hyperboloid1.nb.zip
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.
above: 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.
above: Two hyperboloid sculptures. The left can be made into a modern fruit basket, and the right a paper waste basket. (source: Second Life gallery)
above: Photo of a real basket modeled by hyperboloid.
The silhouette of a rotating dice is a hyperbola.
above: A sculpture emulating a spinning cube. (source: Second Life gallery)
above: 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.
above: «The cooling towers of Didcot Power Station». source ↗. See Cooling tower↗.
above: The Kobe Port Tower↗. Image Source ↗
above: 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.com↗), 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.
Page created: 2004. © 2004 by Xah Lee.
