This will be the last session on 3D Modelling, designed to tie up loose ends. It contents will therefore depend on the challenges encountered in the three first sessions.

Briefing and questions

• 3D Motion (additional notes)
• Quaternions. Ma (2004) Appendix 2.A.

Exercises

1. Review the first two exercises from last week and redo them using homogenous co-ordinates; that is the theatre turntable and the two-boom crane. Check that your calculations match regardless of the method used.

2. Given a rotational matrix \[ \begin{bmatrix} \cos(\pi/6) & -\sin(\pi/6) & 0 \\ \sin(\pi/6) & \cos(\pi/6) & 0 \\ 0 & 0 & 1 \end{bmatrix} \] and a translation \(\vec{v}=[1,0,2]\).

   What are the homogenous matrices describing each of the following operations:
   • rotate by \(R\) and then translate by \(\vec{v}\)
   • translate by \(\vec{v}\) and then rotate by \(R\)

3. Suppose you have rotated by \(R\) and then translated by \(\vec{v}\) as given in the previous exercise. What is the homogeneous matrix to undo this operation?

Exercises from Ma (2004)

1. Exercise 2.11. To calculate eigenvalues and -vectors in Python, you can use numpy.linalg.eig.
2. Exercise 2.7.
   
3. Exercise 2.10.
   
4. Ma (2004:40) Exercise 2.14. Hint: start by drawing
5. Ma (2004:40) Exercise 2.13