"""Define the :class:`~geographiclib.geodesic.Geodesic` class
The ellipsoid parameters are defined by the constructor. The direct and
inverse geodesic problems are solved by
* :meth:`~geographiclib.geodesic.Geodesic.Inverse` Solve the inverse
geodesic problem
* :meth:`~geographiclib.geodesic.Geodesic.Direct` Solve the direct
geodesic problem
* :meth:`~geographiclib.geodesic.Geodesic.ArcDirect` Solve the direct
geodesic problem in terms of spherical arc length
:class:`~geographiclib.geodesicline.GeodesicLine` objects can be created
with
* :meth:`~geographiclib.geodesic.Geodesic.Line`
* :meth:`~geographiclib.geodesic.Geodesic.DirectLine`
* :meth:`~geographiclib.geodesic.Geodesic.ArcDirectLine`
* :meth:`~geographiclib.geodesic.Geodesic.InverseLine`
:class:`~geographiclib.polygonarea.PolygonArea` objects can be created
with
* :meth:`~geographiclib.geodesic.Geodesic.Polygon`
The public attributes for this class are
* :attr:`~geographiclib.geodesic.Geodesic.a`
:attr:`~geographiclib.geodesic.Geodesic.f`
*outmask* and *caps* bit masks are
* :const:`~geographiclib.geodesic.Geodesic.EMPTY`
* :const:`~geographiclib.geodesic.Geodesic.LATITUDE`
* :const:`~geographiclib.geodesic.Geodesic.LONGITUDE`
* :const:`~geographiclib.geodesic.Geodesic.AZIMUTH`
* :const:`~geographiclib.geodesic.Geodesic.DISTANCE`
* :const:`~geographiclib.geodesic.Geodesic.STANDARD`
* :const:`~geographiclib.geodesic.Geodesic.DISTANCE_IN`
* :const:`~geographiclib.geodesic.Geodesic.REDUCEDLENGTH`
* :const:`~geographiclib.geodesic.Geodesic.GEODESICSCALE`
* :const:`~geographiclib.geodesic.Geodesic.AREA`
* :const:`~geographiclib.geodesic.Geodesic.ALL`
* :const:`~geographiclib.geodesic.Geodesic.LONG_UNROLL`
:Example:
>>> from geographiclib.geodesic import Geodesic
>>> # The geodesic inverse problem
... Geodesic.WGS84.Inverse(-41.32, 174.81, 40.96, -5.50)
{'lat1': -41.32,
'a12': 179.6197069334283,
's12': 19959679.26735382,
'lat2': 40.96,
'azi2': 18.825195123248392,
'azi1': 161.06766998615882,
'lon1': 174.81,
'lon2': -5.5}
"""
# geodesic.py
#
# This is a rather literal translation of the GeographicLib::Geodesic class to
# python. See the documentation for the C++ class for more information at
#
# https://geographiclib.sourceforge.io/html/annotated.html
#
# The algorithms are derived in
#
# Charles F. F. Karney,
# Algorithms for geodesics, J. Geodesy 87, 43-55 (2013),
# https://doi.org/10.1007/s00190-012-0578-z
# Addenda: https://geographiclib.sourceforge.io/geod-addenda.html
#
# Copyright (c) Charles Karney (2011-2017) <charles@karney.com> and licensed
# under the MIT/X11 License. For more information, see
# https://geographiclib.sourceforge.io/
######################################################################
import math
from geographiclib.geomath import Math
from geographiclib.constants import Constants
from geographiclib.geodesiccapability import GeodesicCapability
[docs]class Geodesic(object):
"""Solve geodesic problems"""
GEOGRAPHICLIB_GEODESIC_ORDER = 6
nA1_ = GEOGRAPHICLIB_GEODESIC_ORDER
nC1_ = GEOGRAPHICLIB_GEODESIC_ORDER
nC1p_ = GEOGRAPHICLIB_GEODESIC_ORDER
nA2_ = GEOGRAPHICLIB_GEODESIC_ORDER
nC2_ = GEOGRAPHICLIB_GEODESIC_ORDER
nA3_ = GEOGRAPHICLIB_GEODESIC_ORDER
nA3x_ = nA3_
nC3_ = GEOGRAPHICLIB_GEODESIC_ORDER
nC3x_ = (nC3_ * (nC3_ - 1)) // 2
nC4_ = GEOGRAPHICLIB_GEODESIC_ORDER
nC4x_ = (nC4_ * (nC4_ + 1)) // 2
maxit1_ = 20
maxit2_ = maxit1_ + Math.digits + 10
tiny_ = math.sqrt(Math.minval)
tol0_ = Math.epsilon
tol1_ = 200 * tol0_
tol2_ = math.sqrt(tol0_)
tolb_ = tol0_ * tol2_
xthresh_ = 1000 * tol2_
CAP_NONE = GeodesicCapability.CAP_NONE
CAP_C1 = GeodesicCapability.CAP_C1
CAP_C1p = GeodesicCapability.CAP_C1p
CAP_C2 = GeodesicCapability.CAP_C2
CAP_C3 = GeodesicCapability.CAP_C3
CAP_C4 = GeodesicCapability.CAP_C4
CAP_ALL = GeodesicCapability.CAP_ALL
CAP_MASK = GeodesicCapability.CAP_MASK
OUT_ALL = GeodesicCapability.OUT_ALL
OUT_MASK = GeodesicCapability.OUT_MASK
def _SinCosSeries(sinp, sinx, cosx, c):
"""Private: Evaluate a trig series using Clenshaw summation."""
