1

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

3

4 u'''(INTERNAL) base classes C{RhumbBase} and C{RhumbLineBase}, pure Python version of I{Karney}'s

5 C++ classes U{Rhumb<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Rhumb.html>}

6 and U{RhumbLine<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1RhumbLine.html>}

7 from I{GeographicLib versions 2.0} and I{2.2} and I{Karney}'s C++ example U{Rhumb intersect

8<https://SourceForge.net/p/geographiclib/discussion/1026620/thread/2ddc295e/>}.

9

10 Class L{RhumbLineBase} has been enhanced with methods C{Intersecant2}, C{Intersection} and C{PlumbTo}

11 to iteratively find the intersection of a rhumb line and a circle or an other rhumb line, respectively

12 a perpendicular geodesic or other rhumb line.

13

14 For more details, see the C++ U{GeographicLib<https://GeographicLib.SourceForge.io/C++/doc/index.html>}

15 documentation, especially the U{Class List<https://GeographicLib.SourceForge.io/C++/doc/annotated.html>},

16 the background information on U{Rhumb lines<https://GeographicLib.SourceForge.io/C++/doc/rhumb.html>},

17 the utily U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>} and U{Online

18 rhumb line calculations<https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve>}.

19

20 Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2014-2023) and licensed under the MIT/X11

21 License. For more information, see the U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation.

22'''

23# make sure int/int division yields float quotient

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

25

26 from pygeodesy.basics import _copysign, unsigned0, _xinstanceof

27 from pygeodesy.constants import EPS, EPS0, EPS1, INT0, NAN, _over, \

28 _EPSqrt as _TOL, _0_0, _0_01, _1_0, _90_0

29 from pygeodesy.datums import Datum, _earth_datum, _spherical_datum, _WGS84

30 from pygeodesy.errors import IntersectionError, itemsorted, RhumbError, \

31 _xdatum, _xkwds, _xkwds_pop, _Xorder

32# from pygeodesy.etm import ExactTransverseMercator # _MODS

33 from pygeodesy.fmath import euclid, favg, sqrt_a, Fsum

34# from pygeodesy.formy import opposing # _MODS

35# from pygeodesy.fsums import Fsum # from .fmath

36 from pygeodesy.interns import NN, _coincident_, _COMMASPACE_, _Dash, \

37 _dunder_nameof, _parallel_, _too_, _under

38 from pygeodesy.karney import _atan2d, Caps, _CapsBase, _diff182, _fix90, \

39 _norm180, GDict

40# from pygeodesy.ktm import KTransverseMercator, _AlpCoeffs # _MODS

41 from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS

42# from pygeodesy.named import notOverloaded # _MODS

43 from pygeodesy.namedTuples import Distance2Tuple, LatLon2Tuple

44 from pygeodesy.props import deprecated_method, Property, Property_RO, \

45 property_RO, _update_all

46 from pygeodesy.streprs import Fmt, pairs

47 from pygeodesy.units import Float_, Lat, Lon, Meter, Radius_, Int # PYCHOK shared

48 from pygeodesy.utily import acos1, _azireversed, _loneg, sincos2d, sincos2d_, \

49 _unrollon, _Wrap

50 from pygeodesy.vector3d import _intersect3d3, Vector3d # in .Intersection below

51

52 from math import cos, fabs

53

54__all__ = ()

55__version__ = '23.12.03'

56

57_anti_ = _Dash('anti')

58_rls = [] # instances of C{RbumbLine...} to be updated

59_TRIPS = 65 # .Intersection, .PlumbTo, 19+

60

61

62 class _Lat(Lat):

63 '''(INTERNAL) Latitude B{C{lat}}.

64 '''

65 def __init__(self, *lat, **Error_name):

66 kwds = _xkwds(Error_name, clip=0, Error=RhumbError)

67 Lat.__new__(_Lat, *lat, **kwds)

68

69

70 class _Lon(Lon):

71 '''(INTERNAL) Longitude B{C{lon}}.

72 '''

73 def __init__(self, *lon, **Error_name):

74 kwds = _xkwds(Error_name, clip=0, Error=RhumbError)

75 Lon.__new__(_Lon, *lon, **kwds)

76

77

78 def _update_all_rls(r):

79 '''(INTERNAL) Zap cached/memoized C{Property[_RO]}s

80 of any C{RhumbLine} instances tied to the given

81 C{Rhumb} instance B{C{r}}.

82 '''

83 # _xinstanceof(_MODS.rhumb.aux_.RhumbAux, _MODS.rhumb.ekx.Rhumb, r=r)

84 _update_all(r)

85 for rl in _rls: # PYCHOK use weakref?

86 if rl._rhumb is r:

87 _update_all(rl)

88

89

90 class RhumbBase(_CapsBase):

91 '''(INTERNAL) Base class for C{rhumb.aux_.RhumbAux} and C{rhumb.ekx.Rhumb}.

92 '''

93 _datum = _WGS84

94 _exact = True

95 _f_max = _0_01

96 _mTM = 6 # see .TMorder

97

98 def __init__(self, a_earth, f, exact, name):

99 '''New C{RhumbAux} or C{Rhumb}.

100 '''

101 _earth_datum(self, a_earth, f=f, name=name)

102 if not exact:

103 self.exact = False

104 if name:

105 self.name = name

106

107 @Property_RO

108 def a(self):

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

110 '''

111 return self.ellipsoid.a

112

113 equatoradius = a

114

115 def ArcDirect(self, lat1, lon1, azi12, a12, outmask=Caps.LATITUDE_LONGITUDE):

116 '''Solve the I{direct rhumb} problem, optionally with area.

117

118 @arg lat1: Latitude of the first point (C{degrees90}).

119 @arg lon1: Longitude of the first point (C{degrees180}).

120 @arg azi12: Azimuth of the rhumb line (compass C{degrees}).

121 @arg a12: Angle along the rhumb line from the given to the

122 destination point (C{degrees}), can be negative.

123

124 @return: L{GDict} with 2 up to 8 items C{lat2, lon2, a12, S12,

125 lat1, lon1, azi12, s12} with the destination point's

126 latitude C{lat2} and longitude C{lon2} in C{degrees},

127 the rhumb angle C{a12} in C{degrees} and area C{S12}

128 under the rhumb line in C{meter} I{squared}.

129

130 @raise ImportError: Package C{numpy} not found or not installed,

131 only required for area C{S12} when C{B{exact}

132 is True} and L{RhumbAux}.

133

134 @note: If B{C{a12}} is large enough that the rhumb line crosses

135 a pole, the longitude of the second point is indeterminate

136 and C{NAN} is returned for C{lon2} and area C{S12}.

137

138 @note: If the given point is a pole, the cosine of its latitude is

139 taken to be C{sqrt(L{EPS})}. This position is extremely

140 close to the actual pole and allows the calculation to be

141 carried out in finite terms.

142 '''

143 s12 = a12 * self._mpd

144 return self._DirectRhumb(lat1, lon1, azi12, a12, s12, outmask)

145

146 @Property_RO

147 def b(self):

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

149 '''

150 return self.ellipsoid.b

151

152 polaradius = b

153

154 @property

155 def datum(self):

156 '''Get this rhumb's datum (L{Datum}).

157 '''

158 return self._datum

159

160 @datum.setter # PYCHOK setter!

