Matrices

Type of transformation Transformation matrix
Clockwise rotation by an angle θ about the origin.
cos θ sin θ
−sin θ cos θ
Counter-clockwise rotation by an angle θ about the origin.
cos θ −sin θ
sin θ cos θ
Reflection against the x-axis.
1 0
0 −1
Reflection against the y-axis.
−1 0
0 1
Scaling (contraction or dilation) in all x and y direction by a factor k.
k 0
0 k
Horizontal shear (parallel to the x-axis) by a factor m.
1 m
0 1
Horizontal shear (parallel to the y-axis) by a factor m.
1 0
m 1

 

Examples:

  • Rotate point A (2,3) clockwise about the origin by an angle 90°.
    x′
    y′
    =
    cos 90° sin 90°
    −sin 90° cos 90°
    x
    y
    x′
    y′
    =
    0 1
    -1 0
    2
    3
    x′
    y′
    =
    3
    -2
  • Reflect point A (-3,4) against the x-axis
    x′
    y′
    =
    1 0
    0 -1
    x
    y
    x′
    y′
    =
    1 0
    0 -1
    -3
    4
    x′
    y′
    =
    -3
    -4