def compile_2nd_matrix_pseudo_ellipse_torsionless(matrix, Rx2_eigen, theta_x, theta_y):
"""Generate the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param theta_x: The cone opening angle along x.
@type theta_x: float
@param theta_y: The cone opening angle along y.
@type theta_y: float
"""
# The rigid case.
if theta_x == 0.0:
# Set up the matrix as the identity.
matrix[:] = 0.0
for i in range(len(matrix)):
matrix[i, i] = 1.0
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
# The surface area normalisation factor.
fact = 6.0 * pec(theta_x, theta_y)
fact2 = 0.5 * fact
# Invert.
if fact == 0.0:
fact = 1e100
fact2 = 1e100
else:
fact = 1.0 / fact
fact2 = 1.0 / fact2
# Diagonal.
matrix[0, 0] = fact2 * (3.0*pi + quad(part_int_daeg2_pseudo_ellipse_torsionless_00, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[1, 1] = fact * (2.0*pi + quad(part_int_daeg2_pseudo_ellipse_torsionless_11, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[2, 2] = fact * (5.0*pi - quad(part_int_daeg2_pseudo_ellipse_torsionless_22, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[3, 3] = matrix[1, 1]
matrix[4, 4] = fact2 * (3.0*pi + quad(part_int_daeg2_pseudo_ellipse_torsionless_44, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[5, 5] = fact * (5.0*pi - quad(part_int_daeg2_pseudo_ellipse_torsionless_55, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[6, 6] = matrix[2, 2]
matrix[7, 7] = matrix[5, 5]
matrix[8, 8] = fact2 * (2.0*pi - quad(part_int_daeg2_pseudo_ellipse_torsionless_88, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
# Off diagonal set 1.
matrix[0, 4] = matrix[4, 0] = fact2 * (pi + quad(part_int_daeg2_pseudo_ellipse_torsionless_04, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[0, 8] = matrix[8, 0] = fact2 * (2.0*pi + quad(part_int_daeg2_pseudo_ellipse_torsionless_08, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[4, 8] = matrix[8, 4] = fact2 * (2.0*pi + quad(part_int_daeg2_pseudo_ellipse_torsionless_48, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
# Off diagonal set 2.
matrix[1, 3] = matrix[3, 1] = matrix[0, 4]
matrix[2, 6] = matrix[6, 2] = -matrix[0, 8]
matrix[5, 7] = matrix[7, 5] = -matrix[4, 8]
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_1st_matrix_pseudo_ellipse(matrix, R_eigen, theta_x, theta_y, sigma_max):
"""Generate the 1st degree Frame Order matrix for the pseudo-ellipse.
@param matrix: The Frame Order matrix, 1st degree to be populated.
@type matrix: numpy 3D, rank-2 array
@param R_eigen: The eigenframe rotation matrix.
@type R_eigen: numpy 3D, rank-2 array
@param theta_x: The cone opening angle along x.
@type theta_x: float
@param theta_y: The cone opening angle along y.
@type theta_y: float
@param sigma_max: The maximum torsion angle.
@type sigma_max: float
"""
# The surface area normalisation factor.
fact = 2.0 * pec(theta_x, theta_y)
# Invert.
if fact == 0.0:
fact = 1e100
else:
fact = 1.0 / fact
# Sinc value.
sinc_smax = sinc(sigma_max/pi)
# Numerical integration of phi of each element.
matrix[0, 0] = fact * sinc_smax * (2.0*pi + quad(part_int_daeg1_pseudo_ellipse_00, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[1, 1] = fact * sinc_smax * (2.0*pi + quad(part_int_daeg1_pseudo_ellipse_11, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[2, 2] = fact * quad(part_int_daeg1_pseudo_ellipse_22, -pi, pi, args=(theta_x, theta_y), full_output=1)[0]
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, R_eigen)
def compile_1st_matrix_iso_cone(matrix, R_eigen, cone_theta, sigma_max):
"""Generate the 1st degree Frame Order matrix for the isotropic cone.
@param matrix: The Frame Order matrix, 1st degree to be populated.
@type matrix: numpy 3D, rank-2 array
@param R_eigen: The eigenframe rotation matrix.
@type R_eigen: numpy 3D, rank-2 array
@param cone_theta: The cone opening angle.
@type cone_theta: float
@param sigma_max: The maximum torsion angle.
@type sigma_max: float
"""
# Zeros.
matrix[:] = 0.0
# Pre-calculate trig values.
sinc_sigma_max = sinc(sigma_max/pi)
cos_theta = cos(cone_theta)
# Diagonal values.
matrix[0, 0] = sinc_sigma_max * (cos_theta + 3.0)
matrix[1, 1] = matrix[0, 0]
matrix[2, 2] = 2.0*cos_theta + 2.0
# Rotate and return the frame order matrix.
return 0.25 * rotate_daeg(matrix, R_eigen)
def compile_2nd_matrix_pseudo_ellipse_free_rotor(matrix, Rx2_eigen, theta_x, theta_y):
"""Generate the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param theta_x: The cone opening angle along x.
@type theta_x: float
@param theta_y: The cone opening angle along y.
@type theta_y: float
"""
# The surface area normalisation factor.
fact = 1.0 / (6.0 * pec(theta_x, theta_y))
# Diagonal.
matrix[0, 0] = fact * (4.0*pi - quad(part_int_daeg2_pseudo_ellipse_free_rotor_00, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[1, 1] = matrix[3, 3] = fact * 3.0/2.0 * quad(part_int_daeg2_pseudo_ellipse_free_rotor_11, -pi, pi, args=(theta_x, theta_y), full_output=1)[0]
matrix[4, 4] = fact * (4.0*pi - quad(part_int_daeg2_pseudo_ellipse_free_rotor_44, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[8, 8] = fact * (4.0*pi - 2.0*quad(part_int_daeg2_pseudo_ellipse_free_rotor_88, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
# Off diagonal set 1.
matrix[0, 4] = matrix[0, 0]
matrix[4, 0] = matrix[4, 4]
matrix[0, 8] = fact * (4.0*pi + quad(part_int_daeg2_pseudo_ellipse_free_rotor_08, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[8, 0] = fact * (4.0*pi + quad(part_int_daeg2_pseudo_ellipse_free_rotor_80, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[4, 8] = fact * (4.0*pi + quad(part_int_daeg2_pseudo_ellipse_free_rotor_48, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[8, 4] = matrix[8, 0]
# Off diagonal set 2.
matrix[1, 3] = matrix[3, 1] = -matrix[1, 1]
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_1st_matrix_iso_cone(matrix, R_eigen, cone_theta, sigma_max):
"""Generate the 1st degree Frame Order matrix for the isotropic cone.