161 def datum(self, datum):

162 '''Set this rhumb's datum (L{Datum}).

163

164 @raise RhumbError: If C{abs(B{f}} exceeds non-zero C{f_max} and C{exact=False}.

165 '''

166 _xinstanceof(Datum, datum=datum)

167 if self._datum != datum:

168 self._exactest(self.exact, datum.ellipsoid, self.f_max)

169 _update_all_rls(self)

170 self._datum = datum

171

172 def _Direct(self, ll1, azi12, s12, **outmask):

173 '''(INTERNAL) Short-cut version, see .latlonBase.rhumb....

174 '''

175 return self.Direct(ll1.lat, ll1.lon, azi12, s12, **outmask)

176

177 def Direct(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE):

178 '''Solve the I{direct rhumb} problem, optionally with area.

179

180 @arg lat1: Latitude of the first point (C{degrees90}).

181 @arg lon1: Longitude of the first point (C{degrees180}).

182 @arg azi12: Azimuth of the rhumb line (compass C{degrees}).

183 @arg s12: Distance along the rhumb line from the given to

184 the destination point (C{meter}), can be negative.

185

186 @return: L{GDict} with 2 up to 8 items C{lat2, lon2, a12, S12,

187 lat1, lon1, azi12, s12} with the destination point's

188 latitude C{lat2} and longitude C{lon2} in C{degrees},

189 the rhumb angle C{a12} in C{degrees} and area C{S12}

190 under the rhumb line in C{meter} I{squared}.

191

192 @raise ImportError: Package C{numpy} not found or not installed,

193 only required for area C{S12} when C{B{exact}

194 is True} and L{RhumbAux}.

195

196 @note: If B{C{s12}} is large enough that the rhumb line crosses

197 a pole, the longitude of the second point is indeterminate

198 and C{NAN} is returned for C{lon2} and area C{S12}.

199

200 @note: If the given point is a pole, the cosine of its latitude is

201 taken to be C{sqrt(L{EPS})}. This position is extremely

202 close to the actual pole and allows the calculation to be

203 carried out in finite terms.

204 '''

205 a12 = _over(s12, self._mpd)

206 return self._DirectRhumb(lat1, lon1, azi12, a12, s12, outmask)

207

208 def Direct8(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE_AREA):

209 '''Like method L{Rhumb.Direct} but returning a L{Rhumb8Tuple} with area C{S12}.

210 '''

211 return self.Direct(lat1, lon1, azi12, s12, outmask=outmask).toRhumb8Tuple()

212

213 def _DirectLine(self, ll1, azi12, **caps_name):

214 '''(INTERNAL) Short-cut version, see .latlonBase.

215 '''

216 return self.DirectLine(ll1.lat, ll1.lon, azi12, **caps_name)

217

218 def DirectLine(self, lat1, lon1, azi12, **caps_name):

219 '''Define a C{RhumbLine} in terms of the I{direct} rhumb

220 problem to compute several points on a single rhumb line.

221

222 @arg lat1: Latitude of the first point (C{degrees90}).

223 @arg lon1: Longitude of the first point (C{degrees180}).

224 @arg azi12: Azimuth of the rhumb line (compass C{degrees}).

225 @kwarg caps_name: Optional keyword arguments C{B{name}=NN} and

226 C{B{caps}=Caps.STANDARD}, a bit-or'ed combination of

227 L{Caps} values specifying the required capabilities.

228 Include C{Caps.LINE_OFF} if updates to the B{C{rhumb}}

229 should I{not} be reflected in this rhumb line.

230

231 @return: A C{RhumbLine...} instance and invoke its method

232 C{.Position} to compute each point.

233

234 @note: Updates to this rhumb are reflected in the returned

235 rhumb line, unless C{B{caps} |= Caps.LINE_OFF}.

236 '''

237 return self._RhumbLine(self, lat1, lon1, azi12, **caps_name)

238

239 Line = DirectLine # synonyms

240

241 def _DirectRhumb(self, lat1, lon1, azi12, a12, s12, outmask):

242 '''(INTERNAL) See methods C{.ArcDirect} and C{.Direct}.

243 '''

244 rl = self._RhumbLine(self, lat1, lon1, azi12, caps=Caps.LINE_OFF,

245 name=self.name)

246 return rl._Position(a12, s12, outmask | self._debug) # lat2, lon2, S12

247

248 @Property

249 def ellipsoid(self):

250 '''Get this rhumb's ellipsoid (L{Ellipsoid}).

251 '''

252 return self.datum.ellipsoid

253

254 @ellipsoid.setter # PYCHOK setter!

255 def ellipsoid(self, a_earth_f):

256 '''Set this rhumb's ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or

257 L{a_f2Tuple}) or (equatorial) radius and flattening (2-tuple C{(a, f)}).

258

259 @raise RhumbError: If C{abs(B{f}} exceeds non-zero C{f_max} and C{exact=False}.

260 '''

261 self.datum = _spherical_datum(a_earth_f, Error=RhumbError)

262

263 @Property

264 def exact(self):

265 '''Get the I{exact} option (C{bool}).

266 '''

267 return self._exact

268

269 @exact.setter # PYCHOK setter!

270 def exact(self, exact):

271 '''Set the I{exact} option (C{bool}). If C{True}, use I{exact} rhumb

272 expressions, otherwise a series expansion (accurate for oblate or

273 prolate ellipsoids with C{abs(flattening)} below C{f_max}.

274

275 @raise RhumbError: If C{B{exact}=False} and C{abs(flattening})

276 exceeds non-zero C{f_max}.

277

278 @see: Option U{B{-s}<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>}

279 and U{ACCURACY<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html#ACCURACY>}.

280 '''

281 x = bool(exact)

282 if self._exact != x:

283 self._exactest(x, self.ellipsoid, self.f_max)

284 _update_all_rls(self)

285 self._exact = x

286

287 def _exactest(self, exact, ellipsoid, f_max):

288 # Helper for property setters C{ellipsoid}, C{exact} and C{f_max}

289 if fabs(ellipsoid.f) > f_max > 0 and not exact:

290 raise RhumbError(exact=exact, f=ellipsoid.f, f_max=f_max)

291

292 @Property_RO

293 def f(self):

294 '''Get the C{ellipsoid}'s flattening (C{float}).

295 '''

296 return self.ellipsoid.f

297

298 flattening = f

299

300 @property

301 def f_max(self):

302 '''Get the I{max.} flattening (C{float}).

303 '''

304 return self._f_max

305

306 @f_max.setter # PYCHOK setter!

307 def f_max(self, f_max): # PYCHOK no cover

308 '''Set the I{max.} flattening, not to exceed (C{float}).

309

310 @raise RhumbError: If C{exact=False} and C{abs(flattening})

311 exceeds non-zero C{f_max}.

312 '''

313 f = Float_(f_max=f_max, low=_0_0, high=EPS1)

314 if self._f_max != f:

315 self._exactest(self.exact, self.ellipsoid, f)

316 self._f_max = f

317

318 def _Inverse(self, ll1, ll2, wrap, **outmask):

319 '''(INTERNAL) Short-cut version, see .latlonBase.rhumb....

