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Coordinate Rotation in a Cartesian 3-Space

  1. Jun 1, 2008 #1
    I have been trying to derive a set of equations for a new Cartesian coordinate system after a rotation of an original coordinate system. This is what I did:

    1) I transformed the Cartesian coordinates (x,y,z) into spherical coordinates (r,p,q):
    x= r cos(q) cos(p)
    y= r cos(q) sin(p)
    z= r sin (q)

    2) The coordinates are to be rotated by angles of p0 and q0 so that:

    p'= p-p0
    q'= q-q0
    r'= r

    3) Substitution:

    x'= r cos(q-q0) cos (p-p0)
    y'= r cos(q-q0) sin (p-p0)
    z'= r sin(q-q0)

    4) Simplifying and substituting the original values of x, y, and z:

    x'= (r cos(q) cos(q0) + r sin(q) sin(q0)) (cos(p) cos(p0) + sin(p) sin(p0))
    = r cos(q) cos(q0) cos(p) cos(p0) + r cos(q) cos(q0) sin(p) sin(p0) + r sin(q) sin(q0)cos(p) cos(p0) + r sin(q) sin(q0) sin(p) sin(p0)
    = x cos(q0) cos(p0) + y cos(q0) sin(p0) + z sin(p0)cos(p) cos(p0) + z sin(q0) sin(p)
    sin(p0)

    y'=(r cos(q) cos(q0) + r sin(q) sin(q0)) (sin(p) cos(p0) - cos(p) sin(p0))
    =r cos(q) cos(q0) sin(p) cos(p0) - r cos(q) cos(q0) cos(p) sin(p0) + r sin(q) sin(q0)
    sin(p) cos(p0) - r sin(q) sin(q0) cos(p) sin(p0)
    =y cos(q0) cos(p0) - x cos(q0) sin(p0) + z sin(q0) sin(p) cos(p0) - z sin(q0) cos(p)
    sin(p0)

    z'= r sin(q) cos(q0) - r cos(q) sin(q0)
    = z cos(q0) - r cos(q) sin(q0)

    This is as far as I got, but in the equations for x and y, there are still some sin(p)'s and
    cos(p)'s left in there which cannot be evaluated without the original coordinates and I want to find a coordinate-independent set of equations so that the same equations can be used for every point in the original coordinate system.

    My question is: Is there any way to get rid of these sine's and cosine's? Or do you see anything that I could have done wrong or different?

    If you know any simpler equations, please send them to me.

    Thanks.
     
  2. jcsd
  3. Jun 1, 2008 #2
    The spherical coordinates are useful only for the rotation around z-axis. With your notation, that means translation of variable p. For more general rotations, the spherical coordinates are making things only more difficult.

    Here's something about rotations in three dimensions: Elements of SO(3)?
     
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