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# -*- coding: utf-8 -*- 

 

u'''A Python version of I{Karney}'s C++ class U{GeodesicExact 

<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>}. 

 

Classes L{GeodesicExact} and L{GeodesicLineExact} follow the naming, 

methods and return values from I{Karney}s' Python classes C{Geodesic} 

and C{GeodesicLine}, respectively. 

 

Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2008-2022) 

and licensed under the MIT/X11 License. For more information, see the 

U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation. 

''' 

# make sure int/int division yields float quotient 

from __future__ import division as _; del _ # PYCHOK semicolon 

 

# A copy of comments from Karney's C{GeodesicExact.cpp}: 

# 

# This is a reformulation of the geodesic problem. The 

# notation is as follows: 

# - at a general point (no suffix or 1 or 2 as suffix) 

# - phi = latitude 

# - beta = latitude on auxiliary sphere 

# - omega = longitude on auxiliary sphere 

# - lambda = longitude 

# - alpha = azimuth of great circle 

# - sigma = arc length along great circle 

# - s = distance 

# - tau = scaled distance (= sigma at multiples of PI/2) 

# - at northwards equator crossing 

# - beta = phi = 0 

# - omega = lambda = 0 

# - alpha = alpha0 

# - sigma = s = 0 

# - a 12 suffix means a difference, e.g., s12 = s2 - s1. 

# - s and c prefixes mean sin and cos 

 

from pygeodesy.basics import _xinstanceof, _xor, unsigned0 

from pygeodesy.constants import EPS, EPS0, EPS02, MANT_DIG, NAN, PI, _EPSqrt, \ 

_SQRT2_2, isnan, _0_0, _0_001, _0_01, _0_1, _0_5, \ 

_1_0, _N_1_0, _1_75, _2_0, _N_2_0, _2__PI, _3_0, \ 

_4_0, _6_0, _8_0, _16_0, _90_0, _180_0, _1000_0 

# from pygeodesy.datums import _a_ellipsoid # from .karney 

from pygeodesy.fsums import fsum_, fsum1_ 

from pygeodesy.geodesicx.gxbases import _cosSeries, _GeodesicBase, \ 

_sincos12, _sin1cos2, _TINY, _xnC4 

from pygeodesy.geodesicx.gxline import _GeodesicLineExact, _update_glXs 

from pygeodesy.interns import NN, _COMMASPACE_, _DOT_, _UNDER_ 

from pygeodesy.karney import _around, _atan2d, Caps, _cbrt, _copysign, _diff182, \ 

_a_ellipsoid, _EWGS84, GDict, GeodesicError, _fix90, \ 

_hypot, _K_2_0, _norm2, _norm180, _polynomial, \ 

_signBit, _sincos2, _sincos2d, _sincos2de 

from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS 

from pygeodesy.namedTuples import Destination3Tuple, Distance3Tuple 

from pygeodesy.props import deprecated_Property, Property, Property_RO 

from pygeodesy.streprs import Fmt, pairs 

from pygeodesy.utily import atan2d as _atan2d_reverse, unroll180, wrap360 

 

from math import atan2, copysign, cos, degrees, fabs, radians, sqrt 

 

__all__ = () 

__version__ = '22.09.14' 

 

_MAXIT1 = 20 

_MAXIT2 = 10 + _MAXIT1 + MANT_DIG # MANT_DIG == C++ digits 

 

# increased multiplier in defn of _TOL1 from 100 to 200 to fix Inverse 

# case 52.784459512564 0 -52.784459512563990912 179.634407464943777557 

# which otherwise failed for Visual Studio 10 (Release and Debug) 

_TOL0 = EPS 

_TOL1 = _TOL0 * -200 # negative 

_TOL2 = _EPSqrt # == sqrt(_TOL0) 

_TOL3 = _TOL2 * _0_1 

_TOLb = _TOL2 * _TOL0 # Check on bisection interval 

_THR1 = _TOL2 * _1000_0 + _1_0 

 

_TINY3 = _TINY * _3_0 

_TOL08 = _TOL0 * _8_0 

_TOL016 = _TOL0 * _16_0 

 

 

def _atan12(*sincos12, **sineg0): 

'''(INTERNAL) Return C{ang12} in C{radians}. 

''' 

s, c = _sincos12(*sincos12, **sineg0) if sineg0 else \ 

_sincos12(*sincos12) 

return atan2(s, c) 

 

 

def _eTOL2(f): 

# 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. 

t = min(_1_0, _1_0 - f * _0_5) * max(_0_001, fabs(f)) * _0_5 

return _TOL3 / (sqrt(t) if t > EPS02 else EPS0) 

 

 

class _PDict(GDict): 

'''(INTERNAL) Parameters passed around in C{._GDictInverse} and 

optionally returned when C{GeodesicExact.debug} is C{True}. 

''' 

def setsigs(self, ssig1, csig1, ssig2, csig2): 

'''Update the C{sig1} and C{sig2} parameters. 

''' 

self.set_(ssig1=ssig1, csig1=csig1, sncndn1=(ssig1, csig1, self.dn1), # PYCHOK dn1 

ssig2=ssig2, csig2=csig2, sncndn2=(ssig2, csig2, self.dn2)) # PYCHOK dn2 

 

def toGDict(self): # PYCHOK no cover 

'''Return as C{GDict} without attrs C{sncndn1} and C{sncndn2}. 

''' 

def _rest(sncndn1=None, sncndn2=None, **rest): # PYCHOK sncndn* not used 

return GDict(rest) 

 

return _rest(**self) 

 

 

class GeodesicExact(_GeodesicBase): 

'''A pure Python version of I{Karney}'s C++ class U{GeodesicExact 

<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>}, 

modeled after I{Karney}'s Python class U{Geodesic<https://GeographicLib.SourceForge.io/ 

html/python/code.html#module-geographiclib.geodesic>}. 

''' 

_E = _EWGS84 

_nC4 = 30 # default C4order 

 

def __init__(self, a_ellipsoid=_EWGS84, f=None, name=NN, C4order=None, 

C4Order=None): # for backward compatibility 

'''New L{GeodesicExact} instance. 

 

@arg a_ellipsoid: An ellipsoid (L{Ellipsoid}) or datum (L{Datum}) or 

the equatorial radius of the ellipsoid (C{scalar}, 

conventionally in C{meter}), see B{C{f}}. 

@arg f: The flattening of the ellipsoid (C{scalar}) if B{C{a_ellipsoid}} 

is specified as C{scalar}. 

@kwarg name: Optional name (C{str}). 

@kwarg C4order: Optional series expansion order (C{int}), see property 

L{C4order}, default C{30}. 

@kwarg C4Order: DEPRECATED, use keyword argument B{C{C4order}}. 

 

@raise GeodesicError: Invalid B{C{C4order}}. 

''' 

if a_ellipsoid not in (GeodesicExact._E, None): 

self._E = _a_ellipsoid(a_ellipsoid, f, name=name) 

 

if name: 

self.name = name 

if C4order: # XXX private copy, always? 

self.C4order = C4order 

elif C4Order: # for backward compatibility 

self.C4Order = C4Order 

 

@Property_RO 

def a(self): 

'''Get the I{equatorial} radius, semi-axis (C{meter}). 

