Linear Transformation
See also: 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.
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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:
- 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 (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 0 m 1 - Vertical shear (parallel to the y-axis) by a factor m.
1 m 0 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
See also: matrix, Simultaneous Linear Equations
