Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.