Coverage for pygeodesy/rhumbBase.py: 95%

345 statements  

« prev     ^ index     » next       coverage.py v7.2.2, created at 2023-10-11 16:04 -0400

1 

2# -*- coding: utf-8 -*- 

3 

4u'''Base classes C{RhumbBase} and C{RhumbLineBase}, pure Python version of I{Karney}'s C++ 

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

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

7from 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 

10Class L{RhumbLine} has been enhanced with methods C{Intersection} and C{NearestOn} to iteratively 

11find the intersection of two rhumb lines, respectively the nearest point on a rumb line along a 

12geodesic or perpendicular rhumb line from an other point. 

13 

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

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

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

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

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

19 

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

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

22''' 

23# make sure int/int division yields float quotient 

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

25 

26# from pygeodesy.basics import unsigned0, _xinstanceof # from .karney 

27from pygeodesy.constants import EPS, EPS1, INT0, _EPSqrt as _TOL, NAN, \ 

28 _0_0, _0_01, _1_0, _90_0, _over 

29# from pygeodesy.datums import _spherical_datum # from .formy 

30# from pygeodesy.ellipsoids import _EWGS84 # from .karney 

31from pygeodesy.errors import IntersectionError, itemsorted, RhumbError, \ 

32 _xdatum, _xkwds, _Xorder 

33# from pygeodesy.etm import ExactTransverseMercator # _MODS 

34from pygeodesy.fmath import euclid, favg, sqrt_a, fabs, Fsum 

35from pygeodesy.formy import opposing, _spherical_datum 

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

37from pygeodesy.interns import NN, _coincident_, _COMMASPACE_, _Dash, \ 

38 _no_, _parallel_, _too_, _under 

39from pygeodesy.karney import _atan2d, Caps, _CapsBase, _diff182, _fix90, \ 

40 GDict, _norm180, _EWGS84, unsigned0, _xinstanceof 

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

42from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS 

43# from pygeodesy.named import notOverloaded # _MODS 

44from pygeodesy.namedTuples import Distance2Tuple, LatLon2Tuple 

45from pygeodesy.props import deprecated_method, Property, Property_RO, \ 

46 property_RO, _update_all 

47from pygeodesy.streprs import Fmt, pairs, unstr 

48from pygeodesy.units import Float_, Lat, Lon, Meter, Radius_, Int # PYCHOK shared 

49from pygeodesy.utily import atan2d, _azireversed, _loneg, sincos2d, \ 

50 sincos2d_, _unrollon, _Wrap, sincos2_ # PYCHOK shared 

51from pygeodesy.vector3d import _intersect3d3, Vector3d # in .Intersection below 

52 

53# from math import fabs # from .fmath 

54 

55__all__ = () 

56__version__ = '23.10.10' 

57 

58_anti_ = _Dash('anti') 

59_rls = [] # instances of C{RbumbLine...} to be updated 

60_TRIPS = 65 # .Intersection, .NearestOn, 19+ 

61 

62 

63class _Lat(Lat): 

64 '''(INTERNAL) Latitude B{C{lat}}. 

65 ''' 

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

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

68 Lat.__new__(_Lat, *lat, **kwds) 

69 

70 

71class _Lon(Lon): 

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

73 ''' 

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

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

76 Lon.__new__(_Lon, *lon, **kwds) 

77 

78 

79def _update_all_rls(r): 

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

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

82 C{Rhumb} instance B{C{r}}. 

83 ''' 

84 # _xinstanceof(_MODS.rhumbaux.RhumbAux, _MODS.rhumbx.Rhumb, r=r) 

85 _update_all(r) 

86 for rl in _rls: # PYCHOK use weakref? 

87 if rl._rhumb is r: 

88 _update_all(rl) 

89 

90 

91class RhumbBase(_CapsBase): 

92 '''(INTERNAL) Base class for C{rhumbaux.RhumbAux} and C{rhumbx.Rhumb}. 

93 ''' 

94 _E = _EWGS84 

95 _exact = True 

96 _f_max = _0_01 

97 _mTM = 6 # see .TMorder 

98 

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

100 '''New C{rhumbaux.RhumbAux} or C{rhumbx.Rhum}. 

