Homogeneous Transformation Matrix Generator

Visualize and generate 4x4 homogeneous transformation matrices effortlessly.

Input Parameters

Rotation Matrix (3x3)

Enter the values for the 3x3 rotation matrix.

Rotation Matrix: $$\begin{pmatrix} r_{11} & r_{12} & r_{13} \\ r_{21} & r_{22} & r_{23} \\ r_{31} & r_{32} & r_{33} \end{pmatrix}$$

Translation Vector (3D)

Enter the X, Y, and Z components of the translation vector.

Translation Vector: $$\vec{t} = \begin{pmatrix} t_x \\ t_y \\ t_z \end{pmatrix}$$

Understanding Homogeneous Transformation Matrices

In 3D computer graphics and robotics, homogeneous transformation matrices are 4x4 matrices used to represent translations, rotations, and scaling in a unified way. They allow for complex transformations to be represented as a single matrix multiplication, making them highly efficient for calculations involving sequences of transformations.

Rotation Matrix (3x3): Defines the orientation of the transformed space relative to the original space.
Translation Vector (3D): Specifies the displacement of the origin of the original space to the origin of the transformed space.
Homogeneous Coordinates: By using 4x4 matrices and appending a '1' to vectors (making them 4D), translations become linear transformations, simplifying calculations.

This tool helps you generate these matrices, which are fundamental in applications like game development, CAD software, and robotic arm control, for positioning and orienting objects in 3D space.

Homogeneous Transformation Matrix Generator

Input Parameters

Rotation Matrix (3x3)

Translation Vector (3D)

Result

Homogeneous Transformation Matrix (4x4)

3D Visualization

Understanding Homogeneous Transformation Matrices