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{{Short description|Model of n-dimensional hyperbolic geometry}}
[[File:HyperboloidProjection.png|thumb|Red circular arc is geodesic in [[Poincaré disk model|Poincaré disk model]]; it projects to the brown geodesic on the green hyperboloid.]]
[[File:(7,3)-hyperboloid-tiling.webm|thumb|Animation of partial {7,3} hyperbolic tiling of the hyperboloid rotated into the Poincare perspective.]]

In [[Geometry|geometry]], the '''hyperboloid model''', also known as the '''Minkowski model''' after [[Biography:Hermann Minkowski|Hermann Minkowski]], is a model of ''n''-dimensional [[Hyperbolic geometry|hyperbolic geometry]] in which points are represented by points on the forward sheet ''S''<sup>+</sup> of a two-sheeted [[Hyperboloid|hyperboloid]] in (''n''+1)-dimensional [[Physics:Minkowski space|Minkowski space]] or by the displacement [[Pseudo-Euclidean space|vector]]s from the origin to those points, and ''m''-planes are represented by the intersections of (''m''+1)-planes passing through the origin in Minkowski space with ''S''<sup>+</sup> or by [[Exterior algebra|wedge product]]s of ''m'' vectors. Hyperbolic space is [[Embedding|embedded]] isometrically in Minkowski space; that is, the hyperbolic distance function is inherited from Minkowski space, analogous to the way spherical distance is inherited from Euclidean distance when the [[N-sphere|''n''-sphere]] is embedded in (''n''+1)-dimensional Euclidean space.

Other models of hyperbolic space can be thought of as [[Map projection|map projection]]s of ''S''<sup>+</sup>: the [[Beltrami–Klein model]] is the [[Projection (mathematics)|projection]] of ''S''<sup>+</sup> through the origin onto a plane perpendicular to a vector from the origin to specific point in ''S''<sup>+</sup> analogous to the [[Gnomonic projection|gnomonic projection]] of the sphere; the [[Poincaré disk model]] is a projection  of ''S''<sup>+</sup> through a point on the other sheet ''S''<sup>−</sup> onto perpendicular plane, analogous to the [[Stereographic projection|stereographic projection]] of the sphere; the [[Hyperbolic geometry#The Gans model|Gans model]] is the orthogonal projection of ''S''<sup>+</sup> onto a plane perpendicular to a specific point in ''S''<sup>+</sup>, analogous to the [[Orthographic map projection|orthographic projection]]; the [[Band model|band model]] of the hyperbolic plane is a conformal “cylindrical” projection analogous to the [[Mercator projection]] of the sphere; [[Coordinate systems for the hyperbolic plane#Lobachevsky coordinates|Lobachevsky coordinates]] are a cylindrical projection analogous to the [[Equirectangular projection|equirectangular projection]] (longitude, latitude) of the sphere.

== Minkowski quadratic form ==
{{main|Physics:Minkowski space}}
If (''x''<sub>0</sub>, ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) is a vector in the {{nowrap|(''n'' + 1)}}-dimensional coordinate space '''R'''<sup>''n''+1</sup>, the '''Minkowski [[Quadratic form|quadratic form]]''' is defined to be

:<math> Q(x_0, x_1, \ldots, x_n) = -x_0^2 + x_1^2 + \ldots + x_n^2.</math>

The vectors {{nowrap|''v'' ∈ '''R'''<sup>''n''+1</sup>}} such that {{nowrap|1=''Q''(''v'') = -1}} form an ''n''-dimensional [[Hyperboloid|hyperboloid]] ''S'' consisting of two [[Connected space|connected component]]s, or ''sheets'': the forward, or future, sheet ''S''<sup>+</sup>, where ''x''<sub>0</sub>>0 and the backward, or past, sheet ''S''<sup>&minus;</sup>, where ''x''<sub>0</sub><0. The points of the ''n''-dimensional hyperboloid model are the points on the forward sheet ''S''<sup>+</sup>.