320 '''

321 if wrap:

322 ll2 = _unrollon(ll1, _Wrap.point(ll2))

323 return self.Inverse(ll1.lat, ll1.lon, ll2.lat, ll2.lon, **outmask)

324

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

326 '''Solve the I{inverse rhumb} problem.

327

328 @arg lat1: Latitude of the first point (C{degrees90}).

329 @arg lon1: Longitude of the first point (C{degrees180}).

330 @arg lat2: Latitude of the second point (C{degrees90}).

331 @arg lon2: Longitude of the second point (C{degrees180}).

332

333 @return: L{GDict} with 4 to 9 items C{lat1, lon1, lat2, lon2,

334 azi12, azi21, s12, a12, S12}, the rhumb line's azimuth

335 C{azi12} and I{reverse} azimuth C{azi21}, both in

336 compass C{degrees} between C{-180} and C{+180}, the

337 rhumb distance C{s12} and rhumb angle C{a12} between

338 both points in C{meter} respectively C{degrees} and

339 the area C{S12} under the rhumb line in C{meter}

340 I{squared}.

341

342 @raise ImportError: Package C{numpy} not found or not installed,

343 only required for L{RhumbAux} area C{S12}

344 when C{B{exact} is True}.

345

346 @note: The shortest rhumb line is found. If the end points are

347 on opposite meridians, there are two shortest rhumb lines

348 and the East-going one is chosen.

349

350 @note: If either point is a pole, the cosine of its latitude is

351 taken to be C{sqrt(L{EPS})}. This position is extremely

352 close to the actual pole and allows the calculation to be

353 carried out in finite terms.

354 '''

355 r = GDict(lat1=lat1, lon1=lon1, lat2=lat2, lon2=lon2, name=self.name)

356 Cs = Caps

357 if (outmask & Cs.AZIMUTH_DISTANCE_AREA):

358 lon12, _ = _diff182(lon1, lon2, K_2_0=True)

359 y, x, s1, s2 = self._Inverse4(lon12, r, outmask)

360 if (outmask & Cs.AZIMUTH):

361 z = _atan2d(y, x)

362 r.set_(azi12=z, azi21=_azireversed(z))

363 if (outmask & Cs.AREA):

364 S12 = self._S12d(s1, s2, lon12)

365 r.set_(S12=unsigned0(S12)) # like .gx

366 return r

367

368 def _Inverse4(self, lon12, r, outmask): # PYCHOK no cover

369 '''(INTERNAL) I{Must be overloaded}.'''

370 _MODS.named.notOverloaded(self, lon12, r, Caps.toStr(outmask))

371

372 def Inverse8(self, lat1, lon1, azi12, s12, outmask=Caps.AZIMUTH_DISTANCE_AREA):

373 '''Like method L{Rhumb.Inverse} but returning a L{Rhumb8Tuple} with area C{S12}.

374 '''

375 return self.Inverse(lat1, lon1, azi12, s12, outmask=outmask).toRhumb8Tuple()

376

377 def _InverseLine(self, ll1, ll2, wrap, **caps_name):

378 '''(INTERNAL) Short-cut version, see .latlonBase.

379 '''

380 if wrap:

381 ll2 = _unrollon(ll1, _Wrap.point(ll2))

382 return self.InverseLine(ll1.lat, ll1.lon, ll2.lat, ll2.lon, **caps_name)

383

384 def InverseLine(self, lat1, lon1, lat2, lon2, **caps_name):

385 '''Define a C{RhumbLine} in terms of the I{inverse} rhumb problem.

386

387 @arg lat1: Latitude of the first point (C{degrees90}).

388 @arg lon1: Longitude of the first point (C{degrees180}).

389 @arg lat2: Latitude of the second point (C{degrees90}).

390 @arg lon2: Longitude of the second point (C{degrees180}).

391 @kwarg caps_name: Optional keyword arguments C{B{name}=NN} and

392 C{B{caps}=Caps.STANDARD}, a bit-or'ed combination of

393 L{Caps} values specifying the required capabilities.

394 Include C{Caps.LINE_OFF} if updates to the B{C{rhumb}}

395 should I{not} be reflected in this rhumb line.

396

397 @return: A C{RhumbLine...} instance and invoke its method

398 C{ArcPosition} or C{Position} to compute points.

399

400 @note: Updates to this rhumb are reflected in the returned

401 rhumb line, unless C{B{caps} |= Caps.LINE_OFF}.

402 '''

403 r = self.Inverse(lat1, lon1, lat2, lon2, outmask=Caps.AZIMUTH)

404 return self._RhumbLine(self, lat1, lon1, r.azi12, **caps_name)

405

406 @Property_RO

407 def _mpd(self): # PYCHOK no cover

408 '''(INTERNAL) I{Must be overloaded}.'''

409 _MODS.named.notOverloaded(self)

410

411 @property_RO

412 def RAorder(self):

413 '''Get the I{Rhumb Area} order, C{None} always.

414 '''

415 return None

416

417 @property_RO

418 def _RhumbLine(self): # PYCHOK no cover

419 '''(INTERNAL) I{Must be overloaded}.'''

420 _MODS.named.notOverloaded(self, underOK=True)

421

422 def _S12d(self, s1, s2, lon): # PYCHOK no cover

423 '''(INTERNAL) I{Must be overloaded}.'''

424 _MODS.named.notOverloaded(self, s1, s2, lon)

425

426 @Property

427 def TMorder(self):

428 '''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).

429 '''

430 return self._mTM

431

432 @TMorder.setter # PYCHOK setter!

433 def TMorder(self, order):

434 '''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).

435

436 @note: Setting C{TMorder} turns property C{exact} off, but only

437 for L{Rhumb} instances.

438 '''

439 m = _Xorder(_MODS.ktm._AlpCoeffs, RhumbError, TMorder=order)

440 if self._mTM != m:

441 _update_all_rls(self)

442 self._mTM = m

443 if self.exact and isinstance(self, _MODS.rhumb.ekx.Rhumb):

444 self.exact = False

445

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

447 '''Return this C{Rhumb} as string.

448

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

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

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

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

453

454 @return: Tuple items (C{str}).

455 '''

456 d = dict(ellipsoid=self.ellipsoid, RAorder=self.RAorder,

457 exact=self.exact, TMorder=self.TMorder)

458 return sep.join(pairs(itemsorted(d, asorted=False), prec=prec))

459

460

461 class RhumbLineBase(_CapsBase):

462 '''(INTERNAL) Base class for C{rhumb.aux_.RhumbLineAux} and C{rhumb.ekx.RhumbLine}.

463 '''

464 _azi12 = _0_0

465 _calp = _1_0

466# _caps = 0

467# _debug = 0

468# _lat1 = _0_0

469# _lon1 = _0_0

470# _lon12 = _0_0

471 _Rhumb = RhumbBase # compatible C{Rhumb} class

472 _rhumb = None # C{Rhumb} instance

473 _salp = _0_0

474 _talp = _0_0

475

476 def __init__(self, rhumb, lat1, lon1, azi12, caps=Caps.STANDARD, name=NN):

477 '''New C{RhumbLine} or C{RhumbLineAux}.

