def to3llh(self, datum=Datums.WGS84):
'''Convert this (geocentric) Cartesian (x/y/z) point to
(ellipsoidal, geodetic) lat-, longitude and height on
the given datum.
Uses B. R. Bowring’s formulation for μm precision in concise
form: U{'The accuracy of geodetic latitude and height equations'
<https://www.ResearchGate.net/publication/
233668213_The_Accuracy_of_Geodetic_Latitude_and_Height_Equations>},
Survey Review, Vol 28, 218, Oct 1985.
See also Ralph M. Toms U{'An Efficient Algorithm for Geocentric to
Geodetic Coordinate Conversion'<https://www.OSTI.gov/scitech/biblio/110235>},
Sept 1995 and U{'An Improved Algorithm for Geocentric to Geodetic Coordinate
Conversion'<https://www.OSTI.gov/scitech/servlets/purl/231228>},
Apr 1996, from Lawrence Livermore National Laboratory.
@keyword datum: Optional datum to use (L{Datum}).
@return: A L{LatLon3Tuple}C{(lat, lon, height)}.
'''
E = datum.ellipsoid
x, y, z = self.to3xyz()
p = hypot(x, y) # distance from minor axis
r = hypot(p, z) # polar radius
if min(p, r) > EPS:
# parametric latitude (Bowring eqn 17, replaced)
t = (E.b * z) / (E.a * p) * (1 + E.e22 * E.b / r)
c = 1 / hypot1(t)
s = t * c
# geodetic latitude (Bowring eqn 18)
a = atan2(z + E.e22 * E.b * s**3, p - E.e2 * E.a * c**3)
b = atan2(y, x) # ... and longitude
# height above ellipsoid (Bowring eqn 7)
sa, ca = sincos2(a)
# r = E.a / E.e2s(sa) # length of normal terminated by minor axis
# h = p * ca + z * sa - (E.a * E.a / r)
h = fsum_(p * ca, z * sa, -E.a * E.e2s(sa))
a, b = degrees90(a), degrees180(b)
# see <https://GIS.StackExchange.com/questions/28446>
elif p > EPS: # latitude arbitrarily zero
a, b, h = 0.0, degrees180(atan2(y, x)), p - E.a
else: # polar latitude, longitude arbitrarily zero
a, b, h = copysign(90.0, z), 0.0, abs(z) - E.b
return self._xnamed(LatLon3Tuple(a, b, h))
def toNvector(self, datum=Datums.WGS84):
'''Convert this cartesian to an (ellipsoidal) n-vector.
@keyword datum: Optional datum to use (L{Datum}).
@return: The ellipsoidal n-vector (L{Nvector}).
@raise ValueError: The B{C{Cartesian}} at origin.
@example:
>>> from ellipsoidalNvector import LatLon
>>> c = Cartesian(3980581, 97, 4966825)
>>> n = c.toNvector() # (0.62282, 0.000002, 0.78237, +0.24)
'''
if self._Nv is None or datum != self._Nv.datum:
E = datum.ellipsoid
x, y, z = self.to3xyz()
# Kenneth Gade eqn 23
p = (x**2 + y**2) * E.a2_
q = (z**2 * E.e12) * E.a2_
r = fsum_(p, q, -E.e4) / 6
s = (p * q * E.e4) / (4 * r**3)
t = cbrt(fsum_(1, s, sqrt(s * (2 + s))))
u = r * fsum_(1, t, 1 / t)
v = sqrt(u**2 + E.e4 * q)
w = E.e2 * fsum_(u, v, -q) / (2 * v)
k = sqrt(fsum_(u, v, w**2)) - w
if abs(k) < EPS:
raise ValueError('%s: %r' % ('origin', self))
e = k / (k + E.e2)
d = e * hypot(x, y)
t = hypot(d, z)
if t < EPS:
raise ValueError('%s: %r' % ('origin', self))
h = fsum_(k, E.e2, -1) / k * t
s = e / t
self._Nv = Nvector(x * s,
y * s,
z / t,
h=h,
datum=datum,
name=self.name)
return self._Nv
def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **options):
'''Compute the distance between two points using
the U{Equirectangular Approximation / Projection
<https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
@arg lat1: Start latitude (C{degrees}).