The metric on the hyperboloid is<math display="block"> ds^2 = Q(dx_0, dx_1, \ldots, dx_n) = -dx_0^2 + dx_1^2 + \ldots + dx_n^2.</math>The '''Minkowski [[Bilinear form|bilinear form]]''' ''B'' is the [[Polarization identity|polarization]] of the Minkowski quadratic form ''Q'',

:<math>B(\mathbf{u}, \mathbf{v}) = (Q(\mathbf{u}+\mathbf{v}) - Q(\mathbf{u}) - Q(\mathbf{v})) / 2 .</math>

(This is sometimes also written using scalar product notation <math>\mathbf{u}\cdot\mathbf{v}.</math>)
Explicitly,
:<math>B((x_0, x_1, \ldots, x_n), (y_0, y_1, \ldots, y_n)) = -x_0y_0 + x_1 y_1 + \ldots + x_n y_n .</math>

The '''hyperbolic distance''' between two points ''u'' and ''v'' of ''S''<sup>+</sup> is given by the formula

:<math>d(\mathbf{u}, \mathbf{v}) = \operatorname{arcosh}(-B(\mathbf{u}, \mathbf{v})) ,</math>

where {{math|arcosh}} is the [[Inverse function|inverse function]] of hyperbolic cosine.

=== Choice of metric signature ===
The bilinear form <math>B</math> also functions as the [[Physics:Metric tensor|metric tensor]] over the space. In ''n''+1 dimensional Minkowski space, there are two choices for the metric with opposite [[Metric signature|signature]], in the 3-dimensional case either (+, −, −) or (−, +, +).

If the signature (−, +, +) is chosen, then the [[Pseudo-Euclidean space|scalar square]] of chords between distinct points on the same sheet of the hyperboloid will be positive, which more closely aligns with conventional definitions and expectations in mathematics. Then ''n''-dimensional hyperbolic space is a Riemannian space and distance or length can be defined as the square root of the scalar square. If the signature (+, −, −) is chosen, scalar square between distinct points on the hyperboloid will be negative, so various definitions of basic terms must be adjusted, which can be inconvenient. Nonetheless, the signature (+, −, −, −) is also common for describing [[Physics:Spacetime|spacetime]] in physics. (Cf. [[Sign convention]].)

== Straight lines ==
A straight line in hyperbolic ''n''-space is modeled by a [[Geodesic|geodesic]] on the hyperboloid. A geodesic on the hyperboloid is the (non-empty) intersection of the hyperboloid with a two-dimensional linear subspace (including the origin) of the ''n''+1-dimensional Minkowski space. If we take '''u''' and '''v''' to be basis vectors of that linear subspace with
:<math> B (\mathbf{u}, \mathbf{u}) = 1 </math>
:<math> B (\mathbf{v}, \mathbf{v}) = -1 </math>
:<math> B (\mathbf{u}, \mathbf{v}) = B (\mathbf{v}, \mathbf{u}) = 0 </math>
and use ''w'' as a real parameter for points on the geodesic, then
:<math> \mathbf{u} \sinh w + \mathbf{v} \cosh w </math>
will be a point on the geodesic.<ref name=Macfarlane/>

More generally, a ''k''-dimensional "flat" in the hyperbolic ''n''-space will be modeled by the (non-empty) intersection of the hyperboloid with a ''k''+1-dimensional linear subspace (including the origin) of the Minkowski space.

== Isometries ==

The [[Indefinite orthogonal group|indefinite orthogonal group]] O(1,''n''), also called the
(''n''+1)-dimensional [[Lorentz group]], is the [[Lie group]] of  [[Real number|real]] (''n''+1)×(''n''+1) [[Matrix (mathematics)|matrices]] which preserve the Minkowski bilinear form. In a different language, it is
the group of linear [[Isometry|isometries]] of the [[Physics:Minkowski space|Minkowski space]]. In particular, this group preserves the hyperboloid ''S''. Recall that indefinite orthogonal groups have four connected components, corresponding to reversing or preserving the orientation on each subspace (here 1-dimensional and ''n''-dimensional), and form a [[Klein four-group]]. The subgroup of O(1,''n'') that preserves the sign of the first coordinate is the '''orthochronous Lorentz group''', denoted O<sup>+</sup>(1,''n''), and has two components, corresponding to preserving or reversing the orientation of the spatial subspace. Its subgroup SO<sup>+</sup>(1,''n'') consisting of matrices with [[Determinant|determinant]] one is a connected Lie group of dimension ''n''(''n''+1)/2 which acts on ''S''<sup>+</sup> by linear automorphisms and preserves the hyperbolic distance. This action is transitive and the stabilizer of the vector (1,0,...,0) consists of the matrices of the form

:<math>\begin{pmatrix}  
1      & 0 & \ldots & 0 \\
0      &   &        &   \\
\vdots &   & A      &   \\
0      &   &        &   \\
\end{pmatrix}</math>

Where <math> A  </math> belongs to the compact special orthogonal group SO(''n'') (generalizing the [[Rotation group SO(3)|rotation group SO(3)]] for {{nowrap|1=''n'' = 3}}). It follows that the ''n''-dimensional [[Hyperbolic space|hyperbolic space]] can be exhibited as the [[Homogeneous space|homogeneous space]] and a Riemannian symmetric space of rank 1,

: <math> \mathbb{H}^n=\mathrm{SO}^{+}(1,n)/\mathrm{SO}(n).</math>

The group SO<sup>+</sup>(1,''n'') is the full group of orientation-preserving isometries of the ''n''-dimensional hyperbolic space.

