def ellipsoid_shape_func(a, b, c):
phi = np.arccos(c/a)
m = (a**2.-b**2.)/(a**2.-c**2.)
la = a*b*c/((a**2.-b**2.)*np.sqrt(a**2.-c**2.))*(ellipkinc(phi,m)-ellipeinc(phi,m))
lc = b/(b**2.-c**2.)*(b-a*c/np.sqrt(a**2.-c**2.)*ellipeinc(phi,m))
lb = 1.-la-lc
return la, lb, lc
def get_kth_point_on_ellipse(k, n, a, b, accuracy=1e-4):
if a == 0:
return (0, 0)
m = 1 - (b**2 / a**2)
c = get_ellipse_circumference(a, b)
expected = c * (k / n)
# initial guess - as if its a circle
phi = 2 * np.pi * k / n
error = (a * ellipeinc(phi, m) - expected) / c
if error < 0:
phi_under = phi
phi_over = 2 * np.pi
elif error > 0:
phi_under = 0
phi_over = phi
else:
return np.array([a * np.sin(phi), b * np.cos(phi)])
while abs(error) > accuracy:
phi = (phi_over + phi_under) / 2
error = (a * ellipeinc(phi, m) - expected) / c
if error < 0:
phi_under = phi
elif error > 0:
phi_over = phi
else:
return np.array([a * np.sin(phi), b * np.cos(phi)])
return np.array([a * np.sin(phi), b * np.cos(phi)])
def hyperuniform_ellipse(N, a = 1, b = 2):
'''Generate Hyperuniformly-Sampled 2-D Ellipse'''
assert(a < b) # a must be length of minor semi-axis; b major semi-axis
X = [[],[]] # init data list [[x],[y]]
C = np.linspace(0, 1, N) # init color list for plotting
angles = 2*np.pi*np.arange(N)/N
if a != b:
'''Given N points, combine scipy elliptic integral + optimize to find
N equidistant points along ellilpse manifold, then convert to angles'''
e = np.sqrt(1.0 - a**2 / b**2)
tot_size = special.ellipeinc(2.0 * np.pi, e)
arc_size = tot_size / N
arcs = np.arange(N) * arc_size
res = optimize.root(
lambda x: (special.ellipeinc(x, e) - arcs), angles)
angles = res.x
arcs = special.ellipeinc(angles, e)
for t in angles:
X[0].append(a * np.cos(t)) # x
X[1].append(b * np.sin(t)) # y
return np.asarray(X), np.asarray(C)
def get_evenly_distributed_ellipse_coordinates(
self,
number_of_coordinates: int = 50) -> List[Tuple[float, float]]:
"""
Makes use of scipy.special.ellipeinc() which provides the numerical integral along the perimeter of the ellipse,
and scipy.optimize.root() for solving the equal-arcs length equation for the angles.
:param number_of_coordinates: number of evenly distributed points to generate along the ellipsis line
:return: list of tuple's with coordinates along the ellipse line
"""
angles = 2 * np.pi * np.arange(
number_of_coordinates) / number_of_coordinates
e = (1.0 - self.minor_axis**2.0 / self.major_axis**2.0)**0.5
total_size = special.ellipeinc(2.0 * np.pi, e)
arc_size = total_size / number_of_coordinates
arcs = np.arange(number_of_coordinates) * arc_size
res = optimize.root(lambda x: (special.ellipeinc(x, e) - arcs), angles)
angles = res.x
if self.width > self.height:
x_points = list(self.major_axis * np.sin(angles))
y_points = list(self.minor_axis * np.cos(angles))
else:
x_points = list(self.minor_axis * np.cos(angles))
y_points = list(self.major_axis * np.sin(angles))
coordinates = [(point_x + self.x_center, point_y + self.y_center)
for point_x, point_y in zip(x_points, y_points)]
return coordinates
def map_on_ellipse(xq, a=4, b=1, gap_angle=90):
"""Map 2D points on bend ellipse in 3D.
Parameters
----------
xq: (n, 2) array-like
The 2D points
a, b: non-negative, scalar
Ellipsis axis
gap_angle: scalar in [0,360)
Degree of the gap the bend ellipse should have.
E.g. for 0 the data is mapped onto a closed ellipse.
