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
45# from pygeodesy.fmath import cbrt as _cbrt, hypot as hypot_ # from .karney
46from pygeodesy.fsums import fsumf_, fsum1f_
47from pygeodesy.geodesicx.gxbases import _cosSeries, _GeodesicBase, \
48 _sincos12, _sin1cos2, _xnC4
49from pygeodesy.geodesicx.gxline import _GeodesicLineExact, _TINY, _update_glXs
50from pygeodesy.interns import NN, _COMMASPACE_, _DOT_, _UNDER_
51from pygeodesy.karney import _around, _atan2d, Caps, _cbrt, _copysign, _diff182, \
52 _EWGS84, _fix90, GDict, GeodesicError, _hypot, _K_2_0, \
53 _norm2, _norm180, _polynomial, _signBit, _sincos2, \
54 _sincos2d, _sincos2de, _unsigned2, _a_ellipsoid
55from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS
56from pygeodesy.namedTuples import Destination3Tuple, Distance3Tuple
57from pygeodesy.props import deprecated_Property, Property, Property_RO
58from pygeodesy.streprs import Fmt, pairs
59from pygeodesy.utily import atan2d as _atan2d_reverse, _Wrap, wrap360
61from math import atan2, copysign, cos, degrees, fabs, radians, sqrt
63__all__ = ()
64__version__ = '23.07.20'
66_MAXIT1 = 20
67_MAXIT2 = 10 + _MAXIT1 + MANT_DIG # MANT_DIG == C++ digits
69# increased multiplier in defn of _TOL1 from 100 to 200 to fix Inverse
70# case 52.784459512564 0 -52.784459512563990912 179.634407464943777557
71# which otherwise failed for Visual Studio 10 (Release and Debug)
72_TOL0 = EPS
73_TOL1 = _TOL0 * -200 # negative
74_TOL2 = _EPSqrt # == sqrt(_TOL0)
75_TOL3 = _TOL2 * _0_1
76_TOLb = _TOL2 * _TOL0 # Check on bisection interval
77_THR1 = _TOL2 * _1000_0 + _1_0
79_TINY3 = _TINY * _3_0
80_TOL08 = _TOL0 * _8_0
81_TOL016 = _TOL0 * _16_0
84def _atan12(*sincos12, **sineg0):
85 '''(INTERNAL) Return C{ang12} in C{radians}.
86 '''
87 return atan2(*_sincos12(*sincos12, **sineg0))
90def _eTOL2(f):
91 # Using the auxiliary sphere solution with dnm computed at
92 # (bet1 + bet2) / 2, the relative error in the azimuth
93 # consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2.
94 # (Error measured for 1/100 < b/a < 100 and abs(f) >= 1/1000.
96 # For a given f and sig12, the max error occurs for lines
97 # near the pole. If the old rule for computing dnm = (dn1
98 # + dn2)/2 is used, then the error increases by a factor of
99 # 2.) Setting this equal to epsilon gives sig12 = etol2.
101 # Here 0.1 is a safety factor (error decreased by 100) and
102 # max(0.001, abs(f)) stops etol2 getting too large in the
103 # nearly spherical case.
104 t = min(_1_0, _1_0 - f * _0_5) * max(_0_001, fabs(f)) * _0_5
105 return _TOL3 / (sqrt(t) if t > EPS02 else EPS0)
108class _PDict(GDict):
109 '''(INTERNAL) Parameters passed around in C{._GDictInverse} and
110 optionally returned when C{GeodesicExact.debug} is C{True}.
111 '''
112 def setsigs(self, ssig1, csig1, ssig2, csig2):
113 '''Update the C{sig1} and C{sig2} parameters.
114 '''
115 self.set_(ssig1=ssig1, csig1=csig1, sncndn1=(ssig1, csig1, self.dn1), # PYCHOK dn1
116 ssig2=ssig2, csig2=csig2, sncndn2=(ssig2, csig2, self.dn2)) # PYCHOK dn2
118 def toGDict(self): # PYCHOK no cover
119 '''Return as C{GDict} without attrs C{sncndn1} and C{sncndn2}.
120 '''
121 def _rest(sncndn1=None, sncndn2=None, **rest): # PYCHOK sncndn* not used
122 return GDict(rest)
124 return _rest(**self)
127class GeodesicExact(_GeodesicBase):
128 '''A pure Python version of I{Karney}'s C++ class U{GeodesicExact
129 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>},
130 modeled after I{Karney}'s Python class U{geodesic.Geodesic<https://GitHub.com/
131 geographiclib/geographiclib-python>}.
132 '''
133 _E = _EWGS84
134 _nC4 = 30 # default C4order
136 def __init__(self, a_ellipsoid=_EWGS84, f=None, name=NN, C4order=None,
137 C4Order=None): # for backward compatibility
138 '''New L{GeodesicExact} instance.
140 @arg a_ellipsoid: An ellipsoid (L{Ellipsoid}) or datum (L{Datum}) or
141 the equatorial radius of the ellipsoid (C{scalar},
142 conventionally in C{meter}), see B{C{f}}.
143 @arg f: The flattening of the ellipsoid (C{scalar}) if B{C{a_ellipsoid}}
144 is specified as C{scalar}.
145 @kwarg name: Optional name (C{str}).
146 @kwarg C4order: Optional series expansion order (C{int}), see property
147 L{C4order}, default C{30}.
148 @kwarg C4Order: DEPRECATED, use keyword argument B{C{C4order}}.
150 @raise GeodesicError: Invalid B{C{C4order}}.
151 '''
152 if a_ellipsoid not in (GeodesicExact._E, None):
153 self._E = _a_ellipsoid(a_ellipsoid, f, name=name)
155 if name:
156 self.name = name
157 if C4order: # XXX private copy, always?
158 self.C4order = C4order
159 elif C4Order: # for backward compatibility
160 self.C4Order = C4Order
162 @Property_RO
163 def a(self):
164 '''Get the I{equatorial} radius, semi-axis (C{meter}).
165 '''
166 return self.ellipsoid.a
168 def ArcDirect(self, lat1, lon1, azi1, a12, outmask=Caps.STANDARD):
169 '''Solve the I{Direct} geodesic problem in terms of (spherical) arc length.
171 @arg lat1: Latitude of the first point (C{degrees}).
172 @arg lon1: Longitude of the first point (C{degrees}).
173 @arg azi1: Azimuth at the first point (compass C{degrees}).
174 @arg a12: Arc length between the points (C{degrees}), can be negative.
175 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying
176 the quantities to be returned.
178 @return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2,
179 lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1},
180 C{lon1}, C{azi1} and arc length C{a12} always included.
182 @see: C++ U{GeodesicExact.ArcDirect
183 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>}
184 and Python U{Geodesic.ArcDirect<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.
185 '''
186 return self._GDictDirect(lat1, lon1, azi1, True, a12, outmask)
188 def ArcDirectLine(self, lat1, lon1, azi1, a12, caps=Caps.ALL, name=NN):
189 '''Define a L{GeodesicLineExact} in terms of the I{direct} geodesic problem and as arc length.
191 @arg lat1: Latitude of the first point (C{degrees}).
192 @arg lon1: Longitude of the first point (C{degrees}).
193 @arg azi1: Azimuth at the first point (compass C{degrees}).
194 @arg a12: Arc length between the points (C{degrees}), can be negative.
195 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying
196 the capabilities the L{GeodesicLineExact} instance
197 should possess, i.e., which quantities can be
198 returned by calls to L{GeodesicLineExact.Position}
199 and L{GeodesicLineExact.ArcPosition}.
201 @return: A L{GeodesicLineExact} instance.
203 @note: The third point of the L{GeodesicLineExact} is set to correspond
204 to the second point of the I{Inverse} geodesic problem.
206 @note: Latitude B{C{lat1}} should in the range C{[-90, +90]}.
