DAMASK_EICMD/src/rotations.f90

! ###################################################################
! Copyright (c) 2013-2014, Marc De Graef/Carnegie Mellon University
! Modified      2017-2019, Martin Diehl/Max-Planck-Institut für Eisenforschung GmbH
! All rights reserved.
!
! Redistribution and use in source and binary forms, with or without modification, are 
! permitted provided that the following conditions are met:
!
!     - Redistributions of source code must retain the above copyright notice, this list 
!        of conditions and the following disclaimer.
!     - Redistributions in binary form must reproduce the above copyright notice, this 
!        list of conditions and the following disclaimer in the documentation and/or 
!        other materials provided with the distribution.
!     - Neither the names of Marc De Graef, Carnegie Mellon University nor the names 
!        of its contributors may be used to endorse or promote products derived from 
!        this software without specific prior written permission.
!
! THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" 
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! ###################################################################

!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @author Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH
!> @brief rotation storage and conversion
!> @details: rotation is internally stored as quaternion. It cabe inialized from different 
!> represantations and also returns itself in different representations.
!
!  All methods and naming conventions based on Rowenhorst_etal2015
!    Convention 1: coordinate frames are right-handed
!    Convention 2: a rotation angle ω is taken to be positive for a counterclockwise rotation
!                  when viewing from the end point of the rotation axis towards the origin
!    Convention 3: rotations will be interpreted in the passive sense
!    Convention 4: Euler angle triplets are implemented using the Bunge convention,
!                  with the angular ranges as [0, 2π],[0, π],[0, 2π]
!    Convention 5: the rotation angle ω is limited to the interval [0, π]
!    Convention 6: P = -1
!---------------------------------------------------------------------------------------------------

module rotations
  use prec
  use IO
  use math
  use Lambert
  use quaternions
 
  implicit none
  private

  type, public :: rotation
    type(quaternion), private :: q
    contains
      procedure, public :: asQuaternion
      procedure, public :: asEulerAngles
      procedure, public :: asAxisAnglePair
      procedure, public :: asRodriguesFrankVector
      procedure, public :: asRotationMatrix
      !------------------------------------------
      procedure, public :: fromEulerAngles
      procedure, public :: fromAxisAnglePair
      procedure, public :: fromRotationMatrix
      !------------------------------------------
      procedure, private :: rotRot__
      generic,   public  :: operator(*) => rotRot__
      procedure, public  :: rotVector
      procedure, public  :: rotTensor
      procedure, public  :: misorientation
  end type rotation


contains


!---------------------------------------------------------------------------------------------------
! Return rotation in different represenations
!---------------------------------------------------------------------------------------------------
function asQuaternion(self)

  class(rotation), intent(in) :: self
  real(pReal), dimension(4)   :: asQuaternion
 
  asQuaternion = [self%q%w, self%q%x, self%q%y, self%q%z]

end function asQuaternion
!---------------------------------------------------------------------------------------------------
function asEulerAngles(self)
  
  class(rotation), intent(in) :: self
  real(pReal), dimension(3)   :: asEulerAngles
   
  asEulerAngles = qu2eu(self%q)

end function asEulerAngles
!---------------------------------------------------------------------------------------------------
function asAxisAnglePair(self)

  class(rotation), intent(in) :: self
  real(pReal), dimension(4)   :: asAxisAnglePair
 
  asAxisAnglePair = qu2ax(self%q)

end function asAxisAnglePair
!---------------------------------------------------------------------------------------------------
function asRotationMatrix(self)
 
  class(rotation), intent(in) :: self
  real(pReal), dimension(3,3) :: asRotationMatrix
 
  asRotationMatrix = qu2om(self%q)

end function asRotationMatrix
!---------------------------------------------------------------------------------------------------
function asRodriguesFrankVector(self)

  class(rotation), intent(in) :: self
  real(pReal), dimension(4)   :: asRodriguesFrankVector
 
  asRodriguesFrankVector = qu2ro(self%q)
 
end function asRodriguesFrankVector
!---------------------------------------------------------------------------------------------------
function asHomochoric(self)

  class(rotation), intent(in) :: self
  real(pReal), dimension(3)   :: asHomochoric
 