''' 

return self.ellipsoid.a 

 

def ArcDirect(self, lat1, lon1, azi1, a12, outmask=Caps.STANDARD): 

'''Solve the I{Direct} geodesic problem in terms of (spherical) arc length. 

 

@arg lat1: Latitude of the first point (C{degrees}). 

@arg lon1: Longitude of the first point (C{degrees}). 

@arg azi1: Azimuth at the first point (compass C{degrees}). 

@arg a12: Arc length between the points (C{degrees}), can be negative. 

@kwarg outmask: Bit-or'ed combination of L{Caps} values specifying 

the quantities to be returned. 

 

@return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2, 

lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1}, 

C{lon1}, C{azi1} and arc length C{a12} always included. 

 

@see: C++ U{GeodesicExact.ArcDirect 

<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} 

and Python U{Geodesic.ArcDirect<https://GeographicLib.SourceForge.io/C++/doc/python/code.html>}. 

''' 

return self._GDictDirect(lat1, lon1, azi1, True, a12, outmask) 

 

def ArcDirectLine(self, lat1, lon1, azi1, a12, caps): 

'''Define a L{GeodesicLineExact} in terms of the I{direct} geodesic problem and as arc length. 

 

@arg lat1: Latitude of the first point (C{degrees}). 

@arg lon1: Longitude of the first point (C{degrees}). 

@arg azi1: Azimuth at the first point (compass C{degrees}). 

@arg a12: Arc length between the points (C{degrees}), can be negative. 

@kwarg caps: Bit-or'ed combination of L{Caps} values specifying 

the capabilities the L{GeodesicLineExact} instance 

should possess, i.e., which quantities can be 

returned by calls to L{GeodesicLineExact.Position} 

and L{GeodesicLineExact.ArcPosition}. 

 

@return: A L{GeodesicLineExact} instance. 

 

@note: The third point of the L{GeodesicLineExact} is set to correspond 

to the second point of the I{Inverse} geodesic problem. 

 

@note: Latitude B{C{lat1}} should in the range C{[-90, +90]}. 

 

@see: C++ U{GeodesicExact.ArcDirectLine 

<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} and 

Python U{Geodesic.ArcDirectLine<https://GeographicLib.SourceForge.io/C++/doc/python/code.html>}. 

''' 

return self._GenDirectLine(lat1, lon1, azi1, True, a12, caps) 

 

def Area(self, polyline=False, name=NN): 

'''Set up a L{GeodesicAreaExact} to compute area and 

perimeter of a polygon. 

 

@kwarg polyline: If C{True} perimeter only, otherwise 

area and perimeter (C{bool}). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{GeodesicAreaExact} instance. 

 

@note: The B{C{debug}} setting is passed as C{verbose} 

to the returned L{GeodesicAreaExact} instance. 

''' 

gaX = _MODS.geodesicx.GeodesicAreaExact(self, polyline=polyline, 

name=name or self.name) 

if self.debug: 

gaX.verbose = True 

return gaX 

 

@Property_RO 

def b(self): 

'''Get the ellipsoid's I{polar} radius, semi-axis (C{meter}). 

''' 

return self.ellipsoid.b 

 

@Property_RO 

def c2x(self): 

'''Get the ellipsoid's I{authalic} earth radius I{squared} (C{meter} I{squared}). 

''' 

# The Geodesic class substitutes atanh(sqrt(e2)) for asinh(sqrt(ep2)) 

# in the definition of _c2. The latter is more accurate for very 

# oblate ellipsoids (which the Geodesic class does not handle). Of 

# course, the area calculation in GeodesicExact is still based on a 

# series and only holds for moderately oblate (or prolate) ellipsoids. 

return self.ellipsoid.c2x 

 

c2 = c2x # in this particular case 

 

def C4f(self, eps): 

'''Evaluate the C{C4x} coefficients for B{C{eps}}. 

 

@arg eps: Polynomial factor (C{float}). 

 

@return: C{C4order}-Tuple of C{C4x(B{eps})} coefficients. 

''' 

def _c4(nC4, C4x): 

i, x, e = 0, _1_0, eps 

_p = _polynomial 

for r in range(nC4, 0, -1): 

j = i + r 

yield _p(e, C4x, i, j) * x 

x *= e 

i = j 

# assert i == (nC4 * (nC4 + 1)) // 2 

 

return tuple(_c4(self._nC4, self._C4x)) 

 

def _C4f_k2(self, k2): # in ._GDictInverse and gxline._GeodesicLineExact._C4a 

'''(INTERNAL) Compute C{eps} from B{C{k2}} and invoke C{C4f}. 

''' 

return self.C4f(k2 / fsum_(_2_0, sqrt(k2 + _1_0) * _2_0, k2)) 

 

@Property 

def C4order(self): 

'''Get the series expansion order (C{int}, 24, 27 or 30). 

''' 

return self._nC4 

 

@C4order.setter # PYCHOK .setter! 

def C4order(self, order): 

'''Set the series expansion order (C{int}, 24, 27 or 30). 

 

@raise GeodesicError: Invalid B{C{order}}. 

''' 

_xnC4(C4order=order) 

if self._nC4 != order: 

GeodesicExact._C4x._update(self) 

_update_glXs(self) # zap cached _GeodesicLineExact attrs _B41, _C4a 

self._nC4 = order 

 

@deprecated_Property 

def C4Order(self): 

'''DEPRECATED, use property C{C4order}. 

''' 

return self.C4order 

 

@C4Order.setter # PYCHOK .setter! 

def C4Order(self, order): 

'''DEPRECATED, use property C{C4order}. 

''' 

_xnC4(C4Order=order) 

self.C4order = order 

 

def _coeffs(self, nC4): 

'''(INTERNAL) Get the C{C4_24}, C{_27} or C{_30} series coefficients. 

''' 

try: # from pygeodesy.geodesicx._C4_xx import _coeffs_xx as _coeffs 

_C4_xx = _DOT_(_MODS.geodesicx.__name__, NN('_C4_', nC4)) 

_coeffs = _MODS.getattr(_C4_xx, NN('_coeffs_', nC4)) 

except (AttributeError, ImportError, TypeError) as x: 

raise GeodesicError(nC4=nC4, txt=str(x)) 

n = _xnC4(nC4=nC4) 

if len(_coeffs) != n: # double check 

raise GeodesicError(_coeffs=len(_coeffs), _xnC4=n, nC4=nC4) 

return _coeffs 

 

@Property_RO 

def _C4x(self): 

'''Get this ellipsoid's C{C4} coefficients, I{cached} tuple. 

 

@see: Property L{C4order}. 

''' 

# see C4coeff() in GeographicLib.src.GeodesicExactC4.cpp 

def _C4(nC4): 

i, n, cs = 0, self.n, self._coeffs(nC4) 

_p = _polynomial 

for r in range(nC4 + 1, 1, -1): 

for j in range(1, r): 

j = j + i # (j - i - 1) order of polynomial 

yield _p(n, cs, i, j) / cs[j] 

i = j + 1 

# assert i == (nC4 * (nC4 + 1) * (nC4 + 5)) // 6 

 

return tuple(_C4(self._nC4)) # 3rd flattening 

 

def Direct(self, lat1, lon1, azi1, s12, outmask=Caps.STANDARD): 

'''Solve the I{Direct} geodesic problem 

 

@arg lat1: Latitude of the first point (C{degrees}). 

