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> ...not implemented inside quaternion libraries using quaternion exponentiation like the article does, but by computing angles explicitly... How else would yo

> ...not implemented inside quaternion libraries using quaternion exponentiation like the article does, but by computing angles explicitly... How else would you compute quaternion exponentiation? I don't think there's really a dichotomy here. When you compute quaternion exponentiation, one natural way to do it (as with complex numbers) is to think of the quaternions as a real magnitude multiplied by a phase. For complex numbers, the magnitude grows according to exp(x) and the phase evolves according to cos(x)+isin(x). This just falls out of Euler's formula. If you know that the magnitude is 1, you take the exp(x) term out, and you end up with a point that moves in a circle. The same thing applies to quaternions. I'm aware that there are other ways to compute quaternion exponentiation, but this is just a natural way to do it, especially for people who aren't experts in numeric programming.

dahart 5y ago
The article's quat_pow is implemented using 'quat_exp(quat_scale(quat_log(q), n));' where the code example I posted computes the half-angle of the rotation. If you stand back, I'd agree with you that there's an equivalence here, and it could be argued that Euclideanspace's code example is a kind of simplification and flattening of using an exponentiation function. Still, there are real differences between these two implementations, and the article's here looks conceptually simple, while the one people use in practice looks more complicated, and requires knowing how quaternions works. It's natural if you're fluent in quats, but not necessarily intuitive otherwise.
- klodolph 5y ago
  > If you stand back, I'd agree with you that there's an equivalence here, and it could be argued that Euclideanspace's code example is a kind of simplification and flattening of using an exponentiation function. You don't have to stand back that far! They're really quite similar pieces of code. quat_log() is basically a conversion to axis-angle. quat_exp() is basically a conversion from axis-angle back to quaternions. So, the quat_exp(t quat_log(x)) formula, with different names for the functions, is described as: 1. Figure out the angle between the starting and ending position, and the axis of rotation. 2. Vary the angle of rotation smoothly from t=0..1. 3. Convert back from axis-angle to an orientation. The only funny thing here is that the axis-angle encoding of quaternions uses a magnitude which a factor of two away from the angle in radians, so you'll see the sample code you linked to (with SLERP) use variables like "halfTheta" and "cosHalfTheta", where the quat_exp() and quat_log() formulas simply won't name them that way. In the end I think the point of learning more math is so you can see past the differences in naming and recognize when two seemingly different approaches to the same problem are really just two different sets of terminology and names for the same approach.
  - dahart 5y ago
    You're right and it's a great point!

dahart 5y ago

The article's quat_pow is implemented using 'quat_exp(quat_scale(quat_log(q), n));' where the code example I posted computes the half-angle of the rotation. If you stand back, I'd agree with you that there's an equivalence here, and it could be argued that Euclideanspace's code example is a kind of simplification and flattening of using an exponentiation function. Still, there are real differences between these two implementations, and the article's here looks conceptually simple, while the one people use in practice looks more complicated, and requires knowing how quaternions works. It's natural if you're fluent in quats, but not necessarily intuitive otherwise.

klodolph 5y ago
> If you stand back, I'd agree with you that there's an equivalence here, and it could be argued that Euclideanspace's code example is a kind of simplification and flattening of using an exponentiation function. You don't have to stand back that far! They're really quite similar pieces of code. quat_log() is basically a conversion to axis-angle. quat_exp() is basically a conversion from axis-angle back to quaternions. So, the quat_exp(t quat_log(x)) formula, with different names for the functions, is described as: 1. Figure out the angle between the starting and ending position, and the axis of rotation. 2. Vary the angle of rotation smoothly from t=0..1. 3. Convert back from axis-angle to an orientation. The only funny thing here is that the axis-angle encoding of quaternions uses a magnitude which a factor of two away from the angle in radians, so you'll see the sample code you linked to (with SLERP) use variables like "halfTheta" and "cosHalfTheta", where the quat_exp() and quat_log() formulas simply won't name them that way. In the end I think the point of learning more math is so you can see past the differences in naming and recognize when two seemingly different approaches to the same problem are really just two different sets of terminology and names for the same approach.
- dahart 5y ago
  You're right and it's a great point!