In more concrete terms, SO<sup>+</sup>(1,''n'') can be split into ''n''(''n''-1)/2 rotations (formed with a regular Euclidean [[Rotation matrix|rotation matrix]] in the lower-right block) and ''n'' hyperbolic translations, which take the form

:<math>\begin{pmatrix}
\cosh \alpha & \sinh \alpha & 0 & \ldots \\
\sinh \alpha & \cosh \alpha & 0 & \ldots \\
0            & 0            & 1 &        \\
\vdots       & \vdots       &   & \ddots \\
\end{pmatrix}</math>

where <math>\alpha</math> is the distance translated (along the ''x'' axis in this case), and the 2nd row/column can be exchanged with a different pair to change to a translation along a different axis. The general form of a translation in 3 dimensions along the vector <math>(w, x, y, z)</math> is:

:<math>\begin{pmatrix}
w & x                 & y                 & z                 \\
x & \frac{x^2}{w+1}+1 & \frac{yx}{w+1}    & \frac{zx}{w+1}    \\
y & \frac{xy}{w+1}    & \frac{y^2}{w+1}+1 & \frac{zy}{w+1}    \\
z & \frac{xz}{w+1}    & \frac{yz}{w+1}    & \frac{z^2}{w+1}+1 \\
\end{pmatrix}</math> where <math>w = \sqrt{x^2+y^2+z^2+1}</math>.

This extends naturally to more dimensions, and is also the simplified version of a [[Physics:Lorentz transformation#Proper transformations|Lorentz boost]] when you remove the relativity-specific terms.

=== Examples of groups of isometries ===

The group of all isometries of the hyperboloid model is O<sup>+</sup>(1,''n''). Any group of isometries is a subgroup of it.

==== Reflections ====
For two points <math>\mathbf p, \mathbf q \in \mathbb{H}^n, \mathbf p \neq \mathbf q</math>, there is a unique reflection exchanging them.

Let <math>\mathbf u = \frac {\mathbf p - \mathbf q}{\sqrt{Q(\mathbf p - \mathbf q)}}</math>.
Note that <math>Q(\mathbf u) = 1</math>, and therefore <math>u \notin \mathbb{H}^n</math>.

Then

:<math>\mathbf x \mapsto \mathbf x - 2 B(\mathbf x, \mathbf u) \mathbf u</math>

is a reflection that exchanges <math>\mathbf p</math> and <math>\mathbf q</math>.
This is equivalent to the following matrix:

:<math>R = I - 2 \mathbf u \mathbf u^{\operatorname{T}} \begin{pmatrix}
-1 &  0 \\
0 & I \\
\end{pmatrix}</math>

(note the use of [[Block matrix|block matrix]] notation).

Then <math>\{I, R\}</math> is a group of isometries.
All such subgroups are [[Conjugacy class#Conjugacy of subgroups and general subsets|conjugate]].

==== Rotations and reflections ====

:<math>S = \left \{ \begin{pmatrix}  
1 & 0 \\
0 & A \\
\end{pmatrix} : A \in O(n) \right \}</math>

is the group of rotations and reflections that preserve <math>(1, 0, \dots, 0)</math>.
The function <math>A \mapsto \begin{pmatrix}  
1 & 0 \\
0 & A \\
\end{pmatrix}</math> is an [[Group isomorphism|isomorphism]] from [[Orthogonal group|O(''n'')]] to this group.
For any point <math>p</math>, if <math>X</math> is an isometry that maps <math>(1, 0, \dots, 0)</math> to <math>p</math>, then <math>XSX^{-1}</math> is the group of rotations and reflections that preserve <math>p</math>.