208 @see: C++ U{GeodesicExact.ArcDirectLine
209 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} and
210 Python U{Geodesic.ArcDirectLine<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.
211 '''
212 return self._GenDirectLine(lat1, lon1, azi1, True, a12, caps, name=name)
214 def Area(self, polyline=False, name=NN):
215 '''Set up a L{GeodesicAreaExact} to compute area and
216 perimeter of a polygon.
218 @kwarg polyline: If C{True} perimeter only, otherwise
219 area and perimeter (C{bool}).
220 @kwarg name: Optional name (C{str}).
222 @return: A L{GeodesicAreaExact} instance.
224 @note: The B{C{debug}} setting is passed as C{verbose}
225 to the returned L{GeodesicAreaExact} instance.
226 '''
227 gaX = _MODS.geodesicx.GeodesicAreaExact(self, polyline=polyline,
228 name=name or self.name)
229 if self.debug:
230 gaX.verbose = True
231 return gaX
233 @Property_RO
234 def b(self):
235 '''Get the ellipsoid's I{polar} radius, semi-axis (C{meter}).
236 '''
237 return self.ellipsoid.b
239 @Property_RO
240 def c2x(self):
241 '''Get the ellipsoid's I{authalic} earth radius I{squared} (C{meter} I{squared}).
242 '''
243 # The Geodesic class substitutes atanh(sqrt(e2)) for asinh(sqrt(ep2))
244 # in the definition of _c2. The latter is more accurate for very
245 # oblate ellipsoids (which the Geodesic class does not handle). Of
246 # course, the area calculation in GeodesicExact is still based on a
247 # series and only holds for moderately oblate (or prolate) ellipsoids.
248 return self.ellipsoid.c2x
250 c2 = c2x # in this particular case
252 def C4f(self, eps):
253 '''Evaluate the C{C4x} coefficients for B{C{eps}}.
255 @arg eps: Polynomial factor (C{float}).
257 @return: C{C4order}-Tuple of C{C4x(B{eps})} coefficients.
258 '''
259 def _c4(nC4, C4x):
260 i, x, e = 0, _1_0, eps
261 _p = _polynomial
262 for r in range(nC4, 0, -1):
263 j = i + r
264 yield _p(e, C4x, i, j) * x
265 x *= e
266 i = j
267 # assert i == (nC4 * (nC4 + 1)) // 2
269 return tuple(_c4(self._nC4, self._C4x))
271 def _C4f_k2(self, k2): # in ._GDictInverse and gxline._GeodesicLineExact._C4a
272 '''(INTERNAL) Compute C{eps} from B{C{k2}} and invoke C{C4f}.
273 '''
274 return self.C4f(k2 / fsumf_(_2_0, sqrt(k2 + _1_0) * _2_0, k2))
276 @Property
277 def C4order(self):
278 '''Get the series expansion order (C{int}, 24, 27 or 30).
279 '''
280 return self._nC4
282 @C4order.setter # PYCHOK .setter!
283 def C4order(self, order):
284 '''Set the series expansion order (C{int}, 24, 27 or 30).
286 @raise GeodesicError: Invalid B{C{order}}.
287 '''
288 _xnC4(C4order=order)
289 if self._nC4 != order:
290 GeodesicExact._C4x._update(self)
291 _update_glXs(self) # zap cached _GeodesicLineExact attrs _B41, _C4a
292 self._nC4 = order
294 @deprecated_Property
295 def C4Order(self):
296 '''DEPRECATED, use property C{C4order}.
297 '''
298 return self.C4order
300 @C4Order.setter # PYCHOK .setter!
301 def C4Order(self, order):
302 '''DEPRECATED, use property C{C4order}.
303 '''
304 _xnC4(C4Order=order)
305 self.C4order = order
307 @Property_RO
308 def _C4x(self):
309 '''Get this ellipsoid's C{C4} coefficients, I{cached} tuple.
311 @see: Property L{C4order}.
312 '''
313 # see C4coeff() in GeographicLib.src.GeodesicExactC4.cpp
314 def _C4(nC4):
315 i, n, cs = 0, self.n, _C4coeffs(nC4)
316 _p = _polynomial
317 for r in range(nC4 + 1, 1, -1):
318 for j in range(1, r):
319 j = j + i # (j - i - 1) order of polynomial
320 yield _p(n, cs, i, j) / cs[j]
321 i = j + 1
322 # assert i == (nC4 * (nC4 + 1) * (nC4 + 5)) // 6
324 return tuple(_C4(self._nC4)) # 3rd flattening
326 def Direct(self, lat1, lon1, azi1, s12, outmask=Caps.STANDARD):
327 '''Solve the I{Direct} geodesic problem
329 @arg lat1: Latitude of the first point (C{degrees}).
330 @arg lon1: Longitude of the first point (C{degrees}).
331 @arg azi1: Azimuth at the first point (compass C{degrees}).
332 @arg s12: Distance between the points (C{meter}), can be negative.
333 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying
334 the quantities to be returned.
336 @return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2,
337 lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1},
338 C{lon1}, C{azi1} and distance C{s12} always included.
340 @see: C++ U{GeodesicExact.Direct
341 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>}
342 and Python U{Geodesic.Direct<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.
343 '''
344 return self._GDictDirect(lat1, lon1, azi1, False, s12, outmask)
346 def Direct3(self, lat1, lon1, azi1, s12): # PYCHOK outmask
347 '''Return the destination lat, lon and reverse azimuth
348 (final bearing) in C{degrees}.
350 @return: L{Destination3Tuple}C{(lat, lon, final)}.
351 '''
352 r = self._GDictDirect(lat1, lon1, azi1, False, s12, Caps._AZIMUTH_LATITUDE_LONGITUDE)
353 return Destination3Tuple(r.lat2, r.lon2, r.azi2) # no iteration
355 def DirectLine(self, lat1, lon1, azi1, s12, caps=Caps.STANDARD, name=NN):
356 '''Define a L{GeodesicLineExact} in terms of the I{direct} geodesic problem and as distance.
358 @arg lat1: Latitude of the first point (C{degrees}).
359 @arg lon1: Longitude of the first point (C{degrees}).
360 @arg azi1: Azimuth at the first point (compass C{degrees}).
361 @arg s12: Distance between the points (C{meter}), can be negative.
362 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying
363 the capabilities the L{GeodesicLineExact} instance
364 should possess, i.e., which quantities can be
365 returned by calls to L{GeodesicLineExact.Position}.
367 @return: A L{GeodesicLineExact} instance.
369 @note: The third point of the L{GeodesicLineExact} is set to correspond
370 to the second point of the I{Inverse} geodesic problem.
372 @note: Latitude B{C{lat1}} should in the range C{[-90, +90]}.
374 @see: C++ U{GeodesicExact.DirectLine
375 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} and
376 Python U{Geodesic.DirectLine<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.
377 '''
378 return self._GenDirectLine(lat1, lon1, azi1, False, s12, caps, name=name)
380 def _dn(self, sbet, cbet): # in gxline._GeodesicLineExact.__init__
381 '''(INTERNAL) Helper.
382 '''
383 if self.f < 0: # PYCHOK no cover
384 dn = sqrt(_1_0 - cbet**2 * self.e2) / self.f1
385 else:
386 dn = sqrt(_1_0 + sbet**2 * self.ep2)
387 return dn
389 @Property_RO
390 def e2(self):
391 '''Get the ellipsoid's I{(1st) eccentricity squared} (C{float}), M{f * (2 - f)}.
392 '''
393 return self.ellipsoid.e2
395 @Property_RO
396 def _e2a2(self):
397 '''(INTERNAL) Cache M{E.e2 * E.a2}.
398 '''
399 return self.e2 * self.ellipsoid.a2
401 @Property_RO
402 def _e2_f1(self):
403 '''(INTERNAL) Cache M{E.e2 * E.f1}.
404 '''
405 return self.e2 / self.f1
407 @Property_RO
408 def _eF(self):
409 '''(INTERNAL) Get the elliptic function, aka C{.E}.