@param matrix: The Frame Order matrix, 1st degree to be populated.
@type matrix: numpy 3D, rank-2 array
@param R_eigen: The eigenframe rotation matrix.
@type R_eigen: numpy 3D, rank-2 array
@param cone_theta: The cone opening angle.
@type cone_theta: float
@param sigma_max: The maximum torsion angle.
@type sigma_max: float
"""
# Zeros.
matrix[:] = 0.0
# Pre-calculate trig values.
sinc_sigma_max = sinc(sigma_max/pi)
cos_theta = cos(cone_theta)
# Diagonal values.
matrix[0, 0] = sinc_sigma_max * (cos_theta + 3.0)
matrix[1, 1] = matrix[0, 0]
matrix[2, 2] = 2.0*cos_theta + 2.0
# Rotate and return the frame order matrix.
return 0.25 * rotate_daeg(matrix, R_eigen)
def compile_1st_matrix_pseudo_ellipse_free_rotor(matrix, R_eigen, theta_x, theta_y):
"""Generate the 1st degree Frame Order matrix for the free rotor pseudo-ellipse.
@param matrix: The Frame Order matrix, 1st degree to be populated.
@type matrix: numpy 3D, rank-2 array
@param R_eigen: The eigenframe rotation matrix.
@type R_eigen: numpy 3D, rank-2 array
@param theta_x: The cone opening angle along x.
@type theta_x: float
@param theta_y: The cone opening angle along y.
@type theta_y: float
"""
# The surface area normalisation factor.
fact = 2.0 * pec(theta_x, theta_y)
# Invert.
if fact == 0.0:
fact = 1e100
else:
fact = 1.0 / fact
# Numerical integration of phi of each element.
matrix[2, 2] = fact * quad(part_int_daeg1_pseudo_ellipse_free_rotor_22, -pi, pi, args=(theta_x, theta_y), full_output=1)[0]
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, R_eigen)
def compile_2nd_matrix_free_rotor(matrix, Rx2_eigen):
"""Generate the rotated 2nd degree Frame Order matrix for the free rotor model.
The rotor axis is assumed to be parallel to the z-axis in the eigenframe.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
"""
# Zeros.
matrix[:] = 0.0
# Diagonal.
matrix[0, 0] = 0.5
matrix[1, 1] = 0.5
matrix[3, 3] = 0.5
matrix[4, 4] = 0.5
matrix[8, 8] = 1
# Off diagonal set 1.
matrix[0, 4] = matrix[4, 0] = 0.5
# Off diagonal set 2.
matrix[1, 3] = matrix[3, 1] = -0.5
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_1st_matrix_pseudo_ellipse_free_rotor(matrix, R_eigen, theta_x, theta_y):
"""Generate the 1st degree Frame Order matrix for the free rotor pseudo-ellipse.
@param matrix: The Frame Order matrix, 1st degree to be populated.
@type matrix: numpy 3D, rank-2 array
@param R_eigen: The eigenframe rotation matrix.
@type R_eigen: numpy 3D, rank-2 array
@param theta_x: The cone opening angle along x.
@type theta_x: float
@param theta_y: The cone opening angle along y.
@type theta_y: float
"""
# The surface area normalisation factor.
fact = 2.0 * pec(theta_x, theta_y)
# Invert.
if fact == 0.0:
fact = 1e100
else:
fact = 1.0 / fact
# Numerical integration of phi of each element.
matrix[2, 2] = fact * quad(part_int_daeg1_pseudo_ellipse_free_rotor_22, -pi, pi, args=(theta_x, theta_y), full_output=1)[0]
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, R_eigen)
def compile_2nd_matrix_pseudo_ellipse(matrix, Rx2_eigen, theta_x, theta_y, sigma_max):
"""Generate the 2nd degree Frame Order matrix for the pseudo-ellipse.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param theta_x: The cone opening angle along x.
@type theta_x: float
@param theta_y: The cone opening angle along y.
@type theta_y: float
@param sigma_max: The maximum torsion angle.
@type sigma_max: float
"""
# The surface area normalisation factor.
fact = 12.0 * pec(theta_x, theta_y)
# Invert.
if fact == 0.0:
fact = 1e100
else:
fact = 1.0 / fact
# Sigma_max part.
if sigma_max == 0.0:
fact2 = 1e100
else:
fact2 = fact / (2.0 * sigma_max)
# Diagonal.
matrix[0, 0] = fact * (4.0*pi*(sinc(2.0*sigma_max/pi) + 2.0) + quad(part_int_daeg2_pseudo_ellipse_00, -pi, pi, args=(theta_x, theta_y, sigma_max), full_output=1)[0])
matrix[1, 1] = fact * (4.0*pi*sinc(2.0*sigma_max/pi) + quad(part_int_daeg2_pseudo_ellipse_11, -pi, pi, args=(theta_x, theta_y, sigma_max), full_output=1)[0])
matrix[2, 2] = fact * 2.0*sinc(sigma_max/pi) * (5.0*pi - quad(part_int_daeg2_pseudo_ellipse_22, -pi, pi, args=(theta_x, theta_y, sigma_max), full_output=1)[0])
matrix[3, 3] = matrix[1, 1]
matrix[4, 4] = fact * (4.0*pi*(sinc(2.0*sigma_max/pi) + 2.0) + quad(part_int_daeg2_pseudo_ellipse_44, -pi, pi, args=(theta_x, theta_y, sigma_max), full_output=1)[0])
matrix[5, 5] = fact * 2.0*sinc(sigma_max/pi) * (5.0*pi - quad(part_int_daeg2_pseudo_ellipse_55, -pi, pi, args=(theta_x, theta_y, sigma_max), full_output=1)[0])
matrix[6, 6] = matrix[2, 2]
matrix[7, 7] = matrix[5, 5]
matrix[8, 8] = 4.0 * fact * (2.0*pi - quad(part_int_daeg2_pseudo_ellipse_88, -pi, pi, args=(theta_x, theta_y, sigma_max), full_output=1)[0])
