Geometric Linear Transformation (3D)

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 $(x, y, z)$

A

(, , )

B

(, , )

C

(, , )

D

(, , )

E

(, , )

F

(, , )

G

(, , )

H

(, , )

I

(, , )

J

(, , )

Then choose the transformation, enter any parameter if needed (angle, scale factor, etc), and choose the rounding option

Type of transformation

Angle of rotation in degree (enter negative value for anti-clockwise rotation)

Axis of rotation

Reflect against xy-plane

Reflect against xz-plane

Reflect against yz-plane

Reflect against origin

Scaling factor

Shear factor

Shear parallel to the x-axis

Shear parallel to the y-axis

Shear parallel to the z-axis

Rounding:

decimal places

Please report any error to [email protected]

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	$(\begin{matrix} \cos θ & \sin θ \\ - \sin θ & \cos θ \end{matrix})$
Counter-clockwise rotation by an angle $θ$ about the origin	$(\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix})$
Reflection against the $x$ -axis	$(\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})$
Reflection against the $y$ -axis	$(\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix})$
Scaling (contraction or dilation) in both $x$ and $y$ directions by a factor $k$	$(\begin{matrix} k & 0 \\ 0 & k \end{matrix})$
Horizontal shear (parallel to the $x$ -axis) by a factor $m$	$(\begin{matrix} 1 & m \\ 0 & 1 \end{matrix})$
Vertical shear (parallel to the $y$ -axis) by a factor $m$	$(\begin{matrix} 1 & 0 \\ m & 1 \end{matrix})$

Examples:

Rotate point $A (2, 3)$ clockwise about the origin by an angle $90 °$ .
$\begin{aligned} (\begin{matrix} x^{'} \\ y^{'} \end{matrix}) & = (\begin{matrix} \cos 90 ° & \sin 90 ° \\ - \sin 90 ° & \cos 90 ° \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) \\ (\begin{matrix} x^{'} \\ y^{'} \end{matrix}) & = (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) (\begin{matrix} 2 \\ 3 \end{matrix}) \\ (\begin{matrix} x^{'} \\ y^{'} \end{matrix}) & = (\begin{matrix} 3 \\ - 2 \end{matrix}) \end{aligned}$
$A^{'} (3, - 2)$
Reflect point $B (- 3, 4)$ against the $x$ -axis.
$\begin{aligned} (\begin{matrix} x^{'} \\ y^{'} \end{matrix}) & = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) \\ (\begin{matrix} x^{'} \\ y^{'} \end{matrix}) & = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) (\begin{matrix} - 3 \\ 4 \end{matrix}) \\ (\begin{matrix} x^{'} \\ y^{'} \end{matrix}) & = (\begin{matrix} - 3 \\ - 4 \end{matrix}) \end{aligned}$
$B^{'} (- 3, - 4)$

By Jimmy Sie

Geometric Linear Transformation (3D)

Transformation Matrices

Tools

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