@arg lon1: Longitude of the first point (C{degrees}). 

@arg azi1: Azimuth at the first point (compass C{degrees}). 

@arg s12: Distance between the points (C{meter}), can be negative. 

@kwarg outmask: Bit-or'ed combination of L{Caps} values specifying 

the quantities to be returned. 

 

@return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2, 

lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1}, 

C{lon1}, C{azi1} and distance C{s12} always included. 

 

@see: C++ U{GeodesicExact.Direct 

<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} 

and Python U{Geodesic.Direct<https://GeographicLib.SourceForge.io/C++/doc/python/code.html>}. 

''' 

return self._GDictDirect(lat1, lon1, azi1, False, s12, outmask) 

 

def Direct3(self, lat1, lon1, azi1, s12): # PYCHOK outmask 

'''Return the destination lat, lon and reverse azimuth 

(final bearing) in C{degrees}. 

 

@return: L{Destination3Tuple}C{(lat, lon, final)}. 

''' 

r = self._GDictDirect(lat1, lon1, azi1, False, s12, Caps._AZIMUTH_LATITUDE_LONGITUDE) 

return Destination3Tuple(r.lat2, r.lon2, r.azi2) # no iteration 

 

def DirectLine(self, lat1, lon1, azi1, s12, caps=Caps.STANDARD): 

'''Define a L{GeodesicLineExact} in terms of the I{direct} geodesic problem and as distance. 

 

@arg lat1: Latitude of the first point (C{degrees}). 

@arg lon1: Longitude of the first point (C{degrees}). 

@arg azi1: Azimuth at the first point (compass C{degrees}). 

@arg s12: Distance between the points (C{meter}), can be negative. 

@kwarg caps: Bit-or'ed combination of L{Caps} values specifying 

the capabilities the L{GeodesicLineExact} instance 

should possess, i.e., which quantities can be 

returned by calls to L{GeodesicLineExact.Position}. 

 

@return: A L{GeodesicLineExact} instance. 

 

@note: The third point of the L{GeodesicLineExact} is set to correspond 

to the second point of the I{Inverse} geodesic problem. 

 

@note: Latitude B{C{lat1}} should in the range C{[-90, +90]}. 

 

@see: C++ U{GeodesicExact.DirectLine 

<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} and 

Python U{Geodesic.DirectLine<https://GeographicLib.SourceForge.io/C++/doc/python/code.html>}. 

''' 

return self._GenDirectLine(lat1, lon1, azi1, False, s12, caps) 

 

def _dn(self, sbet, cbet): # in gxline._GeodesicLineExact.__init__ 

'''(INTERNAL) Helper. 

''' 

if self.f < 0: # PYCHOK no cover 

dn = sqrt(_1_0 - cbet**2 * self.e2) / self.f1 

else: 

dn = sqrt(_1_0 + sbet**2 * self.ep2) 

return dn 

 

@Property_RO 

def e2(self): 

'''Get the ellipsoid's I{(1st) eccentricity squared} (C{float}), M{f * (2 - f)}. 

''' 

return self.ellipsoid.e2 

 

@Property_RO 

def _e2a2(self): 

'''(INTERNAL) Cache M{E.e2 * E.a2}. 

''' 

return self.e2 * self.ellipsoid.a2 

 

@Property_RO 

def _e2_f1(self): 

'''(INTERNAL) Cache M{E.e2 * E.f1}. 

''' 

return self.e2 / self.f1 

 

@Property_RO 

def _eF(self): 

'''(INTERNAL) Get the elliptic function, aka C{.E}. 

''' 

return _MODS.elliptic.Elliptic(k2=-self.ep2) 

 

def _eF_reset_cHe2_f1(self, x, y): 

'''(INTERNAL) Reset elliptic function and return M{cH * e2 / f1 * ...}. 

''' 

self._eF_reset_k2(x) 

return y * self._eF.cH * self._e2_f1 

 

def _eF_reset_k2(self, x): 

'''(INTERNAL) Reset elliptic function and return C{k2}. 

''' 

ep2 = self.ep2 

k2 = x**2 * ep2 # see .gxline._GeodesicLineExact._eF 

self._eF.reset(k2=-k2, alpha2=-ep2) # kp2, alphap2 defaults 

_update_glXs(self) # zap cached/memoized _GeodesicLineExact attrs 

return k2 

 

@Property_RO 

def ellipsoid(self): 

'''Get the ellipsoid (C{Ellipsoid}). 

''' 

return self._E 

 

@Property_RO 

def ep2(self): 

'''Get the ellipsoid's I{2nd eccentricity squared} (C{float}), M{e2 / (1 - e2)}. 

''' 

return self.ellipsoid.e22 # == self.e2 / self.f1**2 

 

e22 = ep2 # for ellipsoid compatibility 

 

@Property_RO 

def _eTOL2(self): 

'''(INTERNAL) The si12 threshold for "really short". 

''' 

return _eTOL2(self.f) 

 

@Property_RO 

def f(self): 

'''Get the ellipsoid's I{flattening} (C{float}), M{(a - b) / a}, C{0} for spherical, negative for prolate. 

''' 

return self.ellipsoid.f 

 

flattening = f 

 

@Property_RO 

def f1(self): # in .css.CassiniSoldner.reset 

'''Get the ellipsoid's I{1 - flattening} (C{float}). 

''' 

return self.ellipsoid.f1 

 

@Property_RO 

def _f180(self): 

'''(INTERNAL) Cached/memoized. 

''' 

return self.f * _180_0 

 

def _GDictDirect(self, lat1, lon1, azi1, arcmode, s12_a12, outmask=Caps.STANDARD): 

'''(INTERNAL) As C{_GenDirect}, but returning a L{GDict}. 

 

@return: A L{GDict} ... 

''' 

C = outmask if arcmode else (outmask | Caps.DISTANCE_IN) 

glX = self.Line(lat1, lon1, azi1, C | Caps.LINE_OFF) 

return glX._GDictPosition(arcmode, s12_a12, outmask) 

 

def _GDictInverse(self, lat1, lon1, lat2, lon2, outmask=Caps.STANDARD): # MCCABE 33, 41 vars 

'''(INTERNAL) As C{_GenInverse}, but returning a L{GDict}. 

 

@return: A L{GDict} ... 