101 ''' 

102 if f is not None: 

103 self.ellipsoid = a_earth, f 

104 elif a_earth not in (_EWGS84, None): 

105 self.ellipsoid = a_earth 

106 if not exact: 

107 self.exact = False 

108 if name: 

109 self.name = name 

110 

111 @Property_RO 

112 def a(self): 

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

114 ''' 

115 return self.ellipsoid.a 

116 

117 equatoradius = a 

118 

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

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

121 

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

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

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

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

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

127 

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

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

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

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

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

133 

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

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

136 is True} and L{RhumbAux}. 

137 

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

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

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

141 

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

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

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

145 carried out in finite terms. 

146 ''' 

147 s12 = a12 * self._mpd 

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

149 

150 @Property_RO 

151 def b(self): 

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

153 ''' 

154 return self.ellipsoid.b 

155 

156 polaradius = b 

157 

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

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

160 ''' 

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

162 

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

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

165 

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

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

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

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

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

171 

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

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

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

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

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

177 

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

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

180 is True} and L{RhumbAux}. 

181 

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

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

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

185 

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

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

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

189 carried out in finite terms. 

190 ''' 

191 a12 = _over(s12, self._mpd) 

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

193 

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

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

196 ''' 

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

198 

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

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

201 ''' 

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

203 

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

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

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

207 

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

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

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

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

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

213 L{Caps} values specifying the required capabilities. 

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

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

216 

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

218 C{.Position} to compute each point. 

219 

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

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

222 ''' 

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

224 

225 Line = DirectLine # synonyms 

226 

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

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

229 ''' 

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

231 name=self.name) 

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

233 

234 @Property 

235 def ellipsoid(self): 

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

237 ''' 

238 return self._E 

239 

240 @ellipsoid.setter # PYCHOK setter! 

241 def ellipsoid(self, a_earth_f): 

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

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

244 

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

246 ''' 

247 E = _spherical_datum(a_earth_f, Error=RhumbError).ellipsoid 

248 if self._E != E: 

249 self._exactest(self.exact, E, self.f_max) 

250 _update_all_rls(self) 

251 self._E = E 

252 

253 @Property 

254 def exact(self): 

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

256 ''' 

257 return self._exact 

258 

259 @exact.setter # PYCHOK setter! 

260 def exact(self, exact): 

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

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

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

264 

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

266 exceeds non-zero C{f_max}. 

267 

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

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

270 ''' 

271 x = bool(exact) 

272 if self._exact != x: 

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

274 _update_all_rls(self) 

275 self._exact = x 

276 

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

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

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

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

281 

282 @Property_RO 

283 def f(self): 

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

285 ''' 

286 return self.ellipsoid.f 

287 

288 flattening = f 

289 

290 @property 

291 def f_max(self): 

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

293 ''' 

294 return self._f_max 

295 

296 @f_max.setter # PYCHOK setter! 

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

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

299 

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

301 exceeds non-zero C{f_max}. 

302 ''' 

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

304 if self._f_max != f: 

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

306 self._f_max = f 

307 

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

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

310 ''' 

311 if wrap: 

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

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

314 

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

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

317 

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

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

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

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

322 

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

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

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

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

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

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

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

330 I{squared}. 

331 

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

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

334 when C{B{exact} is True}. 

335 

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

337 on opposite meridians, there are two shortest rhumb lines 

338 and the East-going one is chosen. 

339 

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

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

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

343 carried out in finite terms. 

344 ''' 

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

346 Cs = Caps 

347 if (outmask & Cs.AZIMUTH_DISTANCE_AREA): 

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

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

350 if (outmask & Cs.AZIMUTH): 

351 z = _atan2d(y, x) 

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

353 if (outmask & Cs.AREA): 

354 S12 = self._S12d(s1, s2, lon12) 

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

356 return r 

357 

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

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

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

361 

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

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

364 ''' 

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

366 

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

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

369 ''' 

370 if wrap: 

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

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

373 

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

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

376 

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

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

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

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

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

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

383 L{Caps} values specifying the required capabilities. 

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

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

386 

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

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

389 

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

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

392 ''' 

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

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

395 

396 @Property_RO 

397 def _mpd(self): # PYCHOK no cover 

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

399 _MODS.named.notOverloaded(self) 

400 

401 @property_RO 

402 def _RhumbLine(self): # PYCHOK no cover 

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

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

405 

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

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

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

409 

410 @Property 

411 def TMorder(self): 

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

413 ''' 

414 return self._mTM 

415 

416 @TMorder.setter # PYCHOK setter! 