  asHomochoric = qu2ho(self%q)

end function asHomochoric
 
!---------------------------------------------------------------------------------------------------
! Initialize rotation from different representations
!---------------------------------------------------------------------------------------------------
subroutine fromEulerAngles(self,eu,degrees)

  class(rotation), intent(out)          :: self
  real(pReal), dimension(3), intent(in) :: eu
  logical, intent(in), optional         :: degrees

  if (.not. present(degrees)) then
    self%q = eu2qu(eu)
  else
    self%q = eu2qu(merge(eu*INRAD,eu,degrees))
  endif

end subroutine fromEulerAngles
!---------------------------------------------------------------------------------------------------
subroutine fromAxisAnglePair(self,ax,degrees)

  class(rotation), intent(out)          :: self
  real(pReal), dimension(4), intent(in) :: ax
  logical, intent(in), optional         :: degrees

  if (.not. present(degrees)) then
    self%q = ax2qu(ax)
  else
    self%q = ax2qu(merge([ax(1:3),ax(4)*INRAD],ax,degrees))
  endif

end subroutine fromAxisAnglePair
!---------------------------------------------------------------------------------------------------
subroutine fromRotationMatrix(self,om)

  class(rotation), intent(out)            :: self
  real(pReal), dimension(3,3), intent(in) :: om
 
  self%q = om2qu(om)

end subroutine fromRotationMatrix
!---------------------------------------------------------------------------------------------------


!---------------------------------------------------------------------------------------------------
!> @brief: Rotate a rotation
!> ToDo: completly untested
!---------------------------------------------------------------------------------------------------
function rotRot__(self,r) result(rRot)
    
  type(rotation)              :: rRot
  class(rotation), intent(in) :: self,r
  
  rRot = rotation(self%q*r%q)

end function rotRot__


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief rotate a vector passively (default) or actively
!> @details: rotation is based on unit quaternion or rotation matrix (fallback)
!---------------------------------------------------------------------------------------------------
function rotVector(self,v,active) result(vRot)
    
  real(pReal),                 dimension(3) :: vRot
  class(rotation), intent(in)               :: self
  real(pReal),     intent(in), dimension(3) :: v
  logical,         intent(in), optional     :: active
  
  type(quaternion)  :: q
  logical           :: passive
 
  if (present(active)) then
    passive = .not. active
  else
    passive = .true.
  endif
 
  if (dEq(norm2(v),1.0_pReal,tol=1.0e-15_pReal)) then
    if (passive) then
      q = self%q * (quaternion([0.0_pReal, v(1), v(2), v(3)]) * conjg(self%q) )
    else
      q = conjg(self%q) * (quaternion([0.0_pReal, v(1), v(2), v(3)]) * self%q )
    endif
    vRot = [q%x,q%y,q%z]
  else
    if (passive) then
      vRot = matmul(self%asRotationMatrix(),v)
    else
      vRot = matmul(transpose(self%asRotationMatrix()),v)
    endif
  endif

end function rotVector


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief rotate a second rank tensor passively (default) or actively
!> @details: rotation is based on rotation matrix
!---------------------------------------------------------------------------------------------------
function rotTensor(self,m,active) result(mRot)
  
  real(pReal),                 dimension(3,3) :: mRot
  class(rotation), intent(in)                 :: self
  real(pReal),     intent(in), dimension(3,3) :: m
  logical,         intent(in), optional       :: active
   
  logical           :: passive

  if (present(active)) then
    passive = .not. active
  else
    passive = .true.
  endif
 
  if (passive) then
    mRot = matmul(matmul(self%asRotationMatrix(),m),transpose(self%asRotationMatrix()))
  else
    mRot = matmul(matmul(transpose(self%asRotationMatrix()),m),self%asRotationMatrix())
  endif

end function rotTensor


!---------------------------------------------------------------------------------------------------
!> @brief misorientation
!---------------------------------------------------------------------------------------------------
function misorientation(self,other)
  
  type(rotation)              :: misorientation
  class(rotation), intent(in) :: self, other
  
  misorientation%q = conjg(self%q) * other%q                                                        !ToDo: this is the convention used in math

