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Plane with four corner 3-D (x,y,z)

  1. Jul 25, 2005 #1
    If I have a plane with four corner 3-D (x,y,z).. et ceteraa in a coordinate system, how do I translate and rotate it to a new coordinate system orthogonal to it. In other words, do I use the equations/matrices involving sines and cosines to translate the plane?

    For instance...

    I am not sure if I am doing this right, but do I add a fourth axis (x,y,z,w) and use a matrix to rotate it and transform it? Thank you.
  2. jcsd
  3. Jul 26, 2005 #2


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    "Translation" in an ordinary 2 or 3 dimensional coordinate system is not a linear transformation (because it does not map (0,0,0) to itself) and cannot be written as a matrix multiplication. You can, however, use "projective" coordinates. That is, as you said, adding a fourth coordinate, w, (not a fourth axis). For example, to translate a point (x,y,z) by <0, 3, -2> (that is (x,y,z) becomes (x,y+3, z-2)). Write the point as (x,y,z,1) and multiply By the matrix
    [1 0 0 0][x]
    [0 1 0 3][y]
    [0 1 0-2][z]
    [0 0 0 1][1]
    Rotations, about (0,0,0) would use the upper left 3 by 3 area:
    [cos t -sin t 0 0] [x]
    [sin t cos t 0 0] [y]
    [0 0 1 0] [z]
    [0 0 0 1] [1]
    rotates through an angle t about the z-axis.
    In some operations it may be necessary to "renormalize": you may get something like (u, v, w, a) where a is not 1 and must divide each component by a to get back to (x, y, z, 1).
  4. Jul 26, 2005 #3
    Okay, so this is how I did it.
    I have Point P(x,y,z), and I have to rotate it around the z axis. To find the new x,y,z points I use the following equations

    X' = xcos(alpha) + ysin(alpha)
    Y' = -xsin(alpha) + ycos(alpha)

    where alpha is the angle between x and X'

    Then, I have to rotate the point around the X' axis...

    angle beta is the angle from Y' to normal of X'

    Y'' = Y'cos(beta) + Z'sin(beta)
    Z'' = -Y"sin(beta) + Z'cos(beta)
    X'' = X'

    P(X'',Y'',Z'') is the end result. Are my equations correct?

    Basically, I had to rotate some global coordinate axis to a local coordinate axis.

    ._ _ _ _ _ _ > x
    z (pointing out of the page)

    ._ _ _ _ _ _ > y''
    x'' (pointing out of the page)

    Furthermore, are there any equations I can use to project a grid on the axes numerically?

    Thank you
    Last edited: Jul 26, 2005
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