410 '''
411 return _MODS.elliptic.Elliptic(k2=-self.ep2)
413 def _eF_reset_cHe2_f1(self, x, y):
414 '''(INTERNAL) Reset elliptic function and return M{cH * e2 / f1 * ...}.
415 '''
416 self._eF_reset_k2(x)
417 return y * self._eF.cH * self._e2_f1
419 def _eF_reset_k2(self, x):
420 '''(INTERNAL) Reset elliptic function and return C{k2}.
421 '''
422 ep2 = self.ep2
423 k2 = x**2 * ep2 # see .gxline._GeodesicLineExact._eF
424 self._eF.reset(k2=-k2, alpha2=-ep2) # kp2, alphap2 defaults
425 _update_glXs(self) # zap cached/memoized _GeodesicLineExact attrs
426 return k2
428 @Property_RO
429 def ellipsoid(self):
430 '''Get the ellipsoid (C{Ellipsoid}).
431 '''
432 return self._E
434 @Property_RO
435 def ep2(self):
436 '''Get the ellipsoid's I{2nd eccentricity squared} (C{float}), M{e2 / (1 - e2)}.
437 '''
438 return self.ellipsoid.e22 # == self.e2 / self.f1**2
440 e22 = ep2 # for ellipsoid compatibility
442 @Property_RO
443 def _eTOL2(self):
444 '''(INTERNAL) The si12 threshold for "really short".
445 '''
446 return _eTOL2(self.f)
448 @Property_RO
449 def f(self):
450 '''Get the ellipsoid's I{flattening} (C{float}), M{(a - b) / a}, C{0} for spherical, negative for prolate.
451 '''
452 return self.ellipsoid.f
454 flattening = f
456 @Property_RO
457 def f1(self): # in .css.CassiniSoldner.reset
458 '''Get the ellipsoid's I{1 - flattening} (C{float}).
459 '''
460 return self.ellipsoid.f1
462 @Property_RO
463 def _f180(self):
464 '''(INTERNAL) Cached/memoized.
465 '''
466 return self.f * _180_0
468 def _GDictDirect(self, lat1, lon1, azi1, arcmode, s12_a12, outmask=Caps.STANDARD):
469 '''(INTERNAL) As C{_GenDirect}, but returning a L{GDict}.
471 @return: A L{GDict} ...
472 '''
473 C = outmask if arcmode else (outmask | Caps.DISTANCE_IN)
474 glX = self.Line(lat1, lon1, azi1, C | Caps.LINE_OFF)
475 return glX._GDictPosition(arcmode, s12_a12, outmask)
477 def _GDictInverse(self, lat1, lon1, lat2, lon2, outmask=Caps.STANDARD): # MCCABE 33, 41 vars
478 '''(INTERNAL) As C{_GenInverse}, but returning a L{GDict}.
480 @return: A L{GDict} ...
481 '''
482 Cs = Caps
483 if self._debug: # PYCHOK no cover
484 outmask |= Cs._DEBUG_INVERSE & self._debug
485 outmask &= Cs._OUT_MASK # incl. _SALPs_CALPs and _DEBUG_
486 # compute longitude difference carefully (with _diff182):
487 # result is in [-180, +180] but -180 is only for west-going
488 # geodesics, +180 is for east-going and meridional geodesics
489 lon12, lon12s = _diff182(lon1, lon2)
490 # see C{result} from geographiclib.geodesic.Inverse
491 if (outmask & Cs.LONG_UNROLL): # == (lon1 + lon12) + lon12s
492 r = GDict(lon1=lon1, lon2=fsumf_(lon1, lon12, lon12s))
493 else:
494 r = GDict(lon1=_norm180(lon1), lon2=_norm180(lon2))
495 if _K_2_0: # GeographicLib 2.0
496 # make longitude difference positive
497 lon12, lon_ = _unsigned2(lon12)
498 if lon_:
499 lon12s = -lon12s
500 lam12 = radians(lon12)
501 # calculate sincosd(_around(lon12 + correction))
502 slam12, clam12 = _sincos2de(lon12, lon12s)
503 # supplementary longitude difference
504 lon12s = fsumf_(_180_0, -lon12, -lon12s)
505 else: # GeographicLib 1.52
506 # make longitude difference positive and if very close
507 # to being on the same half-meridian, then make it so.
508 if lon12 < 0: # _signBit(lon12)
509 lon_, lon12 = True, -_around(lon12)
510 lon12s = _around(fsumf_(_180_0, -lon12, lon12s))
511 else:
512 lon_, lon12 = False, _around(lon12)
513 lon12s = _around(fsumf_(_180_0, -lon12, -lon12s))
514 lam12 = radians(lon12)
515 if lon12 > _90_0:
516 slam12, clam12 = _sincos2d(lon12s)
517 clam12 = -clam12
518 else:
519 slam12, clam12 = _sincos2(lam12)
520 # If really close to the equator, treat as on equator.
521 lat1 = _around(_fix90(lat1))
522 lat2 = _around(_fix90(lat2))
523 r.set_(lat1=lat1, lat2=lat2)
524 # Swap points so that point with higher (abs) latitude is
525 # point 1. If one latitude is a NAN, then it becomes lat1.
526 swap_ = fabs(lat1) < fabs(lat2) or isnan(lat2)
527 if swap_:
528 lat1, lat2 = lat2, lat1
529 lon_ = not lon_
530 if _signBit(lat1):
531 lat_ = False # note, False
532 else: # make lat1 <= -0
533 lat_ = True # note, True
534 lat1, lat2 = -lat1, -lat2
535 # Now 0 <= lon12 <= 180, -90 <= lat1 <= -0 and lat1 <= lat2 <= -lat1
536 # and lat_, lon_, swap_ register the transformation to bring the
537 # coordinates to this canonical form, where False means no change
538 # made. We make these transformations so that there are few cases
539 # to check, e.g., on verifying quadrants in atan2. In addition,
540 # this enforces some symmetries in the results returned.
542 # Initialize for the meridian. No longitude calculation is
543 # done in this case to let the parameter default to 0.
544 sbet1, cbet1 = self._sinf1cos2d(lat1)
545 sbet2, cbet2 = self._sinf1cos2d(lat2)
546 # If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure
547 # of the |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1),
548 # abs(sbet2) + sbet1 is a better measure. This logic is used
549 # in assigning calp2 in _Lambda6. Sometimes these quantities
550 # vanish and in that case we force bet2 = +/- bet1 exactly. An
551 # example where is is necessary is the inverse problem
552 # 48.522876735459 0 -48.52287673545898293 179.599720456223079643
553 # which failed with Visual Studio 10 (Release and Debug)
554 if cbet1 < -sbet1:
555 if cbet2 == cbet1:
556 sbet2 = copysign(sbet1, sbet2)
557 elif fabs(sbet2) == -sbet1:
558 cbet2 = cbet1
560 p = _PDict(sbet1=sbet1, cbet1=cbet1, dn1=self._dn(sbet1, cbet1),
561 sbet2=sbet2, cbet2=cbet2, dn2=self._dn(sbet2, cbet2))
563 _meridian = _b = True # i.e. not meridian, not b
564 if lat1 == -90 or slam12 == 0:
565 # Endpoints are on a single full meridian,
566 # so the geodesic might lie on a meridian.
567 salp1, calp1 = slam12, clam12 # head to target lon
568 salp2, calp2 = _0_0, _1_0 # then head north
569 # tan(bet) = tan(sig) * cos(alp)
570 p.setsigs(sbet1, calp1 * cbet1, sbet2, calp2 * cbet2)
571 # sig12 = sig2 - sig1
572 sig12 = _atan12(sbet1, p.csig1, sbet2, p.csig2, sineg0=True) # PYCHOK csig*
573 s12x, m12x, _, \
574 M12, M21 = self._Length5(sig12, outmask | Cs.REDUCEDLENGTH, p)
575 # Add the check for sig12 since zero length geodesics
576 # might yield m12 < 0. Test case was
577 # echo 20.001 0 20.001 0 | GeodSolve -i
578 # In fact, we will have sig12 > PI/2 for meridional
579 # geodesic which is not a shortest path.
