See also: Geometric Linear Transformation (3D), matrix, Simultaneous Linear Equations


The calculator below will calculate the image of the points in two-dimensional space after applying the transformation.

First, enter up to 10 points coordinates xy

A
(, )
B
(, )
C
(, )
D
(, )
E
(, )
F
(, )
G
(, )
H
(, )
I
(, )
J
(, )

Then choose the transformation, enter any parameter if needed (angle, scale factor, etc), and choose the rounding option

°
decimal places

Please report any error to [email protected]


Transformation Matrices

The above transformations (rotation, reflection, scaling, and shearing) can be represented by matrices. To find the image of a point, we multiply the transformation matrix by a column vector that represents the point's coordinate.

The transformation matrices are as follows:

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 both x and y directions by a factor k k 0 0 k
Horizontal shear (parallel to the x-axis) by a factor m 1 m 0 1
Vertical shear (parallel to the y-axis) by a factor m 1 0 m 1

Examples:

  • Rotate point A23 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

    A32

  • Reflect point B34 against the x-axis.

    x y = 1 0 0 1 x y x y = 1 0 0 1 3 4 x y = 3 4

    B34

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By Jimmy Sie

See also: Geometric Linear Transformation (3D), matrix, Simultaneous Linear Equations