def _transit(lon1, lon2):
"""Count crossings of prime meridian for AddPoint."""
# Return 1 or -1 if crossing prime meridian in east or west direction.
# Otherwise return zero.
# Compute lon12 the same way as Geodesic::Inverse.
lon1 = Math.AngNormalize(lon1)
lon2 = Math.AngNormalize(lon2)
lon12, _ = Math.AngDiff(lon1, lon2)
cross = (1 if lon1 <= 0 and lon2 > 0 and lon12 > 0 else
(-1 if lon2 <= 0 and lon1 > 0 and lon12 < 0 else 0))
return cross
def transit(lon1, lon2):
# Return 1 or -1 if crossing prime meridian in east or west direction.
# Otherwise return zero.
from geographiclib.geodesic import Geodesic
# Compute lon12 the same way as Geodesic::Inverse.
lon1 = Math.AngNormalize(lon1);
lon2 = Math.AngNormalize(lon2);
lon12 = Math.AngDiff(lon1, lon2);
cross = (1 if lon1 < 0 and lon2 >= 0 and lon12 > 0
else (-1 if lon2 < 0 and lon1 >= 0 and lon12 < 0 else 0))
return cross
def ArcPosition(self, a12, outmask = GeodesicCapability.STANDARD):
"""Find the position on the line given *a12*
:param a12: spherical arc length from the first point to the second
in degrees
:param outmask: the :ref:`output mask <outmask>`
:return: a :ref:`dict`
The default value of *outmask* is STANDARD, i.e., the *lat1*,
*lon1*, *azi1*, *lat2*, *lon2*, *azi2*, *s12*, *a12* entries are
returned.
"""
from geographiclib.geodesic import Geodesic
result = {'lat1': self.lat1,
'lon1': self.lon1 if outmask & Geodesic.LONG_UNROLL else
Math.AngNormalize(self.lon1),
'azi1': self.azi1, 'a12': a12}
a12, lat2, lon2, azi2, s12, m12, M12, M21, S12 = self._GenPosition(
True, a12, outmask)
outmask &= Geodesic.OUT_MASK
if outmask & Geodesic.DISTANCE: result['s12'] = s12
if outmask & Geodesic.LATITUDE: result['lat2'] = lat2
if outmask & Geodesic.LONGITUDE: result['lon2'] = lon2
if outmask & Geodesic.AZIMUTH: result['azi2'] = azi2
if outmask & Geodesic.REDUCEDLENGTH: result['m12'] = m12
if outmask & Geodesic.GEODESICSCALE:
result['M12'] = M12; result['M21'] = M21
if outmask & Geodesic.AREA: result['S12'] = S12
return result
def CheckPosition(lat, lon):
"""Check that lat and lon are legal and return normalized lon"""
if (abs(lat) > 90):
raise ValueError("latitude " + str(lat) + " not in [-90, 90]")
if (lon < -540 or lon >= 540):
raise ValueError("longitude " + str(lon) + " not in [-540, 540)")
return Math.AngNormalize(lon)
def CheckPosition(lat, lon):
"""Check that lat and lon are legal and return normalized lon"""
if abs(lat) > 90:
raise ValueError("latitude " + str(lat) + " not in [-90, 90]")
# if not Math.isfinite(lon):
# raise ValueError("longitude " + str(lon) + " not a finite number")
return Math.AngNormalize(lon)
def Position(self, s12, outmask = GeodesicCapability.STANDARD):
"""Find the position on the line given *s12*
:param s12: the distance from the first point to the second in
meters
:param outmask: the :ref:`output mask <outmask>`
:return: a :ref:`dict`
The default value of *outmask* is STANDARD, i.e., the *lat1*,
*lon1*, *azi1*, *lat2*, *lon2*, *azi2*, *s12*, *a12* entries are
returned. The :class:`~geographiclib.geodesicline.GeodesicLine`
object must have been constructed with the DISTANCE_IN capability.
"""
from geographiclib.geodesic import Geodesic
result = {'lat1': self.lat1,
'lon1': self.lon1 if outmask & Geodesic.LONG_UNROLL else
Math.AngNormalize(self.lon1),
'azi1': self.azi1, 's12': s12}
a12, lat2, lon2, azi2, s12, m12, M12, M21, S12 = self._GenPosition(
False, s12, outmask)
outmask &= Geodesic.OUT_MASK
result['a12'] = a12
if outmask & Geodesic.LATITUDE: result['lat2'] = lat2
if outmask & Geodesic.LONGITUDE: result['lon2'] = lon2
if outmask & Geodesic.AZIMUTH: result['azi2'] = azi2
if outmask & Geodesic.REDUCEDLENGTH: result['m12'] = m12
if outmask & Geodesic.GEODESICSCALE:
result['M12'] = M12; result['M21'] = M21
if outmask & Geodesic.AREA: result['S12'] = S12
return result
def ArcPosition(self, a12,
outmask = GeodesicCapability.LATITUDE |
GeodesicCapability.LONGITUDE | GeodesicCapability.AZIMUTH |
GeodesicCapability.DISTANCE):
"""Return the point a spherical arc length a12 along the geodesic line.
Return a dictionary with (some) of the following entries:
lat1 latitude of point 1
lon1 longitude of point 1
azi1 azimuth of line at point 1
lat2 latitude of point 2
lon2 longitude of point 2
azi2 azimuth of line at point 2
s12 distance from 1 to 2
a12 arc length on auxiliary sphere from 1 to 2
m12 reduced length of geodesic
M12 geodesic scale 2 relative to 1
M21 geodesic scale 1 relative to 2
S12 area between geodesic and equator
outmask determines which fields get included and if outmask is
omitted, then only the basic geodesic fields are computed. The
LONG_UNROLL bit unrolls the longitudes (instead of reducing them to
the range [-180,180)). The mask is an or'ed combination of the
following values
Geodesic.LATITUDE
Geodesic.LONGITUDE
Geodesic.AZIMUTH
Geodesic.DISTANCE
Geodesic.REDUCEDLENGTH
Geodesic.GEODESICSCALE
Geodesic.AREA
Geodesic.ALL (all of the above)
Geodesic.LONG_UNROLL
The default value of outmask is LATITUDE | LONGITUDE | AZIMUTH |
DISTANCE.
