def testHyperbolic(self):
self.assertEqual(math.sinh(5), hyperbolic.sineh_op(5))
self.assertEqual(math.cosh(5), hyperbolic.cosineh_op(5))
self.assertEqual(math.tanh(5), hyperbolic.tangenth_op(5))
self.assertEqual(1. / math.sinh(5), hyperbolic.cosecanth_op(5))
self.assertEqual(1. / math.cosh(5), hyperbolic.secanth_op(5))
self.assertEqual(1. / math.tanh(5), hyperbolic.cotangenth_op(5))
def heisenT(beta):
# Following notation of Xiang (http://arxiv.org/pdf/1201.1144v4.pdf)
W=np.array([[math.sqrt(math.cosh(beta)), math.sqrt(math.sinh(beta))],
[math.sqrt(math.cosh(beta)), -math.sqrt(math.sinh(beta))]])
#T=np.einsum("au,ad,al,ar->udlr",W,W,W,W)
T=np.einsum("al,ar->lr",W,W)
return T
def sinh(x):
""" Return the hyperbolic sine of x. """
if math.isinf(x.real) or math.isinf(x.imag):
# need to raise DomainError if y is +/- infinity and x is not
# a NaN and not -infinity
if math.isinf(x.imag) and not math.isnan(x.real):
raise ValueError("math domain error")
if math.isinf(x.real) and -_INF < x.imag < _INF and x.imag != .0:
if x.real > 0:
_real = math.copysign(_INF, cos(x.imag))
_imag = math.copysign(_INF, sin(x.imag))
else:
_real = -math.copysign(_INF, cos(x.imag))
_imag = math.copysign(_INF, sin(x.imag))
return complex(_real, _imag)
return _SPECIAL_VALUE(x,_sinh_special_values)
if math.fabs(x.real) > _CM_LOG_LARGE_DOUBLE:
x_minus_one = x.real - math.copysign(1.0, x.real)
_real = math.cos(x.imag)*math.sinh(x.imag)*math.e
_imag = math.sin(x.imag)*math.cosh(x.imag)*math.e
else:
_real = math.cos(x.imag)*math.sinh(x.real)
_imag = math.sin(x.imag)*math.cosh(x.real)
if math.isinf(_real) or math.isinf(_imag):
raise OverflowError()
return complex(_real, _imag)
def cosh(x):
"""Return the hyperbolic cosine of x."""
# special treatment for cosh(+/-inf + iy) if y is not a NaN
if isinf(x):
if -_INF < x.imag < _INF and x.imag != .0:
if x.real > 0:
_real = math.copysign(_INF, math.cos(x.imag))
_imag = math.copysign(_INF, math.sin(x.imag))
else:
_real = math.copysign(_INF, math.cos(x.imag))
_imag = -math.copysign(_INF, math.sin(x.imag))
return complex(_real,_imag)
else:
# need to raise math domain error if y is +/- infinity and x is not a NaN
if x.imag != .0 and not math.isnan(x.real):
raise ValueError("math domain error")
return _SPECIAL_VALUE(x,_cosh_special_values)
if math.fabs(x.real) > _CM_LOG_LARGE_DOUBLE:
# deal correctly with cases where cosh(x.real) overflows but
# cosh(z) does not.
x_minus_one = x.real - math.copysign(1.0, x.real)
_real = cos(x.imag) * math.cosh(x_minus_one) * math.e
_imag = sin(x.imag) * math.sinh(x_minus_one) * math.e
else:
_real = math.cos(x.imag) * math.cosh(x.real)
_imag = math.sin(x.imag) * math.sinh(x.real)
ret = complex(_real, _imag)
# detect overflow
if isinf(ret):
raise OverflowError()
return ret
def laplace_Dirichlet_analytic(Ttop, x, y):
""" Function that calculates the analytic solution to the laplace heat equation
d^2(T)/dx^2 + d^2(T)/dy^2 = 0
with Dirichlet boundary conditions:
T(x, 0) = 0
T(0, y) = 0 y<1.5
T(1, y) = 0 y<1.5
T(x, 1.5) = Ttop
on a square grid with xmax = 1 and ymax= 1
Args:
Ttop(float): Temperature at top, T(x, 1.5)
x(float): x-coordinate
y(float): y-coordinate
Returns:
T(float): Temperature at T(x, y)
"""
sum = 0
for n in range(1,50, 1):
lambdan = pi*n
An = 2*((-1)**(n+1)+1)/(lambdan*sinh(1.5*lambdan))
term = An*sinh(lambdan*y )*sin(lambdan*x)
sum += term
return Ttop*sum
def to_wgs84(x, y, twd67=False):
'''
The formula is based on
"https://en.wikipedia.org/wiki/Universal_Transverse_Mercator_coordinate_system"
'''
if twd67:
x, y = twd67_to_twd97(x, y)
x /= 1000.0 # m to km
y /= 1000.0 # m to km
xi = (y - N0) / (k0 * A)
eta = (x - E0) / (k0 * A)
xip = (xi -
b1 * math.sin(2 * xi) * math.cosh(2 * eta) -
b2 * math.sin(4 * xi) * math.cosh(4 * eta) -
b3 * math.sin(6 * xi) * math.cosh(6 * eta))
etap = (eta -
b1 * math.cos(2 * xi) * math.sinh(2 * eta) -
b2 * math.cos(4 * xi) * math.sinh(4 * eta) -
b3 * math.cos(6 * xi) * math.sinh(6 * eta))
chi = math.asin(math.sin(xip) / math.cosh(etap))
lat = math.degrees(
chi +
d1 * math.sin(2 * chi) +
d2 * math.sin(4 * chi) +
d3 * math.sin(6 * chi))
lon = lon0 + math.degrees(
math.atan(math.sinh(etap) / math.cos(xip)))
return (lat, lon)
def pmuboost3d(jets, met, lep):
mw = 80.385
wwwtlv = ROOT.TLorentzVector(1e-9,1e-9,1e-9,1e-9)
mettlv = ROOT.TLorentzVector(1e-9,1e-9,1e-9,1e-9)
leptlv = ROOT.TLorentzVector(1e-9,1e-9,1e-9,1e-9)
for jet in jets:
# if(e.jetPt[ij] > 30.):
tlvaux = ROOT.TLorentzVector(1e-9,1e-9,1e-9,1e-9)
tlvaux.SetPtEtaPhiM(jet['pt'],0.,jet['phi'],0.)
wwwtlv -= tlvaux
mettlv.SetPtEtaPhiM(met['pt'],0.,met['phi'],0.)
leptlv.SetPtEtaPhiM(lep['pt'],lep['eta'],lep['phi'], 0.)
# leptlv.SetPtEtaPhiM(e.muPt[0],e.muEta[0],e.muPhi[0],0.)
# calculate eta(W) estimate; take the + solution
ptl = (wwwtlv - mettlv).Pt()
ptn = mettlv.Pt()
ptw = wwwtlv.Pt()
etal = leptlv.Eta()
A = (mw*mw + ptw*ptw - ptl*ptl - ptn*ptn) / (2.*ptl*ptn)
if(A<1.): A=1.
etaw = asinh((ptl*sinh(etal)+ptn*sinh(etal+acosh(A)))/ptw)
if(abs(etaw)>10.): etaw=10*etaw/abs(etaw) # avoid too large values
wwwtlv.SetPtEtaPhiM(wwwtlv.Pt(),etaw,wwwtlv.Phi(),mw)
# calulate boosted lepton momentum
beta = wwwtlv.BoostVector()
leptlv.Boost(-beta)
return leptlv.P()
def to_latlon(northings, eastings, altitude):
f = 1/298.257223563
n = f / (2 - f)
n_0 = 0.0
k_0 = 0.9996
e_0 = 500.
