Euler angles/Code

Fortran-77 subroutine to compute the Euler factorization of a proper orthogonal 3&times;3 matirx A. Routine returns Euler angles of the factorization

\mathbf{A} = \mathbf{R}_z(\alpha) \mathbf{R}_y(\beta)\mathbf{R}_z(\gamma) $$ Subroutine Euler(A, alpha,beta, gamma)

C    Determine Euler angles of the proper rotation matrix A. C     All conventions according to Biedenharn & Louck C    Author P.E.S. Wormer 1985

implicit real*8(a-h,o-z) parameter (thresh = 1.d-50) dimension A(3,3)

C--Check if matrix is orthogonal with unit determinant. Call Check(A)

C--Get polar angles of third column Call Polar(A(1,3), alpha, beta)

C--Compute gamma Call Comgamma(A, alpha, beta, gamma)

end Subroutine Check(A)

C    Check if matrix is orthogonal with unit determinant.

implicit real*8(a-h,o-z) parameter (thresh = 1.d-12) dimension A(3,3) do i=1,3 do j=1,i-1 t = 0.d0           do k=1,3 t = t + A(i,k)*A(j,k) enddo if (abs(t) .gt. thresh) then write(*,'(1x,a,i1,a,i1,a,d12.5 )') 1        ' Row ',i, ' and ', j, 'non-orthogonal',abs(t) stop 'non-orthogonal' endif enddo t = 0.d0        do k=1,3 t = t + A(i,k)*A(i,k) enddo if (abs(t-1.d0) .gt. thresh) then write(*,'(1x,a,i1,a,d25.15 )') 1     ' Row ',i, ' non-normalized:', t            stop 'non-normalized' endif enddo

t = det(A) if (abs(t-1.d0) .gt. thresh) then if (abs(t+1.d0) .lt. thresh) then write(*,'(//1x,a,1x,d14.6,a)') .    'Non-unit determinant:',t, ' will interchange column 1 and 2' do i=1,3 T = A(i,2) A(i,2) = A(i,1) A(i,1) = T           enddo else write(*,'(//1x,a,d20.10,a)') .     'Non-unit determinant:',t stop ' Determinant' endif endif

end real*8 function det(A)

C    determinant of A

implicit real*8(a-h,o-z) dimension A(3,3), b(3)

B(1) = A(2,2)*A(3,3) - A(3,2)*A(2,3) B(2) = A(3,2)*A(1,3) - A(1,2)*A(3,3) B(3) = A(1,2)*A(2,3) - A(2,2)*A(1,3) det = 0.d0     do i=1,3 det = det + A(i,1)*B(i) enddo end Subroutine Polar(A, alpha, beta)

C    Get polar angles of vector A.

implicit real*8(a-h,o-z) parameter (thresh = 1.d-50) dimension A(3)

R = sqrt( A(1)**2 + A(2)**2 + A(3)**2 ) if ( abs(R) .lt. thresh) then write(*,'(a)') ' zero vector' alpha = 0.d0        beta  = 0.d0         return endif

beta = acos(A(3)/R) cb  = abs (A(3)/R) if ( abs(cb-1.d0) .lt. thresh ) then alpha = 0.d0        return endif alpha = acos( A(1) / (sqrt( A(1)**2 + A(2)**2 ) ) ) if ( A(2) .lt. 0.d0 ) then pi = acos(-1.d0) alpha = 2.d0*pi - alpha endif

end

Subroutine Comgamma(A, alpha,beta, gamma)

C    Compute third Euler angle gamma.

implicit real*8(a-h,o-z) parameter (thresh = 1.d-50) dimension A(3,3), B1(3), B2(3)

cb = cos(beta) sb = sin(beta) ca = cos(alpha) sa = sin(alpha) B1(1) = ca*cb B1(2) = sa*cb B1(3) = -sb B2(1) = -sa B2(2) = ca      B2(3) =  0.d0      cg = 0.d0      sg = 0.d0      do i=1,3 cg = cg + B1(i)*A(i,1) sg = sg + B2(i)*A(i,1) enddo gamma = acos(cg) if ( sg .lt. 0.d0 ) then pi = acos(-1.d0) gamma = 2.d0*pi - gamma endif

end