Coverage for pygeodesy/geodesicx/gxline.py: 97%
230 statements
« prev ^ index » next coverage.py v7.2.2, created at 2023-04-05 15:46 -0400
« prev ^ index » next coverage.py v7.2.2, created at 2023-04-05 15:46 -0400
2# -*- coding: utf-8 -*-
4u'''A pure Python version of I{Karney}'s C++ class U{GeodesicLineExact
5<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicLineExact.html>}.
7Class L{GeodesicLineExact} follows the naming, methods and return
8values from class C{GeodesicLine} from I{Karney}'s Python U{geographiclib
9<https://GeographicLib.SourceForge.io/1.52/python/index.html>}.
11Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2008-2022)
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{GeodesicLineExact.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
39# from pygeodesy.basics import _xinstanceof # from .karney
40from pygeodesy.constants import NAN, _EPSmin, _0_0, _1_0, _180_0, _2__PI
41from pygeodesy.fsums import _COMMASPACE_, fsum_, fsum1_
42from pygeodesy.geodesicx.gxbases import _cosSeries, _GeodesicBase, \
43 _sincos12, _sin1cos2
44# from pygeodesy.interns import _COMMASPACE_ # from .fsums
45from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS
46from pygeodesy.karney import _around, _atan2d, Caps, _copysign, GDict, \
47 _fix90, _K_2_0, _norm2, _norm180, \
48 _sincos2, _sincos2d, _xinstanceof
49from pygeodesy.props import Property_RO, _update_all
50# from pygeodesy.streprs import pairs # from _MODS
51from pygeodesy.utily import atan2d as _atan2d_reverse, sincos2
53from math import atan2, cos, degrees, fabs, floor, radians, sin
55__all__ = ()
56__version__ = '23.04.04'
58_glXs = [] # instances of C{[_]GeodesicLineExact} to be updated
59# underflow guard, we require _TINY * EPS > 0, _TINY + EPS == EPS
60_TINY = _EPSmin
61# assert (_TINY * EPS) > 0 and (_TINY + EPS) == EPS
64def _update_glXs(gX): # see GeodesicExact.C4order and -._ef_reset_k2
65 '''(INTERNAL) Zap cached/memoized C{Property[_RO]}s of
66 any L{GeodesicLineExact} instances tied to the given
67 L{GeodesicExact} instance B{C{gX}}.
68 '''
69 _xinstanceof(gX, _MODS.geodesicx.GeodesicExact)
70 for glX in _glXs: # PYCHOK use weakref?
71 if glX._gX is gX:
72 _update_all(glX)
75class _GeodesicLineExact(_GeodesicBase):
76 '''(INTERNAL) Base class for L{GeodesicLineExact}.
77 '''
78 _a13 = _s13 = NAN
79 _azi1 = _0_0
80 _cchi1 = NAN
81 _dn1 = NAN
82 _gX = None # Exact only
83 _k2 = NAN
84 _lat1 = _lon1 = _0_0
85 _salp0 = _calp0 = NAN
86 _salp1 = _calp1 = NAN
87 _somg1 = _comg1 = NAN
88 _ssig1 = _csig1 = NAN
90 def __init__(self, gX, lat1, lon1, azi1, caps, _debug, *salp1_calp1, **name):
91 '''(INTERNAL) New C{[_]GeodesicLineExact} instance.
92 '''
93 _xinstanceof(gX, _MODS.geodesicx.GeodesicExact)
94 Cs = Caps
95 if _debug: # PYCHOK no cover
96 self._debug |= _debug & Cs._DEBUG_ALL
97 # _CapsBase.debug._update(self)
98 if salp1_calp1:
99 salp1, calp1 = salp1_calp1
100 else:
101 azi1 = _norm180(azi1)
102 # guard against salp0 underflow,
103 # also -0 is converted to +0
104 salp1, calp1 = _sincos2d(_around(azi1))
105 if name:
106 self.name = name
108 self._gX = gX # GeodesicExact only
109 self._lat1 = lat1 = _fix90(lat1)
110 self._lon1 = lon1
111 self._azi1 = azi1
112 self._salp1 = salp1
113 self._calp1 = calp1
114 # allow lat, azimuth and unrolling of lon
115 self._caps = caps | Cs._LINE
117 sbet1, cbet1 = gX._sinf1cos2d(_around(lat1))
118 self._dn1 = gX._dn(sbet1, cbet1)
119 # Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0), with alp0
120 # in [0, pi/2 - |bet1|]. Alt: calp0 = hypot(sbet1, calp1 * cbet1),
121 # but the following is slightly better, consider the case salp1 = 0.
