Coverage for pygeodesy/geodesicx/gx.py: 93%
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2# -*- coding: utf-8 -*-
4u'''A pure Python version of I{Karney}'s C++ class U{GeodesicExact
5<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>}.
7Class L{GeodesicExact} follows the naming, methods and return values
8of class C{Geodesic} from I{Karney}'s Python U{geographiclib
9<https://GitHub.com/geographiclib/geographiclib-python>}.
11Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2008-2023)
12and licensed under the MIT/X11 License. For more information, see the
13U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation.
14'''
15# make sure int/int division yields float quotient
16from __future__ import division as _; del _ # PYCHOK semicolon
18# A copy of comments from Karney's C{GeodesicExact.cpp}:
19#
20# This is a reformulation of the geodesic problem. The
21# notation is as follows:
22# - at a general point (no suffix or 1 or 2 as suffix)
23# - phi = latitude
24# - beta = latitude on auxiliary sphere
25# - omega = longitude on auxiliary sphere
26# - lambda = longitude
27# - alpha = azimuth of great circle
28# - sigma = arc length along great circle
29# - s = distance
30# - tau = scaled distance (= sigma at multiples of PI/2)
31# - at northwards equator crossing
32# - beta = phi = 0
33# - omega = lambda = 0
34# - alpha = alpha0
35# - sigma = s = 0
36# - a 12 suffix means a difference, e.g., s12 = s2 - s1.
37# - s and c prefixes mean sin and cos
39from pygeodesy.basics import _xinstanceof, _xor, unsigned0
40from pygeodesy.constants import EPS, EPS0, EPS02, MANT_DIG, NAN, PI, _EPSqrt, \
41 _SQRT2_2, isnan, _0_0, _0_001, _0_01, _0_1, _0_5, \
42 _1_0, _N_1_0, _1_75, _2_0, _N_2_0, _2__PI, _3_0, \
43 _4_0, _6_0, _8_0, _16_0, _90_0, _180_0, _1000_0
44# from pygeodesy.datums import _a_ellipsoid # from .karney
45from pygeodesy.fsums import fsumf_, fsum1f_
46from pygeodesy.geodesicx.gxbases import _cosSeries, _GeodesicBase, \
47 _sincos12, _sin1cos2, _xnC4
48from pygeodesy.geodesicx.gxline import _GeodesicLineExact, _TINY, _update_glXs
49from pygeodesy.interns import NN, _COMMASPACE_, _DOT_, _UNDER_
50from pygeodesy.karney import _around, _atan2d, Caps, _cbrt, _copysign, _diff182, \
51 _a_ellipsoid, _EWGS84, GDict, GeodesicError, _fix90, \
52 _hypot, _K_2_0, _norm2, _norm180, _polynomial, \
53 _signBit, _sincos2, _sincos2d, _sincos2de, _unsigned2
54from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS
55from pygeodesy.namedTuples import Destination3Tuple, Distance3Tuple
56from pygeodesy.props import deprecated_Property, Property, Property_RO
57from pygeodesy.streprs import Fmt, pairs
58from pygeodesy.utily import atan2d as _atan2d_reverse, _Wrap, wrap360
60from math import atan2, copysign, cos, degrees, fabs, radians, sqrt
62__all__ = ()
63__version__ = '23.05.15'
65_MAXIT1 = 20
66_MAXIT2 = 10 + _MAXIT1 + MANT_DIG # MANT_DIG == C++ digits
68# increased multiplier in defn of _TOL1 from 100 to 200 to fix Inverse
69# case 52.784459512564 0 -52.784459512563990912 179.634407464943777557
70# which otherwise failed for Visual Studio 10 (Release and Debug)
71_TOL0 = EPS
72_TOL1 = _TOL0 * -200 # negative
73_TOL2 = _EPSqrt # == sqrt(_TOL0)
74_TOL3 = _TOL2 * _0_1
75_TOLb = _TOL2 * _TOL0 # Check on bisection interval
76_THR1 = _TOL2 * _1000_0 + _1_0
78_TINY3 = _TINY * _3_0
79_TOL08 = _TOL0 * _8_0
80_TOL016 = _TOL0 * _16_0
83def _atan12(*sincos12, **sineg0):
84 '''(INTERNAL) Return C{ang12} in C{radians}.
85 '''
86 return atan2(*_sincos12(*sincos12, **sineg0))
89def _eTOL2(f):
90 # Using the auxiliary sphere solution with dnm computed at
91 # (bet1 + bet2) / 2, the relative error in the azimuth
92 # consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2.
93 # (Error measured for 1/100 < b/a < 100 and abs(f) >= 1/1000.
95 # For a given f and sig12, the max error occurs for lines
96 # near the pole. If the old rule for computing dnm = (dn1
97 # + dn2)/2 is used, then the error increases by a factor of
98 # 2.) Setting this equal to epsilon gives sig12 = etol2.
100 # Here 0.1 is a safety factor (error decreased by 100) and
101 # max(0.001, abs(f)) stops etol2 getting too large in the
102 # nearly spherical case.
103 t = min(_1_0, _1_0 - f * _0_5) * max(_0_001, fabs(f)) * _0_5
104 return _TOL3 / (sqrt(t) if t > EPS02 else EPS0)
107class _PDict(GDict):
108 '''(INTERNAL) Parameters passed around in C{._GDictInverse} and
109 optionally returned when C{GeodesicExact.debug} is C{True}.
110 '''
111 def setsigs(self, ssig1, csig1, ssig2, csig2):
112 '''Update the C{sig1} and C{sig2} parameters.
113 '''
114 self.set_(ssig1=ssig1, csig1=csig1, sncndn1=(ssig1, csig1, self.dn1), # PYCHOK dn1
115 ssig2=ssig2, csig2=csig2, sncndn2=(ssig2, csig2, self.dn2)) # PYCHOK dn2
117 def toGDict(self): # PYCHOK no cover
118 '''Return as C{GDict} without attrs C{sncndn1} and C{sncndn2}.
119 '''
120 def _rest(sncndn1=None, sncndn2=None, **rest): # PYCHOK sncndn* not used
121 return GDict(rest)
123 return _rest(**self)
126class GeodesicExact(_GeodesicBase):
127 '''A pure Python version of I{Karney}'s C++ class U{GeodesicExact
128 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>},
129 modeled after I{Karney}'s Python class U{geodesic.Geodesic<https://GitHub.com/
130 geographiclib/geographiclib-python>}.
131 '''
132 _E = _EWGS84
133 _nC4 = 30 # default C4order
135 def __init__(self, a_ellipsoid=_EWGS84, f=None, name=NN, C4order=None,
136 C4Order=None): # for backward compatibility
137 '''New L{GeodesicExact} instance.
139 @arg a_ellipsoid: An ellipsoid (L{Ellipsoid}) or datum (L{Datum}) or
140 the equatorial radius of the ellipsoid (C{scalar},
141 conventionally in C{meter}), see B{C{f}}.
142 @arg f: The flattening of the ellipsoid (C{scalar}) if B{C{a_ellipsoid}}
143 is specified as C{scalar}.
144 @kwarg name: Optional name (C{str}).
145 @kwarg C4order: Optional series expansion order (C{int}), see property
146 L{C4order}, default C{30}.
147 @kwarg C4Order: DEPRECATED, use keyword argument B{C{C4order}}.
149 @raise GeodesicError: Invalid B{C{C4order}}.
150 '''
151 if a_ellipsoid not in (GeodesicExact._E, None):
152 self._E = _a_ellipsoid(a_ellipsoid, f, name=name)
154 if name:
155 self.name = name
156 if C4order: # XXX private copy, always?
157 self.C4order = C4order
158 elif C4Order: # for backward compatibility
159 self.C4Order = C4Order
161 @Property_RO
162 def a(self):
163 '''Get the I{equatorial} radius, semi-axis (C{meter}).
164 '''
165 return self.ellipsoid.a
167 def ArcDirect(self, lat1, lon1, azi1, a12, outmask=Caps.STANDARD):
168 '''Solve the I{Direct} geodesic problem in terms of (spherical) arc length.