end function misorientation


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert unit quaternion to rotation matrix
!---------------------------------------------------------------------------------------------------
pure function qu2om(qu) result(om)

  type(quaternion), intent(in)                :: qu
  real(pReal),                 dimension(3,3) :: om
  
  real(pReal)                                 :: qq

  qq = qu%w**2-(qu%x**2 + qu%y**2 + qu%z**2)


  om(1,1) = qq+2.0*qu%x*qu%x
  om(2,2) = qq+2.0*qu%y*qu%y
  om(3,3) = qq+2.0*qu%z*qu%z

  om(1,2) = 2.0*(qu%x*qu%y-qu%w*qu%z)
  om(2,3) = 2.0*(qu%y*qu%z-qu%w*qu%x)
  om(3,1) = 2.0*(qu%z*qu%x-qu%w*qu%y)
  om(2,1) = 2.0*(qu%y*qu%x+qu%w*qu%z)
  om(3,2) = 2.0*(qu%z*qu%y+qu%w*qu%x)
  om(1,3) = 2.0*(qu%x*qu%z+qu%w*qu%y)

  if (P < 0.0) om = transpose(om)

end function qu2om


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert unit quaternion to Euler angles
!---------------------------------------------------------------------------------------------------
pure function qu2eu(qu) result(eu)

  type(quaternion), intent(in)              :: qu
  real(pReal),                 dimension(3) :: eu
  
  real(pReal)                               :: q12, q03, chi, chiInv
  
  q03 = qu%w**2+qu%z**2
  q12 = qu%x**2+qu%y**2
  chi = sqrt(q03*q12)
  
  degenerated: if (dEq0(chi)) then
    eu = merge([atan2(-P*2.0*qu%w*qu%z,qu%w**2-qu%z**2), 0.0_pReal, 0.0_pReal], &
               [atan2(2.0*qu%x*qu%y,qu%x**2-qu%y**2),    PI,        0.0_pReal], &
               dEq0(q12))
  else degenerated
    chiInv = 1.0/chi
    eu = [atan2((-P*qu%w*qu%y+qu%x*qu%z)*chi, (-P*qu%w*qu%x-qu%y*qu%z)*chi ), &
          atan2( 2.0*chi, q03-q12 ), &
          atan2(( P*qu%w*qu%y+qu%x*qu%z)*chi, (-P*qu%w*qu%x+qu%y*qu%z)*chi )]
  endif degenerated
  where(eu<0.0_pReal) eu = mod(eu+2.0_pReal*PI,[2.0_pReal*PI,PI,2.0_pReal*PI])

end function qu2eu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert unit quaternion to axis angle pair
!---------------------------------------------------------------------------------------------------
pure function qu2ax(qu) result(ax)

  type(quaternion), intent(in)              :: qu
  real(pReal),                 dimension(4) :: ax
  
  real(pReal)                               :: omega, s

  if (dEq0(qu%x**2+qu%y**2+qu%z**2)) then
    ax = [ 0.0_pReal, 0.0_pReal, 1.0_pReal, 0.0_pReal ]                                             ! axis = [001]
  elseif (dNeq0(qu%w)) then
    s =  sign(1.0_pReal,qu%w)/sqrt(qu%x**2+qu%y**2+qu%z**2)
    omega = 2.0_pReal * acos(math_clip(qu%w,-1.0_pReal,1.0_pReal))
    ax = [ qu%x*s, qu%y*s, qu%z*s, omega ]
  else
    ax = [ qu%x, qu%y, qu%z, PI ]
  end if

end function qu2ax


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert unit quaternion to Rodrigues vector
!---------------------------------------------------------------------------------------------------
pure function qu2ro(qu) result(ro)
  
  type(quaternion), intent(in)              :: qu
  real(pReal),                 dimension(4) :: ro
  
  real(pReal)                               :: s
  real(pReal), parameter                    :: thr = 1.0e-8_pReal
  
  if (qu%w < thr) then
    ro = [qu%x, qu%y, qu%z, IEEE_value(ro(4),IEEE_positive_inf)]
  else
    s = norm2([qu%x,qu%y,qu%z])
    if (s < thr) then
      ro = [ 0.0_pReal, 0.0_pReal, P, 0.0_pReal]
    else
      ro = [ qu%x/s,  qu%y/s,  qu%z/s, tan(acos(math_clip(qu%w,-1.0_pReal,1.0_pReal)))]
    endif
    