a = 6378.137 #km
big_a = (a / (1 + n)) * (1 + (n**2 / 4) + (n**4 / 64)) #approximately
alpha_1 = .5 * n - (2. / 3.) * n**2 + (5. / 16.) * n**3
alpha_2 = (13./48.) * n**2 - (3./5.) * n**3
alpha_3 = (61./240.) * n**3
beta_1 = (1./2.) * n - (2. / 3.) * n**2 + (37./96.) * n**3
beta_2 = (1./48.) * n**2 + (1. / 15.) * n**3
beta_3 = (17. /480.) * n**3
delta_1 = 2. * n - (2./3.) * n**2 - 2* n**3
delta_2 = (7./3.) * n**2 - (8. / 5.) * n**3
delta_3 = (56./15.) * n**3
psi = (n - n_0) / (k_0 * big_a)
eta = (e - e_0) / (k_0 * big_a)
psi_prime = psi - ((beta_1 * sin(2. * 1 * eta) * cosh(2. * 1 * eta)) +
(beta_2 * sin(2. * 2 * eta) * cosh(2. * 2 * eta)) + (beta_3 * sin(2. * 3 * eta) * cosh(2. * 3 * eta)))
eta_prime = eta - ((beta_1 * cos(2. * 1 * psi) * sinh(2. * 1 * eta)) +
(beta_2 * cos(2. * 2 * psi) * sinh(2. * 2 * eta)) + (beta_3 * cos(2. * 3 * psi) * sinh(2. * 3 * eta)))
sigma_prime = 1. - ((2. * 1 * beta_1 * cos(2. * 1 * psi) * cosh(2. * 1 * eta)) +
(2. * 2 * beta_2 * cos(2. * 2 * psi) * cosh(2. * 2 * eta)) + (2. * 3 * beta_3 * cos(2. * 3 * psi) * cosh(2. * 3 * eta)))
tau_prime = ((2. * 1 * beta_1 * sin(2. * 1 * psi) * sinh(2. * 1 * eta)) +
(2. * 2 * beta_2 * sin(2. * 2 * psi) * sinh(2. * 2 * eta)) + (2. * 3 * beta_3 * sin(2. * 3 * psi) * sinh(2. * 3 * eta)))
chi = asin (sin(psi_prime)/cosh(eta_prime))
phi = chi + (delta_1 * sin(2. * 1 * chi)) + (delta_2 * sin(2. * 2 * chi)) + (delta_3 * sin(2. * 3 * chi))
lambda_0 = 0
lambdu = 0
k = 0
gamma = 0
return None
def lalo_to_xy(la, lo, lo0, E0, ellipsoid):
lo0 = math.radians(lo0)
la = math.radians(la)
lo = math.radians(lo)
e = ellipsoid['e']
k0 = ellipsoid['k0']
h1p = ellipsoid['h1p']
h2p = ellipsoid['h2p']
h3p = ellipsoid['h3p']
h4p = ellipsoid['h4p']
A1 = ellipsoid['A1']
Q = asinh(math.tan(la)) - e * atanh(e * math.sin(la))
be = math.atan(math.sinh(Q))
nnp = atanh(math.cos(be) * math.sin(lo - lo0))
Ep = math.asin(math.sin(be) * math.cosh(nnp))
E1 = h1p * math.sin(2.0 * Ep) * math.cosh(2.0 * nnp)
E2 = h2p * math.sin(4.0 * Ep) * math.cosh(4.0 * nnp)
E3 = h3p * math.sin(6.0 * Ep) * math.cosh(6.0 * nnp)
E4 = h4p * math.sin(8.0 * Ep) * math.cosh(8.0 * nnp)
nn1 = h1p * math.cos(2.0 * Ep) * math.sinh(2.0 * nnp)
nn2 = h2p * math.cos(4.0 * Ep) * math.sinh(4.0 * nnp)
nn3 = h3p * math.cos(6.0 * Ep) * math.sinh(6.0 * nnp)
nn4 = h4p * math.cos(8.0 * Ep) * math.sinh(8.0 * nnp)
E = Ep + E1 + E2 + E3 + E4
nn = nnp + nn1 + nn2 + nn3 + nn4
XY = {}
XY['N'] = A1 * E * k0
XY['E'] = A1 * nn * k0 + E0
return XY
def eccentricAnomaly(e,M):
"""Returns the eccentric anomaly given the eccentricity and mean anomaly of a Keplerian orbit.
Keyword arguments:
e -- the eccentricity
M -- the mean anomaly
"""
if e<1.:
E = M if e<0.8 else math.pi
F = E - e*math.sin(M) - M
for i in range(100):
E = E - F/(1.0-e*math.cos(E))
F = E - e*math.sin(E) - M
if math.fabs(F)<1e-16:
break
E = mod2pi(E)
return E
else:
E = M
F = E - e*math.sinh(E) - M
for i in range(100):
E = E - F/(1.0-e*math.cosh(E))
F = E - e*math.sinh(E) - M
if math.fabs(F)<1e-16:
break
E = mod2pi(E)
return E
def tri_angles(L):
ans = [0, 0, 0]
for i in xrange(3):
top = math.cosh(L[i])*math.cosh(L[(i+2)%3]) - math.cosh( L[(i+1)%3] )
bottom = math.sinh(L[i])*math.sinh(L[(i+2)%3])
ans[i] = math.acos(top/bottom)
return ans
def robustness(self,K):
"""
vals is a list of tuple (eigenvalue,order id)
"""
if len(self.vals) < K or K == 0:
K = len(self.vals)
""" Robustness """
robust = 0.0
""" basic, we assume the lamda_one is large than zero """
lamda_one = self.vals[0]
if lamda_one <= 0:
print "the largest eigenvalue of graph is less than zero, please check the graph"
return
for i in range(0,len(self.vals)):
""" NSC_k """
nsc = 0.0
for j in range(0,K):
lamda_j = self.vals[j]
nsc = nsc + self.vecs[i][j]*self.vecs[i][j]*math.sinh(lamda_j)
r = math.log(self.vecs[i][0]*self.vecs[i][0]) - math.log(nsc / math.sinh(lamda_one))
r *= 0.5
robust = robust + r*r
robust = robust / float(len(self.vals))
robust = math.sqrt(robust)
return robust
def fromwgs84(lat, lng, pkm=False):
"""
Convert coordintes from WGS84 to TWD97
pkm true for Penghu, Kinmen and Matsu area
The latitude and longitude can be in the following formats:
[+/-]DDD°MMM'SSS.SSSS" (unicode)
[+/-]DDD°MMM.MMMM' (unicode)
[+/-]DDD.DDDDD (string, unicode or float)
The returned coordinates are in meters
"""
_lng0 = lng0pkm if pkm else lng0
lat = radians(todegdec(lat))
lng = radians(todegdec(lng))
t = sinh((atanh(sin(lat)) - 2*pow(n,0.5)/(1+n)*atanh(2*pow(n,0.5)/(1+n)*sin(lat))))
epsilonp = atan(t/cos(lng-_lng0))
etap = atan(sin(lng-_lng0) / pow(1+t*t, 0.5))
E = E0 + k0*A*(etap + alpha1*cos(2*1*epsilonp)*sinh(2*1*etap) +
alpha2*cos(2*2*epsilonp)*sinh(2*2*etap) +
alpha3*cos(2*3*epsilonp)*sinh(2*3*etap))
N = N0 + k0*A*(epsilonp + alpha1*sin(2*1*epsilonp)*cosh(2*1*etap) +
alpha2*sin(2*2*epsilonp)*cosh(2*2*etap) +
alpha3*sin(2*3*epsilonp)*cosh(2*3*etap))
return E*1000, N*1000
def test_funcs_multi(self):
pi = math.pi
# sin family
self.assertQuantity(data.sin(Quantity( (0,pi/2), (2,2))), (0,1), (2,0))
self.assertQuantity(data.sinh(Quantity((0,1), (2,2))), (0, math.sinh(1)), (2, math.cosh(1)*2))
self.assertQuantity(data.asin(Quantity((0,0.5), (2,2))), (0,math.asin(0.5)), (2,2/math.sqrt(1-0.5**2)))
self.assertQuantity(data.asinh(Quantity((0,1), (2,2))), (0,math.asinh(1)), (2,2/math.sqrt(1+1**2)))
# cos family
self.assertQuantity(data.cos(Quantity((0,pi/2), (2,2))), (1,0), (0,-2))
self.assertQuantity(data.cosh(Quantity((0,1), (2,2))), (1,math.cosh(1)), (0,math.sinh(1)*2))
self.assertQuantity(data.acos(Quantity((0,0.5), (2,2))), (math.acos(0),math.acos(0.5)), (-2,-2/math.sqrt(1-0.5**2)))
self.assertQuantity(data.acosh(Quantity((2,3), (2,2))), (math.acosh(2), math.acosh(3)), (2/math.sqrt(2**2-1),2/math.sqrt(3**2-1)))
# tan family
self.assertQuantity(data.tan(Quantity((0,1), (2,2))), (0,math.tan(1)), (2,2/math.cos(1)**2))
self.assertQuantity(data.tanh(Quantity((0,1), (2,2))), (0,math.tanh(1)), (2,2/math.cosh(1)**2))
self.assertQuantity(data.atan(Quantity((0,1), (2,2))), (0, math.atan(1)), (2,2/(1+1**2)))
self.assertQuantity(data.atan2(Quantity((0,1), (2,2)), Quantity((1,1), (0,0))), (0,math.atan(1)), (2,2/(1+1**2)))
self.assertQuantity(data.atanh(Quantity((0,0.5), (2,2))), (0,math.atanh(0.5)), (2,2/(1-0.5**2)))
#misc
self.assertQuantity(data.sqrt(Quantity((1,4), (2,2))), (1,2), (1,1/2))
self.assertQuantity(data.exp(Quantity((1,4), (2,2))), (math.e, math.e**4), (2 * math.e,2*math.e**4))
self.assertQuantity(data.log(Quantity((1,4), (2,2))), (0, math.log(4)), (2,1/2))
def testInnerIteration(mesh,dm):
J_inc = 0
N = mesh.n_elems
sig_r = [dm.getND(i).getSigmaR()[0] for i in range(N)]
D = [dm.getND(i).diff_coef[0] for i in range(N)]
Q = np.ones(N)
S = 3
Q = np.ones((N,1))*S
G = dm.n_groups
J_inc = 0.
phi = innerIteration(mesh,Q,dm,J_inc,verbosity=5)
phi_ss = diffSSFixedSource(mesh,D,sig_r,Q,J_inc,tol=1.E-06)
x = mesh.getCellCenters()
from math import sinh, sqrt
rex = mesh.getElement(N-1).rr + 2.*D[0]
L = sqrt(D[0]/sig_r[0])
analytic = lambda r: S*sinh((rex - r)/L)/(4*pi*r*D[0]*sinh(rex/L))
analytic = lambda r: -1.*S*L*L*rex/(r*D[0]*sinh(rex/L))*sinh(r/L) + S*L*L/D[0]
plt.plot(x,phi,'+-b',label='Inner')
plt.plot(x,phi_ss,"o",label='Direct SS Call')
exact = [analytic(i) for i in x]
plt.plot(x,exact,"--r",label="exact")
plt.legend()
plt.show()
exit()
def gridToGeodetic(self, north, east):
"""
Transformation from grid coordinates to geodetic coordinates.
@param north (corresponds to X in RT 90 and N in SWEREF 99.)
@param east (corresponds to Y in RT 90 and E in SWEREF 99.)
