Algorithm 10 Given q0, q1 ∈ , compute a geodesic between them
1:	Initialize a path α between q0 and q1 using Algorithm 9.
2:	repeat
3:	Compute the path velocity ${\alpha }_t\equiv \frac{d\alpha }{dt}$ along α using Algorithm 4.
4:	Calculate the covariant integral (w) of αt using Algorithm 3.
5:	Parallel translate (backward) w(1) along α as w̃ using Algorithm 6.
6:	Compute the gradient of the energy E and project it to 0 as v using Algorithm 7.
7:	Update the path α in the direction v using Algorithm 8.
8:	Compute path energy $E=\frac{1}{2k}{\sum }_0^k\langle {\alpha }_t(\tau ),{\alpha }_t(\tau )\rangle$.
9:	until ‖∇E‖ > ε