---
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)