417 def TMorder(self, order): 

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

419 

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

421 for L{Rhumb} instances. 

422 ''' 

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

424 if self._mTM != m: 

425 _update_all_rls(self) 

426 self._mTM = m 

427 if self.exact and isinstance(self, _MODS.rhumbx.Rhumb): 

428 self.exact = False 

429 

430 def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK no cover 

431 '''I{Must be overloaded}.''' 

432 _MODS.named.notOverloaded(self, prec=prec, sep=sep) 

433 

434 

435class RhumbLineBase(_CapsBase): 

436 '''(INTERNAL) Base class for C{rhumbaux.RhumbLineAux} and C{rhumbx.RhumbLine}. 

437 ''' 

438 _azi12 = _0_0 

439 _calp = _1_0 

440# _caps = 0 

441# _debug = 0 

442# _lat1 = _0_0 

443# _lon1 = _0_0 

444# _lon12 = _0_0 

445 _Rhumb = RhumbBase # compatible C{Rhumb} class 

446 _rhumb = None # C{Rhumb} instance 

447 _salp = _0_0 

448 _talp = _0_0 

449 

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

451 '''New C{RhumbLine}. 

452 ''' 

453 _xinstanceof(self._Rhumb, rhumb=rhumb) 

454 

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

456 self._lon1 = _Lon(lon1= lon1) 

457 self._lon12 = _norm180(self._lon1) 

458 if azi12: # non-zero, non-None 

459 self.azi12 = _norm180(azi12) 

460 

461 n = name or rhumb.name 

462 if n: 

463 self.name=n 

464 

465 self._caps = caps 

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

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

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

469 else: 

470 self._rhumb = rhumb 

471 _rls.append(self) 

472 

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

474 if _rls: # may be empty or None 

475 try: # PYCHOK no cover 

476 _rls.remove(self) 

477 except (TypeError, ValueError): 

478 pass 

479 self._rhumb = None 

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

481 

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

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

484 

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

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

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

488 the quantities to be returned. 

489 

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

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

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

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

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

495 

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

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

498 when C{B{exact} is True}. 

499 

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

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

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

503 

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

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

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

507 carried out in finite terms. 

508 ''' 

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

510 

511 @Property 

512 def azi12(self): 

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

514 ''' 

515 return self._azi12 

516 

517 @azi12.setter # PYCHOK setter! 

518 def azi12(self, azi12): 

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

520 ''' 

521 z = _norm180(azi12) 

522 if self._azi12 != z: 

523 if self._rhumb: 

524 _update_all(self) 

525 self._azi12 = z 

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

527 self._talp = _over(*t) 

528 

529 @property_RO 

530 def azi12_sincos2(self): # PYCHOK no cover 

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

532 ''' 

533 return self._scalp, self._calp 

534 

535 def degrees2m(self, angle): 

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

537 

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

539 

540 @return: Distance (C{meter}). 

541 ''' 

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

543 

544 @deprecated_method 

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

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

547 

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

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

550 ''' 

551 r = self.Inverse(lat, lon) 

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

553 

554 @property_RO 

555 def ellipsoid(self): 

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

557 ''' 

558 return self.rhumb.ellipsoid 

559 

560 @property_RO 

561 def exact(self): 

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

563 ''' 

564 return self.rhumb.exact 

565 

566 def Intersecant2(self, lat0, lon0, radius, **tol_eps): 

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

568 

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

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

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

572 @kwarg tol_eps: Optional keyword arguments, see method 

573 method L{Intersection} for further details. 

574 

575 @return: 2-Tuple C{(P, Q)} with the 2 intersections 

576 (representing a chord), each a L{GDict} from method 

577 L{Intersection} extended to 14 items with C{azi03, 

578 a03, s03, lat3, lon3} for the chord center C{lat3, 

579 lon3} and the angle C{a03}, distance C{s03}, azimuth 

580 C{azi03} and start point C{lat0, lon0} of the rhumb 

581 line perpendicular to this rhumb line. If this 

582 rhumb line is tangential to the circle, both points 

583 are the same L{GDict} instance with rhumb distances 

584 C{s02} and C{s03} near-equal to the B{C{radius}}. 

585 

586 @raise IntersectionError: The circle and this rhumb line 

587 do not intersect. 

