--- title: More Mathematics for 3D Modelling categories: session geometry: margin=2cm fontsize: 12pt --- 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. # Learning Outcomes + Be able to use homogeneous coordinates to model and manipulate motion in 3D # Briefing and questions + [3D Motion]() (additional notes) + Quaternions. Ma (2004) Appendix 2.A. # Exercises 1. Review the stage turntable exercise from [last week](3D Modelling). Use homogeneous co-ordinates to find the global co-ordinates of the actress (item 5). Check that your calculations match with what you did with heterogeneous co-ordinates. 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. Review the crane exercise from [last week](3D Modelling) and redo it using homogenous co-ordinates. Check that your calculations match regardless of the method used. 4. 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. 1. Exercise 2.7. 1. Exercise 2.10. 2. Ma (2004:40) Exercise 2.14. Hint: start by drawing 2. Ma (2004:40) Exercise 2.13 # Debrief I showed this [sample code](Python/homogeneous.py) in class. It shows one way of dealing with homogeneous co-ordinates together with the plotting tools that we have used. + [Sample Solutions](Solutions/3D Modelling Part II)