Returns
-------
xyz: (n, 3) array-like
Mapped 3D points
"""
gap_angle = np.deg2rad(gap_angle) / 2
# Range of the data
x_min = np.min(xq[:, 0])
x_max = np.max(xq[:, 0])
width = x_max - x_min
# Compute perimeter of the open ellipse
a = abs(a)
b = abs(b)
major, minor = max(a, b), min(a, b)
ecc = (1 - minor**2 / major**2)**2
offset = ellipeinc(gap_angle, ecc)
diameter = major * (4 * ellipe(ecc) - offset)
# Scale ellipse to match the span
scale = width / diameter
a *= scale
b *= scale
# Find the angles. We use the canonical parametrization.
x_zeroed = xq[:, 0] - x_min
alpha = (2 * np.pi - 2 * gap_angle) * x_zeroed / (
x_max - x_min) + gap_angle # initial guess
for _ in range(6):
# 6 Newton steps to solve inverse elliptic integral.
# 6 Newton steps ought to be enough for anybody.
f = major * ellipeinc(alpha, ecc) - (x_zeroed + major * offset)
f_prime = major * np.sqrt(1 - ecc * np.sin(alpha)**2)
alpha = alpha - f / f_prime
xy = np.vstack((a * np.cos(alpha), b * np.sin(alpha))).T
z = xq[:, 1] # z-axis is the previous y-axis, we only bend in the xy-plane
xyz = np.hstack((xy, z.reshape(-1, 1)))
return xyz
def ellippi(n, m):
"""
Complete elliptic integral of third kind (simplified version).
.. FIXME: Incorrect, because of non-standard definitions of elliptic integral in the reference
Reference
---------
F. LAMARCHE and C. LEROY, Evaluation of the volume of a sphere with a cylinder by elliptic integrals,
Computer Phys. Comm. 59 (1990) pg. 365
"""
a2 = n
k2 = m
k2_a2 = k2 + a2
phi = np.arcsin(a2 / k2_a2)
Kc = ellipk(k2)
Ec = ellipe(k2)
Ki = ellipkinc(phi, (1. - k2)**1)
Ei = ellipeinc(phi, (1. - k2)**1)
c1 = k2 / k2_a2
c2 = np.sqrt(a2 / (1 + a2) / k2_a2)
return c1 * Kc + c2 * ((Kc - Ec) * Ki + Kc * Ei)
def psi_x(z, x, beta):
"""
Eq.(24) from Ref[1] with argument zeta=0 and no constant factor e*beta**2/2/rho**2.
Note that 'x' here corresponds to 'chi = x/rho',
and 'z' here corresponds to 'xi = z/2/rho' in the paper.
"""
# z = np.float(z)
# x = np.float(x)
kap = kappa(z, x, beta)
alp = alpha(z, x, beta)
arg2 = -4 * (1 + x) / x**2
try:
T1 = (1 / abs(x) / (1 + x) *
((2 + 2 * x + x**2) * ss.ellipkinc(alp, arg2) -
x**2 * ss.ellipeinc(alp, arg2)))
D = kap**2 - beta**2 * (1 + x)**2 * sin(2 * alp)**2
T2 = ((kap**2 - 2 * beta**2 * (1 + x)**2 + beta**2 * (1 + x) *
(2 + 2 * x + x**2) * cos(2 * alp)) / beta / (1 + x) / D)
T3 = -kap * sin(2 * alp) / D
T4 = kap * beta**2 * (1 + x) * sin(2 * alp) * cos(2 * alp) / D
T5 = 1 / abs(x) * ss.ellipkinc(alp,
arg2) # psi_phi without e/rho**2 factor
out = real((T1 + T2 + T3 + T4) - 2 / beta**2 * T5)
except ZeroDivisionError:
out = 0
# print(f"Oops! ZeroDivisionError at (z,x)= ({z:5.2f},{x:5.2f}). Returning 0.")
return np.nan_to_num(out)
def _E_F_field(ellipsoid, kappa, phi):
'''
Calculates the Legendre's normal elliptic integrals of first
and second kinds which are used to calculate the potential
fields outside the triaxial ellipsoid.
Parameters:
* ellipsoid : element of :class:`mesher.TriaxialEllipsoid`.
* lamb: numpy array 1D
Parameter lambda for each point in the ellipsoid system.
* kappa: numpy array 1D
Squared modulus of the elliptic integral.
* phi: numpy array 1D
Amplitude of the elliptic integral.
Returns:
F, E: numpy arrays 1D
Legendre's normal elliptic integrals of first and second kinds.