478 '''

479 _xinstanceof(self._Rhumb, rhumb=rhumb)

480

481 self._lat1 = _Lat(lat1=_fix90(lat1))

482 self._lon1 = _Lon(lon1= lon1)

483 self._lon12 = _norm180(self._lon1)

484 if azi12: # non-zero, non-None

485 self.azi12 = _norm180(azi12)

486

487 n = name or rhumb.name

488 if n:

489 self.name=n

490

491 self._caps = caps

492 self._debug |= (caps | rhumb._debug) & Caps._DEBUG_DIRECT_LINE

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

494 self._rhumb = rhumb.copy(deep=False, name=_under(rhumb.name))

495 else:

496 self._rhumb = rhumb

497 _rls.append(self)

498

499 def __del__(self): # XXX use weakref?

500 if _rls: # may be empty or None

501 try: # PYCHOK no cover

502 _rls.remove(self)

503 except (TypeError, ValueError):

504 pass

505 self._rhumb = None

506 # _update_all(self) # throws TypeError during Python 2 cleanup

507

508 def ArcPosition(self, a12, outmask=Caps.LATITUDE_LONGITUDE):

509 '''Compute a point at a given angular distance on this rhumb line.

510

511 @arg a12: The angle along this rhumb line from its origin to the

512 point (C{degrees}), can be negative.

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

514 the quantities to be returned.

515

516 @return: L{GDict} with 4 to 8 items C{azi12, a12, s12, S12, lat2,

517 lon2, lat1, lon1} with latitude C{lat2} and longitude

518 C{lon2} of the point in C{degrees}, the rhumb distance

519 C{s12} in C{meter} from the start point of and the area

520 C{S12} under this rhumb line in C{meter} I{squared}.

521

522 @raise ImportError: Package C{numpy} not found or not installed,

523 only required for L{RhumbLineAux} area C{S12}

524 when C{B{exact} is True}.

525

526 @note: If B{C{a12}} is large enough that the rhumb line crosses a

527 pole, the longitude of the second point is indeterminate and

528 C{NAN} is returned for C{lon2} and area C{S12}.

529

530 If the first point is a pole, the cosine of its latitude is

531 taken to be C{sqrt(L{EPS})}. This position is extremely

532 close to the actual pole and allows the calculation to be

533 carried out in finite terms.

534 '''

535 return self._Position(a12, self.degrees2m(a12), outmask)

536

537 @Property

538 def azi12(self):

539 '''Get this rhumb line's I{azimuth} (compass C{degrees}).

540 '''

541 return self._azi12

542

543 @azi12.setter # PYCHOK setter!

544 def azi12(self, azi12):

545 '''Set this rhumb line's I{azimuth} (compass C{degrees}).

546 '''

547 z = _norm180(azi12)

548 if self._azi12 != z:

549 if self._rhumb:

550 _update_all(self)

551 self._azi12 = z

552 self._salp, self._calp = t = sincos2d(z) # no NEG0

553 self._talp = _over(*t)

554

555 @property_RO

556 def azi12_sincos2(self): # PYCHOK no cover

557 '''Get the sine and cosine of this rhumb line's I{azimuth} (2-tuple C{(sin, cos)}).

558 '''

559 return self._scalp, self._calp

560

561 @property_RO

562 def datum(self):

563 '''Get this rhumb line's datum (L{Datum}).

564 '''

565 return self.rhumb.datum

566

567 def degrees2m(self, angle):

568 '''Convert an angular distance along this rhumb line to C{meter}.

569

570 @arg angle: Angular distance (C{degrees}).

571

572 @return: Distance (C{meter}).

573 '''

574 return float(angle) * self.rhumb._mpd

575

576 @deprecated_method

577 def distance2(self, lat, lon): # PYCHOK no cover

578 '''DEPRECATED on 23.09.23, use method L{RhumbLineAux.Inverse} or L{RhumbLine.Inverse}.

579

580 @return: A L{Distance2Tuple}C{(distance, initial)} with the C{distance}

581 in C{meter} and C{initial} bearing (azimuth) in C{degrees}.

582 '''

583 r = self.Inverse(lat, lon)

584 return Distance2Tuple(r.s12, r.azi12)

585

586 @property_RO

587 def ellipsoid(self):

588 '''Get this rhumb line's ellipsoid (L{Ellipsoid}).

589 '''

590 return self.rhumb.ellipsoid

591

592 @property_RO

593 def exact(self):

594 '''Get this rhumb line's I{exact} option (C{bool}).

595 '''

596 return self.rhumb.exact

597

598 def Intersecant2(self, lat0, lon0, radius, napier=True, **tol_eps):

599 '''Compute the intersection(s) of this rhumb line and a circle.

600

601 @arg lat0: Latitude of the circle center (C{degrees}).

602 @arg lon0: Longitude of the circle center (C{degrees}).

603 @arg radius: Radius of the circle (C{meter}, conventionally).

604 @kwarg napier: If C{True}, apply I{Napier}'s spherical triangle

605 instead of planar trigonometry (C{bool}).

606 @kwarg tol_eps: Optional keyword arguments, see method

607 method L{Intersection} for further details.

608

609 @return: 2-Tuple C{(P, Q)} with both intersections (representing

610 a rhumb chord), each a L{GDict} from method L{Intersection}

611 extended to 18 items by C{lat3, lon3, azi03, a03, s03}

612 with azimuth C{azi03} of, distance C{a03} in C{degrees}

613 and C{s03} in C{meter} along the rhumb line from the circle

614 C{lat0, lon0} to the chord center C{lat3, lon3}. If this

615 rhumb line is tangential to the circle, both points

616 are the same L{GDict} instance with distances C{s02} and

617 C{s03} near-equal to the B{C{radius}}.

618

619 @raise IntersectionError: The circle and this rhumb line

620 do not intersect.

621

622 @raise UnitError: Invalid B{C{radius}}.

623 '''

624 r = Radius_(radius)

625 p = q = self.PlumbTo(lat0, lon0, exact=None, **tol_eps)

626 a = q.s02

627 t = dict(lat3=q.lat2, lon3=q.lon2, azi03=q.azi02, a03=q.a02, s03=a)

628 if a < r:

629 t.update(iteration=q.iteration, lat0=q.lat1, lon0=q.lon1, # or lat0, lon0

630 name=_dunder_nameof(self.Intersecant2, self.name))

631 if fabs(a) < EPS0: # coincident centers

632 d, h = _0_0, r

633 else:

634 d = q.s12

635 if napier: # Napier rule (R1) cos(b) = cos(c) / cos(a)

636 # <https://WikiPedia.org/wiki/Spherical_trigonometry>

637 m = self.rhumb._mpr

638 h = (acos1(cos(r / m) / cos(a / m)) * m) if m else _0_0

639 else:

640 h = _copysign(sqrt_a(r, a), a)

641 p = q = self.Position(d + h).set_(**t)

642 if h:

643 q = self.Position(d - h).set_(**t)

644 elif a > r:

645 t = _too_(Fmt.distant(a))

646 raise IntersectionError(self, lat0, lon0, radius,

647 txt=t, **tol_eps)

648 else: # tangential

649 q.set_(**t) # == p.set(_**t)

650 return p, q

651

652 def intersection2(self, other, **tol_eps): # PYCHOK no cover

653 '''DEPRECATED on 23.10.10, use method L{Intersection}.'''

654 p = self.Intersection(other, **tol_eps)

655 r = LatLon2Tuple(p.lat2, p.lon2, name=self.intersection2.__name__)

656 r._iteration = p.iteration

657 return r

658

659 def Intersection(self, other, tol=_TOL, **eps):

660 '''I{Iteratively} find the intersection of this and an other rhumb line.

