Understanding Oblique Projection and its Geometry

[fact0ry]

I cannot visualise an oblique projection. I understood the orthogonal one:
The orthogonal projection is P=U$$\cdot$$U^*, where U is an orthonormal matrix (basis of a subspace) : U^*$$\cdot$$U=I .
Now the projection of matrix A on U vectors is: P_A=U^*$$\cdot$$A$$\cdot$$U .

For the orthogonal projection, for a vector case, the geometry is simple: you project a vector x on the direction of a vector v, and the projected vector is the one from the origin to the base of the perpendicular from x on the direction v: vv^*x = vk, where k is the length of the projection (the coordinate of x in v direction), k=v^*x = |v|*|x|*cos(v,x)= x*cos (where |v|=1).

Now an oblique projection is defined as:
OP=WV^* , where V^*W=I. So the oblique projection of A is OP_A=V^*AW.

Can you please explain what an oblique projection is? Geometrically I mean. For a vector case first. I'm guessing the projection of vector x should be the vector in the direction of w but up to the point where the direction of v through x intersects the direction of w. But v^*w = I. I cannot see this in a plane .
Not to mention the projection of a matrix (the column vectors of it I guess). A drawing would be nice :D.

 

[Peeter]

One use of these sorts of projections is for working with non-orthonormal basis vectors. In that case it can be convient to work with pairs $e^i \cdot e_j = \delta^i_j$, where the set of vectors $\{e^i\}$ is called a reciprocal frame. The reciprocal frame can be used to calculate coordinates for any vector in the non-orthonormal basis (the one it is reciprocal to). Roughly speaking, dotting with the reciprocal frame computes the projections onto your non-orthonormal basis.

Exactly the sort of oblique projection matrix products you are using above can be used to calculate the set of reciprocal frame vectors.

 

[fact0ry]

Thanks Peeter for your reply.

So in my example, the non-orthonormal basis would be W and V would be its reciprocal frame?
By multiplying V^*AW, I get the coordinates in the non-orthonormal basis W (in cols(W) directions) by dotting V^*A ?

 

[Peeter]

I wasn't actually sure what you meant by the projection of the matrix.

Last year when I was blundering through these ideas I did write up some notes for myself and have them here if you are interested.

http://sites.google.com/site/peeterjoot/geometric-algebra/oblique_proj.pdf

Page 7, 'Projection using reciprocal frame vectors', explains more completely what I was describing above.