Algorithm 4 Velocity vector field $\frac{d\alpha }{dt}$ for a path α.
1:	Let $\frac{d\alpha }{dt}(0)=0$.
2:	for τ = 1 to k do
3:	$\theta ={\text{cos}}^{-1}\langle \alpha \left(\frac{\tau -1}{k}\right),\alpha \left(\frac{\tau }{k}\right)\rangle$
4:	$f=-\alpha \left(\frac{\tau -1}{k}\right)+\alpha \left(\frac{\tau }{k}\right)\text{cos}(\theta )$
5:	$\frac{d\alpha }{d\tau }\left(\frac{\tau }{k}\right)=\frac{kf\theta }{\sqrt{\langle f,f\rangle }}$
6:	Project $\frac{d\alpha }{d\tau }\left(\frac{\tau }{k}\right)$ into $T_{\alpha \left(\frac{\tau }{k}\right)}(\mathcal{C})$ using Algorithm 2
7:	end for