"""
from geographiclib.geodesic import Geodesic
Geodesic.CheckDistance(a12)
result = {'lat1': self._lat1,
'lon1': self._lon1 if outmask & Geodesic.LONG_UNROLL else
Math.AngNormalize(self._lon1),
'azi1': self._azi1, 'a12': a12}
a12, lat2, lon2, azi2, s12, m12, M12, M21, S12 = self.GenPosition(
True, a12, outmask)
outmask &= Geodesic.OUT_MASK
if outmask & Geodesic.DISTANCE: result['s12'] = s12
if outmask & Geodesic.LATITUDE: result['lat2'] = lat2
if outmask & Geodesic.LONGITUDE: result['lon2'] = lon2
if outmask & Geodesic.AZIMUTH: result['azi2'] = azi2
if outmask & Geodesic.REDUCEDLENGTH: result['m12'] = m12
if outmask & Geodesic.GEODESICSCALE:
result['M12'] = M12; result['M21'] = M21
if outmask & Geodesic.AREA: result['S12'] = S12
return result
def Position(self, s12,
outmask = GeodesicCapability.LATITUDE |
GeodesicCapability.LONGITUDE | GeodesicCapability.AZIMUTH):
"""Return the point a distance s12 along the geodesic line. Return a
dictionary with (some) of the following entries:
lat1 latitude of point 1
lon1 longitude of point 1
azi1 azimuth of line at point 1
lat2 latitude of point 2
lon2 longitude of point 2
azi2 azimuth of line at point 2
s12 distance from 1 to 2
a12 arc length on auxiliary sphere from 1 to 2
m12 reduced length of geodesic
M12 geodesic scale 2 relative to 1
M21 geodesic scale 1 relative to 2
S12 area between geodesic and equator
outmask determines which fields get included and if outmask is
omitted, then only the basic geodesic fields are computed. The
LONG_NOWRAP bit prevents the longitudes being reduced to the range
[-180,180). The mask is an or'ed combination of the following
values
Geodesic.LATITUDE
Geodesic.LONGITUDE
Geodesic.AZIMUTH
Geodesic.REDUCEDLENGTH
Geodesic.GEODESICSCALE
Geodesic.AREA
Geodesic.ALL
Geodesic.LONG_NOWRAP
"""
from geographiclib.geodesic import Geodesic
Geodesic.CheckDistance(s12)
result = {'lat1': self._lat1,
'lon1': self._lon1 if outmask & Geodesic.LONG_NOWRAP else
Math.AngNormalize(self._lon1),
'azi1': self._azi1, 's12': s12}
a12, lat2, lon2, azi2, s12, m12, M12, M21, S12 = self.GenPosition(
False, s12, outmask)
outmask &= Geodesic.OUT_MASK
result['a12'] = a12
if outmask & Geodesic.LATITUDE: result['lat2'] = lat2
if outmask & Geodesic.LONGITUDE: result['lon2'] = lon2
if outmask & Geodesic.AZIMUTH: result['azi2'] = azi2
if outmask & Geodesic.REDUCEDLENGTH: result['m12'] = m12
if outmask & Geodesic.GEODESICSCALE:
result['M12'] = M12; result['M21'] = M21
if outmask & Geodesic.AREA: result['S12'] = S12
return result
def Inverse(self, lat1, lon1, lat2, lon2,
outmask = GeodesicCapability.STANDARD):
"""Solve the inverse geodesic problem
:param lat1: latitude of the first point in degrees
:param lon1: longitude of the first point in degrees
:param lat2: latitude of the second point in degrees
:param lon2: longitude of the second point in degrees
:param outmask: the :ref:`output mask <outmask>`
:return: a :ref:`dict`
Compute geodesic between (*lat1*, *lon1*) and (*lat2*, *lon2*).
The default value of *outmask* is STANDARD, i.e., the *lat1*,
*lon1*, *azi1*, *lat2*, *lon2*, *azi2*, *s12*, *a12* entries are
returned.
"""
a12, s12, salp1,calp1, salp2,calp2, m12, M12, M21, S12 = self._GenInverse(
lat1, lon1, lat2, lon2, outmask)
outmask &= Geodesic.OUT_MASK
if outmask & Geodesic.LONG_UNROLL:
lon12, e = Math.AngDiff(lon1, lon2)
lon2 = (lon1 + lon12) + e
else:
lon2 = Math.AngNormalize(lon2)
result = {'lat1': Math.LatFix(lat1),
'lon1': lon1 if outmask & Geodesic.LONG_UNROLL else
Math.AngNormalize(lon1),
'lat2': Math.LatFix(lat2),
'lon2': lon2}
result['a12'] = a12
if outmask & Geodesic.DISTANCE: result['s12'] = s12
if outmask & Geodesic.AZIMUTH:
result['azi1'] = Math.atan2d(salp1, calp1)
result['azi2'] = Math.atan2d(salp2, calp2)
if outmask & Geodesic.REDUCEDLENGTH: result['m12'] = m12
if outmask & Geodesic.GEODESICSCALE:
result['M12'] = M12; result['M21'] = M21
if outmask & Geodesic.AREA: result['S12'] = S12
return result
def __init__(self, lat0, lon0, h0=0):
self.lat0 = lat0
self.lon0 = geomath.AngNormalize(lon0)
self.h0 = h0
self.origin = (earth_forward(self.lat0, self.lon0, self.h0))
phi = math.radians(lat0)
lam = math.radians(lon0)
sphi = math.sin(phi)
cphi = 0 if abs(self.lat0) == 90 else math.cos(phi)
slam = 0 if self.lon0 == -180 else math.sin(lam)
clam = 0 if abs(self.lon0) == 90 else math.cos(lam)
self.rot = geocentric_rotation(sphi, cphi, slam, clam)
def ArcDirect(self, lat1, lon1, azi1, a12,
outmask = GeodesicCapability.STANDARD):
"""Solve the direct geodesic problem in terms of spherical arc length
:param lat1: latitude of the first point in degrees
:param lon1: longitude of the first point in degrees
:param azi1: azimuth at the first point in degrees
:param a12: spherical arc length from the first point to the second
in degrees
:param outmask: the :ref:`output mask <outmask>`
:return: a :ref:`dict`
Compute geodesic starting at (*lat1*, *lon1*) with azimuth *azi1*
and arc length *a12*. The default value of *outmask* is STANDARD,
i.e., the *lat1*, *lon1*, *azi1*, *lat2*, *lon2*, *azi2*, *s12*,
*a12* entries are returned.
"""
a12, lat2, lon2, azi2, s12, m12, M12, M21, S12 = self._GenDirect(
lat1, lon1, azi1, True, a12, outmask)
outmask &= Geodesic.OUT_MASK
result = {'lat1': Math.LatFix(lat1),
'lon1': lon1 if outmask & Geodesic.LONG_UNROLL else
Math.AngNormalize(lon1),
'azi1': Math.AngNormalize(azi1),
'a12': a12}
if outmask & Geodesic.DISTANCE: result['s12'] = s12
if outmask & Geodesic.LATITUDE: result['lat2'] = lat2
if outmask & Geodesic.LONGITUDE: result['lon2'] = lon2
if outmask & Geodesic.AZIMUTH: result['azi2'] = azi2
if outmask & Geodesic.REDUCEDLENGTH: result['m12'] = m12
if outmask & Geodesic.GEODESICSCALE:
result['M12'] = M12; result['M21'] = M21
if outmask & Geodesic.AREA: result['S12'] = S12
return result
def earth_forward(lat, lon, h):
"""Geocentric::IntForward"""
lon = geomath.AngNormalize(lon)
phi = math.radians(lat)
lam = math.radians(lon)
sphi = math.sin(phi)
cphi = 0 if abs(lat) == 90 else math.cos(phi)
_a = geoconst.WGS84_a
_f = geoconst.WGS84_f
_e2 = _f * (2 - _f)
_e2m = 1 - _e2
n = _a/math.sqrt(1 - _e2 * sphi * sphi)
slam = 0 if lon == -180 else math.sin(lam)
clam = 0 if abs(lon) == 90 else math.cos(lam)
Z = (_e2m * n + h) * sphi
X = (n + h) * cphi
Y = X * slam
X *= clam
return numpy.array([X, Y, Z])
def GenPosition(self, arcmode, s12_a12, outmask):
from geographiclib.geodesic import Geodesic
a12 = lat2 = lon2 = azi2 = s12 = m12 = M12 = M21 = S12 = Math.nan
outmask &= self._caps & Geodesic.OUT_ALL
if not (arcmode or
(self._caps & Geodesic.DISTANCE_IN & Geodesic.OUT_ALL)):