@arg lon1: Start longitude (C{degrees}).
@arg lat2: End latitude (C{degrees}).
@arg lon2: End longitude (C{degrees}).
@kwarg radius: Mean earth radius (C{meter}).
@kwarg options: Optional keyword arguments for function
L{equirectangular_}.
@return: Distance (C{meter}, same units as B{C{radius}}).
@see: Function L{equirectangular_} for more details, the
available B{C{options}}, errors, restrictions and other,
approximate or accurate distance functions.
'''
_, dy, dx, _ = equirectangular_(Lat(lat1, name='lat1'),
Lon(lon1, name='lon1'),
Lat(lat2, name='lat2'),
Lon(lon2, name='lon2'),
**options) # PYCHOK Distance4Tuple
return degrees2m(hypot(dx, dy), radius=radius)
def _r3(a, f):
'''(INTERNAL) Reduced cos, sin, tan.
'''
t = (1 - f) * tan(radians(a))
c = 1 / hypot(1, t)
s = t * c
return c, s, t
def vincentys_(phi2, phi1, lam21):
'''Compute the I{angular} distance between two (spherical) points using
U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
spherical formula.
@arg phi2: End latitude (C{radians}).
@arg phi1: Start latitude (C{radians}).
@arg lam21: Longitudinal delta, M{end-start} (C{radians}).
@return: Angular distance (C{radians}).
@see: Functions L{vincentys}, L{cosineLaw_}, L{equirectangular_},
L{euclidean_}, L{flatLocal_}, L{flatPolar_} and L{haversine_}.
@note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_}
produce equivalent results, but L{vincentys_} is suitable
for antipodal points but slightly more expensive (M{3 cos,
3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_}
(M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and
L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}).
'''
sa1, ca1, sa2, ca2, sb21, cb21 = sincos2(phi1, phi2, lam21)
c = ca2 * cb21
x = sa1 * sa2 + ca1 * c
y = ca1 * sa2 - sa1 * c
return atan2(hypot(ca2 * sb21, y), x)
def to3lld(self, datum=None):
'''Convert this L{Lcc} to a geodetic lat- and longitude.
@keyword datum: Optional datum to use, otherwise use this
B{C{Lcc}}'s conic.datum (C{Datum}).
@return: A L{LatLonDatum3Tuple}C{(lat, lon, datum)}.
@raise TypeError: If B{C{datum}} is not ellipsoidal.
'''
c = self.conic
if datum:
c = c.toDatum(datum)
e = self.easting - c._E0
n = c._r0 - self.northing + c._N0
r_ = copysign(hypot(e, n), c._n)
t_ = pow(r_ / c._aF, c._n_)
x = c._xdef(t_) # XXX c._lon0
while True:
p, x = x, c._xdef(t_ * c._pdef(x))
if abs(x - p) < 1e-9: # XXX EPS too small?
break
# x, y == lon, lat
a = degrees90(x)
b = degrees180((atan(e / n) + c._opt3) * c._n_ + c._lon0)
return LatLonDatum3Tuple(a, b, c.datum)
def flatLocal_(phi2, phi1, lam21, datum=Datums.WGS84):
'''Compute the distance between two (ellipsoidal) points using
the U{ellipsoidal Earth to plane projection
<https://WikiPedia.org/wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
fromula.
@arg phi2: End latitude (C{radians}).
@arg phi1: Start latitude (C{radians}).
@arg lam21: Longitudinal delta, M{end-start} (C{radians}).
@kwarg datum: Optional, (ellipsoidal) datum to use (L{Datum}).
@return: Distance (C{meter}, same units as the B{C{datum}}'s
ellipsoid axes).
