User:Blackbombchu/Sandbox/Conditions of origin rotation

Theorem
A transformation T from R2 to R2 is an origin rotation if and only if
 * There exists real numbers a, b where a2 + b2 = 1 such that for all x, y ∈ R, T(x, y) = (ax - by, ay + bx)

Proof
For every transformation T from R2 to R2, there exists exactly one transformation S from R2 to R2 such that T ∘ C = C ∘ S and S = C-1 ∘ T ∘ C = C ∘ T ∘ C. By definition, T is not an origin rotation unless there exists real numbers a, b such that for all x, y ∈ R, T(x, y) = (ax - by, ay + bx). Suppose there do exist such real numbers a, b for T, then C ∘ T = T-1 ∘ C if and only if S = T-1 but S = T-1 if and only if T ∘ S = I = S ∘ T. T(x, y) = (ax - by, ay + bx) so then S(x, y) = (ax - b(-y), -(a(-y) + bx)) = (ax + by, ay - bx). Finally T ∘ S = (a(ax + by) - b(ay - bx), a(ay - bx) + b(ax + by)) = (a2x + b2x, a2y + b2y) = ((a2 + b2)x, (a2 + b2)y) = (a2x + b2x, a2y + b2y) = (a(ax - by) + b(ay + bx), a(ay + bx) - b(ax - by)) = S ∘ T, but ((a2 + b2)x, (a2 + b2)y) = (x, y) for all x, y if and only if a2 + b2 = 1 so T is an origin rotation if and only if a2 + b2 = 1