Rotation in Three Dimensions - 3D Math Primer for Graphics and Game Development - page 261

Figure 8.12
Interpolating a vector about an arc

Applying some trig to the right triangle with k 1 v 1 as the hypotenuse
(and recalling that v 1 is a unit vector), we see that

sinω = sintω
k 1

,

k 1 = sintω

sinω .

A similar technique to solve for k 0 yields the following result:

k 0 = sin(1 − t)ω
sinω

.

Thus, v t can be expressed as

v t = k 0 v 0 + k 1 v 1 = sin(1 − t)ω
sinω

v 0 + sintω

sinω v 1 .

The same basic idea can be extended into quaternion space, and we can
reformulate the slerp as

slerp( q 0 , q 1 ,t) = sin(1 − t)ω
sinω

q 0 + sintω

Quaternion slerp in
practice

sinω q 1 .
We just need a way to compute ω, the “angle” between the two quater-
nions. As it turns out, an analogy from 2D vector math can be carried into
quaternion space; we can think of the quaternion dot product as returning
cosω.
There are two slight complications. First, the two quaternions q and
− q represent the same orientation, but may produce different results when