  end if

end function qu2ro


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert unit quaternion to homochoric
!---------------------------------------------------------------------------------------------------
pure function qu2ho(qu) result(ho)

  type(quaternion), intent(in)               :: qu
  real(pReal),                  dimension(3) :: ho
  
  real(pReal)                                :: omega, f

  omega = 2.0 * acos(math_clip(qu%w,-1.0_pReal,1.0_pReal))
  
  if (dEq0(omega)) then
    ho = [ 0.0, 0.0, 0.0 ]
  else
    ho = [qu%x, qu%y, qu%z]
    f  = 0.75 * ( omega - sin(omega) )
    ho = ho/norm2(ho)* f**(1.0/3.0)
  end if

end function qu2ho


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert unit quaternion to cubochoric
!---------------------------------------------------------------------------------------------------
function qu2cu(qu) result(cu)
 
  type(quaternion), intent(in)              :: qu
  real(pReal),                 dimension(3) :: cu

  cu = ho2cu(qu2ho(qu))

end function qu2cu


!---------------------------------------------------------------------------------------------------
!> @author Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH
!> @brief convert rotation matrix to cubochoric
!> @details the original formulation (direct conversion) had (numerical?) issues
!---------------------------------------------------------------------------------------------------
pure function om2qu(om) result(qu)

  real(pReal), intent(in), dimension(3,3) :: om
  type(quaternion)                        :: qu

  qu = eu2qu(om2eu(om))

end function om2qu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief orientation matrix to Euler angles
!---------------------------------------------------------------------------------------------------
pure function om2eu(om) result(eu)

  real(pReal), intent(in), dimension(3,3) :: om
  real(pReal),             dimension(3)   :: eu
  real(pReal)                             :: zeta
  
  if (abs(om(3,3)) < 1.0_pReal) then
    zeta = 1.0_pReal/sqrt(1.0_pReal-om(3,3)**2.0_pReal)
    eu = [atan2(om(3,1)*zeta,-om(3,2)*zeta), &
          acos(om(3,3)), &
          atan2(om(1,3)*zeta, om(2,3)*zeta)]
  else 
    eu = [ atan2( om(1,2),om(1,1)), 0.5*PI*(1-om(3,3)),0.0_pReal ]
  end if

  where(eu<0.0_pReal) eu = mod(eu+2.0_pReal*PI,[2.0_pReal*PI,PI,2.0_pReal*PI])
 
end function om2eu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert orientation matrix to axis angle pair
!---------------------------------------------------------------------------------------------------
function om2ax(om) result(ax)

  real(pReal), intent(in)     :: om(3,3)
  real(pReal)                 :: ax(4)
  
  real(pReal)                 :: t
  real(pReal), dimension(3)   :: Wr, Wi
  real(pReal), dimension(10)  :: WORK
  real(pReal), dimension(3,3) :: VR, devNull, o
  integer                     :: INFO, LWORK, i
  
  external :: dgeev,sgeev
  
  o = om
  
  ! first get the rotation angle
  t = 0.5_pReal * (math_trace33(om) - 1.0)
  ax(4) = acos(math_clip(t,-1.0_pReal,1.0_pReal))
  
  if (dEq0(ax(4))) then
    ax(1:3) = [ 0.0, 0.0, 1.0 ]
  else
    ! set some initial LAPACK variables 
    INFO = 0
    ! first initialize the parameters for the LAPACK DGEEV routines
    LWORK = 20   
  