@return (latitude, longitude)
"""
if (self._initialized == False):
return None
deg_to_rad = math.pi / 180
lambda_zero = self._central_meridian * deg_to_rad
xi = (north - self._false_northing) / (self._scale * self._a_roof)
eta = (east - self._false_easting) / (self._scale * self._a_roof)
xi_prim = xi - \
self._delta1*math.sin(2.0*xi) * math.cosh(2.0*eta) - \
self._delta2*math.sin(4.0*xi) * math.cosh(4.0*eta) - \
self._delta3*math.sin(6.0*xi) * math.cosh(6.0*eta) - \
self._delta4*math.sin(8.0*xi) * math.cosh(8.0*eta)
eta_prim = eta - \
self._delta1*math.cos(2.0*xi) * math.sinh(2.0*eta) - \
self._delta2*math.cos(4.0*xi) * math.sinh(4.0*eta) - \
self._delta3*math.cos(6.0*xi) * math.sinh(6.0*eta) - \
self._delta4*math.cos(8.0*xi) * math.sinh(8.0*eta)
phi_star = math.asin(math.sin(xi_prim) / math.cosh(eta_prim))
delta_lambda = math.atan(math.sinh(eta_prim) / math.cos(xi_prim))
lon_radian = lambda_zero + delta_lambda
lat_radian = phi_star + math.sin(phi_star) * math.cos(phi_star) * \
(self._Astar + \
self._Bstar*math.pow(math.sin(phi_star), 2) + \
self._Cstar*math.pow(math.sin(phi_star), 4) + \
self._Dstar*math.pow(math.sin(phi_star), 6))
lat = lat_radian * 180.0 / math.pi
lon = lon_radian * 180.0 / math.pi
return (lat, lon)
def from_utm(easting: float, northing: float,
zone: int, hemisphere: int =1) -> (float, float):
"""
Convert UTM coordinates to decimal latitude and longitude coordinates.
Keyword Arguments:
easting: The easting in m.
northing: The northing in m.
zone: The zone, an integer between 1 and 60, inclusive.
hemisphere: A signed number, where a negative number indicates the
coordinates are located in the southern hemisphere.
Returns:
latitude: The latitude in decimal degrees.
longitude: The longitude in deciaml degrees.
Raises:
OverflowError: The coordinate does not exist.
"""
easting, northing = easting/1000, northing/1000
northing_ = 10000 if hemisphere < 0 else 0
xi_ = xi = (northing - northing_)/(k0*A)
eta_ = eta = (easting - easting_)/(k0*A)
for j in range(1, 4):
p, q = 2*j*xi, 2*j*eta
xi_ -= beta[j - 1]*sin(p)*cosh(q)
eta_ -= beta[j - 1]*cos(p)*sinh(q)
chi = asin(sin(xi_)/cosh(eta_))
latitude = chi + sum(delta[j - 1]*sin(2*j*chi) for j in range(1, 4))
longitude_ = radians(6*zone - 183)
longitude = longitude_ + atan2(sinh(eta_), cos(xi_))
return degrees(latitude), degrees(longitude)
def exact_fn ( x ): # ## EXACT_FN evaluates the exact solution. # from math import sinh value = x - sinh ( x ) / sinh ( 1.0 ) return value
def tileGeometry(self, x, y, z):
n = 2.0 ** z
xmin = x / n * 360.0 - 180.0
xmax = (x + 1) / n * 360.0 - 180
ymin = math.degrees(math.atan(math.sinh(math.pi * (1 - 2 * y / n))))
ymax = math.degrees(math.atan(math.sinh(math.pi * (1 - 2 * (y + 1) / n))))
return QgsGeometry.fromRect(QgsRectangle(xmin, ymin, xmax, ymax))
def sinh(self, other=None):
# Return hiperbolic sine of interval
if other != None:
intv = IReal(self, other)
else:
if type(self) == float or type(self) == str:
intv = IReal(self)
else:
intv = self
if math.sinh(intv.inf) > math.sinh(intv.sup):
inf = max(intv.inf, intv.sup)
sup = min(intv.inf, intv.sup)
else:
inf = intv.inf
sup = intv.sup
ireal.rounding.set_mode(1)
ireal.rounding.set_mode(-1)
ireal.rounding.set_mode(-1)
new_inf = math.sinh(inf)
ireal.rounding.set_mode(1)
new_sup = max(float(IReal('%.16f' % math.sinh(sup)).sup), float(IReal('%.20f' % math.sinh(sup)).sup))
return IReal(new_inf, new_sup)
def c_cosh(x, y):
if not isfinite(x) or not isfinite(y):
if isinf(x) and isfinite(y) and y != 0.:
if x > 0:
real = copysign(INF, math.cos(y))
imag = copysign(INF, math.sin(y))
else:
real = copysign(INF, math.cos(y))
imag = -copysign(INF, math.sin(y))
r = (real, imag)
else:
r = cosh_special_values[special_type(x)][special_type(y)]
# need to raise ValueError if y is +/- infinity and x is not
# a NaN
if isinf(y) and not isnan(x):
raise ValueError("math domain error")
return r
if fabs(x) > CM_LOG_LARGE_DOUBLE:
# deal correctly with cases where cosh(x) overflows but
# cosh(z) does not.
x_minus_one = x - copysign(1., x)
real = math.cos(y) * math.cosh(x_minus_one) * math.e
imag = math.sin(y) * math.sinh(x_minus_one) * math.e
else:
real = math.cos(y) * math.cosh(x)
imag = math.sin(y) * math.sinh(x)
if isinf(real) or isinf(imag):
raise OverflowError("math range error")
return real, imag
def c_sinh(x, y):
# special treatment for sinh(+/-inf + iy) if y is finite and nonzero
if not isfinite(x) or not isfinite(y):
if isinf(x) and isfinite(y) and y != 0.:
if x > 0:
real = copysign(INF, math.cos(y))
imag = copysign(INF, math.sin(y))
else:
real = -copysign(INF, math.cos(y))
imag = copysign(INF, math.sin(y))
r = (real, imag)
else:
r = sinh_special_values[special_type(x)][special_type(y)]
# need to raise ValueError if y is +/- infinity and x is not
# a NaN
if isinf(y) and not isnan(x):
raise ValueError("math domain error")
return r
if fabs(x) > CM_LOG_LARGE_DOUBLE:
x_minus_one = x - copysign(1., x)
real = math.cos(y) * math.sinh(x_minus_one) * math.e
imag = math.sin(y) * math.cosh(x_minus_one) * math.e
else:
real = math.cos(y) * math.sinh(x)
imag = math.sin(y) * math.cosh(x)
if isinf(real) or isinf(imag):
raise OverflowError("math range error")
return real, imag
def cosh(x):
_cosh_special = [
[inf+nanj, None, inf, complex(float("inf"), -0.0), None, inf+nanj, inf+nanj],
[nan+nanj, None, None, None, None, nan+nanj, nan+nanj],
[nan, None, 1, complex(1, -0.0), None, nan, nan],
[nan, None, complex(1, -0.0), 1, None, nan, nan],
[nan+nanj, None, None, None, None, nan+nanj, nan+nanj],
[inf+nanj, None, complex(float("inf"), -0.0), inf, None, inf+nanj, inf+nanj],
[nan+nanj, nan+nanj, nan, nan, nan+nanj, nan+nanj, nan+nanj]
]
z = _make_complex(x)
if not isfinite(z):
if math.isinf(z.imag) and not math.isnan(z.real):
raise ValueError
if math.isinf(z.real) and math.isfinite(z.imag) and z.imag != 0:
if z.real > 0:
return complex(math.copysign(inf, math.cos(z.imag)),
math.copysign(inf, math.sin(z.imag)))
return complex(math.copysign(inf, math.cos(z.imag)),
-math.copysign(inf, math.sin(z.imag)))
return _cosh_special[_special_type(z.real)][_special_type(z.imag)]
if abs(z.real) > _LOG_LARGE_DOUBLE:
x_minus_one = z.real - math.copysign(1, z.real)
ret = complex(e * math.cos(z.imag) * math.cosh(x_minus_one),
e * math.sin(z.imag) * math.sinh(x_minus_one))
else:
ret = complex(math.cos(z.imag) * math.cosh(z.real),
math.sin(z.imag) * math.sinh(z.real))
if math.isinf(ret.real) or math.isinf(ret.imag):
raise OverflowError
return ret
def sinh(x):
_sinh_special = [
[inf+nanj, None, complex(-float("inf"), -0.0), -inf, None, inf+nanj, inf+nanj],
[nan+nanj, None, None, None, None, nan+nanj, nan+nanj],
[nanj, None, complex(-0.0, -0.0), complex(-0.0, 0.0), None, nanj, nanj],
[nanj, None, complex(0.0, -0.0), complex(0.0, 0.0), None, nanj, nanj],
[nan+nanj, None, None, None, None, nan+nanj, nan+nanj],
[inf+nanj, None, complex(float("inf"), -0.0), inf, None, inf+nanj, inf+nanj],
[nan+nanj, nan+nanj, complex(float("nan"), -0.0), nan, nan+nanj, nan+nanj, nan+nanj]
]
z = _make_complex(x)
if not isfinite(z):
if math.isinf(z.imag) and not math.isnan(z.real):
raise ValueError
if math.isinf(z.real) and math.isfinite(z.imag) and z.imag != 0:
if z.real > 0:
return complex(math.copysign(inf, math.cos(z.imag)),
math.copysign(inf, math.sin(z.imag)))
return complex(-math.copysign(inf, math.cos(z.imag)),
math.copysign(inf, math.sin(z.imag)))
return _sinh_special[_special_type(z.real)][_special_type(z.imag)]
if abs(z.real) > _LOG_LARGE_DOUBLE:
x_minus_one = z.real - math.copysign(1, z.real)
return complex(math.cos(z.imag) * math.sinh(x_minus_one) * e,
math.sin(z.imag) * math.cosh(x_minus_one) * e)
return complex(math.cos(z.imag) * math.sinh(z.real),
math.sin(z.imag) * math.cosh(z.real))
def test_sinh(self):
self.assertAlmostEqual(qmath.sinh(self.f1), math.sinh(self.f1))
self.assertAlmostEqual(qmath.sinh(self.c1), cmath.sinh(self.c1))
self.assertAlmostEqual(qmath.sinh(self.c2), cmath.sinh(self.c2))
self.assertAlmostEqual(qmath.sinh(Quat(self.f1)), math.sinh(self.f1))
self.assertAlmostEqual(qmath.sinh(Quat(0, 3)), cmath.sinh(3J))
self.assertAlmostEqual(qmath.sinh(Quat(4, 5)), cmath.sinh(4 + 5J))
self.assertAlmostEqual(qmath.sinh(Quat(0, self.f1)), Quat(0, math.sin(self.f1)))
def levy_harmonic_path(xstart, xend, dtau, N):
x = [xstart]
for k in range(1, N):
dtau_prime = (N - k) * dtau
Ups1 = 1.0 / math.tanh(dtau) + 1.0 / math.tanh(dtau_prime)
Ups2 = x[k-1] / math.sinh(dtau) + xend / math.sinh(dtau_prime)
x.append(random.gauss(Ups2 / Ups1, 1.0 / math.sqrt(Ups1)))
return x
def aux1(x):
'''
Auxiliary function 1.
