def _D2atanhee(self, x, y):
'''(INTERNAL) Function M{D2atanhee(x, y)}, defined as
M{(Datanhee(1, y) - Datanhee(1, x)) / (y - x)}.
'''
e2 = self.datum.ellipsoid.e2
if (abs(x) + abs(y)) * e2 < _0_5:
e = z = _1_0
k = 1
C = Fsum()
S = Fsum()
T = Fsum() # Taylor expansion
for _ in range(_TERMS):
T *= y
p = T.fsum_(z)
z *= x # PYCHOK ;
T *= y
t = T.fsum_(z)
z *= x # PYCHOK ;
e *= e2
k += 2
s, d = S.fsum2_(e * C.fsum_(p, t) / k)
if not d:
break
elif (_1_0 - x):
s = (self._Datanhee(_1_0, y) - self._Datanhee(x, y)) / (_1_0 - x)
else:
raise AlbersError(x=x, y=y, txt=_AlbersBase._D2atanhee.__name__)
return s
def fEinv(self, x):
'''The inverse of the incomplete integral of the second kind.
@arg x: Argument (C{float}).
@return: φ = 1 / E(B{C{x}}, k), such that E(φ, k) = B{C{x}}
(C{float}).
@raise EllipticError: No convergence.
'''
E2 = self.cE * _2_0
n = floor(x / E2 + _0_5)
x -= E2 * n # x now in [-ec, ec)
# linear approximation
phi = PI * x / E2 # phi in [-pi/2, pi/2)
Phi = Fsum(phi)
# first order correction
phi = Phi.fsum_(-self.eps * sin(2 * phi) * _0_5)
# For kp2 close to zero use asin(x/.cE) or J. P. Boyd,
# Applied Math. and Computation 218, 7005-7013 (2012)
# <https://DOI.org/10.1016/j.amc.2011.12.021>
for self._iteration in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC
sn, cn, dn = self._sncndn3(phi)
phi, e = Phi.fsum2_((x - self.fE(sn, cn, dn)) / dn)
if abs(e) < _TolJAC:
if n:
phi = Phi.fsum_(n * PI)
return phi
raise _convergenceError(self.fEinv, x)
def _RF(x, y, z): # used by testElliptic.py
'''Symmetric integral of the first kind C{_RF}.
@return: C{_RF(x, y, z)}.
@see: U{C{_RF} definition<https://DLMF.NIST.gov/19.16.E1>}.
'''
# Carlson, eqs 2.2 - 2.7
m = 1.0
A = fsum_(x, y, z) / 3.0
T = (A, x, y, z)
Q = _Q(A, T, _TolRF)
for _ in range(_TRIPS):
if Q < abs(m * T[0]): # max 6 trips
break
_, _, T = _rsT(T)
m *= 4
else:
raise _convergenceError(_RF, x, y, z)
m *= T[0] # An
x = (A - x) / m
y = (A - y) / m
z = -(x + y)
e2 = x * y - z**2
e3 = x * y * z
# Polynomial is <https://DLMF.NIST.gov/19.36.E1>
# (1 - E2/10 + E3/14 + E2**2/24 - 3*E2*E3/44
# - 5*E2**3/208 + 3*E3**2/104 + E2**2*E3/16)
# converted to Horner form ...
H = Fsum( 6930 * e3, 15015 * e2**2, -16380 * e2, 17160) * e3
H += Fsum(10010 * e2, -5775 * e2**2, -24024) * e2
return H.fsum_(240240) / (240240 * sqrt(T[0]))
def _Horner(e0, e1, e2, e3, e4, e5):
'''(INTERNAL) Horner form for C{_RD} and C{_RJ} below.
'''
# Polynomial is <https://DLMF.NIST.gov/19.36.E2>
# (1 - 3*E2/14 + E3/6 + 9*E2**2/88 - 3*E4/22 - 9*E2*E3/52 + 3*E5/26
# - E2**3/16 + 3*E3**2/40 + 3*E2*E4/20 + 45*E2**2*E3/272
# - 9*(E3*E4+E2*E5)/68)
# converted to Horner form ...
H = Fsum(471240, -540540 * e2) * e5
H += Fsum(612612 * e2, -540540 * e3, -556920) * e4
H += Fsum(306306 * e3, 675675 * e2**2, -706860 * e2, 680680) * e3
H += Fsum(417690 * e2, -255255 * e2**2, -875160) * e2
e = 4084080 * e1
return H.fsum_(4084080, e * e0) / e
def fEinv(self, x):
'''The inverse of the incomplete integral of the second kind.
