1

2# -*- coding: utf-8 -*-

3

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

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

6

7 Class L{GeodesicLineExact} follows the naming, methods and return

8 values from class C{GeodesicLine} from I{Karney}'s Python U{geographiclib

9<https://GeographicLib.SourceForge.io/1.52/python/index.html>}.

10

11 Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2008-2023)

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

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

14'''

15# make sure int/int division yields float quotient

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

17

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

38

39# from pygeodesy.basics import _xinstanceof # from .karney

40 from pygeodesy.constants import NAN, _EPSmin, _0_0, _1_0, _180_0, _2__PI

41 from pygeodesy.fsums import _COMMASPACE_, fsumf_, fsum1f_

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

43 _sincos12, _sin1cos2

44# from pygeodesy.interns import _COMMASPACE_ # from .fsums

45 from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS

46 from pygeodesy.karney import _around, _atan2d, Caps, _copysign, GDict, \

47 _fix90, _K_2_0, _norm2, _norm180, \

48 _sincos2, _sincos2d, _xinstanceof

49 from pygeodesy.props import Property_RO, _update_all

50# from pygeodesy.streprs import pairs # _MODS

51 from pygeodesy.utily import atan2d as _atan2d_reverse, sincos2

52

53 from math import atan2, cos, degrees, fabs, floor, radians, sin

54

55__all__ = ()

56__version__ = '23.06.23'

57

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

62

63

64 def _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(_MODS.geodesicx.GeodesicExact, gX=gX)

70 for glX in _glXs: # PYCHOK use weakref?

71 if glX._gX is gX:

72 _update_all(glX)

73

74

75 class _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

89

90 def __init__(self, gX, lat1, lon1, azi1, caps, _debug, *salp1_calp1, **name):

91 '''(INTERNAL) New C{[_]GeodesicLineExact} instance.

92 '''

93 _xinstanceof(_MODS.geodesicx.GeodesicExact, gX=gX)

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

107

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

116

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.

145

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

154

155 def _update(self, updated, *attrs, **unused):

156 if updated:

157 _update_all(self, *attrs)

158

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

165

166 equatorarc = a1

167

168 @Property_RO

169 def a13(self):

170 '''Get the arc length to reference point 3 (C{degrees}).

171

172 @see: Methods L{Arc} and L{SetArc}.

173 '''

174 return self._a13

175

176 def Arc(self):

177 '''Return the arc length to reference point 3 (C{degrees} or C{NAN}).

178

179 @see: Method L{SetArc} and property L{a13}.

180 '''

181 return self.a13

182

183 def ArcPosition(self, a12, outmask=Caps.STANDARD):

184 '''Find the position on the line given B{C{a12}}.

185

186 @arg a12: Spherical arc length from the first point to the

187 second point (C{degrees}).

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

189 the quantities to be returned.

190

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

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

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

194 except when C{a12=NAN}.

195

196 @note: By default, C{B{outmask}=STANDARD}, meaning thc C{lat1},

197 C{lon1}, C{azi1}, C{lat2}, C{lon2}, C{azi2}, C{s12} and

198 C{a12} entries are returned, except when C{a12=NAN}.

199 '''

200 return self._GDictPosition(True, a12, outmask)

201

202 @Property_RO

203 def azi0(self):

204 '''Get the I{equatorial azimuth}, the azimuth of this geodesic line

205 as it crosses the equator in a northward direction (C{degrees90}).

206 '''

207 return _atan2d(*self.azi0_sincos2) # or NAN

208

209 equatorazimuth = azi0

210

211 @Property_RO

212 def azi0_sincos2(self):

213 '''Get the sine and cosine of the I{equatorial azimuth} (2-tuple C{(sin, cos)}).

214 '''

215 return self._salp0, self._calp0

216

217 @Property_RO

218 def azi1(self):

219 '''Get the azimuth at the first point (compass C{degrees}).

220 '''

221 return self._azi1

222

223 @Property_RO

224 def azi1_sincos2(self):

225 '''Get the sine and cosine of the first point's azimuth (2-tuple C{(sin, cos)}).