  ! call the eigenvalue solver
    call dgeev('N','V',3,o,3,Wr,Wi,devNull,3,VR,3,WORK,LWORK,INFO)
    if (INFO /= 0) call IO_error(0,ext_msg='Error in om2ax DGEEV return not zero')
    i = maxloc(merge(1.0_pReal,0.0_pReal,cEq(cmplx(Wr,Wi,pReal),cmplx(1.0_pReal,0.0_pReal,pReal),tol=1.0e-14_pReal)),dim=1) ! poor substitute for findloc
    ax(1:3) = VR(1:3,i)
    where (                dNeq0([om(2,3)-om(3,2), om(3,1)-om(1,3), om(1,2)-om(2,1)])) &
      ax(1:3) = sign(ax(1:3),-P *[om(2,3)-om(3,2), om(3,1)-om(1,3), om(1,2)-om(2,1)])
  endif

end function om2ax


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert rotation matrix to Rodrigues vector
!---------------------------------------------------------------------------------------------------
pure function om2ro(om) result(ro)

  real(pReal), intent(in), dimension(3,3) :: om
  real(pReal),             dimension(4)   :: ro

  ro = eu2ro(om2eu(om))

end function om2ro


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert rotation matrix to homochoric
!---------------------------------------------------------------------------------------------------
function om2ho(om) result(ho)

  real(pReal), intent(in), dimension(3,3) :: om
  real(pReal),             dimension(3)   :: ho

  ho = ax2ho(om2ax(om))

end function om2ho


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert rotation matrix to cubochoric
!---------------------------------------------------------------------------------------------------
function om2cu(om) result(cu)

  real(pReal), intent(in), dimension(3,3) :: om
  real(pReal),             dimension(3)   :: cu

  cu = ho2cu(om2ho(om))

end function om2cu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief Euler angles to unit quaternion
!---------------------------------------------------------------------------------------------------
pure function eu2qu(eu) result(qu)

  real(pReal),      intent(in), dimension(3) :: eu
  type(quaternion)                           :: qu
  real(pReal),                  dimension(3) :: ee
  real(pReal)                                :: cPhi, sPhi

  ee = 0.5_pReal*eu
  
  cPhi = cos(ee(2))
  sPhi = sin(ee(2))

  qu = quaternion([   cPhi*cos(ee(1)+ee(3)), &
                   -P*sPhi*cos(ee(1)-ee(3)), &
                   -P*sPhi*sin(ee(1)-ee(3)), &
                   -P*cPhi*sin(ee(1)+ee(3))])
  if(qu%w < 0.0_pReal) qu = qu%homomorphed() 

end function eu2qu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief Euler angles to orientation matrix
!---------------------------------------------------------------------------------------------------
pure function eu2om(eu) result(om)
  
  real(pReal), intent(in), dimension(3)   :: eu
  real(pReal),             dimension(3,3) :: om
  
  real(pReal),             dimension(3)   :: c, s      
 
  c = cos(eu)
  s = sin(eu)
 
  om(1,1) =  c(1)*c(3)-s(1)*s(3)*c(2)
  om(1,2) =  s(1)*c(3)+c(1)*s(3)*c(2)
  om(1,3) =  s(3)*s(2)
  om(2,1) = -c(1)*s(3)-s(1)*c(3)*c(2)
  om(2,2) = -s(1)*s(3)+c(1)*c(3)*c(2)
  om(2,3) =  c(3)*s(2)
  om(3,1) =  s(1)*s(2)
  om(3,2) = -c(1)*s(2)
  om(3,3) =  c(2)
 
  where(dEq0(om)) om = 0.0_pReal

end function eu2om


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert euler to axis angle
!---------------------------------------------------------------------------------------------------
pure function eu2ax(eu) result(ax)
 
  real(pReal), intent(in), dimension(3) :: eu
  real(pReal),             dimension(4) :: ax
  
  real(pReal)                           :: t, delta, tau, alpha, sigma
  
  t     = tan(eu(2)*0.5)
  sigma = 0.5*(eu(1)+eu(3))
  delta = 0.5*(eu(1)-eu(3))
  tau   = sqrt(t**2+sin(sigma)**2)
  
  alpha = merge(PI, 2.0*atan(tau/cos(sigma)), dEq(sigma,PI*0.5_pReal,tol=1.0e-15_pReal))
  
  if (dEq0(alpha)) then                                                                             ! return a default identity axis-angle pair
    ax = [ 0.0_pReal, 0.0_pReal, 1.0_pReal, 0.0_pReal ]
  else
    ax(1:3) = -P/tau * [ t*cos(delta), t*sin(delta), sin(sigma) ]                                   ! passive axis-angle pair so a minus sign in front
    ax(4) = alpha
    if (alpha < 0.0) ax = -ax                                                                       ! ensure alpha is positive
  end if
 