# Uninitialized or impossible distance calculation requested
return a12, lat2, lon2, azi2, s12, m12, M12, M21, S12
# Avoid warning about uninitialized B12.
B12 = 0
AB1 = 0
if arcmode:
# Interpret s12_a12 as spherical arc length
sig12 = s12_a12 * Math.degree
s12a = abs(s12_a12)
s12a -= 180 * math.floor(s12a / 180)
ssig12 = 0 if s12a == 0 else math.sin(sig12)
csig12 = 0 if s12a == 90 else math.cos(sig12)
else:
# Interpret s12_a12 as distance
tau12 = s12_a12 / (self._b * (1 + self._A1m1))
s = math.sin(tau12)
c = math.cos(tau12)
# tau2 = tau1 + tau12
B12 = -Geodesic.SinCosSeries(
True, self._stau1 * c + self._ctau1 * s,
self._ctau1 * c - self._stau1 * s, self._C1pa, Geodesic.nC1p_)
sig12 = tau12 - (B12 - self._B11)
ssig12 = math.sin(sig12)
csig12 = math.cos(sig12)
if abs(self._f) > 0.01:
# Reverted distance series is inaccurate for |f| > 1/100, so correct
# sig12 with 1 Newton iteration. The following table shows the
# approximate maximum error for a = WGS_a() and various f relative to
# GeodesicExact.
# erri = the error in the inverse solution (nm)
# errd = the error in the direct solution (series only) (nm)
# errda = the error in the direct solution (series + 1 Newton) (nm)
#
# f erri errd errda
# -1/5 12e6 1.2e9 69e6
# -1/10 123e3 12e6 765e3
# -1/20 1110 108e3 7155
# -1/50 18.63 200.9 27.12
# -1/100 18.63 23.78 23.37
# -1/150 18.63 21.05 20.26
# 1/150 22.35 24.73 25.83
# 1/100 22.35 25.03 25.31
# 1/50 29.80 231.9 30.44
# 1/20 5376 146e3 10e3
# 1/10 829e3 22e6 1.5e6
# 1/5 157e6 3.8e9 280e6
ssig2 = self._ssig1 * csig12 + self._csig1 * ssig12
csig2 = self._csig1 * csig12 - self._ssig1 * ssig12
B12 = Geodesic.SinCosSeries(True, ssig2, csig2, self._C1a,
Geodesic.nC1_)
serr = ((1 + self._A1m1) * (sig12 + (B12 - self._B11)) -
s12_a12 / self._b)
sig12 = sig12 - serr / math.sqrt(1 + self._k2 * Math.sq(ssig2))
ssig12 = math.sin(sig12)
csig12 = math.cos(sig12)
# Update B12 below
# real omg12, lam12, lon12
# real ssig2, csig2, sbet2, cbet2, somg2, comg2, salp2, calp2
# sig2 = sig1 + sig12
ssig2 = self._ssig1 * csig12 + self._csig1 * ssig12
csig2 = self._csig1 * csig12 - self._ssig1 * ssig12
dn2 = math.sqrt(1 + self._k2 * Math.sq(ssig2))
if outmask & (Geodesic.DISTANCE | Geodesic.REDUCEDLENGTH
| Geodesic.GEODESICSCALE):
if arcmode or abs(self._f) > 0.01:
B12 = Geodesic.SinCosSeries(True, ssig2, csig2, self._C1a,
Geodesic.nC1_)
AB1 = (1 + self._A1m1) * (B12 - self._B11)
# sin(bet2) = cos(alp0) * sin(sig2)
sbet2 = self._calp0 * ssig2
# Alt: cbet2 = hypot(csig2, salp0 * ssig2)
cbet2 = math.hypot(self._salp0, self._calp0 * csig2)
if cbet2 == 0:
# I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case
cbet2 = csig2 = Geodesic.tiny_
# tan(omg2) = sin(alp0) * tan(sig2)
somg2 = self._salp0 * ssig2
comg2 = csig2 # No need to normalize
# tan(alp0) = cos(sig2)*tan(alp2)
salp2 = self._salp0
calp2 = self._calp0 * csig2 # No need to normalize
# omg12 = omg2 - omg1
omg12 = math.atan2(somg2 * self._comg1 - comg2 * self._somg1,
comg2 * self._comg1 + somg2 * self._somg1)
if outmask & Geodesic.DISTANCE:
s12 = self._b * (
(1 + self._A1m1) * sig12 + AB1) if arcmode else s12_a12
if outmask & Geodesic.LONGITUDE:
lam12 = omg12 + self._A3c * (sig12 + (Geodesic.SinCosSeries(
True, ssig2, csig2, self._C3a, Geodesic.nC3_ - 1) - self._B31))
lon12 = lam12 / Math.degree
# Use Math.AngNormalize2 because longitude might have wrapped
# multiple times.
lon12 = Math.AngNormalize2(lon12)
lon2 = Math.AngNormalize(self._lon1 + lon12)
if outmask & Geodesic.LATITUDE:
lat2 = math.atan2(sbet2, self._f1 * cbet2) / Math.degree
if outmask & Geodesic.AZIMUTH:
# minus signs give range [-180, 180). 0- converts -0 to +0.
azi2 = 0 - math.atan2(-salp2, calp2) / Math.degree
if outmask & (Geodesic.REDUCEDLENGTH | Geodesic.GEODESICSCALE):
B22 = Geodesic.SinCosSeries(True, ssig2, csig2, self._C2a,
Geodesic.nC2_)
AB2 = (1 + self._A2m1) * (B22 - self._B21)
J12 = (self._A1m1 - self._A2m1) * sig12 + (AB1 - AB2)
if outmask & Geodesic.REDUCEDLENGTH:
# Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure
# accurate cancellation in the case of coincident points.
m12 = self._b * (
(dn2 * (self._csig1 * ssig2) - self._dn1 *
(self._ssig1 * csig2)) - self._csig1 * csig2 * J12)
if outmask & Geodesic.GEODESICSCALE:
t = (self._k2 * (ssig2 - self._ssig1) * (ssig2 + self._ssig1) /
(self._dn1 + dn2))
M12 = csig12 + (t * ssig2 -
csig2 * J12) * self._ssig1 / self._dn1
M21 = csig12 - (t * self._ssig1 -
self._csig1 * J12) * ssig2 / dn2
if outmask & Geodesic.AREA:
B42 = Geodesic.SinCosSeries(False, ssig2, csig2, self._C4a,
Geodesic.nC4_)
# real salp12, calp12
if self._calp0 == 0 or self._salp0 == 0:
# alp12 = alp2 - alp1, used in atan2 so no need to normalized
salp12 = salp2 * self._calp1 - calp2 * self._salp1
calp12 = calp2 * self._calp1 + salp2 * self._salp1
# The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
# salp12 = -0 and alp12 = -180. However this depends on the sign being
# attached to 0 correctly. The following ensures the correct behavior.
if salp12 == 0 and calp12 < 0:
salp12 = Geodesic.tiny_ * self._calp1
calp12 = -1
else:
# tan(alp) = tan(alp0) * sec(sig)
# tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1)
# = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2)
# If csig12 > 0, write
# csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
# else
# csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
# No need to normalize
salp12 = self._calp0 * self._salp0 * (
self._csig1 * (1 - csig12) +
ssig12 * self._ssig1 if csig12 <= 0 else ssig12 *
(self._csig1 * ssig12 / (1 + csig12) + self._ssig1))
calp12 = (Math.sq(self._salp0) +
Math.sq(self._calp0) * self._csig1 * csig2)
S12 = self._c2 * math.atan2(salp12,
calp12) + self._A4 * (B42 - self._B41)
a12 = s12_a12 if arcmode else sig12 / Math.degree
return a12, lat2, lon2, azi2, s12, m12, M12, M21, S12
def _GenPosition(self, arcmode, s12_a12, outmask):
"""Private: General solution of position along geodesic"""
from geographiclib.geodesic import Geodesic
a12 = lat2 = lon2 = azi2 = s12 = m12 = M12 = M21 = S12 = Math.nan
outmask &= self.caps & Geodesic.OUT_MASK
if not (arcmode or (self.caps &
(Geodesic.OUT_MASK & Geodesic.DISTANCE_IN))):
