Editing Hyperboloid

{{short description|Unbounded quadric surface}}
{{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 a '''circular hyperboloid''', is the [[surface (mathematics)|surface]] generated by rotating a [[hyperbola]] around one of its [[Hyperbola#Equation|principal axes]]. A '''hyperboloid''' is the surface 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]] 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 three pairwise [[perpendicular]] [[rotational symmetry|axes of symmetry]], and three pairwise perpendicular [[reflection symmetry|planes of symmetry]].

Given a hyperboloid, one can choose a [[Cartesian coordinate system]] such that the hyperboloid is defined by one of the following equations:
<math display="block"> {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} =  1,</math> 
or
<math display="block"> {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = -1.</math>
The coordinate axes are axes of symmetry of the hyperboloid and the origin is the center of symmetry of the hyperboloid. In any case, the hyperboloid is [[asymptotic]] to the cone of the equations:
<math display="block"> {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 0 .</math>

One has a 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): a '''one-sheet hyperboloid''', also called a '''hyperbolic hyperboloid'''. It is a [[connected set|connected surface]], which has a negative [[Gaussian curvature]] at every point. This implies near every point the intersection of the hyperboloid and its [[tangent plane]] at the point consists of two branches of curve that have distinct tangents at the point. In the case of the one-sheet hyperboloid, these branches of curves are [[line (geometry)|lines]] and thus the one-sheet hyperboloid is a [[doubly ruled]] surface.

In the second case ({{math|−1}} in the right-hand side of the equation): a '''two-sheet hyperboloid''', also called an '''elliptic hyperboloid'''. The surface has two [[connected component (topology)|connected component]]s and a positive Gaussian curvature at every point. 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 display="block">\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 display="block">\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]]

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 display="block">\mathbf 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.

===Generalised equations===
More generally, an arbitrarily oriented hyperboloid, centered at {{math|'''v'''}}, is defined by the equation
<math display="block">(\mathbf{x}-\mathbf{v})^\mathrm{T} A (\mathbf{x}-\mathbf{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.

== Properties ==

=== 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 display="block">g^{\pm}_{\alpha}:
 \mathbf{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 ''[[projective geometry|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]]).

=== Hyperboloid of two sheets {{anchor|Two sheets}}===
[[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]]
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 display="block">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.

===Other properties===

==== Symmetries ====
The hyperboloids with equations 
<math display="block">\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).

==== Curvature ====
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.

==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 display="block">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 display="block">\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>
<blockquote>... 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>}}</blockquote>

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.
 Staatsmijn Emma Koeltoren III - Brunssum - 20260911 - RCE.jpg|The first 1916 patented [[Frederik van Iterson|Van Iterson]] [[cooling tower]] of [[Staatsmijn Emma|DSM Emma]] in [[Heerlen]], [[Netherlands|The Netherlands]], 1918
 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]], [[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, North Rhine-Westphalia|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]]:
<blockquote>... the ''equation of the unit sphere'' {{math|1=''ρ''<sup>2</sup> + 1 = 0}}, and change the vector {{math|''ρ''}} to a ''bivector form'', such as {{math|''σ'' + ''τ'' {{radic|−1}}}}.  The equation of the sphere then breaks up into the system of the two following,
{{block indent | em = 1.5 | text = {{math|1=''σ''<sup>2</sup> &minus; ''τ''<sup>2</sup> + 1 = 0}}, {{math|1='''S'''.''στ'' = 0}};}}
and suggests our considering {{math|''σ''}} and {{math|''τ''}} as two real and rectangular vectors, such that
{{block indent | em = 1.5 | text = {{math|1='''T'''''τ'' = ('''T'''''σ''<sup>2</sup> &minus; 1 )<sup>1/2</sup>}}.}}
Hence it is easy to infer that if we assume {{math|''σ'' {{!!}} ''λ''}}, where {{math|''λ''}} is a vector in a given position, the ''new real vector'' {{math|''σ'' + ''τ''}} will terminate on the surface of a ''double-sheeted and equilateral hyperboloid''; and that if, on the other hand, we assume {{math|''τ'' {{!!}} ''λ''}}, then the locus of the extremity of the real vector {{math|''σ'' + ''τ''}} 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; ...</blockquote>
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 = \left\{ p \; : \; w^2 = x^2 + y^2 + z^2 \right\} </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
{{block indent | em = 1.5 | text = <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==
* [[List of surfaces]]
* [[Ellipsoid]]
* [[Paraboloid]] / [[Hyperbolic paraboloid]]
* [[Regulus (geometry)|Regulus]]
* [[Rotation of axes]]
* {{slink|Split-quaternion|Profile}}
* [[Translation of axes]]
* [[De Sitter space]]
* [[Light cone]]

==References==
{{Reflist}}
{{refbegin}}
* [[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]].
{{refend}}
{{wikiquote}}
{{Commons category|Hyperboloid}}

==External links==
{{wiktionary | hyperboloid}}
*{{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}}

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[[Category:Surfaces]]
[[Category:Quadrics]]
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