==== Translations ====

For any real number <math>t</math>, there is a translation

:<math>L_t = \begin{pmatrix}
\cosh t & \sinh t & 0 \\
\sinh t & \cosh t & 0 \\
0       & 0       & I \\
\end{pmatrix}</math>

This is a translation of distance <math>t</math> in the positive x direction if <math>t \ge 0</math> or of distance <math>-t</math> in the negative x direction if <math>t \le 0</math>.
Any translation of distance <math>t</math> is conjugate to <math>L_t</math> and <math>L_{-t}</math>.
The set <math>\left \{L_t : t \in \mathbb R \right \}</math> is the group of translations through the x-axis, and a group of isometries is conjugate to it if and only if it is a group of isometries through a line.

For example, let's say we want to find the group of translations through a line <math>\overline{\mathbf p \mathbf q}</math>.
Let <math>X</math> be an isometry that maps <math>(1, 0, \dots, 0)</math> to <math>p</math> and let <math>Y</math> be an isometry that fixes <math>p</math> and maps <math>X L_{d(\mathbf p, \mathbf q)} [1, 0, \dots, 0]^{\operatorname{T}}</math> to <math>q</math>.
An example of such a <math>Y</math> is  a reflection exchanging <math>X L_{d(\mathbf p, \mathbf q)} [1, 0, \dots, 0]^{\operatorname{T}}</math> and <math>q</math> (assuming they are different), because they are both the same distance from <math>p</math>.
Then <math>YX</math> is an isometry mapping <math>(1, 0, \dots, 0)</math> to <math>p</math> and a point on the positive x-axis to <math>q</math>.
<math>(YX)L_t(YX)^{-1}</math> is a translation through the line <math>\overline{\mathbf p \mathbf q}</math> of distance <math>|t|</math>.
If <math>t \ge 0</math>, it is in the <math>\overrightarrow{\mathbf p \mathbf q}</math> direction.
If <math>t \le 0</math>, it is in the <math>\overrightarrow{\mathbf q \mathbf p}</math> direction.
<math>\left \{(YX)L_t(YX)^{-1} : t \in \mathbb R \right \}</math> is the group of translations through <math>\overline{\mathbf p \mathbf q}</math>.

==== Symmetries of horospheres ====

Let ''H'' be some [[Horosphere|horosphere]] such that points of the form <math>(w, x, 0, \dots, 0)</math> are inside of it for arbitrarily large ''x''.
For any vector ''b'' in <math>\mathbb R^{n-1}</math>

:<math>\begin{pmatrix}
1 + \frac {\|\mathbf b\|^2} 2 & - \frac {\|\mathbf b\|^2} 2  & \mathbf b^{\operatorname{T}} \\
\frac {\|\mathbf b\|^2} 2     & 1 - \frac{\|\mathbf b\|^2} 2 & \mathbf b^{\operatorname{T}} \\
\mathbf b                     & -\mathbf b                   & I                            \\
\end{pmatrix}</math>

is a hororotation that maps ''H'' to itself.
The set of such hororotations is the group of hororotations preserving ''H''.
All hororotations are conjugate to each other.

For any <math>A</math> in O(''n''-1)

:<math>\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & A \\
\end{pmatrix}</math>

is a rotation or reflection that preserves ''H'' and the x-axis.
These hororotations, rotations, and reflections generate the group of symmetries of ''H''.
The symmetry group of any horosphere is conjugate to it.
They are isomorphic to the [[Euclidean group]] E(''n''-1).

==History==
{{see|History of Lorentz transformations}}
In several papers between 1878-1885, Wilhelm Killing<ref>{{Cite journal|author=Killing, W.|year=1878|orig-year=1877|title=Ueber zwei Raumformen mit constanter positiver Krümmung|journal=Journal für die Reine und Angewandte Mathematik|volume=86|pages=72–83|url= http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002157187}}</ref><ref>{{Cite journal|author=Killing, W.|year=1880|orig-year=1879|title=Die Rechnung in den Nicht-Euklidischen Raumformen|journal=Journal für die Reine und Angewandte Mathematik|volume=89|pages=265–287|url=http://www.digizeitschriften.de/dms/resolveppn/?PID=PPN243919689_0089%7Clog27}}</ref><ref>{{Cite book|author=Killing, W.|year=1885|title=Die nicht-euklidischen Raumformen|location=Leipzig|url=https://archive.org/details/dienichteuklidis00killuoft}}</ref> used the representation he attributed to [[Biography:Karl Weierstrass|Karl Weierstrass]] for Lobachevskian geometry. In particular, he discussed quadratic forms such as <math>k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}</math> or in arbitrary dimensions <math>k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}</math>, where <math>k</math> is the reciprocal measure of curvature, <math>k^{2}=\infty</math> denotes [[Euclidean geometry]], <math>k^{2}>0</math> [[Elliptic geometry|elliptic geometry]], and <math>k^{2}<0</math> hyperbolic geometry.

