Editing Paraboloid (section)

=== Hyperbolic paraboloid ===
[[File:Hyperbolic-paraboloid.svg|thumb|A hyperbolic paraboloid with lines contained in it]]
[[File:Pringles chips.JPG|thumb|[[Pringles]] fried snacks are in the shape of a hyperbolic paraboloid.]]

The hyperbolic paraboloid is a [[doubly ruled surface]]: it contains two families of mutually [[skew lines]]. The lines in each family are parallel to a common plane, but not to each other. Hence the hyperbolic paraboloid is a [[conoid]].

These properties characterize hyperbolic paraboloids and are used in one of the oldest definitions of hyperbolic paraboloids: ''a hyperbolic paraboloid is a surface that may be generated by a moving line that is parallel to a fixed plane and crosses two fixed [[skew lines]]''.

This property makes it simple to manufacture a hyperbolic paraboloid from a variety of materials and for a variety of purposes, from concrete roofs to snack foods. In particular, [[Pringles]] fried snacks resemble a truncated hyperbolic paraboloid.<ref>{{citation|title=Calculus: Early Transcendentals|first1=Dennis G.|last1=Zill|first2=Warren S.|last2=Wright|publisher=Jones & Bartlett Publishers|year=2011|isbn=9781449644482|page=649|url=https://books.google.com/books?id=iHYH_B__ybgC&pg=PA649}}.</ref>

A hyperbolic paraboloid is a [[saddle surface]], as its [[Gauss curvature]] is negative at every point. Therefore, although it is a ruled surface, it is not [[Developable surface|developable]].

From the point of view of [[projective geometry]], a hyperbolic paraboloid is [[one-sheet hyperboloid]] that is [[tangent space|tangent]] to the [[plane at infinity]].

A hyperbolic paraboloid of equation <math>z=axy</math> or <math>z=\tfrac a 2(x^2-y^2)</math> (this is the same [[up to]] a [[rotation of axes]]) may be called a ''rectangular hyperbolic paraboloid'', by analogy with [[rectangular hyperbola]]s.

;Plane sections
[[File:ParabHyper.png|thumb|A hyperbolic paraboloid with hyperbolas and parabolas]]
A plane section of a hyperbolic paraboloid with equation 
<math display="block">z = \frac{x^2}{a^2} - \frac{y^2}{b^2}</math>
can be
* a ''line'', if the plane is parallel to the {{mvar|z}}-axis, and has an equation of the form <math> bx \pm ay+b=0</math>, 
* a ''parabola'', if the plane is parallel to the {{mvar|z}}-axis, and the section is not a line,
* a pair of ''intersecting lines'', if the plane is a [[tangent plane]], 
* a ''hyperbola'', otherwise.
[[File:Hyperbolic_paraboloid.stl|thumb|[[STL (file format)|STL]] hyperbolic paraboloid model]]

====Examples in architecture====

[[Saddle roof]]s are often hyperbolic paraboloids as they are easily constructed from straight sections of material. Some examples:
* [[Philips Pavilion]] Expo '58, Brussels (1958)
* [[IIT Delhi]] - Dogra Hall Roof
* [[St. Mary's Cathedral, Tokyo]], Japan (1964)
* [[St Richard's Church, Ham]], in Ham, London, England (1966)
* [[Cathedral of Saint Mary of the Assumption (San Francisco, California)|Cathedral of Saint Mary of the Assumption]], San Francisco, California, US (1971)
* [[Saddledome]] in Calgary, Alberta, Canada (1983)
* [[Scandinavium]] in Gothenburg, Sweden (1971)
* [[L'Oceanogràfic]] in Valencia, Spain (2003)
* [[London Velopark]], England (2011)
* [[Waterworld, Wrexham|Waterworld Leisure & Activity Centre]], [[Wrexham]], Wales (1970)
* [[Markham Moor Scorer Building|Markham Moor Service Station roof]], A1(southbound), Nottinghamshire, England
* [http://pastvu.com/_p/a/9/e/d/9ed1fc7601f87d453c50cbffa06d9c6f.jpg Cafe "Kometa"], Sokol district, Moscow, Russia (1960). Architect V.Volodin, engineer N.Drozdov. Demolished.

<gallery widths="200px" heights="150px">
W-wa Ochota PKP-WKD.jpg|[[Warszawa Ochota railway station]], an example of a hyperbolic paraboloid structure
Superfície paraboloide hiperbólico - LEMA - UFBA .jpg|Surface illustrating a hyperbolic paraboloid
Restaurante Los Manantiales 07.jpg|Restaurante Los Manantiales, Xochimilco, Mexico
L'Oceanogràfic Valencia 2019 4.jpg|Hyperbolic paraboloid thin-shell roofs at [[L'Oceanogràfic]], Valencia, Spain (taken 2019)
Sam_Scorer%2C_Little_Chef_-_geograph.org.uk_-_173949.jpg|Markham Moor Service Station roof, Nottinghamshire (2009 photo)
</gallery>