Combined Transformation

5.3

Combined Transformation

How to determine the image and object of a combined transformation?

Let \(P\) be the object and \(P^\prime\) be the image, then

To determine the object when an image is given, the transformation of translation needs to be performed in the opposite direction. For example, translation \(\begin{pmatrix} 2\\ -4 \end{pmatrix}\) becomes \(\begin{pmatrix} -2\\ 4 \end{pmatrix}\). For the transformation of rotation, rotation in clockwise direction will become rotation in anticlockwise direction. For the transformation of enlargement, enlargement with a scale factor \(k=2\) will become reciprocal, that is enlargement with scale factor \[\begin{aligned}k=\frac{1}{2}\end{aligned}.\]An object can perform more than one transformation and will produce an image based on the transformations involved. In general, the combination between the transformation \(A\) and transformation \(B\) can be written as transformation \(AB\) or transformation \(BA\) in the order of the desired transformation.

Combined transformation \(AB\) means transformation \(B\) followed by transformation \(A\).

Example 5

The diagram below shows several triangles drawn on a Cartesian plane. It is given that transformation

\(P=\) Translation \(\begin{pmatrix}3\\1\end{pmatrix}\)

\(Q=\) Rotation of \(90^{\circ}\) anticlockwise at centre \((3,4)\)

\(R=\) Enlargement at centre \((8,0)\) with scale factor \(2\)

Determine the image of triangle \(A\) under the combined transformation:

(a) \(\text{P}^2\) (b) \(\text{RQ}\)

Solution:

a) Combined transformation \(\text{P}^2\) means transformation \(\text{P}\) performed 2 times in a row.

b) Combined transformation \(\text{RQ}\) means transformation \(\text{Q}\) followed by transformation \(\text{R}\).