def Inverse(self, lat1, lon1, lat2, lon2, outmask = STANDARD):
"""Solve the inverse geodesic problem. Compute geodesic between (lat1,
lon1) and (lat2, lon2). 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 mask
is an or'ed combination of the following values
Geodesic.AZIMUTH
Geodesic.DISTANCE
Geodesic.STANDARD (all of the above)
Geodesic.REDUCEDLENGTH
Geodesic.GEODESICSCALE
Geodesic.AREA
Geodesic.ALL (all of the above)
Geodesic.LONG_UNROLL
If Geodesic.LONG_UNROLL is set, then lon1 is unchanged and lon2 -
lon1 indicates whether the geodesic is east going or west going.
Otherwise lon1 and lon2 are both reduced to the range [-180,180).
The default value of outmask is STANDARD.
"""
lon1a = Geodesic.CheckPosition(lat1, lon1)
lon2a = Geodesic.CheckPosition(lat2, lon2)
if outmask & Geodesic.LONG_UNROLL:
lon2 = lon1 + Math.AngDiff(lon1a, lon2a)
else:
lon1 = lon1a; lon2 = lon2a
result = {'lat1': lat1, 'lon1': lon1, 'lat2': lat2, 'lon2': lon2}
a12, s12, azi1, azi2, m12, M12, M21, S12 = self.GenInverse(
lat1, lon1a, lat2, lon2a, outmask)
outmask &= Geodesic.OUT_MASK
result['a12'] = a12
if outmask & Geodesic.DISTANCE: result['s12'] = s12
if outmask & Geodesic.AZIMUTH:
result['azi1'] = azi1; 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 _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 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 _GenInverse(self, lat1, lon1, lat2, lon2, outmask):
"""Private: General version of the inverse problem"""
a12 = s12 = m12 = M12 = M21 = S12 = Math.nan # return vals
outmask &= Geodesic.OUT_MASK
# 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, lon12s = Math.AngDiff(lon1, lon2)
# Make longitude difference positive.
lonsign = 1 if lon12 >= 0 else -1
# If very close to being on the same half-meridian, then make it so.
lon12 = lonsign * Math.AngRound(lon12)
lon12s = Math.AngRound((180 - lon12) - lonsign * lon12s)
lam12 = math.radians(lon12)
if lon12 > 90:
slam12, clam12 = Math.sincosd(lon12s); clam12 = -clam12
else:
slam12, clam12 = Math.sincosd(lon12)
# If really close to the equator, treat as on equator.
lat1 = Math.AngRound(Math.LatFix(lat1))
lat2 = Math.AngRound(Math.LatFix(lat2))
# Swap points so that point with higher (abs) latitude is point 1
# If one latitude is a nan, then it becomes lat1.
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
sbet1, cbet1 = Math.sincosd(lat1); sbet1 *= self._f1
# Ensure cbet1 = +epsilon at poles
sbet1, cbet1 = Math.norm(sbet1, cbet1); cbet1 = max(Geodesic.tiny_, cbet1)
sbet2, cbet2 = Math.sincosd(lat2); sbet2 *= self._f1
# Ensure cbet2 = +epsilon at poles
sbet2, cbet2 = Math.norm(sbet2, cbet2); cbet2 = max(Geodesic.tiny_, 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))
# 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 = 1.0; salp2 = 0.0 # 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(0.0, csig1 * ssig2 - ssig1 * csig2),
csig1 * csig2 + ssig1 * ssig2)
s12x, m12x, dummy, M12, M21 = self._Lengths(
self._n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2,
outmask | Geodesic.DISTANCE | Geodesic.REDUCEDLENGTH, 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 | GeodSolve -i
#
# In fact, we will have sig12 > pi/2 for meridional geodesic which is
# not a shortest path.
if sig12 < 1 or m12x >= 0:
if sig12 < 3 * Geodesic.tiny_:
sig12 = m12x = s12x = 0.0
m12x *= self._b
s12x *= self._b
a12 = math.degrees(sig12)
else:
# m12 < 0, i.e., prolate and too close to anti-podal
meridian = False
# end if meridian:
# somg12 > 1 marks that it needs to be calculated
somg12 = 2.0; comg12 = 0.0; omg12 = 0.0
if (not meridian and
sbet1 == 0 and # and sbet2 == 0
# Mimic the way Lambda12 works with calp1 = 0
(self.f <= 0 or lon12s >= self.f * 180)):
# Geodesic runs along equator
calp1 = calp2 = 0.0; salp1 = salp2 = 1.0
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, slam12, clam12, 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 = math.degrees(sig12)
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 = 1.0
salp1b = Geodesic.tiny_; calp1b = -1.0
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
(v, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
eps, domg12, dv) = self._Lambda12(
sbet1, cbet1, dn1, sbet2, cbet2, dn2,
salp1, calp1, slam12, clam12, numit < Geodesic.maxit1_,
C1a, C2a, C3a)
# 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 1) * 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 = Math.norm(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 positive 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 = Math.norm(salp1, calp1)
tripn = False
tripb = (abs(salp1a - salp1) + (calp1a - calp1) < Geodesic.tolb_ or
abs(salp1 - salp1b) + (calp1 - calp1b) < Geodesic.tolb_)
lengthmask = (outmask |
(Geodesic.DISTANCE
if (outmask & (Geodesic.REDUCEDLENGTH |
Geodesic.GEODESICSCALE))
else Geodesic.EMPTY))
s12x, m12x, dummy, M12, M21 = self._Lengths(
eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2,
lengthmask, C1a, C2a)
m12x *= self._b
s12x *= self._b
a12 = math.degrees(sig12)
if outmask & Geodesic.AREA:
# omg12 = lam12 - domg12
sdomg12 = math.sin(domg12); cdomg12 = math.cos(domg12)
somg12 = slam12 * cdomg12 - clam12 * sdomg12
comg12 = clam12 * cdomg12 + slam12 * sdomg12
# end elif not meridian
if outmask & Geodesic.DISTANCE:
s12 = 0.0 + s12x # Convert -0 to 0
if outmask & Geodesic.REDUCEDLENGTH:
m12 = 0.0 + 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 = Math.norm(ssig1, csig1)
ssig2, csig2 = Math.norm(ssig2, csig2)
C4a = list(range(Geodesic.nC4_))
self._C4f(eps, C4a)
B41 = Geodesic._SinCosSeries(False, ssig1, csig1, C4a)
B42 = Geodesic._SinCosSeries(False, ssig2, csig2, C4a)
S12 = A4 * (B42 - B41)
else:
# Avoid problems with indeterminate sig1, sig2 on equator
S12 = 0.0
if not meridian and somg12 > 1:
somg12 = math.sin(omg12); comg12 = math.cos(omg12)
if (not meridian and
# omg12 < 3/4 * pi
comg12 > -0.7071 and # Long difference not too big
sbet2 - sbet1 < 1.75): # Lat difference not 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))
domg12 = 1 + comg12; 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 = -1.0
alp12 = math.atan2(salp12, calp12)
S12 += self._c2 * alp12
S12 *= swapp * lonsign * latsign
# Convert -0 to 0
S12 += 0.0
# 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
return a12, s12, salp1, calp1, salp2, calp2, m12, M12, M21, S12