View source for Hyperboloid

{{distinguish|text=[[hyperbolic paraboloid]], a saddle-like surface}}
{| class=wikitable align=right
|- align=center
|[[File:Hyperboloid1.png|150px]]<br />Hyperboloid of one sheet
|[[File:DoubleCone.png|160px]]<br />[[conical surface]] in between
|[[File:Hyperboloid2.png|150px]]<br />Hyperboloid of two sheets
|}
In [[geometry]], a '''hyperboloid of revolution''', sometimes called '''circular hyperboloid''', is a [[surface (mathematics)|surface]] that may be generated by rotating a [[hyperbola]] around one of its [[Hyperbola#Nomenclature and features|principal axes]]. A '''hyperboloid''' is a surface that may be obtained from a hyperboloid of revolution by deforming it by means of directional [[scaling (geometry)|scaling]]s, or more generally, of an [[affine transformation]].

A hyperboloid is a [[quadric surface]], that is a [[surface (mathematics)|surface]] that may be defined as the [[zero set]] of a [[polynomial]] of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a [[conical surface|cone]] or a [[cylinder]], having a [[central symmetry|center of symmetry]], and intersecting many [[plane (geometry)|planes]] into hyperbolas. A hyperboloid has also three pairwise [[perpendicular]] [[rotational symmetry|axes of symmetry]], and three pairwise perpendicular [[reflection symmetry|planes of symmetry]].

Given a hyperboloid, if one chooses a [[Cartesian coordinate system]] whose axes are axes of symmetry of the hyperboloid, and origin is the center of symmetry of the hyperboloid, then the hyperboloid may be defined by one of the two following equations:
:<math> {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2}= 1,</math> 
or
:<math> {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2}= -1.</math>
Both of these surfaces are [[asymptotic]] to the cone of equation 
:<math> {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 0 .</math>
One has an hyperboloid of revolution if and only if <math>a^2=b^2.</math> Otherwise, the axes are uniquely defined ([[up to]] the exchange of the ''x''-axis and the ''y''-axis.

There are two kinds of hyperboloids. In the first case ({{math|+1}} in the right-hand side of the equation), one has a '''one-sheet hyperboloid''', also called '''hyperbolic hyperboloid'''. It is a [[connected set|connected surface]], which has a negative [[Gaussian curvature]] at every point. This implies that the [[tangent plane]] at any point intersects the hyperboloid at two [[line (geometry)|lines]], and thus that the one-sheet hyperboloid is a [[doubly ruled]] surface.

In the second case ({{math|−1}} in the right-hand side of the equation), one has a '''two-sheet hyperboloid''', also called '''elliptic hyperboloid'''. The surface has two [[connected component (topology)|connected component]]s, and a positive Gaussian curvature at every point. Thus the surface is ''convex'' in the sense that the tangent plane at every point intersects the surface only in this point.

== Parametric representations ==
[[File:Cylinder - hyperboloid - cone.gif|thumb|Animation of a hyperboloid of revolution]]
Cartesian coordinates for the hyperboloids can be defined, similar to [[spherical coordinates]], keeping the [[azimuth]] angle {{math|''θ'' ∈ {{closed-open|0, 2''π''}}}}, but changing inclination {{math|''v''}} into [[hyperbolic trigonometric function]]s:

One-surface hyperboloid: {{math|''v'' ∈ {{open-open|−&infin;, &infin;}}}}
:<math>\begin{align} x&=a \cosh v \cos\theta \\ y&=b \cosh v \sin\theta \\ z&=c \sinh v \end{align}</math>

Two-surface hyperboloid: {{math|''v'' ∈ {{closed-open|0, &infin;}}}}
:<math>\begin{align} x&=a \sinh v \cos\theta \\ y&=b \sinh v \sin\theta \\ z&=\pm c \cosh v \end{align}</math>
[[File:Hyperboloid-1s.svg|thumb|hyperboloid of one sheet: generation by a rotating hyperbola (top) and line (bottom: red or blue)]]
[[File:Hyperbo-1s-cut-all.svg|thumb|hyperboloid of one sheet: plane sections]]