170 @arg lat1: Latitude of the first point (C{degrees}).
171 @arg lon1: Longitude of the first point (C{degrees}).
172 @arg azi1: Azimuth at the first point (compass C{degrees}).
173 @arg a12: Arc length between the points (C{degrees}), can be negative.
174 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying
175 the quantities to be returned.
177 @return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2,
178 lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1},
179 C{lon1}, C{azi1} and arc length C{a12} always included.
181 @see: C++ U{GeodesicExact.ArcDirect
182 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>}
183 and Python U{Geodesic.ArcDirect<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.
184 '''
185 return self._GDictDirect(lat1, lon1, azi1, True, a12, outmask)
187 def ArcDirectLine(self, lat1, lon1, azi1, a12, caps):
188 '''Define a L{GeodesicLineExact} in terms of the I{direct} geodesic problem and as arc length.
190 @arg lat1: Latitude of the first point (C{degrees}).
191 @arg lon1: Longitude of the first point (C{degrees}).
192 @arg azi1: Azimuth at the first point (compass C{degrees}).
193 @arg a12: Arc length between the points (C{degrees}), can be negative.
194 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying
195 the capabilities the L{GeodesicLineExact} instance
196 should possess, i.e., which quantities can be
197 returned by calls to L{GeodesicLineExact.Position}
198 and L{GeodesicLineExact.ArcPosition}.
200 @return: A L{GeodesicLineExact} instance.
202 @note: The third point of the L{GeodesicLineExact} is set to correspond
203 to the second point of the I{Inverse} geodesic problem.
205 @note: Latitude B{C{lat1}} should in the range C{[-90, +90]}.
207 @see: C++ U{GeodesicExact.ArcDirectLine
208 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} and
209 Python U{Geodesic.ArcDirectLine<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.
210 '''
211 return self._GenDirectLine(lat1, lon1, azi1, True, a12, caps)
213 def Area(self, polyline=False, name=NN):
214 '''Set up a L{GeodesicAreaExact} to compute area and
215 perimeter of a polygon.
217 @kwarg polyline: If C{True} perimeter only, otherwise
218 area and perimeter (C{bool}).
219 @kwarg name: Optional name (C{str}).
221 @return: A L{GeodesicAreaExact} instance.
223 @note: The B{C{debug}} setting is passed as C{verbose}
224 to the returned L{GeodesicAreaExact} instance.
225 '''
226 gaX = _MODS.geodesicx.GeodesicAreaExact(self, polyline=polyline,
227 name=name or self.name)
228 if self.debug:
229 gaX.verbose = True
230 return gaX
232 @Property_RO
233 def b(self):
234 '''Get the ellipsoid's I{polar} radius, semi-axis (C{meter}).
235 '''
236 return self.ellipsoid.b
238 @Property_RO
239 def c2x(self):
240 '''Get the ellipsoid's I{authalic} earth radius I{squared} (C{meter} I{squared}).
241 '''
242 # The Geodesic class substitutes atanh(sqrt(e2)) for asinh(sqrt(ep2))
243 # in the definition of _c2. The latter is more accurate for very
244 # oblate ellipsoids (which the Geodesic class does not handle). Of
245 # course, the area calculation in GeodesicExact is still based on a
246 # series and only holds for moderately oblate (or prolate) ellipsoids.
247 return self.ellipsoid.c2x
249 c2 = c2x # in this particular case
251 def C4f(self, eps):
252 '''Evaluate the C{C4x} coefficients for B{C{eps}}.
254 @arg eps: Polynomial factor (C{float}).
256 @return: C{C4order}-Tuple of C{C4x(B{eps})} coefficients.
257 '''
258 def _c4(nC4, C4x):
259 i, x, e = 0, _1_0, eps
260 _p = _polynomial
261 for r in range(nC4, 0, -1):
262 j = i + r
263 yield _p(e, C4x, i, j) * x
264 x *= e
265 i = j
266 # assert i == (nC4 * (nC4 + 1)) // 2
268 return tuple(_c4(self._nC4, self._C4x))
270 def _C4f_k2(self, k2): # in ._GDictInverse and gxline._GeodesicLineExact._C4a
271 '''(INTERNAL) Compute C{eps} from B{C{k2}} and invoke C{C4f}.
272 '''
273 return self.C4f(k2 / fsumf_(_2_0, sqrt(k2 + _1_0) * _2_0, k2))
275 @Property
276 def C4order(self):
277 '''Get the series expansion order (C{int}, 24, 27 or 30).
278 '''
279 return self._nC4
281 @C4order.setter # PYCHOK .setter!
282 def C4order(self, order):
283 '''Set the series expansion order (C{int}, 24, 27 or 30).
285 @raise GeodesicError: Invalid B{C{order}}.
286 '''
287 _xnC4(C4order=order)
288 if self._nC4 != order:
289 GeodesicExact._C4x._update(self)
290 _update_glXs(self) # zap cached _GeodesicLineExact attrs _B41, _C4a
291 self._nC4 = order
293 @deprecated_Property
294 def C4Order(self):
295 '''DEPRECATED, use property C{C4order}.
296 '''
297 return self.C4order
299 @C4Order.setter # PYCHOK .setter!
300 def C4Order(self, order):
301 '''DEPRECATED, use property C{C4order}.
302 '''
303 _xnC4(C4Order=order)
304 self.C4order = order
306 def _coeffs(self, nC4):
307 '''(INTERNAL) Get the C{C4_24}, C{_27} or C{_30} series coefficients.
308 '''
309 try: # from pygeodesy.geodesicx._C4_xx import _coeffs_xx as _coeffs
310 _C4_xx = _DOT_(_MODS.geodesicx.__name__, _UNDER_('_C4', nC4))
311 _coeffs = _MODS.getattr(_C4_xx, _UNDER_('_coeffs', nC4))
312 except (AttributeError, ImportError, TypeError) as x:
313 raise GeodesicError(nC4=nC4, cause=x)
314 n = _xnC4(nC4=nC4)
315 if len(_coeffs) != n: # double check
316 raise GeodesicError(_coeffs=len(_coeffs), _xnC4=n, nC4=nC4)
317 return _coeffs
319 @Property_RO
320 def _C4x(self):
321 '''Get this ellipsoid's C{C4} coefficients, I{cached} tuple.
323 @see: Property L{C4order}.
324 '''
325 # see C4coeff() in GeographicLib.src.GeodesicExactC4.cpp
326 def _C4(nC4):
327 i, n, cs = 0, self.n, self._coeffs(nC4)
328 _p = _polynomial
329 for r in range(nC4 + 1, 1, -1):
330 for j in range(1, r):
331 j = j + i # (j - i - 1) order of polynomial
332 yield _p(n, cs, i, j) / cs[j]
333 i = j + 1
334 # assert i == (nC4 * (nC4 + 1) * (nC4 + 5)) // 6
336 return tuple(_C4(self._nC4)) # 3rd flattening
338 def Direct(self, lat1, lon1, azi1, s12, outmask=Caps.STANDARD):
339 '''Solve the I{Direct} geodesic problem
341 @arg lat1: Latitude of the first point (C{degrees}).
342 @arg lon1: Longitude of the first point (C{degrees}).
343 @arg azi1: Azimuth at the first point (compass C{degrees}).
344 @arg s12: Distance between the points (C{meter}), can be negative.
345 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying
346 the quantities to be returned.
348 @return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2,
349 lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1},
350 C{lon1}, C{azi1} and distance C{s12} always included.
352 @see: C++ U{GeodesicExact.Direct
353 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>}
354 and Python U{Geodesic.Direct<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.
355 '''
356 return self._GDictDirect(lat1, lon1, azi1, False, s12, outmask)
358 def Direct3(self, lat1, lon1, azi1, s12): # PYCHOK outmask
359 '''Return the destination lat, lon and reverse azimuth
360 (final bearing) in C{degrees}.
362 @return: L{Destination3Tuple}C{(lat, lon, final)}.