Break points used to ensure maximal precision::
x
Aux1(x) = -------
sinh(x)
d sinh(x) - x*cosh(x)
--Aux1(x) = -------------------
dx (sinh(x))^2
'''
if isinstance(x, ADVar):
y = x.val
else:
y=float(x);
if y < -_BP0_MISC:
y = -_BP0_MISC
elif y > _BP0_MISC:
y = _BP0_MISC
if y <= _BP0_AUX1:
z = y / math.sinh(y)
elif y <= _BP1_AUX1:
z = 1 - y*y/6.0*(1.0 - 7.0*y*y/60.0)
else:
z = y / math.sinh(y)
if not isinstance(x, ADVar):
return z
r = ADVar()
r.val = z
if y <= _BP4_DAUX1:
pd = 0.0
elif y <= _BP2_DAUX1:
pd = -(1+y)*math.exp(y)
elif y <= _BP0_DAUX1:
tmp = math.sinh(y);
pd = (tmp - y*math.cosh(y))/(tmp*tmp)
elif y <= _BP1_DAUX1:
pd = -y/3.0*(1.0 - 7.0*y*y/30.0)
elif y <= _BP3_DAUX1:
tmp = math.sinh(y)
pd = (z - y*math.cosh(y))/(z*z)
elif y <= _BP5_DAUX1:
pd = (1-y)*math.exp(-y)
else:
pd = 0.0
for i,dx in x.deriv:
r.deriv.append( (i, pd*dx) )
return r
def heat_capacity_v_einstein(T,einstein_T,n):
"""
Heat capacity at constant volume. In J/K/mol
"""
if T <= eps:
return 0.
x = einstein_T/T
C_v = 3.0*n*gas_constant*x*x/4/(math.sinh(x/2.)*math.sinh(x/2.))
return C_v
def t0_xi(): delta_rho = (rho0-rhou)*g rhog = rho0*g mu = mu0 k = wavenumber() return -1.0*(((delta_rho*k**2*mu - k**2*mu*rhog)*D*sinh(D*k)**2 - (delta_rho*k**2*mu - k**2*mu*rhog)*D*cosh(D*k)**2 - (delta_rho*k*mu -\ k*mu*rhog)*sinh(D*k)*cosh(D*k) + sqrt((delta_rho**2*k**2 - 2*delta_rho*k**2*rhog + k**2*rhog**2)*D**2*cosh(D*k)**4 - 2*(delta_rho**2*k -\ 2*delta_rho*k*rhog + k*rhog**2)*D*sinh(D*k)**3*cosh(D*k) + 2*(delta_rho**2*k - 2*delta_rho*k*rhog + k*rhog**2)*D*sinh(D*k)*cosh(D*k)**3 +\ ((delta_rho**2*k**2 + 2*delta_rho*k**2*rhog + k**2*rhog**2)*D**2 + 4*delta_rho*rhog)*sinh(D*k)**4 - (2*(delta_rho**2*k**2 + k**2*rhog**2)*D**2 -\ delta_rho**2 + 2*delta_rho*rhog - rhog**2)*sinh(D*k)**2*cosh(D*k)**2)*k*mu)/(delta_rho*rhog*sinh(D*k)**2))
def __conv_WGS84_SWED_RT90(lat, lon):
"""
Input is lat and lon as two float numbers
Output is X and Y coordinates in RT90
as a tuple of float numbers
The code below converts to/from the Swedish RT90 koordinate
system. The converion functions use "Gauss Conformal Projection
(Transverse Marcator)" Krüger Formulas.
The constanst are for the Swedish RT90-system.
With other constants the conversion should be useful for
other geographical areas.
"""
# Some constants used for conversion to/from Swedish RT90
f = 1.0/298.257222101
e2 = f*(2.0-f)
n = f/(2.0-f)
L0 = math.radians(15.8062845294) # 15 deg 48 min 22.624306 sec
k0 = 1.00000561024
a = 6378137.0 # meter
at = a/(1.0+n)*(1.0+ 1.0/4.0* pow(n, 2)+1.0/64.0*pow(n, 4))
FN = -667.711 # m
FE = 1500064.274 # m
#the conversion
lat_rad = math.radians(lat)
lon_rad = math.radians(lon)
A = e2
B = 1.0/6.0*(5.0*pow(e2, 2) - pow(e2, 3))
C = 1.0/120.0*(104.0*pow(e2, 3) - 45.0*pow(e2, 4))
D = 1.0/1260.0*(1237.0*pow(e2, 4))
DL = lon_rad - L0
E = A + B*pow(math.sin(lat_rad), 2) + \
C*pow(math.sin(lat_rad), 4) + \
D*pow(math.sin(lat_rad), 6)
psi = lat_rad - math.sin(lat_rad)*math.cos(lat_rad)*E
xi = math.atan2(math.tan(psi), math.cos(DL))
eta = atanh(math.cos(psi)*math.sin(DL))
B1 = 1.0/2.0*n - 2.0/3.0*pow(n, 2) + 5.0/16.0*pow(n, 3) + \
41.0/180.0*pow(n, 4)
B2 = 13.0/48.0*pow(n, 2) - 3.0/5.0*pow(n, 3) + 557.0/1440.0*pow(n, 4)
B3 = 61.0/240.0*pow(n, 3) - 103.0/140.0*pow(n, 4)
B4 = 49561.0/161280.0*pow(n, 4)
X = xi + B1*math.sin(2.0*xi)*math.cosh(2.0*eta) + \
B2*math.sin(4.0*xi)*math.cosh(4.0*eta) + \
B3*math.sin(6.0*xi)*math.cosh(6.0*eta) + \
B4*math.sin(8.0*xi)*math.cosh(8.0*eta)
Y = eta + B1*math.cos(2.0*xi)*math.sinh(2.0*eta) + \
B2*math.cos(4.0*xi)*math.sinh(4.0*eta) + \
B3*math.cos(6.0*xi)*math.sinh(6.0*eta) + \
B4*math.cos(8.0*xi)*math.sinh(8.0*eta)
X = X*k0*at + FN
Y = Y*k0*at + FE
return (X, Y)
def rawf(x):
return math.sinh(x)
def _zetaInv0(self, psi, lam):
'''Starting point for C{zetaInv}.
@return: 3-Tuple C{(u, v, trip)}.
@see: C{bool TMExact::zetainv0(real psi, real lam, # radians
real &u, real &v)}.
'''
trip = False
if (psi < -self._e_PI_4 and lam > self._1_e2_PI_2
and psi < lam - self._1_e_PI_2):
# N.B. this branch is normally not taken because psi < 0
# is converted psi > 0 by Forward.
#
# There's a log singularity at w = w0 = Eu.K() + i * Ev.K(),
# corresponding to the south pole, where we have, approximately
#
# psi = _e + i * pi/2 - _e * atanh(cos(i * (w - w0)/(1 + _mu/2)))
#
# Inverting this gives:
h = sinh(1 - psi / self._e)
a = (PI_2 - lam) / self._e
s, c = sincos2(a)
u = self._Eu_K - asinh(s / hypot(c, h)) * self._mu_2_1
v = self._Ev_K - atan2(c, h) * self._mu_2_1
elif (psi < self._e_PI_2 and lam > self._1_e2_PI_2):
# At w = w0 = i * Ev.K(), we have
#
# zeta = zeta0 = i * (1 - _e) * pi/2
# zeta' = zeta'' = 0
#
# including the next term in the Taylor series gives:
#
# zeta = zeta0 - (_mv * _e) / 3 * (w - w0)^3
#
# When inverting this, we map arg(w - w0) = [-90, 0] to
# arg(zeta - zeta0) = [-90, 180]
d = lam - self._1_e_PI_2
r = hypot(psi, d)
# Error using this guess is about 0.21 * (rad/e)^(5/3)
trip = r < self._e_taytol_
# atan2(dlam-psi, psi+dlam) + 45d gives arg(zeta - zeta0)
# in range [-135, 225). Subtracting 180 (since multiplier
# is negative) makes range [-315, 45). Multiplying by 1/3
# (for cube root) gives range [-105, 15). In particular
# the range [-90, 180] in zeta space maps to [-90, 0] in
# w space as required.
r = cbrt(r * self._3_mv_e)
a = atan2(d - psi, psi + d) / 3.0 - PI_4
s, c = sincos2(a)
u = r * c
v = r * s + self._Ev_K
else:
# Use spherical TM, Lee 12.6 -- writing C{atanh(sin(lam) /
# cosh(psi)) = asinh(sin(lam) / hypot(cos(lam), sinh(psi)))}.
# This takes care of the log singularity at C{zeta = Eu.K()},
# corresponding to the north pole.
s, c = sincos2(lam)
h, r = sinh(psi), self._Eu_K / PI_2
# But scale to put 90, 0 on the right place
u = r * atan2(h, c)
v = r * asinh(s / hypot(c, h))
return u, v, trip
def nond_F(x, t):
k = nond_wavenumber()
F0 = nond_eta0()
G0 = nond_xi0()
delta_rho = (rho0 - rhou) * nond_factor() / rho0
zprime = depth / D
T0 = deltaT
alphag = nond_factor() * alpha * T0
rhog = nond_factor(
) # use this as a proxy for the nondimensional factorisation
return ((-0.5*(exp(-delta_rho*rhog*t*sinh(k)**2/((delta_rho*k - k*rhog)*sinh(k)*cosh(k) + delta_rho*k**2 - k**2*rhog -\
sqrt((delta_rho**2 + 2*delta_rho*rhog + rhog**2)*sinh(k)**4 + delta_rho**2*k**2 - 2*delta_rho*k**2*rhog + k**2*rhog**2 + 2*(delta_rho**2*k -\
2*delta_rho*k*rhog + k*rhog**2)*sinh(k)*cosh(k) - (4*delta_rho*k**2*rhog - delta_rho**2 + 2*delta_rho*rhog -\
rhog**2)*sinh(k)**2)*k))/((k*rhog*sinh(k)**2*cosh(k) - k*rhog*cosh(k)**3 + rhog*sinh(k)**3 - rhog*sinh(k)*cosh(k)**2)/((delta_rho +\
rhog)*sinh(k)*cosh(k) - (delta_rho*k + k*rhog)*sinh(k)**2 + (delta_rho*k + k*rhog)*cosh(k)**2 - sqrt(delta_rho**2*k**2*sinh(k)**4 -\
2*delta_rho**2*k**2*sinh(k)**2*cosh(k)**2 + delta_rho**2*k**2*cosh(k)**4 + 2*delta_rho*k**2*rhog*sinh(k)**4 - 2*delta_rho*k**2*rhog*cosh(k)**4\
+ k**2*rhog**2*sinh(k)**4 - 2*k**2*rhog**2*sinh(k)**2*cosh(k)**2 + k**2*rhog**2*cosh(k)**4 - 2*delta_rho**2*k*sinh(k)**3*cosh(k) +\
2*delta_rho**2*k*sinh(k)*cosh(k)**3 + 4*delta_rho*k*rhog*sinh(k)**3*cosh(k) - 4*delta_rho*k*rhog*sinh(k)*cosh(k)**3 -\
2*k*rhog**2*sinh(k)**3*cosh(k) + 2*k*rhog**2*sinh(k)*cosh(k)**3 + delta_rho**2*sinh(k)**2*cosh(k)**2 + 4*delta_rho*rhog*sinh(k)**4 -\
2*delta_rho*rhog*sinh(k)**2*cosh(k)**2 + rhog**2*sinh(k)**2*cosh(k)**2)) - (k*rhog*sinh(k)**2*cosh(k) - k*rhog*cosh(k)**3 + rhog*sinh(k)**3\
- rhog*sinh(k)*cosh(k)**2)/((delta_rho + rhog)*sinh(k)*cosh(k) - (delta_rho*k + k*rhog)*sinh(k)**2 + (delta_rho*k + k*rhog)*cosh(k)**2\
+ sqrt(delta_rho**2*k**2*sinh(k)**4 - 2*delta_rho**2*k**2*sinh(k)**2*cosh(k)**2 + delta_rho**2*k**2*cosh(k)**4 +\
2*delta_rho*k**2*rhog*sinh(k)**4 - 2*delta_rho*k**2*rhog*cosh(k)**4 + k**2*rhog**2*sinh(k)**4 - 2*k**2*rhog**2*sinh(k)**2*cosh(k)**2 +\
k**2*rhog**2*cosh(k)**4 - 2*delta_rho**2*k*sinh(k)**3*cosh(k) + 2*delta_rho**2*k*sinh(k)*cosh(k)**3 + 4*delta_rho*k*rhog*sinh(k)**3*cosh(k)\
- 4*delta_rho*k*rhog*sinh(k)*cosh(k)**3 - 2*k*rhog**2*sinh(k)**3*cosh(k) + 2*k*rhog**2*sinh(k)*cosh(k)**3 +\
delta_rho**2*sinh(k)**2*cosh(k)**2 + 4*delta_rho*rhog*sinh(k)**4 - 2*delta_rho*rhog*sinh(k)**2*cosh(k)**2 +\
rhog**2*sinh(k)**2*cosh(k)**2))) - exp(-delta_rho*rhog*t*sinh(k)**2/((delta_rho*k - k*rhog)*sinh(k)*cosh(k) + delta_rho*k**2 - k**2*rhog +\
sqrt((delta_rho**2 + 2*delta_rho*rhog + rhog**2)*sinh(k)**4 + delta_rho**2*k**2 - 2*delta_rho*k**2*rhog + k**2*rhog**2 + 2*(delta_rho**2*k -\
2*delta_rho*k*rhog + k*rhog**2)*sinh(k)*cosh(k) - (4*delta_rho*k**2*rhog - delta_rho**2 + 2*delta_rho*rhog -\
rhog**2)*sinh(k)**2)*k))/((k*rhog*sinh(k)**2*cosh(k) - k*rhog*cosh(k)**3 + rhog*sinh(k)**3 - rhog*sinh(k)*cosh(k)**2)/((delta_rho +\
rhog)*sinh(k)*cosh(k) - (delta_rho*k + k*rhog)*sinh(k)**2 + (delta_rho*k + k*rhog)*cosh(k)**2 - sqrt(delta_rho**2*k**2*sinh(k)**4 -\
2*delta_rho**2*k**2*sinh(k)**2*cosh(k)**2 + delta_rho**2*k**2*cosh(k)**4 + 2*delta_rho*k**2*rhog*sinh(k)**4 - 2*delta_rho*k**2*rhog*cosh(k)**4\
+ k**2*rhog**2*sinh(k)**4 - 2*k**2*rhog**2*sinh(k)**2*cosh(k)**2 + k**2*rhog**2*cosh(k)**4 - 2*delta_rho**2*k*sinh(k)**3*cosh(k) +\
2*delta_rho**2*k*sinh(k)*cosh(k)**3 + 4*delta_rho*k*rhog*sinh(k)**3*cosh(k) - 4*delta_rho*k*rhog*sinh(k)*cosh(k)**3 -\
2*k*rhog**2*sinh(k)**3*cosh(k) + 2*k*rhog**2*sinh(k)*cosh(k)**3 + delta_rho**2*sinh(k)**2*cosh(k)**2 + 4*delta_rho*rhog*sinh(k)**4 -\
2*delta_rho*rhog*sinh(k)**2*cosh(k)**2 + rhog**2*sinh(k)**2*cosh(k)**2)) - (k*rhog*sinh(k)**2*cosh(k) - k*rhog*cosh(k)**3 + rhog*sinh(k)**3\
- rhog*sinh(k)*cosh(k)**2)/((delta_rho + rhog)*sinh(k)*cosh(k) - (delta_rho*k + k*rhog)*sinh(k)**2 + (delta_rho*k + k*rhog)*cosh(k)**2\
+ sqrt(delta_rho**2*k**2*sinh(k)**4 - 2*delta_rho**2*k**2*sinh(k)**2*cosh(k)**2 + delta_rho**2*k**2*cosh(k)**4 +\
2*delta_rho*k**2*rhog*sinh(k)**4 - 2*delta_rho*k**2*rhog*cosh(k)**4 + k**2*rhog**2*sinh(k)**4 - 2*k**2*rhog**2*sinh(k)**2*cosh(k)**2 +\
k**2*rhog**2*cosh(k)**4 - 2*delta_rho**2*k*sinh(k)**3*cosh(k) + 2*delta_rho**2*k*sinh(k)*cosh(k)**3 + 4*delta_rho*k*rhog*sinh(k)**3*cosh(k)\
- 4*delta_rho*k*rhog*sinh(k)*cosh(k)**3 - 2*k*rhog**2*sinh(k)**3*cosh(k) + 2*k*rhog**2*sinh(k)*cosh(k)**3 +\
delta_rho**2*sinh(k)**2*cosh(k)**2 + 4*delta_rho*rhog*sinh(k)**4 - 2*delta_rho*rhog*sinh(k)**2*cosh(k)**2 +\
rhog**2*sinh(k)**2*cosh(k)**2))))*(G0 + (alphag*k*sinh(k*zprime)*cosh(k) - (alphag*k*zprime*cosh(k*zprime) -\
alphag*sinh(k*zprime))*sinh(k))/(T0*delta_rho*sinh(k)**2)) + (F0 - ((alphag*k*zprime*cosh(k*zprime) -\
alphag*sinh(k*zprime))*sinh(k)*cosh(k) - (alphag*k*zprime*sinh(k*zprime) - alphag*cosh(k*zprime))*sinh(k)**2 -\
alphag*k*sinh(k*zprime))/(T0*rhog*sinh(k)**2))*(((k*rhog*sinh(k)**2*cosh(k) - k*rhog*cosh(k)**3 + rhog*sinh(k)**3 -\
rhog*sinh(k)*cosh(k)**2)/(((k*rhog*sinh(k)**2*cosh(k) - k*rhog*cosh(k)**3 + rhog*sinh(k)**3 - rhog*sinh(k)*cosh(k)**2)/((delta_rho +\
rhog)*sinh(k)*cosh(k) - (delta_rho*k + k*rhog)*sinh(k)**2 + (delta_rho*k + k*rhog)*cosh(k)**2 - sqrt(delta_rho**2*k**2*sinh(k)**4 -\
2*delta_rho**2*k**2*sinh(k)**2*cosh(k)**2 + delta_rho**2*k**2*cosh(k)**4 + 2*delta_rho*k**2*rhog*sinh(k)**4 - 2*delta_rho*k**2*rhog*cosh(k)**4\
+ k**2*rhog**2*sinh(k)**4 - 2*k**2*rhog**2*sinh(k)**2*cosh(k)**2 + k**2*rhog**2*cosh(k)**4 - 2*delta_rho**2*k*sinh(k)**3*cosh(k) +\
2*delta_rho**2*k*sinh(k)*cosh(k)**3 + 4*delta_rho*k*rhog*sinh(k)**3*cosh(k) - 4*delta_rho*k*rhog*sinh(k)*cosh(k)**3 -\
2*k*rhog**2*sinh(k)**3*cosh(k) + 2*k*rhog**2*sinh(k)*cosh(k)**3 + delta_rho**2*sinh(k)**2*cosh(k)**2 + 4*delta_rho*rhog*sinh(k)**4 -\