'''
E = ellipeinc(phi, kappa)
F = ellipkinc(phi, kappa)
return E, F
def _E_F_demag(ellipsoid):
'''
Calculates the Legendre's normal elliptic integrals of first
and second kinds which are used to calculate the demagnetizing
factors.
Parameters:
* ellipsoid : element of :class:`mesher.TriaxialEllipsoid`.
Returns:
F, E : floats
Legendre's normal elliptic integrals of first and second kinds,
respectively.
'''
a = ellipsoid.large_axis
b = ellipsoid.intermediate_axis
c = ellipsoid.small_axis
kappa = (a * a - b * b) / (a * a - c * c)
phi = np.arccos(c / a)
# E = ellipeinc(phi, kappa*kappa)
# F = ellipkinc(phi, kappa*kappa)
E = ellipeinc(phi, kappa)
F = ellipkinc(phi, kappa)
return E, F
def ellippi(n, m):
"""
Complete elliptic integral of third kind (simplified version).
.. FIXME: Incorrect, because of non-standard definitions of elliptic integral in the reference
Reference
---------
F. LAMARCHE and C. LEROY, Evaluation of the volume of a sphere with a cylinder by elliptic integrals,
Computer Phys. Comm. 59 (1990) pg. 365
"""
a2 = n
k2 = m
k2_a2 = k2 + a2
phi = np.arcsin(a2 / k2_a2)
Kc = ellipk(k2)
Ec = ellipe(k2)
Ki = ellipkinc(phi, (1. - k2) ** 1)
Ei = ellipeinc(phi, (1. - k2) ** 1)
c1 = k2 / k2_a2
c2 = np.sqrt(a2 / (1 + a2) / k2_a2)
return c1 * Kc + c2 * ((Kc - Ec) * Ki + Kc * Ei)
def Int3(q, r, R):
'''
Integral I3.
'''
m = 4 * R * r / (q**2 + (r + R)**2)
K0 = ellipk(m) # Complete elliptic integral of the first kind
K1 = ellipk(1 - m)
E0 = ellipe(m) # Complete elliptic integral of the second kind
E1 = ellipe(1 - m)
beta = np.arcsin(q / np.sqrt(q**2 + (R - r)**2))
K2 = ellipkinc(beta,
1 - m) # Incomplete elliptic integral of the first kind
E2 = ellipeinc(beta,
1 - m) # Incomplete elliptic integral of the second kind
Z = E2 - E1 * K2 / K1 # Jacobi zeta function
lamb = K2 / K1 + 2 * K0 * Z / np.pi # Heuman’s lambda function
I3 = -q * np.sqrt(m) * K0 / (2 * np.pi * R * np.sqrt(r * R)) + (
np.heaviside(r - R, 0.5) - np.heaviside(R - r, 0.5)) * lamb / (
2 * R) + np.heaviside(R - r, 0.5) / R
return I3
def _E_F_demag(a, b, c):
'''
Calculates the Legendre's normal elliptic integrals of first
and second kinds which are used to calculate the demagnetizing
factors.
input:
a: float - semi-axis a (in meters).
b: float - semi-axis b (in meters).
c: float - semi-axis c (in meters).
output:
F - Legendre's normal elliptic integrals of first kind.
E - Legendre's normal elliptic integrals of second kind.
'''
assert a > b > c, 'a must be greater than b and b must be greater than c'
assert (a > 0) and (b > 0) and (c > 0), 'a, b and c must \
be all positive'
kappa = np.sqrt(((a * a - b * b) / (a * a - c * c)))
phi = np.arccos(c / a)
E = ellipeinc(phi, kappa * kappa)
F = ellipkinc(phi, kappa * kappa)
return E, F
def _E_F_field(a, b, c, kappa, phi):
'''
Calculates the Legendre's normal elliptic integrals of first
and second kinds which are used to calculate the potential
fields outside the body.
input:
a: float - semi-axis a (in meters).
b: float - semi-axis b (in meters).
c: float - semi-axis c (in meters).
kappa: float - an argument of the elliptic integrals.
phi: numpy array 1D - an argument of the elliptic integrals.
output:
F: numpy array 1D - Legendre's normal elliptic integrals of first kind.
E: numpy array 1D - Legendre's normal elliptic integrals of second kind.