363 '''
364 r = self._GDictDirect(lat1, lon1, azi1, False, s12, Caps._AZIMUTH_LATITUDE_LONGITUDE)
365 return Destination3Tuple(r.lat2, r.lon2, r.azi2) # no iteration
367 def DirectLine(self, lat1, lon1, azi1, s12, caps=Caps.STANDARD):
368 '''Define a L{GeodesicLineExact} in terms of the I{direct} geodesic problem and as distance.
370 @arg lat1: Latitude of the first point (C{degrees}).
371 @arg lon1: Longitude of the first point (C{degrees}).
372 @arg azi1: Azimuth at the first point (compass C{degrees}).
373 @arg s12: Distance between the points (C{meter}), can be negative.
374 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying
375 the capabilities the L{GeodesicLineExact} instance
376 should possess, i.e., which quantities can be
377 returned by calls to L{GeodesicLineExact.Position}.
379 @return: A L{GeodesicLineExact} instance.
381 @note: The third point of the L{GeodesicLineExact} is set to correspond
382 to the second point of the I{Inverse} geodesic problem.
384 @note: Latitude B{C{lat1}} should in the range C{[-90, +90]}.
386 @see: C++ U{GeodesicExact.DirectLine
387 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} and
388 Python U{Geodesic.DirectLine<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.
389 '''
390 return self._GenDirectLine(lat1, lon1, azi1, False, s12, caps)
392 def _dn(self, sbet, cbet): # in gxline._GeodesicLineExact.__init__
393 '''(INTERNAL) Helper.
394 '''
395 if self.f < 0: # PYCHOK no cover
396 dn = sqrt(_1_0 - cbet**2 * self.e2) / self.f1
397 else:
398 dn = sqrt(_1_0 + sbet**2 * self.ep2)
399 return dn
401 @Property_RO
402 def e2(self):
403 '''Get the ellipsoid's I{(1st) eccentricity squared} (C{float}), M{f * (2 - f)}.
404 '''
405 return self.ellipsoid.e2
407 @Property_RO
408 def _e2a2(self):
409 '''(INTERNAL) Cache M{E.e2 * E.a2}.
410 '''
411 return self.e2 * self.ellipsoid.a2
413 @Property_RO
414 def _e2_f1(self):
415 '''(INTERNAL) Cache M{E.e2 * E.f1}.
416 '''
417 return self.e2 / self.f1
419 @Property_RO
420 def _eF(self):
421 '''(INTERNAL) Get the elliptic function, aka C{.E}.
422 '''
423 return _MODS.elliptic.Elliptic(k2=-self.ep2)
425 def _eF_reset_cHe2_f1(self, x, y):
426 '''(INTERNAL) Reset elliptic function and return M{cH * e2 / f1 * ...}.
427 '''
428 self._eF_reset_k2(x)
429 return y * self._eF.cH * self._e2_f1
431 def _eF_reset_k2(self, x):
432 '''(INTERNAL) Reset elliptic function and return C{k2}.
433 '''
434 ep2 = self.ep2
435 k2 = x**2 * ep2 # see .gxline._GeodesicLineExact._eF
436 self._eF.reset(k2=-k2, alpha2=-ep2) # kp2, alphap2 defaults
437 _update_glXs(self) # zap cached/memoized _GeodesicLineExact attrs
438 return k2
440 @Property_RO
441 def ellipsoid(self):
442 '''Get the ellipsoid (C{Ellipsoid}).
443 '''
444 return self._E
446 @Property_RO
447 def ep2(self):
448 '''Get the ellipsoid's I{2nd eccentricity squared} (C{float}), M{e2 / (1 - e2)}.
449 '''
450 return self.ellipsoid.e22 # == self.e2 / self.f1**2
452 e22 = ep2 # for ellipsoid compatibility
454 @Property_RO
455 def _eTOL2(self):
456 '''(INTERNAL) The si12 threshold for "really short".
457 '''
458 return _eTOL2(self.f)
460 @Property_RO
461 def f(self):
462 '''Get the ellipsoid's I{flattening} (C{float}), M{(a - b) / a}, C{0} for spherical, negative for prolate.
463 '''
464 return self.ellipsoid.f
466 flattening = f
468 @Property_RO
469 def f1(self): # in .css.CassiniSoldner.reset
470 '''Get the ellipsoid's I{1 - flattening} (C{float}).
471 '''
472 return self.ellipsoid.f1
474 @Property_RO
475 def _f180(self):
476 '''(INTERNAL) Cached/memoized.
477 '''
478 return self.f * _180_0
480 def _GDictDirect(self, lat1, lon1, azi1, arcmode, s12_a12, outmask=Caps.STANDARD):
481 '''(INTERNAL) As C{_GenDirect}, but returning a L{GDict}.
483 @return: A L{GDict} ...
484 '''
485 C = outmask if arcmode else (outmask | Caps.DISTANCE_IN)
486 glX = self.Line(lat1, lon1, azi1, C | Caps.LINE_OFF)
487 return glX._GDictPosition(arcmode, s12_a12, outmask)
489 def _GDictInverse(self, lat1, lon1, lat2, lon2, outmask=Caps.STANDARD): # MCCABE 33, 41 vars
490 '''(INTERNAL) As C{_GenInverse}, but returning a L{GDict}.
492 @return: A L{GDict} ...
493 '''
494 Cs = Caps
495 if self._debug: # PYCHOK no cover
496 outmask |= Cs._DEBUG_INVERSE & self._debug
497 outmask &= Cs._OUT_MASK # incl. _SALPs_CALPs and _DEBUG_
498 # compute longitude difference carefully (with _diff182):
499 # result is in [-180, +180] but -180 is only for west-going
500 # geodesics, +180 is for east-going and meridional geodesics
501 lon12, lon12s = _diff182(lon1, lon2)
502 # see C{result} from geographiclib.geodesic.Inverse
503 if (outmask & Cs.LONG_UNROLL): # == (lon1 + lon12) + lon12s
504 r = GDict(lon1=lon1, lon2=fsumf_(lon1, lon12, lon12s))
505 else:
506 r = GDict(lon1=_norm180(lon1), lon2=_norm180(lon2))
507 if _K_2_0: # GeographicLib 2.0
508 # make longitude difference positive
509 lon12, lon_ = _unsigned2(lon12)
510 if lon_:
511 lon12s = -lon12s
512 lam12 = radians(lon12)
513 # calculate sincosd(_around(lon12 + correction))
514 slam12, clam12 = _sincos2de(lon12, lon12s)
515 # supplementary longitude difference
516 lon12s = fsumf_(_180_0, -lon12, -lon12s)
517 else: # GeographicLib 1.52
518 # make longitude difference positive and if very close
519 # to being on the same half-meridian, then make it so.
520 if lon12 < 0: # _signBit(lon12)
521 lon_, lon12 = True, -_around(lon12)
522 lon12s = _around(fsumf_(_180_0, -lon12, lon12s))
523 else:
524 lon_, lon12 = False, _around(lon12)
525 lon12s = _around(fsumf_(_180_0, -lon12, -lon12s))
526 lam12 = radians(lon12)
527 if lon12 > _90_0:
528 slam12, clam12 = _sincos2d(lon12s)
529 clam12 = -clam12
530 else:
531 slam12, clam12 = _sincos2(lam12)
532 # If really close to the equator, treat as on equator.
533 lat1 = _around(_fix90(lat1))
534 lat2 = _around(_fix90(lat2))
535 r.set_(lat1=lat1, lat2=lat2)
536 # Swap points so that point with higher (abs) latitude is
537 # point 1. If one latitude is a NAN, then it becomes lat1.
538 swap_ = fabs(lat1) < fabs(lat2) or isnan(lat2)
539 if swap_:
540 lat1, lat2 = lat2, lat1
541 lon_ = not lon_
542 if _signBit(lat1):
543 lat_ = False # note, False
544 else: # make lat1 <= -0
545 lat_ = True # note, True
546 lat1, lat2 = -lat1, -lat2
547 # Now 0 <= lon12 <= 180, -90 <= lat1 <= -0 and lat1 <= lat2 <= -lat1
548 # and lat_, lon_, swap_ register the transformation to bring the
549 # coordinates to this canonical form, where False means no change
550 # made. We make these transformations so that there are few cases
551 # to check, e.g., on verifying quadrants in atan2. In addition,
552 # this enforces some symmetries in the results returned.