580 if m12x >= 0 or sig12 < _1_0:
581 # Need at least 2 to handle 90 0 90 180
582 # Prevent negative s12 or m12 from geographiclib 1.52
583 if sig12 < _TINY3 or (sig12 < _TOL0 and (s12x < 0 or m12x < 0)):
584 sig12 = m12x = s12x = _0_0
585 else:
586 _b = False # apply .b to s12x, m12x
587 _meridian = False
588 C = 1
589 # else: # m12 < 0, prolate and too close to anti-podal
590 # _meridian = True
591 a12 = _0_0 # if _b else degrees(sig12)
593 if _meridian:
594 _b = sbet1 == 0 and (self.f <= 0 or lon12s >= self._f180) # and sbet2 == 0
595 if _b: # Geodesic runs along equator
596 calp1 = calp2 = _0_0
597 salp1 = salp2 = _1_0
598 sig12 = lam12 / self.f1 # == omg12
599 somg12, comg12 = _sincos2(sig12)
600 m12x = self.b * somg12
601 s12x = self.a * lam12
602 M12 = M21 = comg12
603 a12 = lon12 / self.f1
604 C = 2
605 else:
606 # Now point1 and point2 belong within a hemisphere bounded by a
607 # meridian and geodesic is neither meridional or equatorial.
608 p.set_(slam12=slam12, clam12=clam12)
609 # Figure a starting point for Newton's method
610 sig12, salp1, calp1, \
611 salp2, calp2, dnm = self._InverseStart6(lam12, p)
612 if sig12 is None: # use Newton's method
613 # pre-compute the constant _Lambda6 term, once
614 p.set_(bet12=None if cbet2 == cbet1 and fabs(sbet2) == -sbet1 else
615 (((cbet1 + cbet2) * (cbet2 - cbet1)) if cbet1 < -sbet1 else
616 ((sbet1 + sbet2) * (sbet1 - sbet2))))
617 sig12, salp1, calp1, \
618 salp2, calp2, domg12 = self._Newton6(salp1, calp1, p)
619 s12x, m12x, _, M12, M21 = self._Length5(sig12, outmask, p)
620 if (outmask & Cs.AREA):
621 # omg12 = lam12 - domg12
622 s, c = _sincos2(domg12)
623 somg12, comg12 = _sincos12(s, c, slam12, clam12)
624 C = 3 # Newton
625 else: # from _InverseStart6: dnm, salp*, calp*
626 C = 4 # Short lines
627 s, c = _sincos2(sig12 / dnm)
628 m12x = dnm**2 * s
629 s12x = dnm * sig12
630 M12 = M21 = c
631 if (outmask & Cs.AREA):
632 somg12, comg12 = _sincos2(lam12 / (self.f1 * dnm))
634 else: # _meridian is False
635 somg12 = comg12 = NAN
637 r.set_(a12=a12 if _b else degrees(sig12)) # in [0, 180]
639 if (outmask & Cs.DISTANCE):
640 r.set_(s12=unsigned0(s12x if _b else (self.b * s12x)))
642 if (outmask & Cs.REDUCEDLENGTH):
643 r.set_(m12=unsigned0(m12x if _b else (self.b * m12x)))
645 if (outmask & Cs.GEODESICSCALE):
646 if swap_:
647 M12, M21 = M21, M12
648 r.set_(M12=unsigned0(M12),
649 M21=unsigned0(M21))
651 if (outmask & Cs.AREA):
652 S12 = self._InverseArea(_meridian, salp1, calp1,
653 salp2, calp2,
654 somg12, comg12, p)
655 if _xor(swap_, lat_, lon_):
656 S12 = -S12
657 r.set_(S12=unsigned0(S12))
659 if (outmask & (Cs.AZIMUTH | Cs._SALPs_CALPs)):
660 if swap_:
661 salp1, salp2 = salp2, salp1
662 calp1, calp2 = calp2, calp1
663 if _xor(swap_, lon_):
664 salp1, salp2 = -salp1, -salp2
665 if _xor(swap_, lat_):
666 calp1, calp2 = -calp1, -calp2
668 if (outmask & Cs.AZIMUTH):
669 r.set_(azi1=_atan2d(salp1, calp1),
670 azi2=_atan2d_reverse(salp2, calp2, reverse=outmask & Cs.REVERSE2))
671 if (outmask & Cs._SALPs_CALPs):
672 r.set_(salp1=salp1, calp1=calp1,
673 salp2=salp2, calp2=calp2)
675 if (outmask & Cs._DEBUG_INVERSE): # PYCHOK no cover
676 E, eF = self.ellipsoid, self._eF
677 p.set_(C=C, a=self.a, f=self.f, f1=self.f1,
678 e=E.e, e2=self.e2, ep2=self.ep2,
679 c2=E.c2, c2x=self.c2x,
680 eFcD=eF.cD, eFcE=eF.cE, eFcH=eF.cH,
681 eFk2=eF.k2, eFa2=eF.alpha2)
682 p.update(r) # r overrides p
683 r = p.toGDict()
684 return self._iter2tion(r, p)
686 def _GenDirect(self, lat1, lon1, azi1, arcmode, s12_a12, outmask=Caps.STANDARD):
687 '''(INTERNAL) The general I{Inverse} geodesic calculation.
689 @return: L{Direct9Tuple}C{(a12, lat2, lon2, azi2,
690 s12, m12, M12, M21, S12)}.
691 '''
692 r = self._GDictDirect(lat1, lon1, azi1, arcmode, s12_a12, outmask)
693 return r.toDirect9Tuple()
695 def _GenDirectLine(self, lat1, lon1, azi1, arcmode, s12_a12, caps, name=NN):
696 '''(INTERNAL) Helper for C{ArcDirectLine} and C{DirectLine}.
698 @return: A L{GeodesicLineExact} instance.
699 '''
700 azi1 = _norm180(azi1)
701 # guard against underflow in salp0. Also -0 is converted to +0.
702 s, c = _sincos2d(_around(azi1))
703 C = caps if arcmode else (caps | Caps.DISTANCE_IN)
704 return _GeodesicLineExact(self, lat1, lon1, azi1, C,
705 self._debug, s, c, name=name)._GenSet(arcmode, s12_a12)
707 def _GenInverse(self, lat1, lon1, lat2, lon2, outmask=Caps.STANDARD):
708 '''(INTERNAL) The general I{Inverse} geodesic calculation.
710 @return: L{Inverse10Tuple}C{(a12, s12, salp1, calp1, salp2, calp2,
711 m12, M12, M21, S12)}.
712 '''
713 r = self._GDictInverse(lat1, lon1, lat2, lon2, outmask | Caps._SALPs_CALPs)
714 return r.toInverse10Tuple()
716 def Inverse(self, lat1, lon1, lat2, lon2, outmask=Caps.STANDARD):
717 '''Perform the I{Inverse} geodesic calculation.
719 @arg lat1: Latitude of the first point (C{degrees}).
720 @arg lon1: Longitude of the first point (C{degrees}).
721 @arg lat2: Latitude of the second point (C{degrees}).
722 @arg lon2: Longitude of the second point (C{degrees}).
723 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying
724 the quantities to be returned.
726 @return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2,
727 lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1},
728 C{lon1}, C{azi1} and distance C{s12} always included.
730 @note: The third point of the L{GeodesicLineExact} is set to correspond
731 to the second point of the I{Inverse} geodesic problem.
733 @note: Both B{C{lat1}} and B{C{lat2}} should in the range C{[-90, +90]}.
