# Statements(s) > [!NOTE] (1) Parametrization of Hyperbolic Plane \[MA243\] > Let $\mathcal{H}^1$ denote the [[Hyperbolic space|1-dimensional hyperbolic space]] (i.e. the hyperbolic line). Then there is an [[Isometry|isometry]] $\begin{align}f:\mathbb{R}&\to \mathcal{H}^1 \\ t &\mapsto (\cosh(t), \sinh(t)), & t\in [0,\infty).\end{align}$ # Proof(s) ###### Proof of statement 1: Using the hyperbolic pythagorean theorem, $-\cosh^2(t)+\sinh^2t=1$ and so the image of $\mathbb{R}$ under $f$ is in $\mathcal{H}^1$. It follows from [[Inverse of Strictly Monotonic Continuous Real Function Exists, is Strictly Monotonic in the Same Sense, and is Continuous|inverse function theorem]] that $\sinh t$ is bijective and thus $f$ is bijective since the $x_{1}$ coordinate of a point in $\mathcal{H}^1$ is determined uniquely by the $x_{2}$ coordinate. The map is an isometry because of the angle sum formula.