# Off diagonal set 1.
matrix[0, 4] = fact * (4.0*pi*(2.0 - sinc(2.0*sigma_max/pi)) + quad(part_int_daeg2_pseudo_ellipse_04, -pi, pi, args=(theta_x, theta_y, sigma_max), full_output=1)[0])
matrix[4, 0] = fact * (4.0*pi*(2.0 - sinc(2.0*sigma_max/pi)) + quad(part_int_daeg2_pseudo_ellipse_40, -pi, pi, args=(theta_x, theta_y, sigma_max), full_output=1)[0])
matrix[0, 8] = 4.0 * fact * (2.0*pi - quad(part_int_daeg2_pseudo_ellipse_08, -pi, pi, args=(theta_x, theta_y, sigma_max), full_output=1)[0])
matrix[8, 0] = fact * (8.0*pi + quad(part_int_daeg2_pseudo_ellipse_80, -pi, pi, args=(theta_x, theta_y, sigma_max), full_output=1)[0])
matrix[4, 8] = 4.0 * fact * (2.0*pi - quad(part_int_daeg2_pseudo_ellipse_48, -pi, pi, args=(theta_x, theta_y, sigma_max), full_output=1)[0])
matrix[8, 4] = fact * (8.0*pi - quad(part_int_daeg2_pseudo_ellipse_84, -pi, pi, args=(theta_x, theta_y, sigma_max), full_output=1)[0])
# Off diagonal set 2.
matrix[1, 3] = matrix[3, 1] = fact * (4.0*pi*sinc(2.0*sigma_max/pi) + quad(part_int_daeg2_pseudo_ellipse_13, -pi, pi, args=(theta_x, theta_y, sigma_max), full_output=1)[0])
matrix[2, 6] = matrix[6, 2] = -fact * 4.0 * sinc(sigma_max/pi) * (2.0*pi + quad(part_int_daeg2_pseudo_ellipse_26, -pi, pi, args=(theta_x, theta_y, sigma_max), full_output=1)[0])
matrix[5, 7] = matrix[7, 5] = -fact * 4.0 * sinc(sigma_max/pi) * (2.0*pi + quad(part_int_daeg2_pseudo_ellipse_57, -pi, pi, args=(theta_x, theta_y, sigma_max), full_output=1)[0])
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_2nd_matrix_double_rotor(matrix, Rx2_eigen, smax1, smax2):
"""Generate the rotated 2nd degree Frame Order matrix for the double rotor model.
The cone axis is assumed to be parallel to the z-axis in the eigenframe.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param smax1: The maximum torsion angle for the first rotor.
@type smax1: float
@param smax2: The maximum torsion angle for the second rotor.
@type smax2: float
"""
# Zeros.
matrix[:] = 0.0
# Repetitive trig calculations.
sinc_smax1 = sinc(smax1 / pi)
sinc_2smax1 = sinc(2.0 * smax1 / pi)
sinc_2smax1p1 = sinc_2smax1 + 1.0
sinc_2smax1n1 = sinc_2smax1 - 1.0
sinc_smax2 = sinc(smax2 / pi)
sinc_2smax2 = sinc(2.0 * smax2 / pi)
sinc_2smax2p1 = sinc_2smax2 + 1.0
sinc_2smax2n1 = sinc_2smax2 - 1.0
# Diagonal.
matrix[0, 0] = sinc_2smax1 + 1.0
matrix[1, 1] = 2.0 * sinc_smax1 * sinc_smax2
matrix[2, 2] = sinc_smax2 * sinc_2smax1p1
matrix[3, 3] = matrix[1, 1]
matrix[4, 4] = sinc_2smax2p1
matrix[5, 5] = sinc_smax1 * sinc_2smax2p1
matrix[6, 6] = matrix[2, 2]
matrix[7, 7] = matrix[5, 5]
matrix[8, 8] = 0.5 * sinc_2smax1p1 * sinc_2smax2p1
# Off diagonal set 1.
matrix[4, 0] = 0.5 * sinc_2smax1n1 * sinc_2smax2n1
matrix[0, 8] = -sinc_2smax1n1
matrix[8, 0] = -0.5 * sinc_2smax1n1 * sinc_2smax2p1
matrix[4, 8] = -0.5 * sinc_2smax1p1 * sinc_2smax2n1
matrix[8, 4] = -sinc_2smax2n1
# Off diagonal set 2.
matrix[2, 6] = matrix[6, 2] = sinc_smax2 * sinc_2smax1n1
matrix[5, 7] = matrix[7, 5] = sinc_smax1 * sinc_2smax2n1
# Divide by 2.
multiply(0.5, matrix, matrix)
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_2nd_matrix_double_rotor(matrix, Rx2_eigen, smax1, smax2):
"""Generate the rotated 2nd degree Frame Order matrix for the double rotor model.
The cone axis is assumed to be parallel to the z-axis in the eigenframe.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param smax1: The maximum torsion angle for the first rotor.
@type smax1: float
@param smax2: The maximum torsion angle for the second rotor.
