Coverage for pygeodesy/rhumb/bases.py: 94%

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1 

2# -*- coding: utf-8 -*- 

3 

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

5C++ classes 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{RhumbLineBase} has been enhanced with methods C{Intersecant2}, C{Intersection} and C{PlumbTo} 

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

12a perpendicular geodesic or other rhumb line. 

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>} (2014-2024) 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 

26from pygeodesy.basics import _copysign, itemsorted, unsigned0, _xinstanceof 

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

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

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

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

31 _xkwds, _xkwds_pop2, _Xorder 

32# from pygeodesy.etm import ExactTransverseMercator # _MODS 

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

34# from pygeodesy.formy import opposing # _MODS 

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

36from pygeodesy.internals import _DUNDER_nameof, _under 

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

38 _parallel_, _too_ 

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

40 _norm180, GDict 

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

42from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS 

43from pygeodesy.namedTuples import Distance2Tuple, LatLon2Tuple 

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

45 property_RO, _update_all 

46from pygeodesy.streprs import Fmt, pairs 

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

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

49 _unrollon, _Wrap 

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

51 

52from math import cos, fabs 

53 

54__all__ = () 

55__version__ = '24.10.14' 

56 

57_anti_ = _Dash('anti') 

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

59_TRIPS = 129 # .Intersection, .PlumbTo, 19+ 

60 

61 

62class _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 

70class _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 

78def _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 

90class 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, TMorder_name): 

99 '''New C{RhumbAux} or C{Rhumb}. 

100 ''' 

101 if TMorder_name: 

102 M = self._mTM 

103 m, name = _xkwds_pop2(TMorder_name, TMorder=M) 

104 if m != M: 

105 self.TMorder = m 

106 else: 

107 name = {} 

108 _earth_datum(self, a_earth, f=f, **name) 

109 if not exact: 

110 self.exact = False 

111 if name: 

112 self.name = name 

113 

114 @Property_RO 

115 def a(self): 

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

117 ''' 

118 return self.ellipsoid.a 

119 

120 equatoradius = a 

121 

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

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

124 

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

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

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

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

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

130 

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

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

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

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

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

136 

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

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

139 is True} and L{RhumbAux}. 

140 

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

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

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

144 

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

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

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

148 carried out in finite terms. 

149 ''' 

150 s12 = a12 * self._mpd 

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

152 

153 @Property_RO 

154 def b(self): 

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

156 ''' 

157 return self.ellipsoid.b 

158 

159 polaradius = b 

160 

161 @property 

162 def datum(self): 

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

164 ''' 

165 return self._datum 

166 

167 @datum.setter # PYCHOK setter! 

168 def datum(self, datum): 

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

170 

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

172 ''' 

173 _xinstanceof(Datum, datum=datum) 

174 if self._datum != datum: 

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

176 _update_all_rls(self) 

177 self._datum = datum 

178 

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

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

181 ''' 

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

183 

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

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

186 

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

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

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

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

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

192 

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

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

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

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

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

198 

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

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

201 is True} and L{RhumbAux}. 

202 

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

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

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

206 

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

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

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

210 carried out in finite terms. 

211 ''' 

212 a12 = _over(s12, self._mpd) 

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

214 

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

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

217 ''' 

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

219 

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

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

222 ''' 

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

224 

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

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

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

228 

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

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

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

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

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

234 L{Caps} values specifying the required capabilities. 

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

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

237 

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

239 C{.Position} to compute each point. 

240 

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

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

243 ''' 

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

245 

246 Line = DirectLine # synonyms 

247 

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

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

250 ''' 

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

252 name=self.name) 

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

254 

255 @Property 

256 def ellipsoid(self): 

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

258 ''' 

259 return self.datum.ellipsoid 

260 

261 @ellipsoid.setter # PYCHOK setter! 

262 def ellipsoid(self, a_earth_f): 

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

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

265 

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

267 ''' 

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

269 

270 @Property 

271 def exact(self): 

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

273 ''' 

274 return self._exact 

275 

276 @exact.setter # PYCHOK setter! 

