def __compute_t_r(self,n_lobes,lobe_idx,H_in,Hr,d_eval,p4roots,lead_cf):
from pyranha import math
from mpmath import asin, sqrt, ellipf, mpf
assert(n_lobes == 1 or n_lobes == 2)
C = -lead_cf
assert(C > 0)
# First determine if we are integrating in the upper or lower plane.
# NOTE: we don't care about eps, we are only interested in the sign.
if (self.__F1 * math.sin(pt('h_\\ast'))).trim().evaluate(d_eval) > 0:
sign = 1
else:
sign = -1
if n_lobes == 2:
assert(lobe_idx == 0 or lobe_idx == 1)
r0,r1,r2,r3 = p4roots
# k is the same in both cases.
k = sqrt(((r3-r2)*(r1-r0))/((r3-r1)*(r2-r0)))
if lobe_idx == 0:
assert(Hr == r0)
phi = asin(sqrt(((r3-r1)*(H_in-r0))/((r1-r0)*(r3-H_in))))
else:
assert(Hr == r2)
phi = asin(sqrt(((r3-r1)*(H_in-r2))/((H_in-r1)*(r3-r2))))
return -sign * mpf(2) / sqrt(C * (r3 - r1) * (r2 - r0)) * ellipf(phi,k**2)
else:
# TODO: single lobe case.
assert(False)
pass
def _view_cone_calc(lat_geoc, lon_geoc, sat_pos, sat_vel, q_max, m):
"""Semi-private: Performs the viewing cone visibility calculation for the day defined by m.
Note: This function is based on a paper titled "rapid satellite-to-site visibility determination
based on self-adaptive interpolation technique" with some variation to account for interaction
of viewing cone with the satellite orbit.
Args:
lat_geoc (float): site location in degrees at the start of POI
lon_geoc (float): site location in degrees at the start of POI
sat_pos (Vector3D): position of satellite (at the same time as sat_vel)
sat_vel (Vector3D): velocity of satellite (at the same time as sat_pos)
q_max (float): maximum orbital radius
m (int): interval offsets (number of days after initial condition)
Returns:
Returns 4 numbers representing times at which the orbit is tangent to the viewing cone,
Raises:
ValueError: if any of the 4 formulas has a complex answer. This happens when the orbit and
viewing cone do not intersect or only intersect twice.
Note: With more analysis it should be possible to find a correct interval even in the case
where there are only two intersections but this is beyond the current scope of the project.
"""
lat_geoc = (lat_geoc * mp.pi) / 180
lon_geoc = (lon_geoc * mp.pi) / 180
# P vector (also referred to as orbital angular momentum in the paper) calculations
p_unit_x, p_unit_y, p_unit_z = cross(sat_pos, sat_vel) / (mp.norm(sat_pos) * mp.norm(sat_vel))
# Following are equations from Viewing cone section of referenced paper
r_site_magnitude = earth_radius_at_geocetric_lat(lat_geoc)
gamma1 = THETA_NAUGHT + mp.asin((r_site_magnitude * mp.sin((mp.pi / 2) + THETA_NAUGHT)) / q_max)
gamma2 = mp.pi - gamma1
# Note: atan2 instead of atan to get the correct quadrant.
arctan_term = mp.atan2(p_unit_x, p_unit_y)
arcsin_term_gamma, arcsin_term_gamma2 = [(mp.asin((mp.cos(gamma) - p_unit_z * mp.sin(lat_geoc))
/ (mp.sqrt((p_unit_x ** 2) + (p_unit_y ** 2)) *
mp.cos(lat_geoc)))) for gamma in [gamma1, gamma2]]
angle_1 = (arcsin_term_gamma - lon_geoc - arctan_term + 2 * mp.pi * m)
angle_2 = (mp.pi - arcsin_term_gamma - lon_geoc - arctan_term + 2 * mp.pi * m)
angle_3 = (arcsin_term_gamma2 - lon_geoc - arctan_term + 2 * mp.pi * m)
angle_4 = (mp.pi - arcsin_term_gamma2 - lon_geoc - arctan_term + 2 * mp.pi * m)
angles = [angle_1, angle_2, angle_3, angle_4]
# Check for complex answers
if any([not isinstance(angle, mp.mpf) for angle in angles]):
raise ValueError()
# Map all angles to 0 to 2*pi
for idx in range(len(angles)):
while angles[idx] < 0:
angles[idx] += 2 * mp.pi
while angles[idx] > 2 * mp.pi:
angles[idx] -= 2 * mp.pi
# Calculate the corresponding time for each angle and return
return [mp.nint((1 / ANGULAR_VELOCITY_EARTH) * angle) for angle in angles]
def Pinv(self, P):
