Geodesic polar coordinates

According to @needham2021visual page 274, they are the generalization to an arbitrary surface of the idea of spherical coordinates on ${\mathbb{S}}^2$.
If we imagine great circles departing from the north pole in every direction with unit speed, then every point on ${\mathbb{S}}^2$ is characterized by which particle hits this point (labelled by the longitude $\theta$), and when (specified by the latitude $\varphi =t$).

Well, that idea in an arbitrary surface are geodesics polar coordinates $(t,\theta )$.

An important result is Gauss lemma.

Related: Riemann normal coordinates.