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 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. They are called Hyperboloid structure.
The James S McDonnell Planetarium building at the St. Louis Science Center.
The Kobe Port Tower. ❐ ⊕ Source
For more photos, see: Hyperboloid Towers.
Hyperboloid can be used for gear surface, called a Hypoid. (Source: 《Penrose Tiles to Trapdoor Ciphers》 by Martin Gardner. amazon, chapter 15).
Other algebaric surfaces that has cross-sections of conic sections are: ellipsoid, paraboloid, hyperbolic paraboloid, hyperboloid of one sheet, hyperboloid of two sheets.
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