Rotational invariance
In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument. == Mathematics == === Functions === For example, the function f ( x , y ) = x 2 + y 2 {\displaystyle f(x,y)=x^{2}+y^{2}} is invariant under rotations of the plane around the origin, because for a rotated set of coordinates through any angle θ x ′ = x cos θ − y sin θ {\displaystyle x'=x\cos \theta -y\sin \theta } y ′ = x sin θ + y cos θ {\displaystyle y'=x\sin \theta +y\cos \theta } the function, after some cancellation of terms, takes exactly the same form f ( x ′ , y ′ ) = x 2 + y 2 {\displaystyle f(x',y')={x}^{2}+{y}^{2}} The rotation of coordinates can be expressed using matrix form using the rotation matrix, [ x ′ y ′ ] = [ cos θ − sin θ sin θ cos θ ] [ x y ] , {\displaystyle {\begin{bmatrix}x'\\y'\\\end{bmatrix}}={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}{\begin{bmatrix}x\\y\\\end{bmatrix}},} or symbolically, x′ = Rx.