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 __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 __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)