735 @see: C++ U{GeodesicExact.InverseLine
736 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} and
737 Python U{Geodesic.InverseLine<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.
738 '''
739 return self._GDictInverse(lat1, lon1, lat2, lon2, outmask)
741 def Inverse1(self, lat1, lon1, lat2, lon2, wrap=False):
742 '''Return the non-negative, I{angular} distance in C{degrees}.
744 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
745 B{C{lat2}} and B{C{lon2}} (C{bool}).
746 '''
747 # see .FrechetKarney.distance, .HausdorffKarney._distance
748 # and .HeightIDWkarney._distances
749 if wrap:
750 _, lat2, lon2 = _Wrap.latlon3(lat1, lat2, lon2, True) # _Geodesic.LONG_UNROLL
751 return fabs(self._GDictInverse(lat1, lon1, lat2, lon2, Caps._ANGLE_ONLY).a12)
753 def Inverse3(self, lat1, lon1, lat2, lon2): # PYCHOK outmask
754 '''Return the distance in C{meter} and the forward and
755 reverse azimuths (initial and final bearing) in C{degrees}.
757 @return: L{Distance3Tuple}C{(distance, initial, final)}.
758 '''
759 r = self._GDictInverse(lat1, lon1, lat2, lon2, Caps.AZIMUTH_DISTANCE)
760 return Distance3Tuple(r.s12, wrap360(r.azi1), wrap360(r.azi2),
761 iteration=r.iteration)
763 def InverseLine(self, lat1, lon1, lat2, lon2, caps=Caps.STANDARD, name=NN):
764 '''Define a L{GeodesicLineExact} in terms of the I{Inverse} geodesic problem.
766 @arg lat1: Latitude of the first point (C{degrees}).
767 @arg lon1: Longitude of the first point (C{degrees}).
768 @arg lat2: Latitude of the second point (C{degrees}).
769 @arg lon2: Longitude of the second point (C{degrees}).
770 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying
771 the capabilities the L{GeodesicLineExact} instance
772 should possess, i.e., which quantities can be
773 returned by calls to L{GeodesicLineExact.Position}
774 and L{GeodesicLineExact.ArcPosition}.
776 @return: A L{GeodesicLineExact} instance.
778 @note: The third point of the L{GeodesicLineExact} is set to correspond
779 to the second point of the I{Inverse} geodesic problem.
781 @note: Both B{C{lat1}} and B{C{lat2}} should in the range C{[-90, +90]}.
783 @see: C++ U{GeodesicExact.InverseLine
784 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} and
785 Python U{Geodesic.InverseLine<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.
786 '''
787 Cs = Caps
788 r = self._GDictInverse(lat1, lon1, lat2, lon2, Cs._SALPs_CALPs) # No need for AZIMUTH
789 C = (caps | Cs.DISTANCE) if (caps & Cs._DISTANCE_IN_OUT) else caps
790 azi1 = _atan2d(r.salp1, r.calp1)
791 return _GeodesicLineExact(self, lat1, lon1, azi1, C, # ensure a12 is distance
792 self._debug, r.salp1, r.calp1, name=name)._GenSet(True, r.a12)
794 def _InverseArea(self, _meridian, salp1, calp1, # PYCHOK 9 args
795 salp2, calp2,
796 somg12, comg12, p):
797 '''(INTERNAL) Split off from C{_GDictInverse} to reduce complexity/length.
799 @return: Area C{S12}.
800 '''
801 # from _Lambda6: sin(alp1) * cos(bet1) = sin(alp0), calp0 > 0
802 salp0, calp0 = _sin1cos2(salp1, calp1, p.sbet1, p.cbet1)
803 A4 = calp0 * salp0
804 if A4:
805 # from _Lambda6: tan(bet) = tan(sig) * cos(alp)
806 k2 = calp0**2 * self.ep2
807 C4a = self._C4f_k2(k2)
808 B41 = _cosSeries(C4a, *_norm2(p.sbet1, calp1 * p.cbet1))
809 B42 = _cosSeries(C4a, *_norm2(p.sbet2, calp2 * p.cbet2))
810 # multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
811 A4 *= self._e2a2
812 S12 = A4 * (B42 - B41)
813 else: # avoid problems with indeterminate sig1, sig2 on equator
814 A4 = B41 = B42 = k2 = S12 = _0_0
816 if (_meridian and # omg12 < 3/4 * PI
817 comg12 > -_SQRT2_2 and # lon diff not too big
818 (p.sbet2 - p.sbet1) < _1_75): # lat diff not too big
819 # use tan(Gamma/2) = tan(omg12/2) *
820 # (tan(bet1/2) + tan(bet2/2)) /
821 # (tan(bet1/2) * tan(bet2/2) + 1))
822 # with tan(x/2) = sin(x) / (1 + cos(x))
823 dbet1 = p.cbet1 + _1_0
824 dbet2 = p.cbet2 + _1_0
825 domg12 = comg12 + _1_0
826 salp12 = (p.sbet1 * dbet2 + dbet1 * p.sbet2) * somg12
827 calp12 = (p.sbet1 * p.sbet2 + dbet1 * dbet2) * domg12
828 alp12 = _2_0 * atan2(salp12, calp12)
829 else:
830 # alp12 = alp2 - alp1, used in atan2, no need to normalize
831 salp12, calp12 = _sincos12(salp1, calp1, salp2, calp2)
832 # The right thing appears to happen if alp1 = +/-180 and
833 # alp2 = 0, viz salp12 = -0 and alp12 = -180. However,
834 # this depends on the sign being attached to 0 correctly.
835 # Following ensures the correct behavior.
836 if salp12 == 0 and calp12 < 0:
837 alp12 = _copysign(PI, calp1)
838 else:
839 alp12 = atan2(salp12, calp12)
841 p.set_(alp12=alp12, A4=A4, B41=B41, B42=B42, k2=k2)
842 return S12 + self.c2x * alp12
844 def _InverseStart6(self, lam12, p):
845 '''(INTERNAL) Return a starting point for Newton's method in
846 C{salp1} and C{calp1} indicated by C{sig12=None}. If
847 Newton's method doesn't need to be used, return also
848 C{salp2}, C{calp2}, C{dnm} and C{sig12} non-C{None}.
850 @return: 6-Tuple C{(sig12, salp1, calp1, salp2, calp2, dnm)}
851 and C{p.setsigs} updated for Newton, C{sig12=None}.
