In
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
, the
hyperboloid model, also known as the
Minkowski model or the
Lorentz model (after
Hermann MinkowskiHermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity. Life and work :Hermann Minkowski was born...
and
Hendrik LorentzHendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...
), is a model of
ndimensional
hyperbolic geometryIn mathematics, hyperbolic geometry is a nonEuclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
in which points are represented by the points on the forward sheet
S^{+} of a twosheeted
hyperboloid in (
n+1)dimensional
Minkowski spaceIn physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
and
mplanes are represented by the intersections of the (
m+1)planes in Minkowski space with
S^{+}. The hyperbolic distance function admits a simple expression in this model. The hyperboloid model of the
ndimensional hyperbolic space is closely related to the Beltrami–Klein model: both are projective models in the sense that the
isometry groupIn mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation...
is a subgroup of the projective group.
Minkowski quadratic form
If (
x_{0},
x_{1}, …,
x_{n}) is a vector in the (
n+1)dimensional coordinate space
R^{n+1}, the
Minkowski quadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy  3y^2\,\!is a quadratic form in the variables x and y....
is defined to be
The vectors
v∈
R^{n+1} such that
Q(
v) = 1 form an
ndimensional
hyperboloid S consisting of two
connected componentConnected components are part of topology and graph theory, two related branches of mathematics.* For the graphtheoretic concept, see connected component .* In topology: connected component .Implementations:...
s, or
sheets: the forward, or future, sheet
S^{+}, where
x_{0}>0 and the backward, or past, sheet
S^{−}, where
x_{0}<0. The points of the
ndimensional hyperboloid model are the points on the forward sheet
S^{+}.
The
Minkowski bilinear form B is the
polarizationIn mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. Let \x\ \, denote the norm of vector x and \langle x, \ y \rangle \, the inner product of vectors x and y...
of the Minkowski quadratic form
Q,
Explicitly,
.
The
hyperbolic distance between two points
u and
v of
S^{+} is given by the formula
Isometries
The indefinite orthogonal group
O(1,
n), also called the
(
n+1)dimensional
Lorentz groupIn physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
, is the
Lie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
of
realIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
(
n+1)×(
n+1)
matricesIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
which preserve the Minkowski bilinear form. In a different language, it is
the group of linear
isometriesIn mathematics, an isometry is a distancepreserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
of the
Minkowski spaceIn physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
. In particular, this group preserves the hyperboloid
S. The subgroup of
O(1,
n) which preserves the sign of the first coordinate is the
orthochronous Lorentz group, denoted
O^{+}(1,
n). Its subgroup
SO^{+}(1,
n) consisting of matrices with
determinantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
one is a connected Lie group of dimension
n(
n+1)/2 which acts on
S^{+} 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
Where
belongs to the compact special orthogonal group
SO(
n) (generalizing the
rotation groupIn mechanics and geometry, the rotation group is the group of all rotations about the origin of threedimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation ...
for
n=3). It follows that the
ndimensional
hyperbolic spaceIn mathematics, hyperbolic space is a type of nonEuclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
can be exhibited as the
homogeneous spaceIn mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a nonempty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
and a
Riemannian symmetric spaceIn differential geometry, representation theory and harmonic analysis, a symmetric space is a smooth manifold whose group of symmetries contains an inversion symmetry about every point. There are two ways to formulate the inversion symmetry, via Riemannian geometry or via Lie theory...
of rank 1,

In fact, the group
SO^{+}(1,
n) is the full group of orientationpreserving isometries of the
ndimensional hyperbolic space.
History
In 1880
Wilhelm KillingWilhelm Karl Joseph Killing was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and nonEuclidean geometry....
published "Die Rechnung in NichtEuclidischen Raumformen" in
Crelle's JournalCrelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik . History :...
(89:265–87). This work discusses the hyperboloid model in a way that shows the analogy to the hemisphere model. Killing attributes the idea to
Karl WeierstrassKarl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis". Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
in a Berlin seminar some years before. Following on Killing’s attribution, the phrase
Weierstrass coordinates has been associated with elements of the hyperboloid model as follows:
Given an inner product
on R
^{n},
the Weierstrass coordinates of
x ∈ R
^{n} are:
compared to
for the hemispherical model. (See Elena Deza and
Michel DezaMichel Marie Deza is a Soviet and French mathematician, specializing in combinatorics, discrete geometry and graph theory...
(2006)
Dictionary of Distances.)
According to Jeremy Gray (1986),
PoincaréJules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
used the hyperboloid model in his personal notes in 1880. Gray shows where the hyperboloid model is implicit in later writing by Poincaré.
For his part, W. Killing continued to publish on the hyperboloid model, particularly in 1885 in his
Analytic treatment of nonEuclidean spaceforms. Further exposure of the model was given by
Alfred ClebschRudolf Friedrich Alfred Clebsch was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. He subsequently taught in Berlin and Karlsruhe...
and Ferdinand Lindemann in 1891 in
Vorlesungen uber Geometrie, page 524.
The hyperboloid was explored as a
metric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3dimensional Euclidean space...
by
Alexander MacfarlaneAlexander Macfarlane was a Nova Scotia lawyer and political figure. He was a member of the Canadian Senate from 1870 to 1898. His surname also appears as McFarlane in some sources....
in his
Papers in Space Analysis (1894). He noted that points on the hyperboloid could be written
where α is a basis vector orthogonal to the hyperboloid axis. For example, he obtained the hyperbolic law of cosines through use of his
Algebra of PhysicsIn the abstract algebra of algebras over a field, the hyperbolic quaternionq = a + bi + cj + dk, \quad a,b,c,d \in R \!is a mutated quaternion wherei^2 = j^2 = k^2 = +1 \! instead of the usual −1....
.
H. Jansen made the hyperboloid model the explicit focus of his 1909 paper "Representation of hyperbolic geometry on a two sheeted hyperboloid".
In 1993 W.F. Reynolds recounted some of the early history of the model in his article in the
American Mathematical MonthlyThe American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is currently published 10 times each year by the Mathematical Association of America....
.
Being a commonplace model by the twentieth century, it was identified with the
Geschwindigkeitsvectoren (velocity vectors) by
Hermann MinkowskiHermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity. Life and work :Hermann Minkowski was born...
in his
Minkowski spaceIn physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
of 1908. Scott Walter, in his 1999 paper "The NonEuclidean Style of Special Relativity" 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
Vladimir VarićakVladimir Varićak was a Croatian mathematician and theoretical physicist of Serbian descent....
to explain the physics of velocity. In his speech to the German mathematical union in 1912 he referred to Weierstrass coordinates.