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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
! 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.
! ###################################################################
module rotations
use prec
use quaternions
implicit none
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 :: rotVector
procedure, public :: rotTensor
end type rotation
contains
function asQuaternion(this)
class(rotation), intent(in) :: this
real(pReal), dimension(4) :: asQuaternion
asQuaternion = [this%q%w, this%q%x, this%q%y, this%q%z]
end function asQuaternion
function asEulerAngles(this)
class(rotation), intent(in) :: this
real(pReal), dimension(3) :: asEulerAngles
asEulerAngles = qu2eu(this%q)
end function asEulerAngles
function asAxisAnglePair(this)
class(rotation), intent(in) :: this
real(pReal), dimension(4) :: asAxisAnglePair
asAxisAnglePair = qu2ax(this%q)
end function asAxisAnglePair
function asRotationMatrix(this)
class(rotation), intent(in) :: this
real(pReal), dimension(3,3) :: asRotationMatrix
asRotationMatrix = qu2om(this%q)
end function asRotationMatrix
function asRodriguesFrankVector(this)
class(rotation), intent(in) :: this
real(pReal), dimension(4) :: asRodriguesFrankVector
asRodriguesFrankVector = qu2ro(this%q)
end function asRodriguesFrankVector
function asHomochoric(this)
class(rotation), intent(in) :: this
real(pReal), dimension(3) :: asHomochoric
asHomochoric = qu2ho(this%q)
end function asHomochoric
subroutine fromRotationMatrix(this,om)
class(rotation), intent(out) :: this
real(pReal), dimension(3,3), intent(in) :: om
this%q = om2qu(om)
end subroutine
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief rotates a vector passively (default) or actively
!--------------------------------------------------------------------------
function rotVector(this,v,active)
class(rotation), intent(in) :: this
logical, intent(in), optional :: active
real(pReal),intent(in),dimension(3) :: v
real(pReal),dimension(3) :: rotVector
type(quaternion) :: q
if (dEq(norm2(v),1.0_pReal,tol=1.0e-15_pReal)) then
passive: if (merge(.not. active, .true., present(active))) then
q = this%q * (quaternion([0.0_pReal, v(1), v(2), v(3)]) * conjg(this%q) )
else passive
q = conjg(this%q) * (quaternion([0.0_pReal, v(1), v(2), v(3)]) * this%q )
endif passive
rotVector = [q%x,q%y,q%z]
else
passive2: if (merge(.not. active, .true., present(active))) then
rotVector = matmul(this%asRotationMatrix(),v)
else passive2
rotVector = matmul(transpose(this%asRotationMatrix()),v)
endif passive2
endif
end function rotVector
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief rotate a second rank tensor using a rotation matrix, active or passive (single precision)
!--------------------------------------------------------------------------
function rotTensor(this,m,active)
class(rotation), intent(in) :: this
real(pReal),intent(in),dimension(3,3) :: m
logical, intent(in), optional :: active
real(pReal),dimension(3,3) :: rotTensor
passive: if (merge(.not. active, .true., present(active))) then
rotTensor = matmul(matmul(this%asRotationMatrix(),m),transpose(this%asRotationMatrix()))
else passive
rotTensor = matmul(matmul(transpose(this%asRotationMatrix()),m),this%asRotationMatrix())
endif passive
end function rotTensor
!--------------------------------------------------------------------------
!--------------------------------------------------------------------------
! here we start with a series of conversion routines between representations
!--------------------------------------------------------------------------
!--------------------------------------------------------------------------
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief Euler angles to orientation matrix [Morawiec, page 28]
!--------------------------------------------------------------------------
pure function eu2om(eu) result(om)
implicit none
real(pReal), intent(in), dimension(3) :: eu !< Euler angles in radians
real(pReal), dimension(3,3) :: om !< output orientation matrix
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)
use math, only: &
PI
implicit none
real(pReal), intent(in), dimension(3) :: eu !< Euler angles in radians
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) = -epsijk/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)
use, intrinsic :: IEEE_ARITHMETIC, only: &
IEEE_value, &
