Helicoid to catenoid

Two minimal surfaces, the shapes soap films take, joined by a bending that never stretches the surface. Every patch keeps its area and its curvature as the spiral staircase rolls up into the hourglass of a film between two rings.

X = cos θ (sinh v sin u, −sinh v cos u, u) + sin θ (cosh v cos u, cosh v sin u, v)

On mathcurve.com as Hélicoïde et caténoïde