277 def exact(self, exact): 

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

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

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

281 

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

283 exceeds non-zero C{f_max}. 

284 

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

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

287 ''' 

288 x = bool(exact) 

289 if self._exact != x: 

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

291 _update_all_rls(self) 

292 self._exact = x 

293 

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

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

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

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

298 

299 @Property_RO 

300 def f(self): 

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

302 ''' 

303 return self.ellipsoid.f 

304 

305 flattening = f 

306 

307 @property 

308 def f_max(self): 

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

310 ''' 

311 return self._f_max 

312 

313 @f_max.setter # PYCHOK setter! 

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

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

316 

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

318 exceeds non-zero C{f_max}. 

319 ''' 

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

321 if self._f_max != f: 

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

323 self._f_max = f 

324 

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

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

327 ''' 

328 if wrap: 

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

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

331 

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

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

334 

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

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

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

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

339 

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

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

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

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

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

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

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

347 I{squared}. 

348 

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

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

351 when C{B{exact} is True}. 

352 

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

354 on opposite meridians, there are two shortest rhumb lines 

355 and the East-going one is chosen. 

356 

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

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

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

360 carried out in finite terms. 

361 ''' 

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

363 Cs = Caps 

364 if (outmask & Cs.AZIMUTH_DISTANCE_AREA): 

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

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

367 if (outmask & Cs.AZIMUTH): 

368 z = _atan2d(y, x) 

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

370 if (outmask & Cs.AREA): 

371 S12 = self._S12d(s1, s2, lon12) 

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

373 return r 

374 

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

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

377 self._notOverloaded(lon12, r, Caps.toStr(outmask)) # underOK=True 

378 

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

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

381 ''' 

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

383 

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

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

386 ''' 

387 if wrap: 

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

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

390 

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

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

393 

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

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

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

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

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

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

400 L{Caps} values specifying the required capabilities. 

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

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

403 

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

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

406 

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

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

409 ''' 

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

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

412 

413 @Property_RO 

414 def _mpd(self): # PYCHOK no cover 

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

416 _MODS.named.notOverloaded(self) 

417 

418 @property_RO 

419 def RAorder(self): 

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

421 ''' 

422 return None 

423 

424 @property_RO 

425 def _RhumbLine(self): # PYCHOK no cover 

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

427 self._notOverloaded(underOK=True) 

428 

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

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

431 self._notOverloaded(s1, s2, lon) # underOK=True 

432 

433 @Property 

434 def TMorder(self): 

435 '''Get the L{KTransverseMercator} order (C{int}, 4, 5, 6, 7 or 8). 

436 ''' 

437 return self._mTM 

438 

439 @TMorder.setter # PYCHOK setter! 

440 def TMorder(self, order): 

441 '''Set the L{KTransverseMercator} order (C{int}, 4, 5, 6, 7 or 8). 

442 

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

444 for L{Rhumb} instances. 

445 ''' 

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

447 if self._mTM != m: 

448 _update_all_rls(self) 

449 self._mTM = m 

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

451 self.exact = False 

452 

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

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

455 

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

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

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

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

460 

461 @return: Tuple items (C{str}). 

462 ''' 

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

464 exact=self.exact, TMorder=self.TMorder) 

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

466 

467 

468class RhumbLineBase(_CapsBase): 

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

470 ''' 

471 _azi12 = _0_0 

472 _calp = _1_0 

473# _caps = \ 

474# _debug = 0 

475# _lat1 = \ 

476# _lon1 = \ 

477# _lon12 = _0_0 

478 _Rhumb = RhumbBase # compatible C{Rhumb} class 

479 _rhumb = None # C{Rhumb} instance 

480 _salp = \ 

481 _talp = _0_0 

482 

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

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

485 ''' 

486 _xinstanceof(self._Rhumb, rhumb=rhumb) 

487 

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

489 self._lon1 = _Lon(lon1= lon1) 

490 self._lon12 = _norm180(self._lon1) 

491 if azi12: # non-zero, non-None 

492 self.azi12 = _norm180(azi12) 

493 

494 n = name or rhumb.name 

495 if n: 

496 self.name=n 

497 

498 self._caps = caps 

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

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

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

502 else: 

503 self._rhumb = rhumb 

504 _rls.append(self) 

505 

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

507 if _rls: # may be empty or None 

508 try: # PYCHOK no cover 

509 _rls.remove(self) 

510 except (TypeError, ValueError): 

511 pass 

512 self._rhumb = None 

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

514 

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

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

517 

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

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

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

521 the quantities to be returned. 

