Matrices of Composite Transformations (DP IB Applications & Interpretation (AI)): Revision Note

Matrices of Composite 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 GDC 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)