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 to3lld(self, datum=None):
'''Convert this L{Lcc} to a geodetic lat- and longitude.
@keyword datum: Optional datum to use, otherwise use this
I{Lcc}'s conic.datum (C{Datum}).
@return: 3-Tuple (lat, lon, datum) in (C{degrees90},
{degrees180}, I{datum}).
@raise TypeError: If I{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 a, b, c.datum
def to2ll(self, datum=None):
'''Convert this WM coordinate to a geodetic lat- and longitude.
@keyword datum: Optional datum (C{Datum}).
@return: 2-Tuple (lat, lon) in (C{degrees90}, C{degrees180}).
@raise TypeError: Non-ellipsoidal I{datum}.
@raise ValueError: Invalid I{radius}.
'''
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))
# <http://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 degrees90(y), degrees180(x)
def intermediateTo(self, other, fraction, height=None, wrap=False):
'''Locate the point at given fraction between this and an
other point.
@param other: The other point (L{LatLon}).
@param fraction: Fraction between both points (float, 0.0 =
this point, 1.0 = the other point).
@keyword height: Optional height, overriding the fractional
height (C{meter}).
@keyword wrap: Wrap and unroll longitudes (C{bool}).
@return: Intermediate point (L{LatLon}).
@raise TypeError: The I{other} point is not L{LatLon}.
@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)
a1, b1 = self.to2ab()
a2, b2 = other.to2ab()
db, b2 = unrollPI(b1, b2, wrap=wrap)
r = haversine_(a2, a1, db)
sr = sin(r)
if abs(sr) > EPS:
cb1, cb2, ca1, ca2 = map1(cos, b1, b2, a1, a2)
sb1, sb2, sa1, sa2 = map1(sin, b1, b2, a1, a2)
A = sin((1 - fraction) * r) / sr
B = sin(fraction * 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=fraction)
b = favg(b1, b2, f=fraction)
if height is None:
h = self._havg(other, f=fraction)
else:
h = height
return self.classof(degrees90(a), degrees180(b), height=h)
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 Bowring’s (1985) formulation for μm precision in concise
form: U{'The accuracy of geodetic latitude and height equations'
<http://www.ResearchGate.net/publication/
233668213_The_Accuracy_of_Geodetic_Latitude_and_Height_Equations>},
B. R. Bowring, Survey Review, Vol 28, 218, Oct 1985.
See also Ralph M. Toms U{'An Efficient Algorithm for Geocentric to
Geodetic Coordinate Conversion'<http://www.OSTI.gov/scitech/biblio/110235>},
Sept 1995 and U{'An Improved Algorithm for Geocentric to Geodetic Coordinate
Conversion'<http://www.OSTI.gov/scitech/servlets/purl/231228>},
Apr 1996, from Lawrence Livermore National Laboratory.
@keyword datum: Optional datum to use (L{Datum}).
@return: 3-Tuple (lat, lon, heigth) in (C{degrees90},
C{degrees180}, C{meter}).
'''
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)
ca, sa = cos(a), sin(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 <http://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 a, b, h
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 to2ll(self):
'''Convert this vector to (geodetic) lat- and longitude.
@return: 2-Tuple (lat, lon) in (C{degrees90}, C{degrees180}).
@example:
>>> v = Vector3d(0.500, 0.500, 0.707)
>>> a, b = v.to2ll() # 44.99567, 45.0
'''
a, b = self.to2ab()
return degrees90(a), degrees180(b)
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 _ in range(self._iterations):
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('%s, %r' % ('no convergence', self))
t = s1 * ss - c1 * cs * ci
# final bearing (reverse azimuth +/- 180)
r = degrees360(atan2(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
r = self.classof(a, b, height=h, datum=self.datum), r
return r
def to2ll(self):
'''Convert this vector to (geodetic) lat- and longitude.
@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 centroidOf(points, wrap=True, LatLon=None):
'''Determine the centroid of a polygon.
@param points: The polygon points (C{LatLon}[]).
@keyword wrap: Wrap lat-, wrap and unroll longitudes (C{bool}).
@keyword LatLon: Optional (sub-)class to return the centroid
(L{LatLon}) or C{None}.
@return: Centroid location (I{LatLon}) or as 2-tuple (C{lat, lon})
in C{degrees} if I{LatLon} is C{None}.
@raise TypeError: Some I{points} are not C{LatLon}.
@raise ValueError: Insufficient number of I{points} or I{points}
enclose a pole or zero area.
@see: U{Centroid<http://WikiPedia.org/wiki/Centroid#Of_a_polygon>}
and U{Calculating The Area And Centroid Of A Polygon
<http://www.Seas.UPenn.edu/~sys502/extra_materials/
Polygon%20Area%20and%20Centroid.pdf>}.
'''
# setting radius=1 converts degrees to radians
pts = LatLon2psxy(points, closed=True, radius=1, wrap=wrap)
n = len(pts)
A, X, Y = Fsum(), Fsum(), Fsum()
x1, y1, _ = pts[n-1]
for i in range(n):
x2, y2, _ = pts[i]
if wrap and i < (n - 1):
_, x2 = unrollPI(x1, x2, wrap=True)
t = x1 * y2 - x2 * y1
A += t
X += t * (x1 + x2)
Y += t * (y1 + y2)
# XXX more elaborately:
# t1, t2 = x1 * y2, -(x2 * y1)
# A.fadd_(t1, t2)
# X.fadd_(t1 * x1, t1 * x2, t2 * x1, t2 * x2)
# Y.fadd_(t1 * y1, t1 * y2, t2 * y1, t2 * y2)
x1, y1 = x2, y2
A = A.fsum() * 3.0 # 6.0 / 2.0
if abs(A) < EPS:
raise ValueError('polar or zero area: %r' % (pts,))
Y, X = degrees90(Y.fsum() / A), degrees180(X.fsum() / A)
return (Y, X) if LatLon is None else LatLon(Y, X)
def toLatLon(self, LatLon=None, unfalse=True):
'''Convert this UPS coordinate to an (ellipsoidal) geodetic point.
