The involute of catenary starting at the vertex is the tractrix . The radial of catenary is the Kampyle of Eudoxus. In the figure blow, the blue curve is half of catenary. The rainbow lines are the radiuses of osculating circles. A catenary rotating around an axis forms the catenoid, which is a mimimum surface.
Minimum surface are surfaces that connect given curves in space and has the least area.
Catenoid is the minimum surface between two parallel circles on a cylinder or circular cone. You can see minimal surfaces in nature as soap films. Bend wires into two parallel circles and dip them into soap solution. The soap film between the two circles is the catenoid.