def random_gonio():
# make a random rotation axis with a random setting matrix
axis = matrix.col(flex.random_double_point_on_sphere())
fixed_rotation = matrix.sqr((1, 0, 0, 0, 1, 0, 0, 0, 1))
setting_rotation = matrix.sqr(flex.random_double_r3_rotation_matrix())
goniometer = Goniometer(axis, fixed_rotation, setting_rotation)
return goniometer
def test_uniform_rotation_matrix(N=10000, choice=2, verbose=False):
"""
The surface integral of a spherical harmonic function with its conjugate
should be 1. (http://mathworld.wolfram.com/SphericalHarmonic.html, Eq 7)
From Mathematica,
l = 10;
m = 10;
y = SphericalHarmonicY[l, m, \[Theta], \[Phi]];
Integrate[y*Conjugate[y]*Sin[\[Theta]], {\[Theta], 0, Pi}, {\[Phi], 0, 2*Pi}]
should yield 1.
By picking uniformly random points on a sphere, the surface integral can be
numerically approximated.
The results in the comments below are for N = 1 000 000.
"""
if (choice == 0):
# l=1, m=1
# result = (0.883199394206+0j) (0.883824001444+0j)
lm = 1
c = -0.5 * math.sqrt(1.5 / math.pi)
elif (choice == 1):
# l = 5, m = 5
# result = (0.959557841214+0j) (0.959331535539+0j)
lm = 5
c = -(3 / 32) * math.sqrt(77 / math.pi)
else:
# l = 10, m = 10
# result = (0.977753926603+0j) (0.97686871766+0j)
lm = 10
c = (1 / 1024) * math.sqrt(969969 / math.pi)
result = [0.0, 0.0]
for i in range(N):
R = [
matrix.sqr(flex.random_double_r3_rotation_matrix()),
matrix.sqr(flex.random_double_r3_rotation_matrix_arvo_1992())
]
for j in xrange(len(result)):
result[j] += add_point(lm, c, R[j])
# multipy by area at the end, each point has an area of 4pi/N
point_area = 4.0 * math.pi / N # surface area of unit sphere / number of points
for i in xrange(len(result)):
result[i] = point_area * result[i]
if (verbose):
print result[i],
if (verbose):
print
assert (result[0].real > 0.85)
assert (result[0].real < 1.15)
assert (result[1].real > 0.85)
assert (result[1].real < 1.15)
def test_uniform_rotation_matrix(N=10000,choice=2,verbose=False):
"""
The surface integral of a spherical harmonic function with its conjugate
should be 1. (http://mathworld.wolfram.com/SphericalHarmonic.html, Eq 7)
From Mathematica,
l = 10;
m = 10;
y = SphericalHarmonicY[l, m, \[Theta], \[Phi]];
Integrate[y*Conjugate[y]*Sin[\[Theta]], {\[Theta], 0, Pi}, {\[Phi], 0, 2*Pi}]
should yield 1.
By picking uniformly random points on a sphere, the surface integral can be
numerically approximated.
The results in the comments below are for N = 1 000 000.
"""
if (choice == 0):
# l=1, m=1
# result = (0.883199394206+0j) (0.883824001444+0j)
lm = 1
c = -0.5 * math.sqrt(1.5/math.pi)
elif (choice == 1):
# l = 5, m = 5
# result = (0.959557841214+0j) (0.959331535539+0j)
lm = 5
c = -(3/32) * math.sqrt(77/math.pi)
else:
# l = 10, m = 10
# result = (0.977753926603+0j) (0.97686871766+0j)
lm = 10
c = (1/1024) * math.sqrt(969969/math.pi)
result = [ 0.0, 0.0 ]
for i in range(N):
R = [ matrix.sqr(flex.random_double_r3_rotation_matrix()),
matrix.sqr(flex.random_double_r3_rotation_matrix_arvo_1992()) ]
for j in xrange(len(result)):
result[j] += add_point(lm,c,R[j])
# multipy by area at the end, each point has an area of 4pi/N
point_area = 4.0*math.pi/N # surface area of unit sphere / number of points
