def to2ll(self, datum=None):
'''Convert this WM coordinate to a geodetic lat- and longitude.
@keyword datum: Optional datum (C{Datum}).
@return: A L{LatLon2Tuple}C{(lat, lon)}.
@raise TypeError: Non-ellipsoidal B{C{datum}}.
'''
r = self.radius
x = self._x / r
y = 2 * atan(exp(self._y / r)) - PI_2
if datum:
E = datum.ellipsoid
if not E.isEllipsoidal:
raise TypeError('%s not %s: %r' %
('datum', 'ellipsoidal', datum))
# <https://Earth-Info.NGA.mil/GandG/wgs84/web_mercator/
# %28U%29%20NGA_SIG_0011_1.0.0_WEBMERC.pdf>
y = y / r
if E.e:
y -= E.e * atanh(E.e * tanh(y))
y *= E.a
x *= E.a / r
return self._xnamed(LatLon2Tuple(degrees90(y), degrees180(x)))
def latlon2(self, datum=None):
'''Convert this WM coordinate to a lat- and longitude.
@kwarg datum: Optional, ellipsoidal datum (L{Datum},
L{Ellipsoid}, L{Ellipsoid2} or
L{a_f2Tuple}) or C{None}.
@return: A L{LatLon2Tuple}C{(lat, lon)}.
@raise TypeError: Invalid or non-ellipsoidal B{C{datum}}.
@see: Method C{toLatLon}.
'''
r = self.radius
x = self._x / r
y = 2 * atan(exp(self._y / r)) - PI_2
if datum is not None:
E = _ellipsoidal_datum(datum, name=self.name).ellipsoid
if not E.isEllipsoidal:
raise _IsnotError(_ellipsoidal_, datum=datum)
# <https://Earth-Info.NGA.mil/GandG/wgs84/web_mercator/
# %28U%29%20NGA_SIG_0011_1.0.0_WEBMERC.pdf>
y = y / r
if E.e:
y -= E.e * atanh(E.e * tanh(y)) # == E.es_atanh(tanh(y))
y *= E.a
x *= E.a / r
r = LatLon2Tuple(Lat(degrees90(y)), Lon(degrees180(x)))
return self._xnamed(r)
def elevation(self):
'''Get the elevation, tilt of this NED vector in degrees from
horizontal, i.e. tangent to ellipsoid surface (C{degrees90}).
'''
if self._elevation is None:
self._elevation = -degrees90(asin(self.down / self.length))
return self._elevation
def destination(self, distance, bearing, radius=R_M, height=None):
'''Locate the destination from this point after having
travelled the given distance on the given initial bearing.
@arg distance: Distance travelled (C{meter}, same units as
B{C{radius}}).
@arg bearing: Bearing from this point (compass C{degrees360}).
@kwarg radius: Mean earth radius (C{meter}).
@kwarg height: Optional height at destination (C{meter}, same
units a B{C{radius}}).
@return: Destination point (L{LatLon}).
@raise ValueError: Invalid B{C{distance}}, B{C{bearing}},
B{C{radius}} or B{C{height}}.
@example:
>>> p1 = LatLon(51.4778, -0.0015)
>>> p2 = p1.destination(7794, 300.7)
>>> p2.toStr() # '51.5135°N, 000.0983°W'
@JSname: I{destinationPoint}.
'''
a, b = self.philam
r, t = _angular(distance, radius), Bearing_(bearing)
a, b = _destination2(a, b, r, t)
h = self.height if height is None else Height(height)
return self.classof(degrees90(a), degrees180(b), height=h)
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 latlon2(self, datum=None):
'''Convert this WM coordinate to a lat- and longitude.
@kwarg datum: Optional ellipsoidal datum (C{Datum}).
@return: A L{LatLon2Tuple}C{(lat, lon)}.
@raise TypeError: Non-ellipsoidal B{C{datum}}.
@see: Method C{toLatLon}.