@param x: Argument (C{float}).
@return: φ = 1 / E(B{C{x}}, k), such that E(φ, k) = B{C{x}}.
'''
E2 = self.cE * 2.0
n = floor(x / E2 + 0.5)
x -= E2 * n # x now in [-ec, ec)
# linear approximation
phi = PI * x / E2 # phi in [-pi/2, pi/2)
Phi = Fsum(phi)
# first order correction
phi = Phi.fsum_(-self._eps * sin(2 * phi) * 0.5)
# For kp2 close to zero use asin(x/.cE) or J. P. Boyd,
# Applied Math. and Computation 218, 7005-7013 (2012)
# <https://DOI.org/10.1016/j.amc.2011.12.021>
for _ in range(self._trips_): # GEOGRAPHICLIB_PANIC
sn, cn, dn = self._sncndn3(phi)
phi, e = Phi.fsum2_((x - self.fE(sn, cn, dn)) / dn)
if abs(e) < _tolJAC:
return n * PI + phi
raise EllipticError('no %s convergence' % ('fEinv', ))
def toLatLon(self, LatLon=None, datum=Datums.WGS84):
'''Convert this OSGR coordinate to an (ellipsoidal) geodetic
point.
While OS grid references are based on the OSGB36 datum, the
I{Ordnance Survey} have deprecated the use of OSGB36 for
lat-/longitude coordinates (in favour of WGS84). Hence, this
method returns WGS84 by default with OSGB36 as an option,
U{see<https://www.OrdnanceSurvey.co.UK/blog/2014/12/2>}.
I{Note formulation implemented here due to Thomas, Redfearn,
etc. is as published by OS, but is inferior to Krüger as
used by e.g. Karney 2011.}
@kwarg LatLon: Optional ellipsoidal class to return the
geodetic point (C{LatLon}) or C{None}.
@kwarg datum: Optional datum to convert to (L{Datum},
L{Ellipsoid}, L{Ellipsoid2}, L{Ellipsoid2}
or L{a_f2Tuple}).
@return: The geodetic point (B{C{LatLon}}) or a
L{LatLonDatum3Tuple}C{(lat, lon, datum)}
if B{C{LatLon}} is C{None}.
@raise OSGRError: No convergence.
@raise TypeError: If B{C{LatLon}} is not ellipsoidal or
B{C{datum}} is invalid or conversion failed.
@example:
>>> from pygeodesy import ellipsoidalVincenty as eV
>>> g = Osgr(651409.903, 313177.270)
>>> p = g.toLatLon(eV.LatLon) # 52°39′28.723″N, 001°42′57.787″E
>>> # to obtain (historical) OSGB36 lat-/longitude point
>>> p = g.toLatLon(eV.LatLon, datum=Datums.OSGB36) # 52°39′27.253″N, 001°43′04.518″E
'''
if self._latlon:
return self._latlon3(LatLon, datum)
E = self.datum.ellipsoid # _Datums_OSGB36.ellipsoid, Airy130
a_F0 = E.a * _F0
b_F0 = E.b * _F0
e, n = self.easting, self.northing
n_N0 = n - _N0
a, m = _A0, n_N0
sa = Fsum(a)
for self._iteration in range(1, _TRIPS):
a = sa.fsum_(m / a_F0)
m = n_N0 - b_F0 * _M(E.Mabcd, a) # meridional arc
if abs(m) < _10um:
break
else:
t = _dot_(_item_ps(self.classname, self.toStr(prec=-3)),
self.toLatLon.__name__)
raise OSGRError(_no_convergence_, txt=t)
sa, ca = sincos2(a)
s = E.e2s2(sa) # r, v = E.roc2_(sa, _F0)
v = a_F0 / sqrt(s) # nu
r = v * E.e12 / s # rho = a_F0 * E.e12 / pow(s, 1.5) == a_F0 * E.e12 / (s * sqrt(s))
vr = v / r # == s / E.e12
x2 = vr - 1 # η2
ta = tan(a)
v3, v5, v7 = fpowers(v, 7, 3) # PYCHOK false!