# Evaluate
# y = sinp ? sum(c[i] * sin( 2*i * x), i, 1, n) :
# sum(c[i] * cos((2*i+1) * x), i, 0, n-1)
# using Clenshaw summation. N.B. c[0] is unused for sin series
# Approx operation count = (n + 5) mult and (2 * n + 2) add
k = len(c) # Point to one beyond last element
n = k - sinp
ar = 2 * (cosx - sinx) * (cosx + sinx) # 2 * cos(2 * x)
y1 = 0 # accumulators for sum
if n & 1:
k -= 1; y0 = c[k]
else:
y0 = 0
# Now n is even
n = n // 2
while n: # while n--:
n -= 1
# Unroll loop x 2, so accumulators return to their original role
k -= 1; y1 = ar * y0 - y1 + c[k]
k -= 1; y0 = ar * y1 - y0 + c[k]
return ( 2 * sinx * cosx * y0 if sinp # sin(2 * x) * y0
else cosx * (y0 - y1) ) # cos(x) * (y0 - y1)
_SinCosSeries = staticmethod(_SinCosSeries)
def _Astroid(x, y):
"""Private: solve astroid equation."""
# Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k.
# This solution is adapted from Geocentric::Reverse.
p = Math.sq(x)
q = Math.sq(y)
r = (p + q - 1) / 6
if not(q == 0 and r <= 0):
# Avoid possible division by zero when r = 0 by multiplying equations
# for s and t by r^3 and r, resp.
S = p * q / 4 # S = r^3 * s
r2 = Math.sq(r)
r3 = r * r2
# The discriminant of the quadratic equation for T3. This is zero on
# the evolute curve p^(1/3)+q^(1/3) = 1
disc = S * (S + 2 * r3)
u = r
if disc >= 0:
T3 = S + r3
# Pick the sign on the sqrt to maximize abs(T3). This minimizes loss
# of precision due to cancellation. The result is unchanged because
# of the way the T is used in definition of u.
T3 += -math.sqrt(disc) if T3 < 0 else math.sqrt(disc) # T3 = (r * t)^3
# N.B. cbrt always returns the real root. cbrt(-8) = -2.
T = Math.cbrt(T3) # T = r * t
# T can be zero; but then r2 / T -> 0.
u += T + (r2 / T if T != 0 else 0)
else:
# T is complex, but the way u is defined the result is real.
ang = math.atan2(math.sqrt(-disc), -(S + r3))
# There are three possible cube roots. We choose the root which
# avoids cancellation. Note that disc < 0 implies that r < 0.
u += 2 * r * math.cos(ang / 3)
v = math.sqrt(Math.sq(u) + q) # guaranteed positive
# Avoid loss of accuracy when u < 0.
uv = q / (v - u) if u < 0 else u + v # u+v, guaranteed positive
w = (uv - q) / (2 * v) # positive?