# Uninitialized or impossible distance calculation requested
return a12, lat2, lon2, azi2, s12, m12, M12, M21, S12
# Avoid warning about uninitialized B12.
B12 = 0.0
AB1 = 0.0
if arcmode:
# Interpret s12_a12 as spherical arc length
sig12 = math.radians(s12_a12)
ssig12, csig12 = Math.sincosd(s12_a12)
else:
# Interpret s12_a12 as distance
tau12 = s12_a12 / (self._b * (1 + self._A1m1))
s = math.sin(tau12)
c = math.cos(tau12)
# tau2 = tau1 + tau12
B12 = -Geodesic._SinCosSeries(
True, self._stau1 * c + self._ctau1 * s,
self._ctau1 * c - self._stau1 * s, self._C1pa)
sig12 = tau12 - (B12 - self._B11)
ssig12 = math.sin(sig12)
csig12 = math.cos(sig12)
if abs(self.f) > 0.01:
# Reverted distance series is inaccurate for |f| > 1/100, so correct
# sig12 with 1 Newton iteration. The following table shows the
# approximate maximum error for a = WGS_a() and various f relative to
# GeodesicExact.
# erri = the error in the inverse solution (nm)
# errd = the error in the direct solution (series only) (nm)
# errda = the error in the direct solution (series + 1 Newton) (nm)
#
# f erri errd errda
# -1/5 12e6 1.2e9 69e6
# -1/10 123e3 12e6 765e3
# -1/20 1110 108e3 7155
# -1/50 18.63 200.9 27.12
# -1/100 18.63 23.78 23.37
# -1/150 18.63 21.05 20.26
# 1/150 22.35 24.73 25.83
# 1/100 22.35 25.03 25.31
# 1/50 29.80 231.9 30.44
# 1/20 5376 146e3 10e3
# 1/10 829e3 22e6 1.5e6
# 1/5 157e6 3.8e9 280e6
ssig2 = self._ssig1 * csig12 + self._csig1 * ssig12
csig2 = self._csig1 * csig12 - self._ssig1 * ssig12
B12 = Geodesic._SinCosSeries(True, ssig2, csig2, self._C1a)
serr = ((1 + self._A1m1) * (sig12 + (B12 - self._B11)) -
s12_a12 / self._b)
sig12 = sig12 - serr / math.sqrt(1 + self._k2 * Math.sq(ssig2))
ssig12 = math.sin(sig12)
csig12 = math.cos(sig12)
# Update B12 below
# real omg12, lam12, lon12
# real ssig2, csig2, sbet2, cbet2, somg2, comg2, salp2, calp2
# sig2 = sig1 + sig12
ssig2 = self._ssig1 * csig12 + self._csig1 * ssig12
csig2 = self._csig1 * csig12 - self._ssig1 * ssig12
dn2 = math.sqrt(1 + self._k2 * Math.sq(ssig2))
if outmask & (Geodesic.DISTANCE | Geodesic.REDUCEDLENGTH
| Geodesic.GEODESICSCALE):
if arcmode or abs(self.f) > 0.01:
B12 = Geodesic._SinCosSeries(True, ssig2, csig2, self._C1a)
AB1 = (1 + self._A1m1) * (B12 - self._B11)
# sin(bet2) = cos(alp0) * sin(sig2)
sbet2 = self._calp0 * ssig2
# Alt: cbet2 = hypot(csig2, salp0 * ssig2)
cbet2 = math.hypot(self._salp0, self._calp0 * csig2)
if cbet2 == 0:
# I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case
cbet2 = csig2 = Geodesic.tiny_
# tan(alp0) = cos(sig2)*tan(alp2)
salp2 = self._salp0
calp2 = self._calp0 * csig2 # No need to normalize
if outmask & Geodesic.DISTANCE:
s12 = self._b * (
(1 + self._A1m1) * sig12 + AB1) if arcmode else s12_a12
if outmask & Geodesic.LONGITUDE:
# tan(omg2) = sin(alp0) * tan(sig2)
somg2 = self._salp0 * ssig2
comg2 = csig2 # No need to normalize
E = Math.copysign(1, self._salp0) # East or west going?
# omg12 = omg2 - omg1
omg12 = (E *
(sig12 - (math.atan2(ssig2, csig2) -
math.atan2(self._ssig1, self._csig1)) +
(math.atan2(E * somg2, comg2) -
math.atan2(E * self._somg1, self._comg1))) if outmask
& Geodesic.LONG_UNROLL else math.atan2(
somg2 * self._comg1 -
comg2 * self._somg1, comg2 * self._comg1 +
somg2 * self._somg1))
lam12 = omg12 + self._A3c * (sig12 + (Geodesic._SinCosSeries(
True, ssig2, csig2, self._C3a) - self._B31))
lon12 = math.degrees(lam12)
lon2 = (self.lon1 + lon12 if outmask
& Geodesic.LONG_UNROLL else Math.AngNormalize(
Math.AngNormalize(self.lon1) +
Math.AngNormalize(lon12)))
if outmask & Geodesic.LATITUDE:
lat2 = Math.atan2d(sbet2, self._f1 * cbet2)
if outmask & Geodesic.AZIMUTH:
azi2 = Math.atan2d(salp2, calp2)
if outmask & (Geodesic.REDUCEDLENGTH | Geodesic.GEODESICSCALE):
B22 = Geodesic._SinCosSeries(True, ssig2, csig2, self._C2a)
AB2 = (1 + self._A2m1) * (B22 - self._B21)
J12 = (self._A1m1 - self._A2m1) * sig12 + (AB1 - AB2)
if outmask & Geodesic.REDUCEDLENGTH:
# Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure
# accurate cancellation in the case of coincident points.
m12 = self._b * (
(dn2 * (self._csig1 * ssig2) - self._dn1 *
(self._ssig1 * csig2)) - self._csig1 * csig2 * J12)
if outmask & Geodesic.GEODESICSCALE:
t = (self._k2 * (ssig2 - self._ssig1) * (ssig2 + self._ssig1) /
(self._dn1 + dn2))
M12 = csig12 + (t * ssig2 -
csig2 * J12) * self._ssig1 / self._dn1
M21 = csig12 - (t * self._ssig1 -
self._csig1 * J12) * ssig2 / dn2
if outmask & Geodesic.AREA:
B42 = Geodesic._SinCosSeries(False, ssig2, csig2, self._C4a)
# real salp12, calp12
if self._calp0 == 0 or self._salp0 == 0:
# alp12 = alp2 - alp1, used in atan2 so no need to normalize
salp12 = salp2 * self.calp1 - calp2 * self.salp1
calp12 = calp2 * self.calp1 + salp2 * self.salp1
else:
# tan(alp) = tan(alp0) * sec(sig)
# tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1)
# = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2)
# If csig12 > 0, write
# csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
# else
# csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
# No need to normalize
salp12 = self._calp0 * self._salp0 * (
self._csig1 * (1 - csig12) +
ssig12 * self._ssig1 if csig12 <= 0 else ssig12 *
(self._csig1 * ssig12 / (1 + csig12) + self._ssig1))
calp12 = (Math.sq(self._salp0) +
Math.sq(self._calp0) * self._csig1 * csig2)
S12 = (self._c2 * math.atan2(salp12, calp12) + self._A4 *
(B42 - self._B41))
a12 = s12_a12 if arcmode else math.degrees(sig12)
return a12, lat2, lon2, azi2, s12, m12, M12, M21, S12
def CheckAzimuth(azi):
"""Check that azi is legal and return normalized value"""
# if not Math.isfinite(azi):
# raise ValueError("azimuth " + str(azi) + " not a finite number")
return Math.AngNormalize(azi)
def __init__(self, geod, lat1, lon1, azi1, caps = GeodesicCapability.ALL):
"""Construct a GeodesicLine object describing a geodesic line
starting at (lat1, lon1) with azimuth azi1. geod is a Geodesic
object (which embodies the ellipsoid parameters). caps is caps is
an or'ed combination of bit the following values indicating the
capabilities of the returned object
Geodesic.LATITUDE
Geodesic.LONGITUDE
Geodesic.AZIMUTH
Geodesic.DISTANCE
Geodesic.REDUCEDLENGTH
Geodesic.GEODESICSCALE
Geodesic.AREA
Geodesic.DISTANCE_IN
Geodesic.ALL (all of the above)
The default value of caps is ALL.
"""
from geographiclib.geodesic import Geodesic
self._a = geod._a
self._f = geod._f
self._b = geod._b
self._c2 = geod._c2
self._f1 = geod._f1
self._caps = (caps | Geodesic.LATITUDE | Geodesic.AZIMUTH |
Geodesic.LONG_UNROLL)
# Guard against underflow in salp0
self._lat1 = Math.LatFix(lat1)
self._lon1 = lon1
self._azi1 = Math.AngNormalize(azi1)
self._salp1, self._calp1 = Math.sincosd(Math.AngRound(azi1))
# real cbet1, sbet1
sbet1, cbet1 = Math.sincosd(Math.AngRound(lat1)); sbet1 *= self._f1
# Ensure cbet1 = +epsilon at poles
sbet1, cbet1 = Math.norm(sbet1, cbet1); cbet1 = max(Geodesic.tiny_, cbet1)
self._dn1 = math.sqrt(1 + geod._ep2 * Math.sq(sbet1))
# Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0),
self._salp0 = self._salp1 * cbet1 # alp0 in [0, pi/2 - |bet1|]
# Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following
# is slightly better (consider the case salp1 = 0).
self._calp0 = math.hypot(self._calp1, self._salp1 * sbet1)
# Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
# sig = 0 is nearest northward crossing of equator.
# With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
# With bet1 = pi/2, alp1 = -pi, sig1 = pi/2
# With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2
# Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
# With alp0 in (0, pi/2], quadrants for sig and omg coincide.
# No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
# With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi.
self._ssig1 = sbet1; self._somg1 = self._salp0 * sbet1
self._csig1 = self._comg1 = (cbet1 * self._calp1
if sbet1 != 0 or self._calp1 != 0 else 1)
# sig1 in (-pi, pi]
self._ssig1, self._csig1 = Math.norm(self._ssig1, self._csig1)
# No need to normalize
# self._somg1, self._comg1 = Math.norm(self._somg1, self._comg1)
self._k2 = Math.sq(self._calp0) * geod._ep2
eps = self._k2 / (2 * (1 + math.sqrt(1 + self._k2)) + self._k2)
if self._caps & Geodesic.CAP_C1:
self._A1m1 = Geodesic.A1m1f(eps)
self._C1a = list(range(Geodesic.nC1_ + 1))
Geodesic.C1f(eps, self._C1a)
self._B11 = Geodesic.SinCosSeries(
True, self._ssig1, self._csig1, self._C1a)
s = math.sin(self._B11); c = math.cos(self._B11)
# tau1 = sig1 + B11
self._stau1 = self._ssig1 * c + self._csig1 * s
self._ctau1 = self._csig1 * c - self._ssig1 * s
# Not necessary because C1pa reverts C1a
# _B11 = -SinCosSeries(true, _stau1, _ctau1, _C1pa)
if self._caps & Geodesic.CAP_C1p:
self._C1pa = list(range(Geodesic.nC1p_ + 1))
Geodesic.C1pf(eps, self._C1pa)
if self._caps & Geodesic.CAP_C2:
self._A2m1 = Geodesic.A2m1f(eps)
self._C2a = list(range(Geodesic.nC2_ + 1))
Geodesic.C2f(eps, self._C2a)
self._B21 = Geodesic.SinCosSeries(
True, self._ssig1, self._csig1, self._C2a)
if self._caps & Geodesic.CAP_C3:
self._C3a = list(range(Geodesic.nC3_))
geod.C3f(eps, self._C3a)
self._A3c = -self._f * self._salp0 * geod.A3f(eps)
self._B31 = Geodesic.SinCosSeries(
True, self._ssig1, self._csig1, self._C3a)
if self._caps & Geodesic.CAP_C4:
self._C4a = list(range(Geodesic.nC4_))
geod.C4f(eps, self._C4a)
# Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
self._A4 = Math.sq(self._a) * self._calp0 * self._salp0 * geod._e2
self._B41 = Geodesic.SinCosSeries(
False, self._ssig1, self._csig1, self._C4a)
def __init__(self, geod, lat1, lon1, azi1, caps=GeodesicCapability.ALL):
from geographiclib.geodesic import Geodesic
self._a = geod._a
self._f = geod._f
self._b = geod._b
self._c2 = geod._c2
self._f1 = geod._f1
self._caps = caps | Geodesic.LATITUDE | Geodesic.AZIMUTH
# Guard against underflow in salp0
azi1 = Geodesic.AngRound(Math.AngNormalize(azi1))
lon1 = Math.AngNormalize(lon1)
self._lat1 = lat1
self._lon1 = lon1
self._azi1 = azi1
# alp1 is in [0, pi]
alp1 = azi1 * Math.degree
# Enforce sin(pi) == 0 and cos(pi/2) == 0. Better to face the ensuing
# problems directly than to skirt them.
self._salp1 = 0 if azi1 == -180 else math.sin(alp1)
self._calp1 = 0 if abs(azi1) == 90 else math.cos(alp1)
# real cbet1, sbet1, phi
phi = lat1 * Math.degree
# Ensure cbet1 = +epsilon at poles
sbet1 = self._f1 * math.sin(phi)
cbet1 = Geodesic.tiny_ if abs(lat1) == 90 else math.cos(phi)
sbet1, cbet1 = Geodesic.SinCosNorm(sbet1, cbet1)
self._dn1 = math.sqrt(1 + geod._ep2 * Math.sq(sbet1))
# Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0),
self._salp0 = self._salp1 * cbet1 # alp0 in [0, pi/2 - |bet1|]
# Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following
# is slightly better (consider the case salp1 = 0).
self._calp0 = math.hypot(self._calp1, self._salp1 * sbet1)
# Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
# sig = 0 is nearest northward crossing of equator.
# With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
# With bet1 = pi/2, alp1 = -pi, sig1 = pi/2
# With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2
# Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
# With alp0 in (0, pi/2], quadrants for sig and omg coincide.
# No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
# With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi.
self._ssig1 = sbet1
self._somg1 = self._salp0 * sbet1
self._csig1 = self._comg1 = (cbet1 * self._calp1
if sbet1 != 0 or self._calp1 != 0 else 1)
# sig1 in (-pi, pi]
self._ssig1, self._csig1 = Geodesic.SinCosNorm(self._ssig1,
self._csig1)
# No need to normalize
# self._somg1, self._comg1 = Geodesic.SinCosNorm(self._somg1, self._comg1)
self._k2 = Math.sq(self._calp0) * geod._ep2
eps = self._k2 / (2 * (1 + math.sqrt(1 + self._k2)) + self._k2)
if self._caps & Geodesic.CAP_C1:
self._A1m1 = Geodesic.A1m1f(eps)
self._C1a = range(Geodesic.nC1_ + 1)
Geodesic.C1f(eps, self._C1a)
self._B11 = Geodesic.SinCosSeries(True, self._ssig1, self._csig1,
self._C1a, Geodesic.nC1_)
s = math.sin(self._B11)
c = math.cos(self._B11)
# tau1 = sig1 + B11
self._stau1 = self._ssig1 * c + self._csig1 * s
self._ctau1 = self._csig1 * c - self._ssig1 * s
# Not necessary because C1pa reverts C1a
# _B11 = -SinCosSeries(true, _stau1, _ctau1, _C1pa, nC1p_)
if self._caps & Geodesic.CAP_C1p:
self._C1pa = range(Geodesic.nC1p_ + 1)
Geodesic.C1pf(eps, self._C1pa)
if self._caps & Geodesic.CAP_C2:
self._A2m1 = Geodesic.A2m1f(eps)
self._C2a = range(Geodesic.nC2_ + 1)
Geodesic.C2f(eps, self._C2a)
self._B21 = Geodesic.SinCosSeries(True, self._ssig1, self._csig1,
self._C2a, Geodesic.nC2_)
if self._caps & Geodesic.CAP_C3:
self._C3a = range(Geodesic.nC3_)
geod.C3f(eps, self._C3a)
self._A3c = -self._f * self._salp0 * geod.A3f(eps)
self._B31 = Geodesic.SinCosSeries(True, self._ssig1, self._csig1,
self._C3a, Geodesic.nC3_ - 1)
if self._caps & Geodesic.CAP_C4:
self._C4a = range(Geodesic.nC4_)
geod.C4f(eps, self._C4a)
# Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
self._A4 = Math.sq(self._a) * self._calp0 * self._salp0 * geod._e2
self._B41 = Geodesic.SinCosSeries(False, self._ssig1, self._csig1,
self._C4a, Geodesic.nC4_)
def CheckPosition(lat, lon):
if (abs(lat) > 90):
raise ValueError("latitude " + str(lat) + " not in [-90, 90]")
if (lon < -540 or lon >= 540):
raise ValueError("longitude " + str(lon) + " not in [-540, 540)")
return Math.AngNormalize(lon)
def __init__(self,
geod,
lat1,
lon1,
azi1,
caps=GeodesicCapability.STANDARD
| GeodesicCapability.DISTANCE_IN,
salp1=Math.nan,
calp1=Math.nan):
"""Construct a GeodesicLine object
:param geod: a :class:`~geographiclib.geodesic.Geodesic` object
:param lat1: latitude of the first point in degrees
:param lon1: longitude of the first point in degrees
:param azi1: azimuth at the first point in degrees
:param caps: the :ref:`capabilities <outmask>`
This creates an object allowing points along a geodesic starting at
(*lat1*, *lon1*), with azimuth *azi1* to be found. The default
value of *caps* is STANDARD | DISTANCE_IN. The optional parameters
*salp1* and *calp1* should not be supplied; they are part of the
private interface.
"""
from geographiclib.geodesic import Geodesic
self.a = geod.a
"""The equatorial radius in meters (readonly)"""
self.f = geod.f
"""The flattening (readonly)"""
self._b = geod._b
self._c2 = geod._c2
self._f1 = geod._f1
self.caps = (caps | Geodesic.LATITUDE | Geodesic.AZIMUTH
| Geodesic.LONG_UNROLL)
"""the capabilities (readonly)"""
# Guard against underflow in salp0
self.lat1 = Math.LatFix(lat1)
"""the latitude of the first point in degrees (readonly)"""
self.lon1 = lon1
"""the longitude of the first point in degrees (readonly)"""
if Math.isnan(salp1) or Math.isnan(calp1):
self.azi1 = Math.AngNormalize(azi1)
self.salp1, self.calp1 = Math.sincosd(Math.AngRound(azi1))
else:
self.azi1 = azi1
"""the azimuth at the first point in degrees (readonly)"""
self.salp1 = salp1
"""the sine of the azimuth at the first point (readonly)"""
self.calp1 = calp1
"""the cosine of the azimuth at the first point (readonly)"""
# real cbet1, sbet1
sbet1, cbet1 = Math.sincosd(Math.AngRound(lat1))
sbet1 *= self._f1
# Ensure cbet1 = +epsilon at poles
sbet1, cbet1 = Math.norm(sbet1, cbet1)
cbet1 = max(Geodesic.tiny_, cbet1)
self._dn1 = math.sqrt(1 + geod._ep2 * Math.sq(sbet1))
# Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0),
self._salp0 = self.salp1 * cbet1 # alp0 in [0, pi/2 - |bet1|]
# Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following
# is slightly better (consider the case salp1 = 0).
self._calp0 = math.hypot(self.calp1, self.salp1 * sbet1)
# Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
# sig = 0 is nearest northward crossing of equator.
# With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
# With bet1 = pi/2, alp1 = -pi, sig1 = pi/2
# With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2
# Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
# With alp0 in (0, pi/2], quadrants for sig and omg coincide.
# No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
# With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi.
self._ssig1 = sbet1
self._somg1 = self._salp0 * sbet1
self._csig1 = self._comg1 = (cbet1 * self.calp1
if sbet1 != 0 or self.calp1 != 0 else 1)
# sig1 in (-pi, pi]
self._ssig1, self._csig1 = Math.norm(self._ssig1, self._csig1)
# No need to normalize
# self._somg1, self._comg1 = Math.norm(self._somg1, self._comg1)
self._k2 = Math.sq(self._calp0) * geod._ep2
eps = self._k2 / (2 * (1 + math.sqrt(1 + self._k2)) + self._k2)
if self.caps & Geodesic.CAP_C1:
self._A1m1 = Geodesic._A1m1f(eps)
self._C1a = list(range(Geodesic.nC1_ + 1))
Geodesic._C1f(eps, self._C1a)
self._B11 = Geodesic._SinCosSeries(True, self._ssig1, self._csig1,
self._C1a)
s = math.sin(self._B11)
c = math.cos(self._B11)
# tau1 = sig1 + B11
self._stau1 = self._ssig1 * c + self._csig1 * s
self._ctau1 = self._csig1 * c - self._ssig1 * s
# Not necessary because C1pa reverts C1a
# _B11 = -_SinCosSeries(true, _stau1, _ctau1, _C1pa)
if self.caps & Geodesic.CAP_C1p:
self._C1pa = list(range(Geodesic.nC1p_ + 1))
Geodesic._C1pf(eps, self._C1pa)
if self.caps & Geodesic.CAP_C2:
self._A2m1 = Geodesic._A2m1f(eps)
self._C2a = list(range(Geodesic.nC2_ + 1))
Geodesic._C2f(eps, self._C2a)
self._B21 = Geodesic._SinCosSeries(True, self._ssig1, self._csig1,
self._C2a)
if self.caps & Geodesic.CAP_C3:
self._C3a = list(range(Geodesic.nC3_))
geod._C3f(eps, self._C3a)
self._A3c = -self.f * self._salp0 * geod._A3f(eps)
self._B31 = Geodesic._SinCosSeries(True, self._ssig1, self._csig1,
self._C3a)
if self.caps & Geodesic.CAP_C4:
self._C4a = list(range(Geodesic.nC4_))
geod._C4f(eps, self._C4a)
# Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
self._A4 = Math.sq(self.a) * self._calp0 * self._salp0 * geod._e2
self._B41 = Geodesic._SinCosSeries(False, self._ssig1, self._csig1,
self._C4a)
self.s13 = Math.nan
"""the distance between point 1 and point 3 in meters (readonly)"""
self.a13 = Math.nan
"""the arc length between point 1 and point 3 in degrees (readonly)"""
def CheckAzimuth(azi):
if (azi < -540 or azi >= 540):
raise ValueError("azimuth " + str(azi) + " not in [-540, 540)")
return Math.AngNormalize(azi)
def CheckAzimuth(azi):
"""Check that azi is legal and return normalized value"""
if (azi < -540 or azi >= 540):
raise ValueError("azimuth " + str(azi) + " not in [-540, 540)")
return Math.AngNormalize(azi)
def GenInverse(self, lat1, lon1, lat2, lon2, outmask):
"""Private: General version of the inverse problem"""
