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

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)
    • 

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)

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;