# Rearrange expression for k to avoid loss of accuracy due to
# subtraction. Division by 0 not possible because uv > 0, w >= 0.
k = uv / (math.sqrt(uv + Math.sq(w)) + w) # guaranteed positive
else: # q == 0 && r <= 0
# y = 0 with |x| <= 1. Handle this case directly.
# for y small, positive root is k = abs(y)/sqrt(1-x^2)
k = 0
return k
_Astroid = staticmethod(_Astroid)
def _A1m1f(eps):
"""Private: return A1-1."""
coeff = [
1, 4, 64, 0, 256,
]
m = Geodesic.nA1_//2
t = Math.polyval(m, coeff, 0, Math.sq(eps)) / coeff[m + 1]
return (t + eps) / (1 - eps)
_A1m1f = staticmethod(_A1m1f)
def _C1f(eps, c):
"""Private: return C1."""
coeff = [
-1, 6, -16, 32,
-9, 64, -128, 2048,
9, -16, 768,
3, -5, 512,
-7, 1280,
-7, 2048,
]
eps2 = Math.sq(eps)
d = eps
o = 0
for l in range(1, Geodesic.nC1_ + 1): # l is index of C1p[l]
m = (Geodesic.nC1_ - l) // 2 # order of polynomial in eps^2
c[l] = d * Math.polyval(m, coeff, o, eps2) / coeff[o + m + 1]
o += m + 2
d *= eps
_C1f = staticmethod(_C1f)
def _C1pf(eps, c):
"""Private: return C1'"""
coeff = [
205, -432, 768, 1536,
4005, -4736, 3840, 12288,
-225, 116, 384,
-7173, 2695, 7680,
3467, 7680,
38081, 61440,
]
eps2 = Math.sq(eps)
d = eps
o = 0
for l in range(1, Geodesic.nC1p_ + 1): # l is index of C1p[l]
m = (Geodesic.nC1p_ - l) // 2 # order of polynomial in eps^2
c[l] = d * Math.polyval(m, coeff, o, eps2) / coeff[o + m + 1]
o += m + 2
d *= eps
_C1pf = staticmethod(_C1pf)
def _A2m1f(eps):
"""Private: return A2-1"""
coeff = [
-11, -28, -192, 0, 256,
]
m = Geodesic.nA2_//2
t = Math.polyval(m, coeff, 0, Math.sq(eps)) / coeff[m + 1]
return (t - eps) / (1 + eps)
_A2m1f = staticmethod(_A2m1f)
def _C2f(eps, c):
"""Private: return C2"""
coeff = [
1, 2, 16, 32,
35, 64, 384, 2048,
15, 80, 768,
7, 35, 512,
63, 1280,
77, 2048,
]
eps2 = Math.sq(eps)
d = eps
o = 0
for l in range(1, Geodesic.nC2_ + 1): # l is index of C2[l]
m = (Geodesic.nC2_ - l) // 2 # order of polynomial in eps^2
c[l] = d * Math.polyval(m, coeff, o, eps2) / coeff[o + m + 1]
o += m + 2
d *= eps
_C2f = staticmethod(_C2f)
def __init__(self, a, f):
"""Construct a Geodesic object
:param a: the equatorial radius of the ellipsoid in meters
:param f: the flattening of the ellipsoid
An exception is thrown if *a* or the polar semi-axis *b* = *a* (1 -
*f*) is not a finite positive quantity.
"""
self.a = float(a)
"""The equatorial radius in meters (readonly)"""
self.f = float(f)
"""The flattening (readonly)"""
self._f1 = 1 - self.f
self._e2 = self.f * (2 - self.f)
self._ep2 = self._e2 / Math.sq(self._f1) # e2 / (1 - e2)
self._n = self.f / ( 2 - self.f)