852 '''
853 sig12 = None # use Newton
854 salp1 = calp1 = salp2 = calp2 = dnm = NAN
856 # bet12 = bet2 - bet1 in [0, PI)
857 sbet12, cbet12 = _sincos12(p.sbet1, p.cbet1, p.sbet2, p.cbet2)
858 shortline = cbet12 >= 0 and sbet12 < _0_5 and (p.cbet2 * lam12) < _0_5
859 if shortline:
860 # sin((bet1 + bet2)/2)^2 = (sbet1 + sbet2)^2 / (
861 # (cbet1 + cbet2)^2 + (sbet1 + sbet2)^2)
862 t = (p.sbet1 + p.sbet2)**2
863 s = t / ((p.cbet1 + p.cbet2)**2 + t)
864 dnm = sqrt(_1_0 + self.ep2 * s)
865 somg12, comg12 = _sincos2(lam12 / (self.f1 * dnm))
866 else:
867 somg12, comg12 = p.slam12, p.clam12
869 # bet12a = bet2 + bet1 in (-PI, 0], note -sbet1
870 sbet12a, cbet12a = _sincos12(-p.sbet1, p.cbet1, p.sbet2, p.cbet2)
872 c = fabs(comg12) + _1_0 # == (1 - comg12) if comg12 < 0
873 s = somg12**2 / c
874 t = p.sbet1 * p.cbet2 * s
875 salp1 = p.cbet2 * somg12
876 calp1 = (sbet12a - t) if comg12 < 0 else (sbet12 + t)
878 ssig12 = _hypot(salp1, calp1)
879 csig12 = p.sbet1 * p.sbet2 + p.cbet1 * p.cbet2 * comg12
881 if shortline and ssig12 < self._eTOL2: # really short lines
882 t = c if comg12 < 0 else s
883 salp2, calp2 = _norm2(somg12 * p.cbet1,
884 sbet12 - p.cbet1 * p.sbet2 * t)
885 sig12 = atan2(ssig12, csig12) # do not use Newton
887 elif (self._n_0_1 or # Skip astroid calc if too eccentric
888 csig12 >= 0 or ssig12 >= (p.cbet1**2 * self._n6PI)):
889 pass # nothing to do, 0th order spherical approximation OK
891 else:
892 # Scale lam12 and bet2 to x, y coordinate system where antipodal
893 # point is at origin and singular point is at y = 0, x = -1
894 lam12x = atan2(-p.slam12, -p.clam12) # lam12 - PI
895 f = self.f
896 if f < 0: # PYCHOK no cover
897 # ssig1=sbet1, csig1=-cbet1, ssig2=sbet2, csig2=cbet2
898 p.setsigs(p.sbet1, -p.cbet1, p.sbet2, p.cbet2)
899 # if lon12 = 180, this repeats a calculation made in Inverse
900 _, m12b, m0, _, _ = self._Length5(atan2(sbet12a, cbet12a) + PI,
901 Caps.REDUCEDLENGTH, p)
902 t = p.cbet1 * PI # x = dlat, y = dlon
903 x = m12b / (t * p.cbet2 * m0) - _1_0
904 sca = (sbet12a / (x * p.cbet1)) if x < -_0_01 else (-f * t)
905 y = lam12x / sca
906 else: # f >= 0, however f == 0 does not get here
907 sca = self._eF_reset_cHe2_f1(p.sbet1, p.cbet1 * _2_0)
908 x = lam12x / sca # dlon
909 y = sbet12a / (sca * p.cbet1) # dlat
911 if y > _TOL1 and x > -_THR1: # strip near cut
912 if f < 0: # PYCHOK no cover
913 calp1 = max( _0_0, x) if x > _TOL1 else max(_N_1_0, x)
914 salp1 = sqrt(_1_0 - calp1**2)
915 else:
916 salp1 = min( _1_0, -x)
917 calp1 = -sqrt(_1_0 - salp1**2)
918 else:
919 # Estimate alp1, by solving the astroid problem.
920 #
921 # Could estimate alpha1 = theta + PI/2, directly, i.e.,
922 # calp1 = y/k; salp1 = -x/(1+k); for _f >= 0
923 # calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check)
924 #
925 # However, it's better to estimate omg12 from astroid and use
926 # spherical formula to compute alp1. This reduces the mean
927 # number of Newton iterations for astroid cases from 2.24
928 # (min 0, max 6) to 2.12 (min 0, max 5). The changes in the
929 # number of iterations are as follows:
930 #
931 # change percent
932 # 1 5
933 # 0 78
934 # -1 16
935 # -2 0.6
936 # -3 0.04
937 # -4 0.002
938 #
939 # The histogram of iterations is (m = number of iterations
940 # estimating alp1 directly, n = number of iterations
941 # estimating via omg12, total number of trials = 148605):
942 #
943 # iter m n
944 # 0 148 186
945 # 1 13046 13845
946 # 2 93315 102225
947 # 3 36189 32341
948 # 4 5396 7
949 # 5 455 1
950 # 6 56 0
951 #
952 # omg12 is near PI, estimate work with omg12a = PI - omg12
953 k = _Astroid(x, y)
954 sca *= (y * (k + _1_0) / k) if f < 0 else \
955 (x * k / (k + _1_0))
956 s, c = _sincos2(-sca) # omg12a
957 # update spherical estimate of alp1 using omg12 instead of lam12
958 salp1 = p.cbet2 * s
959 calp1 = sbet12a - s * salp1 * p.sbet1 / (c + _1_0) # c = -c
961 # sanity check on starting guess. Backwards check allows NaN through.
962 salp1, calp1 = _norm2(salp1, calp1) if salp1 > 0 else (_1_0, _0_0)
964 return sig12, salp1, calp1, salp2, calp2, dnm
966 def _Lambda6(self, salp1, calp1, diffp, p):
967 '''(INTERNAL) Helper.
969 @return: 6-Tuple C{(lam12, sig12, salp2, calp2, domg12,
970 dlam12} and C{p.setsigs} updated.
971 '''
972 eF = self._eF
973 f1 = self.f1
975 if p.sbet1 == calp1 == 0: # PYCHOK no cover
976 # Break degeneracy of equatorial line
977 calp1 = -_TINY
979 # sin(alp1) * cos(bet1) = sin(alp0), # calp0 > 0
980 salp0, calp0 = _sin1cos2(salp1, calp1, p.sbet1, p.cbet1)
981 # tan(bet1) = tan(sig1) * cos(alp1)
982 # tan(omg1) = sin(alp0) * tan(sig1)
983 # = sin(bet1) * tan(alp1)
984 somg1 = salp0 * p.sbet1
985 comg1 = calp1 * p.cbet1
986 ssig1, csig1 = _norm2(p.sbet1, comg1)
987 # Without normalization we have schi1 = somg1
988 cchi1 = f1 * p.dn1 * comg1
990 # Enforce symmetries in the case abs(bet2) = -bet1.
991 # Need to be careful about this case, since this can
992 # yield singularities in the Newton iteration.
993 # sin(alp2) * cos(bet2) = sin(alp0)
994 salp2 = (salp0 / p.cbet2) if p.cbet2 != p.cbet1 else salp1
995 # calp2 = sqrt(1 - sq(salp2))
996 # = sqrt(sq(calp0) - sq(sbet2)) / cbet2
997 # and subst for calp0 and rearrange to give (choose
998 # positive sqrt to give alp2 in [0, PI/2]).
999 calp2 = fabs(calp1) if p.bet12 is None else (
1000 sqrt((calp1 * p.cbet1)**2 + p.bet12) / p.cbet2)
1001 # tan(bet2) = tan(sig2) * cos(alp2)
1002 # tan(omg2) = sin(alp0) * tan(sig2).
1003 somg2 = salp0 * p.sbet2
1004 comg2 = calp2 * p.cbet2
1005 ssig2, csig2 = _norm2(p.sbet2, comg2)
1006 # without normalization we have schi2 = somg2
1007 cchi2 = f1 * p.dn2 * comg2
1009 # omg12 = omg2 - omg1, limit to [0, PI]
1010 somg12, comg12 = _sincos12(somg1, comg1, somg2, comg2, sineg0=True)
1011 # chi12 = chi2 - chi1, limit to [0, PI]
1012 schi12, cchi12 = _sincos12(somg1, cchi1, somg2, cchi2, sineg0=True)
1014 p.setsigs(ssig1, csig1, ssig2, csig2)
1015 # sig12 = sig2 - sig1, limit to [0, PI]
1016 sig12 = _atan12(ssig1, csig1, ssig2, csig2, sineg0=True)
1018 eta12 = self._eF_reset_cHe2_f1(calp0, salp0) * _2__PI # then ...
1019 eta12 *= fsum1f_(eF.deltaH(*p.sncndn2),
1020 -eF.deltaH(*p.sncndn1), sig12)
1021 # eta = chi12 - lam12
1022 lam12 = _atan12(p.slam12, p.clam12, schi12, cchi12) - eta12
1023 # domg12 = chi12 - omg12 - deta12
1024 domg12 = _atan12( somg12, comg12, schi12, cchi12) - eta12
1026 dlam12 = NAN # dv > 0 in ._Newton6
1027 if diffp:
1028 d = calp2 * p.cbet2
1029 if d:
1030 _, dlam12, _, _, _ = self._Length5(sig12, Caps.REDUCEDLENGTH, p)
1031 dlam12 *= f1 / d
1032 elif p.sbet1:
1033 dlam12 = -f1 * p.dn1 * _2_0 / p.sbet1
1035 # p.set_(deta12=-eta12, lam12=lam12)
1036 return lam12, sig12, salp2, calp2, domg12, dlam12
1038 def _Length5(self, sig12, outmask, p):
1039 '''(INTERNAL) Return M{m12b = (reduced length) / self.b} and
1040 calculate M{s12b = distance / self.b} and M{m0}, the
1041 coefficient of secular term in expression for reduced
1042 length and the geodesic scales C{M12} and C{M21}.