@raise TypeError: Invalid B{C{datum}}.
@note: The meridional and prime_vertical radii of curvature
are taken and scaled at the mean latitude.
@see: Functions L{flatLocal}, L{cosineLaw_}, L{flatPolar_},
L{equirectangular_}, L{euclidean_}, L{haversine_} and
L{vincentys_} and U{local, flat earth approximation
<https://www.edwilliams.org/avform.htm#flat>}.
'''
_xinstanceof(Datum, datum=datum)
m, n = datum.ellipsoid.roc2_((phi2 + phi1) * 0.5, scaled=True)
return hypot(m * (phi2 - phi1), n * lam21)
def flatLocal_(phi2, phi1, lam21, datum=Datums.WGS84):
'''Compute the I{angular} distance between two (ellipsoidal) points using
the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
@arg phi2: End latitude (C{radians}).
@arg phi1: Start latitude (C{radians}).
@arg lam21: Longitudinal delta, M{end-start} (C{radians}).
@kwarg datum: Ellipsoidal datum to use (L{Datum}).
@return: Angular distance (C{radians}).
@raise TypeError: Invalid B{C{datum}}.
@note: The meridional and prime_vertical radii of curvature
are taken and scaled I{at the mean latitude}.
@see: Functions L{flatLocal}/L{hubeny}, L{cosineAndoyerLambert_},
L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
L{flatPolar_}, L{equirectangular_}, L{euclidean_},
L{haversine_}, L{thomas_} and L{vincentys_} and U{local, flat
earth approximation <https://www.EdWilliams.org/avform.htm#flat>}.
'''
_xinstanceof(Datum, datum=datum)
m, n = datum.ellipsoid.roc2_((phi2 + phi1) * 0.5, scaled=True)
return hypot(m * (phi2 - phi1), n * lam21)
def _direct(self, distance, bearing, llr, height=None):
'''(INTERNAL) Direct Vincenty method.
@raise TypeError: The B{C{other}} point is not L{LatLon}.
@raise ValueError: If this and the B{C{other}} point's L{Datum}
ellipsoids are not compatible.
@raise VincentyError: Vincenty fails to converge for the current
L{LatLon.epsilon} and L{LatLon.iterations}
limit.
'''
E = self.ellipsoid()
c1, s1, t1 = _r3(self.lat, E.f)
i = radians(bearing) # initial bearing (forward azimuth)
si, ci = sincos2(i)
s12 = atan2(t1, ci) * 2
sa = c1 * si
c2a = 1 - sa**2
if c2a < EPS:
c2a = 0
A, B = 1, 0
else: # e22 == (a / b)**2 - 1
A, B = _p2(c2a * E.e22)
s = d = distance / (E.b * A)
for self._iteration in range(1, self._iterations + 1):
ss, cs = sincos2(s)
c2sm = cos(s12 + s)
s_, s = s, d + _ds(B, cs, ss, c2sm)
if abs(s - s_) < self._epsilon:
break
else:
raise VincentyError(_no_(_convergence_),
txt=repr(self)) # self.toRepr()
t = s1 * ss - c1 * cs * ci
# final bearing (reverse azimuth +/- 180)
r = atan2b(sa, -t)
if llr:
# destination latitude in [-90, 90)
a = degrees90(
atan2(s1 * cs + c1 * ss * ci, (1 - E.f) * hypot(sa, t)))
# destination longitude in [-180, 180)
b = degrees180(
atan2(ss * si, c1 * cs - s1 * ss * ci) -
_dl(E.f, c2a, sa, s, cs, ss, c2sm) + radians(self.lon))
h = self.height if height is None else height
d = self.classof(a, b, height=h, datum=self.datum)
else:
d = None
return Destination2Tuple(d, r)
def n_xyz2philam(x, y, z):
'''Convert C{n-vector} components to lat- and longitude in C{radians}.
@arg x: X component (C{scalar}).
@arg y: Y component (C{scalar}).
@arg z: Z component (C{scalar}).
