# Edexcel A Level Further Maths: Core Pure:复习笔记2.2.2 Geometric Transformations with Matrices

### 2D Transformations

#### What is meant by a 2D geometric transformation?

• The following transformations can be represented (in 2D) using multiplication of a 2x2 matrix
• 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
• This is given in the formula booklet
• Clockwise through angle θ about the origin
• In both cases
• θ > 0
• θ may be measured in degrees or radians
• Reflection
• In the line
• 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)
• Horizontal stretch (or stretch parallel to the x-axis)
• Scale factor k
• Vertical stretch (or stretch parallel to the y-axis)
• Scale factor k

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

• Let the transformation matrix be
• The image of the point (x, y) after the transformation is (x', y') which can be found by:
• 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)
• (0, 1) is transformed to (b, d)

#### 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 .

a) Find the matrix T that represents a reflection in the line .

b) 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)

#### Exam 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.

a)
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.

b) A square has position matrix .  Tn represents the position matrix of the image square after it has been transformed n times by matrix C.  Find T4

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

### 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),
• 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,
• 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，
• The image of
• The images of  are the 2nd and 3rd columns of T respectively

#### Reflection matrices in 3D

• A reflection in the plane x=0 is given by the 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
• A reflection in the plane z=0 is given by
• 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
• 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
• Notice that the x coordinate is unaffected
• An anticlockwise rotation around the y-axis by angle θ is given by
• 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;

#### Exam 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

#### Worked Example

The 3x3 matrix T, represents an anticlockwise rotation around the x-axis by 120⁰.

a) Find the matrix T.

b) Find the image of point A (2,4,6) under the transformation represented by T.

c) The image of A is now reflected in the plane z=0, and labelled B. Find the coordinates of B.