In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. It can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. Each translation is an isometry.

Matrix representation

A translation cannot be accomplished using a 3-by-3 matrix, so homogeneous coordinates are normally used.

To translate an object by a vector v = (v_x, v_y, v_z), each homogeneous vector p = (p_x, p_y, p_z, 1) would need to be multiplied with this translation matrix:

<math> T_v =

\begin{bmatrix} 1 & 0 & 0 & v_x \\ 0 & 1 & 0 & v_y \\ 0 & 0 & 1 & v_z \\ 0 & 0 & 0 & 1 \end{bmatrix} <math>

As shown below, the multiplication will give the expected result:

<math>

T_v p = \begin{bmatrix} 1 & 0 & 0 & v_x \\ 0 & 1 & 0 & v_y \\ 0 & 0 & 1 & v_z \\ 0 & 0 & 0 & 1 \end{bmatrix}

\begin{bmatrix} p_x \\ p_y \\ p_z \\ 1 \end{bmatrix} = \begin{bmatrix} p_x + v_x \\ p_y + v_y \\ p_z + v_z \\ 1 \end{bmatrix} <math>

The inverse of a translation matrix can be obtained by negating the vector:

<math>T^{-1}_v = T_{-v}<math>

See also

• Translation operator
• Affine transformation
• Translation (physics)