@type smax2: float
"""
# Zeros.
matrix[:] = 0.0
# Repetitive trig calculations.
sinc_smax1 = sinc(smax1/pi)
sinc_2smax1 = sinc(2.0*smax1/pi)
sinc_2smax1p1 = sinc_2smax1 + 1.0
sinc_2smax1n1 = sinc_2smax1 - 1.0
sinc_smax2 = sinc(smax2/pi)
sinc_2smax2 = sinc(2.0*smax2/pi)
sinc_2smax2p1 = sinc_2smax2 + 1.0
sinc_2smax2n1 = sinc_2smax2 - 1.0
# Diagonal.
matrix[0, 0] = sinc_2smax1 + 1.0
matrix[1, 1] = 2.0 * sinc_smax1 * sinc_smax2
matrix[2, 2] = sinc_smax2 * sinc_2smax1p1
matrix[3, 3] = matrix[1, 1]
matrix[4, 4] = sinc_2smax2p1
matrix[5, 5] = sinc_smax1 * sinc_2smax2p1
matrix[6, 6] = matrix[2, 2]
matrix[7, 7] = matrix[5, 5]
matrix[8, 8] = 0.5 * sinc_2smax1p1 * sinc_2smax2p1
# Off diagonal set 1.
matrix[4, 0] = 0.5 * sinc_2smax1n1 * sinc_2smax2n1
matrix[0, 8] = -sinc_2smax1n1
matrix[8, 0] = -0.5 * sinc_2smax1n1 * sinc_2smax2p1
matrix[4, 8] = -0.5 * sinc_2smax1p1 * sinc_2smax2n1
matrix[8, 4] = -sinc_2smax2n1
# Off diagonal set 2.
matrix[2, 6] = matrix[6, 2] = sinc_smax2 * sinc_2smax1n1
matrix[5, 7] = matrix[7, 5] = sinc_smax1 * sinc_2smax2n1
# Divide by 2.
multiply(0.5, matrix, matrix)
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_2nd_matrix_iso_cone(matrix, Rx2_eigen, cone_theta, sigma_max):
"""Generate the rotated 2nd degree Frame Order matrix for the isotropic cone.
The cone axis is assumed to be parallel to the z-axis in the eigenframe.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param cone_theta: The cone opening angle.
@type cone_theta: float
@param sigma_max: The maximum torsion angle.
@type sigma_max: float
"""
# Zeros.
matrix[:] = 0.0
# Repetitive trig calculations.
sinc_smax = sinc(sigma_max/pi)
sinc_2smax = sinc(2.0*sigma_max/pi)
cos_tmax = cos(cone_theta)
cos_tmax2 = cos_tmax**2
# Larger factors.
fact_sinc_2smax = sinc_2smax*(cos_tmax2 + 4.0*cos_tmax + 7.0) / 24.0
fact_cos_tmax2 = (cos_tmax2 + cos_tmax + 4.0) / 12.0
fact_cos_tmax = (cos_tmax + 1.0) / 4.0
# Diagonal.
matrix[0, 0] = fact_sinc_2smax + fact_cos_tmax2
matrix[1, 1] = fact_sinc_2smax + fact_cos_tmax
matrix[2, 2] = sinc_smax * (2.0*cos_tmax2 + 5.0*cos_tmax + 5.0) / 12.0
matrix[3, 3] = matrix[1, 1]
matrix[4, 4] = matrix[0, 0]
matrix[5, 5] = matrix[2, 2]
matrix[6, 6] = matrix[2, 2]
matrix[7, 7] = matrix[2, 2]
matrix[8, 8] = (cos_tmax2 + cos_tmax + 1.0) / 3.0
# Off diagonal set 1.
matrix[0, 4] = matrix[4, 0] = -fact_sinc_2smax + fact_cos_tmax2
matrix[0, 8] = matrix[8, 0] = -(cos_tmax2 + cos_tmax - 2.0) / 6.0
matrix[4, 8] = matrix[8, 4] = matrix[0, 8]
# Off diagonal set 2.
matrix[1, 3] = matrix[3, 1] = fact_sinc_2smax - fact_cos_tmax
matrix[2, 6] = matrix[6, 2] = sinc_smax * (cos_tmax2 + cos_tmax - 2.0) / 6.0
matrix[5, 7] = matrix[7, 5] = matrix[2, 6]
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_2nd_matrix_iso_cone(matrix, Rx2_eigen, cone_theta, sigma_max):
"""Generate the rotated 2nd degree Frame Order matrix for the isotropic cone.
The cone axis is assumed to be parallel to the z-axis in the eigenframe.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param cone_theta: The cone opening angle.
@type cone_theta: float
@param sigma_max: The maximum torsion angle.
@type sigma_max: float
"""
# Zeros.
matrix[:] = 0.0
# Repetitive trig calculations.
sinc_smax = sinc(sigma_max/pi)
sinc_2smax = sinc(2.0*sigma_max/pi)
cos_tmax = cos(cone_theta)
cos_tmax2 = cos_tmax**2
# Larger factors.
fact_sinc_2smax = sinc_2smax*(cos_tmax2 + 4.0*cos_tmax + 7.0) / 24.0
fact_cos_tmax2 = (cos_tmax2 + cos_tmax + 4.0) / 12.0
fact_cos_tmax = (cos_tmax + 1.0) / 4.0
# Diagonal.
matrix[0, 0] = fact_sinc_2smax + fact_cos_tmax2
matrix[1, 1] = fact_sinc_2smax + fact_cos_tmax
matrix[2, 2] = sinc_smax * (2.0*cos_tmax2 + 5.0*cos_tmax + 5.0) / 12.0
matrix[3, 3] = matrix[1, 1]
matrix[4, 4] = matrix[0, 0]
matrix[5, 5] = matrix[2, 2]
matrix[6, 6] = matrix[2, 2]
matrix[7, 7] = matrix[2, 2]
matrix[8, 8] = (cos_tmax2 + cos_tmax + 1.0) / 3.0
# Off diagonal set 1.
matrix[0, 4] = matrix[4, 0] = -fact_sinc_2smax + fact_cos_tmax2
matrix[0, 8] = matrix[8, 0] = -(cos_tmax2 + cos_tmax - 2.0) / 6.0
matrix[4, 8] = matrix[8, 4] = matrix[0, 8]
# Off diagonal set 2.
matrix[1, 3] = matrix[3, 1] = fact_sinc_2smax - fact_cos_tmax
matrix[2, 6] = matrix[6, 2] = sinc_smax * (cos_tmax2 + cos_tmax - 2.0) / 6.0
matrix[5, 7] = matrix[7, 5] = matrix[2, 6]
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_1st_matrix_pseudo_ellipse(matrix, R_eigen, theta_x, theta_y,
sigma_max):
"""Generate the 1st degree Frame Order matrix for the pseudo-ellipse.