522 

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

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

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

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

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

528 

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

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

531 when C{B{exact} is True}. 

532 

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

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

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

536 

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

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

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

540 carried out in finite terms. 

541 ''' 

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

543 

544 @Property 

545 def azi12(self): 

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

547 ''' 

548 return self._azi12 

549 

550 @azi12.setter # PYCHOK setter! 

551 def azi12(self, azi12): 

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

553 ''' 

554 z = _norm180(azi12) 

555 if self._azi12 != z: 

556 if self._rhumb: 

557 _update_all(self) 

558 self._azi12 = z 

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

560 self._talp = _over(*t) 

561 

562 @property_RO 

563 def azi12_sincos2(self): # PYCHOK no cover 

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

565 ''' 

566 return self._scalp, self._calp 

567 

568 @property_RO 

569 def datum(self): 

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

571 ''' 

572 return self.rhumb.datum 

573 

574 def degrees2m(self, angle): 

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

576 

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

578 

579 @return: Distance (C{meter}). 

580 ''' 

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

582 

583 @deprecated_method 

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

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

586 

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

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

589 ''' 

590 r = self.Inverse(lat, lon) 

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

592 

593 @property_RO 

594 def ellipsoid(self): 

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

596 ''' 

597 return self.rhumb.ellipsoid 

598 

599 @property_RO 

600 def exact(self): 

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

602 ''' 

603 return self.rhumb.exact 

604 

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

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

607 

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

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

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

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

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

613 @kwarg tol_eps: Optional keyword arguments, see method 

614 method L{Intersection} for further details. 

615 

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

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

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

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

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

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

622 rhumb line is tangential to the circle, both points 

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

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

625 

626 @raise IntersectionError: The circle and this rhumb line 

627 do not intersect. 

628 

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

630 ''' 

631 r = Radius_(radius) 

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

633 a = q.s02 

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

635 if a < r: 

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

637 name=_DUNDER_nameof(self.Intersecant2, self.name)) 

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

639 d, h = _0_0, r 

640 else: 

641 d = q.s12 

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

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

644 m = self.rhumb._mpr 

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

646 else: 

647 h = _copysign(sqrt_a(r, a), a) 

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

649 if h: 

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

651 elif a > r: 

652 t = _too_(Fmt.distant(a)) 

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

654 txt=t, **tol_eps) 

655 else: # tangential 

656 q.set_(**t) # == p.set(_**t) 

657 return p, q 

658 

659 @deprecated_method 

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

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

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

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

664 r._iteration = p.iteration 

665 return r 

666 

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

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

669 

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

671 @kwarg tol: Tolerance for longitudinal convergence and parallel 

672 error (C{degrees}). 

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

674 

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

676 13 items C{lat1, lon1, azi12, a12, s12, lat2, lon2, lat0, 

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

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

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

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

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

682 See method L{Position} for further details. 

683 

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

685 no intersection for an other reason. 

686 

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

688 L{pygeodesy.intersection3d3}. 

689 

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

691 L{ExactTransverseMercator} or L{KTransverseMercator} 

692 projection and function L{pygeodesy.intersection3d3} in 

693 that domain. 

