The “hyperboloid of one sheet” is a surface of revolution of the curve family hyperbola.
The parametric formula for the Hyperboloid of One Sheet is:
ParametricPlot3D[{Cosh@[u]*Cos@[v], Cosh@[u]*Sin@[v], Sinh@[u]}, {u, -2, 2}, {v, 0, 2*π}]
The u controls the height, and v controls the circular sweep.
A Hyperboloid of one sheet, showing its ruled surface property.
A hyperbola revolving around its transverse axis forms a surface called “hyperboloid of one sheet”. A hyperboloid is a Ruled Surface.
Ruled surfaces are surfaces that for every point on the surface, there is a line on the surface passing it. Or, in other words, a surface generated by a line. If for each point on the surface there are two lines on the surface passing it, then it's called doubly-ruled surface. Hyperboloid is a doubly-ruled surface.
Ruled surfaces also include cylinder and helicoid. There are only 3 doubly ruled surfaces: The hyperboloid, hyperbolic paraboloid, and plane.
Two hyperboloid sculptures. The left can be made into a modern fruit basket, and the right a paper waste basket. (source: Second Life gallery)
Photo of a real basket modeled by hyperboloid.
The silhouette of a rotating dice is a hyperbola.
A sculpture emulating a spinning cube. (source: Second Life gallery)
A photo of a spinning dice. (See Photo of regular polyhedron dices)
Due to its simplicity and beauty, the hyperboloid is often used in architecture for towers. See Hyperboloid structure. The following are some examples.
The cooling tower of Didcot Power Station. Source
The Kobe Port Tower. ⊕ Source
The James S McDonnell Planetarium building at the St. Louis Science Center. Its roof is a hyperboloid structure.
Hyperboloid can be used for gear surface, called a Hypoid. (Source: Penrose Tiles to Trapdoor Ciphers by Martin Gardner. amazon, chapter 15).
See also hyperbola
Other algebaric surfaces that has cross-sections of conic sections are: ellipsoid, paraboloid, hyperbolic paraboloid, hyperboloid of one sheet, hyperboloid of two sheets.