@param matrix: The Frame Order matrix, 1st degree to be populated.
@type matrix: numpy 3D, rank-2 array
@param R_eigen: The eigenframe rotation matrix.
@type R_eigen: numpy 3D, rank-2 array
@param theta_x: The cone opening angle along x.
@type theta_x: float
@param theta_y: The cone opening angle along y.
@type theta_y: float
@param sigma_max: The maximum torsion angle.
@type sigma_max: float
"""
# The surface area normalisation factor.
fact = 2.0 * pec(theta_x, theta_y)
# Invert.
if fact == 0.0:
fact = 1e100
else:
fact = 1.0 / fact
# Sinc value.
sinc_smax = sinc(sigma_max / pi)
# Numerical integration of phi of each element.
matrix[0, 0] = fact * sinc_smax * (2.0 * pi +
quad(part_int_daeg1_pseudo_ellipse_00,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
matrix[1, 1] = fact * sinc_smax * (2.0 * pi +
quad(part_int_daeg1_pseudo_ellipse_11,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
matrix[2, 2] = fact * quad(part_int_daeg1_pseudo_ellipse_22,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0]
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, R_eigen)
def compile_2nd_matrix_pseudo_ellipse_free_rotor(matrix, Rx2_eigen, theta_x, theta_y):
"""Generate the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param theta_x: The cone opening angle along x.
@type theta_x: float
@param theta_y: The cone opening angle along y.
@type theta_y: float
"""
# The surface area normalisation factor.
fact3 = 3.0 * pec(theta_x, theta_y)
fact4 = 4.0 * pec(theta_x, theta_y)
fact6 = 6.0 * pec(theta_x, theta_y)
# Invert.
if fact3 == 0.0:
fact3 = 1e100
fact4 = 1e100
fact6 = 1e100
else:
fact3 = 1.0 / fact3
fact4 = 1.0 / fact4
fact6 = 1.0 / fact6
# Diagonal.
matrix[0, 0] = fact6 * (4.0*pi - quad(part_int_daeg2_pseudo_ellipse_free_rotor_00, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[1, 1] = matrix[3, 3] = fact4 * quad(part_int_daeg2_pseudo_ellipse_free_rotor_11, -pi, pi, args=(theta_x, theta_y), full_output=1)[0]
matrix[4, 4] = fact6 * (4.0*pi - quad(part_int_daeg2_pseudo_ellipse_free_rotor_44, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[8, 8] = fact3 * (2.0*pi - quad(part_int_daeg2_pseudo_ellipse_free_rotor_88, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
# Off diagonal set 1.
matrix[0, 4] = matrix[0, 0]
matrix[4, 0] = matrix[4, 4]
matrix[0, 8] = fact3 * (2.0*pi + quad(part_int_daeg2_pseudo_ellipse_free_rotor_08, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[8, 0] = fact6 * (4.0*pi + quad(part_int_daeg2_pseudo_ellipse_free_rotor_80, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[4, 8] = fact3 * (2.0*pi + quad(part_int_daeg2_pseudo_ellipse_free_rotor_48, -pi, pi, args=(theta_x, theta_y), full_output=1)[0])
matrix[8, 4] = matrix[8, 0]
# Off diagonal set 2.
matrix[1, 3] = matrix[3, 1] = -matrix[1, 1]
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_2nd_matrix_iso_cone_torsionless(matrix, Rx2_eigen, cone_theta):
"""Generate the rotated 2nd degree Frame Order matrix for the torsionless isotropic cone.
The cone axis is assumed to be parallel to the z-axis in the eigenframe.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param cone_theta: The cone opening angle.
@type cone_theta: float
"""
# Zeros.
for i in range(9):
for j in range(9):
matrix[i, j] = 0.0
# Repetitive trig calculations.
cos_tmax = cos(cone_theta)
cos_tmax2 = cos_tmax**2
# Diagonal.
matrix[0, 0] = (3.0*cos_tmax2 + 6.0*cos_tmax + 15.0) / 24.0
matrix[1, 1] = (cos_tmax2 + 10.0*cos_tmax + 13.0) / 24.0
matrix[2, 2] = (4.0*cos_tmax2 + 10.0*cos_tmax + 10.0) / 24.0
matrix[3, 3] = matrix[1, 1]
matrix[4, 4] = matrix[0, 0]
matrix[5, 5] = matrix[2, 2]
matrix[6, 6] = matrix[2, 2]
matrix[7, 7] = matrix[2, 2]
matrix[8, 8] = (cos_tmax2 + cos_tmax + 1.0) / 3.0
# Off diagonal set 1.
matrix[0, 4] = matrix[4, 0] = (cos_tmax2 - 2.0*cos_tmax + 1.0) / 24.0
matrix[0, 8] = matrix[8, 0] = -(cos_tmax2 + cos_tmax - 2.0) / 6.0
matrix[4, 8] = matrix[8, 4] = matrix[0, 8]
# Off diagonal set 2.
matrix[1, 3] = matrix[3, 1] = matrix[0, 4]
matrix[2, 6] = matrix[6, 2] = -matrix[0, 8]
matrix[5, 7] = matrix[7, 5] = -matrix[0, 8]
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_1st_matrix_free_rotor(matrix, R_eigen):
"""Generate the 1st degree Frame Order matrix for the free rotor model.
@param matrix: The Frame Order matrix, 1st degree to be populated.
@type matrix: numpy 3D, rank-2 array
@param R_eigen: The eigenframe rotation matrix.
@type R_eigen: numpy 3D, rank-2 array
"""
# Zeros.
matrix[:] = 0.0
# The single value.
matrix[2, 2] = 1.0
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, R_eigen)
def compile_2nd_matrix_iso_cone_torsionless(matrix, Rx2_eigen, cone_theta):
"""Generate the rotated 2nd degree Frame Order matrix for the torsionless isotropic cone.