== Properties of a hyperboloid of one sheet ==
=== Lines on the surface ===
*A hyperboloid of one sheet contains two pencils of lines. It is a [[doubly ruled surface]]. 
If the hyperboloid has the equation
<math> {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2}= 1</math>  then the lines

:<math>g^{\pm}_{\alpha}:
 \vec{x}(t)=\begin{pmatrix} a\cos\alpha\\ b\sin\alpha\\ 0\end{pmatrix}
  + t\cdot \begin{pmatrix} -a\sin\alpha\\ b\cos\alpha\\ \pm c\end{pmatrix}\ ,\quad t\in \R,\ 0\le \alpha\le 2\pi\  </math>
are contained in the surface.

In case <math>a=b</math> the hyperboloid is a surface of revolution and can be generated by rotating one of the two lines <math>g^{+}_{0}</math> or <math>g^{-}_{0}</math>, which are skew to the rotation axis (see picture). This property is called ''[[Christopher Wren|Wren]]'s theorem''.<ref> K. Strubecker: ''Vorlesungen der Darstellenden Geometrie.'' Vandenhoeck & Ruprecht, Göttingen 1967, p. 218</ref> The more common generation of a one-sheet hyperboloid of revolution is rotating a [[hyperbola]] around its [[Semi-major and semi-minor axes#Hyperbola|semi-minor axis]] (see picture; rotating the hyperbola around its other axis gives a two-sheet hyperbola of revolution).

A hyperboloid of one sheet is ''projectively'' equivalent to a [[hyperbolic paraboloid]].

=== Plane sections ===
For simplicity the plane sections of the ''unit hyperboloid'' with equation <math> \ H_1: x^2+y^2-z^2=1</math> are considered. Because a hyperboloid in general position is an affine image of the unit hyperboloid, the result applies to the general case, too.

*A plane with a slope less than 1 (1 is the slope of the lines on the hyperboloid) intersects <math>H_1</math> in an ''ellipse'',
*A plane with a slope equal to 1 containing the origin intersects <math>H_1</math> in a ''pair of parallel lines'',
*A plane with a slope equal 1 not containing the origin intersects <math>H_1</math> in a ''parabola'',
*A tangential plane intersects <math>H_1</math> in a ''pair of intersecting lines'',
*A non-tangential plane with a slope greater than 1 intersects <math>H_1</math> in a ''hyperbola''.<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/cdg-skript-1998.pdf CDKG: Computerunterstützte Darstellende und Konstruktive Geometrie (TU Darmstadt)] (PDF; 3,4&nbsp;MB), S. 116</ref>
Obviously, any one-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case (see [[circular section]]).

[[File:Hyperboloid-2s.svg|thumb|hyperboloid of two sheets: generation by rotating a hyperbola]]
[[File:Hyperbo-2s-ca.svg|thumb|hyperboloid of two sheets: plane sections]]

== Properties of a hyperboloid of two sheets ==
The hyperboloid of two sheets does ''not'' contain lines. The discussion of plane sections can be performed for the ''unit hyperboloid of two sheets'' with equation
:<math>H_2: \ x^2+y^2-z^2=-1</math>. 
which can be generated by a rotating [[hyperbola]] around one of its axes (the one that cuts the hyperbola)

*A plane with slope less than 1 (1 is the slope of the asymptotes of the generating hyperbola) intersects <math>H_2</math> either in an ''ellipse'' or in a ''point'' or not at all,
*A plane with slope equal to 1 containing the origin (midpoint of the hyperboloid) does ''not intersect'' <math>H_2</math> , 
*A plane with slope equal to 1 not containing the origin intersects <math>H_2</math> in a ''parabola'', 
*A plane with slope greater than 1 intersects <math>H_2</math> in a ''hyperbola''.<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/cdg-skript-1998.pdf CDKG: Computerunterstützte Darstellende und Konstruktive Geometrie (TU Darmstadt)] (PDF; 3,4&nbsp;MB), S. 122</ref>
Obviously, any two-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case (see [[circular section]]).

''Remark:'' A hyperboloid of two sheets is ''projectively'' equivalent to a sphere.

