Homogeneous coordinates
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Rotation of n-dimensional coordinates can be described by an n-by-n matrix. All the components of the matrix are multiplied with one of the coordinates. So the transformation is dependent on the original coordinates. Translation however is actually a vector addition and so independent of the original coordinates (see subarticle about translation).
To be able to also apply a translation by matrix multiplication, an additional dimension is introduced. For three-dimensional space (which is usually regarded in robotics), the homogeneous transformation matrix is a 4-by-4 matrix. consists of the 3-by-3 rotation matrix and a 3-by-1 translation vector combined with the last row of the identity matrix:
Let the components of the rotation matrix and be denoted as follows:
to the standard transformation matrix matrix, where the top left n-by-n submatrix The additional component of vectors is always a 1. So the multiplication for for example results in:
As can be seen in the resulting vector, the components and are independent of the original coordinates and just added to the x- or the y-coordinate, respectively.