The cone axis is assumed to be parallel to the z-axis in the eigenframe.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param cone_theta: The cone opening angle.
@type cone_theta: float
"""
# Zeros.
matrix[:] = 0.0
# Repetitive trig calculations.
cos_tmax = cos(cone_theta)
cos_tmax2 = cos_tmax**2
# Diagonal.
matrix[0, 0] = (3.0 * cos_tmax2 + 6.0 * cos_tmax + 15.0) / 24.0
matrix[1, 1] = (cos_tmax2 + 10.0 * cos_tmax + 13.0) / 24.0
matrix[2, 2] = (4.0 * cos_tmax2 + 10.0 * cos_tmax + 10.0) / 24.0
matrix[3, 3] = matrix[1, 1]
matrix[4, 4] = matrix[0, 0]
matrix[5, 5] = matrix[2, 2]
matrix[6, 6] = matrix[2, 2]
matrix[7, 7] = matrix[2, 2]
matrix[8, 8] = (cos_tmax2 + cos_tmax + 1.0) / 3.0
# Off diagonal set 1.
matrix[0, 4] = matrix[4, 0] = (cos_tmax2 - 2.0 * cos_tmax + 1.0) / 24.0
matrix[0, 8] = matrix[8, 0] = -(cos_tmax2 + cos_tmax - 2.0) / 6.0
matrix[4, 8] = matrix[8, 4] = matrix[0, 8]
# Off diagonal set 2.
matrix[1, 3] = matrix[3, 1] = matrix[0, 4]
matrix[2, 6] = matrix[6, 2] = -matrix[0, 8]
matrix[5, 7] = matrix[7, 5] = -matrix[0, 8]
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_1st_matrix_iso_cone_free_rotor(matrix, R_eigen, tmax):
"""Generate the 1st degree Frame Order matrix for the free rotor isotropic cone.
@param matrix: The Frame Order matrix, 1st degree to be populated.
@type matrix: numpy 3D, rank-2 array
@param R_eigen: The eigenframe rotation matrix.
@type R_eigen: numpy 3D, rank-2 array
@param tmax: The cone opening angle.
@type tmax: float
"""
# Zeros.
matrix[:] = 0.0
# Diagonal values.
matrix[2, 2] = cos(tmax) + 1.0
# Rotate and return the frame order matrix.
return 0.5 * rotate_daeg(matrix, R_eigen)
def compile_1st_matrix_iso_cone_free_rotor(matrix, R_eigen, tmax):
"""Generate the 1st degree Frame Order matrix for the free rotor isotropic cone.
@param matrix: The Frame Order matrix, 1st degree to be populated.
@type matrix: numpy 3D, rank-2 array
@param R_eigen: The eigenframe rotation matrix.
@type R_eigen: numpy 3D, rank-2 array
@param tmax: The cone opening angle.
@type tmax: float
"""
# Zeros.
matrix[:] = 0.0
# Diagonal values.
matrix[2, 2] = cos(tmax) + 1.0
# Rotate and return the frame order matrix.
return 0.5 * rotate_daeg(matrix, R_eigen)
def compile_2nd_matrix_rotor(matrix, Rx2_eigen, smax):
"""Generate the rotated 2nd degree Frame Order matrix for the rotor model.
The cone axis is assumed to be parallel to the z-axis in the eigenframe.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param smax: The maximum torsion angle.
@type smax: float
"""
# Zeros.
for i in range(9):
for j in range(9):
matrix[i, j] = 0.0
# Repetitive trig calculations.
sinc_smax = sinc(smax/pi)
sinc_2smax = sinc(2.0*smax/pi)
# Diagonal.
matrix[0, 0] = (sinc_2smax + 1.0) / 2.0
matrix[1, 1] = matrix[0, 0]
matrix[2, 2] = sinc_smax
matrix[3, 3] = matrix[0, 0]
matrix[4, 4] = matrix[0, 0]
matrix[5, 5] = matrix[2, 2]
matrix[6, 6] = matrix[2, 2]
matrix[7, 7] = matrix[2, 2]
matrix[8, 8] = 1.0
# Off diagonal set 1.
matrix[0, 4] = matrix[4, 0] = -(sinc_2smax - 1.0) / 2.0
# Off diagonal set 2.
matrix[1, 3] = matrix[3, 1] = -matrix[0, 4]
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_2nd_matrix_iso_cone(matrix, Rx2_eigen, cone_theta, sigma_max):
"""Generate the rotated 2nd degree Frame Order matrix for the isotropic cone.
The cone axis is assumed to be parallel to the z-axis in the eigenframe.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param cone_theta: The cone opening angle.
@type cone_theta: float
@param sigma_max: The maximum torsion angle.
@type sigma_max: float
"""
# Populate the Frame Order matrix in the eigenframe.
populate_2nd_eigenframe_iso_cone(matrix, cone_theta, sigma_max)
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_2nd_matrix_rotor(matrix, Rx2_eigen, smax):
"""Generate the rotated 2nd degree Frame Order matrix for the rotor model.
The cone axis is assumed to be parallel to the z-axis in the eigenframe.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param smax: The maximum torsion angle.
@type smax: float
"""
# Zeros.
matrix[:] = 0.0
# Repetitive trig calculations.
sinc_smax = sinc(smax / pi)
sinc_2smax = sinc(2.0 * smax / pi)
# Diagonal.
matrix[0, 0] = (sinc_2smax + 1.0) / 2.0
matrix[1, 1] = matrix[0, 0]
matrix[2, 2] = sinc_smax
matrix[3, 3] = matrix[0, 0]
matrix[4, 4] = matrix[0, 0]
matrix[5, 5] = matrix[2, 2]
matrix[6, 6] = matrix[2, 2]
matrix[7, 7] = matrix[2, 2]
matrix[8, 8] = 1.0
# Off diagonal set 1.
matrix[0, 4] = matrix[4, 0] = -(sinc_2smax - 1.0) / 2.0
# Off diagonal set 2.
matrix[1, 3] = matrix[3, 1] = -matrix[0, 4]
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_2nd_matrix_iso_cone_free_rotor(matrix, Rx2_eigen, s1):
"""Generate the rotated 2nd degree Frame Order matrix for the free rotor isotropic cone.
