information:extraction, quality:corruption
Matrix representation

The combination of a rotation and translation in the plane R can be represented by a certain type of 3×3 matrix known as a homogeneous transform. The 3×3 homogeneous transform is constructed from a 2×2 rotation matrix A (φ) and the 2×1 translation vector d = (d, d), as: = =. {\displaystyle ={\begin{bmatrix}A(\phi)&\mathbf {d} \\\mathbf {0} &1\end{bmatrix}}={\begin{bmatrix}\cos \phi &-\sin \phi &d_{x}\\\sin \phi &\cos \phi &d_{y}\\0&0&1\end{bmatrix}}.} These homogeneous transforms perform rigid transformations on the points in the plane z = 1, that is, on points with coordinates r = (x, y, 1).

In particular, let r define the coordinates of points in a reference frame M coincident with a fixed frame F. Then, when the origin of M is displaced by the translation vector d relative to the origin of F and rotated by the angle φ relative to the x-axis of F, the new coordinates in F of points in M are given by: P = r =. {\displaystyle \mathbf {P} =\mathbf {r} ={\begin{bmatrix}\cos \phi &-\sin \phi &d_{x}\\\sin \phi &\cos \phi &d_{y}\\0&0&1\end{bmatrix}}{\begin{bmatrix}x\\y\\1\end{bmatrix}}.}

Homogeneous transforms represent affine transformations. This formulation is necessary because a translation is not a linear transformation of R. However, using projective geometry, so that R is considered a subset of R, translations become affine linear transformations.