I have two 3D rotations about the origin, represented as $3 \times 3$ orthogonal matrices $M_1$ and $M_2$ (specified by numerical entries), and I would like to interpolate (and compute) a continuous sequence of rotations between $M_1$ and $M_2$. Ideally this interpolation would follow a geodesic on SO($3$).

Assuming this is well-known,
(** Q1**): I would appreciate a pointer to geodesics on SO($3$),
especially suited to my computational task.

(** Q2**) [

*added*]. I wonder if the geodesics identified by Benjamin, Paul Siegel, and Francois Ziegler, which effectively answer

**, can be viewed as equivalent to following a great-circle arc on $S^3$ between unit quaternions representing the two rotations?**

*Q1***Update** (*25Jan2019*): A paper was just published on this
topic:

Novelia, Alyssa, and Oliver M. O’Reilly. "On geodesics of the rotation group $SO(3)$."

Regular and Chaotic Dynamics20, no. 6 (2015): 729-738. Springer link.

**Update** (*26Jan2019*): The paper above raises the interesting question
whether saccadic motions of the eye follow $SO(3)$ geodesics,
an issue possibly dating back
to Helmholtz's investigations in the late $19^{\mathrm{th}}$ century:

von Helmholtz, H., "Über die normalen Bewegungen des menschlichen Auges,"

Archiv für Ophthalmologie, 1863, Vol. 9, No. 2, pp. 153–214; DOI: 10.1007/BF02720895.