Let R be an orthogonal 3×3 matrix with determinant 1; so R represents a general rotation of 3-dimensional space about an axis through the origin. Then there exist Euler angles α‎, β‎, γ‎ in the ranges

such that R = X(α‎)Y(β‎)X(γ‎) where X(θ‎) denotes rotation about the x-axis by θ‎ and Y(θ‎) denotes rotation about the y-axis by θ‎. Euler angles are useful in describing rotations of rigid bodies.