'''
assert a > b > c, 'a must be greater than b and b must be greater than c'
assert (a > 0) and (b > 0) and (c > 0), 'a, b and c must \
be all positive'
E = ellipeinc(phi, kappa * kappa)
F = ellipkinc(phi, kappa * kappa)
return E, F
def old_psi_x(z, x, beta):
"""
Eq.(24) from Ref[1] with argument zeta=0 and no constant factor e*beta**2/2/rho**2.
Note that 'x' here corresponds to 'chi = x/rho',
and 'z' here corresponds to 'xi = z/2/rho' in the paper.
"""
beta2 = beta**2
alp = old_alpha(z, x, beta2)
kap = sqrt(x**2 + 4*(1+x) * sin(alp)**2) # kappa(z, x, beta2) inline
sin2a = sin(2*alp)
cos2a = cos(2*alp)
arg2 = -4 * (1+x) / x**2
ellipkinc = ss.ellipkinc(alp, arg2)
ellipeinc = ss.ellipeinc(alp, arg2)
T1 = (1/abs(x)/(1 + x) * ((2 + 2*x + x**2) * ellipkinc - x**2 * ellipeinc))
D = kap**2 - beta2 * (1 + x)**2 * sin2a**2
T2 = ((kap**2 - 2*beta2 * (1+x)**2 + beta2 * (1+x) * (2 + 2*x + x**2) * cos2a)/ beta/ (1+x)/ D)
T3 = -kap * sin2a / D
T4 = kap * beta2 * (1 + x) * sin2a * cos2a / D
T5 = 1 / abs(x) * ellipkinc # psi_phi without e/rho**2 factor
out = (T1 + T2 + T3 + T4) - 2 / beta2 * T5
return out
def demagEllipsoid(length, width, height, cgs=False):
'''Returns the demag factors of an ellipsoid'''
a = 0.5 * length # Semimajor
b = 0.5 * width # Semiminor
c = 0.5 * height # Semi-more-minor
b1 = b / a
b2 = np.sqrt(1.0 - b1 * b1)
c1 = c / a
c2 = np.sqrt(1.0 - c1 * c1)
d2 = b2 / c2
d1 = np.sqrt(1.0 - d2 * d2)
theta = np.arccos(c1)
m = d2 * d2
E = special.ellipeinc(theta, m) # incomplete elliptic int of second kind
F = special.ellipkinc(theta, m) # incomplete elliptic int of first kind
Nxx = (b1 * c1) / (c2**3 * d2**2) * (F - E)
Nyy = (b1 * c1) / (c2**3 * d2**2 * d1**2) * (E - d1 * d1 * F -
d2 * d2 * c2 * c1 / b1)
Nzz = (b1 * c1) / (c2**3 * d1**2) * (c2 * b1 / c1 - E)
if cgs:
return np.pi * 4.0 * np.array([Nxx, Nyy, Nzz])
else:
return np.array([Nxx, Nyy, Nzz])
def ellipse_points(ellipse, n=12):
c = 0
dtheta = 0
C = ellipse.circumference.evalf()
e = ellipse.eccentricity.evalf()
a = ellipse.hradius
b = ellipse.vradius
points = [0] * n
points[0] = [a.__round__(6), 0]
points[n // 4] = [0, b.__round__(6)]
points[2 * n // 4] = [-a.__round__(6), 0]
points[3 * n // 4] = [0, -b.__round__(6)]
k = 1
while (k < n // 4):
while (c < k * C // n):
c = a * ellipeinc(dtheta, round(
e**2, 16)) # Not rounding causes TypeError at function
dtheta += 0.01
x = a * np.cos(dtheta)
y = b * np.sin(dtheta)
x = x.__round__(6)
y = y.__round__(6)
points[k] = [+x, +y]
points[k + n // 4] = [-x, +y]
points[k + 2 * n // 4] = [-x, -y]
points[k + 3 * n // 4] = [+x, -y]
k += 1
return points
def psi_sx(z, x, beta):
"""
2D longitudinal and transverse potential
Eq. (23) from Ref[1] with no constant factor (e*beta**2/2/rho**2).
Eq.(24) from Ref[1] with argument zeta=0 and no constant factor e*beta**2/2/rho**2.
Note that 'x' here corresponds to 'chi = x/rho',
and 'z' here corresponds to 'xi = z/2/rho' in the paper.
This is the most efficient routine.