694 ''' 

695 _xinstanceof(RhumbLineBase, other=other) 

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

697 try: 

698 if self.others(other) is self: 

699 raise ValueError(_coincident_) 

700 # make invariants and globals locals 

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

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

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

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

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

706 _diff = euclid # approximate length 

707 _i3d3 = _intersect3d3 # NOT .vector3d.intersection3d3 

708 _LL2T = LatLon2Tuple 

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

710 # use halfway point as initial estimate 

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

712 favg(self.lon1, other.lon1)) 

713 for i in range(1, _TRIPS): 

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

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

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

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

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

719 if d < tol: # 19+ trips 

720 break 

721 else: 

722 raise ValueError(Fmt.no_convergence(d, tol)) 

723 

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

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

726 name=_DUNDER_nameof(self.Intersection, self.name)) 

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

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

729 P.set_(azi12= self.azi12, a12=r.a12, s12=r.s12, 

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

731 at=other.azi12 - self.azi12, iteration=i) 

732 except Exception as x: 

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

734 eps=eps, cause=x) 

735 return P 

736 

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

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

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

740 

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

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

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

744 and B{C{lon2}} (C{bool}). 

745 

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

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

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

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

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

751 ''' 

752 if wrap: 

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

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

755 return r 

756 

757 @Property_RO 

758 def isLoxodrome(self): 

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

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

761 

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

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

764 ''' 

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

766 

767 @Property_RO 

768 def lat1(self): 

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

770 ''' 

771 return self._lat1 

772 

773 @Property_RO 

774 def lon1(self): 

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

776 ''' 

777 return self._lon1 

778 

779 @Property_RO 

780 def latlon1(self): 

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

782 ''' 

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

784 

785 def m2degrees(self, distance): 

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

787 

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

789 

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

791 ''' 

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

793 

794 @property_RO 

795 def _mu1(self): # PYCHOK no cover 

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

797 self._notOverloaded(underOK=True) 

798 

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

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

801 self._notOverloaded(mu2) # underOK=True 

802 

803 @deprecated_method 

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

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

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

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

808 name=self.nearestOn4.__name__) 

809 r._iteration = P.iteration 

810 return r 

811 

812 @deprecated_method 

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

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

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

816 

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

818 '''Compute the I{perpendicular} intersection of this rhumb line with a geodesic 

819 from the given point (transcoded from I{Karney}'s C++ U{rhumb-intercept 

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

821 

822 @arg lat0: Latitude of the point on the geodesic (C{degrees}). 

823 @arg lon0: Longitude of the point on the geodesic (C{degrees}). 

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

825 otherwise use an I{exact} C{Geodesic...} from the given point 

826 perpendicular to this rhumb line (C{bool} or C{Geodesic...}), 

827 see method L{geodesic_<pygeodesy.Ellipsoid.geodesic_>}. 

828 @kwarg eps: Optional tolerance (C{EPS}), used only if C{B{exact} is None}, 

829 see function L{intersection3d3<pygeodesy.intersection3d3>}. 

830 @kwarg est: Optionally, an initial estimate for the distance C{s12} of the 

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

832 C{B{exact} is not None}. 

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

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

835 C{not None}. 

836 

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

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

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

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

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

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

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

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

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

846 method L{Position} for further details. 

847 

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

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

850 package not found or not installed. 

851 

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

853 no intersection for some other reason. 

854 

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

856 L{intersection3d3<pygeodesy.intersection3d3>}. 

857 ''' 

858 Cs, tol = Caps, Float_(tol=tol, low=EPS, high=None) 

859 

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

861# if p: 

862# p = (p / (q or _copysign(tol, q))) if isfinite(q) else NAN 

863# return p 

864 

865 if exact is None: 

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

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

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

869 

870 else: # C{rhumb-intercept} 

871 E = self.ellipsoid 

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

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

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

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

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

877 _, s12 = sincos2d(d) 

878 s12 *= r.s12 # signed 

879 else: 

880 s12 = Meter(est=est) 

881 try: 

882 _abs = fabs 

883 _d2 = _diff182 

884 _ErT = E.rocPrimeVertical # aka rocTransverse 

885 _ovr = _over 

886 _S12 = Fsum(s12).fsum2f_ 

887 _scd = sincos2d_ 

888 for i in range(1, _TRIPS): # 9+, suffix 1 == C++ 2, 2 == C++ 3 

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

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

891 d, _ = _d2(self.azi12, r.azi2, K_2_0=True) 

892 s, c, s2, c2 = _scd(d, r.lat2) 

893 c2 *= _ErT(r.lat2) 

894 s *= _ovr(s2 * self._salp, c2) - _ovr(s * r.M21, r.m12) 

895 s12, t = _S12(c / s) # XXX _ovr? 

