Geometric Transformations with Matrices (Edexcel A Level Further Maths: Core Pure)

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Paul

Last updated

2 June 2023

2D Transformations

What is meant by a 2D geometric transformation?

The following transformations can be represented (in 2D) using multiplication of a 2x2 matrix
- rotations (about the origin)
- reflections
- enlargements
- (horizontal) stretches parallel to the x-axis
- (vertical) stretches parallel to the y-axis

What are the matrices for geometric transformations?

Rotation
- Anticlockwise (or counter-clockwise) through angle θ about the origin
  - $(\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix})$
  - This is given in the formula booklet
- Clockwise through angle θ about the origin
  - $(\begin{matrix} \cos θ & \sin θ \\ - \sin θ & \cos θ \end{matrix})$
- In both cases
  - θ > 0
  - θ may be measured in degrees or radians
Reflection
- In the line $y = (\tan θ) x$
  - $(\begin{matrix} \cos 2 θ & \sin 2 θ \\ \sin 2 θ & - \cos 2 θ \end{matrix})$
  - This is given in the formula booklet
- θ may be measured in degrees or radians
- for a reflection in the x-axis, θ = 0° (0 radians)
- for a reflection in the y-axis, θ = 90° (π/2 radians)
Enlargement
- Scale factor k, centre of enlargement at the origin (0, 0)
  - $(\begin{matrix} k & 0 \\ 0 & k \end{matrix})$
Horizontal stretch (or stretch parallel to the x-axis)
- Scale factor k
  - $(\begin{matrix} k & 0 \\ 0 & 1 \end{matrix})$
Vertical stretch (or stretch parallel to the y-axis)
- Scale factor k
  - $(\begin{matrix} 1 & 0 \\ 0 & k \end{matrix})$

How do I find the matrix of a 2D transformation?

Let the transformation matrix be $T = (\begin{matrix} a & b \\ c & d \end{matrix})$
The image of the point (x, y) after the transformation is (x', y') which can be found by:
- $(\begin{matrix} a & b \\ c & d \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} x' \\ y' \end{matrix})$
You can find the values of a, b, c, d by seeing where the points (1, 0) and (0, 1) are transformed
- (1, 0) is transformed to (a, c)
  - $(\begin{matrix} a & b \\ c & d \end{matrix}) (\begin{matrix} 1 \\ 0 \end{matrix}) = (\begin{matrix} a \\ c \end{matrix})$
- (0, 1) is transformed to (b, d)
  - $(\begin{matrix} a & b \\ c & d \end{matrix}) (\begin{matrix} 0 \\ 1 \end{matrix}) = (\begin{matrix} b \\ d \end{matrix})$

Worked example

Triangle PQR has coordinates P(-1, 4), Q(5, 4) and R(2, -1).

The transformation T is a reflection in the line $y = x \sqrt{3}$ .

Find the matrix T that represents a reflection in the line

y = x \sqrt{3}

2-2-2-edx-a-fm-we1a-soltn

Find the position matrix, P’, representing the coordinates of the images of points P, Q and R under the transformation T.

Successive Transformations

The order in which transformations occur can lead to different results – for example a reflection in the x-axis followed by clockwise rotation of 90° is different to rotation first, followed by the reflection.

Therefore, when one transformation is followed by another order is critical.

What is a composite transformation?

A composite function is the result of applying more than one function to a point or set of points

e.g. a rotation, followed by an enlargement

It is possible to find a single composite function matrix that does the same job as applying the individual transformation matrices

How do I find a single matrix representing a composite transformation?

Multiplication of the transformation matrices
However, the order in which the matrices is important
- If the transformation represented by matrix M is applied first, and is then followed by another transformation represented by matrix N
  - the composite matrix is NM
    e. P’ = NMP
    (NM is not necessarily equal to MN)
  - The matrices are applied right to left
  - The composite function matrix is calculated left to right
- Another way to remember this is, starting from P, always pre-multiply by a transformation matrix
  - This is the same as applying composite functions to a value
  - The function (or matrix) furthest to the right is applied first

How do I apply the same transformation matrix more than once?

