1

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

3

4 u'''Wrapper around several C{geomath.Math} functions from I{Karney}'s Python package U{geographiclib

5<https://PyPI.org/project/geographiclib>}, provided that package is installed.

6

7 Methods of the I{wrapped} L{Geodesic<geodesicw.Geodesic>} and L{GeodesicLine<geodesicw.GeodesicLine>}

8 classes return a L{GDict} instance offering access to the C{dict} items either by C{key} or by

9 C{attribute} name.

10

11 With env variable C{PYGEODESY_GEOGRAPHICLIB} left undefined or set to C{"2"}, modules L{geodesicw},

12 L{geodesicx} and this module will use U{GeographicLib 2.0+<https://GeographicLib.SourceForge.io/C++/doc/>}

13 and newer transcoding, otherwise C{1.52} or older. Set C{PYGEODESY_GEOGRAPHICLIB=2.4} to default to the

14 C{Jacobi amplitude} instead of C{Bulirsch}' function in methods L{ExactTransverseMercator.forward

15<pygeodesy.ExactTransverseMercator.forward>} and L{reverse <pygeodesy.ExactTransverseMercator.reverse>}.

16

17 Karney-based functionality

18==========================

19

20 1. The following classes and functions in C{pygeodesy}

21

22 - L{AlbersEqualArea}, L{AlbersEqualArea2}, L{AlbersEqualArea4},

23 L{AlbersEqualAreaCylindrical}, L{AlbersEqualAreaNorth}, L{AlbersEqualAreaSouth} --

24 U{AlbersEqualArea<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AlbersEqualArea.html>}

25

26 - L{AuxAngle}, L{AuxDST}, L{AuxDLat}, L{AuxLat} -- U{AuxAngle

27 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AuxAngle.html>},

28 U{DST<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1DST.html>},

29 U{DAuxLatitude<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1DAuxLatitude.html>},

30 U{AuxLatitude<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AuxLatitude.html>} in I{GeographicLib 2.2+}

31

32 - L{CassiniSoldner} -- U{CassiniSoldner<https://GeographicLib.SourceForge.io/C++/doc/

33 classGeographicLib_1_1CassiniSoldner.html>}

34

35 - L{EcefKarney} -- U{Geocentric<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Geocentric.html>}

36

37 - L{Elliptic} -- U{EllipticFunction<https://GeographicLib.SourceForge.io/C++/doc/

38 classGeographicLib_1_1EllipticFunction.html>}

39

40 - L{EquidistantExact}, L{EquidistantGeodSolve}, L{EquidistantKarney} -- U{AzimuthalEquidistant

41 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AzimuthalEquidistant.html>}

42

43 - L{Etm}, L{ExactTransverseMercator} -- U{TransverseMercatorExact

44 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1TransverseMercatorExact.html>}

45

46 - L{Geodesic}, L{GeodesicLine} -- I{wrapped} U{geodesic.Geodesic<https://PyPI.org/project/geographiclib>},

47 I{wrapped} U{geodesicline.GeodesicLine<https://PyPI.org/project/geographiclib>}

48

49 - L{GeodesicAreaExact}, L{PolygonArea} -- U{PolygonArea<https://GeographicLib.SourceForge.io/C++/doc/

50 classGeographicLib_1_1PolygonAreaT.html>}

51

52 - L{GeodesicExact}, L{GeodesicLineExact} -- U{GeodesicExact<https://GeographicLib.SourceForge.io/C++/doc/

53 classGeographicLib_1_1GeodesicExact.html>}, U{GeodesicLineExact<https://GeographicLib.SourceForge.io/C++/doc/

54 classGeographicLib_1_1GeodesicLineExact.html>}

55

56 - L{GeoidKarney} -- U{Geoid<https://GeographicLib.SourceForge.io/C++/doc/geoid.html>}

57

58 - L{Georef} -- U{Georef<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Georef.html>}

59

60 - L{GnomonicExact}, L{GnomonicGeodSolve}, L{GnomonicKarney} -- U{Gnomonic

61 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Gnomonic.html>}

62

63 - L{Intersector} -- U{Intersect

64 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Intersect.html>} from I{GeographicLib 2.3+}

