(source: relativitet.se)
After seemingly endless searching for terms such as curved cone, hyper-cone etc I am at a loss as to what this shape is called. I believe it is commonly used to depict the curvature of space time.
Also, if you have any relevant information on the geometry of it, such as parametric equations, curved surface area, enclosed area at height z, volume etc that would be much appreciated.
Edit: Funnel.
Any idea how to get the surface area of a disk at height z and volume in a limited domain?
I'll give some background to this problem. I am designing a building modeled on the pillar shape that emerges when a water droplet falls into a body of water, the shape of which is close modeled by this. As part of the design brief it is required that the average floor area (so disks at several discrete heights in z) is adhered to. Surface area and volume are also required for inputs into Building Information Modeling system. So essentially I need to set the parameters to give the building envelope while meeting the average floor area requirements.
Best Answer
The name of the shape depends on the exact curvature in the longitudinal direction and what happens at infinity. You have shown a finite segment, so one cannot say for sure.
If the surface has constant negative Gaussian curvature, it is known as a pseudosphere (the sphere itself has constant positive Gaussian curvature). It has an outer limit where the surface turns parallel to its cross-section, similar to the outermost (largest) circle in the illustration but further down the curve.
If the longitudinal curve is a segment of a parabola or hyperbola, then the shape is an example of a quadric surface. The full curve extends to infinity in both directions.