end function eu2ax


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief Euler angles to Rodrigues vector
!---------------------------------------------------------------------------------------------------
pure function eu2ro(eu) result(ro)

  real(pReal), intent(in), dimension(3) :: eu
  real(pReal),             dimension(4) :: ro
  
  ro = eu2ax(eu)
  if (ro(4) >= PI) then
    ro(4) = IEEE_value(ro(4),IEEE_positive_inf)
  elseif(dEq0(ro(4))) then
    ro = [ 0.0_pReal, 0.0_pReal, P, 0.0_pReal ]
  else
    ro(4) = tan(ro(4)*0.5)
  end if
 
end function eu2ro


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert Euler angles to homochoric
!---------------------------------------------------------------------------------------------------
pure function eu2ho(eu) result(ho)

  real(pReal), intent(in), dimension(3) :: eu
  real(pReal),             dimension(3) :: ho

  ho = ax2ho(eu2ax(eu))

end function eu2ho


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert Euler angles to cubochoric
!---------------------------------------------------------------------------------------------------
function eu2cu(eu) result(cu)

  real(pReal), intent(in), dimension(3) :: eu
  real(pReal),             dimension(3) :: cu

  cu = ho2cu(eu2ho(eu))

end function eu2cu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert axis angle pair to quaternion
!---------------------------------------------------------------------------------------------------
pure function ax2qu(ax) result(qu)
    
  real(pReal),      intent(in), dimension(4) :: ax
  type(quaternion)                           :: qu
  
  real(pReal)                                :: c, s


  if (dEq0(ax(4))) then
    qu = quaternion([ 1.0_pReal, 0.0_pReal, 0.0_pReal, 0.0_pReal ])
  else
    c = cos(ax(4)*0.5)
    s = sin(ax(4)*0.5)
    qu = quaternion([ c, ax(1)*s, ax(2)*s, ax(3)*s ])
  end if

end function ax2qu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert axis angle pair to orientation matrix
!---------------------------------------------------------------------------------------------------
pure function ax2om(ax) result(om)

  real(pReal), intent(in), dimension(4)   :: ax
  real(pReal),             dimension(3,3) :: om
  
  real(pReal)                             :: q, c, s, omc
  integer                                 :: i

  c = cos(ax(4))
  s = sin(ax(4))
  omc = 1.0-c

  forall(i=1:3) om(i,i) = ax(i)**2*omc + c

  q = omc*ax(1)*ax(2)
  om(1,2) = q + s*ax(3)
  om(2,1) = q - s*ax(3)
  
  q = omc*ax(2)*ax(3)
  om(2,3) = q + s*ax(1)
  om(3,2) = q - s*ax(1)
  
  q = omc*ax(3)*ax(1)
  om(3,1) = q + s*ax(2)
  om(1,3) = q - s*ax(2)

  if (P > 0.0) om = transpose(om)

end function ax2om


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert axis angle pair to Euler angles
!---------------------------------------------------------------------------------------------------
pure function ax2eu(ax) result(eu)

  real(pReal), intent(in), dimension(4) :: ax
  real(pReal),             dimension(3) :: eu

  eu = om2eu(ax2om(ax))

end function ax2eu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert axis angle pair to Rodrigues vector
!---------------------------------------------------------------------------------------------------
pure function ax2ro(ax) result(ro)

  real(pReal), intent(in), dimension(4) :: ax
  real(pReal),             dimension(4) :: ro
  
  real(pReal), parameter                :: thr = 1.0E-7
  
  if (dEq0(ax(4))) then
    ro = [ 0.0_pReal, 0.0_pReal, P, 0.0_pReal ]
  else 
    ro(1:3) =  ax(1:3)
    ! we need to deal with the 180 degree case
    ro(4) = merge(IEEE_value(ro(4),IEEE_positive_inf),tan(ax(4)*0.5 ),abs(ax(4)-PI) < thr)
  end if

end function ax2ro


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert axis angle pair to homochoric
!---------------------------------------------------------------------------------------------------
pure function ax2ho(ax) result(ho)

