# Difference between a hyperbolic line and a geodesic

The setting for hyperbolic space in this question will be the upper half plane.

Now I know that to measure the distance between two points $p$ and $q$ in the upper half plane, we take

$\inf \int_\gamma \frac{1}{\Im(z)} |dz|$,

where $\gamma$ is any $C^1$ curve joining these $p$ and $q$. So if $\gamma$ is a curve that realises this infimum, it is a geodesic meaning the triangle inequality for any triple of points $p,q,r$ on $\gamma$, where $q$ is between $p$ and $r$ the triangle inequality is really just an equality.

I was just wondering if there is any difference (in terms of technicalities) between the terms "geodesic" and "hyperbolic line" in the upper half plane, or in any setting of hyperbolic geometry in general.

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