---
title: 3D Mathematics
categories: session
geometry: margin=2cm
fontsize: 12pt
---

+ [Previous: [Introduction]()]-[Up: [Overview]()]-[Next: [3D Objects in Python]()]

# Reading
 
+ Ma (2004) Chapter 2 until Section 2.3 inclusive
+ Ma (2004) Appendix A if necessary to understand Chapter 2.  
+ (Szeliski 2022 Chapter 2 until Section 2.1.3 inclusive)
    - Szeliski is a lot briefer, for better or for worse

# Briefing

1.  [Change of Basis]()

# Exercises

## A stage turntable

Let's try a little 2D exercise before we move to 3D.
Consider a theatre stage with a turntable.

The global co-ordinate system is defined with the origin
at a seat in the middle of the audience.
The $y$-axis points directly towards the stage, and the $x$-axis
is perpendicular, pointing towards the right hand side.

The turntable has its centre at $(0,10)$ in the global co-ordinate system.
The centre of the turntable is also the origin of its local co-ordinate
system.
An actor stands on the turntable at the point $(0,1)$ in the local
co-ordinate system.

1. Draw and annotate the situation.
2. Suppose first that the turntable is turned so that its local $y$-axis
  is aligned with the global one.  Where is the actor located in the globale
  co-ordinate system
3. Suppose the turntable turns clockwise by an angle $\alpha$.
   Write down the corresponding rotation matrix.
4. What becomes the new position of the actor in the global co-ordinate system?
5. An actress was positioned at some local co-ordinates $(x,y)$.
   What is her global position before and after the rotation by $\alpha$.
6. Note that you can calculate the positions of the two actors either directly
   by simple geometric observations and by means of linear algebra and a change
   of basis.  You should try both and compare the results for validation.

## A crane

![Crane illustration](crane2.png)

Imagine a crane or robot arm with two booms as shown in the figure. 
We want to calculate the position of the hand of the arm (i.e. point $A$),
given the position of the two joints.

+ The first boom
    - has its base in the global origin.
    - can rotate around the $z$-axis (the vertical axis).
    - has length $a$ and extends along the $z$-axis.
+ The second boom
    - has its base at the tip of the first boom.
    - has length $b$.
    - is jointed so that it can rotate around the $y$-axis.
      Note that this is the $y$-axis in the local co-ordinate
      system of the first boom, which may rotate in the global co-ordinate system.
+ Write $\alpha$ for the angle of rotation around the base.
  Assume that the boom is in the $xz$-plane when $\alpha=0$. 
+ Write $\beta$ for the angle in the joint.
  Suppose the second boom extends vertically when $\beta=0$.

Note that each joint and boom can be described in a local co-ordinate system
induced by the preceeding joints and booms in the system.

### Step 1.  Some concrete numbers

Suppose $a=1$, $b=2$, and $\alpha=0$.  Let $\beta=\pi/4$ so that the
second boom is horisontal.  Draw this situation and calculate the
global co-ordinates of the hand ($A$).

### General solution

Now we will work with general algebraic values, $a$, $b$, $\alpha$, and $\beta$.
To calculate the position of $A$, we should start with the hand in a local 
co-ordinate system and work backwards towards the base.

### Step 2.  The co-ordinate system of the joint

Point $A$ is the origin in the co-ordinate system of the hand.

1.  Calculate its position of $A$ in the co-ordinate system of the joint.

Note that this comprises a translation along the boom, and a rotation 
by $\alpha$.
You need to specify these two transformations and the order in which 
to apply them, to arrive at the transformation between the two
co-ordinate systems.

Note that each change of co-ordinate system comprises a rotation and a
translation.  If you prefer, you may consider four distinct co-ordinate
systems (instead of two) such that each change of basis makes either 
a rotation *or* a translation.

### Step 3.  The co-ordinate system of the base

1.  Calculate its position of $A$ in the co-ordinate system of the base.

To find $A$ in the global (base) co-ordinate system, we need to
find the transformation between the joint basis and the base basis,
and combine this with the transformation from base 2.

## Properties of the rotation matrix.

This is based on Exercise 2.6 from Ma (2003:38).  

Consider two transformation matrices:

$$
R_1=
\begin{bmatrix}
  \cos \theta & -\sin\theta \\
  \sin\theta & \cos \theta 
\end{bmatrix}
\quad
R_2=
\begin{bmatrix}
  \sin \theta & \cos\theta \\
  \cos\theta & -\sin \theta 
\end{bmatrix}
$$

+ For each matrix, what is the determinant?
+ For each matrix, is it orthogonal?

Consider two points $\vec{a}=(1,1)$ and $\vec{b}=(0,1)$.
Apply each transformation to both points, and draw the six 
points $\vec{a},\vec{b},\vec{a}R_1,\vec{b}R_1,\vec{a}R_2,\vec{b}R_2$
in the same co-ordinate system.

+ Are they both rigid-body transformations (rotation)?
+ If they are not rotations, what are they then?

## Exercises from Ma (2003)

If you have time to spare, when you have done the exercises above,
I recommend the following ones from Ma 2004 page 38ff.

1. Exercise 2.1 a+d.  (See Definition A.12 page 446.)
1. Exercise 2.3.  (See Definition A.13 page 447.)

# Debrief

+ Please ask for solution drafts to be released after the session.