Translation (geometry)
Matrix representation

A translation is an affine transformation with no fixed points. Matrix multiplications always have the origin as a fixed point. Nevertheless, there is a common workaround using homogeneous coordinates to represent a translation of a vector space with matrix multiplication: Write the 3-dimensional vector v = (v x, v y, v z) {\displaystyle \mathbf {v} =(v_{x},v_{y},v_{z})} using 4 homogeneous coordinates as v = (v x, v y, v z, 1) {\displaystyle \mathbf {v} =(v_{x},v_{y},v_{z},1)}.

To translate an object by a vector v {\displaystyle \mathbf {v} }, each homogeneous vector p {\displaystyle \mathbf {p} } (written in homogeneous coordinates) can be multiplied by this translation matrix:

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

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

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

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