Derivation by Composition

• Can derive the matrix for angle-axis rotation by composing basic transformations.
• Rotation given by and .
• Assume that .
• General idea: Map onto one of the canonical axes,
  rotate by , map back.

1. Pick the closest axis to using .
   (Assume we chose the x-axis in the following).
2. Project onto in the xz plane:

   

3. Compute and , where is the angle of with the x-axis.

   

4. Use and to create :

   

5. Rotate onto the xy plane using :

   

6. Compute and , where is the angle of with the x-axis.

   

7. Use and to create :

   

8. Rotate onto the x axis using .
9. Rotate about the x-axis by : .
10. Reverse z-axis rotation: .
11. Reverse y-axis rotation: .

The overall transformation is