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DAMASK_EICMD
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DAMASK_EICMD/src/rotations.f90

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! ###################################################################
! 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"
! AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
! IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
! ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
! LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
! DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
! SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
! CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
! OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE
! USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
! ###################################################################
!---------------------------------------------------------------------------------------------------
!> @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 :: fromRotationMatrix
procedure, public :: fromEulerAngles
!------------------------------------------
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 fromRotationMatrix(self,om)
class(rotation), intent(out) :: self
real(pReal), dimension(3,3), intent(in) :: om
self%q = om2qu(om)
end subroutine
!---------------------------------------------------------------------------------------------------
subroutine fromEulerAngles(self,eu)
class(rotation), intent(out) :: self
real(pReal), dimension(3), intent(in) :: eu
self%q = eu2qu(eu)
end subroutine
!---------------------------------------------------------------------------------------------------
!---------------------------------------------------------------------------------------------------
!> @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)
real(pReal), dimension(3) :: rotVector
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
rotVector = [q%x,q%y,q%z]
else
if (passive) then
rotVector = matmul(self%asRotationMatrix(),v)
else
rotVector = 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)
real(pReal), dimension(3,3) :: rotTensor
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
rotTensor = matmul(matmul(self%asRotationMatrix(),m),transpose(self%asRotationMatrix()))
else
rotTensor = 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
!---------------------------------------------------------------------------------------------------
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
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