IEEE_positive_inf
use math, only: &
PI
implicit none
real(pReal), intent(in), dimension(3) :: eu !< Euler angles in radians
real(pReal), dimension(4) :: ro
ro = eu2ax(eu) ! convert to axis angle representation
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, epsijk, 0.0_pReal ]
else
ro(4) = tan(ro(4)*0.5)
end if
end function eu2ro
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief Euler angles to quaternion
!--------------------------------------------------------------------------
pure function eu2qu(eu) result(qu)
implicit none
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))
! passive quaternion
qu = quaternion([ cPhi*cos(ee(1)+ee(3)), &
-epsijk*sPhi*cos(ee(1)-ee(3)), &
-epsijk*sPhi*sin(ee(1)-ee(3)), &
-epsijk*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 orientation matrix to euler angles
!--------------------------------------------------------------------------
pure function om2eu(om) result(eu)
use math, only: &
PI
implicit none
real(pReal), intent(in), dimension(3,3) :: om !< orientation matrix
real(pReal), dimension(3) :: eu
real(pReal) :: zeta
if (dEq(abs(om(3,3)),1.0_pReal,1.0e-15_pReal)) then
eu = [ atan2( om(1,2),om(1,1)), 0.5*PI*(1-om(3,3)),0.0_pReal ]
else
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)]
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 Axis angle pair to orientation matrix
!--------------------------------------------------------------------------
pure function ax2om(ax) result(om)
implicit none
real(pReal), intent(in), dimension(4) :: ax
real(pReal), dimension(3,3) :: om !< orientation matrix
real(pReal) :: q, c, s, omc
integer(pInt) :: 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 (epsijk > 0.0) om = transpose(om)
end function ax2om
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief Quaternion to Euler angles [Morawiec page 40, with errata !!!! ]
!--------------------------------------------------------------------------
pure function qu2eu(qu) result(eu)
use math, only: &
PI
implicit none
type(quaternion), intent(in) :: qu !< quaternion
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(-epsijk*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((-epsijk*qu%w*qu%y+qu%x*qu%z)*chi, (-epsijk*qu%w*qu%x-qu%y*qu%z)*chi ), &
atan2( 2.0*chi, q03-q12 ), &
atan2(( epsijk*qu%w*qu%y+qu%x*qu%z)*chi, (-epsijk*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 Axis angle pair to homochoric
!--------------------------------------------------------------------------
pure function ax2ho(ax) result(ho)
real(pReal), intent(in), dimension(4) :: ax !< axis angle in degree/radians?
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 Homochoric to axis angle pair
!--------------------------------------------------------------------------
pure function ho2ax(ho) result(ax)
implicit none
real(pReal), intent(in), dimension(3) :: ho !< homochoric coordinates
real(pReal), dimension(4) :: ax
integer(pInt) :: 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 orientation matrix to axis angle
!--------------------------------------------------------------------------
function om2ax(om) result(ax)
use IO, only: &
IO_error
use math, only: &
math_clip, &
math_trace33
implicit none
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(pInt) :: 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
#if (FLOAT==8)
call dgeev('N','V',3,o,3,Wr,Wi,devNull,3,VR,3,WORK,LWORK,INFO)
#elif (FLOAT==4)
call sgeev('N','V',3,o,3,Wr,Wi,devNull,3,VR,3,WORK,LWORK,INFO)
#else
NO SUITABLE PRECISION FOR REAL SELECTED, STOPPING COMPILATION
#endif
if (INFO /= 0) call IO_error(0_pInt,ext_msg='Error in om2ax/(s/d)geev: (S/D)GEEV 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),-epsijk *[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 Rodrigues vector to axis angle pair
!--------------------------------------------------------------------------
pure function ro2ax(ro) result(ax)
use, intrinsic :: IEEE_ARITHMETIC
use math, only: &
PI
implicit none
real(pReal), intent(in), dimension(4) :: ro !< homochoric coordinates
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 axis angle to Rodrigues
!--------------------------------------------------------------------------
pure function ax2ro(ax) result(ro)
use, intrinsic :: IEEE_ARITHMETIC
use math, only: &
PI
implicit none
real(pReal), intent(in), dimension(4) :: ax !< axis angle in degree/radians?