  real(pReal), intent(in), dimension(4) :: ax
  real(pReal),             dimension(3) :: ho
  
  real(pReal)                           :: f
  
  f = 0.75 * ( ax(4) - sin(ax(4)) )
  f = f**(1.0/3.0)
  ho = ax(1:3) * f

end function ax2ho


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert axis angle pair to cubochoric
!---------------------------------------------------------------------------------------------------
function ax2cu(ax) result(cu)

  real(pReal), intent(in), dimension(4) :: ax
  real(pReal),             dimension(3) :: cu

  cu  = ho2cu(ax2ho(ax))

end function ax2cu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert Rodrigues vector to unit quaternion
!---------------------------------------------------------------------------------------------------
pure function ro2qu(ro) result(qu)

  real(pReal),      intent(in), dimension(4) :: ro
  type(quaternion)                           :: qu
  
  qu = ax2qu(ro2ax(ro))

end function ro2qu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert Rodrigues vector to rotation matrix
!---------------------------------------------------------------------------------------------------
pure function ro2om(ro) result(om)

  real(pReal), intent(in), dimension(4)   :: ro
  real(pReal),             dimension(3,3) :: om

  om = ax2om(ro2ax(ro))

end function ro2om


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert Rodrigues vector to Euler angles
!---------------------------------------------------------------------------------------------------
pure function ro2eu(ro) result(eu)

  real(pReal), intent(in), dimension(4) :: ro
  real(pReal),             dimension(3) :: eu
         
  eu = om2eu(ro2om(ro))

end function ro2eu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert Rodrigues vector to axis angle pair
!---------------------------------------------------------------------------------------------------
pure function ro2ax(ro) result(ax)

  real(pReal), intent(in), dimension(4) :: ro
  real(pReal),             dimension(4) :: ax
  
  real(pReal)                           :: ta, angle
  
  ta = ro(4)
  
  if (dEq0(ta))  then 
    ax = [ 0.0, 0.0, 1.0, 0.0 ]
  elseif (.not. IEEE_is_finite(ta)) then
    ax = [ ro(1), ro(2), ro(3), PI ]
  else
    angle = 2.0*atan(ta)
    ta = 1.0/norm2(ro(1:3))
    ax = [ ro(1)/ta, ro(2)/ta, ro(3)/ta, angle ]
  end if

end function ro2ax


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert Rodrigues vector to homochoric
!---------------------------------------------------------------------------------------------------
pure function ro2ho(ro) result(ho)

  real(pReal),        intent(in), dimension(4) :: ro
  real(pReal),                    dimension(3) :: ho
  
  real(pReal)                                  :: f
  
  if (dEq0(norm2(ro(1:3)))) then
    ho = [ 0.0, 0.0, 0.0 ]
  else
    f = merge(2.0*atan(ro(4)) - sin(2.0*atan(ro(4))),PI, IEEE_is_finite(ro(4)))
    ho = ro(1:3) * (0.75_pReal*f)**(1.0/3.0)
  end if

end function ro2ho


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert Rodrigues vector to cubochoric
!---------------------------------------------------------------------------------------------------
function ro2cu(ro) result(cu)

  real(pReal), intent(in), dimension(4) :: ro
  real(pReal),             dimension(3) :: cu

  cu = ho2cu(ro2ho(ro))

end function ro2cu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert homochoric to unit quaternion
!---------------------------------------------------------------------------------------------------
pure function ho2qu(ho) result(qu)

  real(pReal),      intent(in), dimension(3) :: ho
  type(quaternion)                           :: qu

  qu = ax2qu(ho2ax(ho))

end function ho2qu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert homochoric to rotation matrix
!---------------------------------------------------------------------------------------------------
pure function ho2om(ho) result(om)

  real(pReal), intent(in), dimension(3)   :: ho
  real(pReal),             dimension(3,3) :: om

  om = ax2om(ho2ax(ho))

end function ho2om


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert homochoric to Euler angles
!---------------------------------------------------------------------------------------------------
pure function ho2eu(ho) result(eu)

  real(pReal), intent(in), dimension(3) :: ho
  real(pReal),             dimension(3) :: eu

  eu = ax2eu(ho2ax(ho))

