# 3D Mathematics

• 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

# 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

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.)
2. Exercise 2.3. (See Definition A.13 page 447.)