# Reconstructing paths on the sphere from the ratio of acceleration to velocity

Given a path $\gamma:[0,1]\to \mathbb C$, we can determine $\gamma$ from information about its derivatives. For example, $\gamma$ is determined by $\gamma(0), \gamma'(0)$, and $\gamma''(t)$. This just requires integrating twice. More interesting is that we can recover $\gamma$ from $\gamma(0), \gamma'(0)$ and $\frac{\gamma''(t)}{\gamma'(t)}$, since $$\gamma'(t)=\gamma'(0)e^{\int_0^t \frac{\gamma''(t)}{\gamma'(t)}dt}.$$

Is there an analogous procedure for a path on the unit sphere $\gamma:\mathbb R \to S^2\subset \mathbb R^3$? Namely, given the lengths of the velocity vector and the (tangential component of the) acceleration vector, and given the angle between the two vectors (plus appropriate initial conditions), can one reconstruct $\gamma$? Is there an equivalently simple formula to the one in the planar case?

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