588 

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

590 ''' 

591 r = Radius_(radius) 

592 p = q = self.NearestOn(lat0, lon0, exact=None, **tol_eps) 

593 d = q.s02 

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

595 if d < r: 

596 a = sqrt_a(r, d) 

597 s = q.s12 

598 i = q.iteration 

599 n = self.name or self.Intersecant2.__name__ 

600 t.update(lat0=q.lat1, lon0=q.lon1) # or lat0, lon0 

601 p = self.Position(s + a).set_(name=n, **t) 

602 q = self.Position(s - a).set_(name=n, **t) 

603 p._iteration = q._iteration = i 

604 elif d > r: 

605 t = unstr(self.Intersecant2, lat0, lon0, radius=radius, **tol_eps) 

606 raise IntersectionError(_no_(t), txt=_too_(Fmt.distant(d))) 

607 else: # tangential 

608 q.set_(**t) # == p.set(_**t) 

609 return p, q 

610 

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

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

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

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

615 r._iteration = p.iteration 

616 return r 

617 

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

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

620 

621 @arg other: The other rhumb line (C{RhumbLine}) or C{None} for a 

622 rhumb line perpendicular to this one. 

623 @kwarg tol: Tolerance for longitudinal convergence and parallel 

624 error (C{degrees}). 

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

626 

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

628 12 items C{azi12, a12, s12, lat2, lat1, lat0, lon1, lon2, 

629 lon0, azi02, a02, s02} with C{a02} and C{s02} the rhumb 

630 angle and rhumb distance between the intersection C{lat2, 

631 lon2} point and the start point C{lat0, lon0} on the 

632 B{C{other}} rhumb line and the latter's azimuth C{azi02}. 

633 See method L{Position} for further details. 

634 

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

636 no intersection for an other reason. 

637 

638 @see: Methods C{distance2} and C{NearestOn} and function 

639 L{pygeodesy.intersection3d3}. 

640 

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

642 L{ExactTransverseMercator} or L{KTransverseMercator} 

643 projection and function L{pygeodesy.intersection3d3} in 

644 that domain. 

645 ''' 

646 _xinstanceof(RhumbLineBase, other=other) 

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

648 try: 

649 if self.others(other) is self: 

650 raise ValueError(_coincident_) 

651 # make invariants and globals locals 

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

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

654 p = opposing(s_az, o_az, margin=tol) 

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

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

657 _diff = euclid # approximate length 

658 _i3d3 = _intersect3d3 # NOT .vector3d.intersection3d3 

659 _LL2T = LatLon2Tuple 

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

661 # use halfway point as initial estimate 

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

663 favg(self.lon1, other.lon1)) 

664 for i in range(1, _TRIPS): 

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

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

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

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

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

670 if d < tol: # 19 trips 

671 lat, lon = p.lat, p.lon # PYCHOK p... 

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

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

674 p = GDict(azi12=self.azi12, a12=r.a12, s12=r.s12, 

675 lat2=lat, lat1=self.lat1, lat0=other.lat1, 

676 lon2=lon, lon1=self.lon1, lon0=other.lon1, 

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

678 name=self.name or self.Intersection.__name__) 

679 p._iteration = i 

680 return p 

681 else: 

682 raise ValueError(Fmt.no_convergence(d)) 

683 

684 except Exception as x: 

685 t = unstr(self.Intersection, other, tol=tol, eps=eps) 

686 raise IntersectionError(_no_(t), cause=x) 

687 

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

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

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

691 

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

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

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

695 and B{C{lon2}} (C{bool}). 

