AST352L--Spring 1998

Quaternion Rotations

This assignment is DUE on Friday, February 20.

The purpose of this assignment is to give you some practice in quaternion rotations. Since HTML does not have good math capability, I'll have to make some notational adjustments.

I'll use bold face to represent quaternions and plain face for scalars (real numbers, the real part of a quaternion)

Let I, J, K be the unit quaternions in the three directions.

So, IJ=-JI=K, JK=-KJ=I, KI=-IK=J, I²=J²=K²=IJK=-1.

Let a general vector with components x_i, x_j, x_k be written X=x_iI+x_jJ+x_kK.

Let us rotate X through angle f about the following unit quaternions U to obtain X':

1. Calculate X' if U=I.
2. Calculate X' if U=J.
3. Calculate X' if U=K.
4. Compare the results of #1, 2 and 3 with what you would have gotten with the matrix method. That is, verify that the components of the vectors you get with the quaternion method are the same as you would have gotten with the matrix method. Remember, you are rotating the vector here, not the coordinates, whereas the matrices give you the result of rotating the coordinates. Therefore, the angle "phi" of the matrix method in the notes is -f.
5. Calculate X' if U=u_iI+u_jJ+u_kK.
   Note that the sum of the squares of the components of U equals 1.

Important Hint: Remember that

cos²(f/2) = (1+cos f)/2
sin²(f/2) = (1-cos f)/2
2 sin(f/2) cos(f/2) = sin f

Notational Hint: You'll save yourself writing if you write a=cos(f/2), b=sin(f/2) or something similar!