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I apologize if this is a well known problem and I've just missed the answer, but I have searched fairly extensively. Perhaps there is another way to phrase it that I've been missing.

Suppose there are two submanifolds $M_1$ and $M_2$ embedded within $SO(3)$.

$M_1 =\left\{ P_1 e^{\, \mathbf \xi_1 t} | t \in \left[0, ^{2 \pi}/_{|\xi_1|} \right] \right\}$

$M_2 =\left\{ P_2 e^{\, \mathbf \xi_2 t} | t \in \left[0, ^{2 \pi}/_{|\xi_2|} \right] \right\}$

with $P_1, P_2 \subset SO(3)$ and $\mathbf \xi_1, \mathbf \xi_2 \subset \mathfrak{so(3)}$

Is there an analytic way to determine a point on $M_1$ that minimizes

$d(m_1, m_2) \; \forall m_1 \subset M_1, m_2 \subset M_2$

where $d$ is the geodesic distance $d(m_1,m_2)=\| \log m_1 m_2^\top\|$


The specific problem I'm trying to solve is for a rigid body with an initial attitude and a fixed rotation rate, what is the nearest attitude it will reach with a defined roll and pitch so $\xi_1$ would also point only in the direction of $i_z$ if that further constraint helps. I can solve this numerically, but am looking for an analytic solution to try and implement this on a resource constrained platform.

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