'''
r = self.radius
x = self._x / r
y = 2 * atan(exp(self._y / r)) - PI_2
if datum:
_xinstanceof(Datum, datum=datum)
E = datum.ellipsoid
if not E.isEllipsoidal:
raise IsnotError('ellipsoidal', datum=datum)
# <https://Earth-Info.NGA.mil/GandG/wgs84/web_mercator/
# %28U%29%20NGA_SIG_0011_1.0.0_WEBMERC.pdf>
y = y / r
if E.e:
y -= E.e * atanh(E.e * tanh(y)) # == E.es_atanh(tanh(y))
y *= E.a
x *= E.a / r
r = LatLon2Tuple(degrees90(y), degrees180(x))
return self._xnamed(r)
def intermediateTo(self, other, fraction, height=None, wrap=False):
'''Locate the point at given fraction between this and an
other point.
@arg other: The other point (L{LatLon}).
@arg fraction: Fraction between both points (float, between
0.0 for this and 1.0 for the other point).
@kwarg height: Optional height, overriding the fractional
height (C{meter}).
@kwarg wrap: Wrap and unroll longitudes (C{bool}).
@return: Intermediate point (L{LatLon}).
@raise TypeError: The B{C{other}} point is not L{LatLon}.
@raise ValueError: Invalid B{C{fraction}} or B{C{height}}.
@example:
>>> p1 = LatLon(52.205, 0.119)
>>> p2 = LatLon(48.857, 2.351)
>>> p = p1.intermediateTo(p2, 0.25) # 51.3721°N, 000.7073°E
@JSname: I{intermediatePointTo}.
'''
self.others(other)
f = Scalar(fraction, name='fraction')
a1, b1 = self.philam
a2, b2 = other.philam
db, b2 = unrollPI(b1, b2, wrap=wrap)
r = haversine_(a2, a1, db)
sr = sin(r)
if abs(sr) > EPS:
sa1, ca1, sa2, ca2, \
sb1, cb1, sb2, cb2 = sincos2(a1, a2, b1, b2)
A = sin((1 - f) * r) / sr
B = sin(f * r) / sr
x = A * ca1 * cb1 + B * ca2 * cb2
y = A * ca1 * sb1 + B * ca2 * sb2
z = A * sa1 + B * sa2
a = atan2(z, hypot(x, y))
b = atan2(y, x)
else: # points too close
a = favg(a1, a2, f=f)
b = favg(b1, b2, f=f)
if height is None:
h = self._havg(other, f=f)
else:
h = Height(height)
return self.classof(degrees90(a), degrees180(b), height=h)
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 _destination2(a, b, r, t):
'''(INTERNAL) Computes destination lat- and longitude.
@param a: Latitude (C{radians}).
@param b: Longitude (C{radians}).
@param r: Angular distance (C{radians}).
@param t: Bearing (compass C{radians}).
@return: 2-Tuple (lat, lon) of (C{degrees90}, C{degrees180}).
'''
a, b = _destination2_(a, b, r, t)
return degrees90(a), degrees180(b)
def n_xyz2latlon(x, y, z):
'''Convert C{n-vector} components to lat- and longitude in C{degrees}.
@arg x: X component (C{scalar}).
@arg y: Y component (C{scalar}).
@arg z: Z component (C{scalar}).
@return: A L{LatLon2Tuple}C{(lat, lon)}.
@see: Function L{n_xyz2philam}.
'''
a, b = n_xyz2philam(x, y, z) # PYCHOK PhiLam2Tuple
return LatLon2Tuple(degrees90(a), degrees180(b))
def to2ll(self):
'''Convert this vector to (geodetic) lat- and longitude in C{degrees}.
@return: A L{LatLon2Tuple}C{(lat, lon)}.