== Common parametric representation ==
The following parametric representation includes hyperboloids of one sheet, two sheets, and their common boundary cone, each with the <math>z</math>-axis as the axis of symmetry:

<math>\vec x(s,t)=
\left(
\begin{array}{lll}
a \sqrt{s^2+d} \cos t\\
b \sqrt{s^2+d}  \sin t\\
c s
\end{array}
\right)
</math>

*For <math>d>0</math> one obtains a hyperboloid of one sheet, 
*For <math>d<0</math> a hyperboloid of two sheets, and 
*For <math>d=0</math> a double cone.

One can obtain a parametric representation of a hyperboloid with a different coordinate axis as the axis of symmetry by shuffling the position of the <math>c s</math> term to the appropriate component in the equation above.

== Symmetries of a hyperboloid ==
The hyperboloids with equations 
<math>\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 , \quad \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1 \ </math>
are
*''pointsymmetric'' to the origin,
*''symmetric to the coordinate planes'' and
*''rotational symmetric'' to the z-axis and symmetric to any plane containing the z-axis, in case of <math>a=b</math> (hyperboloid of revolution).

== On the curvature of a hyperboloid ==
Whereas the [[Gaussian curvature]] of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive. In spite of its positive curvature, the hyperboloid of two sheets with another suitably chosen metric can also be used as a [[Hyperboloid model|model]] for hyperbolic geometry.

==Generalised equations==
More generally, an arbitrarily oriented hyperboloid, centered at {{math|'''v'''}}, is defined by the equation
:<math>(\mathbf{x-v})^\mathrm{T} A (\mathbf{x-v}) = 1,</math>
where {{math|''A''}} is a [[matrix (mathematics)|matrix]] and {{math|'''x'''}}, {{math|'''v'''}} are [[euclidean vector|vectors]].

The [[eigenvector]]s of {{math|''A''}} define the principal directions of the hyperboloid and the [[eigenvalue]]s of A are the [[Multiplicative inverse|reciprocal]]s of the squares of the semi-axes: <math>{1/a^2}</math>, <math>{1/b^2}  </math> and <math>{1/c^2}</math>. The one-sheet hyperboloid has two positive eigenvalues and one negative eigenvalue. The two-sheet hyperboloid has one positive eigenvalue and two negative eigenvalues.

==In more than three dimensions==
Imaginary hyperboloids are frequently found in mathematics of higher dimensions. For example, in a [[pseudo-Euclidean space]] one has the use of a [[quadratic form]]:
: <math>q(x) = \left(x_1^2+\cdots + x_k^2\right)-\left(x_{k+1}^2+\cdots + x_n^2\right), \quad k < n .</math> 
When {{math|''c''}} is any [[constant (mathematics)|constant]], then the part of the space given by
:<math>\lbrace x \ :\ q(x) = c \rbrace </math>
is called a ''hyperboloid''. The degenerate case corresponds to {{math|1=''c'' = 0}}.

As an example, consider the following passage:<ref>Thomas Hawkins (2000) ''Emergence of the Theory of Lie Groups: an essay in the history of mathematics, 1869—1926'', §9.3 "The Mathematization of Physics at Göttingen", see page 340, Springer {{ISBN|0-387-98963-3}}</ref>
:... the velocity vectors always lie on a surface which Minkowski calls a four-dimensional hyperboloid since, expressed in terms of purely real coordinates {{math|(''y''<sub>1</sub>, ..., ''y''<sub>4</sub>)}}, its equation is {{math|''y''{{su|b=1|p=2}} + ''y''{{su|b=2|p=2}} + ''y''{{su|b=3|p=2}} − ''y''{{su|b=4|p=2}} {{=}} −1}}, analogous to the hyperboloid {{math|''y''{{su|b=1|p=2}} + ''y''{{su|b=2|p=2}} − ''y''{{su|b=3|p=2}} {{=}} −1}} of three-dimensional space.{{refn|Minkowski used the term "four-dimensional hyperboloid" only once, in a posthumously-published typescript and this was non-standard usage, as Minkowski's hyperboloid is a three-dimensional submanifold of a four-dimensional Minkowski space <math>M^4.</math><ref>{{Citation|author=Walter, Scott A.|year=1999|contribution=The non-Euclidean style of Minkowskian relativity|editor=J. Gray|title=The Symbolic Universe: Geometry and Physics 1890-1930|pages=91–127|publisher=Oxford University Press|contribution-url=http://scottwalter.free.fr/papers/1999-symbuniv-walter.html}}</ref>}}

However, the term '''quasi-sphere''' is also used in this context since the sphere and hyperboloid have some commonality (See {{section link||Relation to the sphere}} below).

