Generalization of Euler Angles to N-Dimensional Orthogonal Matrices

An algorithm is presented whereby an N‐dimensional orthogonal matrix can be represented in terms of ½N(N − 1) independent parameters ${\theta }_k^{\mathrm{(\nu )}}\text{[ν=2,3,…,N;k=1,2,…,(ν−1)].}$ The parameters have the character of angles, whose compact domains are defined in a manner such that there exists a one‐to‐one correspondence between the points in the parameter space and the group of orthogonal matrices. Explicit formulas are given which express all matrix elements in terms of the angles, and formulas are given which express the angles in terms of the matrix elements. Special choices of angles give block‐diagonal matrices. For three‐dimensional matrices, the parametrization is equivalent to that of Euler.