Home > Translation (geometry)

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.

1 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:

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

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

2 See also

• Translation operator
• Affine transformation
• Translation (physics)