Translation
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 and thus to make it linear. Thus we write the 3-dimensional vector w = (wx, wy, wz) using 4 homogeneous coordinates as w = (wx, wy, wz, 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:
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).
Read more about this topic: 2D Computer Graphics
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—Paul Goodman (19111972)