# Linear speed of a rotating sphere

1. The problem statement, all variables and given/known data
Hello everyone.
Each minute, I have 3d coordinates of points at the surface of a unit sphere (with center at (0;0;0)) rotating with an axis which can (slightely?) change over time. I want to know the linear speed (s) of this sphere. I don’t know how to find r at each time.

2. Relevant equations

$s=r*\omega$

with $\omega =$ angular speed

3. The attempt at a solution

I found solutions which doesn’t imply directly r :

$\cos{s} = \cos(\theta_1)*\cos(\theta_2) + \sin(\theta_1)*\sin(\theta_2) * \cos(\phi_2-\phi_1)$

with $\theta = colatitude$
and $\phi = longitude$.

Is it a good way to calculate s ?

To have a good linear speed according to positions on the sphere, we also told me to find "dynamically" the plane $\pi$ containing severals points and to project these points on the parallel plane to $\pi$ passing through the center of the sphere. Then, to calculate the angular speed $\omega$.

But it doesn’t work since my linear speed was sometimes higher when points are close together ($\omega~0$) compared at when they describe a small circle ($\omega~0.08 rad/sec$).

Thank you.

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