a12 = s12 = azi1 = azi2 = m12 = M12 = M21 = S12 = Math.nan # return vals
outmask &= Geodesic.OUT_ALL
# Compute longitude difference (AngDiff does this carefully). Result is
# in [-180, 180] but -180 is only for west-going geodesics. 180 is for
# east-going and meridional geodesics.
lon12 = Math.AngDiff(Math.AngNormalize(lon1), Math.AngNormalize(lon2))
# If very close to being on the same half-meridian, then make it so.
lon12 = Geodesic.AngRound(lon12)
# Make longitude difference positive.
lonsign = 1 if lon12 >= 0 else -1
lon12 *= lonsign
# If really close to the equator, treat as on equator.
lat1 = Geodesic.AngRound(lat1)
lat2 = Geodesic.AngRound(lat2)
# Swap points so that point with higher (abs) latitude is point 1
swapp = 1 if abs(lat1) >= abs(lat2) else -1
if swapp < 0:
lonsign *= -1
lat2, lat1 = lat1, lat2
# Make lat1 <= 0
latsign = 1 if lat1 < 0 else -1
lat1 *= latsign
lat2 *= latsign
# Now we have
#
# 0 <= lon12 <= 180
# -90 <= lat1 <= 0
# lat1 <= lat2 <= -lat1
#
# longsign, swapp, latsign register the transformation to bring the
# coordinates to this canonical form. In all cases, 1 means no change was
# made. We make these transformations so that there are few cases to
# check, e.g., on verifying quadrants in atan2. In addition, this
# enforces some symmetries in the results returned.
# real phi, sbet1, cbet1, sbet2, cbet2, s12x, m12x
phi = lat1 * Math.degree
# Ensure cbet1 = +epsilon at poles
sbet1 = self._f1 * math.sin(phi)
cbet1 = Geodesic.tiny_ if lat1 == -90 else math.cos(phi)
sbet1, cbet1 = Geodesic.SinCosNorm(sbet1, cbet1)
phi = lat2 * Math.degree
# Ensure cbet2 = +epsilon at poles
sbet2 = self._f1 * math.sin(phi)
cbet2 = Geodesic.tiny_ if abs(lat2) == 90 else math.cos(phi)
sbet2, cbet2 = Geodesic.SinCosNorm(sbet2, cbet2)
# If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the
# |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is
# a better measure. This logic is used in assigning calp2 in Lambda12.
# Sometimes these quantities vanish and in that case we force bet2 = +/-
# bet1 exactly. An example where is is necessary is the inverse problem
# 48.522876735459 0 -48.52287673545898293 179.599720456223079643
# which failed with Visual Studio 10 (Release and Debug)
if cbet1 < -sbet1:
if cbet2 == cbet1:
sbet2 = sbet1 if sbet2 < 0 else -sbet1
else:
if abs(sbet2) == -sbet1:
cbet2 = cbet1
dn1 = math.sqrt(1 + self._ep2 * Math.sq(sbet1))
dn2 = math.sqrt(1 + self._ep2 * Math.sq(sbet2))
lam12 = lon12 * Math.degree
slam12 = 0 if lon12 == 180 else math.sin(lam12)
clam12 = math.cos(lam12) # lon12 == 90 isn't interesting
# real a12, sig12, calp1, salp1, calp2, salp2
# index zero elements of these arrays are unused
C1a = list(range(Geodesic.nC1_ + 1))
C2a = list(range(Geodesic.nC2_ + 1))
C3a = list(range(Geodesic.nC3_))
meridian = lat1 == -90 or slam12 == 0
if meridian:
# Endpoints are on a single full meridian, so the geodesic might lie on
# a meridian.
calp1 = clam12
salp1 = slam12 # Head to the target longitude
calp2 = 1
salp2 = 0 # At the target we're heading north
# tan(bet) = tan(sig) * cos(alp)
ssig1 = sbet1
csig1 = calp1 * cbet1
ssig2 = sbet2
csig2 = calp2 * cbet2
# sig12 = sig2 - sig1
sig12 = math.atan2(max(csig1 * ssig2 - ssig1 * csig2, 0.0),
csig1 * csig2 + ssig1 * ssig2)
s12x, m12x, dummy, M12, M21 = self.Lengths(
self._n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1,
cbet2, (outmask & Geodesic.GEODESICSCALE) != 0, C1a, C2a)
# Add the check for sig12 since zero length geodesics might yield m12 <
# 0. Test case was
#
# echo 20.001 0 20.001 0 | Geod -i
#
# In fact, we will have sig12 > pi/2 for meridional geodesic which is
# not a shortest path.
if sig12 < 1 or m12x >= 0:
m12x *= self._b
s12x *= self._b
a12 = sig12 / Math.degree
else:
# m12 < 0, i.e., prolate and too close to anti-podal
meridian = False
# end if meridian:
#real omg12
if (not meridian and sbet1 == 0 and # and sbet2 == 0
# Mimic the way Lambda12 works with calp1 = 0
(self._f <= 0 or lam12 <= math.pi - self._f * math.pi)):
# Geodesic runs along equator
calp1 = calp2 = 0
salp1 = salp2 = 1
s12x = self._a * lam12
sig12 = omg12 = lam12 / self._f1
m12x = self._b * math.sin(sig12)
if outmask & Geodesic.GEODESICSCALE:
M12 = M21 = math.cos(sig12)
a12 = lon12 / self._f1
elif not meridian:
# Now point1 and point2 belong within a hemisphere bounded by a
# meridian and geodesic is neither meridional or equatorial.
# Figure a starting point for Newton's method
sig12, salp1, calp1, salp2, calp2, dnm = self.InverseStart(
sbet1, cbet1, dn1, sbet2, cbet2, dn2, lam12, C1a, C2a)
if sig12 >= 0:
# Short lines (InverseStart sets salp2, calp2, dnm)
s12x = sig12 * self._b * dnm
m12x = (Math.sq(dnm) * self._b * math.sin(sig12 / dnm))
if outmask & Geodesic.GEODESICSCALE:
M12 = M21 = math.cos(sig12 / dnm)
a12 = sig12 / Math.degree
omg12 = lam12 / (self._f1 * dnm)
else:
# Newton's method. This is a straightforward solution of f(alp1) =
# lambda12(alp1) - lam12 = 0 with one wrinkle. f(alp) has exactly one
# root in the interval (0, pi) and its derivative is positive at the
# root. Thus f(alp) is positive for alp > alp1 and negative for alp <
# alp1. During the course of the iteration, a range (alp1a, alp1b) is
# maintained which brackets the root and with each evaluation of f(alp)
# the range is shrunk if possible. Newton's method is restarted
# whenever the derivative of f is negative (because the new value of
# alp1 is then further from the solution) or if the new estimate of
# alp1 lies outside (0,pi); in this case, the new starting guess is
# taken to be (alp1a + alp1b) / 2.
# real ssig1, csig1, ssig2, csig2, eps
numit = 0
tripn = tripb = False
# Bracketing range
salp1a = Geodesic.tiny_
calp1a = 1
salp1b = Geodesic.tiny_
calp1b = -1
while numit < Geodesic.maxit2_:
# the WGS84 test set: mean = 1.47, sd = 1.25, max = 16
# WGS84 and random input: mean = 2.85, sd = 0.60
(nlam12, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
eps, omg12, dv) = self.Lambda12(sbet1, cbet1, dn1, sbet2,
cbet2, dn2, salp1, calp1,
numit < Geodesic.maxit1_,
C1a, C2a, C3a)
v = nlam12 - lam12
# 2 * tol0 is approximately 1 ulp for a number in [0, pi].
# Reversed test to allow escape with NaNs
if tripb or not (abs(v) >=
(8 if tripn else 2) * Geodesic.tol0_):
break
# Update bracketing values
if v > 0 and (numit > Geodesic.maxit1_
or calp1 / salp1 > calp1b / salp1b):
salp1b = salp1
calp1b = calp1
elif v < 0 and (numit > Geodesic.maxit1_
or calp1 / salp1 < calp1a / salp1a):
salp1a = salp1
calp1a = calp1
numit += 1
if numit < Geodesic.maxit1_ and dv > 0:
dalp1 = -v / dv
sdalp1 = math.sin(dalp1)
cdalp1 = math.cos(dalp1)
nsalp1 = salp1 * cdalp1 + calp1 * sdalp1
if nsalp1 > 0 and abs(dalp1) < math.pi:
calp1 = calp1 * cdalp1 - salp1 * sdalp1
salp1 = nsalp1
salp1, calp1 = Geodesic.SinCosNorm(salp1, calp1)
# In some regimes we don't get quadratic convergence because
# slope -> 0. So use convergence conditions based on epsilon
# instead of sqrt(epsilon).
tripn = abs(v) <= 16 * Geodesic.tol0_
continue
# Either dv was not postive or updated value was outside legal range.
# Use the midpoint of the bracket as the next estimate. This
# mechanism is not needed for the WGS84 ellipsoid, but it does catch
# problems with more eccentric ellipsoids. Its efficacy is such for
# the WGS84 test set with the starting guess set to alp1 = 90deg:
# the WGS84 test set: mean = 5.21, sd = 3.93, max = 24
# WGS84 and random input: mean = 4.74, sd = 0.99
salp1 = (salp1a + salp1b) / 2
calp1 = (calp1a + calp1b) / 2
salp1, calp1 = Geodesic.SinCosNorm(salp1, calp1)
tripn = False
tripb = (abs(salp1a - salp1) +
(calp1a - calp1) < Geodesic.tolb_
or abs(salp1 - salp1b) +
(calp1 - calp1b) < Geodesic.tolb_)
s12x, m12x, dummy, M12, M21 = self.Lengths(
eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1,
cbet2, (outmask & Geodesic.GEODESICSCALE) != 0, C1a, C2a)
m12x *= self._b
s12x *= self._b
a12 = sig12 / Math.degree
omg12 = lam12 - omg12
# end elif not meridian
if outmask & Geodesic.DISTANCE:
s12 = 0 + s12x # Convert -0 to 0
if outmask & Geodesic.REDUCEDLENGTH:
m12 = 0 + m12x # Convert -0 to 0
if outmask & Geodesic.AREA:
# From Lambda12: sin(alp1) * cos(bet1) = sin(alp0)
salp0 = salp1 * cbet1
calp0 = math.hypot(calp1, salp1 * sbet1) # calp0 > 0
# real alp12
if calp0 != 0 and salp0 != 0:
# From Lambda12: tan(bet) = tan(sig) * cos(alp)
ssig1 = sbet1
csig1 = calp1 * cbet1
ssig2 = sbet2
csig2 = calp2 * cbet2
k2 = Math.sq(calp0) * self._ep2
eps = k2 / (2 * (1 + math.sqrt(1 + k2)) + k2)
# Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0).
A4 = Math.sq(self._a) * calp0 * salp0 * self._e2
ssig1, csig1 = Geodesic.SinCosNorm(ssig1, csig1)
ssig2, csig2 = Geodesic.SinCosNorm(ssig2, csig2)
C4a = list(range(Geodesic.nC4_))
self.C4f(eps, C4a)
B41 = Geodesic.SinCosSeries(False, ssig1, csig1, C4a,
Geodesic.nC4_)
B42 = Geodesic.SinCosSeries(False, ssig2, csig2, C4a,
Geodesic.nC4_)
S12 = A4 * (B42 - B41)
else:
# Avoid problems with indeterminate sig1, sig2 on equator
S12 = 0
if (not meridian and omg12 < 0.75 * math.pi
and # Long difference too big
sbet2 - sbet1 < 1.75): # Lat difference too big
# Use tan(Gamma/2) = tan(omg12/2)
# * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2))
# with tan(x/2) = sin(x)/(1+cos(x))
somg12 = math.sin(omg12)
domg12 = 1 + math.cos(omg12)
dbet1 = 1 + cbet1
dbet2 = 1 + cbet2
alp12 = 2 * math.atan2(
somg12 * (sbet1 * dbet2 + sbet2 * dbet1),
domg12 * (sbet1 * sbet2 + dbet1 * dbet2))
else:
# alp12 = alp2 - alp1, used in atan2 so no need to normalize
salp12 = salp2 * calp1 - calp2 * salp1
calp12 = calp2 * calp1 + salp2 * salp1
# The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
# salp12 = -0 and alp12 = -180. However this depends on the sign
# being attached to 0 correctly. The following ensures the correct
# behavior.
if salp12 == 0 and calp12 < 0:
salp12 = Geodesic.tiny_ * calp1
calp12 = -1
alp12 = math.atan2(salp12, calp12)
S12 += self._c2 * alp12
S12 *= swapp * lonsign * latsign
# Convert -0 to 0
S12 += 0
# Convert calp, salp to azimuth accounting for lonsign, swapp, latsign.
if swapp < 0:
salp2, salp1 = salp1, salp2
calp2, calp1 = calp1, calp2
if outmask & Geodesic.GEODESICSCALE:
M21, M12 = M12, M21
salp1 *= swapp * lonsign
calp1 *= swapp * latsign
salp2 *= swapp * lonsign
calp2 *= swapp * latsign
if outmask & Geodesic.AZIMUTH:
# minus signs give range [-180, 180). 0- converts -0 to +0.
azi1 = 0 - math.atan2(-salp1, calp1) / Math.degree
azi2 = 0 - math.atan2(-salp2, calp2) / Math.degree
# Returned value in [0, 180]
return a12, s12, azi1, azi2, m12, M12, M21, S12
