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3D Rotations using complex numbers

  1. Oct 25, 2013 #1
    I was thinking that if you could use quaternions to rotate an object using quaternion algebra that there might be a way to rotate an object using complex numbers in some fashion. I was looking at quaternion rotation of a vector and the amount of operations seemed to be a lot. Of course it levels out or even out performs matrices with linear interpolation and concatenation of rotations. But anyhow what i decided to do was rotate a vector (x ,y , z) = x + yi + zj

    with the rotation complex numbers
    (a + bi)
    (a + bj)

    a = cos

    b = sin

    Following this algorithm
    (a + bi)(x + yi) = x' + y'i
    (a + bj)(y' + zj) = y'' + z'j

    the new rotated vector is x' + y''i + z'j;
    This works, it rotates the object and preserves the magnitude of the vector.

    What I don't understand is how it is rotating the object.

    When I put in 90 degrees v(1,0,0) goes to v(0,0,1) to (0,-1,0) back to (1,0,0)

    When I put in 90 degrees v(1,1,1) goes to v(-1,-1,1) to (1, -1, -1) to (1,1,1)

    My question is: Is it rotating around an axis? Can it be changed? Is it Euler angles in disguise?

    Why is it the first time it rotates by 90 degrees when I put in 90?

    Then it rotates 180 when I still have 90 degrees in?
  2. jcsd
  3. Oct 25, 2013 #2


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    Science Advisor
    Gold Member

    Sure ... complex numbers are natural for 2D rotations, just as quaternions are natural for 3D rotations.

    But you can also do rotations with matrices; see SO(3).

    The question which you have found, what is the axis of rotation, is to be found in the construction of the above systems.
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