Translation (geometry)

In Euclidean geometry, a translation is moving every point a constant distance in a specified direction. It is one of the rigid motions (other rigid motions include rotation and reflection). A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system.

If v is a fixed vector, then the translation T_v will work as T_v(p) = p + v.

If T is a translation, then the image of a subset A under the function T is the translate of A by T. The translate of A by T_v is often written A + v.

In an Euclidean space, any translation is an isometry. The set of all translations form the translation group T, which is isomorphic to the space itself, and a normal subgroup of Euclidean group E(n ). The quotient group of E(n ) by T is isomorphic to the orthogonal group O(n ):

E(n ) / T ≅ O(n ).

Matrix representation

Since a translation is an affine transformation but not a linear transformation, homogeneous coordinates are normally used to represent the translation operator by a matrix. Thus we write the 3-dimensional vector w = (w_x, w_y, w_z) using 4 homogeneous coordinates as w = (w_x, w_y, w_z, 1).

To translate an object by a vector v, each homogeneous vector p (written in homogeneous coordinates) would need to be multiplied by this translation matrix:

<math> T_{\mathbf{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_{\mathbf{v}} \mathbf{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}= \mathbf{p} + \mathbf{v} . \! </math>

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

<math> T^{-1}_{\mathbf{v}} = T_{-\mathbf{v}} . \! </math>

Similarly, the product of translation matrices is given by adding the vectors:

<math> T_{\mathbf{u}}T_{\mathbf{v}} = T_{\mathbf{u}+\mathbf{v}} . \! </math>

Because addition of vectors is commutative, multiplication of translation matrices is therefore also commutative (unlike multiplication of arbitrary matrices).

See also

• Translation (physics)
• Translational symmetry

External links

• Translation Transform at cut-the-knot
• Geometric Translation (Interactive Animation) at Math Is Fun

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