Parameters
----------
z : array-like
z/(2*rho)
x : array-like
x/rho
beta : float
Relativistic beta
Returns
-------
psi_s, psi_x : tuple(ndarray, ndarray)
"""
# beta**2 appears far more than beta. Use this in internal functions
beta2 = beta**2
alp = alpha(z, x, beta2)
kap = sqrt(x**2 + 4 * (1 + x) * sin(alp)**2) # kappa(z, x, beta2) inline
sin2a = sin(2 * alp)
cos2a = cos(2 * alp)
# psi_s calc
out_psi_s = (cos2a - 1 / (1 + x)) / (kap - beta * (1 + x) * sin2a)
# psi_x calc
arg2 = -4 * (1 + x) / x**2
ellipeinc = ss.ellipeinc(alp, arg2)
ellipkinc = ss.ellipkinc(alp, arg2)
T1 = (1 / abs(x) / (1 + x) *
((2 + 2 * x + x**2) * ellipkinc - x**2 * ellipeinc))
D = kap**2 - beta2 * (1 + x)**2 * sin2a**2
T2 = ((kap**2 - 2 * beta2 * (1 + x)**2 + beta2 * (1 + x) *
(2 + 2 * x + x**2) * cos2a) / beta / (1 + x) / D)
T3 = -kap * sin2a / D
T4 = kap * beta2 * (1 + x) * sin2a * cos2a / D
T5 = 1 / abs(x) * ellipkinc # psi_phi without e/rho**2 factor
out_psi_x = (T1 + T2 + T3 + T4) - 2 / beta2 * T5
return out_psi_s, out_psi_x
def d1(v):
beta = np.arccos((v+1. - np.sqrt(3.)) / (v+1. + np.sqrt(3.)))
sin75 = np.sin(75. * np.pi/180.)
sin75 = sin75*sin75
result = (5./3.) * (v) * (((3.**0.25) * (np.sqrt(1. + v**3.)) * (scs.ellipeinc(beta, sin75)
- (1. / (3.+np.sqrt(3.))) * scs.ellipkinc(beta, sin75)))
+ ((1. - (np.sqrt(3.)+1.)*v*v) / (v+1. + np.sqrt(3.))))
return result
def force_ring(n, m):
qp = dq[n]
qq = dq[m]
rp = p[n]
rq = p[m]
d = rp**2 + rq**2
a = rp + rq
sq_a = (a)**2
s = (rp - rq)
four = 4 * (rp * rq) / sq_a
val = 1 * (1 / (rp * s * np.sqrt(d))) * np.sqrt(d / sq_a) * \
(a * sp.ellipeinc(pi / 4, four) + a * \
sp.ellipeinc(3 * pi / 4, four) + s * \
(sp.ellipkinc(pi / 4, four) + \
sp.ellipkinc(3 * pi / 4, four)))
val = qp * qq * val
return val
def SurfaceAreaTri(La, Lb, Lc, p):
phi = np.arccos(Lc / La)
k = np.sqrt(
(La * La * (Lb * Lb - Lc * Lc)) / (Lb * Lb * (La * La - Lc * Lc)))
Fphik = special.ellipkinc(phi, k)
Ephik = special.ellipeinc(phi, k)
return 2.0 * np.pi * (Lc * Lc / p) + 2.0 * np.pi * (
La * Lb / p) * (Ephik * np.sin(phi) * np.sin(phi) +
Fphik * np.cos(phi) * np.cos(phi)) / np.sin(phi)
def d1(v):
beta = np.arccos((v + 1. - np.sqrt(3.)) / (v + 1. + np.sqrt(3.)))
sin75 = np.sin(75. * np.pi / 180.)
sin75 = sin75**2
return (5. / 3.) * (v) * (
((3.**0.25) * (np.sqrt(1. + v**3.)) *
(scs.ellipeinc(beta, sin75) -
(1. / (3. + np.sqrt(3.))) * scs.ellipkinc(beta, sin75))) +
((1. - (np.sqrt(3.) + 1.) * v * v) / (v + 1. + np.sqrt(3.))))