@return: A L{PhiLam2Tuple}C{(phi, lam)}.
@see: Function L{n_xyz2latlon}.
'''
return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x))
def to2ab(self):
'''Convert this vector to (geodetic) lat- and longitude in C{radians}.
@return: A L{PhiLam2Tuple}C{(phi, lam)}.
@example:
>>> v = Vector3d(0.500, 0.500, 0.707)
>>> a, b = v.to2ab() # 0.785323, 0.785398
'''
a = atan2(self.z, hypot(self.x, self.y))
b = atan2(self.y, self.x)
return self._xnamed(PhiLam2Tuple(a, b))
def _sigmaInv0(self, xi, eta):
'''Starting point for C{sigmaInv}.
@return: 3-Tuple C{(u, v, trip)}.
@see: C{bool TMExact::sigmainv0(real xi, real eta,
real &u, real &v)}.
'''
trip = False
if eta > self._Ev_KE_5_4 or xi < min(- self._Eu_E_1_4,
eta - self._Ev_KE):
# sigma as a simple pole at
# w = w0 = Eu.K() + i * Ev.K()
# and sigma is approximated by
# sigma = (Eu.E() + i * Ev.KE()) + 1/(w - w0)
x = xi - self._Eu_E
y = eta - self._Ev_KE
d = x**2 + y**2
u = self._Eu_K + x / d
v = self._Ev_K - y / d
elif eta > self._Ev_KE or (eta > self._Ev_KE_3_4 and
xi < self._Eu_E_1_4):
# At w = w0 = i * Ev.K(), we have
# sigma = sigma0 = i * Ev.KE()
# sigma' = sigma'' = 0
#
# including the next term in the Taylor series gives:
# sigma = sigma0 - _mv / 3 * (w - w0)^3
#
# When inverting this, we map arg(w - w0) = [-pi/2, -pi/6]
# to arg(sigma - sigma0) = [-pi/2, pi/2]
# mapping arg = [-pi/2, -pi/6] to [-pi/2, pi/2]
d = eta - self._Ev_KE
r = hypot(xi, d)
# Error using this guess is about 0.068 * rad^(5/3)
trip = r < _TAYTOL2
# Map the range [-90, 180] in sigma space to [-90, 0] in
# w space. See discussion in zetainv0 on the cut for ang.
r = cbrt(r * self._3_mv)
a = atan2(d - xi, xi + d) / 3.0 - PI_4
s, c = sincos2(a)
u = r * c
v = r * s + self._Ev_K
else: # use w = sigma * Eu.K/Eu.E (correct in the limit _e -> 0)
r = self._Eu_K / self._Eu_E
u = xi * r
v = eta * r
return u, v, trip
def toLatLon(self, LatLon=None, unfalse=True, **LatLon_kwds):
'''Convert this UPS coordinate to an (ellipsoidal) geodetic point.
@kwarg LatLon: Optional, ellipsoidal class to return the
geodetic point (C{LatLon}) or C{None}.
@kwarg unfalse: Unfalse B{C{easting}} and B{C{northing}}
if falsed (C{bool}).
@kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
arguments, ignored if B{C{LatLon=None}}.
@return: This UPS coordinate (B{C{LatLon}}) or if B{C{LatLon}}
is C{None}, a L{LatLonDatum5Tuple}C{(lat, lon, datum,
convergence, scale)}.
@raise TypeError: If B{C{LatLon}} is not ellipsoidal.
@raise UPSError: Invalid meridional radius or H-value.