@example:
>>> v = Vector3d(0.500, 0.500, 0.707)
>>> a, b = v.to2ll() # 44.99567, 45.0
'''
a, b = self.to2ab()
r = LatLon2Tuple(degrees90(a), degrees180(b))
return self._xnamed(r)
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 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 rhumbMidpointTo(self, other, height=None):
'''Return the (loxodromic) midpoint between this and
an other point.
@arg other: The other point (spherical LatLon).
@kwarg height: Optional height, overriding the mean height
(C{meter}).
@return: The midpoint (spherical C{LatLon}).
@raise TypeError: The I{other} point is not spherical.
@raise ValueError: Invalid B{C{height}}.
@example:
>>> p = LatLon(51.127, 1.338)
>>> q = LatLon(50.964, 1.853)
>>> m = p.rhumb_midpointTo(q)
>>> m.toStr() # '51.0455°N, 001.5957°E'
'''
self.others(other)
# see <https://MathForum.org/library/drmath/view/51822.html>
a1, b1 = self.philam
a2, b2 = other.philam
if abs(b2 - b1) > PI:
b1 += PI2 # crossing anti-meridian
a3 = favg(a1, a2)
b3 = favg(b1, b2)
f1 = tanPI_2_2(a1)
if abs(f1) > EPS:
f2 = tanPI_2_2(a2)
f = f2 / f1
if abs(f) > EPS:
f = log(f)
if abs(f) > EPS:
f3 = tanPI_2_2(a3)
b3 = fsum_(b1 * log(f2), -b2 * log(f1),
(b2 - b1) * log(f3)) / f
h = self._havg(other) if height is None else Height(height)
return self.classof(degrees90(a3), degrees180(b3), height=h)
def toLatLon(self, LatLon=None, datum=None, height=None):
'''Convert this L{Lcc} to an (ellipsoidal) geodetic point.
@kwarg LatLon: Optional, ellipsoidal class to return the
geodetic point (C{LatLon}) or C{None}.
@kwarg datum: Optional datum to use, otherwise use this
B{C{Lcc}}'s conic.datum (L{Datum}, L{Ellipsoid},
L{Ellipsoid2} or L{a_f2Tuple}).
@kwarg height: Optional height for the point, overriding
the default height (C{meter}).
@return: The point (B{C{LatLon}}) or a
L{LatLon4Tuple}C{(lat, lon, height, datum)}
if B{C{LatLon}} is C{None}.
@raise TypeError: If B{C{LatLon}} or B{C{datum}} is
not ellipsoidal or not valid.
'''
if LatLon:
_xsubclassof(_LLEB, LatLon=LatLon)
c = self.conic
if datum not in (None, c.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._lam0
while True:
p, x = x, c._xdef(t_ * c._pdef(x))
if abs(x - p) < 1e-9: # XXX EPS too small?
break
lat = degrees90(x)
lon = degrees180((atan(e / n) + c._opt3) * c._n_ + c._lam0)
h = self.height if height is None else height
d = c.datum
r = LatLon4Tuple(lat, lon, h, d) if LatLon is None else \
LatLon(lat, lon, height=h, datum=d)
return self._xnamed(r)
def rhumbDestination(self, distance, bearing, radius=R_M, height=None):
'''Return the destination point having travelled along a rhumb
(loxodrome) line from this point the given distance on the
given bearing.
@arg distance: Distance travelled (C{meter}, same units as
I{radius}).
@arg bearing: Bearing from this point (compass C{degrees360}).
@kwarg radius: Mean earth radius (C{meter}).
@kwarg height: Optional height, overriding the default height
(C{meter}, same unit as I{radius}).
@return: The destination point (spherical C{LatLon}).
@raise ValueError: Invalid B{C{distance}}, B{C{bearing}},
B{C{radius}} or B{C{height}}.