''' 

if self._debug: # PYCHOK no cover 

outmask |= Caps._DEBUG_INVERSE & self._debug 

outmask &= Caps._OUT_MASK # incl. _SALPs_CALPs and _DEBUG_ 

# compute longitude difference carefully (with _diff182): 

# result is in [-180, +180] but -180 is only for west-going 

# geodesics, +180 is for east-going and meridional geodesics 

lon12, lon12s = _diff182(lon1, lon2) 

# see C{result} from geographiclib.geodesic.Inverse 

if (outmask & Caps.LONG_UNROLL): # == (lon1 + lon12) + lon12s 

r = GDict(lon1=lon1, lon2=fsum_(lon1, lon12, lon12s)) 

else: 

r = GDict(lon1=_norm180(lon1), lon2=_norm180(lon2)) 

if _K_2_0: # GeographicLib 2.0 

# make longitude difference positive 

lon_ = _signBit(lon12) 

if lon_: 

lon12 = -lon12 

lon12s = -lon12s 

lam12 = radians(lon12) 

# calculate sincosd(_around(lon12 + correction)) 

slam12, clam12 = _sincos2de(lon12, lon12s) 

# supplementary longitude difference 

lon12s = fsum_(_180_0, -lon12, -lon12s) 

else: # GeographicLib 1.52 

# make longitude difference positive and if very close 

# to being on the same half-meridian, then make it so. 

if lon12 < 0: # _signBit(lon12) 

lon_, lon12 = True, -_around(lon12) 

lon12s = _around(fsum_(_180_0, -lon12, lon12s)) 

else: 

lon_, lon12 = False, _around(lon12) 

lon12s = _around(fsum_(_180_0, -lon12, -lon12s)) 

lam12 = radians(lon12) 

if lon12 > _90_0: 

slam12, clam12 = _sincos2d(lon12s) 

clam12 = -clam12 

else: 

slam12, clam12 = _sincos2(lam12) 

# If really close to the equator, treat as on equator. 

lat1 = _around(_fix90(lat1)) 

lat2 = _around(_fix90(lat2)) 

r.set_(lat1=lat1, lat2=lat2) 

# Swap points so that point with higher (abs) latitude is 

# point 1. If one latitude is a NAN, then it becomes lat1. 

swap_ = fabs(lat1) < fabs(lat2) or isnan(lat2) 

if swap_: 

lat1, lat2 = lat2, lat1 

lon_ = not lon_ 

if _signBit(lat1): 

lat_ = False # note False! 

else: # make lat1 <= -0 

lat_ = True 

lat1, lat2 = -lat1, -lat2 

# Now 0 <= lon12 <= 180, -90 <= lat1 <= -0 and lat1 <= lat2 <= -lat1 

# and lat_, lon_, swap_ register the transformation to bring the 

# coordinates to this canonical form, where False means no change 

# 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. 

 

# Initialize for the meridian. No longitude calculation is 

# done in this case to let the parameter default to 0. 

sbet1, cbet1 = self._sinf1cos2d(lat1) 

sbet2, cbet2 = self._sinf1cos2d(lat2) 

# 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 _Lambda6. 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 = copysign(sbet1, sbet2) 

elif fabs(sbet2) == -sbet1: 

cbet2 = cbet1 

 

p = _PDict(sbet1=sbet1, cbet1=cbet1, dn1=self._dn(sbet1, cbet1), 

sbet2=sbet2, cbet2=cbet2, dn2=self._dn(sbet2, cbet2)) 

 

_meridian = _b = True # i.e. not meridian, not b 

if lat1 == -90 or slam12 == 0: 

# Endpoints are on a single full meridian, 

# so the geodesic might lie on a meridian. 

salp1, calp1 = slam12, clam12 # head to target lon 

salp2, calp2 = _0_0, _1_0 # then head north 

# tan(bet) = tan(sig) * cos(alp) 

p.setsigs(sbet1, calp1 * cbet1, sbet2, calp2 * cbet2) 

# sig12 = sig2 - sig1 

sig12 = _atan12(sbet1, p.csig1, sbet2, p.csig2, sineg0=True) # PYCHOK csig* 

s12x, m12x, _, \ 

M12, M21 = self._Length5(sig12, outmask | Caps.REDUCEDLENGTH, p) 

# 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 m12x >= 0 or sig12 < _1_0: 

# Need at least 2 to handle 90 0 90 180 

# Prevent negative s12 or m12 from geographiclib 1.52 

if sig12 < _TINY3 or (sig12 < _TOL0 and (s12x < 0 or m12x < 0)): 

sig12 = m12x = s12x = _0_0 

else: 

_b = False # apply .b to s12x, m12x 

_meridian = False 

C = 1 

# else: # m12 < 0, prolate and too close to anti-podal 

# _meridian = True 

a12 = _0_0 # if _b else degrees(sig12) 

 

if _meridian: 

_b = sbet1 == 0 and (self.f <= 0 or lon12s >= self._f180) # and sbet2 == 0 

if _b: # Geodesic runs along equator 

calp1 = calp2 = _0_0 

salp1 = salp2 = _1_0 

sig12 = lam12 / self.f1 # == omg12 

somg12, comg12 = _sincos2(sig12) 

m12x = self.b * somg12 

s12x = self.a * lam12 

M12 = M21 = comg12 

a12 = lon12 / self.f1 

C = 2 

else: 

# Now point1 and point2 belong within a hemisphere bounded by a 

# meridian and geodesic is neither meridional or equatorial. 

p.set_(slam12=slam12, clam12=clam12) 

# Figure a starting point for Newton's method 

sig12, salp1, calp1, \ 

salp2, calp2, dnm = self._InverseStart6(lam12, p) 

if sig12 is None: # use Newton's method 

# pre-compute the constant _Lambda6 term, once 

p.set_(bet12=None if cbet2 == cbet1 and fabs(sbet2) == -sbet1 else 

(((cbet1 + cbet2) * (cbet2 - cbet1)) if cbet1 < -sbet1 else 

((sbet1 + sbet2) * (sbet1 - sbet2)))) 

sig12, salp1, calp1, \ 

salp2, calp2, domg12 = self._Newton6(salp1, calp1, p) 

s12x, m12x, _, M12, M21 = self._Length5(sig12, outmask, p) 

if (outmask & Caps.AREA): 

# omg12 = lam12 - domg12 

s, c = _sincos2(domg12) 

somg12, comg12 = _sincos12(s, c, slam12, clam12) 

C = 3 # Newton 

else: # from _InverseStart6: dnm, salp*, calp* 

C = 4 # Short lines 

s, c = _sincos2(sig12 / dnm) 

m12x = dnm**2 * s 

s12x = dnm * sig12 

M12 = M21 = c 

if (outmask & Caps.AREA): 

somg12, comg12 = _sincos2(lam12 / (self.f1 * dnm)) 

 

else: # _meridian is False 

somg12 = comg12 = NAN 

 

r.set_(a12=a12 if _b else degrees(sig12)) # in [0, 180] 

 

if (outmask & Caps.DISTANCE): 

r.set_(s12=unsigned0(s12x if _b else (self.b * s12x))) 

 

if (outmask & Caps.REDUCEDLENGTH): 

r.set_(m12=unsigned0(m12x if _b else (self.b * m12x))) 

 

if (outmask & Caps.GEODESICSCALE): 

if swap_: 

M12, M21 = M21, M12 

r.set_(M12=unsigned0(M12), 

M21=unsigned0(M21)) 

 

if (outmask & Caps.AREA): 

S12 = self._InverseArea(_meridian, salp1, calp1, 

salp2, calp2, 

somg12, comg12, p) 

if _xor(swap_, lat_, lon_): 

S12 = -S12 

r.set_(S12=unsigned0(S12)) 

 

if (outmask & (Caps.AZIMUTH | Caps._SALPs_CALPs)): 

if swap_: 

salp1, salp2 = salp2, salp1 

calp1, calp2 = calp2, calp1 

if _xor(swap_, lon_): 

salp1, salp2 = -salp1, -salp2 

if _xor(swap_, lat_): 

calp1, calp2 = -calp1, -calp2 

 

if (outmask & Caps.AZIMUTH): 

r.set_(azi1=_atan2d(salp1, calp1), 

azi2=_atan2d_reverse(salp2, calp2, reverse=outmask & Caps.REVERSE2)) 

if (outmask & Caps._SALPs_CALPs): 

r.set_(salp1=salp1, calp1=calp1, 

salp2=salp2, calp2=calp2) 