1044 @return: 5-Tuple C{(s12b, m12b, m0, M12, M21)}.
1045 '''
1046 s12b = m12b = m0 = M12 = M21 = NAN
1048 Cs = Caps
1049 eF = self._eF
1051 # outmask &= Cs._OUT_MASK
1052 if (outmask & Cs.DISTANCE):
1053 # Missing a factor of self.b
1054 s12b = eF.cE * _2__PI * fsum1f_(eF.deltaE(*p.sncndn2),
1055 -eF.deltaE(*p.sncndn1), sig12)
1057 if (outmask & Cs._REDUCEDLENGTH_GEODESICSCALE):
1058 m0x = -eF.k2 * eF.cD * _2__PI
1059 J12 = -m0x * fsum1f_(eF.deltaD(*p.sncndn2),
1060 -eF.deltaD(*p.sncndn1), sig12)
1061 if (outmask & Cs.REDUCEDLENGTH):
1062 m0 = m0x
1063 # Missing a factor of self.b. Add parens around
1064 # (csig1 * ssig2) and (ssig1 * csig2) to ensure
1065 # accurate cancellation for coincident points.
1066 m12b = fsum1f_(p.dn2 * (p.csig1 * p.ssig2),
1067 -p.dn1 * (p.ssig1 * p.csig2),
1068 J12 * (p.csig1 * p.csig2))
1069 if (outmask & Cs.GEODESICSCALE):
1070 M12 = M21 = p.ssig1 * p.ssig2 + \
1071 p.csig1 * p.csig2
1072 t = (p.cbet1 - p.cbet2) * self.ep2 * \
1073 (p.cbet1 + p.cbet2) / (p.dn1 + p.dn2)
1074 M12 += (p.ssig2 * t + p.csig2 * J12) * p.ssig1 / p.dn1
1075 M21 -= (p.ssig1 * t + p.csig1 * J12) * p.ssig2 / p.dn2
1077 return s12b, m12b, m0, M12, M21
1079 def Line(self, lat1, lon1, azi1, caps=Caps.ALL, name=NN):
1080 '''Set up a L{GeodesicLineExact} to compute several points
1081 on a single geodesic.
1083 @arg lat1: Latitude of the first point (C{degrees}).
1084 @arg lon1: Longitude of the first point (C{degrees}).
1085 @arg azi1: Azimuth at the first point (compass C{degrees}).
1086 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying
1087 the capabilities the L{GeodesicLineExact} instance
1088 should possess, i.e., which quantities can be
1089 returnedby calls to L{GeodesicLineExact.Position}
1090 and L{GeodesicLineExact.ArcPosition}.
1092 @return: A L{GeodesicLineExact} instance.
1094 @note: If the point is at a pole, the azimuth is defined by keeping
1095 B{C{lon1}} fixed, writing C{B{lat1} = ±(90 − ε)}, and taking
1096 the limit C{ε → 0+}.
1098 @see: C++ U{GeodesicExact.Line
1099 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>}
1100 and Python U{Geodesic.Line<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.
1101 '''
1102 return _GeodesicLineExact(self, lat1, lon1, azi1, caps, self._debug, name=name)
1104 @Property_RO
1105 def n(self):
1106 '''Get the ellipsoid's I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}.
1107 '''
1108 return self.ellipsoid.n
1110 @Property_RO
1111 def _n_0_1(self):
1112 '''(INTERNAL) Cached once.
1113 '''
1114 return fabs(self.n) > _0_1
1116 @Property_RO
1117 def _n6PI(self):
1118 '''(INTERNAL) Cached once.
1119 '''
1120 return fabs(self.n) * _6_0 * PI
1122 def _Newton6(self, salp1, calp1, p):
1123 '''(INTERNAL) Split off from C{_GDictInverse} to reduce complexity/length.
1125 @return: 6-Tuple C{(sig12, salp1, calp1, salp2, calp2, domg12)}
1126 and C{p.iter} and C{p.trip} updated.
1127 '''
1128 # This is a straightforward solution of f(alp1) = lambda12(alp1) -
1129 # lam12 = 0 with one wrinkle. f(alp) has exactly one root in the
1130 # interval (0, PI) and its derivative is positive at the root.
1131 # Thus f(alp) is positive for alp > alp1 and negative for alp < alp1.
1132 # During the course of the iteration, a range (alp1a, alp1b) is
1133 # maintained which brackets the root and with each evaluation of
1134 # f(alp) the range is shrunk, if possible. Newton's method is
1135 # restarted whenever the derivative of f is negative (because the
1136 # new value of alp1 is then further from the solution) or if the
1137 # new estimate of alp1 lies outside (0,PI); in this case, the new
1138 # starting guess is taken to be (alp1a + alp1b) / 2.
1139 salp1a = salp1b = _TINY
1140 calp1a, calp1b = _1_0, _N_1_0
1141 MAXIT1, TOL0 = _MAXIT1, _TOL0
1142 HALF, TOLb = _0_5, _TOLb
1143 tripb, TOLv = False, TOL0
1144 for i in range(_MAXIT2):
1145 # 1/4 meridian = 10e6 meter and random input,
1146 # estimated max error in nm (nano meter, by
1147 # checking Inverse problem by Direct).
1148 #
1149 # max iterations
1150 # log2(b/a) error mean sd
1151 # -7 387 5.33 3.68
1152 # -6 345 5.19 3.43
1153 # -5 269 5.00 3.05
1154 # -4 210 4.76 2.44
1155 # -3 115 4.55 1.87
1156 # -2 69 4.35 1.38
1157 # -1 36 4.05 1.03
1158 # 0 15 0.01 0.13
1159 # 1 25 5.10 1.53
1160 # 2 96 5.61 2.09
1161 # 3 318 6.02 2.74
1162 # 4 985 6.24 3.22
1163 # 5 2352 6.32 3.44
1164 # 6 6008 6.30 3.45
1165 # 7 19024 6.19 3.30
1166 v, sig12, salp2, calp2, \
1167 domg12, dv = self._Lambda6(salp1, calp1, i < MAXIT1, p)
1169 # 2 * _TOL0 is approximately 1 ulp [0, PI]
1170 # reversed test to allow escape with NaNs
1171 if tripb or fabs(v) < TOLv:
1172 break
1173 # update bracketing values
1174 if v > 0 and (i > MAXIT1 or (calp1 / salp1) > (calp1b / salp1b)):
1175 salp1b, calp1b = salp1, calp1
1176 elif v < 0 and (i > MAXIT1 or (calp1 / salp1) < (calp1a / salp1a)):
1177 salp1a, calp1a = salp1, calp1
1179 if i < MAXIT1 and dv > 0:
1180 dalp1 = -v / dv
1181 if fabs(dalp1) < PI:
1182 s, c = _sincos2(dalp1)
1183 # nalp1 = alp1 + dalp1
1184 s, c = _sincos12(-s, c, salp1, calp1)
1185 if s > 0:
1186 salp1, calp1 = _norm2(s, c)
1187 # in some regimes we don't get quadratic convergence
1188 # because slope -> 0. So use convergence conditions
1189 # based on epsilon instead of sqrt(epsilon)
1190 TOLv = TOL0 if fabs(v) > _TOL016 else _TOL08
1191 continue
1193 # Either dv was not positive or updated value was outside
1194 # legal range. Use the midpoint of the bracket as the next
1195 # estimate. This mechanism is not needed for the WGS84
1196 # ellipsoid, but it does catch problems with more eccentric
1197 # ellipsoids. Its efficacy is such for the WGS84 test set
1198 # with the starting guess set to alp1 = 90 deg: the WGS84
1199 # test set: mean = 5.21, stdev = 3.93, max = 24 and WGS84
1200 # with random input: mean = 4.74, stdev = 0.99
1201 salp1, calp1 = _norm2((salp1a + salp1b) * HALF,
1202 (calp1a + calp1b) * HALF)
1203 tripb = fsum1f_(calp1a, -calp1, fabs(salp1a - salp1)) < TOLb or \
1204 fsum1f_(calp1b, -calp1, fabs(salp1b - salp1)) < TOLb
1205 TOLv = TOL0
1207 else:
1208 raise GeodesicError(Fmt.no_convergence(v, TOLv), txt=repr(self)) # self.toRepr()
1210 p.set_(iter=i, trip=tripb) # like .geodsolve._GDictInvoke: iter NOT iteration!