226 '''

227 return self._salp1, self._calp1

228

229 @Property_RO

230 def _B41(self):

231 '''(INTERNAL) Cached/memoized.

232 '''

233 return _cosSeries(self._C4a, self._ssig1, self._csig1)

234

235 @Property_RO

236 def _C4a(self):

237 '''(INTERNAL) Cached/memoized.

238 '''

239 return self.geodesic._C4f_k2(self._k2)

240

241 @Property_RO

242 def _caps_DISTANCE_IN(self):

243 '''(INTERNAL) Get C{Caps.DISTANCE_IN} and C{_OUT}.

244 '''

245 return self.caps & Caps._DISTANCE_IN_OUT

246

247 @Property_RO

248 def _D0k2(self):

249 '''(INTERNAL) Cached/memoized.

250 '''

251 return self._eF.cD * _2__PI * self._k2

252

253 @Property_RO

254 def _D1(self):

255 '''(INTERNAL) Cached/memoized.

256 '''

257 return self._eF.deltaD(self._ssig1, self._csig1, self._dn1)

258

259 def Distance(self):

260 '''Return the distance to reference point 3 (C{meter} or C{NAN}).

261

262 @see: Method L{SetDistance} and property L{s13}.

263 '''

264 return self.s13

265

266 @Property_RO

267 def _E0b(self):

268 '''(INTERNAL) Cached/memoized.

269 '''

270 return self._eF.cE * _2__PI * self.geodesic.b

271

272 @Property_RO

273 def _E1(self):

274 '''(INTERNAL) Cached/memoized.

275 '''

276 return self._eF.deltaE(self._ssig1, self._csig1, self._dn1)

277

278 @Property_RO

279 def _eF(self):

280 '''(INTERNAL) Cached/memoized C{Elliptic} function.

281 '''

282 # see .gx.GeodesicExact._ef_reset_k2

283 return _MODS.elliptic.Elliptic(k2=-self._k2, alpha2=-self.geodesic.ep2)

284

285 def _GDictPosition(self, arcmode, s12_a12, outmask=Caps.STANDARD): # MCCABE 17

286 '''(INTERNAL) Generate a new position along the geodesic.

287

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

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

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

291 except when C{a12=NAN}.

292 '''

293

294 r = GDict(a12=NAN, s12=NAN) # note both a12 and s12, always

295 if not (arcmode or self._caps_DISTANCE_IN): # PYCHOK no cover

296 return r # Uninitialized or impossible distance requested

297

298 Cs = Caps

299 if self._debug: # PYCHOK no cover

300 outmask |= self._debug & Cs._DEBUG_DIRECT_LINE

301 outmask &= self._caps & Cs._OUT_MASK

302

303 eF = self._eF

304 gX = self.geodesic # ._gX

305

306 if arcmode:

307 # s12_a12 is spherical arc length

308 E2 = _0_0

309 sig12 = radians(s12_a12)

310 if _K_2_0:

311 ssig12, csig12 = sincos2(sig12) # utily, no NEG0

312 else: # PYCHOK no cover

313 a = fabs(s12_a12) # 0 <= fabs(_remainder(s12_a12, _180_0)) <= 90

314 a -= floor(a / _180_0) * _180_0 # 0 <= 0 < 180

315 ssig12 = _0_0 if a == 0 else sin(sig12)

316 csig12 = _0_0 if a == 90 else cos(sig12)

317 else: # s12_a12 is distance

318 t = s12_a12 / self._E0b

319 s, c = _sincos2(t) # tau12

320 # tau2 = tau1 + tau12

321 E2 = -eF.deltaEinv(*_sincos12(-s, c, *self._stau1_ctau1))

322 sig12 = fsum1f_(self._E1, -E2, t) # == t - (E2 - E1)

323 ssig12, csig12 = _sincos2(sig12)

324

325 salp0, calp0 = self._salp0, self._calp0

326 ssig1, csig1 = self._ssig1, self._csig1

327

328 # sig2 = sig1 + sig12

329 ssig2, csig2 = _sincos12(-ssig12, csig12, ssig1, csig1)