from mpmath import ellipf, sqrt, asin, acos, mpc, mpf
Delta = self.Delta
e1, e2, e3 = self.__roots
if self.__ng3:
P = -P
if Delta > 0:
m = (e2 - e3) / (e1 - e3)
retval = (1 / sqrt(e1 - e3)) * ellipf(
asin(sqrt((e1 - e3) / (P - e3))), m=m)
elif Delta < 0:
H2 = (sqrt((e2 - e3) * (e2 - e1))).real
assert (H2 > 0)
m = mpf(1) / mpf(2) - 3 * e2 / (4 * H2)
retval = 1 / (2 * sqrt(H2)) * ellipf(acos(
(e2 - P + H2) / (e2 - P - H2)),
m=m)
else:
g2, g3 = self.__invariants
if g2 == 0 and g3 == 0:
retval = 1 / sqrt(P)
else:
c = e1 / 2
retval = (1 / sqrt(3 * c)) * asin(sqrt((3 * c) / (P + c)))
if self.__ng3:
retval /= mpc(0, 1)
alpha, beta, _, _ = self.reduce_to_fpp(retval)
T1, T2 = self.periods
return T1 * alpha + T2 * beta
def Pinv(self,P): from mpmath import ellipf, sqrt, asin, acos, mpc, mpf Delta = self.Delta e1, e2, e3 = self.__roots if self.__ng3: P = -P if Delta > 0: m = (e2 - e3) / (e1 - e3) retval = (1 / sqrt(e1 - e3)) * ellipf(asin(sqrt((e1 - e3)/(P - e3))),m=m) elif Delta < 0: H2 = (sqrt((e2 - e3) * (e2 - e1))).real assert(H2 > 0) m = mpf(1) / mpf(2) - 3 * e2 / (4 * H2) retval = 1 / (2 * sqrt(H2)) * ellipf(acos((e2-P+H2)/(e2-P-H2)),m=m) else: g2, g3 = self.__invariants if g2 == 0 and g3 == 0: retval = 1 / sqrt(P) else: c = e1 / 2 retval = (1 / sqrt(3 * c)) * asin(sqrt((3 * c)/(P + c))) if self.__ng3: retval /= mpc(0,1) alpha, beta, _, _ = self.reduce_to_fpp(retval) T1, T2 = self.periods return T1 * alpha + T2 * beta
def set_polar_coordinate(self):
from math import asin, cos
self.phi = asin(self.z / self.norm)
sintheta = self.y / (self.norm * cos(self.phi))
self.theta = asin(sintheta)
self.adjust_polar_coordinate()
def set_polar_coordinate(self):
from mpmath import asin, cos, chop
self.phi = asin(self.z / self.norm)
sintheta = chop(self.y / (self.norm * cos(self.phi)))
theta = asin(sintheta)
self.theta = theta
def inverse_sine():
"""Inverse sine function.
The inverse ine or arcsine of x, sin**-1(x), since -1 <= sin(x) <= 1 for
real x, the inverse sine is real-valued for -1 <= x <= 1, or defined
monotonically increasing assuming values between -pi/2 and pi/2.
"""
import mpmath
print(mpmath.asin(-1), mpmath.asin(0), mpmath.asin(1))
def do_approximation(self):
epsilonp = np.sqrt(np.power(10, self.Ap / 10) - 1)
gp = np.power(10, -self.Ap / 20)
k1 = np.sqrt((np.power(10, self.Ap / 10) - 1) /
(np.power(10, self.Ao / 10) - 1))
k = 1 / self.wan
a = mp.asin(1j / epsilonp)
vo = mp.ellipf(1j / epsilonp, k1) / (1j * self.n)
self.n = np.ceil(
special.ellipk(np.sqrt(1 - np.power(k1, 2))) * special.ellipk(k) /
(special.ellipk(k1) * special.ellipk(np.sqrt(1 - np.power(k, 2)))))
for i in range(1, int(np.floor(self.n / 2)) + 1):
cd = mp.ellipfun('cd', (2 * i - 1) / self.n, k)
zero = (1j / (float(k * cd.real) + 1j * float(k * cd.imag)))
self.num = self.num * np.poly1d([1 / zero, 1])
self.num = self.num * np.poly1d([1 / np.conj(zero), 1])
cd = mp.ellipfun('cd', (2 * i - 1) / self.n - 1j * vo, k)
pole = 1j * (float(cd.real) + 1j * float(cd.imag))
if np.real(pole) <= 0:
self.den = self.den * np.poly1d([-1 / pole, 1])
self.den = self.den * np.poly1d([-1 / np.conj(pole), 1])
if np.mod(self.n, 2) == 1:
sn = 1j * mp.ellipfun('sn', 1j * vo, k)
pole = 1j * (float(sn.real) + 1j * float(sn.imag))
if np.real(pole) <= 0:
self.den = self.den * np.poly1d([-1 / pole, 1])
self.den = self.den * np.poly1d([-1 / np.conj(pole), 1])
self.zeroes = np.roots(self.num)
self.poles = np.roots(self.den)
self.aprox_gain = np.power(gp, 1 - (self.n - 2 * np.floor(self.n / 2)))
self.num = self.num * self.aprox_gain
self.norm_sys = signal.TransferFunction(
self.num, self.den) #Filter system is obtained
def get_dtheta_at(self, phi, covered_r):
from math import asin, cos, sqrt, pi
if covered_r >= sqrt(2) * cos(phi):