696 

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

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

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

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

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

702 ''' 

703 if wrap: 

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

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

706 return r 

707 

708 @Property_RO 

709 def isLoxodrome(self): 

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

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

712 

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

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

715 ''' 

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

717 

718 @Property_RO 

719 def lat1(self): 

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

721 ''' 

722 return self._lat1 

723 

724 @Property_RO 

725 def lon1(self): 

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

727 ''' 

728 return self._lon1 

729 

730 @Property_RO 

731 def latlon1(self): 

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

733 ''' 

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

735 

736 def m2degrees(self, distance): 

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

738 

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

740 

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

742 ''' 

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

744 

745 @property_RO 

746 def _mu1(self): # PYCHOK no cover 

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

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

749 

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

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

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

753 

754 @deprecated_method 

755 def nearestOn4(self, lat, lon, **exact_eps_est_tol): 

756 '''DEPRECATED on 23.10.10, use method L{NearestOn}.''' 

757 p = self.NearestOn(lat, lon, **exact_eps_est_tol) 

758 r = _MODS.deprecated.NearestOn4Tuple(p.lat2, p.lon2, p.s12, p.azi02, 

759 name=self.nearestOn4.__name__) 

760 r._iteration = p.iteration 

761 return r 

762 

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

764 '''I{Iteratively} locate the point on this rhumb line nearest to the 

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

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

767 

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

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

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

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

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

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

774 @kwarg eps: Optional tolerance for L{pygeodesy.intersection3d3} 

775 (C{EPS}), used only if C{B{exact} is None}. 

776 @kwarg est: Optional, initial estimate for the distance C{s13} of 

777 the intersection I{along} this rhumb line (C{meter}), 

778 used only if C{B{exact} is not None}. 

779 @kwarg tol: Longitudinal convergence tolerance (C{degrees}) or the 

780 distance tolerance (C(meter)) when C{B{exact} is None}, 

781 respectively C{not None}. 

782 

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

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

785 C{None}, a L{GDict} of 13 items C{azi12, a12, s12, lat2, lat1, 

786 lon2, lon1, a02, s02, azi0, azi2, lat0, lon0} with C{a02} and 

787 C{s02} the angle respectively distance from the given C{lat0, 

788 lon0} to the nearest C{lat2, lon2} point along the specified 

789 C{geodetic} and C{azi0} and C{azi2} the (forward) azimuth at 

790 the given respectively the nearest point. The latter is always 

791 perpendicular to this rhumb line's azimuth. See method 

792 L{Position} for further details. 

793 

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

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

796 package not found or not installed. 

797 

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

799 no intersection for an other reason. 

800 

801 @note: The (forward) azimuth C{azi2} at the nearest point is 

802 always perpendicular to this rhumb line's azimuth. 

803 

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

805 and function L{pygeodesy.intersection3d3}. 

806 ''' 

807 Cs = Caps 

808 if exact is None: 

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

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

811 p = self.Intersection(rl, tol=tol, eps=eps) 

812 

813 else: # C{rhumb-intercept} 

814 azi = self.azi12 

815 szi = self._salp 

816 E = self.ellipsoid 

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

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

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

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

821 d, _ = _diff182(r.azi2, azi, K_2_0=True) 

822 _, s12 = sincos2d(d) 

823 s12 *= r.s12 # signed 

824 else: 

825 s12 = Meter(est=est) 

826 try: 

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

828 # def _over(p, q): # see @note at RhumbLine[Aux].Position 

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

830 

831 _ErT = E.rocPrimeVertical # aka rocTransverse 

832 _S12 = Fsum(s12).fsum2_ 

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

834 p = self.Position(s12) # outmask = Cs.LATITUDE_LONGITUDE 

835 r = _gI(lat0, lon0, p.lat2, p.lon2, outmask=gm) 

836 d, _ = _diff182(azi, r.azi2, K_2_0=True) 

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

838 c2 *= _ErT(r.lat2) 

839 s *= _over(s2 * szi, c2) - _over(s * r.M21, r.m12) 

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

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

842 break 

843 p.set_(azi0=r.azi1, azi2=r.azi2, a02=r.a12, s02=r.s12, 

844 lat0=lat0, lon0=lon0, name=self.name or 

845 self.NearestOn.__name__) 

846 p._iteration = i 

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

848 t = unstr(self.NearestOn, lat0, lon0, tol=tol, exact=exact, 

849 iteration=i, eps=eps, est=est) 

850 raise IntersectionError(_no_(t), cause=x) 

851 

852 return p 

853 

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

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

856 

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

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

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

860 the quantities to be returned. 