for i in xrange(len(result)):
result[i] = point_area * result[i]
if (verbose):
print result[i],
if (verbose):
print
assert(result[0].real > 0.85)
assert(result[0].real < 1.15)
assert(result[1].real > 0.85)
assert(result[1].real < 1.15)
def exercise_axis_and_angle(axis_range=2,
angle_max_division=12,
angle_min_power=-30):
from_matrix = scitbx.math.r3_rotation_axis_and_angle_from_matrix(
r=[1, 0, 0, 0, 1, 0, 0, 0, 1])
assert approx_equal(from_matrix.axis, [1 / 3**0.5] * 3)
assert approx_equal(from_matrix.angle(deg=True), 0)
assert approx_equal(from_matrix.angle(deg=False), 0)
from_matrix = scitbx.math.r3_rotation_axis_and_angle_from_matrix(r=[0] * 9)
assert approx_equal(from_matrix.axis, [0, 0, 1])
assert approx_equal(from_matrix.angle(deg=True), 90)
assert approx_equal(from_matrix.angle(deg=False), math.pi / 2)
#
angles = []
for d in range(1, angle_max_division + 1):
angles.append(360 / d)
angles.append(-360 / d)
for p in range(-angle_min_power + 1):
angles.append(10**(-p))
angles.append(-10**(-p))
hex_orth = matrix.sqr([
8.7903631196301042, -4.3951815598150503, 0, 0, 7.6126777700894994, 0,
0, 0, 14.943617303371177
])
for u in range(-axis_range, axis_range + 1):
for v in range(-axis_range, axis_range + 1):
for w in range(axis_range + 1):
for axis in [(u, v, w), (hex_orth * matrix.col(
(u, v, w))).elems]:
for angle in angles:
try:
r = scitbx.math.r3_rotation_axis_and_angle_as_matrix(
axis=axis, angle=angle, deg=True)
except RuntimeError:
assert axis == (0, 0, 0)
try:
scitbx.math.r3_rotation_axis_and_angle_as_unit_quaternion(
axis=axis, angle=angle, deg=True)
except RuntimeError:
pass
else:
raise Exception_expected
else:
q = scitbx.math.r3_rotation_axis_and_angle_as_unit_quaternion(
axis=axis, angle=angle, deg=True)
assert approx_equal(abs(matrix.col(q)), 1)
rq = scitbx.math.r3_rotation_unit_quaternion_as_matrix(
q=q)
assert approx_equal(rq, r)
from_matrix = scitbx.math.r3_rotation_axis_and_angle_from_matrix(
r=r)
rr = from_matrix.as_matrix()
assert approx_equal(rr, r)
qq = from_matrix.as_unit_quaternion()
assert approx_equal(abs(matrix.col(qq)), 1)
rq = scitbx.math.r3_rotation_unit_quaternion_as_matrix(
q=qq)
assert approx_equal(rq, r)
qq = scitbx.math.r3_rotation_matrix_as_unit_quaternion(
r=r)
assert approx_equal(abs(matrix.col(qq)), 1)
rq = scitbx.math.r3_rotation_unit_quaternion_as_matrix(
q=qq)
assert approx_equal(rq, r)
rm = matrix.sqr(r)
assert rm.is_r3_rotation_matrix()
qm = rm.r3_rotation_matrix_as_unit_quaternion()
assert approx_equal(abs(qm), 1)
assert approx_equal(qm, qq)
rqmm = qm.unit_quaternion_as_r3_rotation_matrix()
assert approx_equal(rqmm, r)
for deg in [False, True]:
rr = scitbx.math.r3_rotation_axis_and_angle_as_matrix(
axis=from_matrix.axis,
angle=from_matrix.angle(deg=deg),
deg=deg,
min_axis_length=1 - 1.e-5)
assert approx_equal(rr, r)
qq = scitbx.math.r3_rotation_axis_and_angle_as_unit_quaternion(
axis=from_matrix.axis,
angle=from_matrix.angle(deg=deg),
deg=deg,
min_axis_length=1 - 1.e-5)
qq = from_matrix.as_unit_quaternion()
assert approx_equal(abs(matrix.col(qq)), 1)
rq = scitbx.math.r3_rotation_unit_quaternion_as_matrix(
q=qq)
assert approx_equal(rq, r)
for conv in [
scitbx.math.
r3_rotation_axis_and_angle_as_matrix,
scitbx.math.
