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.

First, enter up to 10 points coordinates Then choose the transformation and any parameter if needed (angle, scale factor, etc)
A, Transformation type:

B,
C,
D,
E,
F,
G,
H,
I,
J,

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

Please report any error to [email protected]. Thank you.



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 θsin θ
−sin θcos θ
 
  
Counter-clockwise rotation by an angle θ about the origin.
  
 
cos θ−sin θ
sin θcos θ
 
  
Reflection against the x-axis.
  
 
10
0−1
 
  
Reflection against the y-axis.
  
 
−10
01
 
  
Scaling (contraction or dilation) in all x and y direction by a factor k.
  
 
k0
0k
 
  
Horizontal shear (parallel to the x-axis) by a factor m.
  
 
1m
01
 
  
Horizontal shear (parallel to the y-axis) by a factor m.
  
 
10
m1
 
  

Examples:

By Jimmy Sie

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