122 self._salp0, self._calp0 = _sin1cos2(salp1, calp1, sbet1, cbet1)
123 self._k2 = self._calp0**2 * gX.ep2
124 # Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
125 # sig = 0 is nearest northward crossing of equator.
126 # With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
127 # With bet1 = pi/2, alp1 = -pi, sig1 = pi/2
128 # With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2
129 # Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
130 # With alp0 in (0, pi/2], quadrants for sig and omg coincide.
131 # No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
132 # With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi.
133 self._somg1 = sbet1 * self._salp0
134 self._comg1 = c = (cbet1 * calp1) if (sbet1 or calp1) else _1_0
135 # Without normalization we have schi1 = somg1.
136 self._cchi1 = gX.f1 * self._dn1 * c
137 self._ssig1, self._csig1 = _norm2(sbet1, c) # sig1 in (-pi, pi]
138 # _norm2(somg1, comg1) # no need to normalize!
139 # _norm2(schi1?, cchi1) # no need to normalize!
140 if not (caps & Cs.LINE_OFF):
141 _glXs.append(self)
142 # no need to pre-compute other attrs based on _Caps.X. All are
143 # Property_RO's, computed once and cached/memoized until reset
144 # when C4order is changed or Elliptic function reset is invoked.
146 def __del__(self): # XXX use weakref?
147 if _glXs: # may be empty or None
148 try: # PYCHOK no cover
149 _glXs.remove(self)
150 except (TypeError, ValueError):
151 pass
152 self._gX = None
153 # _update_all(self) # throws TypeError during Python 2 cleanup
155 def _update(self, updated, *attrs, **unused):
156 if updated:
157 _update_all(self, *attrs)
159 @Property_RO
160 def a1(self):
161 '''Get the I{equatorial arc} (C{degrees}), the arc length between
162 the northward equatorial crossing and the first point.
163 '''
164 return _atan2d(self._ssig1, self._csig1) # or NAN
166 equatorarc = a1
168 @Property_RO
169 def a13(self):
170 '''Get (spherical) arc length from the first to the reference point (C{degrees}).
172 @see: Method L{SetArc}.
173 '''
174 return self._a13
176 def ArcPosition(self, a12, outmask=Caps.STANDARD):
177 '''Find the position on the line given B{C{a12}}.
179 @arg a12: Spherical arc length from the first point to the
180 second point (C{degrees}).
181 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying
182 the quantities to be returned.
184 @return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2,
185 lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1},
186 C{lon1}, C{azi1} and arc length C{a12} always included,
187 except when C{a12=NAN}.
189 @note: By default, C{B{outmask}=STANDARD}, meaning thc C{lat1},
190 C{lon1}, C{azi1}, C{lat2}, C{lon2}, C{azi2}, C{s12} and
191 C{a12} entries are returned, except when C{a12=NAN}.
192 '''
193 return self._GDictPosition(True, a12, outmask)
195 @Property_RO
196 def azi0(self):
197 '''Get the I{equatorial azimuth}, the azimuth of this geodesic line
198 as it crosses the equator in a northward direction (C{degrees90}).
199 '''
200 return _atan2d(*self.azi0_sincos2) # or NAN
202 equatorazimuth = azi0
204 @Property_RO
205 def azi0_sincos2(self):
206 '''Get the sine and cosine of the I{equatorial azimuth} (2-tuple C{(sin, cos)}).
207 '''
208 return self._salp0, self._calp0
210 @Property_RO
211 def azi1(self):
212 '''Get the azimuth at the first point (compass C{degrees}).
