Hyperboloid

<h2 id="parametric-representations">Parametric representations</h2>

Cartesian coordinates for the hyperboloids can be defined, similar to <a href="/facts/Spherical_coordinates/5cVnNof3">spherical coordinates</a>, keeping the <a href="/facts/Azimuth/kEyt9nJ3">azimuth</a> angle θ ∈ [0, 2π), but changing inclination v into <a href="/facts/Hyperbolic_trigonometric_function/Y4Cb5S35">hyperbolic trigonometric functions</a>:
One-surface hyperboloid: v ∈ (−∞, ∞)

x
              
              
                
                =
                a
                cosh
                ⁡
                v
                cos
                ⁡
                θ
              
            
            
              
                y
              
              
                
                =
                b
                cosh
                ⁡
                v
                sin
                ⁡
                θ
              
            
            
              
                z
              
              
                
                =
                c
                sinh
                ⁡
                v
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}x&=a\cosh v\cos \theta \\y&=b\cosh v\sin \theta \\z&=c\sinh v\end{aligned}}}

Two-surface hyperboloid: v ∈ [0, ∞)

x
              
              
                
                =
                a
                sinh
                ⁡
                v
                cos
                ⁡
                θ
              
            
            
              
                y
              
              
                
                =
                b
                sinh
                ⁡
                v
                sin
                ⁡
                θ
              
            
            
              
                z
              
              
                
                =
                ±
                c
                cosh
                ⁡
                v
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}x&=a\sinh v\cos \theta \\y&=b\sinh v\sin \theta \\z&=\pm c\cosh v\end{aligned}}}

The following parametric representation includes hyperboloids of one sheet, two sheets, and their common boundary cone, each with the 
 
 
 
 z
 
 
 {\displaystyle z}
 
-axis as the axis of symmetry:

x
        
        (
        s
        ,
        t
        )
        =
        
          (
          
            
              
                
                  a
                  
                    
                      
                        s
                        
                          2
                        
                      
                      +
                      d
                    
                  
                  cos
                  ⁡
                  t
                
              
              
                
                  b
                  
                    
                      
                        s
                        
                          2
                        
                      
                      +
                      d
                    
                  
                  sin
                  ⁡
                  t
                
              
              
                
                  c
                  s
                
              
            
          
          )
        
      
    
    {\displaystyle \mathbf {x} (s,t)=\left({\begin{array}{lll}a{\sqrt {s^{2}+d}}\cos t\\b{\sqrt {s^{2}+d}}\sin t\\cs\end{array}}\right)}

<ul><li>For 
 
 
 
 d
 >
 0
 
 
 {\displaystyle d>0}
 
 one obtains a hyperboloid of one sheet,</li>
<li>For 
 
 
 
 d
 <
 0
 
 
 {\displaystyle d<0}
 
 a hyperboloid of two sheets, and</li>
<li>For 
 
 
 
 d
 =
 0
 
 
 {\displaystyle d=0}
 
 a double cone.</li></ul>
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 
 
 
 
 c
 s
 
 
 {\displaystyle cs}
 
 term to the appropriate component in the equation above.

<h3>Generalised equations</h3>
More generally, an arbitrarily oriented hyperboloid, centered at v, is defined by the equation

(
        
          x
        
        −
        
          v
        
        
          )
          
            
              T
            
          
        
        A
        (
        
          x
        
        −
        
          v
        
        )
        =
        1
        ,
      
    
    {\displaystyle (\mathbf {x} -\mathbf {v} )^{\mathrm {T} }A(\mathbf {x} -\mathbf {v} )=1,}

where A is a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> and x, v are <a href="/facts/Euclidean_vector/bCLrdksy">vectors</a>.
The <a href="/facts/Eigenvector/8TjEoT8u">eigenvectors</a> of A define the principal directions of the hyperboloid and the <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> of A are the <a href="/facts/Multiplicative_inverse/yczdYyCw">reciprocals</a> of the squares of the semi-axes: 
 
 
 
 
 1
 
 /
 
 
 a
 
 2
 
 
 
 
 
 {\displaystyle {1/a^{2}}}
 
, 
 
 
 
 
 1
 
 /
 
 
 b
 
 2
 
 
 
 
 
 {\displaystyle {1/b^{2}}}
 
 and 
 
 
 
 
 1
 
 /
 
 
 c
 
 2
 
 
 
 
 
 {\displaystyle {1/c^{2}}}
 
. The one-sheet hyperboloid has two positive eigenvalues and one negative eigenvalue. The two-sheet hyperboloid has one positive eigenvalue and two negative eigenvalues.

