Linear Transformation (2D)
See also: Geometric Linear Transformation (3D), matrix, Simultaneous Linear Equations
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.
Please see below for transformation matrices used in the above transformations.
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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. |
| |||||||||||||
| Counter-clockwise rotation by an angle θ about the origin. |
| |||||||||||||
| Reflection against the x-axis. |
| |||||||||||||
| Reflection against the y-axis. |
| |||||||||||||
| Scaling (contraction or dilation) in all x and y direction by a factor k. |
| |||||||||||||
| Horizontal shear (parallel to the x-axis) by a factor m. |
| |||||||||||||
| Horizontal shear (parallel to the y-axis) by a factor m. |
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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
See also: Geometric Linear Transformation (3D), matrix, Simultaneous Linear Equations
