Catenary Equation of a plane (3D) A catenary equation models a curve supported by two points, when solely acted on by gravity. The common formula is given as $y= a \cosh(\frac{x}{a})$ where $a$ is a constant regulating the steepness of the curve. My question is, is there a similar equation for modelling a structure in 3D. For example, if I had some points $(x,y)$ (perhaps defined on a circle, but not necessarily), can I create a parabolic structure as in this figure? Paraboloid of Revolution Note: This is not the same as question Catenary equation in 3D, which is asking about a catenary curve in a 3D space, I am looking for how a 3D structure can be modelled.

If you are looking for the equation of the surface generated by the rotation of a catenary around its axis of symmetry (taken as $z$ axis), then it is simply: $$ z=a\cosh\left({\sqrt{x^2+y^2}\over a}\right). $$