Apparently you can associate Octonions with two balls, one with trice the radius of the other, the smaller ball rolling on the surface of the larger one. Kind of obscure a connection, but interesting nonetheless:
http://math.ucr.edu/home/baez/octonions/Quarternions are
kind of like the bi-vector part plus the scalar part of 3D Geometric Algebra. (
kind of, because I think the sign convention is a little different, but the whole thing behaves essentially the same.)
In Geometric Algebra, you basically get all the combinatoric variations for sections in your space. For 3D space you get a total of 2
3=8:
- 1 0D component (scalar)
- 3 1D components (the spacial directions)
- 3 2D components (the three perpendicular planes that can be built from them)
- 1 3D component (representing the volume of the space, also acting like an imaginary unit and a "dual" operator - it turns directions int planes and vice-versa, for instance
The three 2D plane-components are signed and can be put together to describe an arbitrary directed sine of a 3D spacial rotation. (This is also true for Quaternions: the ijk components are equivalents of the sine but for 3D spacial rotations.)
The scalar component, meanwhile, stores the rotation's cosine (also true for Quaternions)
(I am neglecting the size of the four components here: It's only true for Quaternions and Multivectors that are normalized. If they aren't normalized, you'll additionally scale up/down your points with 0 at the origin.)
You can see that these things are correct from how you can find the sine and cosine of the angle between two vectors in your usual tried and true vector algebra:
 = \frac{ v_1 \cdot v_2 }{ \left| v_1 \right| \left| v_2 \right| )
 = \frac{v_1 \times v_2}{ \left| v_1 \right| \left| v_2 \right|)
The difference between the Quarternion- and the GA- convention are simply whether it's that formula for the sine or there should be an extra negative sign. (Cross products anti-commute so changing the order of arguments changes the sign. This is a generalization of the fact that the sine is an anti-symmetric function: Plugging in a negative angle is the same as plugging in the positive one and changing the sign.)
The reason the cosine is unaffected by this is simply because the dot-product commutes - changing the order of arguments changes nothing at all (this, in turn, is a generalization of the cosine being symmetric: plugging in a negative angle changes nothing at all)
The end-effect this has on a given rotation simply is that one turns things clock-wise, the other counter-clockwise. - I can never remember which does which though. It's a trivial change anyway.
I personally really like Geometric Algebra for its clarity. I never got these facts about Quaternions before I learned them through a Geometric Algebra lens. Quaternions are by far not as obscure as they seem once you understand this.
January 05, 2010, 12:36:20 PM
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