## Linear Transformation (2D)

### Linear Transformation Calculator (2D) — Rotation, reflection, scaling, and shear

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 Then choose the transformation and any parameter if needed (angle, scale factor, etc) A , Transformation type: .. choose one .. Rotation about the origin Reflection Scaling (Contraction/Dilation) Shear B , C , D , E , F , G , H , I , J ,

Please see below for transformation matrices used in the above transformations.

### 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 transformationTransformation 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