Fortran77 subroutine for rotation about an arbitrary axis

quaternion_rotation.F

c                 Copyright (c) 2014 by Billy Wayne McCann
c                        All rights reserved.
c
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c     
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c
c-----------------------------------------------------------------------
c-----------------------------------------------------------------------
c
c     This subroutine translates a system of coordinates such that a
c     a given vector within the coordinates originates at the origin
c     (0,0,0) and is aligned to the positive or negative z-axis.  
c
c     P A S S E D  V A R I A B L E S :
c
c     integer n         number of points in the cartesian coordinate arrays
c     real    x, y, z   input cartesian coordinate arrays 
c     integer ii        point to be set at the origin
c     integer jj        point to be aligned to +- z-axis
c     integer idir      +1 = positive z-axis, -1 = negative z-axis
c
c     I N T E R N A L  V A R I A B L E S :
c
c     real    xo,yo,zo  output cartesian coordinate arrays
c-----------------------------------------------------------------------
c-----------------------------------------------------------------------
      subroutine alignz(n, x, y, z, ii, jj, idir, xo, yo, zo)
  
      implicit none

      real x(n), y(n), z(n)
      real xt(n),yt(n),zt(n)
      real xo(n),yo(n),zo(n)

      integer i, ii, jj, n, idir
      real xi, yi, zi
      real init_vector(3), jj_vector(3)
      real z_axis(3)
      real rotaxis(3)
      real rx, ry, rz
      real Q1(4), Q2(4), Q2CONJ(4), Q3(4), Q4(4)
      real magq,magv,phi,PI
c
c     Initialize
c
      jj_vector    = 0.0
      init_vector  = 0.0
      rotaxis      = 0.0
      rx           = 0.0
      ry           = 0.0
      rz           = 0.0
      Q1           = 0.0
      Q2           = 0.0
      Q2CONJ       = 0.0
      Q3           = 0.0
      Q4           = 0.0
      magq         = 0.0
      magv         = 0.0
      PI           = acos(-1.0e0)
c
c     idir determines whether to align to the positive or negative z
c     axis
c
      z_axis = [ 0.0, 0.0, idir*1.0e0 ]
c
c     Remember x(ii), as it will be the origin in the output xo,yo,zo:
c
      xi = x(ii)
      yi = y(ii)
      zi = z(ii)
c
c     Move ii to origin, translating all atoms with it.
c
      do i=1, n
          xt(i) = x(i) - xi
          yt(i) = y(i) - yi
          zt(i) = z(i) - zi
      end do
c
c     Find the angle between jj and the z-axis, as well as the axis of
c     rotation.
c
      jj_vector = [ xt(jj), yt(jj), zt(jj) ]

      magv = sqrt(dot_product(jj_vector, jj_vector)) 
     &     * sqrt(dot_product(z_axis, z_axis))

      phi = acos(dot_product(jj_vector, z_axis)/magv)

      rotaxis = [ z_axis(2)*jj_vector(3) - z_axis(3)*jj_vector(2),
     &            z_axis(3)*jj_vector(1) - z_axis(1)*jj_vector(3),
     &            z_axis(1)*jj_vector(2) - z_axis(2)*jj_vector(1) 
     &          ]

      magv = sqrt(dot_product(rotaxis, rotaxis))
      rotaxis = rotaxis / magv  !normalize
c
c     Assign x,y,z components of rotation axis their own variables
c
      rx = rotaxis(1)
      ry = rotaxis(2)
      rz = rotaxis(3)

c
c     Rotate the cartesion coordinates phi degrees about an arbitrary 
c     axis using a quaternion.
c
c     1. Let Q1 be the initial quaternion containing the initial vector
c
c     2. The rotation quaternion is :
c
c              Q2 = ( cos(phi/2), rx*sin(phi/2), 
c                    ry*sin(phi/2)), rz*sin(phi/2) )
c
c     3. The rotated vector is then the last three components of the
c        quaternion multiplication
c
c              Q2 x Q1 x Q2*
c
c        where Q2* is the complex conjugate of Q2, herein labeled 
c        Q2CONJ.  
c
c     To solve this, the quaternion multiplication is broken down into
c     two steps:
c
c             1. Q1 x Q2CONJ ==> Q3
c             2. Q2 x Q3     ==> Q4
c
c     The new cartesian coordinates are then
c
c              Q4(2:4)     ==> x, y, z
c
      Q2(1) = cos(phi/2.)
      Q2(2) = rx*sin(phi/2.)
      Q2(3) = ry*sin(phi/2.)
      Q2(4) = rz*sin(phi/2.)

      Q2CONJ = [ Q2(1), -1.0e0*Q2(2), -1.0e0*Q2(3), -1.0e0*Q2(4) ]

      magq = sqrt( Q2(1)**2 + Q2(2)**2 + Q2(3)**2 + Q2(4)**2 )
      Q2 = Q2 / magq    !normalize

      do i=1, n
          Q1 = [0.0, xt(i), yt(i), zt(i) ]

          Q3(1) = Q1(1)*Q2CONJ(1) - Q1(2)*Q2CONJ(2)
     &          - Q1(3)*Q2CONJ(3) - Q1(4)*Q2CONJ(4)
          Q3(2) = Q1(1)*Q2CONJ(2) + Q1(2)*Q2CONJ(1)
     &          - Q1(3)*Q2CONJ(4) + Q1(4)*Q2CONJ(3)
          Q3(3) = Q1(1)*Q2CONJ(3) + Q1(2)*Q2CONJ(4)
     &          + Q1(3)*Q2CONJ(1) - Q1(4)*Q2CONJ(2)
          Q3(4) = Q1(1)*Q2CONJ(4) - Q1(2)*Q2CONJ(3)
     &          + Q1(3)*Q2CONJ(2) + Q1(4)*Q2CONJ(1)
  
          Q4(1) = Q2(1)*Q3(1) - Q2(2)*Q3(2) - Q2(3)*Q3(3) - Q2(4)*Q3(4)
          Q4(2) = Q2(1)*Q3(2) + Q2(2)*Q3(1) - Q2(3)*Q3(4) + Q2(4)*Q3(3)
          Q4(3) = Q2(1)*Q3(3) + Q2(2)*Q3(4) + Q2(3)*Q3(1) - Q2(4)*Q3(2)
          Q4(4) = Q2(1)*Q3(4) - Q2(2)*Q3(3) + Q2(3)*Q3(2) + Q2(4)*Q3(1)

          xo(i) = Q4(2)
          yo(i) = Q4(3)
          zo(i) = Q4(4)
      end do
  
      end subroutine alignz