# In this case, the confidence radius is so large such that it covers the whole orbit and the north pole.
dtheta = pi
else:
dtheta = 2 * asin((.5 * covered_r) / cos(phi))
return dtheta
def one_side(x,y):
m = ((-1 + x)*(1 + y))/((1 + x)*(-1 + y))
k = sqrt(m)
u = asin(1/k)
EE = ellipe(m)
EF = re(ellipf(u,m))
n = (-1 + x)/(-1 + y)
EPI = ellippi(n/m,1/m)/k
#EPI = ellippi(n,u,m)
return re(-(EE*(1 + x)*(-1 + y) + (x - y)*(EF + EPI*(x-y) + EF*y))/sqrt(((1 + x)*(1 - y))))
def equiv_sin(x, range1, range2, radians=None, degrees=None, accuracy=2):
mpmath.mp.dps = accuracy + 1
range1, range2 = TrigEquivAngle.deg_or_rad([range1, range2], degrees)
firstSinAngle = mpmath.asin(x)
secondSinAngle = mpmath.pi - firstSinAngle
angles = TrigEquivAngle.create_equiv_angle_list(
[firstSinAngle, secondSinAngle], range1, range2, 2 * mpmath.pi)
return (TrigEquivAngle.clean_angles_rad_deg(angles, radians, degrees,
accuracy))
def go_down(phi, r):
# It returns the orbit new_phi which is below phi and has distance r
# (on the sphere on which our directions are located)
from math import asin, sin, pi
if sin(phi) - r >= -1:
new_phi = asin(sin(phi) - r)
else:
new_phi = -pi / 2
return new_phi
def __compute_tau_eta(self): # Gradshtein 3.131.5. from mpmath import sqrt, asin, ellipf u = self.__init_coordinates[1]**2 / 2 c,b,a = self.__roots_eta k = asin(sqrt(((a - c) * (u - b)) / ((a - b) * (u - c)))) p = sqrt((a - b) / (a - c)) retval = 1 / sqrt(8 * self.eps) * (2 / sqrt(a - c)) * ellipf(k,p**2) if self.__init_momenta[1] >= 0: return -abs(retval) else: return abs(retval)
def __compute_tau_eta(self):
# Gradshtein 3.131.5.
from mpmath import sqrt, asin, ellipf
u = self.__init_coordinates[1]**2 / 2
c, b, a = self.__roots_eta
k = asin(sqrt(((a - c) * (u - b)) / ((a - b) * (u - c))))
p = sqrt((a - b) / (a - c))
retval = 1 / sqrt(8 * self.eps) * (2 / sqrt(a - c)) * ellipf(k, p**2)
if self.__init_momenta[1] >= 0:
return -abs(retval)
else:
return abs(retval)
def __compute_tau_xi(self):
from mpmath import sqrt, asin, ellipf, mpc, atan
if self.bound:
# Gradshtein 3.131.3.
u = self.__init_coordinates[0]**2 / 2
c, b, a = self.__roots_xi
gamma = asin(sqrt((u - c) / (b - c)))
q = sqrt((b - c) / (a - c))
# NOTE: here it's q**2 instead of q because of the difference in
# convention between G and mpmath :(
# NOTE: the external factor 1 / sqrt(8 * self.eps) comes from the fact
# that G gives the formula for a polynomial normalised by its leading coefficient.
retval = 1 / sqrt(8 * self.eps) * (2 / sqrt(a - c)) * ellipf(
gamma, q**2)
else:
# Here we will need two cases: one for when the other two roots are real
# but negative, one for when the other two roots are imaginary.
if isinstance(self.__roots_xi[1], mpc):
# G 3.138.7.
m = self.__roots_xi[1].real
n = abs(self.__roots_xi[1].imag)