2*delta_rho*rhog*sinh(k)**2*cosh(k)**2 + rhog**2*sinh(k)**2*cosh(k)**2)) - (k*rhog*sinh(k)**2*cosh(k) - k*rhog*cosh(k)**3 + rhog*sinh(k)**3\
- rhog*sinh(k)*cosh(k)**2)/((delta_rho + rhog)*sinh(k)*cosh(k) - (delta_rho*k + k*rhog)*sinh(k)**2 + (delta_rho*k + k*rhog)*cosh(k)**2\
+ sqrt(delta_rho**2*k**2*sinh(k)**4 - 2*delta_rho**2*k**2*sinh(k)**2*cosh(k)**2 + delta_rho**2*k**2*cosh(k)**4 +\
2*delta_rho*k**2*rhog*sinh(k)**4 - 2*delta_rho*k**2*rhog*cosh(k)**4 + k**2*rhog**2*sinh(k)**4 - 2*k**2*rhog**2*sinh(k)**2*cosh(k)**2 +\
k**2*rhog**2*cosh(k)**4 - 2*delta_rho**2*k*sinh(k)**3*cosh(k) + 2*delta_rho**2*k*sinh(k)*cosh(k)**3 + 4*delta_rho*k*rhog*sinh(k)**3*cosh(k)\
- 4*delta_rho*k*rhog*sinh(k)*cosh(k)**3 - 2*k*rhog**2*sinh(k)**3*cosh(k) + 2*k*rhog**2*sinh(k)*cosh(k)**3 +\
delta_rho**2*sinh(k)**2*cosh(k)**2 + 4*delta_rho*rhog*sinh(k)**4 - 2*delta_rho*rhog*sinh(k)**2*cosh(k)**2 +\
rhog**2*sinh(k)**2*cosh(k)**2)))*((delta_rho + rhog)*sinh(k)*cosh(k) - (delta_rho*k + k*rhog)*sinh(k)**2 + (delta_rho*k +\
k*rhog)*cosh(k)**2 + sqrt(delta_rho**2*k**2*sinh(k)**4 - 2*delta_rho**2*k**2*sinh(k)**2*cosh(k)**2 + delta_rho**2*k**2*cosh(k)**4 +\
2*delta_rho*k**2*rhog*sinh(k)**4 - 2*delta_rho*k**2*rhog*cosh(k)**4 + k**2*rhog**2*sinh(k)**4 - 2*k**2*rhog**2*sinh(k)**2*cosh(k)**2 +\
k**2*rhog**2*cosh(k)**4 - 2*delta_rho**2*k*sinh(k)**3*cosh(k) + 2*delta_rho**2*k*sinh(k)*cosh(k)**3 + 4*delta_rho*k*rhog*sinh(k)**3*cosh(k)\
- 4*delta_rho*k*rhog*sinh(k)*cosh(k)**3 - 2*k*rhog**2*sinh(k)**3*cosh(k) + 2*k*rhog**2*sinh(k)*cosh(k)**3 +\
delta_rho**2*sinh(k)**2*cosh(k)**2 + 4*delta_rho*rhog*sinh(k)**4 - 2*delta_rho*rhog*sinh(k)**2*cosh(k)**2 +\
rhog**2*sinh(k)**2*cosh(k)**2))) + 1)*exp(-delta_rho*rhog*t*sinh(k)**2/((delta_rho*k - k*rhog)*sinh(k)*cosh(k) + delta_rho*k**2 - k**2*rhog\
+ sqrt((delta_rho**2 + 2*delta_rho*rhog + rhog**2)*sinh(k)**4 + delta_rho**2*k**2 - 2*delta_rho*k**2*rhog + k**2*rhog**2 + 2*(delta_rho**2*k\
- 2*delta_rho*k*rhog + k*rhog**2)*sinh(k)*cosh(k) - (4*delta_rho*k**2*rhog - delta_rho**2 + 2*delta_rho*rhog - rhog**2)*sinh(k)**2)*k)) -\
(k*rhog*sinh(k)**2*cosh(k) - k*rhog*cosh(k)**3 + rhog*sinh(k)**3 -\
rhog*sinh(k)*cosh(k)**2)*exp(-delta_rho*rhog*t*sinh(k)**2/((delta_rho*k - k*rhog)*sinh(k)*cosh(k) + delta_rho*k**2 - k**2*rhog -\
sqrt((delta_rho**2 + 2*delta_rho*rhog + rhog**2)*sinh(k)**4 + delta_rho**2*k**2 - 2*delta_rho*k**2*rhog + k**2*rhog**2 + 2*(delta_rho**2*k -\
2*delta_rho*k*rhog + k*rhog**2)*sinh(k)*cosh(k) - (4*delta_rho*k**2*rhog - delta_rho**2 + 2*delta_rho*rhog -\
rhog**2)*sinh(k)**2)*k))/(((k*rhog*sinh(k)**2*cosh(k) - k*rhog*cosh(k)**3 + rhog*sinh(k)**3 - rhog*sinh(k)*cosh(k)**2)/((delta_rho +\
rhog)*sinh(k)*cosh(k) - (delta_rho*k + k*rhog)*sinh(k)**2 + (delta_rho*k + k*rhog)*cosh(k)**2 - sqrt(delta_rho**2*k**2*sinh(k)**4 -\
2*delta_rho**2*k**2*sinh(k)**2*cosh(k)**2 + delta_rho**2*k**2*cosh(k)**4 + 2*delta_rho*k**2*rhog*sinh(k)**4 - 2*delta_rho*k**2*rhog*cosh(k)**4\
+ k**2*rhog**2*sinh(k)**4 - 2*k**2*rhog**2*sinh(k)**2*cosh(k)**2 + k**2*rhog**2*cosh(k)**4 - 2*delta_rho**2*k*sinh(k)**3*cosh(k) +\
2*delta_rho**2*k*sinh(k)*cosh(k)**3 + 4*delta_rho*k*rhog*sinh(k)**3*cosh(k) - 4*delta_rho*k*rhog*sinh(k)*cosh(k)**3 -\
2*k*rhog**2*sinh(k)**3*cosh(k) + 2*k*rhog**2*sinh(k)*cosh(k)**3 + delta_rho**2*sinh(k)**2*cosh(k)**2 + 4*delta_rho*rhog*sinh(k)**4 -\
2*delta_rho*rhog*sinh(k)**2*cosh(k)**2 + rhog**2*sinh(k)**2*cosh(k)**2)) - (k*rhog*sinh(k)**2*cosh(k) - k*rhog*cosh(k)**3 + rhog*sinh(k)**3\
- rhog*sinh(k)*cosh(k)**2)/((delta_rho + rhog)*sinh(k)*cosh(k) - (delta_rho*k + k*rhog)*sinh(k)**2 + (delta_rho*k + k*rhog)*cosh(k)**2\
+ sqrt(delta_rho**2*k**2*sinh(k)**4 - 2*delta_rho**2*k**2*sinh(k)**2*cosh(k)**2 + delta_rho**2*k**2*cosh(k)**4 +\
2*delta_rho*k**2*rhog*sinh(k)**4 - 2*delta_rho*k**2*rhog*cosh(k)**4 + k**2*rhog**2*sinh(k)**4 - 2*k**2*rhog**2*sinh(k)**2*cosh(k)**2 +\
k**2*rhog**2*cosh(k)**4 - 2*delta_rho**2*k*sinh(k)**3*cosh(k) + 2*delta_rho**2*k*sinh(k)*cosh(k)**3 + 4*delta_rho*k*rhog*sinh(k)**3*cosh(k)\
- 4*delta_rho*k*rhog*sinh(k)*cosh(k)**3 - 2*k*rhog**2*sinh(k)**3*cosh(k) + 2*k*rhog**2*sinh(k)*cosh(k)**3 +\
delta_rho**2*sinh(k)**2*cosh(k)**2 + 4*delta_rho*rhog*sinh(k)**4 - 2*delta_rho*rhog*sinh(k)**2*cosh(k)**2 +\
rhog**2*sinh(k)**2*cosh(k)**2)))*((delta_rho + rhog)*sinh(k)*cosh(k) - (delta_rho*k + k*rhog)*sinh(k)**2 + (delta_rho*k +\
k*rhog)*cosh(k)**2 + sqrt(delta_rho**2*k**2*sinh(k)**4 - 2*delta_rho**2*k**2*sinh(k)**2*cosh(k)**2 + delta_rho**2*k**2*cosh(k)**4 +\
2*delta_rho*k**2*rhog*sinh(k)**4 - 2*delta_rho*k**2*rhog*cosh(k)**4 + k**2*rhog**2*sinh(k)**4 - 2*k**2*rhog**2*sinh(k)**2*cosh(k)**2 +\
k**2*rhog**2*cosh(k)**4 - 2*delta_rho**2*k*sinh(k)**3*cosh(k) + 2*delta_rho**2*k*sinh(k)*cosh(k)**3 + 4*delta_rho*k*rhog*sinh(k)**3*cosh(k)\
- 4*delta_rho*k*rhog*sinh(k)*cosh(k)**3 - 2*k*rhog**2*sinh(k)**3*cosh(k) + 2*k*rhog**2*sinh(k)*cosh(k)**3 +\
delta_rho**2*sinh(k)**2*cosh(k)**2 + 4*delta_rho*rhog*sinh(k)**4 - 2*delta_rho*rhog*sinh(k)**2*cosh(k)**2 +\
rhog**2*sinh(k)**2*cosh(k)**2)))) + ((alphag*k*zprime*cosh(k*zprime) - alphag*sinh(k*zprime))*sinh(k)*cosh(k) -\
(alphag*k*zprime*sinh(k*zprime) - alphag*cosh(k*zprime))*sinh(k)**2 - alphag*k*sinh(k*zprime))/(T0*rhog*sinh(k)**2)))*cos(k*x)
def funcl(x):
return -(a / x) * mt.cosh(a * x) + (x**-2) * mt.sinh(
a * x) + (b / x) * mt.cosh(b * x) - (x**-2) * mt.sinh(b * x)
print(math.pow(x, y)) print(math.acos(0.55)) print(math.asin(0.55)) print(math.atan(0.55)) print(math.atan2(0.55, 0.55)) print(math.cos(x)) print(math.hypot(x, y)) print(math.sin(x)) print(math.tan(x)) print(math.degrees(x)) print(math.radians(x)) print(math.acosh(x)) print(math.asinh(x)) print(math.atan(0.39)) print(math.cosh(x)) print(math.sinh(x)) print(math.tanh(x)) print(math.erf(x)) print(math.erfc(x)) print(math.gamma(x)) print(math.lgamma(x)) print(math.pi) print(math.e) # Random module print(random.seed(3)) print(random.getstate()) state = random.getstate() print(random.setstate(state)) print(random.getrandbits(43))