661

662 @arg other: The other rhumb line (C{RhumbLine}).

663 @kwarg tol: Tolerance for longitudinal convergence and parallel

664 error (C{degrees}).

665 @kwarg eps: Tolerance for L{pygeodesy.intersection3d3} (C{EPS}).

666

667 @return: The intersection point, a L{Position}-like L{GDict} with

668 13 items C{lat1, lon1, azi12, a12, s12, lat2, lon2, lat0,

669 lon0, azi02, a02, s02, at} with the rhumb angle C{a02}

670 and rhumb distance C{s02} between the start point C{lat0,

671 lon0} of the B{C{other}} rhumb line and the intersection

672 C{lat2, lon2}, the azimuth C{azi02} of the B{C{other}}

673 rhumb line and the angle C{at} between both rhumb lines.

674 See method L{Position} for further details.

675

676 @raise IntersectionError: No convergence for this B{C{tol}} or

677 no intersection for an other reason.

678

679 @see: Methods C{distance2} and C{PlumbTo} and function

680 L{pygeodesy.intersection3d3}.

681

682 @note: Each iteration involves a round trip to this rhumb line's

683 L{ExactTransverseMercator} or L{KTransverseMercator}

684 projection and function L{pygeodesy.intersection3d3} in

685 that domain.

686 '''

687 _xinstanceof(RhumbLineBase, other=other)

688 _xdatum(self.rhumb, other.rhumb, Error=RhumbError)

689 try:

690 if self.others(other) is self:

691 raise ValueError(_coincident_)

692 # make invariants and globals locals

693 _s_3d, s_az = self._xTM3d, self.azi12

694 _o_3d, o_az = other._xTM3d, other.azi12

695 p = _MODS.formy.opposing(s_az, o_az, margin=tol)

696 if p is not None: # == t in (True, False)

697 raise ValueError(_anti_(_parallel_) if p else _parallel_)

698 _diff = euclid # approximate length

699 _i3d3 = _intersect3d3 # NOT .vector3d.intersection3d3

700 _LL2T = LatLon2Tuple

701 _xTMr = self.xTM.reverse # ellipsoidal or spherical

702 # use halfway point as initial estimate

703 p = _LL2T(favg(self.lat1, other.lat1),

704 favg(self.lon1, other.lon1))

705 for i in range(1, _TRIPS):

706 v = _i3d3(_s_3d(p), s_az, # point + bearing

707 _o_3d(p), o_az, useZ=False, **eps)[0]

708 t = _xTMr(v.x, v.y, lon0=p.lon) # PYCHOK Reverse4Tuple

709 d = _diff(t.lon - p.lon, t.lat) # PYCHOK t.lat + p.lat - p.lat

710 p = _LL2T(t.lat + p.lat, t.lon) # PYCHOK t.lon + p.lon = lon0

711 if d < tol: # 19 trips

712 break

713 else:

714 raise ValueError(Fmt.no_convergence(d))

715

716 n = _dunder_nameof(self.Intersection, self.name)

717 r = self.Inverse( p.lat, p.lon, outmask=Caps.DISTANCE)

718 t = other.Inverse(p.lat, p.lon, outmask=Caps.DISTANCE)

719 P = GDict(lat1=self.lat1, lat2=p.lat, lat0=other.lat1,

720 lon1=self.lon1, lon2=p.lon, lon0=other.lon1,

721 azi12= self.azi12, a12=r.a12, s12=r.s12,

722 azi02=other.azi12, a02=t.a12, s02=t.s12,

723 at=other.azi12 - self.azi12, name=n)

724 P._iteration = i # .set_(iteration=i, ...) only

725 except Exception as x:

726 raise IntersectionError(self, other, tol=tol,

727 eps=eps, cause=x)

728 return P

729

730 def Inverse(self, lat2, lon2, wrap=False, **outmask):

731 '''Return the rhumb angle, distance, azimuth, I{reverse} azimuth, etc. of

732 a rhumb line between the given point and this rhumb line's start point.

733

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

735 @arg lon2: Longitude of the points (C{degrees}).

736 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}

737 and B{C{lon2}} (C{bool}).

738

739 @return: L{GDict} with 8 items C{a12, s12, azi12, azi21, lat1, lon1,

740 lat2, lon2}, the rhumb angle C{a12} and rhumb distance C{s12}

741 between both points in C{degrees} respectively C{meter}, the

742 rhumb line's azimuth C{azi12} and I{reverse} azimuth C{azi21}

743 both in compass C{degrees} between C{-180} and C{+180}.

744 '''

745 if wrap:

746 _, lat2, lon2 = _Wrap.latlon3(self.lon1, _fix90(lat2), lon2, wrap)

747 r = self.rhumb.Inverse(self.lat1, self.lon1, lat2, lon2, **outmask)

748 return r

749

750 @Property_RO

751 def isLoxodrome(self):

752 '''Is this rhumb line a meridional (C{None}), a parallel

753 (C{False}) or a C{True} loxodrome?

754

755 @see: I{Osborne's} U{2.5 Rumb lines and loxodromes

756 <https://Zenodo.org/record/35392>}, page 37.

757 '''

758 return bool(self._salp) if self._calp else None

759

760 @Property_RO

761 def lat1(self):

762 '''Get this rhumb line's latitude (C{degrees90}).

763 '''

764 return self._lat1

765

766 @Property_RO

767 def lon1(self):

768 '''Get this rhumb line's longitude (C{degrees180}).

769 '''

770 return self._lon1

771

772 @Property_RO

773 def latlon1(self):

774 '''Get this rhumb line's lat- and longitude (L{LatLon2Tuple}C{(lat, lon)}).

775 '''

776 return LatLon2Tuple(self.lat1, self.lon1)

777

778 def m2degrees(self, distance):

779 '''Convert a distance along this rhumb line to an angular distance.

780

781 @arg distance: Distance (C{meter}).

782

783 @return: Angular distance (C{degrees}).

784 '''

785 return _over(float(distance), self.rhumb._mpd)

786

787 @property_RO

788 def _mu1(self): # PYCHOK no cover

789 '''(INTERNAL) I{Must be overloaded}.'''

790 _MODS.named.notOverloaded(self, underOK=True)

791

792 def _mu2lat(self, mu2): # PYCHOK no cover

793 '''(INTERNAL) I{Must be overloaded}.'''

794 _MODS.named.notOverloaded(self, mu2, underOK=True)

795

796 @deprecated_method

797 def nearestOn4(self, lat0, lon0, **exact_eps_est_tol): # PYCHOK no cover

798 '''DEPRECATED on 23.10.10, use method L{PlumbTo}.'''

799 P = self.PlumbTo(lat0, lon0, **exact_eps_est_tol)

800 r = _MODS.deprecated.classes.NearestOn4Tuple(P.lat2, P.lon2, P.s12, P.azi02,

801 name=self.nearestOn4.__name__)

802 r._iteration = P.iteration

803 return r

804

805 @deprecated_method

806 def NearestOn(self, lat0, lon0, **exact_eps_est_tol): # PYCHOK no cover

807 '''DEPRECATED on 23.10.30, use method L{PlumbTo}.'''

