# Definition(s) > [!NOTE] Definition 1 (Hyperboloid Model of Hyperbolic Space) > Let $n$ be a positive natural number. The $n$-dimensional hyperbolic space is $(\mathcal{H}^{n}, d_{\mathcal{H}^{n}})$ where $\begin{align} \mathcal{H}^{n} &= \{ \mathbf{x} \in \mathbb{R}^{n+1}: -x_{1}^2 +x_{2}^2 +x_{3}^2 + \dots + x_{n+1}^2 = -1, x_{1}>0 \} \\ &= \{ \mathbf{x} \in \mathbb{R}^{n+1} : ||\mathbf{x}||_{L}=i, x_{1} >0 \} \end{align}$and $d_{\mathcal{H}^n}: \mathcal{H}^n\to \mathbb{R}$ is defined by $d_{\mathcal{H}^n}: (\mathbf{x},\mathbf{y}) \mapsto \cosh^{-1} \left( \frac{\langle \mathbf{x}, \mathbf{y} \rangle_{L}}{ \Vert\mathbf{ x} \Vert_{L} \Vert \mathbf{y} \Vert_{L} } \right)=\cosh^{-1}(-\langle \mathbf{x}, \mathbf{y} \rangle_{L})$ where $\cosh$ is a [[Hyperbolic Trigonometric Functions|hyperbolic trig function]] and noting that $\begin{align} \langle \mathbf{x},\mathbf{y}\rangle_{L}&=\mathbf{x}^TJ_{n}\mathbf{y} & J_{n} = \begin{pmatrix} -1 & \mathbf{0} \\ \mathbf{0} & I_{n-1} \end{pmatrix} \\ &= -x_{1}y_{1} + x_{2}y_{2} + x_{3}y_{3}+\dots +x_{n+1}y_{n+1} \end{align}$is called the *Lorentz inner product* and $\Vert \mathbf{x} \Vert_{L}=\sqrt{ \langle \mathbf{x}, \mathbf{x}\rangle_{L} }$is called the *Lorentz norm*. > [!Example] Example > The hyperbolic line $\mathcal{H}^1: \sqrt{ x_{1}^2+x_{2}^2 }=i, x_{1}>0$ is the upper half of the hyperbola: > ![[Pasted image 20241109132941.png|100]] > (vertical axis is $x_{1}$; horizontal is $x_{2}$). It follows from hyperbolic trig addition formulae that the hyperbolic line is parametrised by $(\cosh t, \sinh t)$, $t\in[0,\infty)$. # Properties(s) # Application(s) **More examples**: # Reference(s)