896 if _abs(t) < tol: # or _abs(c) < EPS 

897 break 

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

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

900 name=_DUNDER_nameof(self.PlumbTo, self.name)) 

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

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

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

904 

905 return P 

906 

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

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

909 

910 @arg s12: The distance along this rhumb line from its origin to the point 

911 (C{meters}), can be negative. 

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

913 quantities to be returned. 

914 

915 @return: L{GDict} with 4 to 8 items C{azi12, a12, s12, S12, lat2, lat1, 

916 lon2, lon1} with latitude C{lat2} and longitude C{lon2} of the 

917 point in C{degrees}, the rhumb angle C{a12} in C{degrees} from 

918 the start point of and the area C{S12} under this rhumb line 

919 in C{meter} I{squared}. 

920 

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

922 only for L{RhumbLineAux} area C{S12} when C{B{exact} 

923 is True}. 

924 

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

926 longitude of the second point is indeterminate and C{NAN} is returned 

927 for C{lon2} and area C{S12}. 

928 

929 If the first point is a pole, the cosine of its latitude is taken to 

930 be C{sqrt(L{EPS})}. This position is extremely close to the actual 

931 pole and allows the calculation to be carried out in finite terms. 

932 ''' 

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

934 

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

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

937 ''' 

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

939 Cs = Caps 

940 if (outmask & Cs.LATITUDE_LONGITUDE_AREA): 

941 if a12 or s12: 

942 mu12 = self._calp * a12 

943 mu2 = self._mu1 + mu12 

944 if fabs(mu2) > 90: # past pole 

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

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

947 mu2 = _norm180(_loneg(mu2)) 

948 lat2 = self._mu2lat(mu2) 

949 lon2 = S12 = NAN 

950 else: 

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

952 if (outmask & Cs.AREA): 

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

954 S12 = unsigned0(S12) # like .gx 

955# else: 

956# S12 = None # unused 

957 if (outmask & Cs.LONGITUDE): 

958 if (outmask & Cs.LONG_UNROLL): 

959 lon2 += self.lon1 

960 else: 

961 lon2 = _norm180(self._lon12 + lon2) 

962 else: # coincident 

963 lat2, lon2 = self.latlon1 

964 S12 = _0_0 

965 

966 if (outmask & Cs.AREA): 

967 r.set_(S12=S12) 

968 if (outmask & Cs.LATITUDE): 

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

970 if (outmask & Cs.LONGITUDE): 

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

972 return r 

973 

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

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

976 self._notOverloaded(a12, s12, mu2, mu12) # underOK=True 

977 

978 @Property_RO 

979 def rhumb(self): 

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

981 ''' 

982 return self._rhumb 

983 

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

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

986 

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

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

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

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

991 

992 @return: C{RhumbLine} (C{str}). 

993 ''' 

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

995 azi12=self.azi12, exact=self.exact, 

996 TMorder=self.TMorder, xTM=self.xTM) 

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

998 

999 @property_RO 

1000 def TMorder(self): 

1001 '''Get this rhumb line's I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8). 

1002 ''' 

1003 return self.rhumb.TMorder 

1004 

1005 @Property_RO 

1006 def xTM(self): 

1007 '''Get this rhumb line's I{Transverse Mercator} projection (L{ExactTransverseMercator} 

1008 if I{exact} and I{ellipsoidal}, otherwise L{KTransverseMercator} for C{TMorder}). 