808 return self.PlumbTo(lat0, lon0, **exact_eps_est_tol)

809

810 def PlumbTo(self, lat0, lon0, exact=None, eps=EPS, est=None, tol=_TOL):

811 '''Compute the I{perpendicular} intersection of this rumb line with a geodesic

812 from the given point, in part transcoded from I{Karney}'s C++ U{rhumb-intercept

813 <https://SourceForge.net/p/geographiclib/discussion/1026620/thread/2ddc295e/>}.

814

815 @arg lat0: Latitude of the point (C{degrees}).

816 @arg lon0: Longitude of the point (C{degrees}).

817 @kwarg exact: If C{None}, use a rhumb line perpendicular to this rhumb

818 line, otherwise use an I{exact} C{Geodesic...} from the

819 given point perpendicular to this rhumb line (C{bool} or

820 C{Geodesic...}), see method L{Ellipsoid.geodesic_}.

821 @kwarg eps: Optional tolerance for L{pygeodesy.intersection3d3} (C{EPS}),

822 used only if C{B{exact} is None}.

823 @kwarg est: Optional, initial estimate for the distance C{s12} of the

824 intersection I{along} this rhumb line (C{meter}), used only

825 if C{B{exact} is not None}.

826 @kwarg tol: Longitudinal convergence tolerance (C{degrees}) or distance

827 tolerance (C(meter)) when C{B{exact} is None}, respectively

828 C{not None}.

829

830 @return: The intersection point on this rhumb line, a L{GDict} from method

831 L{Intersection} if B{C{exact}=None}. If B{C{exact}} is not C{None},

832 a L{Position}-like L{GDict} of 13 items C{azi12, a12, s12, lat2,

833 lat1, lat0, lon2, lon1, lon0, azi0, a02, s02, at} with distance

834 C{a02} in C{degrees} and C{s02} in C{meter} between the given point

835 C{lat0, lon0} and the intersection C{lat2, lon2}, geodesic azimuth

836 C{azi0} at the given point and the (perpendicular) angle C{at}

837 between the geodesic and this rhumb line at the intersection. The

838 I{geodesic} azimuth at the intersection is C{(at + azi12)}. See

839 method L{Position} for further details.

840

841 @raise ImportError: I{Karney}'s U{geographiclib

842 <https://PyPI.org/project/geographiclib>}

843 package not found or not installed.

844

845 @raise IntersectionError: No convergence for this B{C{eps}} or no

846 intersection for some other reason.

847

848 @see: Methods C{distance2}, C{Intersecant2} and C{Intersection}

849 and function L{pygeodesy.intersection3d3}.

850 '''

851 Cs = Caps

852 if exact is None:

853 z = _norm180(self.azi12 + _90_0) # perpendicular azimuth

854 rl = RhumbLineBase(self.rhumb, lat0, lon0, z, caps=Cs.LINE_OFF)

855 P = self.Intersection(rl, tol=tol, eps=eps)

856

857 else: # C{rhumb-intercept}

858 E = self.ellipsoid

859 _gI = E.geodesic_(exact=exact).Inverse

860 gm = Cs.STANDARD | Cs._REDUCEDLENGTH_GEODESICSCALE # ^ Cs.DISTANCE_IN

861 if est is None: # get an estimate from the "perpendicular" geodesic

862 r = _gI(self.lat1, self.lon1, lat0, lon0, outmask=Cs.AZIMUTH_DISTANCE)

863 d, _ = _diff182(r.azi2, self.azi12, K_2_0=True)

864 _, s12 = sincos2d(d)

865 s12 *= r.s12 # signed

866 else:

867 s12 = Meter(est=est)

868 try:

869 tol = Float_(tol=tol, low=EPS, high=None)

870 # def _over(p, q): # see @note at method C{.Position}

871 # return (p / (q or _copysign(tol, q))) if isfinite(q) else NAN

872

873 _ErT = E.rocPrimeVertical # aka rocTransverse

874 _S12 = Fsum(s12).fsum2_

875 for i in range(1, _TRIPS): # suffix 1 == C++ 2, 2 == C++ 3

876 P = self.Position(s12) # outmask = Cs.LATITUDE_LONGITUDE

877 r = _gI(lat0, lon0, P.lat2, P.lon2, outmask=gm)

878 d, _ = _diff182(self.azi12, r.azi2, K_2_0=True)

879 s, c, s2, c2 = sincos2d_(d, r.lat2)

880 c2 *= _ErT(r.lat2)

881 s *= _over(s2 * self._salp, c2) - _over(s * r.M21, r.m12)

882 s12, t = _S12(c / s) # XXX _over?

883 if fabs(t) < tol: # or fabs(c) < EPS

884 break

885 P.set_(azi0=r.azi1, a02=r.a12, s02=r.s12, # azi2=r.azi2,

886 lat0=lat0, lon0=lon0, iteration=i, at=r.azi2 - self.azi12,

887 name=_dunder_nameof(self.PlumbTo, self.name))

888 except Exception as x: # Fsum(NAN) Value-, ZeroDivisionError

889 raise IntersectionError(lat0, lon0, tol=tol, exact=exact,

890 eps=eps, est=est, iteration=i, cause=x)

891

892 return P

893

894 def Position(self, s12, outmask=Caps.LATITUDE_LONGITUDE):

895 '''Compute a point at a given distance on this rhumb line.

896

897 @arg s12: The distance along this rhumb line from its origin to

898 the point (C{meters}), can be negative.

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

900 the quantities to be returned.

901

902 @return: L{GDict} with 4 to 8 items C{azi12, a12, s12, S12, lat2,

903 lat1, lon2, lon1} with latitude C{lat2} and longitude

904 C{lon2} of the point in C{degrees}, the rhumb angle C{a12}

905 in C{degrees} from the start point of and the area C{S12}

906 under this rhumb line in C{meter} I{squared}.

907

908 @raise ImportError: Package C{numpy} not found or not installed,

909 only required for L{RhumbLineAux} area C{S12}

910 when C{B{exact} is True}.

911

912 @note: If B{C{s12}} is large enough that the rhumb line crosses a

913 pole, the longitude of the second point is indeterminate and

914 C{NAN} is returned for C{lon2} and area C{S12}.

915

916 If the first point is a pole, the cosine of its latitude is

917 taken to be C{sqrt(L{EPS})}. This position is extremely

918 close to the actual pole and allows the calculation to be

919 carried out in finite terms.

920 '''

921 return self._Position(self.m2degrees(s12), s12, outmask)

922

923 def _Position(self, a12, s12, outmask):

924 '''(INTERNAL) C{Arc-/Position} helper.