# Three ways to define a grid (inpired by http://mapproxy.org/docs/1.8.0/configuration.html#id6):
# - submit a list of resolutions > "resolutions": [32,16,8,4] (This parameters override the others)
# - submit just "resFactor", initial res is computed such as at zoom level zero, 1 tile covers whole bounding box
# - submit "resFactor" and "initRes"
# About Web Mercator
# Technically, the Mercator projection is defined for any latitude up to (but not including)
# 90 degrees, but it makes sense to cut it off sooner because it grows exponentially with
# increasing latitude. The logic behind this particular cutoff value, which is the one used
# by Google Maps, is that it makes the projection square. That is, the rectangle is equal in
# the X and Y directions. In this case the maximum latitude attained must correspond to y = w/2.
# y = 2*pi*R / 2 = pi*R --> y/R = pi
# lat = atan(sinh(y/R)) = atan(sinh(pi))
# wm_origin = (-20037508, 20037508) with 20037508 = GRS80.perimeter / 2
cutoff_lat = math.atan(math.sinh(math.pi)) * 180 / math.pi #= 85.05112°
GRIDS = {
"WM": {
"name": 'Web Mercator',
"description": 'Global grid in web mercator projection',
"CRS": 'EPSG:3857',
"bbox": [-180, -cutoff_lat, 180, cutoff_lat], #w,s,e,n
"bboxCRS": 'EPSG:4326',
#"bbox": [-20037508, -20037508, 20037508, 20037508],
#"bboxCRS": 3857,
"tileSize": 256,
"originLoc": "NW", #North West or South West
"resFactor": 2
},
"WGS84": {
def toUtm(latlon, lon=None, datum=None, Utm=Utm, name='', cmoff=True):
'''Convert a lat-/longitude point to a UTM coordinate.
@param latlon: Latitude (C{degrees}) or an (ellipsoidal)
geodetic C{LatLon} point.
@keyword lon: Optional longitude (C{degrees} or C{None}).
@keyword datum: Optional datum for this UTM coordinate,
overriding I{latlon}'s datum (C{Datum}).
@keyword Utm: Optional (sub-)class to use for the UTM
coordinate (L{Utm}) or C{None}.
@keyword name: Optional I{Utm} name (C{str}).
@keyword cmoff: Offset longitude from zone's central meridian,
apply false easting and false northing (C{bool}).
@return: The UTM coordinate (L{Utm}) or a 6-tuple (zone, easting,
northing, band, convergence, scale) if I{Utm} is C{None}
or I{cmoff} is C{False}.
@raise TypeError: If I{latlon} is not ellipsoidal.
@raise RangeError: If I{lat} is outside the valid UTM bands or if
I{lat} or I{lon} outside the valid range and
I{rangerrrors} set to C{True}.
@raise ValueError: If I{lon} value is missing or if I{latlon}
is invalid.
@note: Implements Karney’s method, using 8-th order Krüger series,
giving results accurate to 5 nm (or better) for distances
up to 3900 km from the central meridian.
@example:
>>> p = LatLon(48.8582, 2.2945) # 31 N 448251.8 5411932.7
>>> u = toUtm(p) # 31 N 448252 5411933
>>> p = LatLon(13.4125, 103.8667) # 48 N 377302.4 1483034.8
>>> u = toUtm(p) # 48 N 377302 1483035
'''
try:
lat, lon = latlon.lat, latlon.lon
if not isinstance(latlon, _LLEB):
raise TypeError('%s not %s: %r' % ('latlon', 'ellipsoidal', latlon))
if not name: # use latlon.name
name = _nameof(latlon) or name # PYCHOK no effect
d = datum or latlon.datum
except AttributeError:
lat, lon = parseDMS2(latlon, lon)
d = datum or Datums.WGS84
E = d.ellipsoid
z, B, a, b = _toZBab4(lat, lon, cmoff)
# easting, northing: Karney 2011 Eq 7-14, 29, 35
cb, sb, tb = cos(b), sin(b), tan(b)
T = tan(a)
T12 = hypot1(T)
S = sinh(E.e * atanh(E.e * T / T12))
T_ = T * hypot1(S) - S * T12
H = hypot(T_, cb)
y = atan2(T_, cb) # ξ' ksi
x = asinh(sb / H) # η' eta
A0 = _K0 * E.A
Ks = _Kseries(E.AlphaKs, x, y) # Krüger series
y = Ks.ys(y) * A0 # ξ
x = Ks.xs(x) * A0 # η
if cmoff:
# C.F.F. Karney, "Test data for the transverse Mercator projection (2009)",
# <http://GeographicLib.SourceForge.io/html/transversemercator.html> and
# <http://Zenodo.org/record/32470#.W4LEJS2ZON8>
x += _FalseEasting # make x relative to false easting
if y < 0:
y += _FalseNorthing # y relative to false northing in S
# convergence: Karney 2011 Eq 23, 24
p_ = Ks.ps(1)
q_ = Ks.qs(0)
c = degrees(atan(T_ / hypot1(T_) * tb) + atan2(q_, p_))
# scale: Karney 2011 Eq 25
s = E.e2s(sin(a)) * T12 / H * (A0 / E.a * hypot(p_, q_))
if cmoff and Utm is not None:
h = 'S' if a < 0 else 'N' # hemisphere
return _xnamed(Utm(z, h, x, y, band=B, datum=d,
convergence=c, scale=s), name)
else: # zone, easting, northing, band, convergence and scale
return z, x, y, B, c, s
def compute_delta_phi(r, rp):
return math.atan(math.tanh(r / K) / math.sinh(rp / K))
def func_sinh(x):
"""Return sinh(x) where x is in radians"""
# use the function from the 'math' library
return sinh(x)
def _marcToLat(x):
return math.degrees(math.atan(math.sinh(x)))
def compute_delta_theta(r, rp, phi):
return math.atan(math.tanh(r / K) / (math.sinh(rp / K) * math.sinh(phi)))
def num2deg(xtile: int, ytile: int, zoom: int) -> Tuple[float, float]:
n = 2.0**zoom
lon_deg = xtile / n * 360.0 - 180.0
lat_rad = math.atan(math.sinh(math.pi * (1 - 2 * ytile / n)))
lat_deg = math.degrees(lat_rad)
return (lon_deg, lat_deg)
def sinh(self):
self.result = False
self.current = math.sinh(math.radians(float(txtDisplay.get())))
self.display(self.current)
def func(x):
return -(1 / x) * (mt.sinh(a * x) - mt.sinh(b * x)) - l
def tile2ll(x, y, z):
n = 2**z
lon = x / n * 360.0 - 180.0
lat = atan(sinh(π * (1 - 2 * y / n))) * 180.0 / π
return [lon, lat]
def csch(x):
return 1 / math.sinh(x)
def b(x1,x2):
return sinh(As*ksi(x1,x2))/(As*ksi(x1,x2));
def lum_4(gen_z, omm, oml, omk, c_0, h0, n_sn, dlum):
#integrating for first z point (needs to be done seperately to avoid indexing errors)
cd, err = quad(integrand, 0, gen_z[0], args=(omm, oml, omk))
const = (abs(omk))**0.5 #defining constant for dlum calculation
#calculating dlum for respecive omk values
if omk < 0:
dlum[0] = (c_0 /
(h0 * const)) * (1.0 + gen_z[0]) * math.sin(const * cd)
elif omk > 0:
dlum[0] = (c_0 /
(h0 * const)) * (1.0 + gen_z[0]) * math.sinh(const * cd)
else:
dlum[0] = (c_0 / h0) * (1.0 + gen_z[0]) * cd
#loop over rest of z values (different calculation for varying omk values)
if omk < 0:
for i in range(1, n_sn): #looping through all z values
if gen_z[i] == gen_z[
i -
1]: #if statment campares new point to last, if equal assign same dlum value (saves coputational cost)
dlum[i] = dlum[i - 1]
else:
temp = quad(
integrand, gen_z[i - 1], gen_z[i], args=(
omm, oml,
omk)) #integrates area between current and next point
cd = cd + temp[
0] #cumulative uses previous calculations for next
dlum[i] = (c_0 / (h0 * const)) * (1.0 + gen_z[i]) * math.sin(
const * cd)
elif omk > 0:
for i in range(1, n_sn):
if gen_z[i] == gen_z[i - 1]:
dlum[i] = dlum[i - 1]
else:
temp = quad(integrand,
gen_z[i - 1],
gen_z[i],
args=(omm, oml, omk))
cd = cd + temp[0]
dlum[i] = (c_0 / (h0 * const)) * (1.0 + gen_z[i]) * math.sinh(
const * cd)
else:
for i in range(1, n_sn):
if gen_z[i] == gen_z[i - 1]:
dlum[i] = dlum[i - 1]
else:
temp = quad(integrand,
gen_z[i - 1],
gen_z[i],
args=(omm, oml, omk))
cd = cd + temp[0]
dlum[i] = (c_0 / h0) * (1.0 + gen_z[i]) * cd
return dlum
def uEF(k, L, x, A, U_o):
u_exact = [
((sinh(k * (L - x)) + sinh(k * x)) / sinh(k * L) - 1) * A / k**2 +
U_o * sinh(k * (L - x)) / sinh(k * L) for x in x[1:-1]
]
return (u_exact)
def cdf(dist, a, *args, **kwargs):
G = dist.cdf(*args, **kwargs)
return 2 * math.exp(a) / (math.exp(2 * a) - 1) * math.sinh(a * G)
def toLatLon(self, LatLon=None):
'''Convert this UTM coordinate to an (ellipsoidal) geodetic point.
