Hyperboloid model

; it projects to the brown geodesic on the green hyperboloid. In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet S + of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors 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 + or by wedge products of m vectors. Hyperbolic space is 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 is embedded in (n+1)-dimensional Euclidean space.…

; it projects to the brown geodesic on the green hyperboloid.]] In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet S + of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors 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 + or by wedge products of m vectors. Hyperbolic space is 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 is embedded in (n+1)-dimensional Euclidean space. Other models of hyperbolic space can be thought of as map projections of S + : the Beltrami–Klein model is the projection of S + through the origin onto a plane perpendicular to a vector from the origin to specific point in S + analogous to the gnomonic projection of the sphere; the Poincaré disk model is a projection of S + through a point on the other sheet S − onto perpendicular plane, analogous to the stereographic projection of the sphere; the Gans model is the orthogonal projection of S + onto a plane perpendicular to a specific point in S + , analogous to the orthographic projection; the band model of the hyperbolic plane is a conformal “cylindrical” projection analogous to the Mercator projection of the sphere; Lobachevsky coordinates are a cylindrical projection analogous to the equirectangular projection (longitude, latitude) of the sphere. Q(x_0, x_1, \ldots, x_n) = -x_0^2 + x_1^2 + \ldots + x_n^2. The vectors such that form an n-dimensional hyperboloid S consisting of two connected components, or sheets: the forward, or future, sheet S + , where x 0 >0 and the backward, or past, sheet S − , where x 0 + . The metric on the hyperboloid is ds^2 = Q(dx_0, dx_1, \ldots, dx_n) = -dx_0^2 + dx_1^2 + \ldots + dx_n^2. The Minkowski bilinear form B is the polarization of the Minkowski quadratic form Q, B(\mathbf{u}, \mathbf{v}) = (Q(\mathbf{u}+\mathbf{v}) - Q(\mathbf{u}) - Q(\mathbf{v})) / 2 . (This is sometimes also written using scalar product notation \mathbf{u}\cdot\mathbf{v}. ) Explicitly, 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 . The hyperbolic distance between two points u and v of S + is given by the formula d(\mathbf{u}, \mathbf{v}) = \operatorname{arcosh}(-B(\mathbf{u}, \mathbf{v})) , where is the inverse function of hyperbolic cosine. B (\mathbf{u}, \mathbf{u}) = 1 B (\mathbf{v}, \mathbf{v}) = -1 B (\mathbf{u}, \mathbf{v}) = B (\mathbf{v}, \mathbf{u}) = 0 and use w as a real parameter for points on the geodesic, then \mathbf{u} \sinh w + \mathbf{v} \cosh w will be a point on the geodesic. 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. \mathbb{H}^n=\mathrm{SO}^{+}(1,n)/\mathrm{SO}(n). The group SO + (1,n) is the full group of orientation-preserving isometries of the n-dimensional hyperbolic space. In more concrete terms, SO + (1,n) can be split into n(n−1)/2 rotations (formed with a regular Euclidean rotation matrix in the lower-right block) and n hyperbolic translations, which take the form \begin{pmatrix} \cosh \alpha & \sinh \alpha & 0 & \cdots \\[2mu] \sinh \alpha & \cosh \alpha & 0 & \cdots \\[2mu] 0 & 0 & 1 & \\[-7mu] \vdots & \vdots & & \ddots \\ \end{pmatrix} where \alpha 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 (w, x, y, z) is: \begin{pmatrix} w & x & y & z \\[2mu] x & \ \dfrac{x^2}{w+1}+1 & \dfrac{yx}{w+1} & \dfrac{zx}{w+1} \\[2mu] y & \dfrac{xy}{w+1} & \,\dfrac{y^2}{w+1}+1 & \dfrac{zy}{w+1} \\[2mu] z & \dfrac{xz}{w+1} & \dfrac{yz}{w+1} & \dfrac{z^2}{w+1}+1 \end{pmatrix}_{\vphantom|}, where . This extends naturally to more dimensions, and is also the simplified version of a Lorentz boost when you remove the relativity-specific terms. Let \mathbf u = \frac {\mathbf p - \mathbf q}{\sqrt{Q(\mathbf p - \mathbf q)}} . Note that Q(\mathbf u) = 1 , and therefore u \notin \mathbb{H}^n . Then \mathbf x \mapsto \mathbf x - 2 B(\mathbf x, \mathbf u) \mathbf u is a reflection that exchanges \mathbf p and \mathbf q . This is equivalent to the following matrix: R = I - 2 \mathbf u \mathbf u^{\operatorname{T}} \begin{pmatrix} -1 & 0 \\ 0 & I \\ \end{pmatrix} (note the use of block matrix notation). Then \{I, R\} is a group of isometries. All such subgroups are conjugate. S = \left \{ \begin{pmatrix} 1 & 0 \\ 0 & A \\ \end{pmatrix} : A \in O(n) \right \} is the group of rotations and reflections that preserve (1, 0, \dots, 0) . The function A \mapsto \begin{pmatrix} 1 & 0 \\ 0 & A \\ \end{pmatrix} is an isomorphism from O(n) to this group. For any point p , if X is an isometry that maps (1, 0, \dots, 0) to p , then XSX^{-1} is the group of rotations and reflections that preserve p . L_t = \begin{pmatrix} \cosh t & \sinh t & 0 \\ \sinh t & \cosh t & 0 \\ 0 & 0 & I \\ \end{pmatrix} = e^\begin{pmatrix} 0 & t & 0 \\ t & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} (The expression on the RHS is a matrix exponential.) This is a translation of distance t in the positive x direction if t \ge 0 or of distance -t in the negative x direction if t \le 0 . Any translation of distance t is conjugate to L_t and L_{-t} . The set \left \{L_t : t \in \mathbb R \right \} 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 \overline{\mathbf p \mathbf q} . Let X be an isometry that maps (1, 0, \dots, 0) to p and let Y be an isometry that fixes p and maps X L_{d(\mathbf p, \mathbf q)} [1, 0, \dots, 0]^{\operatorname{T}} to q . An example of such a Y is a reflection exchanging X L_{d(\mathbf p, \mathbf q)} [1, 0, \dots, 0]^{\operatorname{T}} and q (assuming they are different), because they are both the same distance from p . Then YX is an isometry mapping (1, 0, \dots, 0) to p and a point on the positive x-axis to q . (YX)L_t(YX)^{-1} is a translation through the line \overline{\mathbf p \mathbf q} of distance |t| . If t \ge 0 , it is in the \overrightarrow{\mathbf p \mathbf q} direction. If t \le 0 , it is in the \overrightarrow{\mathbf q \mathbf p} direction. \left \{(YX)L_t(YX)^{-1} : t \in \mathbb R \right \} is the group of translations through \overline{\mathbf p \mathbf q} . \end{pmatrix} = e^\begin{pmatrix} 0 & 0 & \mathbf b^{\operatorname{T}} \\ 0 & 0 & \mathbf b^{\operatorname{T}} \\ \mathbf b & -\mathbf b & 0 \end{pmatrix} 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 A in O(n−1) \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & A \\ \end{pmatrix} 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

