Ken Shoemake’s 1985 Siggraph paper “Animation Rotation with Quaternion Curves”, that brought quaternions to computer graphics, covered this. The idea is to use

Ken Shoemake’s 1985 Siggraph paper “Animation Rotation with Quaternion Curves”, that brought quaternions to computer graphics, covered this. The idea is to use quaternions as control points in a spline the same way you would use 3d points in a spline. You could have a series of quaternion orientations, and connect them with C_2 continuity by using a connected series of piecewise cubic Bezier splines. The abstract mentions it: “This paper gives one answer by presenting a new kind of spline curve, created on a sphere, suitable for smoothly in-hetweening (i.e. interpolating) sequences of arbitrary rotations.” And the final punch line is section 4.3, then you can work through the details in the earlier sections. https://www.cs.cmu.edu/~kiranb/animation/p245-shoemake.pdf https://www.cs.cmu.edu/~kiranb/animation/p245-shoemake.pdf