@example:
>>> p = LatLon(51.127, 1.338)
>>> q = p.rhumbDestination(40300, 116.7) # 50.9642°N, 001.8530°E
@JSname: I{rhumbDestinationPoint}
'''
r = _angular(distance, radius)
a1, b1 = self.philam
sb, cb = sincos2(Bearing_(bearing))
da = r * cb
a2 = a1 + da
# normalize latitude if past pole
if a2 > PI_2:
a2 = PI - a2
elif a2 < -PI_2:
a2 = -PI - a2
dp = log(tanPI_2_2(a2) / tanPI_2_2(a1))
# E-W course becomes ill-conditioned with 0/0
q = (da / dp) if abs(dp) > EPS else cos(a1)
b2 = (b1 + r * sb / q) if abs(q) > EPS else b1
h = self.height if height is None else Height(height)
return self.classof(degrees90(a2), degrees180(b2), height=h)
def rhumbDestination(self, distance, bearing, radius=R_M, height=None):
'''Return the destination point having travelled along a rhumb
(loxodrome) line from this point the given distance on the
given bearing.
@param distance: Distance travelled (C{meter}, same units as
I{radius}).
@param bearing: Bearing from this point (compass C{degrees360}).
@keyword radius: Optional, mean earth radius (C{meter}).
@keyword height: Optional height, overriding the default
height (C{meter}, same unit as I{radius}).
@return: The destination point (spherical C{LatLon}).
@example:
>>> p = LatLon(51.127, 1.338)
>>> q = p.rhumbDestination(40300, 116.7) # 50.9642°N, 001.8530°E
@JSname: I{rhumbDestinationPoint}
'''
a1, b1 = self.to2ab()
r = float(distance) / float(radius) # angular distance in radians
sb, cb = sincos2d(bearing)
da = r * cb
a2 = a1 + da
# normalize latitude if past pole
if a2 > PI_2:
a2 = PI - a2
elif a2 < -PI_2:
a2 = -PI - a2
dp = log(tanPI_2_2(a2) / tanPI_2_2(a1))
# E-W course becomes ill-conditioned with 0/0
q = (da / dp) if abs(dp) > EPS else cos(a1)
b2 = (b1 + r * sb / q) if abs(q) > EPS else b1
h = self.height if height is None else height
return self.classof(degrees90(a2), degrees180(b2), height=h)
def maxLat(self, bearing):
'''Return the maximum latitude reached when travelling
on a great circle on given bearing from this point
based on Clairaut's formula.
The maximum latitude is independent of longitude
and the same for all points on a given latitude.
Negate the result for the minimum latitude (on the
Southern hemisphere).
@param bearing: Initial bearing (compass C{degrees360}).
@return: Maximum latitude (C{degrees90}).
@JSname: I{maxLatitude}.
'''
a, _ = self.to2ab()
m = acos1(abs(sin(radians(bearing)) * cos(a)))
return degrees90(m)
def midpointTo(self, other, height=None, wrap=False):
'''Find the midpoint between this and an other point.
@arg other: The other point (L{LatLon}).
@kwarg height: Optional height for midpoint, overriding
the mean height (C{meter}).
@kwarg wrap: Wrap and unroll longitudes (C{bool}).
@return: Midpoint (L{LatLon}).
@raise TypeError: The B{C{other}} point is not L{LatLon}.
@raise ValueError: Invalid B{C{height}}.
@example:
>>> p1 = LatLon(52.205, 0.119)
>>> p2 = LatLon(48.857, 2.351)
>>> m = p1.midpointTo(p2) # '50.5363°N, 001.2746°E'
'''
self.others(other)
# see <https://MathForum.org/library/drmath/view/51822.html>
a1, b1 = self.philam
a2, b2 = other.philam
db, b2 = unrollPI(b1, b2, wrap=wrap)
sa1, ca1, sa2, ca2, sdb, cdb = sincos2(a1, a2, db)
x = ca2 * cdb + ca1
y = ca2 * sdb
a = atan2(sa1 + sa2, hypot(x, y))
b = atan2(y, x) + b1
if height is None:
h = self._havg(other)
else:
h = Height(height)
return self.classof(degrees90(a), degrees180(b), height=h)
def maxLat(self, bearing):
'''Return the maximum latitude reached when travelling
on a great circle on given bearing from this point
based on Clairaut's formula.