real(pReal), dimension(4) :: ro
real(pReal), parameter :: thr = 1.0E-7
if (dEq0(ax(4))) then
ro = [ 0.0_pReal, 0.0_pReal, epsijk, 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 to quaternion
!--------------------------------------------------------------------------
pure function ax2qu(ax) result(qu)
implicit none
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 rodrigues to homochoric
!--------------------------------------------------------------------------
pure function ro2ho(ro) result(ho)
use, intrinsic :: IEEE_ARITHMETIC
use math, only: &
PI
implicit none
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 a quaternion to a 3x3 matrix
!--------------------------------------------------------------------------
pure function qu2om(qu) result(om)
implicit none
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 (epsijk < 0.0) om = transpose(om)
end function qu2om
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert a 3x3 rotation matrix to a unit quaternion (see Morawiec, page 37)
!--------------------------------------------------------------------------
function om2qu(om) result(qu)
implicit none
real(pReal), intent(in), dimension(3,3) :: om
type(quaternion) :: qu
real(pReal), dimension(4) :: qu_A
real(pReal), dimension(4) :: s
s = [+om(1,1) +om(2,2) +om(3,3) +1.0_pReal, &
+om(1,1) -om(2,2) -om(3,3) +1.0_pReal, &
-om(1,1) +om(2,2) -om(3,3) +1.0_pReal, &
-om(1,1) -om(2,2) +om(3,3) +1.0_pReal]
qu_A = sqrt(max(s,0.0_pReal))*0.5_pReal*[1.0_pReal,epsijk,epsijk,epsijk]
qu_A = qu_A/norm2(qu_A)
if(any(dEq(abs(qu_A),1.0_pReal,1.0e-15_pReal))) &
where (.not.(dEq(abs(qu_A),1.0_pReal,1.0e-15_pReal))) qu_A = 0.0_pReal
if (om(3,2) < om(2,3)) qu_A(2) = -qu_A(2)
if (om(1,3) < om(3,1)) qu_A(3) = -qu_A(3)
if (om(2,1) < om(1,2)) qu_A(4) = -qu_A(4)
qu = quaternion(qu_A)
!qu_A = om2ax(om)
!if(any(qu_A(1:3) * [qu%x,qu%y,qu%z] < 0.0)) print*, 'sign error'
end function om2qu
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert quaternion to axis angle
!--------------------------------------------------------------------------
pure function qu2ax(qu) result(ax)
use math, only: &
PI
implicit none
type(quaternion), intent(in) :: qu
real(pReal), dimension(4) :: ax
real(pReal) :: omega, s
omega = 2.0 * acos(qu%w)
! if the angle equals zero, then we return the rotation axis as [001]
if (dEq0(omega)) then
ax = [ 0.0, 0.0, 1.0, 0.0 ]
elseif (dNeq0(qu%w)) then
s = sign(1.0_pReal,qu%w)/sqrt(qu%x**2+qu%y**2+qu%z**2)
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 quaternion to Rodrigues
!--------------------------------------------------------------------------
pure function qu2ro(qu) result(ro)
use, intrinsic :: IEEE_ARITHMETIC
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])
ro = merge ( [ 0.0_pReal, 0.0_pReal, epsijk, 0.0_pReal] , &
[ qu%x/s, qu%y/s, qu%z/s, tan(acos(qu%w))], &
s < thr)
end if
end function qu2ro
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert quaternion to homochoric
!--------------------------------------------------------------------------
pure function qu2ho(qu) result(ho)
implicit none
type(quaternion), intent(in) :: qu
real(pReal), dimension(3) :: ho
real(pReal) :: omega, f
omega = 2.0 * acos(qu%w)
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 homochoric to cubochoric
!--------------------------------------------------------------------------
function ho2cu(ho) result(cu)
use Lambert, only: LambertBallToCube
implicit none
real(pReal), intent(in), dimension(3) :: ho
real(pReal), dimension(3) :: cu
cu = LambertBallToCube(ho)
end function ho2cu
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert cubochoric to homochoric
!--------------------------------------------------------------------------
function cu2ho(cu) result(ho)
use Lambert, only: LambertCubeToBall
implicit none
real(pReal), intent(in), dimension(3) :: cu
real(pReal), dimension(3) :: ho
ho = LambertCubeToBall(cu)
end function cu2ho
!--------------------------------------------------------------------------
!--------------------------------------------------------------------------
! and here are a bunch of transformation routines that are derived from the others
!--------------------------------------------------------------------------
!--------------------------------------------------------------------------
!--------------------------------------------------------------------------
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief Rodrigues vector to Euler angles
!--------------------------------------------------------------------------
pure function ro2eu(ro) result(eu)
implicit none
real(pReal), intent(in), dimension(4) :: ro !< Rodrigues vector
real(pReal), dimension(3) :: eu
eu = om2eu(ro2om(ro))
end function ro2eu
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert euler to homochoric
!--------------------------------------------------------------------------
pure function eu2ho(eu) result(ho)
implicit none
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 orientation matrix to Rodrigues
!--------------------------------------------------------------------------
pure function om2ro(om) result(ro)
implicit none