== Hyperboloid structures ==
{{main|Hyperboloid structure}}
One-sheeted hyperboloids are used in construction, with the structures called [[hyperboloid structure]]s. A hyperboloid is a [[doubly ruled surface]]; thus, it can be built with straight steel beams, producing a strong structure at a lower cost than other methods. Examples include [[cooling tower]]s, especially of [[power station]]s, and [[list of hyperboloid structures|many other structures]].

<gallery caption="Gallery of one sheet hyperboloid structures" widths = 150px heights = 200px>
 Adziogol hyperboloid Lighthouse by Vladimir Shukhov 1911.jpg|The [[Adziogol Lighthouse]], [[Ukraine]], 1911.
 Kobe port tower11s3200.jpg|[[Kobe Port Tower]], [[Japan]], 1963.
 Mcdonnell planetarium slsc.jpg|[[Saint Louis Science Center]]'s James S. [[McDonnell Planetarium]], [[St. Louis]], [[Missouri]], 1963.
 Newcastle International Airport Control Tower.jpg|[[Newcastle International Airport]] control tower, [[Newcastle upon Tyne]], [[England]], 1967.
 Jested 002.JPG|[[Ještěd Tower|Ještěd Transmission Tower]], [[Czech Republic]], 1968.
 Catedral1 Rodrigo Marfan.jpg|[[Cathedral of Brasília|Cathedral]] of [[Brasília]], [[Brazil]], 1970.
 Ciechanow_water_tower.jpg|[[Ciechanów#Monuments|Hyperboloid water tower]] with [[Toroid (geometry)|toroidal]] tank, [[Ciechanów]], [[Poland]], 1972.
 Toronto - ON - Roy Thomson Hall.jpg|[[Roy Thomson Hall]], [[Toronto]], [[Canada]], 1982.
 Thtr300 kuehlturm.jpg|The [[THTR-300]] [[cooling tower]] for the now decommissioned [[Thorium fuel cycle|thorium]] [[nuclear reactor]] in [[Hamm]]-Uentrop, [[Germany]], 1983. 
 Bridge over Corporation Street - geograph.org.uk - 809089.jpg|The [[Corporation Street Bridge]], [[Manchester]], [[England]], 1999.
 Killesberg Tower.jpg|The [[Killesberg Tower|Killesberg]] observation tower, [[Stuttgart]], [[Germany]], 2001.
 BMW-Welt at night 2.JPG|[[BMW Welt]], (BMW World), museum and event venue, [[Munich]], [[Germany]], 2007.
 Canton tower in asian games opening ceremony.jpg|The [[Canton Tower]], [[China]], 2010.
 Les Essarts-le-Roi Château d'eau.JPG|The [[Essarts-le-Roi]] water tower, [[France]].
</gallery>