The maximum latitude is independent of longitude
and the same for all points on a given latitude.
Negate the result for the minimum latitude (on the
Southern hemisphere).
@arg bearing: Initial bearing (compass C{degrees360}).
@return: Maximum latitude (C{degrees90}).
@raise ValueError: Invalid B{C{bearing}}.
@JSname: I{maxLatitude}.
'''
m = acos1(abs(sin(Bearing_(bearing)) * cos(self.phi)))
return degrees90(m)
def par2(self):
'''Get the 2nd standard parallel (C{degrees90}).
'''
return degrees90(self._par2)
def par1(self):
'''Get the 1st standard parallel (C{degrees90}).
'''
return degrees90(self._par1)
def lat0(self):
'''Get the origin latitude (C{degrees90}).
'''
return degrees90(self._phi0)
def _destination1(bearing):
a, b = _destination2(a1, b1, r1, bearing)
return _latlon3(degrees90(a), degrees180(b), h, intersections2, LatLon,
**LatLon_kwds)
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)
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 intersection(start1,
end1,
start2,
end2,
height=None,
wrap=False,
LatLon=LatLon,
**LatLon_kwds):
'''Compute the intersection point of two paths both defined
by two points or a start point and bearing from North.
@arg start1: Start point of the first path (L{LatLon}).
@arg end1: End point ofthe first path (L{LatLon}) or
the initial bearing at the first start point
(compass C{degrees360}).
@arg start2: Start point of the second path (L{LatLon}).
@arg end2: End point of the second path (L{LatLon}) or
the initial bearing at the second start point
(compass C{degrees360}).
@kwarg height: Optional height for the intersection point,
overriding the mean height (C{meter}).
@kwarg wrap: Wrap and unroll longitudes (C{bool}).
@kwarg LatLon: Optional class to return the intersection
point (L{LatLon}) or C{None}.
@kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
arguments, ignored if B{C{LatLon=None}}.
@return: The intersection point (B{C{LatLon}}) or a
L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}}
is C{None}. An alternate intersection point might
be the L{antipode} to the returned result.
@raise TypeError: A B{C{start}} or B{C{end}} point not L{LatLon}.
@raise ValueError: Intersection is ambiguous or infinite or
the paths are parallel, coincident or null
or invalid B{C{height}}.
@example:
>>> p = LatLon(51.8853, 0.2545)
>>> s = LatLon(49.0034, 2.5735)
>>> i = intersection(p, 108.547, s, 32.435) # '50.9078°N, 004.5084°E'
'''
_Trll.others(start1, name='start1')
_Trll.others(start2, name='start2')
hs = [start1.height, start2.height]
a1, b1 = start1.philam
a2, b2 = start2.philam
db, b2 = unrollPI(b1, b2, wrap=wrap)
r12 = haversine_(a2, a1, db)
if abs(r12) < EPS: # [nearly] coincident points
a, b = favg(a1, a2), favg(b1, b2)
# see <https://www.EdWilliams.org/avform.htm#Intersection>
elif isscalar(end1) and isscalar(end2): # both bearings
sa1, ca1, sa2, ca2, sr12, cr12 = sincos2(a1, a2, r12)
x1, x2 = (sr12 * ca1), (sr12 * ca2)
if abs(x1) < EPS or abs(x2) < EPS:
raise ValueError('intersection %s: %r vs %r' % ('parallel',
(start1, end1),
(start2, end2)))
# handle domain error for equivalent longitudes,
# see also functions asin_safe and acos_safe at
# <https://www.EdWilliams.org/avform.htm#Math>
t1, t2 = map1(acos1, (sa2 - sa1 * cr12) / x1, (sa1 - sa2 * cr12) / x2)
