Closest point of one SO(3) submanifold onto another submanifold

Question

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?

This question was posted four days ago at Math.SE, but with no answer or comments.

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

Is there an analytic way to determine the pair of points on $M_1$ and $M_2$ that minimizes the geodesic distance between them.

$\arg \min d(m_1, m_2)= \| \log m_1 m_2^\top\|\; : \; m_1 \in M_1, m_2 \in M_2$

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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.

Answer 1

I'm assuming you meant $P_1, P_2 \in SO(3)$, and that $M_1$ and $M_2$ are 1-parameter subgroups. Assuming you want the biinvariant metric on $SO(3)$, the 1-parameter subgroups are also geodesics. If I were doing it, I'd lift them to the double cover (the 3-sphere), where they're great circles, apply an isometry to both of them so that the first moves to standard position, say the $xy$-plane (where $z=w=0$), and find the closest points there. Then undo your isometry.