1211 return sig12, salp1, calp1, salp2, calp2, domg12
1213 Polygon = Area # for C{geographiclib} compatibility
1215 def _sinf1cos2d(self, lat):
1216 '''(INTERNAL) Helper, also for C{_G_GeodesicLineExact}.
1217 '''
1218 sbet, cbet = _sincos2d(lat)
1219 # ensure cbet1 = +epsilon at poles; doing the fix on beta means
1220 # that sig12 will be <= 2*tiny for two points at the same pole
1221 sbet, cbet = _norm2(sbet * self.f1, cbet)
1222 return sbet, max(_TINY, cbet)
1224 def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature
1225 '''Return this C{GeodesicExact} as string.
1227 @kwarg prec: The C{float} precision, number of decimal digits (0..9).
1228 Trailing zero decimals are stripped for B{C{prec}} values
1229 of 1 and above, but kept for negative B{C{prec}} values.
1230 @kwarg sep: Separator to join (C{str}).
1232 @return: Tuple items (C{str}).
1233 '''
1234 d = dict(ellipsoid=self.ellipsoid, C4order=self.C4order)
1235 return sep.join(pairs(d, prec=prec))
1238class GeodesicLineExact(_GeodesicLineExact):
1239 '''A pure Python version of I{Karney}'s C++ class U{GeodesicLineExact
1240 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicLineExact.html>},
1241 modeled after I{Karney}'s Python class U{geodesicline.GeodesicLine<https://GitHub.com/
1242 geographiclib/geographiclib-python>}.
1243 '''
1245 def __init__(self, geodesic, lat1, lon1, azi1, caps=Caps.STANDARD, name=NN):
1246 '''New L{GeodesicLineExact} instance, allowing points to be found along
1247 a geodesic starting at C{(B{lat1}, B{lon1})} with azimuth B{C{azi1}}.
1249 @arg geodesic: The geodesic to use (L{GeodesicExact}).
1250 @arg lat1: Latitude of the first point (C{degrees}).
1251 @arg lon1: Longitude of the first point (C{degrees}).
1252 @arg azi1: Azimuth at the first points (compass C{degrees}).
1253 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying
1254 the capabilities the L{GeodesicLineExact} instance
1255 should possess, i.e., which quantities can be
1256 returned by calls to L{GeodesicLineExact.Position}
1257 and L{GeodesicLineExact.ArcPosition}.
1258 @kwarg name: Optional name (C{str}).
1260 @raise TypeError: Invalid B{C{geodesic}}.
1261 '''
1262 _xinstanceof(GeodesicExact, geodesic=geodesic)
1263 if (caps & Caps.LINE_OFF): # copy to avoid updates
1264 geodesic = geodesic.copy(deep=False, name=NN(_UNDER_, geodesic.name))
1265# _update_all(geodesic)
1266 _GeodesicLineExact.__init__(self, geodesic, lat1, lon1, azi1, caps, 0, name=name)
1269def _Astroid(x, y):
1270 '''(INTERNAL) Solve M{k^4 + 2 * k^3 - (x^2 + y^2 - 1)
1271 * k^2 - (2 * k + 1) * y^2 = 0} for positive root k.
1272 '''
1273 p = x**2
1274 q = y**2
1275 r = fsumf_(_1_0, q, p, _N_2_0)
1276 if r > 0 or q:
1277 # avoid possible division by zero when r = 0
1278 # by multiplying s and t by r^3 and r, resp.
1279 S = p * q / _4_0 # S = r^3 * s
1280 if r:
1281 r = r / _6_0 # /= chokes PyChecker
1282 r3 = r**3
1283 T3 = r3 + S
1284 # discriminant of the quadratic equation for T3 is
1285 # zero on the evolute curve p^(1/3) + q^(1/3) = 1
1286 d = (r3 + T3) * S
1287 if d < 0:
1288 # T is complex, but u is defined for a real result
1289 a = atan2(sqrt(-d), -T3) / _3_0
1290 # There are 3 possible cube roots, choose the one which
1291 # avoids cancellation. Note d < 0 implies that r < 0.
1292 u = (cos(a) * _2_0 + _1_0) * r
1293 else:
1294 # pick the sign on the sqrt to maximize abs(T3) to
1295 # minimize loss of precision due to cancellation.
1296 if d:
1297 T3 += _copysign(sqrt(d), T3) # T3 = (r * t)^3
1298 # _cbrt always returns the real root, _cbrt(-8) = -2
1299 u = _cbrt(T3) # T = r * t
1300 if u: # T can be zero; but then r2 / T -> 0
1301 u += r**2 / u
1302 u += r
1303 elif S: # d == T3**2 == S**2: sqrt(d) == abs(S) == abs(T3)
1304 u = _cbrt(S * _2_0) # == T3 + _copysign(abs(S), T3)
1305 else:
1306 u = _0_0
1307 v = _hypot(u, y) # sqrt(u**2 + q)
1308 # avoid loss of accuracy when u < 0
1309 u = (q / (v - u)) if u < 0 else (v + u)
1310 w = (u - q) / (v + v) # positive?
1311 # rearrange expression for k to avoid loss of accuracy due to
1312 # subtraction, division by 0 impossible because u > 0, w >= 0
1313 k = u / (sqrt(w**2 + u) + w) # guaranteed positive
1315 else: # q == 0 && r <= 0
1316 # y = 0 with |x| <= 1. Handle this case directly, for
1317 # y small, positive root is k = abs(y) / sqrt(1 - x^2)
1318 k = _0_0
1320 return k
1323def _C4coeffs(nC4): # in .geodesicx.__main__
1324 '''(INTERNAL) Get the C{C4_24}, C{_27} or C{_30} series coefficients.
1325 '''
1326 try: # from pygeodesy.geodesicx._C4_xx import _coeffs_xx as _coeffs
1327 _C4_xx = _DOT_(_MODS.geodesicx.__name__, _UNDER_('_C4', nC4))
1328 _coeffs = _MODS.getattr(_C4_xx, _UNDER_('_coeffs', nC4))
1329 except (AttributeError, ImportError, TypeError) as x:
1330 raise GeodesicError(nC4=nC4, cause=x)
1331 n = _xnC4(nC4=nC4)
1332 if len(_coeffs) != n: # double check
1333 raise GeodesicError(_coeffs=len(_coeffs), _xnC4=n, nC4=nC4)
1334 return _coeffs
1337__all__ += _ALL_DOCS(GeodesicExact, GeodesicLineExact)
1339# **) MIT License
1340#
1341# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.
1342#
1343# Permission is hereby granted, free of charge, to any person obtaining a
1344# copy of this software and associated documentation files (the "Software"),
1345# to deal in the Software without restriction, including without limitation
1346# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1347# and/or sell copies of the Software, and to permit persons to whom the
1348# Software is furnished to do so, subject to the following conditions:
1349#
1350# The above copyright notice and this permission notice shall be included
1351# in all copies or substantial portions of the Software.
1352#
1353# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1354# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1355# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1356# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1357# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1358# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1359# OTHER DEALINGS IN THE SOFTWARE.