1009 ''' 

1010 E = self.ellipsoid 

1011 # ExactTransverseMercator doesn't handle spherical earth models 

1012 return _MODS.etm.ExactTransverseMercator(E) if self.exact and E.isEllipsoidal else \ 

1013 _MODS.ktm.KTransverseMercator(E, TMorder=self.TMorder) 

1014 

1015 def _xTM3d(self, latlon0, z=INT0, V3d=Vector3d): 

1016 '''(INTERNAL) C{xTM.forward} this C{latlon1} to C{V3d} with B{C{latlon0}} 

1017 as current intersection estimate and central meridian. 

1018 ''' 

1019 t = self.xTM.forward(self.lat1 - latlon0.lat, self.lon1, lon0=latlon0.lon) 

1020 return V3d(t.easting, t.northing, z) 

1021 

1022 

1023class _PseudoRhumbLine(RhumbLineBase): 

1024 '''(INTERNAL) Pseudo-rhumb line for a geodesic (line), see C{geodesicw._PlumbTo}. 

1025 ''' 

1026 def __init__(self, gl, name=NN): 

1027 R = RhumbBase(gl.geodesic.ellipsoid, None, True, name) 

1028 RhumbLineBase.__init__(self, R, gl.lat1, gl.lon1, 0, caps=Caps.LINE_OFF) 

1029 self._azi1 = self.azi12 = gl.azi1 

1030 self._gl = gl 

1031 self._gD = gl.geodesic.Direct 

1032 

1033 def PlumbTo(self, lat0, lon0, **exact_eps_est_tol): # PYCHOK signature 

1034 P = RhumbLineBase.PlumbTo(self, lat0, lon0, **exact_eps_est_tol) 

1035 z, P = _xkwds_pop2(P, azi12=None) 

1036 P.set_(azi1=self._gl.azi1, azi2=z) 

1037 return P # geodesic L{Position} 

1038 

1039 def Position(self, s12, **unused): # PYCHOK signature 

1040 r = self._gD(self.lat1, self.lon1, self._azi1, s12) 

1041 self._azi1 = r.azi1 

1042 self.azi12 = z = r.azi2 

1043 self._salp, _ = sincos2d(z) 

1044 return r.set_(azi12=z) 

1045 

1046 

1047__all__ += _ALL_DOCS(RhumbBase, RhumbLineBase) 

1048 

1049if __name__ == '__main__': 

1050 

1051 from pygeodesy import printf, Rhumb as Rh, RhumbAux as Ah 

1052 from pygeodesy.basics import _zip 

1053 from pygeodesy.ellipsoids import _EWGS84 

1054 

1055 Al = Ah(_EWGS84).Line(30, 0, 45) 

1056 Rl = Rh(_EWGS84).Line(30, 0, 45) 

1057 

1058 for i in range(1, 10): 

1059 s = .5e6 + 1e6 / i 

1060 a = Al.Position(s).lon2 

1061 r = Rl.Position(s).lon2 

1062 e = (fabs(a - r) / a) if a else 0 

1063 printf('# Position.lon2 %.14f vs %.14f, diff %g', r, a, e) 

1064 

1065 for exact in (None, False, True): 

1066 for est in (None, 1e6): 

1067 a = Al.PlumbTo(60, 0, exact=exact, est=est) 

1068 r = Rl.PlumbTo(60, 0, exact=exact, est=est) 

1069 printf('# %s, iteration=%s, exact=%s, est=%s\n# %s, iteration=%s', 

1070 a.toRepr(), a.iteration, exact, est, 

1071 r.toRepr(), r.iteration, nl=1) 

1072 

1073 NE_=(71.688899882813, 0.2555198244234, 44095641862956.11) 

1074 LHR=(77.7683897102557, 5771083.38332803, 37395209100030.39) 

1075 NRT=(-92.38888798169965, 12782581.067684170, -63760642939072.50) 

1076 

1077 def _ref(fmt, r3, x3): 

1078 e3 = [] 

1079 for r, x in _zip(r3, x3): # strict=True 

1080 e = fabs(r - x) / fabs(x) 

1081 e3.append('%.g' % (e,)) 