213 '''
214 return self._azi1
216 @Property_RO
217 def azi1_sincos2(self):
218 '''Get the sine and cosine of the first point's azimuth (2-tuple C{(sin, cos)}).
219 '''
220 return self._salp1, self._calp1
222 @Property_RO
223 def _B41(self):
224 '''(INTERNAL) Cached/memoized.
225 '''
226 return _cosSeries(self._C4a, self._ssig1, self._csig1)
228 @Property_RO
229 def _C4a(self):
230 '''(INTERNAL) Cached/memoized.
231 '''
232 return self.geodesic._C4f_k2(self._k2)
234 @Property_RO
235 def _caps_DISTANCE_IN(self):
236 '''(INTERNAL) Get C{Caps.DISTANCE_IN} and C{_OUT}.
237 '''
238 return self.caps & Caps._DISTANCE_IN_OUT
240 @Property_RO
241 def _D0k2(self):
242 '''(INTERNAL) Cached/memoized.
243 '''
244 return self._eF.cD * _2__PI * self._k2
246 @Property_RO
247 def _D1(self):
248 '''(INTERNAL) Cached/memoized.
249 '''
250 return self._eF.deltaD(self._ssig1, self._csig1, self._dn1)
252 @Property_RO
253 def _E0b(self):
254 '''(INTERNAL) Cached/memoized.
255 '''
256 return self._eF.cE * _2__PI * self.geodesic.b
258 @Property_RO
259 def _E1(self):
260 '''(INTERNAL) Cached/memoized.
261 '''
262 return self._eF.deltaE(self._ssig1, self._csig1, self._dn1)
264 @Property_RO
265 def _eF(self):
266 '''(INTERNAL) Cached/memoized C{Elliptic} function.
267 '''
268 # see .gx.GeodesicExact._ef_reset_k2
269 return _MODS.elliptic.Elliptic(k2=-self._k2, alpha2=-self.geodesic.ep2)
271 def _GDictPosition(self, arcmode, s12_a12, outmask): # MCCABE 17
272 '''(INTERNAL) Generate a new position along the geodesic.
274 @return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2,
275 lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1},
276 C{lon1}, C{azi1} and arc length C{a12} always included,
277 except when C{a12=NAN}.
278 '''
280 r = GDict(a12=NAN, s12=NAN) # note both a12 and s12, always
281 if not (arcmode or self._caps_DISTANCE_IN): # PYCHOK no cover
282 return r # Uninitialized or impossible distance requested
284 Cs = Caps
285 if self._debug: # PYCHOK no cover
286 outmask |= self._debug & Cs._DEBUG_DIRECT_LINE
287 outmask &= self._caps & Cs._OUT_MASK
289 eF = self._eF
290 gX = self.geodesic # ._gX
292 if arcmode:
293 # s12_a12 is spherical arc length
294 E2 = _0_0
295 sig12 = radians(s12_a12)
296 if _K_2_0:
297 ssig12, csig12 = sincos2(sig12) # utily, no NEG0
298 else: # PYCHOK no cover
299 a = fabs(s12_a12) # 0 <= fabs(_remainder(s12_a12, _180_0)) <= 90
300 a -= floor(a / _180_0) * _180_0 # 0 <= 0 < 180
301 ssig12 = _0_0 if a == 0 else sin(sig12)
302 csig12 = _0_0 if a == 90 else cos(sig12)
303 else: # s12_a12 is distance
304 t = s12_a12 / self._E0b
305 s, c = _sincos2(t) # tau12
306 # tau2 = tau1 + tau12
307 E2 = -eF.deltaEinv(*_sincos12(-s, c, *self._stau1_ctau1))
308 sig12 = fsum1_(self._E1, -E2, t) # == t - (E2 - E1)