self._b = self.a * self._f1
# authalic radius squared
self._c2 = (Math.sq(self.a) + Math.sq(self._b) *
(1 if self._e2 == 0 else
(Math.atanh(math.sqrt(self._e2)) if self._e2 > 0 else
math.atan(math.sqrt(-self._e2))) /
math.sqrt(abs(self._e2))))/2
# The sig12 threshold for "really short". Using the auxiliary sphere
# solution with dnm computed at (bet1 + bet2) / 2, the relative error in
# the azimuth consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2.
# (Error measured for 1/100 < b/a < 100 and abs(f) >= 1/1000. For a given
# f and sig12, the max error occurs for lines near the pole. If the old
# rule for computing dnm = (dn1 + dn2)/2 is used, then the error increases
# by a factor of 2.) Setting this equal to epsilon gives sig12 = etol2.
# Here 0.1 is a safety factor (error decreased by 100) and max(0.001,
# abs(f)) stops etol2 getting too large in the nearly spherical case.
self._etol2 = 0.1 * Geodesic.tol2_ / math.sqrt( max(0.001, abs(self.f)) *
min(1.0, 1-self.f/2) / 2 )
if not(Math.isfinite(self.a) and self.a > 0):
raise ValueError("Equatorial radius is not positive")
if not(Math.isfinite(self._b) and self._b > 0):
raise ValueError("Polar semi-axis is not positive")
self._A3x = list(range(Geodesic.nA3x_))
self._C3x = list(range(Geodesic.nC3x_))
self._C4x = list(range(Geodesic.nC4x_))
self._A3coeff()
self._C3coeff()
self._C4coeff()
def _A3coeff(self):
"""Private: return coefficients for A3"""
coeff = [
-3, 128,
-2, -3, 64,
-1, -3, -1, 16,
3, -1, -2, 8,
1, -1, 2,
1, 1,
]
o = 0; k = 0
for j in range(Geodesic.nA3_ - 1, -1, -1): # coeff of eps^j
m = min(Geodesic.nA3_ - j - 1, j) # order of polynomial in n
self._A3x[k] = Math.polyval(m, coeff, o, self._n) / coeff[o + m + 1]
k += 1
o += m + 2
def _C3coeff(self):
"""Private: return coefficients for C3"""
coeff = [
3, 128,
2, 5, 128,
-1, 3, 3, 64,
-1, 0, 1, 8,
-1, 1, 4,
5, 256,
1, 3, 128,
-3, -2, 3, 64,
1, -3, 2, 32,
7, 512,
-10, 9, 384,
5, -9, 5, 192,
7, 512,
-14, 7, 512,
21, 2560,
]
o = 0; k = 0
for l in range(1, Geodesic.nC3_): # l is index of C3[l]
for j in range(Geodesic.nC3_ - 1, l - 1, -1): # coeff of eps^j
m = min(Geodesic.nC3_ - j - 1, j) # order of polynomial in n
self._C3x[k] = Math.polyval(m, coeff, o, self._n) / coeff[o + m + 1]
k += 1
o += m + 2
def _C4coeff(self):
"""Private: return coefficients for C4"""
coeff = [
97, 15015,
1088, 156, 45045,
-224, -4784, 1573, 45045,
-10656, 14144, -4576, -858, 45045,
64, 624, -4576, 6864, -3003, 15015,
100, 208, 572, 3432, -12012, 30030, 45045,
1, 9009,
-2944, 468, 135135,
5792, 1040, -1287, 135135,
5952, -11648, 9152, -2574, 135135,
-64, -624, 4576, -6864, 3003, 135135,
8, 10725,
1856, -936, 225225,
-8448, 4992, -1144, 225225,
-1440, 4160, -4576, 1716, 225225,
-136, 63063,
1024, -208, 105105,
3584, -3328, 1144, 315315,
-128, 135135,
-2560, 832, 405405,
128, 99099,
]
o = 0; k = 0
for l in range(Geodesic.nC4_): # l is index of C4[l]
for j in range(Geodesic.nC4_ - 1, l - 1, -1): # coeff of eps^j
m = Geodesic.nC4_ - j - 1 # order of polynomial in n
self._C4x[k] = Math.polyval(m, coeff, o, self._n) / coeff[o + m + 1]
k += 1
o += m + 2
def _A3f(self, eps):
"""Private: return A3"""
# Evaluate A3
return Math.polyval(Geodesic.nA3_ - 1, self._A3x, 0, eps)
def _C3f(self, eps, c):
"""Private: return C3"""
# Evaluate C3
# Elements c[1] thru c[nC3_ - 1] are set
mult = 1
o = 0
for l in range(1, Geodesic.nC3_): # l is index of C3[l]
m = Geodesic.nC3_ - l - 1 # order of polynomial in eps
mult *= eps
c[l] = mult * Math.polyval(m, self._C3x, o, eps)
o += m + 1
def _C4f(self, eps, c):
"""Private: return C4"""
# Evaluate C4 coeffs by Horner's method
# Elements c[0] thru c[nC4_ - 1] are set
mult = 1
o = 0
for l in range(Geodesic.nC4_): # l is index of C4[l]
m = Geodesic.nC4_ - l - 1 # order of polynomial in eps
c[l] = mult * Math.polyval(m, self._C4x, o, eps)
o += m + 1
mult *= eps
# return s12b, m12b, m0, M12, M21
def _Lengths(self, eps, sig12,
ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2, outmask,
# Scratch areas of the right size
C1a, C2a):
"""Private: return a bunch of lengths"""