ta2, ta4, ta6 = fpowers(ta**2, 3) # PYCHOK false!
tar = ta / r
V4 = (a, tar / (2 * v), tar / (24 * v3) * fdot(
(1, 3, -9), 5 + x2, ta2, ta2 * x2), tar / (720 * v5) * fdot(
(61, 90, 45), 1, ta2, ta4))
csa = 1.0 / ca
X5 = (_B0, csa / v, csa / (6 * v3) * fsum_(vr, ta2, ta2),
csa / (120 * v5) * fdot(
(5, 28, 24), 1, ta2, ta4), csa / (5040 * v7) * fdot(
(61, 662, 1320, 720), 1, ta2, ta4, ta6))
d, d2, d3, d4, d5, d6, d7 = fpowers(e - _E0, 7) # PYCHOK false!
a = fdot(V4, 1, -d2, d4, -d6)
b = fdot(X5, 1, d, -d3, d5, -d7)
r = _LLEB(degrees90(a),
degrees180(b),
datum=self.datum,
name=self.name)
r._iteration = self._iteration # only ellipsoidal LatLon
self._latlon = r
return self._latlon3(LatLon, datum)
def toLatLon(self, LatLon=None, datum=Datums.WGS84):
'''Convert this OSGR coordinate to an (ellipsoidal) geodetic
point.
I{Note formulation implemented here due to Thomas, Redfearn,
etc. is as published by OS, but is inferior to Krüger as
used by e.g. Karney 2011.}
@keyword LatLon: Optional ellipsoidal (sub-)class to return
the point (C{LatLon}) or C{None}.
@keyword datum: Optional datum to use (C{Datum}).
@return: The geodetic point (B{C{LatLon}}) or a
L{LatLonDatum3Tuple}C{(lat, lon, datum)}
if B{C{LatLon}} is C{None}.
@raise TypeError: If B{C{LatLon}} is not ellipsoidal or if
B{C{datum}} conversion failed.
@example:
>>> from pygeodesy import ellipsoidalVincenty as eV
>>> g = Osgr(651409.903, 313177.270)
>>> p = g.toLatLon(eV.LatLon) # 52°39′28.723″N, 001°42′57.787″E
>>> # to obtain (historical) OSGB36 lat-/longitude point
>>> p = g.toLatLon(eV.LatLon, datum=Datums.OSGB36) # 52°39′27.253″N, 001°43′04.518″E
'''
if self._latlon:
return self._latlon3(LatLon, datum)
E = _OSGB36.ellipsoid # Airy130
a_F0 = E.a * _F0
b_F0 = E.b * _F0
e, n = self._easting, self._northing
n_N0 = n - _N0
a, M = _A0, 0
sa = Fsum(a)
while True:
t = n_N0 - M
if t < _10um:
break
a = sa.fsum_(t / a_F0)
M = b_F0 * _M(E.Mabcd, a)
sa, ca = sincos2(a)
s = E.e2s2(sa) # v, r = E.roc2_(sa, _F0)
v = a_F0 / sqrt(s) # nu
r = v * E.e12 / s # rho
vr = v / r # == s / E.e12
x2 = vr - 1 # η2
ta = tan(a)
v3, v5, v7 = fpowers(v, 7, 3) # PYCHOK false!
ta2, ta4, ta6 = fpowers(ta**2, 3) # PYCHOK false!
tar = ta / r
V4 = (a,
tar / ( 2 * v),
tar / ( 24 * v3) * fdot((1, 3, -9), 5 + x2, ta2, ta2 * x2),
tar / (720 * v5) * fdot((61, 90, 45), 1, ta2, ta4))
csa = 1.0 / ca
X5 = (_B0,
csa / v,
csa / ( 6 * v3) * fsum_(vr, ta, ta),
csa / ( 120 * v5) * fdot((5, 28, 24), 1, ta2, ta4),
csa / (5040 * v7) * fdot((61, 662, 1320, 720), ta, ta2, ta4, ta6))
d, d2, d3, d4, d5, d6, d7 = fpowers(e - _E0, 7) # PYCHOK false!
a = fdot(V4, 1, -d2, d4, -d6)
b = fdot(X5, 1, d, -d3, d5, -d7)
self._latlon = _LLEB(degrees90(a), degrees180(b), datum=_OSGB36, name=self.name)
return self._latlon3(LatLon, datum)