if sin(db) > 0:
t12, t21 = t1, PI2 - t2
else:
t12, t21 = PI2 - t1, t2
t13, t23 = map1(radiansPI2, end1, end2)
x1, x2 = map1(
wrapPI,
t13 - t12, # angle 2-1-3
t21 - t23) # angle 1-2-3
sx1, cx1, sx2, cx2 = sincos2(x1, x2)
if sx1 == 0 and sx2 == 0: # max(abs(sx1), abs(sx2)) < EPS
raise ValueError('intersection %s: %r vs %r' % ('infinite',
(start1, end1),
(start2, end2)))
sx3 = sx1 * sx2
# if sx3 < 0:
# raise ValueError('intersection %s: %r vs %r' % ('ambiguous',
# (start1, end1), (start2, end2)))
x3 = acos1(cr12 * sx3 - cx2 * cx1)
r13 = atan2(sr12 * sx3, cx2 + cx1 * cos(x3))
a, b = _destination2(a1, b1, r13, t13)
# choose antipode for opposing bearings
if _xb(a1, b1, end1, a, b, wrap) < 0 or \
_xb(a2, b2, end2, a, b, wrap) < 0:
a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple
else: # end point(s) or bearing(s)
x1, d1 = _x3d2(start1, end1, wrap, '1', hs)
x2, d2 = _x3d2(start2, end2, wrap, '2', hs)
x = x1.cross(x2)
if x.length < EPS: # [nearly] colinear or parallel paths
raise ValueError('intersection %s: %r vs %r' % ('colinear',
(start1, end1),
(start2, end2)))
a, b = x.philam
# choose intersection similar to sphericalNvector
d1 = _xdot(d1, a1, b1, a, b, wrap)
if d1:
d2 = _xdot(d2, a2, b2, a, b, wrap)
if (d2 < 0 and d1 > 0) or (d2 > 0 and d1 < 0):
a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple
h = fmean(hs) if height is None else Height(height)
return _latlon3(degrees90(a), degrees180(b), h, intersection, LatLon,
**LatLon_kwds)
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 reverse(self, xyz, y=None, z=None, **no_M): # PYCHOK unused M
'''Convert from geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)}
transcribed from I{Chris Veness}' U{JavaScript
<https://www.Movable-Type.co.UK/scripts/geodesy/docs/latlon-ellipsoidal.js.html>}.
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.
@arg xyz: Either an L{Ecef9Tuple}, an C{(x, y, z)} 3-tuple or C{scalar}
ECEF C{x} coordinate in C{meter}.
@kwarg y: ECEF C{y} coordinate in C{meter} for C{scalar} B{C{xyz}} and B{C{z}}.
@kwarg z: ECEF C{z} coordinate in C{meter} for C{scalar} B{C{xyz}} and B{C{y}}.
@kwarg no_M: Rotation matrix C{M} not available.
@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)}
with geodetic coordinates C{(lat, lon, height)} for the given
geocentric ones C{(x, y, z)}, case C{C}, L{EcefMatrix} C{M}
always C{None} and C{datum} if available.
@raise EcefError: If B{C{xyz}} not L{Ecef9Tuple} or C{scalar} C{x}
or B{C{y}} and/or B{C{z}} not C{scalar} for C{scalar}
B{C{xyz}}.
@see: 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, both from Lawrence Livermore National Laboratory (LLNL).
'''
x, y, z, name = _xyzn4(xyz, y, z, Error=EcefError)
E = self.ellipsoid
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))
C, lat, lon = 1, degrees90(a), degrees180(b)
# see <https://GIS.StackExchange.com/questions/28446>
elif p > EPS: # lat arbitrarily zero
C, lat, lon, h = 2, 0.0, degrees180(atan2(y, x)), p - E.a
else: # polar lat, lon arbitrarily zero
C, lat, lon, h = 3, copysign(90.0, z), 0.0, abs(z) - E.b
r = Ecef9Tuple(x, y, z, lat, lon, h, C, None, self.datum)
return self._xnamed(r, name)