554 # Initialize for the meridian. No longitude calculation is
555 # done in this case to let the parameter default to 0.
556 sbet1, cbet1 = self._sinf1cos2d(lat1)
557 sbet2, cbet2 = self._sinf1cos2d(lat2)
558 # If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure
559 # of the |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1),
560 # abs(sbet2) + sbet1 is a better measure. This logic is used
561 # in assigning calp2 in _Lambda6. Sometimes these quantities
562 # vanish and in that case we force bet2 = +/- bet1 exactly. An
563 # example where is is necessary is the inverse problem
564 # 48.522876735459 0 -48.52287673545898293 179.599720456223079643
565 # which failed with Visual Studio 10 (Release and Debug)
566 if cbet1 < -sbet1:
567 if cbet2 == cbet1:
568 sbet2 = copysign(sbet1, sbet2)
569 elif fabs(sbet2) == -sbet1:
570 cbet2 = cbet1
572 p = _PDict(sbet1=sbet1, cbet1=cbet1, dn1=self._dn(sbet1, cbet1),
573 sbet2=sbet2, cbet2=cbet2, dn2=self._dn(sbet2, cbet2))
575 _meridian = _b = True # i.e. not meridian, not b
576 if lat1 == -90 or slam12 == 0:
577 # Endpoints are on a single full meridian,
578 # so the geodesic might lie on a meridian.
579 salp1, calp1 = slam12, clam12 # head to target lon
580 salp2, calp2 = _0_0, _1_0 # then head north
581 # tan(bet) = tan(sig) * cos(alp)
582 p.setsigs(sbet1, calp1 * cbet1, sbet2, calp2 * cbet2)
583 # sig12 = sig2 - sig1
584 sig12 = _atan12(sbet1, p.csig1, sbet2, p.csig2, sineg0=True) # PYCHOK csig*
585 s12x, m12x, _, \
586 M12, M21 = self._Length5(sig12, outmask | Cs.REDUCEDLENGTH, p)
587 # Add the check for sig12 since zero length geodesics
588 # might yield m12 < 0. Test case was
589 # echo 20.001 0 20.001 0 | GeodSolve -i
590 # In fact, we will have sig12 > PI/2 for meridional
591 # geodesic which is not a shortest path.
592 if m12x >= 0 or sig12 < _1_0:
593 # Need at least 2 to handle 90 0 90 180
594 # Prevent negative s12 or m12 from geographiclib 1.52
595 if sig12 < _TINY3 or (sig12 < _TOL0 and (s12x < 0 or m12x < 0)):
596 sig12 = m12x = s12x = _0_0
597 else:
598 _b = False # apply .b to s12x, m12x
599 _meridian = False
600 C = 1
601 # else: # m12 < 0, prolate and too close to anti-podal
602 # _meridian = True
603 a12 = _0_0 # if _b else degrees(sig12)
605 if _meridian:
606 _b = sbet1 == 0 and (self.f <= 0 or lon12s >= self._f180) # and sbet2 == 0
607 if _b: # Geodesic runs along equator
608 calp1 = calp2 = _0_0
609 salp1 = salp2 = _1_0
610 sig12 = lam12 / self.f1 # == omg12
611 somg12, comg12 = _sincos2(sig12)
612 m12x = self.b * somg12
613 s12x = self.a * lam12
614 M12 = M21 = comg12
615 a12 = lon12 / self.f1
616 C = 2
617 else:
618 # Now point1 and point2 belong within a hemisphere bounded by a
619 # meridian and geodesic is neither meridional or equatorial.
620 p.set_(slam12=slam12, clam12=clam12)
621 # Figure a starting point for Newton's method
622 sig12, salp1, calp1, \
623 salp2, calp2, dnm = self._InverseStart6(lam12, p)
624 if sig12 is None: # use Newton's method
625 # pre-compute the constant _Lambda6 term, once
626 p.set_(bet12=None if cbet2 == cbet1 and fabs(sbet2) == -sbet1 else
627 (((cbet1 + cbet2) * (cbet2 - cbet1)) if cbet1 < -sbet1 else
628 ((sbet1 + sbet2) * (sbet1 - sbet2))))
629 sig12, salp1, calp1, \
630 salp2, calp2, domg12 = self._Newton6(salp1, calp1, p)
631 s12x, m12x, _, M12, M21 = self._Length5(sig12, outmask, p)
632 if (outmask & Cs.AREA):
633 # omg12 = lam12 - domg12
634 s, c = _sincos2(domg12)
635 somg12, comg12 = _sincos12(s, c, slam12, clam12)
636 C = 3 # Newton
637 else: # from _InverseStart6: dnm, salp*, calp*
638 C = 4 # Short lines
639 s, c = _sincos2(sig12 / dnm)
640 m12x = dnm**2 * s
641 s12x = dnm * sig12
642 M12 = M21 = c
643 if (outmask & Cs.AREA):
644 somg12, comg12 = _sincos2(lam12 / (self.f1 * dnm))
646 else: # _meridian is False
647 somg12 = comg12 = NAN
649 r.set_(a12=a12 if _b else degrees(sig12)) # in [0, 180]
651 if (outmask & Cs.DISTANCE):
652 r.set_(s12=unsigned0(s12x if _b else (self.b * s12x)))
654 if (outmask & Cs.REDUCEDLENGTH):
655 r.set_(m12=unsigned0(m12x if _b else (self.b * m12x)))
657 if (outmask & Cs.GEODESICSCALE):
658 if swap_:
659 M12, M21 = M21, M12
660 r.set_(M12=unsigned0(M12),
661 M21=unsigned0(M21))
663 if (outmask & Cs.AREA):
664 S12 = self._InverseArea(_meridian, salp1, calp1,
665 salp2, calp2,
666 somg12, comg12, p)
667 if _xor(swap_, lat_, lon_):
668 S12 = -S12
669 r.set_(S12=unsigned0(S12))
671 if (outmask & (Cs.AZIMUTH | Cs._SALPs_CALPs)):
672 if swap_:
673 salp1, salp2 = salp2, salp1
674 calp1, calp2 = calp2, calp1
675 if _xor(swap_, lon_):
676 salp1, salp2 = -salp1, -salp2
677 if _xor(swap_, lat_):
678 calp1, calp2 = -calp1, -calp2
680 if (outmask & Cs.AZIMUTH):
681 r.set_(azi1=_atan2d(salp1, calp1),
682 azi2=_atan2d_reverse(salp2, calp2, reverse=outmask & Cs.REVERSE2))
683 if (outmask & Cs._SALPs_CALPs):
684 r.set_(salp1=salp1, calp1=calp1,
685 salp2=salp2, calp2=calp2)
687 if (outmask & Cs._DEBUG_INVERSE): # PYCHOK no cover
688 E, eF = self.ellipsoid, self._eF
689 p.set_(C=C, a=self.a, f=self.f, f1=self.f1,
690 e=E.e, e2=self.e2, ep2=self.ep2,
691 c2=E.c2, c2x=self.c2x,
692 eFcD=eF.cD, eFcE=eF.cE, eFcH=eF.cH,
693 eFk2=eF.k2, eFa2=eF.alpha2)
694 p.update(r) # r overrides p
695 r = p.toGDict()
696 return self._iter2tion(r, p)
698 def _GenDirect(self, lat1, lon1, azi1, arcmode, s12_a12, outmask):
699 '''(INTERNAL) The general I{Inverse} geodesic calculation.
701 @return: L{Direct9Tuple}C{(a12, lat2, lon2, azi2,
702 s12, m12, M12, M21, S12)}.
703 '''
704 r = self._GDictDirect(lat1, lon1, azi1, arcmode, s12_a12, outmask)
705 return r.toDirect9Tuple()
707 def _GenDirectLine(self, lat1, lon1, azi1, arcmode, s12_a12, caps):
708 '''(INTERNAL) Helper for C{ArcDirectLine} and C{DirectLine}.
710 @return: A L{GeodesicLineExact} instance.
