Editing Paraboloid

{{Short description|Quadric surface with one axis of symmetry and no center of symmetry}}
{{more citations needed|date=June 2020}}
[[Image:Paraboloid of Revolution.svg|thumb|right|Paraboloid of revolution]]
In [[geometry]], a '''paraboloid''' is a [[quadric surface]] that has exactly one [[axial symmetry|axis of symmetry]] and no [[central symmetry|center of symmetry]]. The term "paraboloid" is derived from [[parabola]], which refers to a [[conic section]] that has a similar property of symmetry.

Every [[plane section]] of a paraboloid made by a plane [[Parallel (geometry)#A line and a plane|parallel]] to the axis of symmetry is a parabola. The paraboloid is '''hyperbolic''' if every other plane section is either a [[hyperbola]], or two crossing lines (in the case of a section by a tangent plane). The paraboloid is '''elliptic''' if every other nonempty plane section is either an [[ellipse]], or a single point (in the case of a section by a tangent plane). A paraboloid is either elliptic or hyperbolic.

Equivalently, a paraboloid may be defined as a quadric surface that is not a [[cylinder]], and has an [[implicit surface|implicit equation]] whose part of degree two may be factored over the [[complex number]]s into two different linear factors. The paraboloid is hyperbolic if the factors are real; elliptic if the factors are [[complex conjugate]].

An elliptic paraboloid is shaped like an oval cup and has a [[maximum]] or minimum point when its axis is vertical. In a suitable [[coordinate system]] with three axes {{math|''x''}}, {{math|''y''}}, and {{math|''z''}}, it can be represented by the equation<ref>{{cite book |title=Thomas' Calculus 11th ed. |last=Thomas |first=George B. |author2=Maurice D. Weir |author3=Joel Hass |author3-link=Joel Hass |author4=Frank R. Giordiano |year=2005 |publisher= Pearson Education, Inc |isbn=0-321-18558-7 |page=892}}</ref>
<math display="block">z = \frac{x^2}{a^2} + \frac{y^2}{b^2}.</math>
where {{math|''a''}} and {{math|''b''}} are constants that dictate the level of curvature in the {{math|''xz''}} and {{math|''yz''}} planes respectively. In this position, the elliptic paraboloid opens upward.

[[Image:HyperbolicParaboloid.svg|thumb|right|Hyperbolic paraboloid]]
A hyperbolic paraboloid (not to be confused with a [[hyperboloid]]) is a [[doubly ruled surface]] shaped like a [[saddle]]. In a suitable coordinate system, a hyperbolic paraboloid can be represented by the equation<ref name="Weisstein">Weisstein, Eric W. "Hyperbolic Paraboloid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HyperbolicParaboloid.html</ref><ref>{{cite book |title=Thomas' Calculus 11th ed. |last=Thomas |first=George B.|author2=Maurice D. Weir |author3=Joel Hass |author4=Frank R. Giordiano |year=2005 |publisher= Pearson Education, Inc |isbn=0-321-18558-7 |page=896}}</ref>
<math display="block">z = \frac{y^2}{b^2} - \frac{x^2}{a^2}.</math>
In this position, the hyperbolic paraboloid opens downward along the {{math|''x''}}-axis and upward along the {{math|''y''}}-axis (that is, the parabola in the plane {{math|''x'' {{=}} 0}} opens upward and the parabola in the plane {{math|''y'' {{=}} 0}} opens downward).

Any paraboloid (elliptic or hyperbolic) is a [[Translation surface (differential geometry)|translation surface]], as it can be generated by a moving parabola directed by a second parabola.

== 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>

== Cylinder between pencils of elliptic and hyperbolic paraboloids ==
[[File:Parabol-el-zy-hy-s.svg|400px|thumb|elliptic paraboloid, parabolic cylinder, hyperbolic paraboloid]]

The [[pencil (mathematics)|pencil]] of elliptic paraboloids
<math display="block">z=x^2 + \frac{y^2}{b^2}, \ b>0, </math> 
and the pencil of hyperbolic paraboloids
<math display="block">z=x^2 - \frac{y^2}{b^2}, \ b>0, </math> 
approach the same surface 
<math display="block"> z=x^2</math>
for <math> b \rightarrow \infty</math>,
which is a ''parabolic cylinder'' (see image).

== Curvature ==
The elliptic paraboloid, parametrized simply as
<math display="block">\vec \sigma(u,v) = \left(u, v, \frac{u^2}{a^2} + \frac{v^2}{b^2}\right) </math>
has [[Gaussian curvature]]
<math display="block">K(u,v) = \frac{4}{a^2 b^2 \left(1 + \frac{4u^2}{a^4} + \frac{4v^2}{b^4}\right)^2}</math>
and [[mean curvature]]
<math display="block">H(u,v) = \frac{a^2 + b^2 + \frac{4u^2}{a^2} + \frac{4v^2}{b^2}}{a^2 b^2 \sqrt{\left(1 + \frac{4u^2}{a^4} + \frac{4v^2}{b^4}\right)^3}}</math>
which are both always positive, have their maximum at the origin, become smaller as a point on the surface moves further away from the origin, and tend asymptotically to zero as the said point moves infinitely away from the origin.