# Only real root is the first one.
a = self.__roots_xi[0]
u = self.__init_coordinates[0]**2 / 2
p = sqrt((m - a)**2 + n**2)
retval = 1 / sqrt(8 * self.eps) * (1 / sqrt(p)) * ellipf(
2 * atan(sqrt((u - a) / p)), (p + m - a) / (2 * p))
else:
# G 3.131.7.
u = self.__init_coordinates[0]**2 / 2
c, b, a = self.__roots_xi
mu = asin(sqrt((u - a) / (u - b)))
q = sqrt((b - c) / (a - c))
retval = 1 / sqrt(8 * self.eps) * (2 / sqrt(a - c)) * ellipf(
mu, q**2)
# Fix the sign according to the initial momentum.
if self.__init_momenta[0] >= 0:
return -abs(retval)
else:
return abs(retval)
def __compute_tau_xi(self): from mpmath import sqrt, asin, ellipf, mpc, atan if self.bound: # Gradshtein 3.131.3. u = self.__init_coordinates[0]**2 / 2 c,b,a = self.__roots_xi gamma = asin(sqrt((u - c)/(b - c))) q = sqrt((b - c)/(a - c)) # NOTE: here it's q**2 instead of q because of the difference in # convention between G and mpmath :( # NOTE: the external factor 1 / sqrt(8 * self.eps) comes from the fact # that G gives the formula for a polynomial normalised by its leading coefficient. retval = 1 / sqrt(8 * self.eps) * (2 / sqrt(a - c)) * ellipf(gamma,q**2) else: # Here we will need two cases: one for when the other two roots are real # but negative, one for when the other two roots are imaginary. if isinstance(self.__roots_xi[1],mpc): # G 3.138.7. m = self.__roots_xi[1].real n = abs(self.__roots_xi[1].imag) # Only real root is the first one. a = self.__roots_xi[0] u = self.__init_coordinates[0]**2 / 2 p = sqrt((m - a)**2 + n**2) retval = 1 / sqrt(8 * self.eps) * (1 / sqrt(p)) * ellipf(2 * atan(sqrt((u - a) / p)),(p + m - a) / (2*p)) else: # G 3.131.7. u = self.__init_coordinates[0]**2 / 2 c,b,a = self.__roots_xi mu = asin(sqrt((u - a)/(u - b))) q = sqrt((b - c)/(a - c)) retval = 1 / sqrt(8 * self.eps) * (2 / sqrt(a - c)) * ellipf(mu,q**2) # Fix the sign according to the initial momentum. if self.__init_momenta[0] >= 0: return -abs(retval) else: return abs(retval)
def elliptic_core_g(x,y): x = x/2 y = y/2 factor = (cos(x)/sin(y) + sin(y)/cos(x) - (cos(y)/tan(y)/cos(x) + sin(x)*tan(x)/sin(y)))/pi k = tan(x)/tan(y) m = k*k n = (sin(x)/sin(y))*(sin(x)/sin(y)) u = asin(tan(y)/tan(x)) complete = ellipk(m) - ellippi(n, m) incomplete = ellipf(u,m) - ellippi(n/k/k,1/m)/k #incomplete = ellipf(u,m) - ellippi(n,u,m) return re(1.0 - factor*(incomplete + complete))
def calc_abel(k, zeta, eta):
k1 = sqrt(1 - k**2)
a = k1 + complex(0, 1) * k
b = k1 - complex(0, 1) * k
abel_tmp = map(lambda zetai : \
complex(0, 1) * 1/(complex64(ellipk(k**2))*2*b) \
* complex64(ellipf( asin( (zetai )/a), mfrom(k=a/b))) \
- taufrom(k=k)/2,
zeta)
abel = []
for i in range(0, 4, 1):
abel.append(abel_select(k, abel_tmp[i], eta[i]))
return abel
def calc_abel(k, zeta, eta):
k1 = sqrt(1-k**2)
a=k1+complex(0, 1)*k
b=k1-complex(0, 1)*k
abel_tmp = map(lambda zetai : \
complex(0, 1) * 1/(complex64(ellipk(k**2))*2*b) \
* complex64(ellipf( asin( (zetai )/a), mfrom(k=a/b))) \
- taufrom(k=k)/2,
zeta)
abel = []
for i in range(0, 4, 1):
abel.append(abel_select(k, abel_tmp[i], eta[i]))
return abel
def which_cannot_cover_with_half_radius(self, v: NormalVector, shift=0):
from math import asin, pi
d = self.checked_discrepancy_upper_bound
r = v.expected_radius
half_r = r / 2
distance_to_center = .86 * r
larger_r = distance_to_center + r
phi_arround_pole = asin(r_to_dot(distance_to_center))
