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 x, d y), as: [ T ( ϕ ϕ , d ) ] = [ A ( ϕ ϕ ) d 0 1 ] = [ cos ⁡ ⁡ ϕ ϕ − − sin ⁡ ⁡ ϕ ϕ d x sin ⁡ ⁡ ϕ ϕ cos ⁡ ⁡ ϕ ϕ d y 0 0 1 ] . These homogeneous transforms perform rigid transformations on the points in the plane z = 1, that is, on points with coordinates r = (x, y, 1).