 

if (outmask & Caps._DEBUG_INVERSE): # PYCHOK no cover 

E, eF = self.ellipsoid, self._eF 

p.set_(C=C, a=self.a, f=self.f, f1=self.f1, 

e=E.e, e2=self.e2, ep2=self.ep2, 

c2=E.c2, c2x=self.c2x, 

eFcD=eF.cD, eFcE=eF.cE, eFcH=eF.cH, 

eFk2=eF.k2, eFa2=eF.alpha2) 

p.update(r) # r overrides p 

r = p.toGDict() 

return self._iter2tion(r, p) 

 

def _GenDirect(self, lat1, lon1, azi1, arcmode, s12_a12, outmask): 

'''(INTERNAL) The general I{Inverse} geodesic calculation. 

 

@return: L{Direct9Tuple}C{(a12, lat2, lon2, azi2, 

s12, m12, M12, M21, S12)}. 

''' 

r = self._GDictDirect(lat1, lon1, azi1, arcmode, s12_a12, outmask) 

return r.toDirect9Tuple() 

 

def _GenDirectLine(self, lat1, lon1, azi1, arcmode, s12_a12, caps): 

'''(INTERNAL) Helper for C{ArcDirectLine} and C{DirectLine}. 

 

@return: A L{GeodesicLineExact} instance. 

''' 

azi1 = _norm180(azi1) 

# guard against underflow in salp0. Also -0 is converted to +0. 

s, c = _sincos2d(_around(azi1)) 

C = caps if arcmode else (caps | Caps.DISTANCE_IN) 

return _GeodesicLineExact(self, lat1, lon1, azi1, C, 

self._debug, s, c)._GenSet(arcmode, s12_a12) 

 

def _GenInverse(self, lat1, lon1, lat2, lon2, outmask): 

'''(INTERNAL) The general I{Inverse} geodesic calculation. 

 

@return: L{Inverse10Tuple}C{(a12, s12, salp1, calp1, salp2, calp2, 

m12, M12, M21, S12)}. 

''' 

r = self._GDictInverse(lat1, lon1, lat2, lon2, outmask | Caps._SALPs_CALPs) 

return r.toInverse10Tuple() 

 

def Inverse(self, lat1, lon1, lat2, lon2, outmask=Caps.STANDARD): 

'''Perform the I{Inverse} geodesic calculation. 

 

@arg lat1: Latitude of the first point (C{degrees}). 

@arg lon1: Longitude of the first point (C{degrees}). 

@arg lat2: Latitude of the second point (C{degrees}). 

@arg lon2: Longitude of the second point (C{degrees}). 

@kwarg outmask: Bit-or'ed combination of L{Caps} values specifying 

the quantities to be returned. 

 

@return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2, 

lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1}, 

C{lon1}, C{azi1} and distance C{s12} always included. 

 

@note: The third point of the L{GeodesicLineExact} is set to correspond 

to the second point of the I{Inverse} geodesic problem. 

 

@note: Both B{C{lat1}} and B{C{lat2}} should in the range C{[-90, +90]}. 

 

@see: C++ U{GeodesicExact.InverseLine 

<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} and 

Python U{Geodesic.InverseLine<https://GeographicLib.SourceForge.io/C++/doc/python/code.html>}. 

''' 

return self._GDictInverse(lat1, lon1, lat2, lon2, outmask) 

 

def Inverse1(self, lat1, lon1, lat2, lon2, wrap=False): 

'''Return the non-negative, I{angular} distance in C{degrees}. 

''' 

# see .FrechetKarney.distance, .HausdorffKarney._distance 

# and .HeightIDWkarney._distances 

_, lon2 = unroll180(lon1, lon2, wrap=wrap) # self.LONG_UNROLL 

return fabs(self._GDictInverse(lat1, lon1, lat2, lon2, Caps._ANGLE_ONLY).a12) 

 

def Inverse3(self, lat1, lon1, lat2, lon2): # PYCHOK outmask 

'''Return the distance in C{meter} and the forward and 

reverse azimuths (initial and final bearing) in C{degrees}. 

 

@return: L{Distance3Tuple}C{(distance, initial, final)}. 

''' 

r = self._GDictInverse(lat1, lon1, lat2, lon2, Caps.AZIMUTH_DISTANCE) 

return Distance3Tuple(r.s12, wrap360(r.azi1), wrap360(r.azi2), 

iteration=r.iteration) 

 

def InverseLine(self, lat1, lon1, lat2, lon2, caps=Caps.STANDARD): 

'''Define a L{GeodesicLineExact} in terms of the I{Inverse} geodesic problem. 

 

@arg lat1: Latitude of the first point (C{degrees}). 

@arg lon1: Longitude of the first point (C{degrees}). 

@arg lat2: Latitude of the second point (C{degrees}). 

@arg lon2: Longitude of the second point (C{degrees}). 

@kwarg caps: Bit-or'ed combination of L{Caps} values specifying 

the capabilities the L{GeodesicLineExact} instance 

should possess, i.e., which quantities can be 

returned by calls to L{GeodesicLineExact.Position} 

and L{GeodesicLineExact.ArcPosition}. 

 

@return: A L{GeodesicLineExact} instance. 

 

@note: The third point of the L{GeodesicLineExact} is set to correspond 

to the second point of the I{Inverse} geodesic problem. 

 

@note: Both B{C{lat1}} and B{C{lat2}} should in the range C{[-90, +90]}. 

 

@see: C++ U{GeodesicExact.InverseLine 

<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} and 

Python U{Geodesic.InverseLine<https://GeographicLib.SourceForge.io/C++/doc/python/code.html>}. 

''' 

r = self._GDictInverse(lat1, lon1, lat2, lon2, Caps._SALPs_CALPs) # No need for AZIMUTH 

C = (caps | Caps.DISTANCE) if (caps & Caps._DISTANCE_IN_OUT) else caps 

azi1 = _atan2d(r.salp1, r.calp1) 

return _GeodesicLineExact(self, lat1, lon1, azi1, C, # ensure a12 is distance 

self._debug, r.salp1, r.calp1)._GenSet(True, r.a12) 

 

def _InverseArea(self, _meridian, salp1, calp1, # PYCHOK 9 args 

salp2, calp2, 

somg12, comg12, p): 

'''(INTERNAL) Split off from C{_GDictInverse} to reduce complexity/length. 

 

@return: Area C{S12}. 