711 '''
712 azi1 = _norm180(azi1)
713 # guard against underflow in salp0. Also -0 is converted to +0.
714 s, c = _sincos2d(_around(azi1))
715 C = caps if arcmode else (caps | Caps.DISTANCE_IN)
716 return _GeodesicLineExact(self, lat1, lon1, azi1, C,
717 self._debug, s, c)._GenSet(arcmode, s12_a12)
719 def _GenInverse(self, lat1, lon1, lat2, lon2, outmask):
720 '''(INTERNAL) The general I{Inverse} geodesic calculation.
722 @return: L{Inverse10Tuple}C{(a12, s12, salp1, calp1, salp2, calp2,
723 m12, M12, M21, S12)}.
724 '''
725 r = self._GDictInverse(lat1, lon1, lat2, lon2, outmask | Caps._SALPs_CALPs)
726 return r.toInverse10Tuple()
728 def Inverse(self, lat1, lon1, lat2, lon2, outmask=Caps.STANDARD):
729 '''Perform the I{Inverse} geodesic calculation.
731 @arg lat1: Latitude of the first point (C{degrees}).
732 @arg lon1: Longitude of the first point (C{degrees}).
733 @arg lat2: Latitude of the second point (C{degrees}).
734 @arg lon2: Longitude of the second point (C{degrees}).
735 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying
736 the quantities to be returned.
738 @return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2,
739 lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1},
740 C{lon1}, C{azi1} and distance C{s12} always included.
742 @note: The third point of the L{GeodesicLineExact} is set to correspond
743 to the second point of the I{Inverse} geodesic problem.
745 @note: Both B{C{lat1}} and B{C{lat2}} should in the range C{[-90, +90]}.
747 @see: C++ U{GeodesicExact.InverseLine
748 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} and
749 Python U{Geodesic.InverseLine<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.
750 '''
751 return self._GDictInverse(lat1, lon1, lat2, lon2, outmask)
753 def Inverse1(self, lat1, lon1, lat2, lon2, wrap=False):
754 '''Return the non-negative, I{angular} distance in C{degrees}.
756 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
757 B{C{lat2}} and B{C{lon2}} (C{bool}).
758 '''
759 # see .FrechetKarney.distance, .HausdorffKarney._distance
760 # and .HeightIDWkarney._distances
761 if wrap:
762 _, lat2, lon2 = _Wrap.latlon3(lat1, lat2, lon2, True) # _Geodesic.LONG_UNROLL
763 return fabs(self._GDictInverse(lat1, lon1, lat2, lon2, Caps._ANGLE_ONLY).a12)
765 def Inverse3(self, lat1, lon1, lat2, lon2): # PYCHOK outmask
766 '''Return the distance in C{meter} and the forward and
767 reverse azimuths (initial and final bearing) in C{degrees}.
769 @return: L{Distance3Tuple}C{(distance, initial, final)}.
770 '''
771 r = self._GDictInverse(lat1, lon1, lat2, lon2, Caps.AZIMUTH_DISTANCE)
772 return Distance3Tuple(r.s12, wrap360(r.azi1), wrap360(r.azi2),
773 iteration=r.iteration)
775 def InverseLine(self, lat1, lon1, lat2, lon2, caps=Caps.STANDARD):
776 '''Define a L{GeodesicLineExact} in terms of the I{Inverse} geodesic problem.
778 @arg lat1: Latitude of the first point (C{degrees}).
779 @arg lon1: Longitude of the first point (C{degrees}).
780 @arg lat2: Latitude of the second point (C{degrees}).
781 @arg lon2: Longitude of the second point (C{degrees}).
782 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying
783 the capabilities the L{GeodesicLineExact} instance
784 should possess, i.e., which quantities can be
785 returned by calls to L{GeodesicLineExact.Position}
786 and L{GeodesicLineExact.ArcPosition}.
788 @return: A L{GeodesicLineExact} instance.
790 @note: The third point of the L{GeodesicLineExact} is set to correspond
791 to the second point of the I{Inverse} geodesic problem.
793 @note: Both B{C{lat1}} and B{C{lat2}} should in the range C{[-90, +90]}.
795 @see: C++ U{GeodesicExact.InverseLine
796 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} and
797 Python U{Geodesic.InverseLine<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.
798 '''
799 Cs = Caps
800 r = self._GDictInverse(lat1, lon1, lat2, lon2, Cs._SALPs_CALPs) # No need for AZIMUTH
801 C = (caps | Cs.DISTANCE) if (caps & Cs._DISTANCE_IN_OUT) else caps
802 azi1 = _atan2d(r.salp1, r.calp1)
803 return _GeodesicLineExact(self, lat1, lon1, azi1, C, # ensure a12 is distance
804 self._debug, r.salp1, r.calp1)._GenSet(True, r.a12)
806 def _InverseArea(self, _meridian, salp1, calp1, # PYCHOK 9 args
807 salp2, calp2,
808 somg12, comg12, p):
809 '''(INTERNAL) Split off from C{_GDictInverse} to reduce complexity/length.
811 @return: Area C{S12}.
812 '''
813 # from _Lambda6: sin(alp1) * cos(bet1) = sin(alp0), calp0 > 0
814 salp0, calp0 = _sin1cos2(salp1, calp1, p.sbet1, p.cbet1)
815 A4 = calp0 * salp0
816 if A4:
817 # from _Lambda6: tan(bet) = tan(sig) * cos(alp)
818 k2 = calp0**2 * self.ep2
819 C4a = self._C4f_k2(k2)
820 B41 = _cosSeries(C4a, *_norm2(p.sbet1, calp1 * p.cbet1))
821 B42 = _cosSeries(C4a, *_norm2(p.sbet2, calp2 * p.cbet2))
822 # multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
823 A4 *= self._e2a2
824 S12 = A4 * (B42 - B41)
825 else: # avoid problems with indeterminate sig1, sig2 on equator
826 A4 = B41 = B42 = k2 = S12 = _0_0
828 if (_meridian and # omg12 < 3/4 * PI
829 comg12 > -_SQRT2_2 and # lon diff not too big
830 (p.sbet2 - p.sbet1) < _1_75): # lat diff not too big
831 # use tan(Gamma/2) = tan(omg12/2) *
832 # (tan(bet1/2) + tan(bet2/2)) /
833 # (tan(bet1/2) * tan(bet2/2) + 1))
834 # with tan(x/2) = sin(x) / (1 + cos(x))
835 dbet1 = p.cbet1 + _1_0
836 dbet2 = p.cbet2 + _1_0
837 domg12 = comg12 + _1_0
838 salp12 = (p.sbet1 * dbet2 + dbet1 * p.sbet2) * somg12
839 calp12 = (p.sbet1 * p.sbet2 + dbet1 * dbet2) * domg12
840 alp12 = _2_0 * atan2(salp12, calp12)
841 else:
842 # alp12 = alp2 - alp1, used in atan2, no need to normalize
843 salp12, calp12 = _sincos12(salp1, calp1, salp2, calp2)
844 # The right thing appears to happen if alp1 = +/-180 and
845 # alp2 = 0, viz salp12 = -0 and alp12 = -180. However,
846 # this depends on the sign being attached to 0 correctly.
847 # Following ensures the correct behavior.
848 if salp12 == 0 and calp12 < 0:
849 alp12 = _copysign(PI, calp1)
850 else:
851 alp12 = atan2(salp12, calp12)
853 p.set_(alp12=alp12, A4=A4, B41=B41, B42=B42, k2=k2)
854 return S12 + self.c2x * alp12
856 def _InverseStart6(self, lam12, p):
857 '''(INTERNAL) Return a starting point for Newton's method in
858 C{salp1} and C{calp1} indicated by C{sig12=None}. If
859 Newton's method doesn't need to be used, return also
860 C{salp2}, C{calp2}, C{dnm} and C{sig12} non-C{None}.
862 @return: 6-Tuple C{(sig12, salp1, calp1, salp2, calp2, dnm)}
863 and C{p.setsigs} updated for Newton, C{sig12=None}.