r3_rotation_axis_and_angle_as_unit_quaternion
]:
try:
conv(axis=from_matrix.axis,
angle=from_matrix.angle(deg=deg),
deg=deg,
min_axis_length=1 + 1.e-5)
except RuntimeError:
pass
else:
raise Exception_expected
#
for i_trial in range(100):
r = flex.random_double_r3_rotation_matrix()
from_matrix = scitbx.math.r3_rotation_axis_and_angle_from_matrix(r=r)
rr = from_matrix.as_matrix()
assert approx_equal(rr, r)
assert approx_equal(math.cos(from_matrix.angle()),
(r[0] + r[4] + r[8] - 1) / 2)
q = matrix.col(from_matrix.as_unit_quaternion())
assert approx_equal(abs(q), 1)
rq = scitbx.math.r3_rotation_unit_quaternion_as_matrix(q=q)
assert approx_equal(rq, r)
rq = matrix.col(q).unit_quaternion_as_r3_rotation_matrix()
assert approx_equal(rq, r)
def exercise_axis_and_angle(
axis_range=2,
angle_max_division=12,
angle_min_power=-30):
from_matrix = scitbx.math.r3_rotation_axis_and_angle_from_matrix(
r=[1,0,0,0,1,0,0,0,1])
assert approx_equal(from_matrix.axis, [1/3**0.5]*3)
assert approx_equal(from_matrix.angle(deg=True), 0)
assert approx_equal(from_matrix.angle(deg=False), 0)
from_matrix = scitbx.math.r3_rotation_axis_and_angle_from_matrix(r=[0]*9)
assert approx_equal(from_matrix.axis, [0,0,1])
assert approx_equal(from_matrix.angle(deg=True), 90)
assert approx_equal(from_matrix.angle(deg=False), math.pi/2)
#
angles = []
for d in xrange(1,angle_max_division+1):
angles.append(360/d)
angles.append(-360/d)
for p in xrange(-angle_min_power+1):
angles.append(10**(-p))
angles.append(-10**(-p))
hex_orth = matrix.sqr([
8.7903631196301042, -4.3951815598150503, 0,
0, 7.6126777700894994, 0,
0, 0, 14.943617303371177])
for u in xrange(-axis_range, axis_range+1):
for v in xrange(-axis_range, axis_range+1):
for w in xrange(axis_range+1):
for axis in [(u,v,w), (hex_orth*matrix.col((u,v,w))).elems]:
for angle in angles:
try:
r = scitbx.math.r3_rotation_axis_and_angle_as_matrix(
axis=axis, angle=angle, deg=True)
except RuntimeError:
assert axis == (0,0,0)
try:
scitbx.math.r3_rotation_axis_and_angle_as_unit_quaternion(
axis=axis, angle=angle, deg=True)
except RuntimeError: pass
else: raise Exception_expected
else:
q = scitbx.math.r3_rotation_axis_and_angle_as_unit_quaternion(
axis=axis, angle=angle, deg=True)
assert approx_equal(abs(matrix.col(q)), 1)
rq = scitbx.math.r3_rotation_unit_quaternion_as_matrix(q=q)
assert approx_equal(rq, r)
from_matrix = scitbx.math.r3_rotation_axis_and_angle_from_matrix(
r=r)
rr = from_matrix.as_matrix()
assert approx_equal(rr, r)
qq = from_matrix.as_unit_quaternion()
assert approx_equal(abs(matrix.col(qq)), 1)
rq = scitbx.math.r3_rotation_unit_quaternion_as_matrix(q=qq)
assert approx_equal(rq, r)
qq = scitbx.math.r3_rotation_matrix_as_unit_quaternion(r=r)
assert approx_equal(abs(matrix.col(qq)), 1)
rq = scitbx.math.r3_rotation_unit_quaternion_as_matrix(q=qq)
assert approx_equal(rq, r)
rm = matrix.sqr(r)
assert rm.is_r3_rotation_matrix()
qm = rm.r3_rotation_matrix_as_unit_quaternion()
assert approx_equal(abs(qm), 1)
assert approx_equal(qm, qq)
rqmm = qm.unit_quaternion_as_r3_rotation_matrix()
assert approx_equal(rqmm, r)
for deg in [False, True]:
rr = scitbx.math.r3_rotation_axis_and_angle_as_matrix(
axis=from_matrix.axis,
angle=from_matrix.angle(deg=deg),
deg=deg,
min_axis_length=1-1.e-5)
assert approx_equal(rr, r)
qq = scitbx.math.r3_rotation_axis_and_angle_as_unit_quaternion(
axis=from_matrix.axis,
angle=from_matrix.angle(deg=deg),
deg=deg,
min_axis_length=1-1.e-5)
qq = from_matrix.as_unit_quaternion()
assert approx_equal(abs(matrix.col(qq)), 1)
rq = scitbx.math.r3_rotation_unit_quaternion_as_matrix(q=qq)
assert approx_equal(rq, r)
for conv in [
scitbx.math.r3_rotation_axis_and_angle_as_matrix,
scitbx.math.r3_rotation_axis_and_angle_as_unit_quaternion]:
try:
conv(
axis=from_matrix.axis,
angle=from_matrix.angle(deg=deg),
deg=deg,
min_axis_length=1+1.e-5)
except RuntimeError: pass
else: raise Exception_expected
#
for i_trial in xrange(100):
r = flex.random_double_r3_rotation_matrix()
from_matrix = scitbx.math.r3_rotation_axis_and_angle_from_matrix(r=r)
rr = from_matrix.as_matrix()
assert approx_equal(rr, r)
assert approx_equal(math.cos(from_matrix.angle()), (r[0]+r[4]+r[8]-1)/2)
q = matrix.col(from_matrix.as_unit_quaternion())
assert approx_equal(abs(q), 1)
rq = scitbx.math.r3_rotation_unit_quaternion_as_matrix(q=q)
assert approx_equal(rq, r)
rq = matrix.col(q).unit_quaternion_as_r3_rotation_matrix()
assert approx_equal(rq, r)