Interpolating rotations with SLERP

Naive interpolation of rotation matrices does not produce a rotation matrix. That is, if R₁ and R₂ are rotation (orthogonal) matrices and 0 < t < 1, then

is not in general a rotation matrix.

You can represent rotations with unit quaternions rather than orthogonal matrices (see details here), so a reasonable approach might be to interpolate between the rotations represented by unit quaternions q₁ and q₂ using

but this has a similar problem: the quaternion above is not a unit quaternion.

One way to patch this up would be to normalize the expression above, dividing by its norm. That would indeed produce unit quaternions, and hence correspond to rotations. However, uniformly varying t from 0 to 1 does not produce a uniform rotation.

The solution, first developed by Ken Shoemake [1], is to use spherical linear interpolation or SLERP.

Let θ be the angle between q₁ and q₂. Then the spherical linear interpolation between q₁ and q₂ is given by

Now q(t) is a unit quaternion, and uniformly increasing t from 0 to 1 creates a uniform rotation.

[1] Ken Shoemake. “Animating Rotation with Quaternion Curves.” SIGGRAPH 1985.