<h2 id="properties">Properties</h2>
<h3>Hyperboloid of one sheet</h3>
<h4>Lines on the surface</h4>
<ul><li>A hyperboloid of one sheet contains two pencils of lines. It is a <a href="/facts/Doubly_ruled_surface/uAGM9b1X">doubly ruled surface</a>.</li></ul>
If the hyperboloid has the equation

x
 
 2
 
 
 
 a
 
 2
 
 
 
 
 +
 
 
 
 y
 
 2
 
 
 
 b
 
 2
 
 
 
 
 −
 
 
 
 z
 
 2
 
 
 
 c
 
 2
 
 
 
 
 =
 1
 
 
 {\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=1}
 
 then the lines

 
 
 
 
 g
 
 α
 
 
 ±
 
 
 :
 
 x
 
 (
 t
 )
 =
 
 
 (
 
 
 
 a
 cos
 ⁡
 α
 
 
 
 
 b
 sin
 ⁡
 α
 
 
 
 
 0
 
 
 
 )
 
 
 +
 t
 ⋅
 
 
 (
 
 
 
 −
 a
 sin
 ⁡
 α
 
 
 
 
 b
 cos
 ⁡
 α
 
 
 
 
 ±
 c
 
 
 
 )
 
 
  
 ,
 
 t
 ∈
 
 R
 
 ,
  
 0
 ≤
 α
 ≤
 2
 π
  
 
 
 {\displaystyle g_{\alpha }^{\pm }:\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 \mathbb {R} ,\ 0\leq \alpha \leq 2\pi \ }

are contained in the surface.
In case 
 
 
 
 a
 =
 b
 
 
 {\displaystyle a=b}
 
 the hyperboloid is a surface of revolution and can be generated by rotating one of the two lines 
 
 
 
 
 g
 
 0
 
 
 +
 
 
 
 
 {\displaystyle g_{0}^{+}}
 
 or 
 
 
 
 
 g
 
 0
 
 
 −
 
 
 
 
 {\displaystyle g_{0}^{-}}
 
, which are skew to the rotation axis (see picture). This property is called <a href="/facts/Christopher_Wren/yKSh7QUa">Wren</a>'s theorem.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> The more common generation of a one-sheet hyperboloid of revolution is rotating a <a href="/facts/Hyperbola/6R6ER0mY">hyperbola</a> around its <a href="/facts/Semi-major_and_semi-minor_axes/l7JES2wo">semi-minor axis</a> (see picture; rotating the hyperbola around its other axis gives a two-sheet hyperbola of revolution).
A hyperboloid of one sheet is <a href="/facts/Projective_geometry/rHKSaEbx">projectively</a> equivalent to a <a href="/facts/Hyperbolic_paraboloid/AUdRV2QH">hyperbolic paraboloid</a>.

<h4>Plane sections</h4>
For simplicity the plane sections of the unit hyperboloid with equation 
 
 
 
  
 
 H
 
 1
 
 
 :
 
 x
 
 2
 
 
 +
 
 y
 
 2
 
 
 −
 
 z
 
 2
 
 
 =
 1
 
 
 {\displaystyle \ H_{1}:x^{2}+y^{2}-z^{2}=1}
 
 are considered. Because a hyperboloid in general position is an affine image of the unit hyperboloid, the result applies to the general case, too.

<ul><li>A plane with a slope less than 1 (1 is the slope of the lines on the hyperboloid) intersects 
 
 
 
 
 H
 
 1
 
 
 
 
 {\displaystyle H_{1}}
 
 in an ellipse,</li>
<li>A plane with a slope equal to 1 containing the origin intersects 
 
 
 
 
 H
 
 1
 
 
 
 
 {\displaystyle H_{1}}
 
 in a pair of parallel lines,</li>
<li>A plane with a slope equal 1 not containing the origin intersects 
 
 
 
 
 H
 
 1
 
 
 
 
 {\displaystyle H_{1}}
 
 in a parabola,</li>
<li>A tangential plane intersects 
 
 
 
 
 H
 
 1
 
 
 
 
 {\displaystyle H_{1}}
 
 in a pair of intersecting lines,</li>
<li>A non-tangential plane with a slope greater than 1 intersects 
 
 
 
 
 H
 
 1
 
 
 
 
 {\displaystyle H_{1}}
 
 in a hyperbola.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li></ul>
Obviously, any one-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case (see <a href="/facts/Circular_section/hddscubM">circular section</a>).