309 ssig12, csig12 = _sincos2(sig12)
311 salp0, calp0 = self._salp0, self._calp0
312 ssig1, csig1 = self._ssig1, self._csig1
314 # sig2 = sig1 + sig12
315 ssig2, csig2 = _sincos12(-ssig12, csig12, ssig1, csig1)
316 dn2 = eF.fDelta(ssig2, csig2)
317 # sin(bet2) = cos(alp0) * sin(sig2) and
318 # cbet2 = hypot(salp0, calp0 * csig2). Alt:
319 # cbet2 = hypot(csig2, salp0 * ssig2)
320 sbet2, cbet2 = _sin1cos2(calp0, salp0, csig2, ssig2)
321 if cbet2 == 0: # salp0 = 0, csig2 = 0, break degeneracy
322 cbet2 = csig2 = _TINY
323 # tan(alp0) = cos(sig2) * tan(alp2)
324 salp2 = salp0
325 calp2 = calp0 * csig2 # no need to normalize
327 if (outmask & Cs.DISTANCE):
328 if arcmode: # or f_0_01
329 E2 = eF.deltaE(ssig2, csig2, dn2)
330 # AB1 = _E0 * (E2 - _E1)
331 # s12 = _b * (_E0 * sig12 + AB1)
332 # = _b * _E0 * (sig12 + (E2 - _E1))
333 # = _b * _E0 * (E2 - _E1 + sig12)
334 s12 = self._E0b * fsum1_(E2, -self._E1, sig12)
335 else:
336 s12 = s12_a12
337 r.set_(s12=s12)
339 if (outmask & Cs._DEBUG_DIRECT_LINE): # PYCHOK no cover
340 r.set_(sig12=sig12, dn2=dn2, b=gX.b, e2=gX.e2, f1=gX.f1,
341 E0b=self._E0b, E1=self._E1, E2=E2, eFk2=eF.k2, eFa2=eF.alpha2)
343 if (outmask & Cs.LONGITUDE):
344 schi1 = self._somg1
345 cchi1 = self._cchi1
346 schi2 = ssig2 * salp0
347 cchi2 = gX.f1 * dn2 * csig2 # schi2 = somg2 without normalization
348 lam12 = salp0 * self._H0e2_f1 * fsum1_(eF.deltaH(ssig2, csig2, dn2),
349 -self._H1, sig12)
350 if (outmask & Cs.LONG_UNROLL):
351 t = _copysign(_1_0, salp0) # east-going?
352 tchi1 = t * schi1
353 tchi2 = t * schi2
354 chi12 = t * fsum1_(atan2(ssig1, csig1), -atan2(ssig2, csig2),
355 atan2(tchi2, cchi2), -atan2(tchi1, cchi1), sig12)
356 lon2 = self.lon1 + degrees(chi12 - lam12)
357 else:
358 chi12 = atan2(*_sincos12(schi1, cchi1, schi2, cchi2))
359 lon2 = _norm180(self._lon1_norm180 + _norm180(degrees(chi12 - lam12)))
360 r.set_(lon2=lon2)
361 if (outmask & Cs._DEBUG_DIRECT_LINE): # PYCHOK no cover
362 r.set_(ssig2=ssig2, chi12=chi12, H0e2_f1=self._H0e2_f1,
363 csig2=csig2, lam12=lam12, H1=self._H1)
365 if (outmask & Cs.LATITUDE):
366 r.set_(lat2=_atan2d(sbet2, gX.f1 * cbet2))
368 if (outmask & Cs.AZIMUTH):
369 r.set_(azi2=_atan2d_reverse(salp2, calp2, reverse=outmask & Cs.REVERSE2))
371 if (outmask & Cs._REDUCEDLENGTH_GEODESICSCALE):
372 dn1 = self._dn1
373 J12 = self._D0k2 * fsum_(eF.deltaD(ssig2, csig2, dn2), -self._D1, sig12)
374 if (outmask & Cs._DEBUG_DIRECT_LINE): # PYCHOK no cover
375 r.set_(ssig1=ssig1, dn1=dn1, D0k2=self._D0k2,
376 csig1=csig1, J12=J12, D1=self._D1)
377 if (outmask & Cs.REDUCEDLENGTH):
378 # Add parens around (csig1 * ssig2) and (ssig1 * csig2) to
379 # ensure accurate cancellation in the case of coincident points.