864 '''
865 sig12 = None # use Newton
866 salp1 = calp1 = salp2 = calp2 = dnm = NAN
868 # bet12 = bet2 - bet1 in [0, PI)
869 sbet12, cbet12 = _sincos12(p.sbet1, p.cbet1, p.sbet2, p.cbet2)
870 shortline = cbet12 >= 0 and sbet12 < _0_5 and (p.cbet2 * lam12) < _0_5
871 if shortline:
872 # sin((bet1 + bet2)/2)^2 = (sbet1 + sbet2)^2 / (
873 # (cbet1 + cbet2)^2 + (sbet1 + sbet2)^2)
874 t = (p.sbet1 + p.sbet2)**2
875 s = t / ((p.cbet1 + p.cbet2)**2 + t)
876 dnm = sqrt(_1_0 + self.ep2 * s)
877 somg12, comg12 = _sincos2(lam12 / (self.f1 * dnm))
878 else:
879 somg12, comg12 = p.slam12, p.clam12
881 # bet12a = bet2 + bet1 in (-PI, 0], note -sbet1
882 sbet12a, cbet12a = _sincos12(-p.sbet1, p.cbet1, p.sbet2, p.cbet2)
884 c = fabs(comg12) + _1_0 # == (1 - comg12) if comg12 < 0
885 s = somg12**2 / c
886 t = p.sbet1 * p.cbet2 * s
887 salp1 = p.cbet2 * somg12
888 calp1 = (sbet12a - t) if comg12 < 0 else (sbet12 + t)
890 ssig12 = _hypot(salp1, calp1)
891 csig12 = p.sbet1 * p.sbet2 + p.cbet1 * p.cbet2 * comg12
893 if shortline and ssig12 < self._eTOL2: # really short lines
894 t = c if comg12 < 0 else s
895 salp2, calp2 = _norm2(somg12 * p.cbet1,
896 sbet12 - p.cbet1 * p.sbet2 * t)
897 sig12 = atan2(ssig12, csig12) # do not use Newton
899 elif (self._n_0_1 or # Skip astroid calc if too eccentric
900 csig12 >= 0 or ssig12 >= (p.cbet1**2 * self._n6PI)):
901 pass # nothing to do, 0th order spherical approximation OK
903 else:
904 # Scale lam12 and bet2 to x, y coordinate system where antipodal
905 # point is at origin and singular point is at y = 0, x = -1
906 lam12x = atan2(-p.slam12, -p.clam12) # lam12 - PI
907 f = self.f
908 if f < 0: # PYCHOK no cover
909 # ssig1=sbet1, csig1=-cbet1, ssig2=sbet2, csig2=cbet2
910 p.setsigs(p.sbet1, -p.cbet1, p.sbet2, p.cbet2)
911 # if lon12 = 180, this repeats a calculation made in Inverse
912 _, m12b, m0, _, _ = self._Length5(atan2(sbet12a, cbet12a) + PI,
913 Caps.REDUCEDLENGTH, p)
914 t = p.cbet1 * PI # x = dlat, y = dlon
915 x = m12b / (t * p.cbet2 * m0) - _1_0
916 sca = (sbet12a / (x * p.cbet1)) if x < -_0_01 else (-f * t)
917 y = lam12x / sca
918 else: # f >= 0, however f == 0 does not get here
919 sca = self._eF_reset_cHe2_f1(p.sbet1, p.cbet1 * _2_0)
920 x = lam12x / sca # dlon
921 y = sbet12a / (sca * p.cbet1) # dlat
923 if y > _TOL1 and x > -_THR1: # strip near cut
924 if f < 0: # PYCHOK no cover
925 calp1 = max( _0_0, x) if x > _TOL1 else max(_N_1_0, x)
926 salp1 = sqrt(_1_0 - calp1**2)
927 else:
928 salp1 = min( _1_0, -x)
929 calp1 = -sqrt(_1_0 - salp1**2)
930 else:
931 # Estimate alp1, by solving the astroid problem.
932 #
933 # Could estimate alpha1 = theta + PI/2, directly, i.e.,
934 # calp1 = y/k; salp1 = -x/(1+k); for _f >= 0
935 # calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check)
936 #
937 # However, it's better to estimate omg12 from astroid and use
938 # spherical formula to compute alp1. This reduces the mean
939 # number of Newton iterations for astroid cases from 2.24
940 # (min 0, max 6) to 2.12 (min 0, max 5). The changes in the
941 # number of iterations are as follows:
942 #
943 # change percent
944 # 1 5
945 # 0 78
946 # -1 16
947 # -2 0.6
948 # -3 0.04
949 # -4 0.002
950 #
951 # The histogram of iterations is (m = number of iterations
952 # estimating alp1 directly, n = number of iterations
953 # estimating via omg12, total number of trials = 148605):
954 #
955 # iter m n
956 # 0 148 186
957 # 1 13046 13845
958 # 2 93315 102225
959 # 3 36189 32341
960 # 4 5396 7
961 # 5 455 1
962 # 6 56 0
963 #
964 # omg12 is near PI, estimate work with omg12a = PI - omg12
965 k = _Astroid(x, y)
966 sca *= (y * (k + _1_0) / k) if f < 0 else \
967 (x * k / (k + _1_0))
968 s, c = _sincos2(-sca) # omg12a
969 # update spherical estimate of alp1 using omg12 instead of lam12
970 salp1 = p.cbet2 * s
971 calp1 = sbet12a - s * salp1 * p.sbet1 / (c + _1_0) # c = -c
973 # sanity check on starting guess. Backwards check allows NaN through.
974 salp1, calp1 = _norm2(salp1, calp1) if salp1 > 0 else (_1_0, _0_0)
976 return sig12, salp1, calp1, salp2, calp2, dnm
978 def _Lambda6(self, salp1, calp1, diffp, p):
979 '''(INTERNAL) Helper.
981 @return: 6-Tuple C{(lam12, sig12, salp2, calp2, domg12,
982 dlam12} and C{p.setsigs} updated.
983 '''
984 eF = self._eF
985 f1 = self.f1
987 if p.sbet1 == calp1 == 0: # PYCHOK no cover
988 # Break degeneracy of equatorial line
989 calp1 = -_TINY
991 # sin(alp1) * cos(bet1) = sin(alp0), # calp0 > 0
992 salp0, calp0 = _sin1cos2(salp1, calp1, p.sbet1, p.cbet1)
993 # tan(bet1) = tan(sig1) * cos(alp1)
994 # tan(omg1) = sin(alp0) * tan(sig1)
995 # = sin(bet1) * tan(alp1)
996 somg1 = salp0 * p.sbet1
997 comg1 = calp1 * p.cbet1
998 ssig1, csig1 = _norm2(p.sbet1, comg1)
999 # Without normalization we have schi1 = somg1
1000 cchi1 = f1 * p.dn1 * comg1
1002 # Enforce symmetries in the case abs(bet2) = -bet1.
1003 # Need to be careful about this case, since this can
1004 # yield singularities in the Newton iteration.
1005 # sin(alp2) * cos(bet2) = sin(alp0)
1006 salp2 = (salp0 / p.cbet2) if p.cbet2 != p.cbet1 else salp1
1007 # calp2 = sqrt(1 - sq(salp2))
1008 # = sqrt(sq(calp0) - sq(sbet2)) / cbet2
1009 # and subst for calp0 and rearrange to give (choose
1010 # positive sqrt to give alp2 in [0, PI/2]).
1011 calp2 = fabs(calp1) if p.bet12 is None else (
1012 sqrt((calp1 * p.cbet1)**2 + p.bet12) / p.cbet2)
1013 # tan(bet2) = tan(sig2) * cos(alp2)
1014 # tan(omg2) = sin(alp0) * tan(sig2).