# Return s12b, m12b, m0, M12, M21, where
# m12b = (reduced length)/_b; s12b = distance/_b,
# m0 = coefficient of secular term in expression for reduced length.
outmask &= Geodesic.OUT_MASK
# outmask & DISTANCE: set s12b
# outmask & REDUCEDLENGTH: set m12b & m0
# outmask & GEODESICSCALE: set M12 & M21
s12b = m12b = m0 = M12 = M21 = Math.nan
if outmask & (Geodesic.DISTANCE | Geodesic.REDUCEDLENGTH |
Geodesic.GEODESICSCALE):
A1 = Geodesic._A1m1f(eps)
Geodesic._C1f(eps, C1a)
if outmask & (Geodesic.REDUCEDLENGTH | Geodesic.GEODESICSCALE):
A2 = Geodesic._A2m1f(eps)
Geodesic._C2f(eps, C2a)
m0x = A1 - A2
A2 = 1 + A2
A1 = 1 + A1
if outmask & Geodesic.DISTANCE:
B1 = (Geodesic._SinCosSeries(True, ssig2, csig2, C1a) -
Geodesic._SinCosSeries(True, ssig1, csig1, C1a))
# Missing a factor of _b
s12b = A1 * (sig12 + B1)
if outmask & (Geodesic.REDUCEDLENGTH | Geodesic.GEODESICSCALE):
B2 = (Geodesic._SinCosSeries(True, ssig2, csig2, C2a) -
Geodesic._SinCosSeries(True, ssig1, csig1, C2a))
J12 = m0x * sig12 + (A1 * B1 - A2 * B2)
elif outmask & (Geodesic.REDUCEDLENGTH | Geodesic.GEODESICSCALE):
# Assume here that nC1_ >= nC2_
for l in range(1, Geodesic.nC2_):
C2a[l] = A1 * C1a[l] - A2 * C2a[l]
J12 = m0x * sig12 + (Geodesic._SinCosSeries(True, ssig2, csig2, C2a) -
Geodesic._SinCosSeries(True, ssig1, csig1, C2a))
if outmask & Geodesic.REDUCEDLENGTH:
m0 = m0x
# Missing a factor of _b.
# Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure
# accurate cancellation in the case of coincident points.
m12b = (dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) -
csig1 * csig2 * J12)
if outmask & Geodesic.GEODESICSCALE:
csig12 = csig1 * csig2 + ssig1 * ssig2
t = self._ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2)
M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1
M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2
return s12b, m12b, m0, M12, M21
# return sig12, salp1, calp1, salp2, calp2, dnm
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
# return lam12, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2, eps,
# domg12, dlam12
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)
# return a12, s12, salp1, calp1, salp2, calp2, m12, M12, M21, S12
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
[docs] 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
# return a12, lat2, lon2, azi2, s12, m12, M12, M21, S12
def _GenDirect(self, lat1, lon1, azi1, arcmode, s12_a12, outmask):
"""Private: General version of direct problem"""
from geographiclib.geodesicline import GeodesicLine
# Automatically supply DISTANCE_IN if necessary
if not arcmode: outmask |= Geodesic.DISTANCE_IN
line = GeodesicLine(self, lat1, lon1, azi1, outmask)
return line._GenPosition(arcmode, s12_a12, outmask)
[docs] def Direct(self, lat1, lon1, azi1, s12,
outmask = GeodesicCapability.STANDARD):
"""Solve the direct geodesic problem
: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 s12: the distance from the first point to the second in
meters
:param outmask: the :ref:`output mask <outmask>`
:return: a :ref:`dict`
Compute geodesic starting at (*lat1*, *lon1*) with azimuth *azi1*
and length *s12*. 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, False, s12, 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),
's12': s12}
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
[docs] 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
[docs] def Line(self, lat1, lon1, azi1,
caps = GeodesicCapability.STANDARD |
GeodesicCapability.DISTANCE_IN):
"""Return a GeodesicLine 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>`
:return: a :class:`~geographiclib.geodesicline.GeodesicLine`
This allows points along a geodesic starting at (*lat1*, *lon1*),
with azimuth *azi1* to be found. The default value of *caps* is
STANDARD | DISTANCE_IN, allowing direct geodesic problem to be
solved.