330 dn2 = eF.fDelta(ssig2, csig2)

331 # sin(bet2) = cos(alp0) * sin(sig2) and

332 # cbet2 = hypot(salp0, calp0 * csig2). Alt:

333 # cbet2 = hypot(csig2, salp0 * ssig2)

334 sbet2, cbet2 = _sin1cos2(calp0, salp0, csig2, ssig2)

335 if cbet2 == 0: # salp0 = 0, csig2 = 0, break degeneracy

336 cbet2 = csig2 = _TINY

337 # tan(alp0) = cos(sig2) * tan(alp2)

338 salp2 = salp0

339 calp2 = calp0 * csig2 # no need to normalize

340

341 if (outmask & Cs.DISTANCE):

342 if arcmode: # or f_0_01

343 E2 = eF.deltaE(ssig2, csig2, dn2)

344 # AB1 = _E0 * (E2 - _E1)

345 # s12 = _b * (_E0 * sig12 + AB1)

346 # = _b * _E0 * (sig12 + (E2 - _E1))

347 # = _b * _E0 * (E2 - _E1 + sig12)

348 s12 = self._E0b * fsum1f_(E2, -self._E1, sig12)

349 else:

350 s12 = s12_a12

351 r.set_(s12=s12)

352

353 if (outmask & Cs._DEBUG_DIRECT_LINE): # PYCHOK no cover

354 r.set_(sig12=sig12, dn2=dn2, b=gX.b, e2=gX.e2, f1=gX.f1,

355 E0b=self._E0b, E1=self._E1, E2=E2, eFk2=eF.k2, eFa2=eF.alpha2)

356

357 if (outmask & Cs.LONGITUDE):

358 schi1 = self._somg1

359 cchi1 = self._cchi1

360 schi2 = ssig2 * salp0

361 cchi2 = gX.f1 * dn2 * csig2 # schi2 = somg2 without normalization

362 lam12 = salp0 * self._H0e2_f1 * fsum1f_(eF.deltaH(ssig2, csig2, dn2),

363 -self._H1, sig12)

364 if (outmask & Cs.LONG_UNROLL):

365 t = _copysign(_1_0, salp0) # east-going?

366 tchi1 = t * schi1

367 tchi2 = t * schi2

368 chi12 = t * fsum1f_(atan2(ssig1, csig1), -atan2(ssig2, csig2),

369 atan2(tchi2, cchi2), -atan2(tchi1, cchi1), sig12)

370 lon2 = self.lon1 + degrees(chi12 - lam12)

371 else:

372 chi12 = atan2(*_sincos12(schi1, cchi1, schi2, cchi2))

373 lon2 = _norm180(self._lon1_norm180 + _norm180(degrees(chi12 - lam12)))

374 r.set_(lon2=lon2)

375 if (outmask & Cs._DEBUG_DIRECT_LINE): # PYCHOK no cover

376 r.set_(ssig2=ssig2, chi12=chi12, H0e2_f1=self._H0e2_f1,

377 csig2=csig2, lam12=lam12, H1=self._H1)

378

379 if (outmask & Cs.LATITUDE):

380 r.set_(lat2=_atan2d(sbet2, gX.f1 * cbet2))

381

382 if (outmask & Cs.AZIMUTH):

383 r.set_(azi2=_atan2d_reverse(salp2, calp2, reverse=outmask & Cs.REVERSE2))

384

385 if (outmask & Cs._REDUCEDLENGTH_GEODESICSCALE):

386 dn1 = self._dn1

387 J12 = self._D0k2 * fsumf_(eF.deltaD(ssig2, csig2, dn2), -self._D1, sig12)

388 if (outmask & Cs._DEBUG_DIRECT_LINE): # PYCHOK no cover

389 r.set_(ssig1=ssig1, dn1=dn1, D0k2=self._D0k2,

390 csig1=csig1, J12=J12, D1=self._D1)

391 if (outmask & Cs.REDUCEDLENGTH):

392 # Add parens around (csig1 * ssig2) and (ssig1 * csig2) to

393 # ensure accurate cancellation in the case of coincident points.