'''
if self._latlon and self._latlon_args == unfalse:
return self._latlon5(LatLon)
E = self.datum.ellipsoid # XXX vs LatLon.datum.ellipsoid
x, y = self.to2en(falsed=not unfalse)
r = hypot(x, y)
t = (r / (2 * self.scale0 * E.a / E.es_c)) if r > 0 else EPS**2
t = E.es_tauf((1 / t - t) * 0.5)
if self._pole == _N_:
a, b, c = atan(t), atan2(x, -y), 1
else:
a, b, c = -atan(t), atan2(x, y), -1
a, b = degrees90(a), degrees180(b)
if not self._band:
self._band = _Band(a, b)
if not self._hemisphere:
self._hemisphere = _hemi(a)
ll = _LLEB(a, b, datum=self._datum, name=self.name)
ll._convergence = b * c # gamma
ll._scale = _scale(E, r, t) if r > 0 else self.scale0
self._latlon_to(ll, unfalse)
return self._latlon5(LatLon, **LatLon_kwds)
def toLatLon(self, LatLon=None, eps=EPS, unfalse=True, **LatLon_kwds):
'''Convert this UTM coordinate to an (ellipsoidal) geodetic point.
@kwarg LatLon: Optional, ellipsoidal class to return the
geodetic point (C{LatLon}) or C{None}.
@kwarg eps: Optional convergence limit, L{EPS} or above
(C{float}).
@kwarg unfalse: Unfalse B{C{easting}} and B{C{northing}}
if falsed (C{bool}).
@kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
arguments, ignored if B{C{LatLon=None}}.
@return: This UTM coordinate (B{C{LatLon}}) or if B{C{LatLon}}
is C{None}, a L{LatLonDatum5Tuple}C{(lat, lon, datum,
convergence, scale)}.
@raise TypeError: If B{C{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 eps < EPS:
eps = EPS # less doesn't converge
if self._latlon and self._latlon_args == (eps, unfalse):
return self._latlon5(LatLon)
E = self.datum.ellipsoid # XXX vs LatLon.datum.ellipsoid
x, y = self.to2en(falsed=not unfalse)
# from Karney 2011 Eq 15-22, 36
A0 = self.scale0 * E.A
if A0 < EPS:
raise self._Error(meridional=A0)
x /= A0 # η eta
y /= A0 # ξ ksi
K = _Kseries(E.BetaKs, x, y) # Krüger series
y = -K.ys(-y) # ξ'
x = -K.xs(-x) # η'
shx = sinh(x)
sy, cy = sincos2(y)
H = hypot(shx, cy)
if H < EPS:
raise self._Error(H=H)
T = t0 = sy / H # τʹ
S = Fsum(T)
q = 1.0 / E.e12
P = 7 # -/+ toggle trips
d = 1.0 + eps
while abs(d) > eps and P > 0:
p = -d # previous d, toggled
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, d = S.fsum2_(d) # τi, (τi - τi-1)
if d == p: # catch -/+ toggling of d
P -= 1
# else:
# P = 0
a = atan(T) # lat
b = atan2(shx, cy)
if unfalse and self.falsed:
b += radians(_cmlon(self.zone))
ll = _LLEB(degrees90(a), degrees180(b), datum=self.datum, name=self.name)
# convergence: Karney 2011 Eq 26, 27
p = -K.ps(-1)
q = K.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_to(ll, eps, unfalse)
return self._latlon5(LatLon, **LatLon_kwds)
def toUtm8(latlon, lon=None, datum=None, Utm=Utm, falsed=True, name=NN,
zone=None, **cmoff):
'''Convert a lat-/longitude point to a UTM coordinate.
@arg latlon: Latitude (C{degrees}) or an (ellipsoidal)
geodetic C{LatLon} point.
@kwarg lon: Optional longitude (C{degrees}) or C{None}.
@kwarg datum: Optional datum for this UTM coordinate,
overriding B{C{latlon}}'s datum (L{Datum},
L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}).
@kwarg Utm: Optional class to return the UTM coordinate
(L{Utm}) or C{None}.
@kwarg falsed: False both easting and northing (C{bool}).
@kwarg name: Optional B{C{Utm}} name (C{str}).
@kwarg zone: Optional UTM zone to enforce (C{int} or C{str}).
@kwarg cmoff: DEPRECATED, use B{C{falsed}}. Offset longitude
from the zone's central meridian (C{bool}).
