to a helicoid...