''' 

# from _Lambda6: sin(alp1) * cos(bet1) = sin(alp0), calp0 > 0 

salp0, calp0 = _sin1cos2(salp1, calp1, p.sbet1, p.cbet1) 

A4 = calp0 * salp0 

if A4: 

# from _Lambda6: tan(bet) = tan(sig) * cos(alp) 

k2 = calp0**2 * self.ep2 

C4a = self._C4f_k2(k2) 

B41 = _cosSeries(C4a, *_norm2(p.sbet1, calp1 * p.cbet1)) 

B42 = _cosSeries(C4a, *_norm2(p.sbet2, calp2 * p.cbet2)) 

# multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0) 

A4 *= self._e2a2 

S12 = A4 * (B42 - B41) 

else: # avoid problems with indeterminate sig1, sig2 on equator 

A4 = B41 = B42 = k2 = S12 = _0_0 

 

if (_meridian and # omg12 < 3/4 * PI 

comg12 > -_SQRT2_2 and # lon diff not too big 

(p.sbet2 - p.sbet1) < _1_75): # lat diff not too big 

# use tan(Gamma/2) = tan(omg12/2) * 

# (tan(bet1/2) + tan(bet2/2)) / 

# (tan(bet1/2) * tan(bet2/2) + 1)) 

# with tan(x/2) = sin(x) / (1 + cos(x)) 

dbet1 = p.cbet1 + _1_0 

dbet2 = p.cbet2 + _1_0 

domg12 = comg12 + _1_0 

salp12 = (p.sbet1 * dbet2 + dbet1 * p.sbet2) * somg12 

calp12 = (p.sbet1 * p.sbet2 + dbet1 * dbet2) * domg12 

alp12 = _2_0 * atan2(salp12, calp12) 

else: 

# alp12 = alp2 - alp1, used in atan2, no need to normalize 

salp12, calp12 = _sincos12(salp1, calp1, salp2, calp2) 

# 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. 

# Following ensures the correct behavior. 

if salp12 == 0 and calp12 < 0: 

alp12 = _copysign(PI, calp1) 

else: 

alp12 = atan2(salp12, calp12) 

 

p.set_(alp12=alp12, A4=A4, B41=B41, B42=B42, k2=k2) 

return S12 + self.c2x * alp12 

 

def _InverseStart6(self, lam12, p): 

'''(INTERNAL) Return a starting point for Newton's method in 

C{salp1} and C{calp1} indicated by C{sig12=None}. If 

Newton's method doesn't need to be used, return also 

C{salp2}, C{calp2}, C{dnm} and C{sig12} non-C{None}. 

 

@return: 6-Tuple C{(sig12, salp1, calp1, salp2, calp2, dnm)} 

and C{p.setsigs} updated for Newton, C{sig12=None}. 

''' 

sig12 = None # use Newton 

salp1 = calp1 = salp2 = calp2 = dnm = NAN 

 

# bet12 = bet2 - bet1 in [0, PI) 

sbet12, cbet12 = _sincos12(p.sbet1, p.cbet1, p.sbet2, p.cbet2) 

shortline = cbet12 >= 0 and sbet12 < _0_5 and (p.cbet2 * lam12) < _0_5 

if shortline: 

# sin((bet1 + bet2)/2)^2 = (sbet1 + sbet2)^2 / ( 

# (cbet1 + cbet2)^2 + (sbet1 + sbet2)^2) 

t = (p.sbet1 + p.sbet2)**2 

s = t / ((p.cbet1 + p.cbet2)**2 + t) 

dnm = sqrt(_1_0 + self.ep2 * s) 

somg12, comg12 = _sincos2(lam12 / (self.f1 * dnm)) 

else: 

somg12, comg12 = p.slam12, p.clam12 

 

# bet12a = bet2 + bet1 in (-PI, 0], note -sbet1 

sbet12a, cbet12a = _sincos12(-p.sbet1, p.cbet1, p.sbet2, p.cbet2) 

 

c = fabs(comg12) + _1_0 # == (1 - comg12) if comg12 < 0 

s = somg12**2 / c 

t = p.sbet1 * p.cbet2 * s 

salp1 = p.cbet2 * somg12 

calp1 = (sbet12a - t) if comg12 < 0 else (sbet12 + t) 

 

ssig12 = _hypot(salp1, calp1) 

csig12 = p.sbet1 * p.sbet2 + p.cbet1 * p.cbet2 * comg12 

 

if shortline and ssig12 < self._eTOL2: # really short lines 

t = c if comg12 < 0 else s 

salp2, calp2 = _norm2(somg12 * p.cbet1, 

sbet12 - p.cbet1 * p.sbet2 * t) 

sig12 = atan2(ssig12, csig12) # do not use Newton 

 

elif (self._n_0_1 or # Skip astroid calc if too eccentric 

csig12 >= 0 or ssig12 >= (p.cbet1**2 * self._n6PI)): 

pass # nothing to do, 0th order spherical approximation OK 

 

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 

lam12x = atan2(-p.slam12, -p.clam12) # lam12 - PI 

f = self.f 

if f < 0: # PYCHOK no cover 

# ssig1=sbet1, csig1=-cbet1, ssig2=sbet2, csig2=cbet2 

p.setsigs(p.sbet1, -p.cbet1, p.sbet2, p.cbet2) 

# if lon12 = 180, this repeats a calculation made in Inverse 

_, m12b, m0, _, _ = self._Length5(atan2(sbet12a, cbet12a) + PI, 

Caps.REDUCEDLENGTH, p) 

t = p.cbet1 * PI # x = dlat, y = dlon 

x = m12b / (t * p.cbet2 * m0) - _1_0 

sca = (sbet12a / (x * p.cbet1)) if x < -_0_01 else (-f * t) 

y = lam12x / sca 

else: # f >= 0, however f == 0 does not get here 

sca = self._eF_reset_cHe2_f1(p.sbet1, p.cbet1 * _2_0) 

x = lam12x / sca # dlon 

y = sbet12a / (sca * p.cbet1) # dlat 

 

if y > _TOL1 and x > -_THR1: # strip near cut 

if f < 0: # PYCHOK no cover 

calp1 = max(_0_0, x) if x > _TOL1 else max(_N_1_0, x) 

salp1 = sqrt(_1_0 - calp1**2) 

else: 

salp1 = min(_1_0, -x) 

calp1 = -sqrt(_1_0 - salp1**2) 

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 

# 

# omg12 is near PI, estimate work with omg12a = PI - omg12 

k = _Astroid(x, y) 

sca *= (y * (k + _1_0) / k) if f < 0 else \ 

(x * k / (k + _1_0)) 

s, c = _sincos2(-sca) # omg12a 

# update spherical estimate of alp1 using omg12 instead of lam12 

salp1 = p.cbet2 * s 

calp1 = sbet12a - s * salp1 * p.sbet1 / (c + _1_0) # c = -c 

 

# sanity check on starting guess. Backwards check allows NaN through. 

salp1, calp1 = _norm2(salp1, calp1) if salp1 > 0 else (_1_0, _0_0) 

 

return sig12, salp1, calp1, salp2, calp2, dnm 

 

def _Lambda6(self, salp1, calp1, diffp, p): 

'''(INTERNAL) Helper. 

 

@return: 6-Tuple C{(lam12, sig12, salp2, calp2, domg12, 

dlam12} and C{p.setsigs} updated. 