<h3>Hyperboloid of two sheets </h3>

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

H
          
            2
          
        
        :
         
        
          x
          
            2
          
        
        +
        
          y
          
            2
          
        
        −
        
          z
          
            2
          
        
        =
        −
        1.
      
    
    {\displaystyle H_{2}:\ x^{2}+y^{2}-z^{2}=-1.}

which can be generated by a rotating <a href="/facts/Hyperbola/6R6ER0mY">hyperbola</a> around one of its axes (the one that cuts the hyperbola)

<ul><li>A plane with slope less than 1 (1 is the slope of the asymptotes of the generating hyperbola) intersects 
 
 
 
 
 H
 
 2
 
 
 
 
 {\displaystyle H_{2}}
 
 either in an ellipse or in a point or not at all,</li>
<li>A plane with slope equal to 1 containing the origin (midpoint of the hyperboloid) does not intersect 
 
 
 
 
 H
 
 2
 
 
 
 
 {\displaystyle H_{2}}
 
,</li>
<li>A plane with slope equal to 1 not containing the origin intersects 
 
 
 
 
 H
 
 2
 
 
 
 
 {\displaystyle H_{2}}
 
 in a parabola,</li>
<li>A plane with slope greater than 1 intersects 
 
 
 
 
 H
 
 2
 
 
 
 
 {\displaystyle H_{2}}
 
 in a hyperbola.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li></ul>
Obviously, any two-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case (see <a href="/facts/Circular_section/hddscubM">circular section</a>).
Remark: A hyperboloid of two sheets is projectively equivalent to a sphere.

<h3>Other properties</h3>
<h4>Symmetries</h4>
The hyperboloids with equations

x
              
                2
              
            
            
              a
              
                2
              
            
          
        
        +
        
          
            
              y
              
                2
              
            
            
              b
              
                2
              
            
          
        
        −
        
          
            
              z
              
                2
              
            
            
              c
              
                2
              
            
          
        
        =
        1
        ,
        
        
          
            
              x
              
                2
              
            
            
              a
              
                2
              
            
          
        
        +
        
          
            
              y
              
                2
              
            
            
              b
              
                2
              
            
          
        
        −
        
          
            
              z
              
                2
              
            
            
              c
              
                2
              
            
          
        
        =
        −
        1
      
    
    {\displaystyle {\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}

are

<ul><li>pointsymmetric to the origin,</li>
<li>symmetric to the coordinate planes and</li>
<li>rotational symmetric to the z-axis and symmetric to any plane containing the z-axis, in case of 
 
 
 
 a
 =
 b
 
 
 {\displaystyle a=b}
 
 (hyperboloid of revolution).</li></ul>
<h4>Curvature</h4>
Whereas the <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian curvature</a> 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 <a href="/facts/Hyperboloid_model/L79z8k4Q">model</a> for hyperbolic geometry.

<h2 id="in-more-than-three-dimensions">In more than three dimensions</h2>
Imaginary hyperboloids are frequently found in mathematics of higher dimensions. For example, in a <a href="/facts/Pseudo-Euclidean_space/UW6sbGeK">pseudo-Euclidean space</a> one has the use of a <a href="/facts/Quadratic_form/l2n1jXPe">quadratic form</a>:

q
 (
 x
 )
 =
 
 (
 
 
 x
 
 1
 
 
 2
 
 
 +
 ⋯
 +
 
 x
 
 k
 
 
 2
 
 
 
 )
 
 −
 
 (
 
 
 x
 
 k
 +
 1
 
 
 2
 
 
 +
 ⋯
 +
 
 x
 
 n
 
 
 2
 
 
 
 )
 
 ,
 
 k
 <
 n
 .
 