925 '''

926 r = GDict(azi12=self.azi12, a12=a12, s12=s12, name=self.name)

927 Cs = Caps

928 if (outmask & Cs.LATITUDE_LONGITUDE_AREA):

929 if a12 or s12:

930 mu12 = self._calp * a12

931 mu2 = self._mu1 + mu12

932 if fabs(mu2) > 90: # past pole

933 mu2 = _norm180(mu2) # reduce to [-180, 180)

934 if fabs(mu2) > 90: # point on anti-meridian

935 mu2 = _norm180(_loneg(mu2))

936 lat2 = self._mu2lat(mu2)

937 lon2 = S12 = NAN

938 else:

939 lat2, lon2, S1, S2 = self._Position4(a12, mu2, s12, mu12)

940 if (outmask & Cs.AREA):

941 S12 = self.rhumb._S12d(S1, S2, lon2)

942 S12 = unsigned0(S12) # like .gx

943# else:

944# S12 = None # unused

945 if (outmask & Cs.LONGITUDE):

946 if (outmask & Cs.LONG_UNROLL):

947 lon2 += self.lon1

948 else:

949 lon2 = _norm180(self._lon12 + lon2)

950 else: # coincident

951 lat2, lon2 = self.latlon1

952 S12 = _0_0

953

954 if (outmask & Cs.AREA):

955 r.set_(S12=S12)

956 if (outmask & Cs.LATITUDE):

957 r.set_(lat2=lat2, lat1=self.lat1)

958 if (outmask & Cs.LONGITUDE):

959 r.set_(lon2=lon2, lon1=self.lon1)

960 return r

961

962 def _Position4(self, a12, mu2, s12, mu12): # PYCHOK no cover

963 '''(INTERNAL) I{Must be overloaded}.'''

964 _MODS.named.notOverloaded(self, a12, s12, mu2, mu12)

965

966 @Property_RO

967 def rhumb(self):

968 '''Get this rhumb line's rhumb (L{RhumbAux} or L{Rhumb}).

969 '''

970 return self._rhumb

971

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

973 '''Return this C{RhumbLine} as string.

974

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

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

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

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

979

980 @return: C{RhumbLine} (C{str}).

981 '''

982 d = dict(rhumb=self.rhumb, lat1=self.lat1, lon1=self.lon1,

983 azi12=self.azi12, exact=self.exact,

984 TMorder=self.TMorder, xTM=self.xTM)

985 return sep.join(pairs(itemsorted(d, asorted=False), prec=prec))

986

987 @property_RO

988 def TMorder(self):

989 '''Get this rhumb line's I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).

990 '''

991 return self.rhumb.TMorder

992

993 @Property_RO

994 def xTM(self):

995 '''Get this rhumb line's I{Transverse Mercator} projection (L{ExactTransverseMercator}

996 if I{exact} and I{ellipsoidal}, otherwise L{KTransverseMercator} for C{TMorder}).

997 '''

998 E = self.ellipsoid

999 # ExactTransverseMercator doesn't handle spherical earth models

1000 return _MODS.etm.ExactTransverseMercator(E) if self.exact and E.isEllipsoidal else \

1001 _MODS.ktm.KTransverseMercator(E, TMorder=self.TMorder)

1002

1003 def _xTM3d(self, latlon0, z=INT0, V3d=Vector3d):

1004 '''(INTERNAL) C{xTM.forward} this C{latlon1} to C{V3d} with B{C{latlon0}}

1005 as current intersection estimate and central meridian.

1006 '''

1007 t = self.xTM.forward(self.lat1 - latlon0.lat, self.lon1, lon0=latlon0.lon)

1008 return V3d(t.easting, t.northing, z)

1009

1010

1011 class _PseudoRhumbLine(RhumbLineBase):

1012 '''(INTERNAL) Pseudo-rhumb line for a geodesic (line), see C{geodesicw._PlumbTo}.

1013 '''

1014 def __init__(self, gl, name=NN):

1015 R = RhumbBase(gl.geodesic.ellipsoid, None, True, name)

1016 RhumbLineBase.__init__(self, R, gl.lat1, gl.lon1, 0, caps=Caps.LINE_OFF)

1017 self._azi1 = self.azi12 = gl.azi1

1018 self._gl = gl

1019 self._gD = gl.geodesic.Direct

1020

1021 def PlumbTo(self, lat0, lon0, **exact_eps_est_tol): # PYCHOK signature

1022 P = RhumbLineBase.PlumbTo(self, lat0, lon0, **exact_eps_est_tol)

1023 P.set_(azi1=self._gl.azi1, azi2=_xkwds_pop(P, azi12=None))

1024 return P # geodesic L{Position}

1025

1026 def Position(self, s12, **unused): # PYCHOK signature

1027 r = self._gD(self.lat1, self.lon1, self._azi1, s12)

1028 self._azi1 = r.azi1

1029 self.azi12 = z = r.azi2

1030 self._salp, _ = sincos2d(z)

1031 return r.set_(azi12=z)

1032

1033

1034__all__ += _ALL_DOCS(RhumbBase, RhumbLineBase)

1035

1036 if __name__ == '__main__':

1037

1038 from pygeodesy import printf, Rhumb as Rh, RhumbAux as Ah

1039 from pygeodesy.basics import _zip

1040 from pygeodesy.ellipsoids import _EWGS84

1041

1042 Al = Ah(_EWGS84).Line(30, 0, 45)

1043 Rl = Rh(_EWGS84).Line(30, 0, 45)

1044

1045 for i in range(1, 10):

1046 s = .5e6 + 1e6 / i

1047 a = Al.Position(s).lon2

1048 r = Rl.Position(s).lon2

1049 e = (fabs(a - r) / a) if a else 0

1050 printf('# Position.lon2 %.14f vs %.14f, diff %g', r, a, e)

1051

1052 for exact in (None, False, True):

1053 for est in (None, 1e6):

1054 a = Al.PlumbTo(60, 0, exact=exact, est=est)

1055 r = Rl.PlumbTo(60, 0, exact=exact, est=est)

1056 printf('# %s, iteration=%s, exact=%s, est=%s\n# %s, iteration=%s',

1057 a.toRepr(), a.iteration, exact, est,

1058 r.toRepr(), r.iteration, nl=1)

1059

1060 NE_=(71.688899882813, 0.2555198244234, 44095641862956.11)

1061 LHR=(77.7683897102557, 5771083.38332803, 37395209100030.39)

1062 NRT=(-92.38888798169965, 12782581.067684170, -63760642939072.50)

1063

1064 def _ref(fmt, r3, x3):

1065 e3 = []

1066 for r, x in _zip(r3, x3): # strict=True

1067 e = fabs(r - x) / fabs(x)

1068 e3.append('%.g' % (e,))

1069 printf((fmt % r3) + ', rel errors: ' + ', '.join(e3))

1070

1071 for R in (Ah, Rh): # <https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve -p 9> version 2.2

1072 rh = R(exact=True) # WGS84 default

1073 printf('# %r', rh, nl=1)

1074 r = rh.Direct8(40.6, -73.8, 51, 5.5e6) # from JFK about NE

1075 _ref('# JFK NE lat2=%.12f, lon2=%.12f, S12=%.1f', (r.lat2, r.lon2, r.S12), NE_)

1076 r = rh.Inverse8(40.6, -73.8, 51.6, -0.5) # JFK to LHR

1077 _ref('# JFK-LHR azi12=%.12f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), LHR)

1078 r = rh.Inverse8(40.6, -73.8, 35.8, 140.3) # JFK to Tokyo Narita

1079 _ref('# JFK-NRT azi12=%.12f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), NRT)

