Geometric Transformations with Matrices (Edexcel A Level Further Maths): Revision Note

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

    • 

    • 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)

    • 

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)

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 is 

    • The images of  and  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 

  • 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;