Algorithm 2.

Given two curves q₀ and q₁, compute a geodesic in $\mathcal{S}$

Find the geodesic between q0 and q1 in $\mathcal{C}∕({\mathbb{S}}^1\times SO\left(n\right))$ using the approach outlined in Sec. 3.2. This also yields the tangent vector αt(1) at q1. Let {vi}, i = 1, …, d be the Fourier basis for Tid($\mathcal{D}$). Project the vector αt (1) on Tq1(${\mathcal{D}}_{q1}$) using Eqn. 8. if ||π(u)||2 < ε then Stop. end if Form the tangent vector $\mathbf{g}\in T_{id}\left(\mathcal{D}\right)$ as, $\mathbf{g}={\sum }_{i=1}^d\langle {\alpha }_{\mathbf{t}}\left(1\right),{\varphi }_{\ast }\left(v_i\right)\rangle v_i$. Compute the flow on $\mathcal{D}$ at id, such that $\tilde{\gamma }={\Psi }_{∊}(id,\mathbf{g})=id-∊\mathbf{g}$. Set $q_1=q_1\cdot \tilde{\gamma }=\sqrt{\gamma \prime }qo\tilde{\gamma }$. Go to Step 1.