380 r.set_(m12=gX.b * fsum1_(dn2 * (csig1 * ssig2),
381 -dn1 * (ssig1 * csig2),
382 -J12 * (csig1 * csig2)))
383 if (outmask & Cs.GEODESICSCALE):
384 t = self._k2 * (ssig2 - ssig1) * (ssig2 + ssig1) / (dn2 + dn1)
385 r.set_(M12=csig12 + ssig1 * (t * ssig2 - csig2 * J12) / dn1,
386 M21=csig12 - ssig2 * (t * ssig1 - csig1 * J12) / dn2)
388 if (outmask & Cs.AREA):
389 A4 = salp0 * calp0
390 if A4:
391 # tan(alp) = tan(alp0) * sec(sig)
392 # tan(alp2-alp1) = (tan(alp2) - tan(alp1)) / (tan(alp2) * tan(alp1) + 1)
393 # = calp0 * salp0 * (csig1 - csig2) / (salp0^2 + calp0^2 * csig1 * csig2)
394 # If csig12 > 0, write
395 # csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
396 # else
397 # csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
398 # No need to normalize
399 salp12 = (((ssig12 * csig1 / (_1_0 + csig12) + ssig1) * ssig12) if csig12 > 0 else
400 (csig1 * (_1_0 - csig12) + ssig1 * ssig12)) * A4
401 calp12 = salp0**2 + calp0**2 * csig1 * csig2
402 A4 *= gX._e2a2
403 B41 = self._B41
404 B42 = _cosSeries(self._C4a, ssig2, csig2)
405 S12 = (B42 - B41) * A4
406 else:
407 S12 = A4 = B41 = B42 = _0_0
408 # alp12 = alp2 - alp1, used in atan2 so no need to normalize
409 salp12, calp12 = _sincos12(self._salp1, self._calp1, salp2, calp2)
410 # We used to include some patch up code that purported to deal
411 # with nearly meridional geodesics properly. However, this turned
412 # out to be wrong once salp1 = -0 was allowed (via InverseLine).
413 # In fact, the calculation of {s,c}alp12 was already correct
414 # (following the IEEE rules for handling signed zeros). So,
415 # the patch up code was unnecessary (as well as dangerous).
416 if (outmask & Cs._DEBUG_DIRECT_LINE): # PYCHOK no cover
417 r.set_(salp12=salp12, salp0=salp0, B41=B41, A4=A4,
418 calp12=calp12, calp0=calp0, B42=B42, c2=gX.c2)
419 S12 += gX.c2 * atan2(salp12, calp12)
420 r.set_(S12=S12)
422 r.set_(a12=s12_a12 if arcmode else degrees(sig12),
423 lat1=self.lat1, # == _fix90(lat1)
424 lon1=self.lon1 if (outmask & Cs.LONG_UNROLL) else self._lon1_norm180,
425 azi1=_norm180(self.azi1))
426 return r
428 def _GenPosition(self, arcmode, s12_a12, outmask):
429 '''(INTERNAL) Generate a new position along the geodesic.
431 @return: L{Direct9Tuple}C{(a12, lat2, lon2, azi2,
432 s12, m12, M12, M21, S12)}.
433 '''
434 r = self._GDictPosition(arcmode, s12_a12, outmask)
435 return r.toDirect9Tuple()
437 def _GenSet(self, arcmode, s13_a13):
438 '''(INTERNAL) Aka C++ C{GenSetDistance}.
439 '''
440 if arcmode:
441 self.SetArc(s13_a13)
442 else:
443 self.SetDistance(s13_a13)
444 return self # for gx.GeodesicExact.InverseLine and -._GenDirectLine
446 @Property_RO
447 def geodesic(self):
448 '''Get the I{exact} geodesic (L{GeodesicExact}).
449 '''
450 assert isinstance(self._gX, _MODS.geodesicx.GeodesicExact)
451 return self._gX
453 @Property_RO
454 def _H0e2_f1(self):
455 '''(INTERNAL) Cached/memoized.
456 '''
457 return self._eF.cH * _2__PI * self.geodesic._e2_f1
459 @Property_RO
460 def _H1(self):
461 '''(INTERNAL) Cached/memoized.
462 '''
463 return self._eF.deltaH(self._ssig1, self._csig1, self._dn1)
465 @Property_RO
466 def lat1(self):
467 '''Get the latitude of the first point (C{degrees}).
