It's called "Geometric Algebra" ... On 6/6/13, Henry Baker <hbaker1@pipeline.com> wrote:
The vector cross product in 3D is a very curious construct.
Gibbs found quaternions to be too limiting for his applications, and so he split their product into two pieces -- the dot product & the cross product:
If p,q are 'vector' quaternions (whose scalar component is zero), then
pq = pxq - p.q
Cross products are a kind of 'incomplete' determinant:
| i j k| |ux uy uz| = uxv |vx vy vz|
where i, j, k are formal variables to be substituted later to form the complete 'triple product' via the dot product:
|wx wy wz| |ux uy uz| = w.(uxv) |vx vy vz|
But suppose we didn't know (or care) what the _interpretation_ of uxv was; suppose we only wanted to compute a bunch of 3x3 determinants whose 2nd & 3rd rows were always the constants [ux,uy,uz] and [vx,vy,vz].
Then the computation of these determinants is speeded up by the concept of cross products, because we've already done most of the work by pre-computing the determinant minors. So, in effect, the cross product has acted like a 'cache' for these pre-computed results.
Since 3x3 determinants compute _signed_ areas, they are extremely useful in computer graphics to compute whether some point falls 'above' or 'below' some line segment.
The cross product effectively pre-computes the information from the line segment so that we can test many other points against this line segment.
This also works for higher dimensions. We can test whether a point p is 'above' or 'below' the plane determined by the triangle u,v,w with the following determinant:
|px py pz 1| |ux uy uz 1| |vx vy vz 1| = D |wx wy wz 1|
But we often have to perform this test with a large number of different p's.
This means that if we could _precompute_ the appropriate minors and gather this information up into a 4D 'vector', we could do each test with a simple 4D dot product.
D =
|uy uz 1| |vy vz 1| * px - |wy wz 1|
|ux uz 1| |vx vz 1| * py + |wx wz 1|
|ux uy 1| |vx vy 1| * pz - |wx wy 1|
|ux uy uz| |vx vy vz| = |wx wy wz|
[px,py,pz,1].Dxyz1
where Dxyz1 is the appropriate 4D vector of pre-computed minors.
Note that we don't need to know or care about any interpretation of this vector, other than that it is an 'incomplete' determinant.
Once we have the notion of an 'incomplete' determinant, we can come up with macro code for incomplete determinants of more general kinds, by including formal variables for various arbitrary entries in a determinant.
We leave it as an exercise for some CS major to automatically 'compile' such an incomplete determinant by expanding-by-minors so as to minimize the number of operations left to be performed once the formal variables become bound.
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