The angles (called Euler angles or Tait–Bryan angles) are defined by the following procedure of successive rotations: Given two arbitrary coordinate frames A and B, consider a temporary frame T that initially coincides with A. In order to make T align with B, we first rotate T an angle x about its x-axis (common axis for both A and T). Secondly, T is rotated an angle y about the NEW y-axis of T. Finally, T is rotated an angle z about its NEWEST z-axis. The final orientation of T now coincides with the orientation of B. The signs of the angles are given by the directions of the axes and the right hand rule.




a 3x3 rotation matrix (direction cosine matrix) such that the relation between a vector v decomposed in A and B is given by: v_A = R_AB * v_B


x,y,z Angles of rotation about new axes (rad)


Kenneth Gade A Nonsingular Horizontal Position Representation. The Journal of Navigation, Volume 63, Issue 03, pp 395-417, July 2010.

See also


R_AB <- matrix( c( 0.9980212 , 0.05230407, -0.0348995 , -0.05293623, 0.99844556, -0.01744177, 0.03393297, 0.01925471, 0.99923861), nrow = 3, ncol = 3, byrow = TRUE) R2xyz(R_AB)
#> [1] 0.01745329 -0.03490659 -0.05235987