861 

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

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

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

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

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

867 

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

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

870 when C{B{exact} is True}. 

871 

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

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

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

875 

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

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

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

879 carried out in finite terms. 

880 ''' 

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

882 

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

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

885 ''' 

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

887 Cs = Caps 

888 if (outmask & Cs.LATITUDE_LONGITUDE_AREA): 

889 if a12 or s12: 

890 mu12 = self._calp * a12 

891 mu2 = self._mu1 + mu12 

892 if fabs(mu2) > 90: # past pole 

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

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

895 mu2 = _norm180(_loneg(mu2)) 

896 lat2 = self._mu2lat(mu2) 

897 lon2 = S12 = NAN 

898 else: 

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

900 if (outmask & Cs.AREA): 

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

902 S12 = unsigned0(S12) # like .gx 

903# else: 

904# S12 = None # unused 

905 if (outmask & Cs.LONGITUDE): 

906 if (outmask & Cs.LONG_UNROLL): 

907 lon2 += self.lon1 

908 else: 

909 lon2 = _norm180(self._lon12 + lon2) 

910 else: # coincident 

911 lat2, lon2 = self.latlon1 

912 S12 = _0_0 

913 

914 if (outmask & Cs.AREA): 

915 r.set_(S12=S12) 

916 if (outmask & Cs.LATITUDE): 

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

918 if (outmask & Cs.LONGITUDE): 

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

920 return r 

921 

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

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

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

925 

926 @Property_RO 

927 def rhumb(self): 

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

929 ''' 

930 return self._rhumb 

931 

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

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

934 

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

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

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

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

939 

940 @return: C{RhumbLine} (C{str}). 

941 ''' 

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

943 azi12=self.azi12, exact=self.exact, 

944 TMorder=self.TMorder, xTM=self.xTM) 

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

946 

947 @property_RO 

948 def TMorder(self): 

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

950 ''' 

951 return self.rhumb.TMorder 

952 

953 @Property_RO 

954 def xTM(self): 

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

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

957 ''' 

958 E = self.ellipsoid 

959 # ExactTransverseMercator doesn't handle spherical earth models 

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

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

962 

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

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

965 as current intersection estimate and central meridian. 

966 ''' 

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

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

969 

970 

971__all__ += _ALL_DOCS(RhumbBase, RhumbLineBase) 

972 

973if __name__ == '__main__': 

974 

975 from pygeodesy import printf, Rhumb as R, RhumbAux as A 

976 

977 A = A(_EWGS84).Line(30, 0, 45) 

978 R = R(_EWGS84).Line(30, 0, 45) 

979 

980 for i in range(1, 10): 

981 s = .5e6 + 1e6 / i 

982 a = A.Position(s).lon2 

983 r = R.Position(s).lon2 

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

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

986 

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

988 for est in (None, 1e6): 

989 a = A.NearestOn(60, 0, exact=exact, est=est) 

990 r = R.NearestOn(60, 0, exact=exact, est=est) 

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

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

993 r.toRepr(), r.iteration, nl=1) 

994 

995# % python3 -m pygeodesy.rhumbBase 

996 

997# Position.lon2 11.61455846901637 vs 11.61455846901637, diff 6.11769e-16 

998# Position.lon2 7.58982302826842 vs 7.58982302826842, diff 1.17022e-16 

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

1000# Position.lon2 5.63938995325146 vs 5.63938995325146, diff 4.72486e-16 

1001# Position.lon2 5.25385527435707 vs 5.25385527435707, diff 1.69053e-16 

1002# Position.lon2 4.99764604290380 vs 4.99764604290380, diff 3.55439e-16 

1003# Position.lon2 4.81503363740473 vs 4.81503363740473, diff 1.84459e-16 

1004# Position.lon2 4.67828821748836 vs 4.67828821748835, diff 5.69553e-16 

1005# Position.lon2 4.57205667906283 vs 4.57205667906283, diff 1.94262e-16 

1006 

1007# Intersection(a02=17.798332, a12=19.521356, azi02=135.0, azi12=45.0, lat1=30.0, lat2=45.0, lon1=0.0, lon2=15.830286, name='Intersection', s02=1977981.142985, s12=2169465.957531), iteration=9, exact=None, est=None 

