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 _Lambda12(self, sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
slam120, clam120, diffp,
# Scratch areas of the right size
C1a, C2a, C3a):
"""Private: Solve hybrid problem"""
if sbet1 == 0 and calp1 == 0:
# Break degeneracy of equatorial line. This case has already been
# handled.
calp1 = -Geodesic.tiny_
# sin(alp1) * cos(bet1) = sin(alp0)
salp0 = salp1 * cbet1
calp0 = math.hypot(calp1, salp1 * sbet1) # calp0 > 0
# real somg1, comg1, somg2, comg2, lam12
# tan(bet1) = tan(sig1) * cos(alp1)
# tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1)
ssig1 = sbet1; somg1 = salp0 * sbet1
csig1 = comg1 = calp1 * cbet1
ssig1, csig1 = Math.norm(ssig1, csig1)
# Math.norm(somg1, comg1); -- don't need to normalize!
# Enforce symmetries in the case abs(bet2) = -bet1. Need to be careful
# about this case, since this can yield singularities in the Newton
# iteration.
# sin(alp2) * cos(bet2) = sin(alp0)
salp2 = salp0 / cbet2 if cbet2 != cbet1 else salp1
# calp2 = sqrt(1 - sq(salp2))
# = sqrt(sq(calp0) - sq(sbet2)) / cbet2
# and subst for calp0 and rearrange to give (choose positive sqrt
# to give alp2 in [0, pi/2]).
calp2 = (math.sqrt(Math.sq(calp1 * cbet1) +
((cbet2 - cbet1) * (cbet1 + cbet2) if cbet1 < -sbet1
else (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2
if cbet2 != cbet1 or abs(sbet2) != -sbet1 else abs(calp1))
# tan(bet2) = tan(sig2) * cos(alp2)
# tan(omg2) = sin(alp0) * tan(sig2).
ssig2 = sbet2; somg2 = salp0 * sbet2
csig2 = comg2 = calp2 * cbet2
ssig2, csig2 = Math.norm(ssig2, csig2)
# Math.norm(somg2, comg2); -- don't need to normalize!
# sig12 = sig2 - sig1, limit to [0, pi]
sig12 = math.atan2(max(0.0, csig1 * ssig2 - ssig1 * csig2),
csig1 * csig2 + ssig1 * ssig2)
# omg12 = omg2 - omg1, limit to [0, pi]
somg12 = max(0.0, comg1 * somg2 - somg1 * comg2)
comg12 = comg1 * comg2 + somg1 * somg2
# eta = omg12 - lam120
eta = math.atan2(somg12 * clam120 - comg12 * slam120,
comg12 * clam120 + somg12 * slam120)
# real B312
k2 = Math.sq(calp0) * self._ep2
eps = k2 / (2 * (1 + math.sqrt(1 + k2)) + k2)
self._C3f(eps, C3a)
B312 = (Geodesic._SinCosSeries(True, ssig2, csig2, C3a) -
Geodesic._SinCosSeries(True, ssig1, csig1, C3a))
domg12 = -self.f * self._A3f(eps) * salp0 * (sig12 + B312)
lam12 = eta + domg12
if diffp:
if calp2 == 0:
dlam12 = - 2 * self._f1 * dn1 / sbet1
else:
dummy, dlam12, dummy, dummy, dummy = self._Lengths(
eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2,
Geodesic.REDUCEDLENGTH, C1a, C2a)
dlam12 *= self._f1 / (calp2 * cbet2)
else:
dlam12 = Math.nan
return (lam12, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2, eps,
domg12, dlam12)
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
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 _InverseStart(self, sbet1, cbet1, dn1, sbet2, cbet2, dn2,
lam12, slam12, clam12,
# Scratch areas of the right size
C1a, C2a):
"""Private: Find a starting value for Newton's method."""