end function ho2eu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert homochoric to axis angle pair
!---------------------------------------------------------------------------------------------------
pure function ho2ax(ho) result(ax)

  real(pReal), intent(in), dimension(3) :: ho
  real(pReal),             dimension(4) :: ax
  
  integer                               :: i
  real(pReal)                           :: hmag_squared, s, hm
  real(pReal), parameter, dimension(16) :: &
    tfit = [ 1.0000000000018852_pReal,      -0.5000000002194847_pReal, & 
            -0.024999992127593126_pReal,    -0.003928701544781374_pReal, & 
            -0.0008152701535450438_pReal,   -0.0002009500426119712_pReal, & 
            -0.00002397986776071756_pReal,  -0.00008202868926605841_pReal, & 
            +0.00012448715042090092_pReal,  -0.0001749114214822577_pReal, & 
            +0.0001703481934140054_pReal,   -0.00012062065004116828_pReal, & 
            +0.000059719705868660826_pReal, -0.00001980756723965647_pReal, & 
            +0.000003953714684212874_pReal, -0.00000036555001439719544_pReal ]
  
  ! normalize h and store the magnitude
  hmag_squared = sum(ho**2.0_pReal)
  if (dEq0(hmag_squared)) then
    ax = [ 0.0_pReal, 0.0_pReal, 1.0_pReal, 0.0_pReal ]
  else
    hm = hmag_squared
  
  ! convert the magnitude to the rotation angle
    s = tfit(1) + tfit(2) * hmag_squared
    do i=3,16
      hm = hm*hmag_squared
      s  = s + tfit(i) * hm
    end do
    ax = [ho/sqrt(hmag_squared), 2.0_pReal*acos(s)]
  end if

end function ho2ax


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert homochoric to Rodrigues vector
!---------------------------------------------------------------------------------------------------
pure function ho2ro(ho) result(ro)

  real(pReal), intent(in), dimension(3) :: ho
  real(pReal),             dimension(4) :: ro

  ro = ax2ro(ho2ax(ho))

end function ho2ro


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert homochoric to cubochoric
!---------------------------------------------------------------------------------------------------
function ho2cu(ho) result(cu)

  real(pReal), intent(in), dimension(3) :: ho
  real(pReal),             dimension(3) :: cu

  cu = Lambert_BallToCube(ho)

end function ho2cu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert cubochoric to unit quaternion
!---------------------------------------------------------------------------------------------------
function cu2qu(cu) result(qu)

  real(pReal), intent(in), dimension(3) :: cu
  type(quaternion)                      :: qu

  qu = ho2qu(cu2ho(cu))

end function cu2qu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert cubochoric to rotation matrix
!---------------------------------------------------------------------------------------------------
function cu2om(cu) result(om)

  real(pReal), intent(in), dimension(3)   :: cu
  real(pReal),             dimension(3,3) :: om

  om = ho2om(cu2ho(cu))

end function cu2om


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert cubochoric to Euler angles
!---------------------------------------------------------------------------------------------------
function cu2eu(cu) result(eu)

  real(pReal), intent(in), dimension(3) :: cu
  real(pReal),             dimension(3) :: eu

  eu = ho2eu(cu2ho(cu))

end function cu2eu


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert cubochoric to axis angle pair
!---------------------------------------------------------------------------------------------------
function cu2ax(cu) result(ax)

  real(pReal), intent(in), dimension(3) :: cu
  real(pReal),             dimension(4) :: ax

  ax = ho2ax(cu2ho(cu))

end function cu2ax


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert cubochoric to Rodrigues vector
!---------------------------------------------------------------------------------------------------
function cu2ro(cu) result(ro)

  real(pReal), intent(in), dimension(3) :: cu
  real(pReal),             dimension(4) :: ro

  ro = ho2ro(cu2ho(cu))

end function cu2ro


!---------------------------------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert cubochoric to homochoric
!---------------------------------------------------------------------------------------------------
function cu2ho(cu) result(ho)

  real(pReal), intent(in), dimension(3) :: cu
  real(pReal),             dimension(3) :: ho

  ho = Lambert_CubeToBall(cu)

end function cu2ho


end module rotations