The cone axis is assumed to be parallel to the z-axis in the eigenframe. In this model, the three order parameters are defined as::
S1 = S2,
S3 = 0
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param s1: The cone order parameter.
@type s1: float
"""
# Populate the Frame Order matrix in the eigenframe.
populate_2nd_eigenframe_iso_cone_free_rotor(matrix, s1)
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_1st_matrix_rotor(matrix, R_eigen, sigma_max):
"""Generate the 1st degree Frame Order matrix for the rotor model.
@param matrix: The Frame Order matrix, 1st degree to be populated.
@type matrix: numpy 3D, rank-2 array
@param R_eigen: The eigenframe rotation matrix.
@type R_eigen: numpy 3D, rank-2 array
@param sigma_max: The maximum torsion angle.
@type sigma_max: float
"""
# Zeros.
matrix[:] = 0.0
# The sinc value.
val = sinc(sigma_max / pi)
# Diagonal values.
matrix[0, 0] = val
matrix[1, 1] = val
matrix[2, 2] = 1.0
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, R_eigen)
def compile_1st_matrix_double_rotor(matrix, R_eigen, smax1, smax2):
"""Generate the 1st degree Frame Order matrix for the double rotor model.
@param matrix: The Frame Order matrix, 1st degree to be populated.
@type matrix: numpy 3D, rank-2 array
@param R_eigen: The eigenframe rotation matrix.
@type R_eigen: numpy 3D, rank-2 array
@param smax1: The maximum torsion angle for the first rotor.
@type smax1: float
@param smax2: The maximum torsion angle for the second rotor.
@type smax2: float
"""
# Repetitive trig calculations.
sinc_smax1 = sinc(smax1 / pi)
sinc_smax2 = sinc(smax2 / pi)
# Numerical integration of phi of each element.
matrix[0, 0] = sinc_smax1
matrix[1, 1] = sinc_smax2
matrix[2, 2] = sinc_smax1 * sinc_smax2
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, R_eigen)
def compile_2nd_matrix_iso_cone_free_rotor(matrix, Rx2_eigen, tmax):
"""Generate the rotated 2nd degree Frame Order matrix for the free rotor isotropic cone.
The cone axis is assumed to be parallel to the z-axis in the eigenframe.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param tmax: The cone opening angle.
@type tmax: float
"""
# Zeros.
matrix[:] = 0.0
# Repetitive trig calculations.
cos_tmax = cos(tmax)
cos_tmax2 = cos_tmax**2
# Diagonal.
matrix[0, 0] = matrix[4, 4] = (cos_tmax2 + cos_tmax + 4.0) / 12.0
matrix[1, 1] = matrix[3, 3] = (cos_tmax + 1.0) / 4.0
matrix[8, 8] = (cos_tmax2 + cos_tmax + 1.0) / 3.0
# Off diagonal set 1.
matrix[0, 4] = matrix[4, 0] = matrix[0, 0]
matrix[0, 8] = matrix[8, 0] = -(cos_tmax2 + cos_tmax - 2.0) / 6.0
matrix[4, 8] = matrix[8, 4] = matrix[0, 8]
# Off diagonal set 2.
matrix[1, 3] = matrix[3, 1] = -matrix[1, 1]
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_1st_matrix_rotor(matrix, R_eigen, sigma_max):
"""Generate the 1st degree Frame Order matrix for the rotor model.
@param matrix: The Frame Order matrix, 1st degree to be populated.
@type matrix: numpy 3D, rank-2 array
@param R_eigen: The eigenframe rotation matrix.
@type R_eigen: numpy 3D, rank-2 array
@param sigma_max: The maximum torsion angle.
@type sigma_max: float
"""
# Zeros.
matrix[:] = 0.0
# The sinc value.
val = sinc(sigma_max/pi)
# Diagonal values.
matrix[0, 0] = val
matrix[1, 1] = val
matrix[2, 2] = 1.0
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, R_eigen)
def compile_1st_matrix_double_rotor(matrix, R_eigen, smax1, smax2):
"""Generate the 1st degree Frame Order matrix for the double rotor model.
@param matrix: The Frame Order matrix, 1st degree to be populated.
@type matrix: numpy 3D, rank-2 array
@param R_eigen: The eigenframe rotation matrix.
@type R_eigen: numpy 3D, rank-2 array
@param smax1: The maximum torsion angle for the first rotor.
@type smax1: float
@param smax2: The maximum torsion angle for the second rotor.
@type smax2: float
"""
# Repetitive trig calculations.
sinc_smax1 = sinc(smax1/pi)
sinc_smax2 = sinc(smax2/pi)
# Numerical integration of phi of each element.
matrix[0, 0] = sinc_smax1
matrix[1, 1] = sinc_smax2
matrix[2, 2] = sinc_smax1 * sinc_smax2
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, R_eigen)
def compile_2nd_matrix_pseudo_ellipse_torsionless(matrix, Rx2_eigen, theta_x,
theta_y):
"""Generate the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param theta_x: The cone opening angle along x.
@type theta_x: float
@param theta_y: The cone opening angle along y.