The hyperbolic paraboloid,<ref name="Weisstein" /> when parametrized as
<math display="block">\vec \sigma (u,v) = \left(u, v, \frac{u^2}{a^2} - \frac{v^2}{b^2}\right) </math>
has Gaussian curvature
<math display="block">K(u,v) = \frac{-4}{a^2 b^2 \left(1 + \frac{4u^2}{a^4} + \frac{4v^2}{b^4}\right)^2} </math>
and mean curvature
<math display="block">H(u,v) = \frac{-a^2 + b^2 - \frac{4u^2}{a^2} + \frac{4v^2}{b^2}}{a^2 b^2 \sqrt{\left(1 + \frac{4u^2}{a^4} + \frac{4v^2}{b^4}\right)^3}}. </math>

== Geometric representation of multiplication table ==
If the hyperbolic paraboloid
<math display="block">z = \frac{x^2}{a^2} - \frac{y^2}{b^2}</math>
is rotated by an angle of {{math|{{sfrac|π|4}}}} in the {{math|+''z''}} direction (according to the [[right hand rule]]), the result is the surface
<math display="block">z = \left(\frac{x^2 + y^2}{2}\right) \left(\frac{1}{a^2} - \frac{1}{b^2}\right) + xy \left(\frac{1}{a^2} + \frac{1}{b^2}\right)</math>
and if {{math|''a'' {{=}} ''b''}} then this simplifies to
<math display="block">z = \frac{2xy}{a^2}.</math>
Finally, letting {{math|''a'' {{=}} {{sqrt|2}}}}, we see that the hyperbolic paraboloid
<math display="block">z = \frac{x^2 - y^2}{2}.</math>
is congruent to the surface
<math display="block">z = xy</math>
which can be thought of as the geometric representation (a three-dimensional [[nomogram|nomograph]], as it were) of a [[multiplication table]].

The two paraboloidal {{math|'''R'''{{sup|2}} → '''R'''}} functions
<math display="block">z_1 (x,y) = \frac{x^2 - y^2}{2}</math>
and
<math display="block">z_2 (x,y) = xy</math>
are [[harmonic conjugate]]s, and together form the [[analytic function]]
<math display="block">f(z) = \frac{z^2}{2} = f(x + yi) = z_1 (x,y) + i z_2 (x,y)</math>
which is the [[analytic continuation]] of the {{math|'''R''' → '''R'''}} parabolic function {{math|1=''f''(''x'') = {{sfrac|''x''{{sup|2}}|2}}}}.

== Dimensions of a paraboloidal dish ==
The dimensions of a symmetrical paraboloidal dish are related by the equation
<math display="block">4FD = R^2,</math>
where {{math|''F''}} is the focal length, {{math|''D''}} is the depth of the dish (measured along the axis of symmetry from the vertex to the plane of the rim), and {{math|''R''}} is the radius of the rim. They must all be in the same [[unit of length]]. If two of these three lengths are known, this equation can be used to calculate the third.

A more complex calculation is needed to find the diameter of the dish ''measured along its surface''. This is sometimes called the "linear diameter", and equals the diameter of a flat, circular sheet of material, usually metal, which is the right size to be cut and bent to make the dish. Two intermediate results are useful in the calculation: {{math|''P'' {{=}} 2''F''}} (or the equivalent: {{math|''P'' {{=}} {{sfrac|''R''{{sup|2}}|2''D''}}}}) and {{math|''Q'' {{=}} {{sqrt|''P''{{sup|2}} + ''R''{{sup|2}}}}}}, where {{math|''F''}}, {{math|''D''}}, and {{math|''R''}} are defined as above. The diameter of the dish, measured along the surface, is then given by
<math display="block">\frac{RQ}{P} + P \ln\left(\frac{R+Q}{P}\right),</math>
where {{math|ln ''x''}} means the [[natural logarithm]] of {{math|''x''}}, i.e. its logarithm to base {{math|''[[e (mathematical constant)|e]]''}}.

The volume of the dish, the amount of liquid it could hold if the rim were horizontal and the vertex at the bottom (e.g. the capacity of a paraboloidal [[wok]]), is given by
<math display="block">\frac{\pi}{2} R^2 D,</math>
where the symbols are defined as above. This can be compared with the formulae for the volumes of a [[Cylinder (geometry)|cylinder]] ({{math|π''R''{{sup|2}}''D''}}), a [[sphere|hemisphere]] ({{math|{{sfrac|2π|3}}''R''{{sup|2}}''D''}}, where {{math|''D'' {{=}} ''R''}}), and a [[Cone (geometry)|cone]] ({{math|{{sfrac|π|3}}''R''{{sup|2}}''D''}}). {{math|π''R''{{sup|2}}}} is the aperture area of the dish, the area enclosed by the rim, which is proportional to the amount of sunlight a reflector dish can intercept. The surface area of a parabolic dish can be found using the area formula for a [[surface of revolution#Area formula|surface of revolution]] which gives
<math display="block">A = \frac{\pi R\left(\sqrt{(R^2+4D^2)^3}-R^3\right)}{6D^2}.</math>

== See also ==
{{Portal|Mathematics}}
* {{annotated link|Ellipsoid}}
* {{annotated link|Hyperboloid}}
* {{annotated link|Parabolic loudspeaker}}
* {{annotated link|Parabolic reflector}}

== References ==
{{Reflist}}

==External links==
*{{Commons category-inline}}

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[[Category:Geometric shapes]]
[[Category:Surfaces]]
[[Category:Quadrics]]
[[Category:Parabolas]]