# Return a starting point for Newton's method in salp1 and calp1 (function
# value is -1). If Newton's method doesn't need to be used, return also
# salp2 and calp2 and function value is sig12.
sig12 = -1; salp2 = calp2 = dnm = Math.nan # Return values
# bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0]
sbet12 = sbet2 * cbet1 - cbet2 * sbet1
cbet12 = cbet2 * cbet1 + sbet2 * sbet1
# Volatile declaration needed to fix inverse cases
# 88.202499451857 0 -88.202499451857 179.981022032992859592
# 89.262080389218 0 -89.262080389218 179.992207982775375662
# 89.333123580033 0 -89.333123580032997687 179.99295812360148422
# which otherwise fail with g++ 4.4.4 x86 -O3
sbet12a = sbet2 * cbet1
sbet12a += cbet2 * sbet1
shortline = cbet12 >= 0 and sbet12 < 0.5 and cbet2 * lam12 < 0.5
if shortline:
sbetm2 = Math.sq(sbet1 + sbet2)
# sin((bet1+bet2)/2)^2
# = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2)
sbetm2 /= sbetm2 + Math.sq(cbet1 + cbet2)
dnm = math.sqrt(1 + self._ep2 * sbetm2)
omg12 = lam12 / (self._f1 * dnm)
somg12 = math.sin(omg12); comg12 = math.cos(omg12)
else:
somg12 = slam12; comg12 = clam12
salp1 = cbet2 * somg12
calp1 = (
sbet12 + cbet2 * sbet1 * Math.sq(somg12) / (1 + comg12) if comg12 >= 0
else sbet12a - cbet2 * sbet1 * Math.sq(somg12) / (1 - comg12))
ssig12 = math.hypot(salp1, calp1)
csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12
if shortline and ssig12 < self._etol2:
# really short lines
salp2 = cbet1 * somg12
calp2 = sbet12 - cbet1 * sbet2 * (Math.sq(somg12) / (1 + comg12)
if comg12 >= 0 else 1 - comg12)
salp2, calp2 = Math.norm(salp2, calp2)
# Set return value
sig12 = math.atan2(ssig12, csig12)
elif (abs(self._n) >= 0.1 or # Skip astroid calc if too eccentric
csig12 >= 0 or
ssig12 >= 6 * abs(self._n) * math.pi * Math.sq(cbet1)):
# Nothing to do, zeroth order spherical approximation is OK
pass
else:
# Scale lam12 and bet2 to x, y coordinate system where antipodal point
# is at origin and singular point is at y = 0, x = -1.
# real y, lamscale, betscale
# Volatile declaration needed to fix inverse case
# 56.320923501171 0 -56.320923501171 179.664747671772880215
# which otherwise fails with g++ 4.4.4 x86 -O3
# volatile real x
lam12x = math.atan2(-slam12, -clam12)
if self.f >= 0: # In fact f == 0 does not get here
# x = dlong, y = dlat
k2 = Math.sq(sbet1) * self._ep2
eps = k2 / (2 * (1 + math.sqrt(1 + k2)) + k2)
lamscale = self.f * cbet1 * self._A3f(eps) * math.pi
betscale = lamscale * cbet1
x = lam12x / lamscale
y = sbet12a / betscale
else: # _f < 0
# x = dlat, y = dlong
cbet12a = cbet2 * cbet1 - sbet2 * sbet1
bet12a = math.atan2(sbet12a, cbet12a)
# real m12b, m0, dummy
# In the case of lon12 = 180, this repeats a calculation made in
# Inverse.
dummy, m12b, m0, dummy, dummy = self._Lengths(
self._n, math.pi + bet12a, sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
cbet1, cbet2, Geodesic.REDUCEDLENGTH, C1a, C2a)
x = -1 + m12b / (cbet1 * cbet2 * m0 * math.pi)
betscale = (sbet12a / x if x < -0.01
else -self.f * Math.sq(cbet1) * math.pi)
lamscale = betscale / cbet1
y = lam12x / lamscale
if y > -Geodesic.tol1_ and x > -1 - Geodesic.xthresh_:
# strip near cut
if self.f >= 0:
salp1 = min(1.0, -x); calp1 = - math.sqrt(1 - Math.sq(salp1))
else:
calp1 = max((0.0 if x > -Geodesic.tol1_ else -1.0), x)
salp1 = math.sqrt(1 - Math.sq(calp1))
else:
# Estimate alp1, by solving the astroid problem.
#
# Could estimate alpha1 = theta + pi/2, directly, i.e.,
# calp1 = y/k; salp1 = -x/(1+k); for _f >= 0
# calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check)
#
# However, it's better to estimate omg12 from astroid and use
# spherical formula to compute alp1. This reduces the mean number of
# Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12
# (min 0 max 5). The changes in the number of iterations are as
# follows:
#
# change percent
# 1 5
# 0 78
# -1 16
# -2 0.6
# -3 0.04
# -4 0.002
#
# The histogram of iterations is (m = number of iterations estimating
# alp1 directly, n = number of iterations estimating via omg12, total
# number of trials = 148605):
#
# iter m n
# 0 148 186
# 1 13046 13845
# 2 93315 102225
# 3 36189 32341
# 4 5396 7
# 5 455 1
# 6 56 0
#
# Because omg12 is near pi, estimate work with omg12a = pi - omg12
k = Geodesic._Astroid(x, y)
omg12a = lamscale * ( -x * k/(1 + k) if self.f >= 0
else -y * (1 + k)/k )
somg12 = math.sin(omg12a); comg12 = -math.cos(omg12a)
# Update spherical estimate of alp1 using omg12 instead of lam12
salp1 = cbet2 * somg12
calp1 = sbet12a - cbet2 * sbet1 * Math.sq(somg12) / (1 - comg12)
# Sanity check on starting guess. Backwards check allows NaN through.
if not (salp1 <= 0):
salp1, calp1 = Math.norm(salp1, calp1)
else:
salp1 = 1; calp1 = 0
return sig12, salp1, calp1, salp2, calp2, dnm
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 |
Geodesic.LONG_UNROLL)
# Guard against underflow in salp0
azi1 = Math.AngRound(Math.AngNormalize(azi1))
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 = Math.norm(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 = 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 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_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 = Math.AngDiff(Math.AngNormalize(lon1), Math.AngNormalize(lon2))
# If very close to being on the same half-meridian, then make it so.
lon12 = Math.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 = Math.AngRound(lat1)
lat2 = Math.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 = Math.norm(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 = Math.norm(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 | 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:
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 = 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 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 = Math.norm(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 = 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
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
def Lambda12(self, sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1, diffp,
# Scratch areas of the right size
C1a, C2a, C3a):
"""Private: Solve hybrid problem"""
if sbet1 == 0 and calp1 == 0:
# Break degeneracy of equatorial line. This case has already been
# handled.
calp1 = -Geodesic.tiny_
# sin(alp1) * cos(bet1) = sin(alp0)
salp0 = salp1 * cbet1
calp0 = math.hypot(calp1, salp1 * sbet1) # calp0 > 0
# real somg1, comg1, somg2, comg2, omg12, lam12
# tan(bet1) = tan(sig1) * cos(alp1)
# tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1)
ssig1 = sbet1; somg1 = salp0 * sbet1
csig1 = comg1 = calp1 * cbet1
ssig1, csig1 = Math.norm(ssig1, csig1)
# Math.norm(somg1, comg1); -- don't need to normalize!
# Enforce symmetries in the case abs(bet2) = -bet1. Need to be careful
# about this case, since this can yield singularities in the Newton
# iteration.
# sin(alp2) * cos(bet2) = sin(alp0)
salp2 = salp0 / cbet2 if cbet2 != cbet1 else salp1
# calp2 = sqrt(1 - sq(salp2))
# = sqrt(sq(calp0) - sq(sbet2)) / cbet2
# and subst for calp0 and rearrange to give (choose positive sqrt
# to give alp2 in [0, pi/2]).
calp2 = (math.sqrt(Math.sq(calp1 * cbet1) +
((cbet2 - cbet1) * (cbet1 + cbet2) if cbet1 < -sbet1
else (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2
if cbet2 != cbet1 or abs(sbet2) != -sbet1 else abs(calp1))
# tan(bet2) = tan(sig2) * cos(alp2)
# tan(omg2) = sin(alp0) * tan(sig2).
ssig2 = sbet2; somg2 = salp0 * sbet2
csig2 = comg2 = calp2 * cbet2
ssig2, csig2 = Math.norm(ssig2, csig2)