"""
from geographiclib.geodesicline import GeodesicLine
return GeodesicLine(self, lat1, lon1, azi1, caps)
def _GenDirectLine(self, lat1, lon1, azi1, arcmode, s12_a12,
caps = GeodesicCapability.STANDARD |
GeodesicCapability.DISTANCE_IN):
"""Private: general form of DirectLine"""
from geographiclib.geodesicline import GeodesicLine
# Automatically supply DISTANCE_IN if necessary
if not arcmode: caps |= Geodesic.DISTANCE_IN
line = GeodesicLine(self, lat1, lon1, azi1, caps)
if arcmode:
line.SetArc(s12_a12)
else:
line.SetDistance(s12_a12)
return line
[docs] def DirectLine(self, lat1, lon1, azi1, s12,
caps = GeodesicCapability.STANDARD |
GeodesicCapability.DISTANCE_IN):
"""Define a GeodesicLine object in terms of the direct geodesic
problem specified 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 s12: the distance from the first point to the second in
meters
:param caps: the :ref:`capabilities <outmask>`
:return: a :class:`~geographiclib.geodesicline.GeodesicLine`
This function sets point 3 of the GeodesicLine to correspond to
point 2 of the direct geodesic problem. The default value of *caps*
is STANDARD | DISTANCE_IN, allowing direct geodesic problem to be
solved.
"""
return self._GenDirectLine(lat1, lon1, azi1, False, s12, caps)
[docs] def ArcDirectLine(self, lat1, lon1, azi1, a12,
caps = GeodesicCapability.STANDARD |
GeodesicCapability.DISTANCE_IN):
"""Define a GeodesicLine object in terms of the direct geodesic
problem specified 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 caps: the :ref:`capabilities <outmask>`
:return: a :class:`~geographiclib.geodesicline.GeodesicLine`
This function sets point 3 of the GeodesicLine to correspond to
point 2 of the direct geodesic problem. The default value of *caps*
is STANDARD | DISTANCE_IN, allowing direct geodesic problem to be
solved.
"""
return self._GenDirectLine(lat1, lon1, azi1, True, a12, caps)
[docs] def InverseLine(self, lat1, lon1, lat2, lon2,
caps = GeodesicCapability.STANDARD |
GeodesicCapability.DISTANCE_IN):
"""Define a GeodesicLine object in terms of the invese 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 caps: the :ref:`capabilities <outmask>`
:return: a :class:`~geographiclib.geodesicline.GeodesicLine`
This function sets point 3 of the GeodesicLine to correspond to
point 2 of the inverse geodesic problem. The default value of *caps*
is STANDARD | DISTANCE_IN, allowing direct geodesic problem to be
solved.
"""
from geographiclib.geodesicline import GeodesicLine
a12, _, salp1, calp1, _, _, _, _, _, _ = self._GenInverse(
lat1, lon1, lat2, lon2, 0)
azi1 = Math.atan2d(salp1, calp1)
if caps & (Geodesic.OUT_MASK & Geodesic.DISTANCE_IN):
caps |= Geodesic.DISTANCE
line = GeodesicLine(self, lat1, lon1, azi1, caps, salp1, calp1)
line.SetArc(a12)
return line
[docs] def Polygon(self, polyline = False):
"""Return a PolygonArea object
:param polyline: if True then the object describes a polyline
instead of a polygon
:return: a :class:`~geographiclib.polygonarea.PolygonArea`
"""
from geographiclib.polygonarea import PolygonArea
return PolygonArea(self, polyline)
EMPTY = GeodesicCapability.EMPTY
"""No capabilities, no output."""
LATITUDE = GeodesicCapability.LATITUDE
"""Calculate latitude *lat2*."""
LONGITUDE = GeodesicCapability.LONGITUDE
"""Calculate longitude *lon2*."""
AZIMUTH = GeodesicCapability.AZIMUTH
"""Calculate azimuths *azi1* and *azi2*."""
DISTANCE = GeodesicCapability.DISTANCE
"""Calculate distance *s12*."""
STANDARD = GeodesicCapability.STANDARD
"""All of the above."""
DISTANCE_IN = GeodesicCapability.DISTANCE_IN
"""Allow distance *s12* to be used as input in the direct geodesic
problem."""
REDUCEDLENGTH = GeodesicCapability.REDUCEDLENGTH
"""Calculate reduced length *m12*."""
GEODESICSCALE = GeodesicCapability.GEODESICSCALE
"""Calculate geodesic scales *M12* and *M21*."""
AREA = GeodesicCapability.AREA
"""Calculate area *S12*."""
ALL = GeodesicCapability.ALL
"""All of the above."""
LONG_UNROLL = GeodesicCapability.LONG_UNROLL
"""Unroll longitudes, rather than reducing them to the range
[-180d,180d].
"""
Geodesic.WGS84 = Geodesic(Constants.WGS84_a, Constants.WGS84_f)
"""Instantiation for the WGS84 ellipsoid"""