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 = [ T ( ϕ ϕ , d ) ] r = [ cos ⁡ ⁡ ϕ ϕ − − sin ⁡ ⁡ ϕ ϕ d x sin ⁡ ⁡ ϕ ϕ cos ⁡ ⁡ ϕ ϕ d y 0 0 1 ] [ x y 1 ] .