According to Jeremy Gray (1986),<ref>''Linear differential equations and group theory from Riemann to Poincaré'' (pages 271,2)</ref> [[Biography:Henri Poincaré|Poincaré]] used the hyperboloid model in his personal notes in 1880. Poincaré published his results in 1881, in which he discussed the invariance of the quadratic form <math>\xi^{2}+\eta^{2}-\zeta^{2}=-1</math>.<ref>{{Cite journal|author=Poincaré, H.|year=1881|title=Sur les applications de la géométrie non-euclidienne à la théorie des formes quadratiques|journal=Association Française Pour l'Avancement des Sciences|volume=10|pages=132–138|url=http://henripoincarepapers.univ-lorraine.fr/chp/hp-pdf/hp1881af.pdf}}</ref> Gray shows where the hyperboloid model is implicit in later writing by Poincaré.<ref>See also Poincaré: ''On the fundamental hypotheses of geometry'' 1887 Collected works vol.11, 71-91 and referred to in the book of B.A. Rosenfeld ''A History of Non-Euclidean Geometry'' p.266 in English version (Springer 1988).</ref>

Also [[Biography:Homersham Cox (mathematician)|Homersham Cox]] in 1882<ref>{{Cite journal|author=Cox, H.|year=1881|title=Homogeneous coordinates in imaginary geometry and their application to systems of forces|journal=The Quarterly Journal of Pure and Applied Mathematics|volume=18|issue=70|pages=178–192|url=http://resolver.sub.uni-goettingen.de/purl?PPN600494829_0018}}</ref><ref>{{Cite journal|author=Cox, H.|year=1882|orig-year=1881|title=Homogeneous coordinates in imaginary geometry and their application to systems of forces (continued)|journal=The Quarterly Journal of Pure and Applied Mathematics|volume=18|issue=71|pages=193–215|url=http://resolver.sub.uni-goettingen.de/purl?PPN600494829_0018}}</ref> used Weierstrass coordinates (without using this name) satisfying the relation <math>z^{2}-x^{2}-y^{2}=1</math> as well as <math>w^{2}-x^{2}-y^{2}-z^{2}=1</math>.

Further exposure of the model was given by [[Biography:Alfred Clebsch|Alfred Clebsch]] and Ferdinand Lindemann in 1891 discussing the relation <math>x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}</math> and <math>x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}</math>.<ref>{{Cite book|author=Lindemann, F.|year=1891|orig-year=1890|title=Vorlesungen über Geometrie von Clebsch II|location=Leipzig|url=https://archive.org/details/vorlesungenber21cleb|page=[https://archive.org/details/vorlesungenber21cleb/page/524 524]}}</ref>

Weierstrass coordinates were also used by Gérard (1892),<ref>{{Cite book|author=Gérard, L. |year=1892|title=Sur la géométrie non-Euclidienne|location=Paris|publisher=Gauthier-Villars|url=https://archive.org/details/surlagomtrienon01grgoog}}</ref> [[Biography:Felix Hausdorff|Felix Hausdorff]] (1899),<ref>{{Cite journal|author=Hausdorff, F.|year=1899|title=Analytische Beiträge zur nichteuklidischen Geometrie|journal=Leipziger Math.-Phys. Berichte|volume=51|pages=161–214|hdl=2027/hvd.32044092889328}}</ref> Frederick S. Woods (1903)],<ref>{{Cite journal|author=Woods, F. S.|year=1905|orig-year=1903|title=Forms of non-Euclidean space|journal=The Boston Colloquium: Lectures on Mathematics for the Year 1903|pages=[https://archive.org/details/in.ernet.dli.2015.161268/page/n40 31]–74|url=https://archive.org/details/in.ernet.dli.2015.161268}}</ref> Heinrich Liebmann (1905).<ref>{{Cite book|author=Liebmann, H. |year=1905|orig-year=1904|title=Nichteuklidische Geometrie|location=Leipzig|publisher=Göschen|url=https://archive.org/details/nichteuklidisch00liebgoog}}</ref>