394 r.set_(m12=gX.b * fsum1f_(dn2 * (csig1 * ssig2),

395 -dn1 * (ssig1 * csig2),

396 -J12 * (csig1 * csig2)))

397 if (outmask & Cs.GEODESICSCALE):

398 t = self._k2 * (ssig2 - ssig1) * (ssig2 + ssig1) / (dn2 + dn1)

399 r.set_(M12=csig12 + ssig1 * (t * ssig2 - csig2 * J12) / dn1,

400 M21=csig12 - ssig2 * (t * ssig1 - csig1 * J12) / dn2)

401

402 if (outmask & Cs.AREA):

403 A4 = salp0 * calp0

404 if A4:

405 # tan(alp) = tan(alp0) * sec(sig)

406 # tan(alp2-alp1) = (tan(alp2) - tan(alp1)) / (tan(alp2) * tan(alp1) + 1)

407 # = calp0 * salp0 * (csig1 - csig2) / (salp0^2 + calp0^2 * csig1 * csig2)

408 # If csig12 > 0, write

409 # csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)

410 # else

411 # csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1

412 # No need to normalize

413 salp12 = (((ssig12 * csig1 / (_1_0 + csig12) + ssig1) * ssig12) if csig12 > 0 else

414 (csig1 * (_1_0 - csig12) + ssig1 * ssig12)) * A4

415 calp12 = salp0**2 + calp0**2 * csig1 * csig2

416 A4 *= gX._e2a2

417 B41 = self._B41

418 B42 = _cosSeries(self._C4a, ssig2, csig2)

419 S12 = (B42 - B41) * A4

420 else:

421 S12 = A4 = B41 = B42 = _0_0

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

423 salp12, calp12 = _sincos12(self._salp1, self._calp1, salp2, calp2)

424 # We used to include some patch up code that purported to deal

425 # with nearly meridional geodesics properly. However, this turned

426 # out to be wrong once salp1 = -0 was allowed (via InverseLine).

427 # In fact, the calculation of {s,c}alp12 was already correct

428 # (following the IEEE rules for handling signed zeros). So,

429 # the patch up code was unnecessary (as well as dangerous).

430 if (outmask & Cs._DEBUG_DIRECT_LINE): # PYCHOK no cover

431 r.set_(salp12=salp12, salp0=salp0, B41=B41, A4=A4,

432 calp12=calp12, calp0=calp0, B42=B42, c2=gX.c2)

433 S12 += gX.c2 * atan2(salp12, calp12)

434 r.set_(S12=S12)

435

436 r.set_(a12=s12_a12 if arcmode else degrees(sig12),

437 lat1=self.lat1, # == _fix90(lat1)

438 lon1=self.lon1 if (outmask & Cs.LONG_UNROLL) else self._lon1_norm180,

439 azi1=_norm180(self.azi1))

440 return r

441

442 def _GenPosition(self, arcmode, s12_a12, outmask):

443 '''(INTERNAL) Generate a new position along the geodesic.

444

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

446 s12, m12, M12, M21, S12)}.

447 '''

448 r = self._GDictPosition(arcmode, s12_a12, outmask)

449 return r.toDirect9Tuple()

450

451 def _GenSet(self, arcmode, s13_a13):

452 '''(INTERNAL) Aka C++ C{GenSetDistance}.

453 '''

454 if arcmode:

455 self.SetArc(s13_a13)

456 else:

457 self.SetDistance(s13_a13)

458 return self # for gx.GeodesicExact.InverseLine and -._GenDirectLine

459

460 @Property_RO

461 def geodesic(self):

462 '''Get the I{exact} geodesic (L{GeodesicExact}).

463 '''

464 assert isinstance(self._gX, _MODS.geodesicx.GeodesicExact)

465 return self._gX

466

467 @Property_RO

468 def _H0e2_f1(self):

469 '''(INTERNAL) Cached/memoized.

470 '''

471 return self._eF.cH * _2__PI * self.geodesic._e2_f1

472

473 @Property_RO

474 def _H1(self):

475 '''(INTERNAL) Cached/memoized.