 
 {\displaystyle q(x)=\left(x_{1}^{2}+\cdots +x_{k}^{2}\right)-\left(x_{k+1}^{2}+\cdots +x_{n}^{2}\right),\quad k<n.}
 
 
When c is any <a href="/facts/Constant_(mathematics)/SxlxTWFT">constant</a>, then the part of the space given by

{
        x
         
        :
         
        q
        (
        x
        )
        =
        c
        }
      
    
    {\displaystyle \lbrace x\ :\ q(x)=c\rbrace }

is called a hyperboloid. The degenerate case corresponds to c = 0.
As an example, consider the following passage:<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<blockquote>... the velocity vectors always lie on a surface which Minkowski calls a four-dimensional hyperboloid since, expressed in terms of purely real coordinates (y1, ..., y4), its equation is y21 + y22 + y23 − y24 = −1, analogous to the hyperboloid y21 + y22 − y23 = −1 of three-dimensional space.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></blockquote>
However, the term quasi-sphere is also used in this context since the sphere and hyperboloid have some commonality (See § Relation to the sphere below).

<h2 id="hyperboloid-structures">Hyperboloid structures</h2>
Main article: <a href="/facts/Hyperboloid_structure/xVfUDnhV">Hyperboloid structure</a>
One-sheeted hyperboloids are used in construction, with the structures called <a href="/facts/Hyperboloid_structure/xVfUDnhV">hyperboloid structures</a>. A hyperboloid is a <a href="/facts/Doubly_ruled_surface/uAGM9b1X">doubly ruled surface</a>; thus, it can be built with straight steel beams, producing a strong structure at a lower cost than other methods. Examples include <a href="/facts/Cooling_tower/rnTgbsSA">cooling towers</a>, especially of <a href="/facts/Power_station/vYeVY1fD">power stations</a>, and <a href="/facts/List_of_hyperboloid_structures/HVg0Jjln">many other structures</a>.

<h2 id="relation-to-the-sphere">Relation to the sphere</h2>
In 1853 <a href="/facts/William_Rowan_Hamilton/U86CqUlt">William Rowan Hamilton</a> published his Lectures on Quaternions which included presentation of <a href="/facts/Biquaternion/YsI2czSu">biquaternions</a>. The following passage from page 673 shows how Hamilton uses biquaternion algebra and vectors from <a href="/facts/Quaternion/T3tnu7mX">quaternions</a> to produce hyperboloids from the equation of a <a href="/facts/Sphere/jywYLNM2">sphere</a>:

<blockquote>... the equation of the unit sphere ρ2 + 1 = 0, and change the vector ρ to a bivector form, such as σ + τ √−1. The equation of the sphere then breaks up into the system of the two following,
σ2 − τ2 + 1 = 0, S.στ = 0;
and suggests our considering σ and τ as two real and rectangular vectors, such that

Tτ = (Tσ2 − 1 )1/2.
Hence it is easy to infer that if we assume σ || λ, where λ is a vector in a given position, the new real vector σ + τ will terminate on the surface of a double-sheeted and equilateral hyperboloid; and that if, on the other hand, we assume τ || λ, then the locus of the extremity of the real vector 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 S is the operator giving the scalar part of a quaternion, and T is the "tensor", now called <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a>, of a quaternion.
A modern view of the unification of the sphere and hyperboloid uses the idea of a <a href="/facts/Conic_section/teqgLptX">conic section</a> as a <a href="/facts/Conic_section/teqgLptX">slice of a quadratic form</a>. Instead of a <a href="/facts/Conical_surface/n6h4xmLu">conical surface</a>, one requires conical <a href="/facts/Hypersurface/cTvtLVsV">hypersurfaces</a> in <a href="/facts/Four-dimensional_space/vFQd0M7T">four-dimensional space</a> with points p = (w, x, y, z) ∈ R4 determined by <a href="/facts/Quadratic_form/l2n1jXPe">quadratic forms</a>. First consider the conical hypersurface

<ul><li>
 
 
 
 P
 =
 
 {
 
 p
 
 :
 
 
 w
 
 2
 
 
 =
 
 x
 
 2
 
 
 +
 
 y
 
 2
 
 
 +
 
 z
 
 2
 
 
 
 }
 
 
 
 {\displaystyle P=\left\{p\;:\;w^{2}=x^{2}+y^{2}+z^{2}\right\}}
 
 and</li>
<li>
 
 
 
 
 H
 
 r
 
 
 =
 {
 p
  
 :
  
 w
 =
 r
 }
 ,
 
 
 {\displaystyle H_{r}=\lbrace p\ :\ w=r\rbrace ,}
 
 which is a <a href="/facts/Hyperplane/2s2ycYYd">hyperplane</a>.</li></ul>
Then 
 
 
 
 P
 ∩
 
 H
 
 r
 
 
 