dtheta = 2 * pi / self.cap_covering_number
candidates = [NormalVector(theta=0, phi=pi / 2)]
candidates += [
NormalVector(phi=phi_arround_pole, theta=shift + (i * dtheta))
for i in range(self.cap_covering_number)
]
rotation = get_rotation_matrix_north_pole_to(theta=v.theta, phi=v.phi)
cannot_cover = []
for cand_v in candidates:
cand_v.rotate(rotation)
discrepancy, projection = self.get_discrepancy_data(cand_v)
if discrepancy >= d:
return "found higher discrepancy at", cand_v, candidates
else:
cr = self.get_confidence_radius(d=d,
discrepancy=discrepancy,
projection=projection)
cand_v.set_discrepancy_data(discrepancy=discrepancy,
confidence_radius=cr,
expected_radius=half_r)
if cr >= larger_r:
return "all covered by", cand_v, candidates
elif cr < half_r:
cannot_cover += [cand_v]
return "cannot list", cannot_cover, candidates
def test_on_line(k, x0, x1, y, z, partition_size):
k1 = sqrt(1 - k**2)
a = k1 + complex(0, 1) * k
b = k1 - complex(0, 1) * k
x_step = (x1 - x0) / partition_size
for i in range(0, partition_size):
x = x0 + i * x_step
zeta = calc_zeta(k, x, y, z)
eta = calc_eta(k, x, y, z)
abel = calc_abel(k, zeta, eta)
mu = calc_mu(k, x, y, z, zeta, abel)
print "zeta/a", zeta[0] / a, zeta[1] / a, zeta[2] / a, zeta[3] / a, '\n'
print "eta", eta[0], eta[1], eta[2], eta[3], '\n'
print "abel", abel[0], abel[1], abel[2], abel[3], '\n'
abel_tmp= map(lambda zetai : \
complex(0, 1) * 1/(complex64(ellipk(k**2))*2*b) \
* complex64(ellipf( asin( (zetai )/a), mfrom(k=a/b))) \
- taufrom(k=k)/2,
zeta)
print "abel_tmp", abel_tmp[0], abel_tmp[1], abel_tmp[2], abel_tmp[
3], '\n'
print abel[0] + conj(abel[2]), abel[1] + conj(abel[3]), '\n'
print -taufrom(k=k) / 2, '\n'
for l in range(0, 4):
tmp = abs(complex64(calc_eta_by_theta(k, abel[l])) - eta[l])
print tmp
value = higgs_squared(k, x, y, z)
print "higgs", higgs_squared(k, x, y, z)
if (value > 1.0 or value < 0.0):
print "Exception"
print '\n'
# print mu[0]+mu[2], mu[1]+mu[3], '\n'
return
def test_on_line(k, x0, x1, y, z, partition_size):
k1 = sqrt(1-k**2)
a=k1+complex(0, 1)*k
b=k1-complex(0, 1)*k
x_step = (x1 - x0) / partition_size
for i in range(0, partition_size):
x=x0+i*x_step
zeta = calc_zeta(k ,x, y, z)
eta = calc_eta(k, x, y, z)
abel = calc_abel(k, zeta, eta)
mu = calc_mu(k, x, y, z, zeta, abel)
print "zeta/a", zeta[0]/a, zeta[1]/a, zeta[2]/a, zeta[3]/a, '\n'
print "eta", eta[0], eta[1], eta[2], eta[3], '\n'
print "abel", abel[0], abel[1],abel[2],abel[3],'\n'
abel_tmp= map(lambda zetai : \
complex(0, 1) * 1/(complex64(ellipk(k**2))*2*b) \
* complex64(ellipf( asin( (zetai )/a), mfrom(k=a/b))) \
- taufrom(k=k)/2,
zeta)
print "abel_tmp", abel_tmp[0], abel_tmp[1],abel_tmp[2],abel_tmp[3],'\n'
print abel[0]+conj(abel[2]), abel[1]+conj(abel[3]), '\n'
print - taufrom(k=k)/2, '\n'
for l in range(0,4):
tmp= abs(complex64(calc_eta_by_theta(k, abel[l])) - eta[l])
print tmp
value = higgs_squared(k, x, y, z)
print "higgs", higgs_squared(k,x,y,z)
if (value > 1.0 or value <0.0):
print "Exception"
print '\n'
# print mu[0]+mu[2], mu[1]+mu[3], '\n'
return
def calc_lambda_r(r, r1, r2, r3, r4, En):
"""
Mino time as a function of r (which in turn is a function of psi)
Parameters:
r (float): radius
r1 (float): radial root
r2 (float): radial root
r3 (float): radial root
r4 (float): radial root
En (float): energy
Returns:
lambda (float)
"""
kr = ((r1 - r2) * (r3 - r4)) / ((r1 - r3) * (r2 - r4))
