Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Paraboloid
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
== Properties and applications == === Elliptic paraboloid === [[File:Paraboloid-3dmesh.png|thumb|right|[[Polygon mesh]] of a circular paraboloid]] [[File:Paraboloid3d.png|thumb|right|Circular paraboloid]] In a suitable [[Cartesian coordinate system]], an elliptic paraboloid has the equation <math display="block">z = \frac{x^2}{a^2}+\frac{y^2}{b^2}.</math> If {{math|1=''a'' = ''b''}}, an elliptic paraboloid is a ''circular paraboloid'' or ''paraboloid of revolution''. It is a [[surface of revolution]] obtained by revolving a [[parabola]] around its axis. A circular paraboloid contains circles. This is also true in the general case (see [[Circular section]]). From the point of view of [[projective geometry]], an elliptic paraboloid is an [[ellipsoid]] that is [[tangent space|tangent]] to the [[plane at infinity]]. ; Plane sections The plane sections of an elliptic paraboloid can be: * a ''parabola'', if the plane is parallel to the axis, * a ''point'', if the plane is a [[tangent plane]]. * an ''ellipse'' or ''empty'', otherwise. ====Parabolic reflector==== {{main|Parabolic reflector|parabolic antenna}} On the axis of a circular paraboloid, there is a point called the [[Focus (optics)|''focus'']] (or ''focal point''), such that, if the paraboloid is a mirror, light (or other waves) from a point source at the focus is reflected into a parallel beam, parallel to the axis of the paraboloid. This also works the other way around: a parallel beam of light that is parallel to the axis of the paraboloid is concentrated at the focal point. For a proof, see {{slink|Parabola|Proof of the reflective property}}. Therefore, the shape of a circular paraboloid is widely used in [[astronomy]] for parabolic reflectors and parabolic antennas. The surface of a rotating liquid is also a circular paraboloid. This is used in [[liquid-mirror telescope]]s and in making solid telescope mirrors (see [[rotating furnace]]). <gallery widths="200px" heights="180px"> Parabola with focus and arbitrary line.svg|Parallel rays coming into a circular paraboloidal mirror are reflected to the focal point, {{math|F}}, or ''vice versa'' Erdfunkstelle Raisting 2a.jpg|Parabolic reflector Centrifugal 0.PNG|Rotating water in a glass </gallery> === 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>
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)