1008# Intersection(a02=17.798332, a12=19.521356, azi02=135.0, azi12=45.0, lat1=30.0, lat2=45.0, lon1=0.0, lon2=15.830286, name='Intersection', s02=1977981.142985, s12=2169465.957531), iteration=9 

1009 

1010# Intersection(a02=17.798332, a12=19.521356, azi02=135.0, azi12=45.0, lat1=30.0, lat2=45.0, lon1=0.0, lon2=15.830286, name='Intersection', s02=1977981.142985, s12=2169465.957531), iteration=9, exact=None, est=1000000.0 

1011# Intersection(a02=17.798332, a12=19.521356, azi02=135.0, azi12=45.0, lat1=30.0, lat2=45.0, lon1=0.0, lon2=15.830286, name='Intersection', s02=1977981.142985, s12=2169465.957531), iteration=9 

1012 

1013# NearestOn(a02=17.978107, a12=27.74256, azi0=113.73626, azi12=45.0, azi2=135.0, lat1=30.0, lat2=49.634582, lon1=0.0, lon2=25.767876, name='NearestOn', s02=1997960.116871, s12=3083112.636236), iteration=5, exact=False, est=None 

1014# NearestOn(a02=17.978107, a12=27.74256, azi0=113.73626, azi12=45.0, azi2=135.0, lat1=30.0, lat2=49.634582, lon1=0.0, lon2=25.767876, name='NearestOn', s02=1997960.116871, s12=3083112.636236), iteration=5 

1015 

1016# NearestOn(a02=17.978107, a12=27.74256, azi0=113.73626, azi12=45.0, azi2=135.0, lat1=30.0, lat2=49.634582, lon1=0.0, lon2=25.767876, name='NearestOn', s02=1997960.116871, s12=3083112.636236), iteration=7, exact=False, est=1000000.0 

1017# NearestOn(a02=17.978107, a12=27.74256, azi0=113.73626, azi12=45.0, azi2=135.0, lat1=30.0, lat2=49.634582, lon1=0.0, lon2=25.767876, name='NearestOn', s02=1997960.116871, s12=3083112.636236), iteration=7 

1018 

1019# NearestOn(a02=17.978107, a12=27.74256, azi0=113.73626, azi12=45.0, azi2=135.0, lat1=30.0, lat2=49.634582, lon1=0.0, lon2=25.767876, name='NearestOn', s02=1997960.116871, s12=3083112.636236), iteration=5, exact=True, est=None 

1020# NearestOn(a02=17.978107, a12=27.74256, azi0=113.73626, azi12=45.0, azi2=135.0, lat1=30.0, lat2=49.634582, lon1=0.0, lon2=25.767876, name='NearestOn', s02=1997960.116871, s12=3083112.636236), iteration=5 

1021 

1022# NearestOn(a02=17.978107, a12=27.74256, azi0=113.73626, azi12=45.0, azi2=135.0, lat1=30.0, lat2=49.634582, lon1=0.0, lon2=25.767876, name='NearestOn', s02=1997960.116871, s12=3083112.636236), iteration=7, exact=True, est=1000000.0 

1023# NearestOn(a02=17.978107, a12=27.74256, azi0=113.73626, azi12=45.0, azi2=135.0, lat1=30.0, lat2=49.634582, lon1=0.0, lon2=25.767876, name='NearestOn', s02=1997960.116871, s12=3083112.636236), iteration=7 

1024 

1025# **) MIT License 

1026# 

1027# Copyright (C) 2022-2023 -- mrJean1 at Gmail -- All Rights Reserved. 

1028# 

1029# Permission is hereby granted, free of charge, to any person obtaining a 

1030# copy of this software and associated documentation files (the "Software"), 

1031# to deal in the Software without restriction, including without limitation 

1032# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1033# and/or sell copies of the Software, and to permit persons to whom the 

1034# Software is furnished to do so, subject to the following conditions: 

1035# 

1036# The above copyright notice and this permission notice shall be included 

1037# in all copies or substantial portions of the Software. 

1038# 

1039# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1040# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1041# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1042# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1043# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1044# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1045# OTHER DEALINGS IN THE SOFTWARE.