def gendata(phis, alps):
l = '%23s%23s%23s\n' % ('phi', 'k', 'E') # phi [rad]
for i, alp in enumerate(alps):
k = np.sin(d2r(alp))
m = k**2.0
for j, phi in enumerate(phis):
p = d2r(phi)
r = sp.ellipeinc(p, m)
l += '%23.15e%23.15e%23.15e\n' % (p, k, r)
return l
def calc_t_r(qr, r1, r2, r3, r4, En, Lz, aa, M=1):
"""
delta_t_r in Drasco and Hughes (2005)
Parameters:
qr (float)
r1 (float): radial root
r2 (float): radial root
r3 (float): radial root
r4 (float): radial root
En (float): energy
Lz (float): angular momentum
aa (float): spin
Keyword Args:
M (float): mass
Returns:
t_r (float)
"""
psi_r = calc_psi_r(qr, r1, r2, r3, r4)
kr = ((r1 - r2) * (r3 - r4)) / ((r1 - r3) * (r2 - r4))
rp = M + sqrt(-(aa**2) + M**2)
rm = M - sqrt(-(aa**2) + M**2)
hr = (r1 - r2) / (r1 - r3)
hp = ((r1 - r2) * (r3 - rp)) / ((r1 - r3) * (r2 - rp))
hm = ((r1 - r2) * (r3 - rm)) / ((r1 - r3) * (r2 - rm))
return -((En * (
(-4 * (r2 - r3) *
(-(((-2 * aa**2 + (4 - (aa * Lz) / En) * rm) *
((qr * float(ellippi(hm, kr))) / pi -
float(ellippi(hm, psi_r, kr)))) /
((r2 - rm) *
(r3 - rm))) +
((-2 * aa**2 + (4 - (aa * Lz) / En) * rp) *
((qr * float(ellippi(hp, kr))) / pi - float(ellippi(hp, psi_r, kr)))
) /
((r2 - rp) * (r3 - rp)))) /
(-rm + rp) + 4 * (r2 - r3) *
((qr * float(ellippi(hr, kr))) / pi - float(ellippi(hr, psi_r, kr)))
+
(r2 - r3) *
(r1 + r2 + r3 + r4) *
((qr * float(ellippi(hr, kr))) / pi - float(ellippi(hr, psi_r, kr))) +
(r1 - r3) *
(r2 - r4) *
((qr * ellipe(kr)) / pi - ellipeinc(psi_r, kr) +
(hr * cos(psi_r) * sin(psi_r) * sqrt(1 - kr * sin(psi_r)**2)) /
(1 - hr * sin(psi_r)**2)))) / sqrt(
(1 - En**2) * (r1 - r3) * (r2 - r4)))
def generate_ellipse_angles(num, a, b):
angles = 2 * numpy.pi * numpy.arange(num) / num
if a < b:
e = float((1.0 - a**2 / b**2)**0.5)
tot_size = ellipeinc(2.0 * numpy.pi, e)
arc_size = tot_size / num
arcs = numpy.arange(num) * arc_size
res = optimize.root(lambda x: (ellipeinc(x, e) - arcs), angles)
angles = res.x
elif b < a:
e = float((1.0 - b**2 / a**2)**0.5)
tot_size = ellipeinc(2.0 * numpy.pi, e)
arc_size = tot_size / num
arcs = numpy.arange(num) * arc_size
res = optimize.root(lambda x: (ellipeinc(x, e) - arcs), angles)
angles = numpy.pi / 2 + res.x
else:
numpy.arange(0, 2 * numpy.pi, 2 * numpy.pi / num)
return angles
def test_ellint_2():
if NumCpp.NO_USE_BOOST and not NumCpp.STL_SPECIAL_FUNCTIONS:
return
a = np.random.rand(1).item()
b = np.random.rand(1).item()
assert (roundScaler(NumCpp.ellint_2_Scaler(a, b), NUM_DECIMALS_ROUND) ==
roundScaler(sp.ellipeinc(b, a**2), NUM_DECIMALS_ROUND))
shapeInput = np.random.randint(20, 100, [2, ])
shape = NumCpp.Shape(shapeInput[0].item(), shapeInput[1].item())
aArray = NumCpp.NdArray(shape)
bArray = NumCpp.NdArray(shape)
a = np.random.rand(shape.rows, shape.cols)
b = np.random.rand(shape.rows, shape.cols)
aArray.setArray(a)
bArray.setArray(b)
assert np.array_equal(roundArray(NumCpp.ellint_2_Array(aArray, bArray), NUM_DECIMALS_ROUND),
roundArray(sp.ellipeinc(b, a**2), NUM_DECIMALS_ROUND))
def d1(v):
"""
d1(v) = D(a)/a where D is growth function see. Einsenstein 1997
"""
beta = np.arccos((v+1.-np.sqrt(3.))/(v+1.+np.sqrt(3.)))