''' 

eF = self._eF 

f1 = self.f1 

 

if p.sbet1 == 0 and calp1 == 0: # PYCHOK no cover 

# Break degeneracy of equatorial line 

calp1 = -_TINY 

 

# sin(alp1) * cos(bet1) = sin(alp0), # calp0 > 0 

salp0, calp0 = _sin1cos2(salp1, calp1, p.sbet1, p.cbet1) 

# tan(bet1) = tan(sig1) * cos(alp1) 

# tan(omg1) = sin(alp0) * tan(sig1) 

# = sin(bet1) * tan(alp1) 

somg1 = salp0 * p.sbet1 

comg1 = calp1 * p.cbet1 

ssig1, csig1 = _norm2(p.sbet1, comg1) 

# Without normalization we have schi1 = somg1 

cchi1 = f1 * p.dn1 * comg1 

 

# 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 / p.cbet2) if p.cbet2 != p.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 = fabs(calp1) if p.bet12 is None else ( 

sqrt((calp1 * p.cbet1)**2 + p.bet12) / p.cbet2) 

# tan(bet2) = tan(sig2) * cos(alp2) 

# tan(omg2) = sin(alp0) * tan(sig2). 

somg2 = salp0 * p.sbet2 

comg2 = calp2 * p.cbet2 

ssig2, csig2 = _norm2(p.sbet2, comg2) 

# without normalization we have schi2 = somg2 

cchi2 = f1 * p.dn2 * comg2 

 

# omg12 = omg2 - omg1, limit to [0, PI] 

somg12, comg12 = _sincos12(somg1, comg1, somg2, comg2, sineg0=True) 

# chi12 = chi2 - chi1, limit to [0, PI] 

schi12, cchi12 = _sincos12(somg1, cchi1, somg2, cchi2, sineg0=True) 

 

p.setsigs(ssig1, csig1, ssig2, csig2) 

# sig12 = sig2 - sig1, limit to [0, PI] 

sig12 = _atan12(ssig1, csig1, ssig2, csig2, sineg0=True) 

 

eta12 = self._eF_reset_cHe2_f1(calp0, salp0) * _2__PI # then ... 

eta12 *= fsum1_(eF.deltaH(*p.sncndn2), 

-eF.deltaH(*p.sncndn1), sig12) 

# eta = chi12 - lam12 

lam12 = _atan12(p.slam12, p.clam12, schi12, cchi12) - eta12 

# domg12 = chi12 - omg12 - deta12 

domg12 = _atan12( somg12, comg12, schi12, cchi12) - eta12 

 

dlam12 = NAN # dv > 0 in ._Newton6 

if diffp: 

d = calp2 * p.cbet2 

if d: 

_, dlam12, _, _, _ = self._Length5(sig12, Caps.REDUCEDLENGTH, p) 

dlam12 *= f1 / d 

elif p.sbet1: 

dlam12 = -f1 * p.dn1 * _2_0 / p.sbet1 

 

# p.set_(deta12=-eta12, lam12=lam12) 

return lam12, sig12, salp2, calp2, domg12, dlam12 

 

def _Length5(self, sig12, outmask, p): 

'''(INTERNAL) Return M{m12b = (reduced length) / self.b} and 

calculate M{s12b = distance / self.b} and M{m0}, the 

coefficient of secular term in expression for reduced 

length and the geodesic scales C{M12} and C{M21}. 

 

@return: 5-Tuple C{(s12b, m12b, m0, M12, M21)}. 

''' 

s12b = m12b = m0 = M12 = M21 = NAN 

 

eF = self._eF 

 

# outmask &= Caps._OUT_MASK 

if (outmask & Caps.DISTANCE): 

# Missing a factor of self.b 

s12b = eF.cE * _2__PI * fsum1_(eF.deltaE(*p.sncndn2), 

-eF.deltaE(*p.sncndn1), sig12) 

 

if (outmask & Caps._REDUCEDLENGTH_GEODESICSCALE): 

m0x = -eF.k2 * eF.cD * _2__PI 

J12 = -m0x * fsum1_(eF.deltaD(*p.sncndn2), 

-eF.deltaD(*p.sncndn1), sig12) 

if (outmask & Caps.REDUCEDLENGTH): 

m0 = m0x 

# Missing a factor of self.b. Add parens around 

# (csig1 * ssig2) and (ssig1 * csig2) to ensure 

# accurate cancellation for coincident points. 

m12b = fsum1_(p.dn2 * (p.csig1 * p.ssig2), 

-p.dn1 * (p.ssig1 * p.csig2), 

J12 * (p.csig1 * p.csig2)) 

if (outmask & Caps.GEODESICSCALE): 

M12 = M21 = p.ssig1 * p.ssig2 + \ 

p.csig1 * p.csig2 

t = (p.cbet1 - p.cbet2) * self.ep2 * \ 

(p.cbet1 + p.cbet2) / (p.dn1 + p.dn2) 

M12 += (p.ssig2 * t + p.csig2 * J12) * p.ssig1 / p.dn1 

M21 -= (p.ssig1 * t + p.csig1 * J12) * p.ssig2 / p.dn2 

 

return s12b, m12b, m0, M12, M21 

 

def Line(self, lat1, lon1, azi1, caps=Caps.ALL): 

'''Set up a L{GeodesicLineExact} to compute several points 

on a single geodesic. 

 

@arg lat1: Latitude of the first point (C{degrees}). 

@arg lon1: Longitude of the first point (C{degrees}). 

@arg azi1: Azimuth at the first point (compass C{degrees}). 

@kwarg caps: Bit-or'ed combination of L{Caps} values specifying 

the capabilities the L{GeodesicLineExact} instance 

should possess, i.e., which quantities can be 

returnedby calls to L{GeodesicLineExact.Position} 

and L{GeodesicLineExact.ArcPosition}. 

 

@return: A L{GeodesicLineExact} instance. 

 

@note: If the point is at a pole, the azimuth is defined by keeping 

B{C{lon1}} fixed, writing C{B{lat1} = ±(90 − ε)}, and taking 

the limit C{ε → 0+}. 

 

@see: C++ U{GeodesicExact.Line 

<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} 

and Python U{Geodesic.Line<https://GeographicLib.SourceForge.io/C++/doc/python/code.html>}. 

''' 

return _GeodesicLineExact(self, lat1, lon1, azi1, caps, self._debug) 

 

@Property_RO 

def n(self): 

'''Get the ellipsoid's I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}. 

''' 

return self.ellipsoid.n 

 

@Property_RO 

def _n_0_1(self): 

'''(INTERNAL) Cached once. 

''' 

return fabs(self.n) > _0_1 

 

@Property_RO 

def _n6PI(self): 

'''(INTERNAL) Cached once. 

''' 

return fabs(self.n) * _6_0 * PI 

 

def _Newton6(self, salp1, calp1, p): 

'''(INTERNAL) Split off from C{_GDictInverse} to reduce complexity/length. 

 

@return: 6-Tuple C{(sig12, salp1, calp1, salp2, calp2, domg12)} 

and C{p.iter} and C{p.trip} updated. 