476 '''

477 return self._eF.deltaH(self._ssig1, self._csig1, self._dn1)

478

479 @Property_RO

480 def lat1(self):

481 '''Get the latitude of the first point (C{degrees}).

482 '''

483 return self._lat1

484

485 @Property_RO

486 def lon1(self):

487 '''Get the longitude of the first point (C{degrees}).

488 '''

489 return self._lon1

490

491 @Property_RO

492 def _lon1_norm180(self):

493 '''(INTERNAL) Cached/memoized.

494 '''

495 return _norm180(self._lon1)

496

497 def Position(self, s12, outmask=Caps.STANDARD):

498 '''Find the position on the line given B{C{s12}}.

499

500 @arg s12: Distance from the first point to the second (C{meter}).

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

502 the quantities to be returned.

503

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

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

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

507 except when C{a12=NAN}.

508

509 @note: By default, C{B{outmask}=STANDARD}, meaning thc C{lat1},

510 C{lon1}, C{azi1}, C{lat2}, C{lon2}, C{azi2}, C{s12} and

511 C{a12} entries are returned, except when C{a12=NAN}.

512

513 @note: This L{GeodesicLineExact} instance must have been

514 constructed with capability C{Caps.DISTANCE_IN} set.

515 '''

516 return self._GDictPosition(False, s12, outmask)

517

518 @Property_RO

519 def s13(self):

520 '''Get the distance to reference point 3 (C{meter} or C{NAN}).

521

522 @see: Methods L{Distance} and L{SetDistance}.

523 '''

524 return self._s13

525

526 def SetArc(self, a13):

527 '''Set reference point 3 in terms relative to the first point.

528

529 @arg a13: Spherical arc length from the first to the reference

530 point (C{degrees}).

531

532 @return: The distance C{s13} (C{meter}) between the first and

533 the reference point or C{NAN}.

534 '''

535 self._a13 = a13

536 self._s13 = s13 = self._GDictPosition(True, a13, Caps.DISTANCE).s12

537 _update_all(self)

538 return s13

539

540 def SetDistance(self, s13):

541 '''Set reference point 3 in terms relative to the first point.

542

543 @arg s13: Distance from the first to the reference point (C{meter}).

544

545 @return: The arc length C{a13} (C{degrees}) between the first

546 and the reference point or C{NAN}.

547 '''

548 self._s13 = s13

549 self._a13 = a13 = self._GDictPosition(False, s13, 0).a12

550 _update_all(self)

551 return a13 # NAN for GeodesicLineExact without Cap.DISTANCE_IN

552

553 @Property_RO

554 def _stau1_ctau1(self):

555 '''(INTERNAL) Cached/memoized.

556 '''

557 s, c = _sincos2(self._E1)

558 # tau1 = sig1 + B11

559 return _sincos12(-s, c, self._ssig1, self._csig1)

560 # unnecessary because Einv inverts E

561 # return -self._eF.deltaEinv(stau1, ctau1)

562

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

564 '''Return this C{GeodesicLineExact} as string.

565

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

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

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

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

570

571 @return: C{GeodesicLineExact} (C{str}).

572 '''

573 d = dict(geodesic=self.geodesic,

574 lat1=self.lat1, lon1=self.lon1, azi1=self.azi1,

575 a13=self.a13, s13=self.s13)

576 return sep.join(_MODS.streprs.pairs(d, prec=prec))

577

578

579__all__ += _ALL_DOCS(_GeodesicLineExact)

580

581# **) MIT License

582#

583# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.

584#

585# Permission is hereby granted, free of charge, to any person obtaining a

586# copy of this software and associated documentation files (the "Software"),

587# to deal in the Software without restriction, including without limitation

588# the rights to use, copy, modify, merge, publish, distribute, sublicense,

589# and/or sell copies of the Software, and to permit persons to whom the

590# Software is furnished to do so, subject to the following conditions:

591#

592# The above copyright notice and this permission notice shall be included

593# in all copies or substantial portions of the Software.

594#

595# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

596# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

597# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

598# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

599# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

600# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

601# OTHER DEALINGS IN THE SOFTWARE.