# Math.norm(somg2, comg2); -- don't need to normalize!
# sig12 = sig2 - sig1, limit to [0, pi]
sig12 = math.atan2(max(csig1 * ssig2 - ssig1 * csig2, 0.0),
csig1 * csig2 + ssig1 * ssig2)
# omg12 = omg2 - omg1, limit to [0, pi]
omg12 = math.atan2(max(comg1 * somg2 - somg1 * comg2, 0.0),
comg1 * comg2 + somg1 * somg2)
# real B312, h0
k2 = Math.sq(calp0) * self._ep2
eps = k2 / (2 * (1 + math.sqrt(1 + k2)) + k2)
self.C3f(eps, C3a)
B312 = (Geodesic.SinCosSeries(True, ssig2, csig2, C3a) -
Geodesic.SinCosSeries(True, ssig1, csig1, C3a))
h0 = -self._f * self.A3f(eps)
domg12 = salp0 * h0 * (sig12 + B312)
lam12 = omg12 + domg12
if diffp:
if calp2 == 0:
dlam12 = - 2 * self._f1 * dn1 / sbet1
else:
dummy, dlam12, dummy, dummy, dummy = self.Lengths(
eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2,
False, C1a, C2a)
dlam12 *= self._f1 / (calp2 * cbet2)
else:
dlam12 = Math.nan
return (lam12, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2, eps,
domg12, dlam12)
def InverseStart(self, sbet1, cbet1, dn1, sbet2, cbet2, dn2, lam12,
# Scratch areas of the right size
C1a, C2a):
"""Private: Find a starting value for Newton's method."""
# Return a starting point for Newton's method in salp1 and calp1 (function
# value is -1). If Newton's method doesn't need to be used, return also
# salp2 and calp2 and function value is sig12.
sig12 = -1; salp2 = calp2 = dnm = Math.nan # Return values
# bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0]
sbet12 = sbet2 * cbet1 - cbet2 * sbet1
cbet12 = cbet2 * cbet1 + sbet2 * sbet1
# Volatile declaration needed to fix inverse cases
# 88.202499451857 0 -88.202499451857 179.981022032992859592
# 89.262080389218 0 -89.262080389218 179.992207982775375662
# 89.333123580033 0 -89.333123580032997687 179.99295812360148422
# which otherwise fail with g++ 4.4.4 x86 -O3
sbet12a = sbet2 * cbet1
sbet12a += cbet2 * sbet1
shortline = cbet12 >= 0 and sbet12 < 0.5 and cbet2 * lam12 < 0.5
omg12 = lam12
if shortline:
sbetm2 = Math.sq(sbet1 + sbet2)
# sin((bet1+bet2)/2)^2
# = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2)
sbetm2 /= sbetm2 + Math.sq(cbet1 + cbet2)
dnm = math.sqrt(1 + self._ep2 * sbetm2)
omg12 /= self._f1 * dnm
somg12 = math.sin(omg12); comg12 = math.cos(omg12)
salp1 = cbet2 * somg12
calp1 = (
sbet12 + cbet2 * sbet1 * Math.sq(somg12) / (1 + comg12) if comg12 >= 0
else sbet12a - cbet2 * sbet1 * Math.sq(somg12) / (1 - comg12))
ssig12 = math.hypot(salp1, calp1)
csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12
if shortline and ssig12 < self._etol2:
# really short lines
salp2 = cbet1 * somg12
calp2 = sbet12 - cbet1 * sbet2 * (Math.sq(somg12) / (1 + comg12)
if comg12 >= 0 else 1 - comg12)
salp2, calp2 = Math.norm(salp2, calp2)
# Set return value
sig12 = math.atan2(ssig12, csig12)
elif (abs(self._n) >= 0.1 or # Skip astroid calc if too eccentric
csig12 >= 0 or
ssig12 >= 6 * abs(self._n) * math.pi * Math.sq(cbet1)):
# Nothing to do, zeroth order spherical approximation is OK
pass
else:
# Scale lam12 and bet2 to x, y coordinate system where antipodal point
# is at origin and singular point is at y = 0, x = -1.
# real y, lamscale, betscale
# Volatile declaration needed to fix inverse case
# 56.320923501171 0 -56.320923501171 179.664747671772880215
# which otherwise fails with g++ 4.4.4 x86 -O3
# volatile real x
if self._f >= 0: # In fact f == 0 does not get here
# x = dlong, y = dlat
k2 = Math.sq(sbet1) * self._ep2
eps = k2 / (2 * (1 + math.sqrt(1 + k2)) + k2)
lamscale = self._f * cbet1 * self.A3f(eps) * math.pi
betscale = lamscale * cbet1
x = (lam12 - math.pi) / lamscale
y = sbet12a / betscale
else: # _f < 0
# x = dlat, y = dlong
cbet12a = cbet2 * cbet1 - sbet2 * sbet1
bet12a = math.atan2(sbet12a, cbet12a)
# real m12b, m0, dummy
# In the case of lon12 = 180, this repeats a calculation made in
# Inverse.
dummy, m12b, m0, dummy, dummy = self.Lengths(
self._n, math.pi + bet12a, sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
cbet1, cbet2, False, C1a, C2a)
x = -1 + m12b / (cbet1 * cbet2 * m0 * math.pi)
betscale = (sbet12a / x if x < -0.01
else -self._f * Math.sq(cbet1) * math.pi)
lamscale = betscale / cbet1
y = (lam12 - math.pi) / lamscale
if y > -Geodesic.tol1_ and x > -1 - Geodesic.xthresh_:
# strip near cut
if self._f >= 0:
salp1 = min(1.0, -x); calp1 = - math.sqrt(1 - Math.sq(salp1))
else:
calp1 = max((0.0 if x > -Geodesic.tol1_ else -1.0), x)
salp1 = math.sqrt(1 - Math.sq(calp1))
else:
# Estimate alp1, by solving the astroid problem.
#
# Could estimate alpha1 = theta + pi/2, directly, i.e.,
# calp1 = y/k; salp1 = -x/(1+k); for _f >= 0
# calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check)
#
# However, it's better to estimate omg12 from astroid and use
# spherical formula to compute alp1. This reduces the mean number of
# Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12
# (min 0 max 5). The changes in the number of iterations are as
# follows:
#
# change percent
# 1 5
# 0 78
# -1 16
# -2 0.6
# -3 0.04
# -4 0.002
#
# The histogram of iterations is (m = number of iterations estimating
# alp1 directly, n = number of iterations estimating via omg12, total
# number of trials = 148605):
#
# iter m n
# 0 148 186
# 1 13046 13845
# 2 93315 102225
# 3 36189 32341
# 4 5396 7
# 5 455 1
# 6 56 0
#
# Because omg12 is near pi, estimate work with omg12a = pi - omg12
k = Geodesic.Astroid(x, y)
omg12a = lamscale * ( -x * k/(1 + k) if self._f >= 0
else -y * (1 + k)/k )
somg12 = math.sin(omg12a); comg12 = -math.cos(omg12a)
# Update spherical estimate of alp1 using omg12 instead of lam12
salp1 = cbet2 * somg12
calp1 = sbet12a - cbet2 * sbet1 * Math.sq(somg12) / (1 - comg12)
# Sanity check on starting guess. Backwards check allows NaN through.
if not (salp1 <= 0):
salp1, calp1 = Math.norm(salp1, calp1)
else:
salp1 = 1; calp1 = 0
return sig12, salp1, calp1, salp2, calp2, dnm
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)