Quaternions in 2D

[Frank C.]

I'd like to continue this discussion after being scolded in the "How to handle entity rendering" thread for using quaternions to handle rotations in 2D.

First, I totally understand the objections and the fact that quaternions are overkill for 2D, but as worked to replace my rotation handling with scalar angles I ran into a major problem with interpolation. Here's a quote from the originating thread:

Originally Posted by warmi

If you are dealing with 2d then there is no interpolation problem whatsoever ...

Au contraire! The problem is that angles going from -π to π (or 0 to π.2 depending on your preference) can't simply be linearly interpolated when tweening. You need to find the shortest path and adjust the angles to compensate. Even when I thought I had this working I got bad artifacts, and the code was horribly ugly.

After my frustrations with scalar angles I came up with an interim solution: Use a half-quaternion (I'm making up words again, there's probably a real name for this - complex? spinor?) Anyway, I just took my quaternion code and cut it down to the Z and W components, since X and Y were always zero in the 2D cases. The math is greatly simplified, and the interpolation automagically picks the best/shortest path just like a quaternion.

So, anyone care to share a decent scalar angle interpolation function? Or should I just be happy with the half-quaternion deal I got going?

[Ingemar]

Doesn't a quaternion in 2D simply reduce to ordinary complex numbers?

[AnotherJake]

Originally Posted by Frank C.

After my frustrations with scalar angles I came up with an interim solution: Use a half-quaternion (I'm making up words again, there's probably a real name for this - complex? spinor?) Anyway, I just took my quaternion code and cut it down to the Z and W components, since X and Y were always zero in the 2D cases. The math is greatly simplified, and the interpolation automagically picks the best/shortest path just like a quaternion.

I am terrible at math, but I think Ingemar is right, since ordinary complex numbers should be 2D. As I recall reading about it sometime in my foggy past, the dude that came up with quaternions did what you did, but in reverse -- he was trying to figure out how to extend complex numbers from 2D to 3D. You just trimmed it back down from 3D to 2D, which I *think*, should essentially be the same equation as for a 2D rotation matrix. So theoretically (at least in my limited understanding), what you're doing sounds correct.

[Skorche]

Yep. In the 2D case you just use complex numbers and complex multiplication. Very simple. I use it all over the place in Chipmunk to make things fast and simple.

[Frank C.]

Originally Posted by AnotherJake

I am terrible at math...

Ditto - and even more so...

Thanks for the explanation guys - I recall reading that quaternions were "extended" complex numbers but I never realized how useful plain old complex numbers were for 2D stuff. I should've paid more attention to the Chipmunk source!

[Skorche]

Yep, it's pretty simple. Chipmunk uses the following two functions:

static inline cpVect
cpvrotate(const cpVect v1, const cpVect v2)
{
        return cpv(v1.x*v2.x - v1.y*v2.y, v1.x*v2.y + v1.y*v2.x);
}

static inline cpVect
cpvunrotate(const cpVect v1, const cpVect v2)
{
        return cpv(v1.x*v2.x + v1.y*v2.y, v1.y*v2.x - v1.x*v2.y);
}

Rotation (and inverse) of v1 using v2 as a rotation angle. v2 should be normalized unless you also want it to scale as well.

[Frank C.]

So if I understand correctly, v2 in those cpvrotate functions are just normals (direction vectors)? That's useful in its own right but I ended up handling things a little differently.

I found this page that pretty much describes it: http://www.euclideanspace.com/maths/...orms/index.htm. I'm using the "Alternative (spinor representation)" and I've settled on calling these "spinors" in my math library to avoid ambiguous usage.