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Comment by nighthawk454 | Hacker News Reader

The easiest way for me to conceptualize it is to think of it as orientation + rotation. 3 dims for the orientation vector to get the object facing/pointing the right way, then a further 1 dim/var for rotation about that axis. For a total of 4 variables/dimensions

3blue1brown and Ben Eater did a series of interactive videos on the subject that can be explored:

https://eater.net/quaternions

My favorite demo on this point is this one: https://eater.net/quaternions/video/rotation

Where they have a toggle to go between `a + bi + cj + dk` quaternion notation and an equivalent formulation in terms of 3d orientation vector plus rotation angle as `cos(θ) + sin(θ)*(ai + bj + ck)`

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> The easiest way for me to conceptualize it is to think of it as orientation + rotation. 3 dims for the orientation vector to get the object facing/pointing the right way, then a further 1 dim/var for rotation about that axis. For a total of 4 variables/dimensions

Something's missing. Orientation in 3D space is a two-dimensional quantity; you would never need three dimensions to express it. The third dimension has to be providing some additional information, like a magnitude.

Only if you’re describing orientation as two orthogonal rotations. I’m saying think of it like a ‘pointing’ vector that defines the axis of rotation. And such a vector does require 3 components in 3d space

It’s possible to point in 3D space with two rotation components. For example, the first two components of a UV texture.

But I agree it is helpful to think of quaternions as direction and spin.

Yes, a three-dimensional vector is a combination of a 3D orientation and a magnitude. An orientation by itself doesn't have three dimensions. The surface of a sphere is a two-dimensional space.

> Only if you’re describing orientation as two orthogonal rotations.

No, the space has the dimensionality it has. You may choose to describe a 3D orientation with more than two numbers, but you won't stop it from being a two-dimensional quantity that way. If you use more than two numbers, those numbers will fail to be independent of each other.

Orientation would conventionally be a member of SO(3), which is a 3-dimensional manifold.

Your comment is essentially correct if you replace the word "orientation" with "direction", though.

Direction in 3D has two degrees of freedom, orientation has three.

I think a better word than orientation is "axis". They are conceptually similar to axis-angle, with some added sugar to make an algebra

What happens if rotation is zero? Won't (ai +bj+ ck) get multiplied by zero then, and the orientation information is lost?

If you click through to the interactive example, you can try it! You'll see that the if the above equations are `q`, then full formula to rotate a vector `v` is `v' = q * v * q^-1`

Sorry, what you have stated is not very clear. If you use cos(theta) + sin(theta)*(ai + bj + ck) representation as you mentioned, what does happen to the orientation information when sin(theta) becomes zero

When theta=0, you have

  q = cos(0) + sin(0)*(ai+bj+ck) = 1 + 0 = 1

That means applying the "rotation" to the vector gives

  v' = q * v * q^-1 = 1 * v * 1^-1 = v

So the output ("rotated") vector is the same as the input, as you would expect for a rotation by 0.

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