 
 {\displaystyle P\cap H_{r}}
 
 is the sphere with radius r. On the other hand, the conical hypersurface

Q
 =
 {
 p
  
 :
  
 
 w
 
 2
 
 
 +
 
 z
 
 2
 
 
 =
 
 x
 
 2
 
 
 +
 
 y
 
 2
 
 
 }
 
 
 {\displaystyle Q=\lbrace p\ :\ w^{2}+z^{2}=x^{2}+y^{2}\rbrace }
 
 provides that 
 
 
 
 Q
 ∩
 
 H
 
 r
 
 
 
 
 {\displaystyle Q\cap H_{r}}
 
 is a hyperboloid.
In the theory of <a href="/facts/Quadratic_form/l2n1jXPe">quadratic forms</a>, a unit <a href="/facts/Quasi-sphere/DQ3207yK">quasi-sphere</a> is the subset of a quadratic space X consisting of the x ∈ X such that the quadratic norm of x is one.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/List_of_surfaces/033mpY7o">List of surfaces</a></li>
<li><a href="/facts/Ellipsoid/pe9vm0Sh">Ellipsoid</a></li>
<li><a href="/facts/Paraboloid/AUdRV2QH">Paraboloid</a> / <a href="/facts/Hyperbolic_paraboloid/AUdRV2QH">Hyperbolic paraboloid</a></li>
<li><a href="/facts/Regulus_(geometry)/M77LIboa">Regulus</a></li>
<li><a href="/facts/Rotation_of_axes/lgzWdxpn">Rotation of axes</a></li>
<li><a href="/facts/Split-quaternion/1nltLGbk">Split-quaternion § Profile</a></li>
<li><a href="/facts/Translation_of_axes/3dbW7KB0">Translation of axes</a></li>
<li><a href="/facts/De_Sitter_space/gwIURvVf">De Sitter space</a></li>
<li><a href="/facts/Light_cone/GwJ0Yhjf">Light cone</a></li></ul>

<ul><li><a href="/facts/Wilhelm_Blaschke/NQPEocq3">Wilhelm Blaschke</a> (1948) Analytische Geometrie, Kapital V: "Quadriken", Wolfenbutteler Verlagsanstalt.</li>
<li>David A. Brannan, M. F. Esplen, & Jeremy J Gray (1999) Geometry, pp. 39–41 <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>.</li>
<li><a href="/facts/H._S._M._Coxeter/D2l5jWGG">H. S. M. Coxeter</a> (1961) Introduction to Geometry, p. 130, <a href="/facts/John_Wiley_%26_Sons/53g3LqJc">John Wiley & Sons</a>.</li></ul>

Wikiquote has quotations related to Hyperboloid.

Wikimedia Commons has media related to Hyperboloid.

<h2 id="external-links">External links</h2>

Look up hyperboloid in Wiktionary, the free dictionary.

<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Hyperboloid.html">"Hyperboloid"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/One-SheetedHyperboloid.html">"One-sheeted hyperboloid"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Two-SheetedHyperboloid.html">"Two-sheeted hyperboloid"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/EllipticHyperboloid.html">"Elliptic Hyperboloid"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li></ul></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">K. Strubecker: Vorlesungen der Darstellenden Geometrie. Vandenhoeck & Ruprecht, Göttingen 1967, p. 218 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">CDKG: Computerunterstützte Darstellende und Konstruktive Geometrie (TU Darmstadt) (PDF; 3,4 MB), S. 116 <a href="http://www.mathematik.tu-darmstadt.de/~ehartmann/cdg-skript-1998.pdf" target="_blank">http://www.mathematik.tu-darmstadt.de/~ehartmann/cdg-skript-1998.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">CDKG: Computerunterstützte Darstellende und Konstruktive Geometrie (TU Darmstadt) (PDF; 3,4 MB), S. 122 <a href="http://www.mathematik.tu-darmstadt.de/~ehartmann/cdg-skript-1998.pdf" target="_blank">http://www.mathematik.tu-darmstadt.de/~ehartmann/cdg-skript-1998.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">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 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">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 
 
 
 
 
 M
 
 4
 
 
 .
 
 
 {\displaystyle M^{4}.}
 
[5] <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Ian R. Porteous (1995) Clifford Algebras and the Classical Groups, pages 22, 24 & 106, Cambridge University Press ISBN 0-521-55177-3 <a href="/wiki/Ian_R._Porteous" target="_blank">/wiki/Ian_R._Porteous</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Hyperboloid open-in-new

Hyperboloid