The hyperboloid was explored as a [[Metric space|metric space]] by [[Biography:Alexander Macfarlane|Alexander Macfarlane]] in his ''Papers in Space Analysis'' (1894). He noted that points on the hyperboloid could be written as
:<math>\cosh A + \alpha  \sinh A,</math>
where &alpha; is a basis vector orthogonal to the hyperboloid axis. For example, he obtained the [[Hyperbolic law of cosines|hyperbolic law of cosines]] through use of his 
[[Hyperbolic quaternion|Algebra of Physics]].<ref name=Macfarlane>[[Biography:Alexander Macfarlane|Alexander Macfarlane]] (1894) ''[https://archive.org/details/principlesalgeb01macfgoog Papers on Space Analysis]'', B. Westerman, New York, weblink from archive.org</ref>

H. Jansen made the hyperboloid model the explicit focus of his 1909 paper "Representation of hyperbolic geometry on a two sheeted hyperboloid".<ref>''Abbildung hyperbolische Geometrie auf ein zweischaliges Hyperboloid'' Mitt. Math. Gesellsch Hamburg 4:409&ndash;440.</ref>  
In 1993 W.F. Reynolds recounted some of the early history of the model in his article in the [[American Mathematical Monthly]].<ref>Reynolds, William F. (1993) "Hyperbolic geometry on a hyperboloid", [[American Mathematical Monthly]] 100:442&ndash;55, [https://www.jstor.org/stable/2324297 Jstor link]</ref>

Being a commonplace model by the twentieth century, it was identified with the ''Geschwindigkeitsvectoren'' (velocity vectors) by [[Biography:Hermann Minkowski|Hermann Minkowski]] in his 1907 Göttingen lecture 'The Relativity Principle'. Scott Walter, in his 1999 paper "The Non-Euclidean Style of Minkowskian Relativity"<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> recalls Minkowski's awareness, but traces the lineage of the model to Hermann Helmholtz rather than Weierstrass and Killing.

In the early years of relativity the hyperboloid model was used by [[Biography:Vladimir Varićak|Vladimir Varićak]] to explain the physics of velocity. In his speech to the German mathematical union in 1912 he referred to Weierstrass coordinates.<ref>{{Citation | author=Varićak, V. | year=1912| title=On the Non-Euclidean Interpretation of the Theory of Relativity| journal=Jahresbericht der Deutschen Mathematiker-Vereinigung | volume =21| pages =103–127| title-link=s:On the Non-Euclidean Interpretation of the Theory of Relativity}}
</ref>

==See also==

* [[Poincaré disk model]]
* [[Hyperbolic quaternion]]s

==Notes and references==
{{Reflist}}

* {{Citation | last1=Alekseevskij | first1=D.V. | last2=Vinberg | first2=E.B. | last3=Solodovnikov | first3=A.S. | title=Geometry of Spaces of Constant Curvature | publisher=Springer-Verlag | location=Berlin, New York | series=Encyclopaedia of Mathematical Sciences | isbn=978-3-540-52000-9 | year=1993}}
* {{Citation | last1=Anderson | first1=James | title=Hyperbolic Geometry  | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Springer Undergraduate Mathematics Series | isbn=978-1-85233-934-0 | year=2005}}
* {{Citation | last1=Ratcliffe | first1=John G. | title=Foundations of hyperbolic manifolds | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94348-0 | year=1994 | url-access=registration | url=https://archive.org/details/foundationsofhyp0000ratc }}, Chapter 3
* [[Biography:Miles Reid|Miles Reid]] & Balázs Szendröi (2005) ''Geometry and Topology'', Figure 3.10, p 45, {{wipe|Cambridge University Press}}, {{ISBN|0-521-61325-6}}, {{MathSciNet|id=2194744}}.
* {{Citation | last1=Ryan | first1=Patrick J. | title=Euclidean and non-Euclidean geometry: An analytical approach | publisher={{wipe|Cambridge University Press}} | location=Cambridge, London, New York, New Rochelle, Melbourne, Sydney | isbn=978-0-521-25654-4 | year=1986 }}
* {{Cite web|last=Parkkonen|first=Jouni|date=|title=HYPERBOLIC GEOMETRY|url=http://users.jyu.fi/~parkkone/RG2012/HypGeom.pdf|archive-url=|archive-date=|access-date=September 5, 2020|website=}}

[[Category:Multi-dimensional geometry]]
[[Category:Hyperbolic geometry]]
[[Category:Minkowski spacetime]]

{{Sourceattribution|Hyperboloid model}}