1015 somg2 = salp0 * p.sbet2
1016 comg2 = calp2 * p.cbet2
1017 ssig2, csig2 = _norm2(p.sbet2, comg2)
1018 # without normalization we have schi2 = somg2
1019 cchi2 = f1 * p.dn2 * comg2
1021 # omg12 = omg2 - omg1, limit to [0, PI]
1022 somg12, comg12 = _sincos12(somg1, comg1, somg2, comg2, sineg0=True)
1023 # chi12 = chi2 - chi1, limit to [0, PI]
1024 schi12, cchi12 = _sincos12(somg1, cchi1, somg2, cchi2, sineg0=True)
1026 p.setsigs(ssig1, csig1, ssig2, csig2)
1027 # sig12 = sig2 - sig1, limit to [0, PI]
1028 sig12 = _atan12(ssig1, csig1, ssig2, csig2, sineg0=True)
1030 eta12 = self._eF_reset_cHe2_f1(calp0, salp0) * _2__PI # then ...
1031 eta12 *= fsum1f_(eF.deltaH(*p.sncndn2),
1032 -eF.deltaH(*p.sncndn1), sig12)
1033 # eta = chi12 - lam12
1034 lam12 = _atan12(p.slam12, p.clam12, schi12, cchi12) - eta12
1035 # domg12 = chi12 - omg12 - deta12
1036 domg12 = _atan12( somg12, comg12, schi12, cchi12) - eta12
1038 dlam12 = NAN # dv > 0 in ._Newton6
1039 if diffp:
1040 d = calp2 * p.cbet2
1041 if d:
1042 _, dlam12, _, _, _ = self._Length5(sig12, Caps.REDUCEDLENGTH, p)
1043 dlam12 *= f1 / d
1044 elif p.sbet1:
1045 dlam12 = -f1 * p.dn1 * _2_0 / p.sbet1
1047 # p.set_(deta12=-eta12, lam12=lam12)
1048 return lam12, sig12, salp2, calp2, domg12, dlam12
1050 def _Length5(self, sig12, outmask, p):
1051 '''(INTERNAL) Return M{m12b = (reduced length) / self.b} and
1052 calculate M{s12b = distance / self.b} and M{m0}, the
1053 coefficient of secular term in expression for reduced
1054 length and the geodesic scales C{M12} and C{M21}.
1056 @return: 5-Tuple C{(s12b, m12b, m0, M12, M21)}.
1057 '''
1058 s12b = m12b = m0 = M12 = M21 = NAN
1060 Cs = Caps
1061 eF = self._eF
1063 # outmask &= Cs._OUT_MASK
1064 if (outmask & Cs.DISTANCE):
1065 # Missing a factor of self.b
1066 s12b = eF.cE * _2__PI * fsum1f_(eF.deltaE(*p.sncndn2),
1067 -eF.deltaE(*p.sncndn1), sig12)
1069 if (outmask & Cs._REDUCEDLENGTH_GEODESICSCALE):
1070 m0x = -eF.k2 * eF.cD * _2__PI
1071 J12 = -m0x * fsum1f_(eF.deltaD(*p.sncndn2),
1072 -eF.deltaD(*p.sncndn1), sig12)
1073 if (outmask & Cs.REDUCEDLENGTH):
1074 m0 = m0x
1075 # Missing a factor of self.b. Add parens around
1076 # (csig1 * ssig2) and (ssig1 * csig2) to ensure
1077 # accurate cancellation for coincident points.
1078 m12b = fsum1f_(p.dn2 * (p.csig1 * p.ssig2),
1079 -p.dn1 * (p.ssig1 * p.csig2),
1080 J12 * (p.csig1 * p.csig2))
1081 if (outmask & Cs.GEODESICSCALE):
1082 M12 = M21 = p.ssig1 * p.ssig2 + \
1083 p.csig1 * p.csig2
1084 t = (p.cbet1 - p.cbet2) * self.ep2 * \
1085 (p.cbet1 + p.cbet2) / (p.dn1 + p.dn2)
1086 M12 += (p.ssig2 * t + p.csig2 * J12) * p.ssig1 / p.dn1
1087 M21 -= (p.ssig1 * t + p.csig1 * J12) * p.ssig2 / p.dn2
1089 return s12b, m12b, m0, M12, M21
1091 def Line(self, lat1, lon1, azi1, caps=Caps.ALL):
1092 '''Set up a L{GeodesicLineExact} to compute several points
1093 on a single geodesic.
1095 @arg lat1: Latitude of the first point (C{degrees}).
1096 @arg lon1: Longitude of the first point (C{degrees}).
1097 @arg azi1: Azimuth at the first point (compass C{degrees}).
1098 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying
1099 the capabilities the L{GeodesicLineExact} instance
1100 should possess, i.e., which quantities can be
1101 returnedby calls to L{GeodesicLineExact.Position}
1102 and L{GeodesicLineExact.ArcPosition}.
1104 @return: A L{GeodesicLineExact} instance.
1106 @note: If the point is at a pole, the azimuth is defined by keeping
1107 B{C{lon1}} fixed, writing C{B{lat1} = ±(90 − ε)}, and taking
1108 the limit C{ε → 0+}.
1110 @see: C++ U{GeodesicExact.Line
1111 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>}
1112 and Python U{Geodesic.Line<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.
1113 '''
1114 return _GeodesicLineExact(self, lat1, lon1, azi1, caps, self._debug)
1116 @Property_RO
1117 def n(self):
1118 '''Get the ellipsoid's I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}.
1119 '''
1120 return self.ellipsoid.n
1122 @Property_RO
1123 def _n_0_1(self):
1124 '''(INTERNAL) Cached once.
1125 '''
1126 return fabs(self.n) > _0_1
1128 @Property_RO
1129 def _n6PI(self):
1130 '''(INTERNAL) Cached once.
1131 '''
1132 return fabs(self.n) * _6_0 * PI
1134 def _Newton6(self, salp1, calp1, p):
1135 '''(INTERNAL) Split off from C{_GDictInverse} to reduce complexity/length.
1137 @return: 6-Tuple C{(sig12, salp1, calp1, salp2, calp2, domg12)}
1138 and C{p.iter} and C{p.trip} updated.
1139 '''
1140 # This is a straightforward solution of f(alp1) = lambda12(alp1) -
1141 # lam12 = 0 with one wrinkle. f(alp) has exactly one root in the
1142 # interval (0, PI) and its derivative is positive at the root.
1143 # Thus f(alp) is positive for alp > alp1 and negative for alp < alp1.
1144 # During the course of the iteration, a range (alp1a, alp1b) is
1145 # maintained which brackets the root and with each evaluation of
1146 # f(alp) the range is shrunk, if possible. Newton's method is
1147 # restarted whenever the derivative of f is negative (because the
1148 # new value of alp1 is then further from the solution) or if the
1149 # new estimate of alp1 lies outside (0,PI); in this case, the new
1150 # starting guess is taken to be (alp1a + alp1b) / 2.
1151 salp1a = salp1b = _TINY
1152 calp1a, calp1b = _1_0, _N_1_0
1153 MAXIT1, TOL0 = _MAXIT1, _TOL0
1154 HALF, TOLb = _0_5, _TOLb
1155 tripb, TOLv = False, TOL0
1156 for i in range(_MAXIT2):
1157 # 1/4 meridian = 10e6 meter and random input,
1158 # estimated max error in nm (nano meter, by
1159 # checking Inverse problem by Direct).