@return: The UTM coordinate (B{C{Utm}}) or if B{C{Utm}} is
C{None} or not B{C{falsed}}, a L{UtmUps8Tuple}C{(zone,
hemipole, easting, northing, band, datum, convergence,
scale)}. The C{hemipole} is the C{'N'|'S'} hemisphere.
@raise RangeError: If B{C{lat}} outside the valid UTM bands or
if B{C{lat}} or B{C{lon}} outside the valid
range and L{rangerrors} set to C{True}.
@raise TypeError: Invalid B{C{datum}} or B{C{latlon}} not ellipsoidal.
@raise UTMError: Invalid B{C{zone}}.
@raise ValueError: If B{C{lon}} value is missing or if
B{C{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
'''
z, B, lat, lon, d, f, name = _to7zBlldfn(latlon, lon, datum,
falsed, name, zone,
UTMError, **cmoff)
d = _ellipsoidal_datum(d, name=name)
E = d.ellipsoid
a, b = radians(lat), radians(lon)
# easting, northing: Karney 2011 Eq 7-14, 29, 35
sb, cb = sincos2(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 = E.A * getattr(Utm, '_scale0', _K0) # Utm is class or None
K = _Kseries(E.AlphaKs, x, y) # Krüger series
y = K.ys(y) * A0 # ξ
x = K.xs(x) * A0 # η
# convergence: Karney 2011 Eq 23, 24
p_ = K.ps(1)
q_ = K.qs(0)
c = degrees(atan(T_ / hypot1(T_) * tan(b)) + atan2(q_, p_))
# scale: Karney 2011 Eq 25
k = E.e2s(sin(a)) * T12 / H * (A0 / E.a * hypot(p_, q_))
return _toXtm8(Utm, z, lat, x, y,
B, d, c, k, f, name, latlon, EPS)
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 _Lat_s_c3(name, s, c):
L = Lat_(atan2d(s, c), name=name, Error=AlbersError)
r = _1_0 / Float_(hypot(s, c), name=name, Error=AlbersError)
return L, s * r, c * r
def reverse(self, x, y, lon0=0, name=NN, LatLon=None, **LatLon_kwds):
'''Convert an east- and northing location to geodetic lat- and longitude.
@arg x: Easting of the location (C{meter}).
@arg y: Northing of the location (C{meter}).
@kwarg lon0: Optional central meridian longitude (C{degrees}).
@kwarg name: Optional name for the location (C{str}).
@kwarg LatLon: Class to use (C{LatLon}) or C{None}.
@kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
arguments, ignored if C{B{LatLon}=None}.
@return: The geodetic (C{LatLon}) or if B{C{LatLon}} is C{None} an
L{Albers7Tuple}C{(x, y, lat, lon, gamma, scale, datum)}.
@note: The origin latitude is returned by C{property lat0}. No
false easting or northing is added. The returned value of
C{lon} is in the range C{[-180..180] degrees} and C{lat}
is in the range C{[-90..90] degrees}. If the given
B{C{x}} or B{C{y}} point is outside the valid projected
space the nearest pole is returned.
'''
x = Meter(x=x)
y = Meter(y=y)
k0 = self._k0
n0 = self._n0
k0n0 = self._k0n0
nrho0 = self._nrho0
txi0 = self._txi0
y_ = self._sign * y
nx = k0n0 * x
ny = k0n0 * y_
y1 = nrho0 - ny
drho = den = nrho0 + hypot(nx,
y1) # 0 implies origin with polar aspect
if den:
drho = fsum_(x * nx, -_2_0 * y_ * nrho0, y_ * ny) * k0 / den
# dsxia = scxi0 * dsxi
dsxia = -self._scxi0 * (_2_0 * nrho0 + n0 * drho) * drho / self._qZa2
t = _1_0 - dsxia * (_2_0 * txi0 + dsxia)
txi = (txi0 + dsxia) / (sqrt(t) if t > _EPS__4 else _EPS__2)
ta = self._tanf(txi)
lat = atand(ta * self._sign)
th = atan2(nx, y1)
lon = degrees(th / self._k02n0 if n0 else x / (y1 * k0))
if lon0:
lon += _norm180(lon0)
lon = _norm180(lon)
if LatLon is None:
g = degrees360(self._sign * th)
if den:
k0 = self._azik(nrho0 + n0 * drho, ta)
r = Albers7Tuple(x, y, lat, lon, g, k0, self.datum)
else:
kwds = _xkwds(LatLon_kwds, datum=self.datum)
r = LatLon(lat, lon, **kwds)
return self._xnamed(r, name=name)
def _inverse(self, other, azis, wrap):
'''(INTERNAL) Inverse Vincenty method.