# if r1 == r2:
# # circular orbit
# print('Circular orbits currently do not work.')
# return 0
yr = sqrt(((r - r2) * (r1 - r3)) / ((r1 - r2) * (r - r3)))
F_asin = ellipf(asin(yr), kr)
return (2 * F_asin) / (sqrt(1 - En * En) * sqrt((r1 - r3) * (r2 - r4)))
def make_polar_coordinate(self,
n,
is_twisted=True,
precision=30,
log_scale=0):
# This function distribute points with polar coordinate on sphere. This algorithm is O (n^2).
# It also returns a phi above which we know north pole is a local maximum.
from mpmath import mp, sin, cos, sqrt, pi, floor, mpf, asin
from numpy import array, cross
from numpy.linalg import norm
from os import remove
mp.dps = precision
remove(self.points_address)
if is_twisted:
name = "twisted_polar_coordinates_" + str(n)
self.point_set = PointSet(name=name, address=self.output_folder)
else:
name = "polar_coordinates_" + str(n)
self.point_set = PointSet(name=name, address=self.output_folder)
self.points_address = self.point_set.point_file_address
if not float(Point._log_scale) == log_scale:
Point.change_log_scale(log_scale)
print("Point scale is", Point.get_scale())
if not Point._precision == precision:
Point.change_precision(precision)
phis = [0] * (n - 1)
m = [0] * (n - 1)
for j in range(n - 1):
phis[j] = ((pi * (j + 1)) / n) - (pi / 2)
m[j] = int(floor(mpf(".5") + sqrt(3) * n * cos(phis[j])))
for j in range(n - 1):
for i in range(m[j]):
if is_twisted:
if j + 1 < n / 2:
shift = (2 * pi * (j + 1)) / (n * m[j])
else:
shift = (2 * pi * (1 - mpf(j + 1) / mpf(n))) / m[j]
else:
shift = 0
theta = ((2 * pi * i) / m[j]) + shift
new_point = Point(theta=theta, phi=phis[j])
self.point_set.add_point(new_point)
north_pole = Point(theta=0, phi=pi / 2)
south_pole = Point(theta=0, phi=-pi / 2)
self.point_set.add_point(north_pole)
self.point_set.add_point(south_pole)
self.min_discrepancy_distance = 1 / self.point_set.size
# In below we get the confidence phi around the north pole
highest_phi = -1
for i in range(int(n / 2) - 1, n - 2):
phi1 = phis[i]
phi2 = phis[i + 1]
v1 = array([mpf(0), cos(phi1), mpf(0)])
v2 = array([2 * cos(phi1), mpf(0), sin(phi2) - sin(phi1)])
normal_vector = cross(v1, v2)
normal_vector /= norm(normal_vector)
highest_phi = max(highest_phi, asin(abs(normal_vector[2])))
return float(highest_phi)
def phiinv(x):
return mpmath.asin(mpmath.sqrt(x)) / (2.0 * mpmath.pi)
def m(rx, theta):
return mpmath.power(
mpmath.sin(mpmath.power(2, rx) * mpmath.asin(mpmath.sqrt(theta))), 2)
import mpmath as math
from units import *
from copy import copy
def sqrt(x):
if hasattr(x,'unit'):
return math.sqrt(x.number) | x.unit**0.5
return math.sqrt(x)
#trigonometric convenience functions which are "unit aware"
sin=lambda x: math.sin(1.*x)
cos=lambda x: math.cos(1.*x)
tan=lambda x: math.tan(1.*x)
cot=lambda x: math.cot(1.*x)
asin=lambda x: math.asin(x) | rad
acos=lambda x: math.acos(x) | rad
atan=lambda x: math.atan(x) | rad
atan2=lambda x,y: math.atan2(x,y) | rad
#~ cos=math.cos
#~ sin=math.sin
#~ tan=math.tan
#~ cosh=math.cosh
#~ sinh=math.sinh
#~ tanh=math.tanh
#~ acos=math.acos
#~ asin=math.asin
#~ atan=math.atan
acosh=math.acosh
asinh=math.asinh
'cbrt': ['primitive', [lambda x, y: mp.cbrt(x), None]],
'root': ['primitive', [lambda x, y: mp.root(x, y[0]), None]], # y's root
'unitroots': ['primitive', [lambda x, y: Vector(mp.unitroots(x)),
None]], #
'hypot': ['primitive', [lambda x, y: mp.hypot(x, y[0]),
None]], # sqrt(x**2+y**2)
#
'sin': ['primitive', [lambda x, y: mp.sin(x), None]],
'cos': ['primitive', [lambda x, y: mp.cos(x), None]],
'tan': ['primitive', [lambda x, y: mp.tan(x), None]],
'sinpi': ['primitive', [lambda x, y: mp.sinpi(x), None]], #sin(x * pi)
'cospi': ['primitive', [lambda x, y: mp.cospi(x), None]],
'sec': ['primitive', [lambda x, y: mp.sec(x), None]],
'csc': ['primitive', [lambda x, y: mp.csc(x), None]],
'cot': ['primitive', [lambda x, y: mp.cot(x), None]],
'asin': ['primitive', [lambda x, y: mp.asin(x), None]],
'acos': ['primitive', [lambda x, y: mp.acos(x), None]],
'atan': ['primitive', [lambda x, y: mp.atan(x), None]],
'atan2': ['primitive', [lambda x, y: mp.atan2(y[0], x), None]],