1082 printf((fmt % r3) + ', rel errors: ' + ', '.join(e3)) 

1083 

1084 for R in (Ah, Rh): # <https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve -p 9> version 2.2 

1085 rh = R(exact=True) # WGS84 default 

1086 printf('# %r', rh, nl=1) 

1087 r = rh.Direct8(40.6, -73.8, 51, 5.5e6) # from JFK about NE 

1088 _ref('# JFK NE lat2=%.12f, lon2=%.12f, S12=%.1f', (r.lat2, r.lon2, r.S12), NE_) 

1089 r = rh.Inverse8(40.6, -73.8, 51.6, -0.5) # JFK to LHR 

1090 _ref('# JFK-LHR azi12=%.12f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), LHR) 

1091 r = rh.Inverse8(40.6, -73.8, 35.8, 140.3) # JFK to Tokyo Narita 

1092 _ref('# JFK-NRT azi12=%.12f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), NRT) 

1093 

1094# % python3.10 -m pygeodesy3.rhumb.Bases 

1095 

1096# Position.lon2 11.61455846901637 vs 11.61455846901637, diff 3.05885e-16 

1097# Position.lon2 7.58982302826842 vs 7.58982302826842, diff 2.34045e-16 

1098# Position.lon2 6.28526067416369 vs 6.28526067416369, diff 2.82623e-16 

1099# Position.lon2 5.63938995325146 vs 5.63938995325146, diff 1.57495e-16 

1100# Position.lon2 5.25385527435707 vs 5.25385527435707, diff 0 

1101# Position.lon2 4.99764604290380 vs 4.99764604290380, diff 8.88597e-16 

1102# Position.lon2 4.81503363740473 vs 4.81503363740473, diff 1.84459e-16 

1103# Position.lon2 4.67828821748836 vs 4.67828821748835, diff 5.69553e-16 

1104# Position.lon2 4.57205667906283 vs 4.57205667906283, diff 5.82787e-16 

1105 

1106# 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 

1107# 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 

1108 

1109# 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 

1110# 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 

1111 

1112# 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 

1113# 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 

1114 

1115# 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 

1116# 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 

1117 

1118# 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 

1119# 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 

1120 

1121# 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 

1122# 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 

1123 

1124# 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) 

1125# JFK NE lat2=71.688899882813, lon2=0.255519824423, S12=44095641862956.1, rel errors: 4e-16, 2e-13, 4e-16 

1126# JFK-LHR azi12=77.768389710256, s12=5771083.383 S12=37395209100030.4, rel errors: 5e-16, 3e-16, 8e-16 

1127# JFK-NRT azi12=-92.388887981700, s12=12782581.068 S12=-63760642939072.5, rel errors: 0, 1e-16, 7e-16 

1128 

1129# 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) 

1130# JFK NE lat2=71.688899882813, lon2=0.255519824423, S12=44095641862956.1, rel errors: 2e-16, 1e-13, 5e-16 

1131# JFK-LHR azi12=77.768389710256, s12=5771083.383 S12=37395209100030.4, rel errors: 4e-16, 3e-16, 6e-16 

1132# JFK-NRT azi12=-92.388887981700, s12=12782581.068 S12=-63760642939072.5, rel errors: 0, 1e-16, 1e-16 

1133 

1134# **) MIT License 

1135# 

1136# Copyright (C) 2022-2025 -- mrJean1 at Gmail -- All Rights Reserved. 

1137# 

1138# Permission is hereby granted, free of charge, to any person obtaining a 

1139# copy of this software and associated documentation files (the "Software"), 

1140# to deal in the Software without restriction, including without limitation 

1141# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1142# and/or sell copies of the Software, and to permit persons to whom the 

1143# Software is furnished to do so, subject to the following conditions: 

1144# 

1145# The above copyright notice and this permission notice shall be included 

1146# in all copies or substantial portions of the Software. 

1147# 

1148# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1149# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1150# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1151# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1152# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1153# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1154# OTHER DEALINGS IN THE SOFTWARE.