65

66 - L{JacobiConformal} -- U{JacobiConformal

67 <https://geographiclib.sourceforge.io/C++/doc/classGeographicLib_1_1experimental_1_1JacobiConformal.html>}

68

69 - L{KTransverseMercator} - U{TransverseMercator

70 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1TransverseMercator.html>}

71

72 - L{LocalCartesian}, L{Ltp} -- U{LocalCartesian<https://GeographicLib.SourceForge.io/C++/doc/

73 classGeographicLib_1_1LocalCartesian.html>}

74

75 - L{Osgr} -- U{OSGB<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1OSGB.html>}

76

77 - L{rhumb.aux_}, L{RhumbAux}, L{RhumbLineAux} -- U{Rhumb

78 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Rhumb.html>} and U{RhumbLine

79 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1RhumbLine.html>} from I{GeographicLib 2.2+}

80

81 - L{rhumb.ekx}, L{Rhumb}, L{RhumbLine} -- U{Rhumb

82 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Rhumb.html>},

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

84 U{TransverseMercator<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1TransverseMercator.html>}

85 from I{GeographicLib 2.0}

86

87 - L{Ups} -- U{PolarStereographic<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1PolarStereographic.html>}

88

89 - L{Utm} -- U{TransverseMercator<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1TransverseMercator.html>}

90

91 - L{UtmUps}, L{Epsg} -- U{UTMUPS<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1UTMUPS.html>}

92

93 - L{atan1d}, L{atan2d}, L{sincos2}, L{sincos2d}, L{tand} -- U{geomath.Math

94 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Math.html>}

95

96 are I{transcoded} from C++ classes in I{Karney}'s U{GeographicLib<https://GeographicLib.SourceForge.io/C++/doc/annotated.html>}.

97

98 2. These C{pygeodesy} modules and classes

99

100 - L{ellipsoidalGeodSolve}, L{ellipsoidalKarney}, L{geodesici}, L{geodsolve}, L{karney}, L{rhumb.solve}

101 - L{EquidistantKarney}, L{FrechetKarney}, L{GeodesicSolve}, L{GeodesicLineSolve}, L{GnomonicGeodSolve},

102 L{GnomonicKarney}, L{HeightIDWkarney}, L{Intersectool}

103

104 are or use I{wrappers} around I{Karney}'s Python U{geographiclib<https://PyPI.org/project/geographiclib>} or

105 C++ utility U{GeodSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>},

106 U{IntersectTool<https://GeographicLib.SourceForge.io/C++/doc/IntersectTool.1.html>} or

107 U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>}.

108

109 3. All C{pygeodesy} functions and methods to compute I{ellipsoidal} intersections, nearest points and trilaterations

110

111 - L{ellipsoidalExact.intersection3}, L{ellipsoidalExact.intersections2}, L{ellipsoidalExact.nearestOn},

112 L{ellipsoidalExact.LatLon.intersection3}, L{ellipsoidalExact.LatLon.intersections2},

113 L{ellipsoidalExact.LatLon.nearestOn}, L{ellipsoidalExact.LatLon.trilaterate5}

114

115 - L{ellipsoidalKarney.intersection3}, L{ellipsoidalKarney.intersections2}, L{ellipsoidalKarney.nearestOn},

116 L{ellipsoidalKarney.LatLon.intersection3}, L{ellipsoidalKarney.LatLon.intersections2},

117 L{ellipsoidalKarney.LatLon.nearestOn}, L{ellipsoidalKarney.LatLon.trilaterate5}

118

119 - L{ellipsoidalVincenty.intersection3}, L{ellipsoidalVincenty.intersections2}, L{ellipsoidalVincenty.nearestOn},

120 L{ellipsoidalVincenty.LatLon.intersection3}, L{ellipsoidalVincenty.LatLon.intersections2},

121 L{ellipsoidalVincenty.LatLon.nearestOn}, L{ellipsoidalVincenty.LatLon.trilaterate5}

122

123 - L{RhumbLineAux.Intersection} and L{RhumbLine.Intersection}

124

125 are implementations of I{Karney}'s iterative solution posted under U{The B{ellipsoidal} case

126<https://GIS.StackExchange.com/questions/48937/calculating-intersection-of-two-circles>} and in paper U{Geodesics

127 on an ellipsoid of revolution<https://ArXiv.org/pdf/1102.1215.pdf>} (pp 20-21, section B{14. MARITIME BOUNDARIES}).