@raise TypeError: The B{C{other}} point is not L{LatLon}.
@raise ValueError: If this and the B{C{other}} point's L{Datum}
ellipsoids are not compatible.
@raise VincentyError: Vincenty fails to converge for the current
L{LatLon.epsilon} and L{LatLon.iterations}
limit and/or if this and the B{C{other}}
point are coincident or near-antipodal.
'''
E = self.ellipsoids(other)
c1, s1, _ = _r3(self.lat, E.f)
c2, s2, _ = _r3(other.lat, E.f)
c1c2, s1c2 = c1 * c2, s1 * c2
c1s2, s1s2 = c1 * s2, s1 * s2
dl, _ = unroll180(self.lon, other.lon, wrap=wrap)
ll = dl = radians(dl)
for self._iteration in range(1, self._iterations + 1):
ll_ = ll
sll, cll = sincos2(ll)
ss = hypot(c2 * sll, c1s2 - s1c2 * cll)
if ss < EPS: # coincident or antipodal, ...
if self.isantipodeTo(other, eps=self._epsilon):
t = '%r %sto %r' % (self, _antipodal_, other)
raise VincentyError(_ambiguous_, txt=t)
# return zeros like Karney, but unlike Veness
return Distance3Tuple(0.0, 0, 0)
cs = s1s2 + c1c2 * cll
s = atan2(ss, cs)
sa = c1c2 * sll / ss
c2a = 1 - sa**2
if abs(c2a) < EPS:
c2a = 0 # equatorial line
ll = dl + E.f * sa * s
else:
c2sm = cs - 2 * s1s2 / c2a
ll = dl + _dl(E.f, c2a, sa, s, cs, ss, c2sm)
if abs(ll - ll_) < self._epsilon:
break
# # omitted and applied only after failure to converge below, see footnote
# # under Inverse at <https://WikiPedia.org/wiki/Vincenty's_formulae>
# # <https://GitHub.com/ChrisVeness/geodesy/blob/master/latlon-vincenty.js>
# elif abs(ll) > PI and self.isantipodeTo(other, eps=self._epsilon):
# raise VincentyError('%s, %r %sto %r' % ('ambiguous', self,
# _antipodal_, other))
else:
t = _antipodal_ if self.isantipodeTo(other,
eps=self._epsilon) else NN
t = _SPACE_(repr(self), NN(t, _to_), repr(other))
raise VincentyError(_no_(_convergence_), txt=t)
if c2a: # e22 == (a / b)**2 - 1
A, B = _p2(c2a * E.e22)
s = A * (s - _ds(B, cs, ss, c2sm))
b = E.b
# if self.height or other.height:
# b += self._havg(other)
d = b * s
if azis: # forward and reverse azimuth
sll, cll = sincos2(ll)
f = atan2b(c2 * sll, c1s2 - s1c2 * cll)
r = atan2b(c1 * sll, -s1c2 + c1s2 * cll)
else:
f = r = 0
return Distance3Tuple(d, f, r)
def _Lat_s_c3(s, c, name):
L = _Lat_(atan2d(s, c), name=name)
r = 1.0 / Float_(hypot(s, c), Error=AlbersError, name=name)
return L, s * r, c * r