# 2 axis angles difference ?

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Well i just encountered following problem: For example I have base rotation of object (angle1 , x1 , y1 , z1). Then object is rotated to the new angle (angle2 , x2 , y2 , z2) , note that x1!=x2, y1 != y2 and z1 != z2 . What I need to find is the difference between angles , and move the object by this difference. How can I do that ? Thanks for any help.

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I don't understand your problem. Please, describe it in more detail. What do you mean by "base rotation"? What do "angle1", "x1", "y1" and "z1" mean?

A particular example with concrete numbers might help too.

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This can be done using quaternions:
quaternion q represents the base rotation/current rotation
quaternion r is the new axis-angle, the one you wish to rotate to
quaternion d is the difference between them, so that
q·d = r
q-1·q·d = q-1·r
d = q-1·r
where q-1 is the inverse of q. You can then build a rotation matrix from this, or transform the point directly using
p' = d · p · d-1
p - vector before transformation
p' - vector after transformation
(note: to multiply a vector with a quaternion, you simply see it as a quaternion with a real component of 0)

For unit quaternions, calculating the inverse simplifies to the conjugate
d = q*·r
p' = d · p · d*

Another way of doing it would be to unrotate (angle1, x1, y1, z1) and then rotate by (angle2, x2, y2, z2). Or to just store the unrotated coordinates.

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There are multiple solutions to this problem, but here is one solution.

Calculate the normal to the plane formed by the two axis. That is the cross product of the unit vectors of (x1,y1,z1) and (x2,y2,z2). Then calculate the angle between the two vectors. The dot product of two unit vectors is the cosine of that angle. If you didn't know, the unit vector is a vector which has a length of 1.0.

Pseudocode:
v1 = (x1, y1, z1) and v2 = (x2, y2, z2) are assumed to have a length of 1.0
v = v1 x v2
( now v is (y1*z2 - z1*y2, z1*x2 - x1*z2, x1*y1 - y1*x2) )
a = acos(v1 - v2)
( a = acos(x1*x2+y1*y2+z1*z2) )
note that a is in radians, per default. OpenGL wants degrees as parameter.
now rotate a degrees around axis v

I did not double check all the formulas, but MathWorld is your friend.

As said above, Quaternions can be used to do the trick. So can many other methods. Quaternions are probably the most powerful, but they require some extra coding if you need just a simple rotation.

-Richardo

EDIT: I am sorry, this is not quite what you wanted. This is a rotation that maps a vector to another vector. Use quaternions. there are plenty of quaternion tutorials on the net. I particularly like the matrix&quaternion faq in the articles section of gamedev.net

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