==Relation to the sphere==
In 1853 [[William Rowan Hamilton]] published his ''Lectures on Quaternions'' which included presentation of [[biquaternion]]s.  The following passage from page 673 shows how Hamilton uses biquaternion algebra and vectors from [[quaternion]]s to produce hyperboloids from the equation of a [[sphere]]:
::... the ''equation of the unit sphere'' {{math|1=''&rho;''<sup>2</sup> + 1 = 0}}, and change the vector {{math|''&rho;''}} to a ''bivector form'', such as {{math|''&sigma;'' + ''&tau;'' {{radic|−1}}}}.  The equation of the sphere then breaks up into the system of the two following,
:::{{math|1=''&sigma;''<sup>2</sup> &minus; ''&tau;''<sup>2</sup> + 1 = 0}}, {{math|1='''S'''.''&sigma;&tau;'' = 0}};
::and suggests our considering {{math|''&sigma;''}} and {{math|''&tau;''}} as two real and rectangular vectors, such that
:::{{math|1='''T'''''&tau;'' = ('''T'''''&sigma;''<sup>2</sup> &minus; 1 )<sup>1/2</sup>}}.
::Hence it is easy to infer that if we assume {{math|''&sigma;'' <math>\parallel</math> ''&lambda;''}}, where {{math|''&lambda;''}} is a vector in a given position, the ''new real vector'' {{math|''&sigma;'' + ''&tau;''}} will terminate on the surface of a ''double-sheeted and equilateral hyperboloid''; and that if, on the other hand, we assume {{math|''&tau;'' <math>\parallel</math> ''&lambda;''}}, then the locus of the extremity of the real vector {{math|''&sigma;'' + ''&tau;''}} will be an ''equilateral but single-sheeted hyperboloid''. The study of these two hyperboloids is, therefore, in this way connected very simply, through biquaternions, with the study of the sphere; ...
In this passage {{math|'''S'''}} is the operator giving the scalar part of a quaternion, and {{math|'''T'''}} is the "tensor", now called [[norm (mathematics)|norm]], of a quaternion.

A modern view of the unification of the sphere and hyperboloid uses the idea of a [[conic section]] as a [[conic section#As slice of quadratic form|slice of a quadratic form]]. Instead of a [[conical surface]], one requires conical [[hypersurface]]s in [[four-dimensional space]] with points {{math|1=''p'' = (''w'', ''x'', ''y'', ''z'') &isin; '''R'''<sup>4</sup>}} determined by [[quadratic form]]s. First consider the conical hypersurface
:<math>P = \lbrace p \ : \ w^2 = x^2 + y^2 + z^2 \rbrace </math> and
:<math>H_r = \lbrace p \ :\  w = r \rbrace ,</math> which is a [[hyperplane]].
Then <math>P \cap H_r</math> is the sphere with radius {{math|''r''}}. On the other hand, the conical hypersurface
:<math>Q = \lbrace p \ :\  w^2 + z^2 = x^2 + y^2 \rbrace</math> provides that <math>Q \cap H_r</math> is a hyperboloid.

In the theory of [[quadratic form]]s, a '''unit [[quasi-sphere]]''' is the subset of a quadratic space {{math|''X''}} consisting of the {{math|''x'' &isin; ''X''}} such that the quadratic norm of {{math|''x''}} is one.<ref>[[Ian R. Porteous]] (1995) ''Clifford Algebras and the Classical Groups'', pages 22, 24 & 106, [[Cambridge University Press]] {{ISBN|0-521-55177-3}}</ref>

==See also==
[[File:Vyksa Shukhov tower.ogv|thumb|250px|right|[[Vladimir Shukhov|Shukhov]] hyperboloid tower (1898) in [[Vyksa]], Russia]]
* [[de Sitter space]]
* [[Ellipsoid]]
* [[List of surfaces]]
* [[Paraboloid]] / [[Hyperbolic paraboloid]]
* [[Regulus (geometry)|Regulus]]
* [[Rotation of axes]]
* {{slink|Split-quaternion|Profile}}
* [[Translation of axes]]

==References==
{{Reflist}}

* [[Wilhelm Blaschke]] (1948) ''Analytische Geometrie'', Kapital V: "Quadriken", Wolfenbutteler Verlagsanstalt.
* David A. Brannan, M. F. Esplen, & Jeremy J Gray (1999) ''Geometry'', pp.&nbsp;39&ndash;41 [[Cambridge University Press]]. 
* [[H. S. M. Coxeter]] (1961) ''Introduction to Geometry'', p.&nbsp;130, [[John Wiley & Sons]].

{{wikiquote}}
{{Commons category|Hyperboloid}}

==External links==
*{{MathWorld |title=Hyperboloid |urlname=Hyperboloid}}
** {{mathworld |urlname=One-SheetedHyperboloid |title=One-sheeted hyperboloid}}
** {{mathworld |urlname=Two-SheetedHyperboloid |title=Two-sheeted hyperboloid}}
**{{MathWorld |title=Elliptic Hyperboloid |urlname=EllipticHyperboloid}}

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