---
tag: 809288G50Y
kind: example
name: Hyperboloid model
msc: [51M10, 53A35]
instance_of: [7CBT4SM83Y]
uses: [GN4N94GQ0Z, 0SQZMXB6GP, M2900DMEW6, 036696EMG2, 85MM2EA0TK, 1WN9B18H86]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperboloid_model
same_as:
  wikidata: Q3318095
  wikipedia: Hyperboloid model
---

## Example

Let $n \ge 1$ and give $\R^{n+1}$, with coordinates $x = (x_0, x_1, \ldots, x_n)$, the symmetric [[bilinear form|0SQZMXB6GP]]
$$B(x, y) = x_0 y_0 - x_1 y_1 - \cdots - x_n y_n.$$
The *hyperboloid model* of hyperbolic $n$-space is the upper sheet of a hyperboloid of two sheets,
$$H^n = \set{x \in \R^{n+1} : B(x, x) = 1, \ x_0 > 0},$$
with the distance $d(x, y) = \operatorname{arcosh} B(x, y)$.
On each tangent space of $H^n$ the form $-B$ is positive definite, and the resulting [[Riemannian manifold|7CBT4SM83Y]] has constant sectional curvature $-1$.
Its geodesics, the lines of [[hyperbolic geometry|GN4N94GQ0Z]], are the nonempty intersections of $H^n$ with 2-dimensional linear subspaces of $\R^{n+1}$.

Check for $n = 1$: the curve $\gamma(t) = (\cosh t, \sinh t)$ lies on $H^1$, since $\cosh^2 t - \sinh^2 t = 1$ and $\cosh t > 0$.
Its velocity is $\gamma'(t) = (\sinh t, \cosh t)$, and $-B(\gamma', \gamma') = \cosh^2 t - \sinh^2 t = 1$, so $t$ is arc length.
The formula gives $d(\gamma(0), \gamma(t)) = \operatorname{arcosh}(\cosh t) = \abs{t}$, in agreement.

## Remarks

The isometries of $H^n$ are the linear maps of $\R^{n+1}$ that preserve $B$ and map the upper sheet to itself, the subgroup $O^+(1, n)$ of the [[indefinite orthogonal group|M2900DMEW6]].
So isometries act by matrices, which makes computations in this model linear algebra.
The central projection $x \mapsto (x_1/x_0, \ldots, x_n/x_0)$ maps $H^n$ onto the [[Beltrami–Klein model|1WN9B18H86]].
The projection $x \mapsto (x_1, \ldots, x_n)/(1 + x_0)$ from the point $(-1, 0, \ldots, 0)$ maps $H^n$ onto the [[Poincaré disk model|85MM2EA0TK]] when $n = 2$, and onto the Poincaré ball model in general.
For $n = 3$, $\R^4$ with the form $B$ is Minkowski spacetime, and $H^3$ is the set of unit future-pointing timelike vectors.

## Explanations

### L1 human

Take the bowl-shaped sheet $x_0 = \sqrt{1 + x_1^2 + x_2^2}$ in space, and measure lengths on it with a modified rule in which the vertical direction counts negatively.
With this rule the bowl becomes the hyperbolic plane.
Hyperbolic lines are the curves cut out of the bowl by planes through the origin, and rotations and Lorentz boosts move the bowl onto itself.
This is the hyperbolic analogue of the round sphere, which is the set of points at distance $1$ from the origin for the usual form $x_0^2 + x_1^2 + x_2^2$.