1080

1081# % python3 -m pygeodesy.rhumb.bases

1082

1083# Position.lon2 11.61455846901637 vs 11.61455846901637, diff 3.05885e-16

1084# Position.lon2 7.58982302826842 vs 7.58982302826842, diff 2.34045e-16

1085# Position.lon2 6.28526067416369 vs 6.28526067416369, diff 2.82623e-16

1086# Position.lon2 5.63938995325146 vs 5.63938995325146, diff 1.57495e-16

1087# Position.lon2 5.25385527435707 vs 5.25385527435707, diff 0

1088# Position.lon2 4.99764604290380 vs 4.99764604290380, diff 8.88597e-16

1089# Position.lon2 4.81503363740473 vs 4.81503363740473, diff 1.84459e-16

1090# Position.lon2 4.67828821748836 vs 4.67828821748835, diff 5.69553e-16

1091# Position.lon2 4.57205667906283 vs 4.57205667906283, diff 5.82787e-16

1092

1093# Intersection(a02=17.798332, a12=19.521356, at=90.0, azi02=135.0, azi12=45.0, lat0=60.0, lat1=30.0, lat2=45.0, lon0=0.0, lon1=0.0, lon2=15.830286, name='Intersection', s02=1977981.142985, s12=2169465.957531), iteration=9, exact=None, est=None

1094# Intersection(a02=17.798332, a12=19.521356, at=90.0, azi02=135.0, azi12=45.0, lat0=60.0, lat1=30.0, lat2=45.0, lon0=0.0, lon1=0.0, lon2=15.830286, name='Intersection', s02=1977981.142985, s12=2169465.957531), iteration=9

1095

1096# Intersection(a02=17.798332, a12=19.521356, at=90.0, azi02=135.0, azi12=45.0, lat0=60.0, lat1=30.0, lat2=45.0, lon0=0.0, lon1=0.0, lon2=15.830286, name='Intersection', s02=1977981.142985, s12=2169465.957531), iteration=9, exact=None, est=1000000.0

1097# Intersection(a02=17.798332, a12=19.521356, at=90.0, azi02=135.0, azi12=45.0, lat0=60.0, lat1=30.0, lat2=45.0, lon0=0.0, lon1=0.0, lon2=15.830286, name='Intersection', s02=1977981.142985, s12=2169465.957531), iteration=9

1098

1099# PlumbTo(a02=17.967658, a12=27.74256, at=90.0, azi0=113.73626, azi12=45.0, lat0=60, lat1=30.0, lat2=49.634582, lon0=0, lon1=0.0, lon2=25.767876, name='PlumbTo', s02=1997960.116871, s12=3083112.636236), iteration=5, exact=False, est=None

1100# PlumbTo(a02=17.967658, a12=27.74256, at=90.0, azi0=113.73626, azi12=45.0, lat0=60, lat1=30.0, lat2=49.634582, lon0=0, lon1=0.0, lon2=25.767876, name='PlumbTo', s02=1997960.116871, s12=3083112.636236), iteration=5

1101

1102# PlumbTo(a02=17.967658, a12=27.74256, at=90.0, azi0=113.73626, azi12=45.0, lat0=60, lat1=30.0, lat2=49.634582, lon0=0, lon1=0.0, lon2=25.767876, name='PlumbTo', s02=1997960.116871, s12=3083112.636236), iteration=7, exact=False, est=1000000.0

1103# PlumbTo(a02=17.967658, a12=27.74256, at=90.0, azi0=113.73626, azi12=45.0, lat0=60, lat1=30.0, lat2=49.634582, lon0=0, lon1=0.0, lon2=25.767876, name='PlumbTo', s02=1997960.116871, s12=3083112.636236), iteration=7

1104

1105# PlumbTo(a02=17.967658, a12=27.74256, at=90.0, azi0=113.73626, azi12=45.0, lat0=60, lat1=30.0, lat2=49.634582, lon0=0, lon1=0.0, lon2=25.767876, name='PlumbTo', s02=1997960.116871, s12=3083112.636236), iteration=5, exact=True, est=None

1106# PlumbTo(a02=17.967658, a12=27.74256, at=90.0, azi0=113.73626, azi12=45.0, lat0=60, lat1=30.0, lat2=49.634582, lon0=0, lon1=0.0, lon2=25.767876, name='PlumbTo', s02=1997960.116871, s12=3083112.636236), iteration=5

1107

1108# PlumbTo(a02=17.967658, a12=27.74256, at=90.0, azi0=113.73626, azi12=45.0, lat0=60, lat1=30.0, lat2=49.634582, lon0=0, lon1=0.0, lon2=25.767876, name='PlumbTo', s02=1997960.116871, s12=3083112.636236), iteration=7, exact=True, est=1000000.0

1109# PlumbTo(a02=17.967658, a12=27.74256, at=90.0, azi0=113.73626, azi12=45.0, lat0=60, lat1=30.0, lat2=49.634582, lon0=0, lon1=0.0, lon2=25.767876, name='PlumbTo', s02=1997960.116871, s12=3083112.636236), iteration=7

1110

1111# RhumbAux(RAorder=None, TMorder=6, ellipsoid=Ellipsoid(name='WGS84', a=6378137, b=6356752.31424518, f_=298.25722356, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e21=0.99330562, e22=0.0067395, e32=0.00335843, A=6367449.14582341, L=10001965.72931272, R1=6371008.77141506, R2=6371007.18091847, R3=6371000.79000916, Rbiaxial=6367453.63451633, Rtriaxial=6372797.5559594), exact=True)

1112# JFK NE lat2=71.688899882813, lon2=0.255519824423, S12=44095641862956.1, rel errors: 4e-16, 2e-13, 4e-16

1113# JFK-LHR azi12=77.768389710256, s12=5771083.383 S12=37395209100030.4, rel errors: 5e-16, 3e-16, 8e-16

1114# JFK-NRT azi12=-92.388887981700, s12=12782581.068 S12=-63760642939072.5, rel errors: 0, 1e-16, 7e-16

1115

1116# Rhumb(RAorder=6, TMorder=6, ellipsoid=Ellipsoid(name='WGS84', a=6378137, b=6356752.31424518, f_=298.25722356, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e21=0.99330562, e22=0.0067395, e32=0.00335843, A=6367449.14582341, L=10001965.72931272, R1=6371008.77141506, R2=6371007.18091847, R3=6371000.79000916, Rbiaxial=6367453.63451633, Rtriaxial=6372797.5559594), exact=True)

1117# JFK NE lat2=71.688899882813, lon2=0.255519824423, S12=44095641862956.1, rel errors: 2e-16, 1e-13, 5e-16

1118# JFK-LHR azi12=77.768389710256, s12=5771083.383 S12=37395209100030.4, rel errors: 4e-16, 3e-16, 6e-16

1119# JFK-NRT azi12=-92.388887981700, s12=12782581.068 S12=-63760642939072.5, rel errors: 0, 1e-16, 1e-16

1120

1121# **) MIT License

1122#

1123# Copyright (C) 2022-2024 -- mrJean1 at Gmail -- All Rights Reserved.

1124#

1125# Permission is hereby granted, free of charge, to any person obtaining a

1126# copy of this software and associated documentation files (the "Software"),

1127# to deal in the Software without restriction, including without limitation

1128# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1129# and/or sell copies of the Software, and to permit persons to whom the

1130# Software is furnished to do so, subject to the following conditions:

1131#

1132# The above copyright notice and this permission notice shall be included

1133# in all copies or substantial portions of the Software.

1134#

1135# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1136# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1137# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1138# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1139# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1140# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1141# OTHER DEALINGS IN THE SOFTWARE.