'asec': ['primitive', [lambda x, y: mp.asec(x), None]],
'acsc': ['primitive', [lambda x, y: mp.acsc(x), None]],
'acot': ['primitive', [lambda x, y: mp.acot(x), None]],
'sinc': ['primitive', [lambda x, y: mp.sinc(x), None]],
'sincpi': ['primitive', [lambda x, y: mp.sincpi(x), None]],
'degrees': ['primitive', [lambda x, y: mp.degrees(x),
None]], #radian - >degree
'radians': ['primitive', [lambda x, y: mp.radians(x),
None]], #degree - >radian
#
'exp': ['primitive', [lambda x, y: mp.exp(x), None]],
'expj': ['primitive', [lambda x, y: mp.expj(x), None]], #exp(x*i)
def get_angle_delta(self):
if mpmath.almosteq(self.getUncertainty(), 1, epsilon):
return mpmath.mpf("0")
else:
return mpmath.asin(self.getBelief() / self.get_length_to_uncertainty())
def asin(op):
expr = mpmath.asin(op)
return convertFromRadians(expr)
# ------------------------------------------------------
# Now the actual code
# ------------------------------------------------------
# make this a command line input parameter so it can be set
# automatically based on the legth of the points array
#r = 8 # 2^{-r} is our precision
# concatenate the first r bits of each phiinv(y) to omega represented as a string (omegastr)
omegastr = ""
for n, x, y in points:
omegastr += float2binary(phiinv(y))[:r]
print("Omega = ", omegastr)
omega = binary2float(omegastr) # convert omega to a mp real
theta = phi(omega) # compute theta
print("Theta = ", theta)
m.asin_sqrt_theta = mpmath.asin(
mpmath.sqrt(theta)) # IK: needs to be computed only once
# now run the model to recover, print the fitted values
for n, x, y in points:
ymodel = m(r * n, theta)
# these fitted values may then be plotted
print(x, y, float(ymodel))
def eval(self, z):
return mpmath.asin(z)
def arcsin_degrees(x):
"""Return arcsine of x in degrees."""
return normalized_degrees_from_radians(asin(x))
def zeta_r(_P: float, r: float, M: float) -> float:
"""Calculate the elliptic integral argument Zeta_r for a given value of P and r"""
Qvar = Q(_P, M)
a = (Qvar - _P + 2 * M + (4 * M * _P) / r) / (Qvar - _P + (6 * M))
s = mpmath.asin(mpmath.sqrt(a))
return s
def ft(x):
return mpmath.asin(x)
def eval(self, z):
return mpmath.asin(z)
def zeta_inf(_P: float, M: float, tol: float = 1e-6) -> float:
"""Calculate Zeta_inf for elliptic integral F(Zeta_inf, k)"""
Qvar = Q(_P, M) # Q variable, only call to function once
arg = (Qvar - _P + 2 * M) / (Qvar - _P + 6 * M)
z_inf = mpmath.asin(mpmath.sqrt(arg))
return z_inf
def is_discrepancy_less_than(self,
d,
highest_phi=None,
lowest_phi=0,
first_theta=0,
last_theta=None,
dtheta_method='dynamic'):
from math import pi, asin
from time import time
start = time()
if highest_phi:
self.highest_phi = highest_phi
else:
self.highest_phi = pi / 2
if last_theta:
self.last_theta = last_theta
else:
self.last_theta = 2 * pi
self.lowest_phi = lowest_phi
self.first_theta = first_theta
phi_check = 0 <= self.lowest_phi and self.lowest_phi < self.highest_phi and self.highest_phi <= pi / 2
if not phi_check:
print("There is an issue in phi constrains")
return False
theta_check = 0 <= self.first_theta and self.first_theta < self.last_theta and self.last_theta <= 2 * pi
if not theta_check:
print("There is an issue in theta constrains")
return False
self.reset_discrepancy_parameters()
self.is_complete_orbit = (self.first_theta == 0) and (self.last_theta
== 2 * pi)
self.checked_discrepancy_upper_bound = d
# making directions file
title = [
"x", "y", "z", "theta", "phi", "norm", "discrepancy", "conf. r",
"covered radius", "covering type"
]
self.write_title(title_list=title,
file_address=self.directions_file_address)
# making cover cap file
title = []
self.write_title(title_list=title,
file_address=self.cover_cap_file_address)
# making orbit file
title = [
"phi", "expected radius", "dtheta", "covered phi", "fd", "ld", "#d"
]
self.write_title(title_list=title,
file_address=self.orbits_file_address,
is_orbit_file=True)
if self.highest_phi == pi / 2:
could_add = self.add_direction(theta=0,
phi=pi / 2,
d=d,
expected_radius=0)
# We can put a positive expected radius
if not could_add:
print("The algorithm cannot handle the north pole!")