1160 #
1161 # max iterations
1162 # log2(b/a) error mean sd
1163 # -7 387 5.33 3.68
1164 # -6 345 5.19 3.43
1165 # -5 269 5.00 3.05
1166 # -4 210 4.76 2.44
1167 # -3 115 4.55 1.87
1168 # -2 69 4.35 1.38
1169 # -1 36 4.05 1.03
1170 # 0 15 0.01 0.13
1171 # 1 25 5.10 1.53
1172 # 2 96 5.61 2.09
1173 # 3 318 6.02 2.74
1174 # 4 985 6.24 3.22
1175 # 5 2352 6.32 3.44
1176 # 6 6008 6.30 3.45
1177 # 7 19024 6.19 3.30
1178 v, sig12, salp2, calp2, \
1179 domg12, dv = self._Lambda6(salp1, calp1, i < MAXIT1, p)
1181 # 2 * _TOL0 is approximately 1 ulp [0, PI]
1182 # reversed test to allow escape with NaNs
1183 if tripb or fabs(v) < TOLv:
1184 break
1185 # update bracketing values
1186 if v > 0 and (i > MAXIT1 or (calp1 / salp1) > (calp1b / salp1b)):
1187 salp1b, calp1b = salp1, calp1
1188 elif v < 0 and (i > MAXIT1 or (calp1 / salp1) < (calp1a / salp1a)):
1189 salp1a, calp1a = salp1, calp1
1191 if i < MAXIT1 and dv > 0:
1192 dalp1 = -v / dv
1193 if fabs(dalp1) < PI:
1194 s, c = _sincos2(dalp1)
1195 # nalp1 = alp1 + dalp1
1196 s, c = _sincos12(-s, c, salp1, calp1)
1197 if s > 0:
1198 salp1, calp1 = _norm2(s, c)
1199 # in some regimes we don't get quadratic convergence
1200 # because slope -> 0. So use convergence conditions
1201 # based on epsilon instead of sqrt(epsilon)
1202 TOLv = TOL0 if fabs(v) > _TOL016 else _TOL08
1203 continue
1205 # Either dv was not positive or updated value was outside
1206 # legal range. Use the midpoint of the bracket as the next
1207 # estimate. This mechanism is not needed for the WGS84
1208 # ellipsoid, but it does catch problems with more eccentric
1209 # ellipsoids. Its efficacy is such for the WGS84 test set
1210 # with the starting guess set to alp1 = 90 deg: the WGS84
1211 # test set: mean = 5.21, stdev = 3.93, max = 24 and WGS84
1212 # with random input: mean = 4.74, stdev = 0.99
1213 salp1, calp1 = _norm2((salp1a + salp1b) * HALF,
1214 (calp1a + calp1b) * HALF)
1215 tripb = fsum1f_(calp1a, -calp1, fabs(salp1a - salp1)) < TOLb or \
1216 fsum1f_(calp1b, -calp1, fabs(salp1b - salp1)) < TOLb
1217 TOLv = TOL0
1219 else:
1220 raise GeodesicError(Fmt.no_convergence(v, TOLv), txt=repr(self)) # self.toRepr()
1222 p.set_(iter=i, trip=tripb) # like .geodsolve._GDictInvoke: iter NOT iteration!
1223 return sig12, salp1, calp1, salp2, calp2, domg12
1225 Polygon = Area # for C{geographiclib} compatibility
1227 def _sinf1cos2d(self, lat):
1228 '''(INTERNAL) Helper, also for C{_G_GeodesicLineExact}.
1229 '''
1230 sbet, cbet = _sincos2d(lat)
1231 # ensure cbet1 = +epsilon at poles; doing the fix on beta means
1232 # that sig12 will be <= 2*tiny for two points at the same pole
1233 sbet, cbet = _norm2(sbet * self.f1, cbet)
1234 return sbet, max(_TINY, cbet)
1236 def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature
1237 '''Return this C{GeodesicExact} as string.
1239 @kwarg prec: The C{float} precision, number of decimal digits (0..9).
1240 Trailing zero decimals are stripped for B{C{prec}} values
1241 of 1 and above, but kept for negative B{C{prec}} values.
1242 @kwarg sep: Separator to join (C{str}).
1244 @return: Tuple items (C{str}).
1245 '''
1246 d = dict(ellipsoid=self.ellipsoid, C4order=self.C4order)
1247 return sep.join(pairs(d, prec=prec))
1250class GeodesicLineExact(_GeodesicLineExact):
1251 '''A pure Python version of I{Karney}'s C++ class U{GeodesicLineExact
1252 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicLineExact.html>},
1253 modeled after I{Karney}'s Python class U{geodesicline.GeodesicLine<https://GitHub.com/
1254 geographiclib/geographiclib-python>}.
1255 '''
1257 def __init__(self, geodesic, lat1, lon1, azi1, caps=Caps.STANDARD, name=NN):
1258 '''New L{GeodesicLineExact} instance, allowing points to be found along
1259 a geodesic starting at C{(B{lat1}, B{lon1})} with azimuth B{C{azi1}}.
1261 @arg geodesic: The geodesic to use (L{GeodesicExact}).
1262 @arg lat1: Latitude of the first point (C{degrees}).
1263 @arg lon1: Longitude of the first point (C{degrees}).
1264 @arg azi1: Azimuth at the first points (compass C{degrees}).
1265 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying
1266 the capabilities the L{GeodesicLineExact} instance
1267 should possess, i.e., which quantities can be
1268 returned by calls to L{GeodesicLineExact.Position}
1269 and L{GeodesicLineExact.ArcPosition}.
1270 @kwarg name: Optional name (C{str}).
1272 @raise TypeError: Invalid B{C{geodesic}}.
1273 '''
1274 _xinstanceof(GeodesicExact, geodesic=geodesic)
1275 if (caps & Caps.LINE_OFF): # copy to avoid updates
1276 geodesic = geodesic.copy(deep=False, name=NN(_UNDER_, geodesic.name))
1277# _update_all(geodesic)
1278 _GeodesicLineExact.__init__(self, geodesic, lat1, lon1, azi1, caps, 0, name=name)
1281def _Astroid(x, y):
1282 '''(INTERNAL) Solve M{k^4 + 2 * k^3 - (x^2 + y^2 - 1 ) * k^2 -
1283 (2 * k + 1) * y^2 = 0} for positive root k.
1284 '''
1285 p = x**2
1286 q = y**2
1287 r = fsumf_(_1_0, q, p, _N_2_0)
1288 if q or r > 0:
1289 r = r / _6_0 # /= chokes PyChecker
1290 # avoid possible division by zero when r = 0
1291 # by multiplying s and t by r^3 and r, resp.
1292 S = p * q / _4_0 # S = r^3 * s
1293 r3 = r**3
1294 T3 = r3 + S
1295 # discriminant of the quadratic equation for T3 is
1296 # zero on the evolute curve p^(1/3) + q^(1/3) = 1
1297 d = S * (S + r3 * _2_0)
1298 if d < 0:
1299 # T is complex, but u is defined for a real result
1300 a = atan2(sqrt(-d), -T3) / _3_0
1301 # There are 3 possible cube roots, choose the one which
1302 # avoids cancellation. Note d < 0 implies that r < 0.
1303 u = (cos(a) * _2_0 + _1_0) * r
1304 else:
1305 # pick the sign on the sqrt to maximize abs(T3) to
1306 # minimize loss of precision due to cancellation.
1307 if d:
1308 T3 += _copysign(sqrt(d), T3) # T3 = (r * t)^3
1309 # _cbrt always returns the real root, _cbrt(-8) = -2
1310 u = _cbrt(T3) # T = r * t
1311 if u: # T can be zero; but then r2 / T -> 0
1312 u += r**2 / u
1313 u += r
1314 v = _hypot(u, y) # sqrt(u**2 + q)
1315 # avoid loss of accuracy when u < 0
1316 u = (q / (v - u)) if u < 0 else (v + u)
1317 w = (u - q) / (v + v) # positive?
1318 # rearrange expression for k to avoid loss of accuracy due to
1319 # subtraction, division by 0 impossible because u > 0, w >= 0
1320 k = u / (sqrt(w**2 + u) + w) # guaranteed positive
1322 else: # q == 0 && r <= 0
1323 # y = 0 with |x| <= 1. Handle this case directly, for
1324 # y small, positive root is k = abs(y) / sqrt(1 - x^2)
1325 k = _0_0
1327 return k
1330__all__ += _ALL_DOCS(GeodesicExact, GeodesicLineExact)
1332# **) MIT License
1333#
1334# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.
1335#
1336# Permission is hereby granted, free of charge, to any person obtaining a
1337# copy of this software and associated documentation files (the "Software"),
1338# to deal in the Software without restriction, including without limitation
1339# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1340# and/or sell copies of the Software, and to permit persons to whom the
1341# Software is furnished to do so, subject to the following conditions:
1342#
1343# The above copyright notice and this permission notice shall be included
1344# in all copies or substantial portions of the Software.
1345#
1346# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1347# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1348# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1349# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1350# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1351# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1352# OTHER DEALINGS IN THE SOFTWARE.