468 '''
469 return self._lat1
471 @Property_RO
472 def lon1(self):
473 '''Get the longitude of the first point (C{degrees}).
474 '''
475 return self._lon1
477 @Property_RO
478 def _lon1_norm180(self):
479 '''(INTERNAL) Cached/memoized.
480 '''
481 return _norm180(self._lon1)
483 def Position(self, s12, outmask=Caps.STANDARD):
484 '''Find the position on the line given B{C{s12}}.
486 @arg s12: Distance from the first point to the second (C{meter}).
487 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying
488 the quantities to be returned.
490 @return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2,
491 lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1},
492 C{lon1}, C{azi1} and arc length C{a12} always included,
493 except when C{a12=NAN}.
495 @note: By default, C{B{outmask}=STANDARD}, meaning thc C{lat1},
496 C{lon1}, C{azi1}, C{lat2}, C{lon2}, C{azi2}, C{s12} and
497 C{a12} entries are returned, except when C{a12=NAN}.
499 @note: This L{GeodesicLineExact} instance must have been
500 constructed with capability C{Caps.DISTANCE_IN} set.
501 '''
502 return self._GDictPosition(False, s12, outmask)
504 @Property_RO
505 def s13(self):
506 '''Get the distance from the first to the reference point (C{meter}).
508 @see: Method L{SetDistance}.
509 '''
510 return self._s13
512 def SetArc(self, a13):
513 '''Set reference point 3 in terms of distance to the first point.
515 @arg a13: Spherical arc length from the first to the reference
516 point (C{degrees}).
518 @return: The distance C{s13} (C{meter}) between the first and
519 the reference point or C{NAN}.
520 '''
521 self._a13 = a13
522 self._s13 = s13 = self._GDictPosition(True, a13, Caps.DISTANCE).s12
523 return s13
525 def SetDistance(self, s13):
526 '''Set reference point 3 in terms of distance to the first point.
528 @arg s13: Distance from the first to the reference point (C{meter}).
530 @return: The arc length C{a13} (C{degrees}) between the first
531 and the reference point or C{NAN}.
532 '''
533 self._s13 = s13
534 self._a13 = a13 = self._GDictPosition(False, s13, 0).a12
535 return a13 # NAN for GeodesicLineExact without Cap.DISTANCE_IN
537 @Property_RO
538 def _stau1_ctau1(self):
539 '''(INTERNAL) Cached/memoized.
540 '''
541 s, c = _sincos2(self._E1)
542 # tau1 = sig1 + B11
543 return _sincos12(-s, c, self._ssig1, self._csig1)
544 # unnecessary because Einv inverts E
545 # return -self._eF.deltaEinv(stau1, ctau1)
547 def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature
548 '''Return this C{GeodesicLineExact} as string.
550 @kwarg prec: The C{float} precision, number of decimal digits (0..9).
551 Trailing zero decimals are stripped for B{C{prec}} values
552 of 1 and above, but kept for negative B{C{prec}} values.
553 @kwarg sep: Separator to join (C{str}).
555 @return: C{GeodesicLineExact} (C{str}).
556 '''
557 d = dict(geodesic=self.geodesic,
558 lat1=self.lat1, lon1=self.lon1, azi1=self.azi1,
559 a13=self.a13, s13=self.s13)
560 return sep.join(_MODS.streprs.pairs(d, prec=prec))
563__all__ += _ALL_DOCS(_GeodesicLineExact)
565# **) MIT License
566#
567# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.
568#
569# Permission is hereby granted, free of charge, to any person obtaining a
570# copy of this software and associated documentation files (the "Software"),
571# to deal in the Software without restriction, including without limitation
572# the rights to use, copy, modify, merge, publish, distribute, sublicense,
573# and/or sell copies of the Software, and to permit persons to whom the
574# Software is furnished to do so, subject to the following conditions:
575#
576# The above copyright notice and this permission notice shall be included
577# in all copies or substantial portions of the Software.
578#
579# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
580# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
581# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
582# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
583# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
584# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
585# OTHER DEALINGS IN THE SOFTWARE.