sin75 = np.sin(75.*np.pi/180.)
sin75 = sin75**2
ans = (5./3.)*(v)*(((3.**0.25)*(np.sqrt(1.+v**3.))*(scs.ellipeinc(beta,sin75)\
-(1./(3.+np.sqrt(3.)))*scs.ellipkinc(beta,sin75)))\
+((1.-(np.sqrt(3.)+1.)*v*v)/(v+1.+np.sqrt(3.))))
return ans
def ellipeOsc(x,m):
"""
Compute the oscillating part of elliptic function ?th kind, E
Args:
x (real): elliptic function arugment
m (real): elliptic function modulus
Returns:
real
"""
return ellipeinc(x , m) - 2*x*ellipe(m)/(np.pi)
def ellipsoid_area(ellipsoid):
"""
Compute the surface of a define ellipsoid with its half-axes (a, b, c)
"""
c, b, a = sorted([ellipsoid['a'], ellipsoid['b'], ellipsoid['c']])
if a == b == c:
area = 4*np.pi*a**2
else:
phi = np.arccos(c/a)
m = (a**2 * (b**2 - c**2)) / (b**2 * (a**2 - c**2))
temp = ellipeinc(phi, m)*np.sin(phi)**2 + ellipkinc(phi, m)*np.cos(phi)**2
area = 2*np.pi*(c**2 + a*b*temp/np.sin(phi))
return area
def precomputeEllipticArc(grid_size, ratio):
"""
Precomputes the incomplete elliptic integrals on a grid of theta so that
one can find the adress on a ellipse (x, y coordinates) for a given arc
length, :math:`k = \\sqrt{1 - b^2/a^2}` where b is semi-minor and a is
semi-major axis of the ellipse
*Args:*
grid_size:
The number of points (on a=1 ellipse) where the
incomplete elliptic integrals is computed
ratio:
b/a (semi-minor/semi-major) of Ellipse
*Returns:*
all_phi:
the grid of angles (ellipse parameter) on which the arc
lengths are evaluated
all_path_lengths:
the evaluated arc lengths
"""
k = np.sqrt(1- ratio**2)
phi = np.linspace(0, np.pi/2, grid_size+1)
ellip = scsp.ellipeinc(phi, k)
# now the phi runs from 0 - pi/2 owing to symmetry of the ellipse
# compute the arc length of the ellipse along the perimeter for phi from
# 0 to 2pi, weave together 0-pi/2, and then pi/2-0 etc.
max_val = ellip[-1]
ellip = np.delete(ellip, -1)
all_path_length = np.zeros(4*(grid_size) + 1 ) # four quarters of 0 - (pi/2 - grid)
all_phi = np.zeros(4*(grid_size) + 1) # the last value is 2*pi
all_path_length[0:grid_size] = ellip #the last element is for pi/2
all_path_length[(grid_size):(2*grid_size)] = (max_val - ellip[::-1]) + max_val #weaving requres reversing,
#owing to the nature of arc length on an ellipse
all_path_length[(2*grid_size):(3*grid_size)] = ellip + 2*max_val
all_path_length[(3*grid_size):(4*grid_size)] = (max_val - ellip[::-1]) + 3*max_val
all_path_length[-1] = 4*max_val
for i in range(4):
all_phi[i*(grid_size):(i+1)*(grid_size)] = phi[:-1] + i*phi[-1]
all_phi[-1] = 2*np.pi
return all_phi, all_path_length
def growth_factor(z, Omega_m):
a = 1.0/(1.0+z)
v = (1.0+z)*(Omega_m/(1.0-Omega_m))**(1.0/3.0)
phi = np.arccos((v+1.0-3.0**0.5)/(v+1.0+3.0**0.5))
m = (np.sin(75.0/180.0* np.pi))**2.0
part1c = 3.0**0.25 * (1.0+ v**3.0)**0.5
# first elliptic integral
F_elliptic = special.ellipkinc(phi, m)
# second elliptic integral
Se_elliptic = special.ellipeinc(phi, m)
part1 = part1c * ( Se_elliptic - 1.0/(3.0+3.0**0.5)*F_elliptic)
part2 = (1.0 - (3.0**0.5 + 1.0)*v*v)/(v+1.0+3.0**0.5)
d_1 = 5.0/3.0*v*(part1 + part2)
# if a goes to 0, use d_11, when z=1100, d_1 is close to d_11
# d_11 = 1.0 - 2.0/11.0/v**3.0 + 16.0/187.0/v**6.0
return a*d_1
def calculate_ellipse_arc(ellipse, t):
"""
:param ellipse:
:param t:
:return:
"""
phi = EllipticIntegralHelper.convert_phi_to_t(t)
ellipse_arc_from_y_axis = ellipse.semi_x_axis * spsp.ellipeinc(
phi, ellipse.m)
ellipse_arc = ellipse.circumference / 4 - ellipse_arc_from_y_axis
return ellipse_arc
def surface_area(self):
"""float: Get the surface area."""
