Linear Transformation (3D)

See also: Geometric Linear Transformation (2D), matrix, Simultaneous Linear Equations

Linear Transformation Calculator (3D) — Rotation, reflection, scaling

The calculator below will calculate the image of the points in three-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 contact@idomaths.com. Thank you.

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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:

Rotation

Type of transformationTransformation matrix
Clockwise rotation by an angle θ about the positive x-axis
or
Anti-clockwise rotation by an angle θ about the negative x-axis
  
 
100
0cos θsin θ
0−sin θcos θ
 
  
Clockwise rotation by an angle θ about the negative x-axis
or
Anti-clockwise rotation by an angle θ about the positive x-axis
  
 
100
0cos θ−sin θ
0sin θcos θ
 
  
Clockwise rotation by an angle θ about the positive y-axis
or
Anti-clockwise rotation by an angle θ about the negative y-axis
  
 
cos θ0−sin θ
010
sin θ0cos θ
 
  
Clockwise rotation by an angle θ about the negative y-axis
or
Anti-clockwise rotation by an angle θ about the positive y-axis
  
 
cos θ0sin θ
010
−sin θ0cos θ
 
  
Clockwise rotation by an angle θ about the positive z-axis
or
Anti-clockwise rotation by an angle θ about the negative z-axis
  
 
cos θsin θ0
−sin θcos θ0
001
 
  
Clockwise rotation by an angle θ about the negative z-axis
or
Anti-clockwise rotation by an angle θ about the positive z-axis
  
 
cos θ−sin θ0
sin θcos θ0
001
 
  

Reflection

Type of transformationTransformation matrix
Reflection against the xy-plane
  
 
100
010
00−1
 
  
Reflection against the xz-plane
  
 
100
0−10
001
 
  
Reflection against the yz-plane
  
 
−100
010
001
 
  
Reflection against the origin
  
 
-100
0-10
00-1
 
  

Scaling (Contraction or Dilation)

Type of transformationTransformation matrix
Scaling in all x, y, and z direction by a factor k
  
 
k00
0k0
00k
 
  

Shear

Type of transformationTransformation matrix
Shear parallel to the x-axis by a factor m
  
 
1mm
010
001
 
  
Shear parallel to the y-axis by a factor m
  
 
100
m1m
001
 
  
Shear parallel to the z-axis by a factor m
  
 
100
010
mm1
 
  

Orthogonal Projection

Type of transformationTransformation matrix
Orthogonal projection on the xy-plane
  
 
100
010
000
 
  
Orthogonal projection on the xz-plane
  
 
100
000
001
 
  
Orthogonal projection on the yz-plane
  
 
000
010
001
 
  

Examples:

See also: Geometric Linear Transformation (2D), matrix, Simultaneous Linear Equations