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

Matrices of geometric transformations

How do I find the matrix for a transformation?

• You be asked to find the matrix transformation given an object and its image

  • You need to know three pairs of coordinates of the object and its image

• You can just substitute coordinates of the object $\left(x,y\right)$ and coordinates of the image $\left(x\text{'},y\text{'}\right)$ into the equation $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}e\\ f\end{array}\right)=\left(\begin{array}{c}x\text{'}\\ y\text{'}\end{array}\right)$

  • Each pair of coordinates gives you two equations

  • You can solve the equations to find the unknown values

Transformations with translations

• It is easiest if you know the images of $\left(0,0\right)$, $\left(1,0\right)$ and $\left(0,1\right)$

  • The image of $\left(0,0\right)$ is $\left(e,f\right)$

  • The image of $\left(1,0\right)$ is $\left(a+e,c+f\right)$

  • The image of $\left(0,1\right)$ is $\left(b+e,d+f\right)$

• For example, suppose (0, 0) ⇾ (2, 5), (1, 0) ⇾ (3, -1) and (0, 1) ⇾ (5, 0)

  • $\left(2,5\right)$ is the image of the origin so this is the translation vector

    • Subtract this from the image of the other two points

  • The matrix transformation is $\left(\begin{array}{cc}1& 3\\ -6& -5\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}2\\ 5\end{array}\right)$

Transformations with no translations

• It is easier if you know there are no translations

  • This means that the origin does not change under the transformation

• In these cases, you can quickly find the transformation matrix by seeing where $\left(1,0\right)$ and $\left(0,1\right)$ are mapped to

  • These coordinates are the columns of the matrices

    • $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}1\\ 0\end{array}\right)=\left(\begin{array}{c}a\\ c\end{array}\right)$

    • $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}0\\ 1\end{array}\right)=\left(\begin{array}{c}b\\ d\end{array}\right)$

• For example, suppose (0, 0) ⇾ (0, 0), (1, 0) ⇾ (3, -1) and (0, 1) ⇾ (5, 0)

  • The transformation matrix is $\left(\begin{array}{cc}3& 5\\ -1& 0\end{array}\right)$

What are the matrices for common geometric transformations?

• Most of the following transformation matrices are given in the formula booklet

  • The translation one is not given

• If you are given a line of reflection in the form $y=mx$

  • then solve $\tan \theta =m$