return False
else:
confidence_r = self.directions[0].confidence_radius
covered_phi = asin(r_to_dot(confidence_r))
else:
covered_phi = self.highest_phi
while covered_phi > self.lowest_phi:
phi, expected_radius, mindtheta = self.get_phi_expected_r_dtheta(
d=d, covered_phi=covered_phi)
# if expected_radius is 0 we should stop, or do something
if expected_radius == 0:
self.write_report(start=start, is_covered=False)
return False
theta = self.first_theta
minind = self.direction_number + 1
should_continue = True
while should_continue:
could_add = self.add_direction(theta=theta,
phi=phi,
d=d,
expected_radius=expected_radius)
if not could_add:
last_direction, discrepancy, projection = self.create_direction(
theta=theta, phi=phi)
cr = self.get_confidence_radius(d=d,
discrepancy=discrepancy,
projection=projection)
last_direction.directional_discrepancy = discrepancy
last_direction.confidence_radius = cr
self.write_report(start=start,
is_covered=False,
found_higher_disrepancy=True)
return False
direction = self.directions[-1]
dtheta = self.get_dtheta(direction=direction,
mindtheta=mindtheta,
dtheta_method=dtheta_method)
# The below lets the cover go one direction after last_theta if the orbit is not complete
if not self.is_complete_orbit:
should_continue = theta < self.last_theta
theta += dtheta
if self.is_complete_orbit:
should_continue = theta < self.last_theta
self.orbits_number += 1
maxind = self.direction_number
covered_phi = self.get_covered_phi(minind=minind,
maxind=maxind,
phi=phi,
dtheta=dtheta,
dtheta_method=dtheta_method)
self.write_orbit_data(phi=phi,
expected_r=expected_radius,
dtheta=dtheta,
minind=minind,
maxind=maxind,
coveredphi=covered_phi)
cover_cap_index_file_add = self.output_folder + "cover_cap_indices.txt"
first_round_run_time = time() - start
cover_cap_start = time()
answer = True
not_covered_with_cover_cap = []
for ind in self.not_covered_directions:
direction = self.directions[ind]
could_cover, found_higher_discrepancy, calls_num = self.cover_cap(
direction)
if found_higher_discrepancy:
self.write_report(start=start,
is_covered=False,
found_higher_disrepancy=True)
return False
elif not could_cover:
answer = False
not_covered_with_cover_cap += [ind]
self.calls_list.append(calls_num)
cover_cap_run_time = time() - cover_cap_start
cover_cap_ind = open(cover_cap_index_file_add, mode="w+", buffering=1)
cover_cap_ind.write("number\t" + "direction_ind\t" +
"# of covercap\t\n")
for i in range(len(self.not_covered_directions)):
cover_cap_ind.write(
str(i) + "\t" + str(self.not_covered_directions[i]) + "\t" +
str(self.calls_list[i]) + "\t\n")
cover_cap_ind.close()
if not answer:
issue_file_add = self.output_folder + "not_covered_directions.txt"
file = open(issue_file_add, "w+")
file.write("The following directions could not be covered\n")
file.close()
write_comma_separated(List=not_covered_with_cover_cap,
address=issue_file_add)
self.write_report(start=start,
is_covered=answer,
problematic_directions=not_covered_with_cover_cap,
first_round_run_time=first_round_run_time,
cover_cap_run_time=cover_cap_run_time)
return answer