@type theta_y: float
"""
# The rigid case.
if theta_x == 0.0:
# Set up the matrix as the identity.
matrix[:] = 0.0
for i in range(len(matrix)):
matrix[i, i] = 1.0
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
# The surface area normalisation factor.
fact = 6.0 * pec(theta_x, theta_y)
fact2 = 0.5 * fact
# Invert.
if fact == 0.0:
fact = 1e100
fact2 = 1e100
else:
fact = 1.0 / fact
fact2 = 1.0 / fact2
# Diagonal.
matrix[0, 0] = fact2 * (3.0 * pi +
quad(part_int_daeg2_pseudo_ellipse_torsionless_00,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
matrix[1, 1] = fact * (2.0 * pi +
quad(part_int_daeg2_pseudo_ellipse_torsionless_11,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
matrix[2, 2] = fact * (5.0 * pi -
quad(part_int_daeg2_pseudo_ellipse_torsionless_22,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
matrix[3, 3] = matrix[1, 1]
matrix[4, 4] = fact2 * (3.0 * pi +
quad(part_int_daeg2_pseudo_ellipse_torsionless_44,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
matrix[5, 5] = fact * (5.0 * pi -
quad(part_int_daeg2_pseudo_ellipse_torsionless_55,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
matrix[6, 6] = matrix[2, 2]
matrix[7, 7] = matrix[5, 5]
matrix[8, 8] = fact2 * (2.0 * pi -
quad(part_int_daeg2_pseudo_ellipse_torsionless_88,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
# Off diagonal set 1.
matrix[0, 4] = matrix[
4,
0] = fact2 * (pi + quad(part_int_daeg2_pseudo_ellipse_torsionless_04,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
matrix[0, 8] = matrix[8, 0] = fact2 * (
2.0 * pi + quad(part_int_daeg2_pseudo_ellipse_torsionless_08,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
matrix[4, 8] = matrix[8, 4] = fact2 * (
2.0 * pi + quad(part_int_daeg2_pseudo_ellipse_torsionless_48,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
# Off diagonal set 2.
matrix[1, 3] = matrix[3, 1] = matrix[0, 4]
matrix[2, 6] = matrix[6, 2] = -matrix[0, 8]
matrix[5, 7] = matrix[7, 5] = -matrix[4, 8]
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
def compile_2nd_matrix_pseudo_ellipse(matrix, Rx2_eigen, theta_x, theta_y,
sigma_max):
"""Generate the 2nd degree Frame Order matrix for the pseudo-ellipse.
@param matrix: The Frame Order matrix, 2nd degree to be populated.
@type matrix: numpy 9D, rank-2 array
@param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
@type Rx2_eigen: numpy 9D, rank-2 array
@param theta_x: The cone opening angle along x.
@type theta_x: float
@param theta_y: The cone opening angle along y.
@type theta_y: float
@param sigma_max: The maximum torsion angle.
@type sigma_max: float
"""
# The rigid case.
if theta_x == 0.0 and sigma_max == 0.0:
# Set up the matrix as the identity.
matrix[:] = 0.0
for i in range(len(matrix)):
matrix[i, i] = 1.0
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)
# The surface area normalisation factor.
fact = 12.0 * pec(theta_x, theta_y)
# Invert.
if fact == 0.0:
fact = 1e100
else:
fact = 1.0 / fact
# Repetitive calculations.
sinc_smax = sinc(sigma_max / pi)
sinc_2smax = sinc(2.0 * sigma_max / pi)
# Diagonal.
matrix[0, 0] = fact * (4.0 * pi * (sinc_2smax + 2.0) +
quad(part_int_daeg2_pseudo_ellipse_00,
-pi,
pi,
args=(theta_x, theta_y, sinc_2smax),
full_output=1)[0])
matrix[1, 1] = fact * (4.0 * pi * sinc_2smax +
quad(part_int_daeg2_pseudo_ellipse_11,
-pi,
pi,
args=(theta_x, theta_y, sinc_2smax),
full_output=1)[0])
matrix[2, 2] = fact * 2.0 * sinc_smax * (
5.0 * pi - quad(part_int_daeg2_pseudo_ellipse_22,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
matrix[3, 3] = matrix[1, 1]
matrix[4, 4] = fact * (4.0 * pi * (sinc_2smax + 2.0) +
quad(part_int_daeg2_pseudo_ellipse_44,
-pi,
pi,
args=(theta_x, theta_y, sinc_2smax),
full_output=1)[0])
matrix[5, 5] = fact * 2.0 * sinc_smax * (
5.0 * pi - quad(part_int_daeg2_pseudo_ellipse_55,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
matrix[6, 6] = matrix[2, 2]
matrix[7, 7] = matrix[5, 5]
matrix[8,
8] = 4.0 * fact * (2.0 * pi - quad(part_int_daeg2_pseudo_ellipse_88,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
# Off diagonal set 1.
matrix[0, 4] = fact * (4.0 * pi * (2.0 - sinc_2smax) +
quad(part_int_daeg2_pseudo_ellipse_04,
-pi,
pi,
args=(theta_x, theta_y, sinc_2smax),
full_output=1)[0])
matrix[4, 0] = fact * (4.0 * pi * (2.0 - sinc_2smax) +
quad(part_int_daeg2_pseudo_ellipse_40,
-pi,
pi,
args=(theta_x, theta_y, sinc_2smax),
full_output=1)[0])
matrix[0,
8] = 4.0 * fact * (2.0 * pi - quad(part_int_daeg2_pseudo_ellipse_08,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
matrix[8, 0] = fact * (8.0 * pi + quad(part_int_daeg2_pseudo_ellipse_80,
-pi,
pi,
args=(theta_x, theta_y, sinc_2smax),
full_output=1)[0])
matrix[4,
8] = 4.0 * fact * (2.0 * pi - quad(part_int_daeg2_pseudo_ellipse_48,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
matrix[8, 4] = fact * (8.0 * pi - quad(part_int_daeg2_pseudo_ellipse_84,
-pi,
pi,
args=(theta_x, theta_y, sinc_2smax),
full_output=1)[0])
# Off diagonal set 2.
matrix[1,
3] = matrix[3,
1] = fact * (4.0 * pi * sinc_2smax +
quad(part_int_daeg2_pseudo_ellipse_13,
-pi,
pi,
args=(theta_x, theta_y, sinc_2smax),
full_output=1)[0])
matrix[2, 6] = matrix[6, 2] = -fact * 4.0 * sinc_smax * (
2.0 * pi + quad(part_int_daeg2_pseudo_ellipse_26,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
matrix[5, 7] = matrix[7, 5] = -fact * 4.0 * sinc_smax * (
2.0 * pi + quad(part_int_daeg2_pseudo_ellipse_57,
-pi,
pi,
args=(theta_x, theta_y),
full_output=1)[0])
# Rotate and return the frame order matrix.
return rotate_daeg(matrix, Rx2_eigen)