In several papers between 1878–1885, Wilhelm Killing used the representation he attributed to Karl Weierstrass for Lobachevskian geometry. In particular, he discussed quadratic forms such as k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2} or in arbitrary dimensions k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2} , where k is the reciprocal measure of curvature, k^{2}=\infty denotes Euclidean geometry, k^{2}>0 elliptic geometry, and k^{2} hyperbolic geometry. According to Jeremy Gray (1986), Poincaré used the hyperboloid model in his personal notes in 1880. Poincaré published his results in 1881¹, in which he discussed the invariance of the quadratic form \xi^{2}+\eta^{2}-\zeta^{2}=-1 . Gray shows where the hyperboloid model is implicit in later writing by Poincaré. Also Homersham Cox in 1882 used Weierstrass coordinates (without using this name) satisfying the relation z^{2}-x^{2}-y^{2}=1 as well as w^{2}-x^{2}-y^{2}-z^{2}=1 . Further exposure of the model was given by Alfred Clebsch and Ferdinand Lindemann in 1891 discussing the relation x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2} and x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2} . Weierstrass coordinates were also used by Gérard (1892), Felix Hausdorff (1899), Frederick S. Woods (1903)], Heinrich Liebmann (1905). The hyperboloid was explored as a metric space by Alexander Macfarlane in his Papers in Space Analysis (1894). He noted that points on the hyperboloid could be written as \cosh A + \alpha \sinh A, 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 Physics. 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 Monthly. Being a commonplace model by the twentieth century, it was identified with the Geschwindigkeitsvectoren (velocity vectors) by Hermann Minkowski in his 1907 Göttingen lecture 'The Relativity Principle'. Scott Walter, in his 1999 paper "The Non-Euclidean Style of Minkowskian 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 Varićak to explain the physics of velocity. In his speech to the German mathematical union in 1912 he referred to Weierstrass coordinates.

See also

Poincaré disk model Hyperbolic quaternions

Notes and references

, Chapter 3 Miles Reid & Balázs Szendröi (2005) Geometry and Topology, Figure 3.10, p 45, Cambridge University Press, , .

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