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Oblique projection

  1. Apr 3, 2009 #1
    I cannot visualise an oblique projection. I understood the orthogonal one:
    The orthogonal projection is P=U[tex]\cdot[/tex]U*, where U is an orthonormal matrix (basis of a subspace) : U*[tex]\cdot[/tex]U=I .
    Now the projection of matrix A on U vectors is: PA=U*[tex]\cdot[/tex]A[tex]\cdot[/tex]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 OPA=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 :confused: .
    Not to mention the projection of a matrix (the column vectors of it I guess). A drawing would be nice :D.
     
  2. jcsd
  3. Apr 3, 2009 #2
    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 [itex] e^i \cdot e_j = \delta^i_j[/itex], where the set of vectors [itex]\{e^i\}[/itex] 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.
     
  4. Apr 6, 2009 #3
    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 ?
     
  5. Apr 6, 2009 #4
    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.
     
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