''' 

# 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. 

salp1a = salp1b = _TINY 

calp1a, calp1b = _1_0, _N_1_0 

tripb, TOLv = False, _TOL0 

for i in range(_MAXIT2): 

# 1/4 meridian = 10e6 meter and random input, 

# estimated max error in nm (nano meter, by 

# checking Inverse problem by Direct). 

# 

# max iterations 

# log2(b/a) error mean sd 

# -7 387 5.33 3.68 

# -6 345 5.19 3.43 

# -5 269 5.00 3.05 

# -4 210 4.76 2.44 

# -3 115 4.55 1.87 

# -2 69 4.35 1.38 

# -1 36 4.05 1.03 

# 0 15 0.01 0.13 

# 1 25 5.10 1.53 

# 2 96 5.61 2.09 

# 3 318 6.02 2.74 

# 4 985 6.24 3.22 

# 5 2352 6.32 3.44 

# 6 6008 6.30 3.45 

# 7 19024 6.19 3.30 

v, sig12, salp2, calp2, \ 

domg12, dv = self._Lambda6(salp1, calp1, i < _MAXIT1, p) 

 

# 2 * _TOL0 is approximately 1 ulp [0, PI] 

# reversed test to allow escape with NaNs 

if tripb or fabs(v) < TOLv: 

break 

# update bracketing values 

if v > 0 and (i > _MAXIT1 or (calp1 / salp1) > (calp1b / salp1b)): 

salp1b, calp1b = salp1, calp1 

elif v < 0 and (i > _MAXIT1 or (calp1 / salp1) < (calp1a / salp1a)): 

salp1a, calp1a = salp1, calp1 

 

if i < _MAXIT1 and dv > 0: 

dalp1 = -v / dv 

if fabs(dalp1) < PI: 

s, c = _sincos2(dalp1) 

# nalp1 = alp1 + dalp1 

s, c = _sincos12(-s, c, salp1, calp1) 

if s > 0: 

salp1, calp1 = _norm2(s, c) 

# in some regimes we don't get quadratic convergence 

# because slope -> 0. So use convergence conditions 

# based on epsilon instead of sqrt(epsilon) 

TOLv = _TOL0 if fabs(v) > _TOL016 else _TOL08 

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 = 90 deg: the WGS84 

# test set: mean = 5.21, sd = 3.93, max = 24 and WGS84 with 

# random input: mean = 4.74, sd = 0.99 

salp1, calp1 = _norm2((salp1a + salp1b) * _0_5, 

(calp1a + calp1b) * _0_5) 

tripb = fsum1_(calp1a, -calp1, fabs(salp1a - salp1)) < _TOLb or \ 

fsum1_(calp1b, -calp1, fabs(salp1b - salp1)) < _TOLb 

TOLv = _TOL0 

 

else: 

raise GeodesicError(Fmt.no_convergence(v, TOLv), txt=repr(self)) # self.toRepr() 

 

p.set_(iter=i, trip=tripb) # like .geodsolve._GDictInvoke: iter NOT iteration! 

return sig12, salp1, calp1, salp2, calp2, domg12 

 

Polygon = Area # for C{geographiclib} compatibility 

 

def _sinf1cos2d(self, lat): 

'''(INTERNAL) Helper, also for C{_G_GeodesicLineExact}. 

''' 

sbet, cbet = _sincos2d(lat) 

# ensure cbet1 = +epsilon at poles; doing the fix on beta means 

# that sig12 will be <= 2*tiny for two points at the same pole 

sbet, cbet = _norm2(sbet * self.f1, cbet) 

return sbet, max(_TINY, cbet) 

 

def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature 

'''Return this C{GeodesicExact} as string. 

 

@kwarg prec: The C{float} precision, number of decimal digits (0..9). 

Trailing zero decimals are stripped for B{C{prec}} values 

of 1 and above, but kept for negative B{C{prec}} values. 

@kwarg sep: Separator to join (C{str}). 

 

@return: Tuple items (C{str}). 

''' 

d = dict(ellipsoid=self.ellipsoid, C4order=self.C4order) 

return sep.join(pairs(d, prec=prec)) 

 

 

class GeodesicLineExact(_GeodesicLineExact): 

'''A pure Python version of I{Karney}'s C++ class U{GeodesicLineExact 

<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicLineExact.html>}, 

modeled after I{Karney}'s Python class U{GeodesicLine<https://GeographicLib.SourceForge.io/ 

html/python/code.html#module-geographiclib.geodesicline>}. 

''' 

 

def __init__(self, geodesic, lat1, lon1, azi1, caps=Caps.STANDARD, name=NN): 

'''New L{GeodesicLineExact} instance, allowing points to be found along 

a geodesic starting at C{(B{lat1}, B{lon1})} with azimuth B{C{azi1}}. 

 

@arg geodesic: The geodesic to use (L{GeodesicExact}). 

@arg lat1: Latitude of the first point (C{degrees}). 

@arg lon1: Longitude of the first point (C{degrees}). 

@arg azi1: Azimuth at the first points (compass C{degrees}). 

@kwarg caps: Bit-or'ed combination of L{Caps} values specifying 

the capabilities the L{GeodesicLineExact} instance 

should possess, i.e., which quantities can be 

returned by calls to L{GeodesicLineExact.Position} 

and L{GeodesicLineExact.ArcPosition}. 

@kwarg name: Optional name (C{str}). 

 

@raise TypeError: Invalid B{C{geodesic}}. 

''' 

_xinstanceof(GeodesicExact, geodesic=geodesic) 

if (caps & Caps.LINE_OFF): # copy to avoid updates 

geodesic = geodesic.copy(deep=False, name=NN(_UNDER_, geodesic.name)) 

# _update_all(geodesic) 

_GeodesicLineExact.__init__(self, geodesic, lat1, lon1, azi1, caps, 0, name=name) 

 

 

def _Astroid(x, y): 

'''(INTERNAL) Solve M{k^4 + 2 * k^3 - (x^2 + y^2 - 1 ) * k^2 - 

(2 * k + 1) * y^2 = 0} for positive root k. 

''' 

p = x**2 

q = y**2 

r = fsum_(_1_0, q, p, _N_2_0, floats=True) 

if q or r > 0: 

r = r / _6_0 # /= chokes PyChecker 

# avoid possible division by zero when r = 0 

# by multiplying s and t by r^3 and r, resp. 

S = p * q / _4_0 # S = r^3 * s 

r3 = r**3 

T3 = r3 + S 

# discriminant of the quadratic equation for T3 is 

# zero on the evolute curve p^(1/3) + q^(1/3) = 1 

d = S * (S + r3 * _2_0) 

if d < 0: 

# T is complex, but u is defined for a real result 

a = atan2(sqrt(-d), -T3) / _3_0 

# There are 3 possible cube roots, choose the one which 

# avoids cancellation. Note d < 0 implies that r < 0. 

u = (cos(a) * _2_0 + _1_0) * r 

else: 

# pick the sign on the sqrt to maximize abs(T3) to 

# minimize loss of precision due to cancellation. 

if d: 

T3 += _copysign(sqrt(d), T3) # T3 = (r * t)^3 

# _cbrt always returns the real root, _cbrt(-8) = -2 

u = _cbrt(T3) # T = r * t 

if u: # T can be zero; but then r2 / T -> 0 

u += r**2 / u 

u += r 

v = _hypot(u, y) # sqrt(u**2 + q) 

# avoid loss of accuracy when u < 0 

u = (q / (v - u)) if u < 0 else (v + u) 

w = (u - q) / (v + v) # positive? 

# rearrange expression for k to avoid loss of accuracy due to 

# subtraction, division by 0 impossible because u > 0, w >= 0 

k = u / (sqrt(w**2 + u) + 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_0 

 

return k 

 

 

__all__ += _ALL_DOCS(GeodesicExact, GeodesicLineExact) 

 

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# 

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