If a transformation, represented by the matrix T, is applied twice we would write the composite transformation matrix as T²
- T² = TT
This would be the case for any number of repeated applications
- T⁵ would be the matrix for five applications of a transformation
A calculator can quickly calculate T², T⁵, etc
Problems may involve considering patterns and sequences formed by repeated applications of a transformation
- The coordinates of point(s) follow a particular pattern
  - (20, 16) – (10, 8) – (5, 4) – (2.5, 2) …
- The area of a shape increases/decreases by a constant factor with each application

e.g. if one transformation doubles the area then three applications will increase the (original) area eight times (2³)

Examiner Tip

When performing multiple transformations on a set of points, make sure you put your transformation matrices in the correct order, you can check this in an exam but sketching a diagram and checking that the transformed point ends up where it should
You may be asked to show your workings but you can still check that you have performed you matrix multiplication correctly by putting it through your calculator

Worked example

The matrix E represents an enlargement with scale factor 0.25, centred on the origin.
The matrix R represents a rotation, 90° anticlockwise about the origin.

Find the matrix, C, that represents a rotation, 90° anticlockwise about the origin followed by an enlargement of scale factor 0.25, centred on the origin.

2-2-2-edx-a-fm-we2a-soltn

A square has position matrix

T_{0} = (\begin{matrix} 0 & 0 & 256 & 256 \\ 0 & 256 & 256 & 0 \end{matrix})

. T_n represents the position matrix of the image square after it has been transformed n times by matrix C. Find T₄

Find the single transformation matrix that would map T₄ to T₀.

2-2-2-edx-a-fm-we2c-soltn

3D Transformations

Transforming 3D coordinates with matrices

We can apply transformations to coordinates in 3D the same way that we apply them in 2D
We do this by multiplying the transformation matrix by the position vector we wish to transform
Rather than transforming 2x1 matrices (2D position vectors) we are now transforming 3x1 matrices (3D position vectors), $[\begin{matrix} x \\ y \\ z \end{matrix}]$

You can group together coordinates into a larger position matrix

For example, all four vertices of a rectangle in 3D can become a 3x4 position matrix, $[\begin{matrix} x_{1} & x_{2} & x_{3} & x_{4} \\ y_{1} & y_{2} & y_{3} & y_{4} \\ z_{1} & z_{2} & z_{3} & z_{4} \end{matrix}]$
This is helpful as you can transform the entire shape in one matrix multiplication

3D transformations will be confined to

A reflection in one of x=0, y=0, or z=0
A rotation about one of the coordinate axes

As with 2D transformations, the transformation matrix describes how the unit vectors in each direction (i, j, and k) are mapped

For a transformation matrix T, $[\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}]$

The image of $[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}]$ is $[\begin{matrix} a \\ d \\ g \end{matrix}]$
The images of $[\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}]$ and $[\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]$ are the 2^nd and 3^rd columns of T respectively

Reflection matrices in 3D

A reflection in the plane x=0 is given by the matrix $[\begin{matrix} - 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

Notice that the first column, the image of i, is multiplied by -1, or ‘mirrored’, whilst j and k stay the same

A reflection in the plane y=0 is given by $[\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$
A reflection in the plane z=0 is given by $[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 1 \end{matrix}]$
You are not given these transformation matrices in the formula book

Rotation matrices in 3D

An anticlockwise rotation around the z-axis by angle θ is given by $[\begin{matrix} \cos θ & - \sin θ & 0 \\ \sin θ & \cos θ & 0 \\ 0 & 0 & 1 \end{matrix}]$

Notice that the z coordinate of a point would be unchanged by this transformation
This makes it equivalent to a rotation around the origin in 2D

Therefore the top left corner of this matrix is the same as the 2x2 matrix for an anticlockwise rotation around the origin, given in the formula book

An anticlockwise rotation around the x-axis by angle θ is given by $[\begin{matrix} 1 & 0 & 0 \\ 0 & \cos θ & - \sin θ \\ 0 & \sin θ & \cos θ \end{matrix}]$

Notice that the x coordinate is unaffected

An anticlockwise rotation around the y-axis by angle θ is given by $[\begin{matrix} \cos θ & 0 & \sin θ \\ 0 & 1 & 0 \\ - \sin θ & 0 & \cos θ \end{matrix}]$

Notice that the y coordinate is unaffected

When we describe an “anticlockwise rotation around the x/y/z-axis” this is from the perspective of standing on the positive axis in question, looking towards the origin
You are not given these transformation matrices in the formula book

You are however given the matrix for an anticlockwise rotation about the origin in 2D, which may be useful; $[\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}]$

rotation-around-the-y-axis-in-3d-matrix-fixed

Examiner Tip

When describing a rotation, remember to state the axis and direction of rotation
Use your calculator where possible for matrix multiplication, or checking your answer to matrix multiplication if required to show working