# Implemented from this example:
# https://www.johndcook.com/blog/2014/07/06/ellipsoid-surface-area/
# It requires that a >= b >= c, so we sort the principal axes:
c, b, a = sorted([self.a, self.b, self.c])
if a > c:
phi = np.arccos(c / a)
m = (a**2 * (b**2 - c**2)) / (b**2 * (a**2 - c**2))
elliptic_part = ellipeinc(phi, m) * np.sin(phi)**2
elliptic_part += ellipkinc(phi, m) * np.cos(phi)**2
elliptic_part /= np.sin(phi)
else:
elliptic_part = 1
result = 2 * np.pi * (c**2 + a * b * elliptic_part)
return result
def area(rad):
"""Area of a spheroid
Arguments
---------
rad : dict
{'a', 'b', 'c'} Equatorial1, Equatorial2, and polar radius
"""
a, b, c = rad['a'], rad['b'], rad['c']
if (a != c) and (b != c): #spheroid
k = a**2*(b**2-c**2) / (b**2*(a**2-c**2))
phi = np.arccos(c/a)
tmp = ellipeinc(phi, k)*sin(phi)**2 + ellipkinc(phi, k)*cos(phi)**2
out = 2*pi*c**2 + tmp*2*pi*a*b/sin(phi)
else: #sphere
out = 4*pi*a**2
return out
def ellipeinc_(f, k):
return ellipeinc(f, k*k)
def sin_length(phi,k,omega): ## The length of a sinusoid is a classic calculus problem that falls into an elliptic integral. m = 1 + k**2 * omega**2 return ellipeinc(pi/2+omega*phi, 1-1/m )*sqrt(m)/omega
def fun(x): return np.sqrt(37)/6.0 * ellipeinc(6*x, 36.0/37.0);
def fun(x): return ellipeinc(x,0.5);
a1b2 = 1 - (e*e)
a1b = math.sqrt(a1b2)
### Case 2: b is semi-major axis of length 1; b = 1 > a
b1b = b1b2 = a1a
b1a2 = a1b2
b1a = a1b
### Append Cases 1 and 2 results to elis list
elis.append(
(subpct ### Percent of PI/2 to use for upper limit
,theta ### Upper limit Eccentric Anomaly angle
,e ### Eccentricity a.k.a. Elliptic Modulus
,a1b ### semi-minor axis b for a = 1 > b
,eli(a1a,a1b,substep) ### Case 1 arc length from a for b < a = 1
,ellipeinc(math.pi/2,e2)-ellipeinc(math.pi/2-theta,e2) ### Case 1 actual
,eli(b1a,b1b,substep) ### Case 2 arc length from a for b = 1 > a
,ellipeinc(theta,e2) ### Case 2 actual
,)
)
########################################################################
### Print out results
for subpct,theta,e,a1b,a1eli,Ecompl,b1eli,E in elis:
print('%s%3d %16.16f %10.8f %12.10f %12.10f %12.10f %12.4e %12.10f %12.10f %12.4e'
%(e==0. and '\n' or ''
,subpct ### Upper limit, percent of PI/2
,theta ### Upper limit Ecc. Anom. angle
,e ### Eccentricity
,a1b ### semi-minor axis
,a1eli ### Case 1 numerical result
from __future__ import division, print_function
from math import pi
from scipy.special import ellipeinc
import sys
J = float(sys.argv[1])
Gamma = 1
lam = J/(2*Gamma)
theta = 4*lam/(1+lam)**2
print(ellipeinc(pi/2,theta)/(pi/2),1+lam)
print("{:.15f}".format(-Gamma*2/pi*(1+lam)*ellipeinc(pi/2,theta)))
