At the north pole, for $v=(v_1,v_2,0)$ and $r=|v|$, $\exp_p(v)=\cos r\,p+(\sin r/r)v$. Polar coordinates give $(\sin r\cos\theta,\sin r\sin\theta,\cos r)$. They are one-to-one for $0<r<\pi$, covering the sphere minus the two poles as a coordinate chart, with the north pole added as the polar origin; the south pole is the cut locus.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 26I / b / Solution

 Solution (source code)