Trivia Q: the "w" in a Quaternion?

Quaternions work fine whether I grasp them or not, but I'm just curious: what is the "w"?

Quaternion
Quaternion (float x, float y, float z, float w)

(I looked up quaternion at Wikipedia and not only did I fail to learn anything as a result, I actually forgot several useful facts I used to know.)

 

Quaternions are a form of imaginary number where it is assumed that -1 has three square roots (i, j, k) and not one (i) as per the standard complex numbers.

See:

http://en.wikipedia.org/wiki/Quaternion

(Oops you already did.)

x, y, z, w are basically coefficients:

Since ijk = -1 and rules such as ij = k (etc.) convert all products of i, j, and k back into one of them, any quaternion can be expressed as:

q = x + yi + zj + wk

So the answer is that y, z, and w are all abstract. (i.e. w is just as weird as y and z). x is the "real" part.

Now if you go back to your high school math you may recall that an ordinary imaginary number happens to act a lot like a vector, so you can think of x + yi as being an orientation about a single axis, plus a magnitude.

If the magnitude of the vector is 1, then you can think of it as simply representing a direction. E.g. (0,1) is straight up the y axis.

Quaternions do for orientation about three axes what complex numbers to for orientation about one axis. (And, again, your quaternions will be magnitude 1, or "normalized").

 

Great--I still don't know what w is, and now I don't know what y and z are either. Anyone care to take x away from me?

Actually, your summary is all the answer I was hoping for--thanks!

Imaginary rotations are probably perfect, since I have an imaginary game.

 

Rofl... very nice...

-Jeremy

 

I've read a bit math info found through google and wikipedia about quaternions. (At least I hope I understood it correctly. Please whip me hard and long if I prove to be wrong )

The parts have this meaning:
x: is the cosine of the amount of rotation.
(y ,z ,w): is the axis of rotaion.

Note that the quaternion must be normalized, ie. the magnitude of the quaternion must be 1. ( x^2+y^2+z^2+w^2=1 )

So in order to answer the original question, the w of a Quaternion corresponds with the z part of the axis of rotation. (as does z to y and y to x). Confusing? Yes.

 

Interesting--that made even more sense!

 

Okay... I'm in for a spanking for being wring then, right?

 

Cool would be if i wouldn't have to care about quaterions at all, no matter if i understand them or not.

 

That's exactly the point.

I've done some weird matrices, rotations, etc in my time. I never understood them.

David did the gooball handling - if there ever was a rotation-heavy game, that's it. He doesn't get it either, AFAIK.

Quaternions are just rotations. Think of them as that.

 

Hi Nicholas,

i know what they are but i hardly never had to deal with them directly...

Ahm by the way you smokestuff looks nice. Brute force textures with different alphas your throwing around? I once saw very nice volumetric clouds in a demo. Had no time to think about how it was done so i just looked at it and enjoyed it...

At least time to fire a question now: How are volumetric clouds done?


Regards,

taumel

 

What I learned from GooBall (and am still trying to forget): When you're working with rotations, you need exactly three (3) functions:

Quaternion.AngleAxis - Creates a rotation which rotates X degrees around an axis.
Quaternion.FromToRotation - Creates a rotation which rotates from some direction to another. Usually you use this to rotate a transform so that one of its axes, for example the y-axis - follows a target direction in world space.
Quaternion.LookRotation - Creates a rotation that looks along forward with the the head upwards along some direction.

The result of all these functions is a Quaternion (read: a rotation). Assign it to transform.rotation and see instant action. Or use Quaternion.Slerp to glide towards them (Slerp is a high-precision version of Lerp for Quaternions).

It's all in the docs: http://otee.dk/Documentation/ScriptReference/Quaternion.html#AxisAngle


... I agree there should be a cookbook, with examples of these far-most important operations. Some day


d.

 

The USE of Quaternions is well-documented--plus Transform.RotateAround is a favorite of mine And all the "slerps" have been real time-savers too. I'm used to writing that stuff by hand.

Now, understanding the THEORY is irrelevant, but I got curious And thanks to everyone, I'm understanding them just enough to satisfy my curiosity.

I wonder how much progress I could have made on my game instead of pondering that w?

I will now open the floor to further irrelevant theory discussion--volumetric clouds?

If that's like the yellow sun example someone posted, then I guess it's just a cluster of a lot of alpha textures.