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

Matrices of Geometric Transformations

What is meant by a 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

• The following transformations can be represented (in 2D) using addition of a 2x1 matrix

  • translations

What are the matrices for geometric transformations?

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

• Rotation

  • Anticlockwise (or counter-clockwise) through angle θ about the origin

    • 

  • Clockwise through angle θ about the origin

    • 

  • In both cases

    • θ > 0

    • θ may be measured in degrees or radians

• Reflection

  • In the line 

    • 

  • 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

    • 

• Translation (vector)

  • p units in the (positive) x-direction

  • q units in the (positive) y direction

    • 

    • This is not given in the formula booklet

How do I solve problems involving geometric transformations?

• The matrix equations involved in problems will be of the form

  • P’=AP or

  • P’=AP+b where b is a translation vector

    • (sometimes called an affine transformation)

  • where

    • P is the position vector of the object coordinates

    • P’ is the position vector of the image coordinates

    • A is the transformation matrix

    • b is a translation vector

• Problems may ask you to

  • find the coordinates of point(s) on the image

  • find the coordinates of point(s) on the object using an inverse matrix (A^-1)

  • deduce/identify a matrix corresponding to one of the common geometric transformations

    • E.g. Find the matrix of a rotation of 45° clockwise about the origin