Both circular rotation [b][i]R[/i][/b] and hyperbolic rotation (squeeze) [b][i]H[/i][/b] can be decomposed into triples of parabolic rotations (shears) [b][i]S[sub]x[/sub][/i][/b], [b][i]S[sub]y[/sub][/i][/b].[br][b]x-y-x triples:[br][/b][list][*][math]R(θ)=S_x(−\tan(θ/2))⋅S_y(\sin(θ))⋅S_x(−\tan(θ/2))[/math][br][/*][*][math]H(ψ)=S_x(\tanh(ψ/2))⋅S_y(\sinh(ψ))⋅S_x(\tanh(ψ/2))[/math][br][/*][/list][b]y-x-y triples:[br][/b][list][*][math]R(θ)=S_y(\tan(θ/2))⋅S_x(-\sin(θ))⋅S_y(\tan(θ/2))[/math][br][/*][*][math]H(ψ)=S_y(\tanh(ψ/2))⋅S_x(\sinh(ψ))⋅S_y(\tanh(ψ/2))[/math][br][/*][/list]These are so beautiful with symmetries![br][list][*]H = [b]tanh ⋅ sinh ⋅ tanh[/b] for both triples[/*][*]R = (-/+)[b]tan[/b] ⋅ (+/-)[b]sin[/b] ⋅ (-/+)[b]tan[/b] for x-t-x / y-x-y[/*][/list]Here [math]S_x(k_x)[/math], [math]S_y(k_y)[/math] are X & Y shears with shear factors [i]k[sub]x[/sub], k[sub]y[/sub][/i] (parabolic angles), and [i]R[/i]([i]θ[/i]), [i]H[/i]([i]ψ[/i]) are circular and hyperbolic rotations with angles [i]θ[/i], [i]ψ[/i].