@keyword LatLon: Optional, ellipsoidal (sub-)class to use
for the point (C{LatLon}) or C{None}.
@return: This UTM coordinate as (I{LatLon}) or 5-tuple
(lat, lon, datum, convergence, scale) if I{LatLon}
is C{None}.
@raise TypeError: If I{LatLon} is not ellipsoidal.
@raise UTMError: Invalid meridional radius or H-value.
@example:
>>> u = Utm(31, 'N', 448251.795, 5411932.678)
>>> from pygeodesy import ellipsoidalVincenty as eV
>>> ll = u.toLatLon(eV.LatLon) # 48°51′29.52″N, 002°17′40.20″E
'''
if self._latlon:
return self._latlon5(LatLon)
E = self._datum.ellipsoid # XXX vs LatLon.datum.ellipsoid
x = self._easting - _FalseEasting # relative to central meridian
y = self._northing
if self._hemi == 'S': # relative to equator
y -= _FalseNorthing
# from Karney 2011 Eq 15-22, 36
A0 = _K0 * E.A
if A0 < EPS:
raise UTMError('%s invalid: %r' % ('meridional', E.A))
x /= A0 # η eta
y /= A0 # ξ ksi
Ks = _Kseries(E.BetaKs, x, y) # Krüger series
y = -Ks.ys(-y) # ξ'
x = -Ks.xs(-x) # η'
shx = sinh(x)
cy, sy = cos(y), sin(y)
H = hypot(shx, cy)
if H < EPS:
raise UTMError('%s invalid: %r' % ('H', H))
T = t0 = sy / H # τʹ
q = 1.0 / E.e12
d = 1
sd = Fsum(T)
# toggles on +/-1.12e-16 eg. 31 N 400000 5000000
while abs(d) > EPS: # 1e-12
h = hypot1(T)
s = sinh(E.e * atanh(E.e * T / h))
t = T * hypot1(s) - s * h
d = (t0 - t) / hypot1(t) * (q + T**2) / h
T = sd.fsum_(d) # τi
a = atan(T) # lat
b = atan2(shx, cy) + radians(_cmlon(self._zone))
ll = _LLEB(degrees90(a), degrees180(b), datum=self._datum, name=self.name)
# convergence: Karney 2011 Eq 26, 27
p = -Ks.ps(-1)
q = Ks.qs(0)
ll._convergence = degrees(atan(tan(y) * tanh(x)) + atan2(q, p))
# scale: Karney 2011 Eq 28
ll._scale = E.e2s(sin(a)) * hypot1(T) * H * (A0 / E.a / hypot(p, q))
self._latlon = ll
return self._latlon5(LatLon)
def yfun(t):
return s_y_max / 2 * math.cosh(t / T_c) + T_c * vy_0 * math.sinh(
t / T_c)
def acoth(x):
return math.cosh(x) / math.sinh(x)
def xfun(t):
return -s_x / 2 * math.cosh(t / T_c) + T_c * vx_0 * math.sinh(t / T_c)
'cbf',
None,
"test",
None,
params.distance,
verbose=True,
header_only=True)
img = FormatCBFCspadInMemory(cbf)
beam = BeamFactory.simple(params.wavelength).get_s0()
beam_center = (0, 0)
data = numpy.zeros((11840, 194))
if annulus:
inner = params.distance * math.tan(
2 * math.sinh(params.wavelength / (2 * params.annulus_inner)))
outer = params.distance * math.tan(
2 * math.sinh(params.wavelength / (2 * params.annulus_outer)))
print "Inner (mm):", inner
print "Outer (mm):", outer
if params.resolution is not None:
radius = params.distance * math.tan(
2 * math.sinh(params.wavelength / (2 * params.resolution)))
print "Radius (mm):", radius
print "Panel:",
sys.stdout.flush()
for p_id, panel in enumerate(img.get_detector()):
print p_id,
sys.stdout.flush()
for y in xrange(185):
def get_com_locus(theta):
s0 = 0.04
s_x = s0 * math.cos(theta)
s_y = s0 * math.sin(theta)
s_y_max = s_y / 2 + 0.12
s_y_min = -s_y / 2 + 0.12
z = 0.3
g = 9.8
T_c = math.sqrt(z / g)
T = 1
vx_0 = s_x / 2 * (math.cosh(T / T_c) + 1) / (T_c * math.sinh(T / T_c))
vy_0 = (s_y_min / 2 -
s_y_max / 2 * math.cosh(T / T_c)) / (T_c * math.sinh(T / T_c))
def xfun(t):
return -s_x / 2 * math.cosh(t / T_c) + T_c * vx_0 * math.sinh(t / T_c)
def yfun(t):
return s_y_max / 2 * math.cosh(t / T_c) + T_c * vy_0 * math.sinh(
t / T_c)
# locus for steps with even number
comx_b1 = [0.0] * N
comy_b1 = [0.0] * N
t_b1 = [0.0] * N
for i in range(N):
t_b1[i] = (i + 1) * T / (N - 1)
comx_b1[i] = xfun(t_b1[i] - T / (N - 1))
comy_b1[i] = yfun(t_b1[i] - T / (N - 1))
locus_b1 = (comx_b1, comy_b1, t_b1)
# Locus for steps with odd number
comx_b2 = comx_b1
comy_b2 = comy_b1[-1::-1]
t_b2 = t_b1
locus_b2 = (comx_b2, comy_b2, t_b2)
# locus for the first step
comx_s = [0.0] * N * 2
comy_s = [0.0] * N * 2
t_s = [0.0] * N * 2
def yfun_s(t):
return (-vy_0 / (2 * T)**6 - 3 * s_y / 2 /
(2 * T)**7) * t**7 + (3.5 * s_y / 2 / (2 * T)**6 + vy_0 /
(2 * T)**5) * t**6 + 0.06
def xfun_s(t):
if t > T:
return s_x / 2 / math.cosh(T / T_c) * math.cosh((t - T) / T_c)
else:
return s_x / 2 / math.cosh(T / T_c) * (t / T)**2
for i in range(N * 2):
t_s[i] = (i + 1) * 2 * T / (2 * N - 1)
comy_s[i] = yfun_s(t_s[i] - 2 * T / (2 * N - 1))
comx_s[i] = xfun_s(t_s[i] - 2 * T / (2 * N - 1))
locus_s = (comx_s, comy_s, t_s)
# locus for the last step
comx_e = [-x for x in comx_s][-1::-1]
comy_e = comy_s[-1::-1]
locus_e = (comx_e, comy_e, t_s)
return locus_b1, locus_b2, locus_s, locus_e
pass
def exactFiFunc(x):
return math.sinh(2 * x + 0.45)
def method_sinh(self, space, value):
try:
res = math.sinh(value)
except OverflowError:
res = rfloat.copysign(rfloat.INFINITY, value)
return space.newfloat(res)
def num2deg(xtile, ytile, zoom):
n = 2.0**zoom
lon_deg = xtile / n * 360.0 - 180.0
lat_rad = math.atan(math.sinh(math.pi * (1 - 2 * ytile / n)))
lat_deg = math.degrees(lat_rad)
return (lat_deg, lon_deg)
epsilon = 0.00000001
maxIter = 50
print("Q1. Determinando o Comprimento de L através dos Métodos: \n")
print("Método da Bisseção")
(houveErro, raiz, k) = bissecao(l_bissecao,
a,
b,
epsilon,
maxIter,
mostraTabela=True)
if houveErro:
print("O Método da Bisseção retornou um erro.")
if raiz is not None:
print("Raiz encontrada: %s" % raiz)
L = 2 * raiz * sinh(50 / (2 * raiz))
print("Valor de L: %s" % L)
#print("O número de iterações: %s"%k)
'''
Raiz da Equação: 390.75826058909297
Interações: 28
E o Valor de L encontrado foi: 50.034117037925476
'''
def l_false_pos(x):
return x * (cosh(50 / (2 * x)) - 1) - 0.80