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 orientation matrix to homochoric
!--------------------------------------------------------------------------
function om2ho(om) result(ho)
implicit none
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 axis angle to euler
!--------------------------------------------------------------------------
pure function ax2eu(ax) result(eu)
implicit none
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 rodrigues to orientation matrix
!--------------------------------------------------------------------------
pure function ro2om(ro) result(om)
implicit none
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 to quaternion
!--------------------------------------------------------------------------
pure function ro2qu(ro) result(qu)
implicit none
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 homochoric to euler
!--------------------------------------------------------------------------
pure function ho2eu(ho) result(eu)
implicit none
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 orientation matrix
!--------------------------------------------------------------------------
pure function ho2om(ho) result(om)
implicit none
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 Rodrigues
!--------------------------------------------------------------------------
pure function ho2ro(ho) result(ro)
implicit none
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 quaternion
!--------------------------------------------------------------------------
pure function ho2qu(ho) result(qu)
implicit none
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 euler angles to cubochoric
!--------------------------------------------------------------------------
function eu2cu(eu) result(cu)
implicit none
real(pReal), intent(in), dimension(3) :: eu !< Bunge-Euler angles in radians
real(pReal), dimension(3) :: cu
cu = ho2cu(eu2ho(eu))
end function eu2cu
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert orientation matrix to cubochoric
!--------------------------------------------------------------------------
function om2cu(om) result(cu)
implicit none
real(pReal), intent(in), dimension(3,3) :: om !< rotation matrix
real(pReal), dimension(3) :: cu
cu = ho2cu(om2ho(om))
end function om2cu
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert axis angle to cubochoric
!--------------------------------------------------------------------------
function ax2cu(ax) result(cu)
implicit none
real(pReal), intent(in), dimension(4) :: ax !< axis angle in degree/radians?
real(pReal), dimension(3) :: cu
cu = ho2cu(ax2ho(ax))
end function ax2cu
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert Rodrigues to cubochoric
!--------------------------------------------------------------------------
function ro2cu(ro) result(cu)
implicit none
real(pReal), intent(in), dimension(4) :: ro !< Rodrigues vector
real(pReal), dimension(3) :: cu
cu = ho2cu(ro2ho(ro))
end function ro2cu
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert quaternion to cubochoric
!--------------------------------------------------------------------------
function qu2cu(qu) result(cu)
implicit none
type(quaternion), intent(in) :: qu ! unit quaternion
real(pReal), dimension(3) :: cu
cu = ho2cu(qu2ho(qu))
end function qu2cu
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert cubochoric to euler angles
!--------------------------------------------------------------------------
function cu2eu(cu) result(eu)
implicit none
real(pReal), intent(in), dimension(3) :: cu ! cubochoric?
real(pReal), dimension(3) :: eu
eu = ho2eu(cu2ho(cu))
end function cu2eu
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert cubochoric to orientation matrix
!--------------------------------------------------------------------------
function cu2om(cu) result(om)
implicit none
real(pReal), intent(in), dimension(3) :: cu ! cubochoric?
real(pReal), dimension(3,3) :: om
om = ho2om(cu2ho(cu))
end function cu2om
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert cubochoric to axis angle
!--------------------------------------------------------------------------
function cu2ax(cu) result(ax)
implicit none
real(pReal), intent(in), dimension(3) :: cu ! cubochoric?
real(pReal), dimension(4) :: ax
ax = ho2ax(cu2ho(cu))
end function cu2ax
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert cubochoric to Rodrigues
!--------------------------------------------------------------------------
function cu2ro(cu) result(ro)
implicit none
real(pReal), intent(in), dimension(3) :: cu ! cubochoric?
real(pReal), dimension(4) :: ro
ro = ho2ro(cu2ho(cu))
end function cu2ro
!--------------------------------------------------------------------------
!> @author Marc De Graef, Carnegie Mellon University
!> @brief convert cubochoric to quaternion
!--------------------------------------------------------------------------
function cu2qu(cu) result(qu)
implicit none
real(pReal), intent(in), dimension(3) :: cu ! cubochoric?
type(quaternion) :: qu ! cubochoric?
qu = ho2qu(cu2ho(cu))
end function cu2qu
end module rotations
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