Catenoid

catenoid 36be6
catenoid 36be6

Catenoid is a surface of revolution of the curve Catenary. Catenoid is also the first known minimal surface.

catenoid[u_, v_] := {Cosh[u] Cos[v], Cosh[u] Sin[v], u};
ParametricPlot3D[catenoid[u, v], {u, -2, 2}, {v, 0, 2 Pi},
 PlotLabel -> "Catenoid", PlotStyle -> White]
helicoid 36efb
helicoid 36efb 
helicoid[u_, v_] := {Sinh[u]  Cos[v], Sinh[u]  Sin[v], v};
ParametricPlot3D[helicoid[u, v], {u, -2, 2}, {v, 0, 2  Pi},
PlotStyle -> White,
 PlotLabel -> "Helicoid (isometric parametrization)"]

Catenary, Catenoid