2

I have a problem about spherical linear interpolation, or slerp for short.

As linked, Wikipedia gives the following formula for an interpolation between quaternions q0 and q1 with a t parameter (between 0 and 1):

Quaternion formula

My misunderstanding here is about the t exponent. Before this formula, the same page displays:

qt = costΩ + v * sintΩ

But I'm not sure this is the formula to take to complete this one, nor how to find this Ω angle.

Aracthor
  • 145

1 Answers1

2

Assuming $|q_0|=|q_1|$, the quotient $q_1q_0^{-1}$ is a unit quaternion. Thus, it has a polar form $$ q_1q_0^{-1}=\exp(\Omega\mathbf{v})=\cos\Omega+\sin\Omega\,\mathbf{v} $$

where $\Omega\in[0,\pi)$ (assuming $q_1\ne\pm q_0$ so $\Omega\ne\pi$) is a convex angle and $\mathbf{v}$ is a unit vector (hence also a square root of $-1$, which is why Euler's formula applies). From there you can define

$$ (q_1q_0^{-1})^t=\exp(t\Omega\mathbf{v})=\cos(t\Omega)+\sin(t\Omega)\mathbf{v} $$

for $0\le t\le1$.

anon
  • 151,657