@keyword LatLon: Optional, ellipsoidal (sub-)class to return
the point (C{LatLon}) or C{None}.
@keyword unfalse: Unfalse I{easting} and I{northing} if falsed
(C{bool}).
@return: This UPS coordinate as (I{LatLon}) or 5-tuple
(C{lat, lon, datum, convergence, scale}) if
I{LatLon} is C{None}.
@raise TypeError: If I{LatLon} is not ellipsoidal.
@raise UPSError: Invalid meridional radius or H-value.
'''
if self._latlon:
return self._latlon5(LatLon)
E = self.datum.ellipsoid # XXX vs LatLon.datum.ellipsoid
f = self.falsed if unfalse else 0
x = self.easting - f
y = self.northing - f
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 = ll
return self._latlon5(LatLon)
def rhumbMidpointTo(self, other, height=None):
'''Return the (loxodromic) midpoint between this and
an other point.
@param other: The other point (spherical LatLon).
@keyword height: Optional height, overriding the mean height
(C{meter}).
@return: The midpoint (spherical C{LatLon}).
@raise TypeError: The I{other} point is not spherical.
@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.to2ab()
a2, b2 = other.to2ab()
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
return self.classof(degrees90(a3), degrees180(b3), height=h)
def _destination2(a, b, r, t):
'''(INTERNAL) Computes destination lat-/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}).
'''
# see <http://www.EdWilliams.org/avform.htm#LL>
ca, cr, ct = map1(cos, a, r, t)
sa, sr, st = map1(sin, a, r, t)
a = asin(ct * sr * ca + cr * sa)
d = atan2(st * sr * ca, cr - sa * sin(a))
# note, use addition, not subtraction of d (since
# in EdWilliams.org/avform.htm W is + and E is -)
return degrees90(a), degrees180(b + d)
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.
@param other: The other point (L{LatLon}).
@keyword height: Optional height for midpoint, overriding
the mean height (C{meter}).
@keyword wrap: Wrap and unroll longitudes (C{bool}).
@return: Midpoint (L{LatLon}).
@raise TypeError: The I{other} point is not L{LatLon}.
@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 <http://MathForum.org/library/drmath/view/51822.html>
a1, b1 = self.to2ab()
a2, b2 = other.to2ab()
db, b2 = unrollPI(b1, b2, wrap=wrap)
ca1, ca2, cdb = map1(cos, a1, a2, db)
sa1, sa2, sdb = map1(sin, a1, a2, db)
x = ca2 * cdb + ca1
y = ca2 * sdb
a = atan2(sa1 + sa2, hypot(x, y))
b = atan2(y, x) + b1
h = self._havg(other) if height is None else height
return self.classof(degrees90(a), degrees180(b), height=h)
def par2(self):
'''Get the 2nd standard parallel (C{degrees90}).
'''
return degrees90(self._par2)
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 (I{LatLon}) or 3-tuple (C{degrees90},
C{degrees180}, I{datum}) if I{LatLon} is C{None}.
@raise TypeError: If I{LatLon} is not ellipsoidal or if
I{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 = 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):
'''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 toLatLon(self, LatLon=None, eps=EPS, unfalse=True):
'''Convert this UTM coordinate to an (ellipsoidal) geodetic point.
@keyword LatLon: Optional, ellipsoidal (sub-)class to return
the point (C{LatLon}) or C{None}.
@keyword eps: Optional convergence limit, L{EPS} or above
(C{float}).
@keyword unfalse: Unfalse I{easting} and I{northing} if falsed
(C{bool}).
@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 eps < EPS:
eps = EPS # less doesn't converge
if self._latlon and self._latlon_eps == eps:
return self._latlon5(LatLon)
E = self._datum.ellipsoid # XXX vs LatLon.datum.ellipsoid
x = self._easting
y = self._northing
if unfalse and self._falsed:
x -= _FalseEasting # relative to central meridian
if self._hemisphere == '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)
sy, cy = sincos2(y)
H = hypot(shx, cy)
if H < EPS:
raise UTMError('%s invalid: %r' % ('H', H))
d = 1.0 + eps
q = 1.0 / E.e12
T = t0 = sy / H # τʹ
sd = Fsum(T)
while abs(d) > eps:
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 = sd.fsum2_(d) # τi, (τi - τi-1)
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, self._latlon_eps = ll, eps
return self._latlon5(LatLon)
def toLatLon(self, LatLon=None, eps=EPS, unfalse=True):
'''Convert this UTM coordinate to an (ellipsoidal) geodetic point.
@keyword LatLon: Optional, ellipsoidal (sub-)class to return
the point (C{LatLon}) or C{None}.
@keyword eps: Optional convergence limit, L{EPS} or above
(C{float}).
@keyword unfalse: Unfalse B{C{easting}} and B{C{northing}}
if falsed (C{bool}).
@return: This UTM coordinate as (B{C{LatLon}}) or a
L{LatLonDatum5Tuple}C{(lat, lon, datum,
convergence, scale)} if B{C{LatLon}} is C{None}.
@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('%s invalid: %r' % ('meridional', E.A))
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('%s invalid: %r' % ('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)
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._lat0)