128

129 4. The C{pygeodesy} methods to compute I{ellipsoidal} intersections and nearest points

130

131 - L{RhumbLineAux.Intersecant2}, L{RhumbLineAux.PlumbTo}, L{RhumbLine.Intersecant2} and L{RhumbLine.PlumbTo}

132

133 are I{transcoded} of I{Karney}'s iterative C++ function U{rhumb-intercept

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

135

136 5. Spherical functions

137

138 - L{pygeodesy.excessKarney_}, L{sphericalTrigonometry.areaOf}

139

140 in C{pygeodesy} are based on I{Karney}'s post U{Area of a spherical polygon

141<https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}, 3rd Answer.

142'''

143# make sure int/int division yields float quotient, see .basics

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

145

146 from pygeodesy.basics import _copysign, isint, neg, unsigned0, \

147 _xgeographiclib, _zip

148 from pygeodesy.constants import NAN, _isfinite as _math_isfinite, _0_0, \

149 _1_16th, _1_0, _2_0, _180_0, _N_180_0, _360_0

150 from pygeodesy.errors import GeodesicError, _ValueError, _xkwds

151 from pygeodesy.fmath import cbrt, fremainder, norm2 # Fhorner, Fsum

152 from pygeodesy.internals import _getenv, _popen2, _PYGEODESY, _version_info

153 from pygeodesy.interns import NN, _a12_, _area_, _azi1_, _azi2_, _azi12_, \

154 _composite_, _lat1_, _lat2_, _lon1_, _lon2_, \

155 _m12_, _M12_, _M21_, _number_, _s12_, _S12_, \

156 _SPACE_, _UNDER_, _X_, _1_, _2_, _BAR_ # PYCHOK used!

157 from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS

158 from pygeodesy.named import ADict, _NamedBase, _NamedTuple, notImplemented, _Pass

159 from pygeodesy.props import deprecated_method, Property_RO, property_RO, \

160 property_ROnce

161 from pygeodesy.units import Azimuth as _Azi, Degrees as _Deg, Lat, Lon, \

162 Meter as _M, Meter2 as _M2, Number_

163 from pygeodesy.utily import atan2d, sincos2d, tand, _unrollon, fabs

164

165# from math import fabs # from .utily

166

167__all__ = _ALL_LAZY.karney

168__version__ = '24.10.14'

169

170_2_4_ = '2.4'

171_K_2_0 = _getenv(_PYGEODESY(_xgeographiclib, 1), _2_)

172_K_2_4 = _K_2_0 == _2_4_

173_K_2_0 = _K_2_4 or (_K_2_0 == _2_)

174_perimeter_ = 'perimeter'

175

176

177 class _GTuple(_NamedTuple): # in .testNamedTuples

178 '''(INTERNAL) Helper.

179 '''

180 def toGDict(self, **updates): # NO name=NN

181 '''Convert this C{*Tuple} to a L{GDict}.

182

183 @kwarg updates: Optional items to apply (C{name=value} pairs)

184 '''

185 r = GDict(_zip(self._Names_, self)) # strict=True

186 if updates:

187 r.update(updates)

188 if self._iteration is not None:

189 r._iteration = self._iteration

190 return r

191

192

193 class _Lat(Lat):

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

195 '''

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

197 kwds = _xkwds(Error_name, clip=0, Error=GeodesicError)

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

199

200

201 class _Lon(Lon):

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

203 '''

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

205 kwds = _xkwds(Error_name, clip=0, Error=GeodesicError)

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

207

208

209 class Area3Tuple(_NamedTuple): # in .geodesicx.gxarea

210 '''3-Tuple C{(number, perimeter, area)} with the C{number}

211 of points of the polygon or polyline, the C{perimeter} in

212 C{meter} and the C{area} in C{meter} I{squared}.

213 '''

214 _Names_ = (_number_, _perimeter_, _area_)

215 _Units_ = ( Number_, _M, _M2)

216

217

218 class Caps(object):

219 '''(INTERNAL) Overriden by C{Caps} below.

220 '''

221 EMPTY = 0 # formerly aka NONE

222 _CAP_1 = 1 << 0 # for goedesici/-w

223 _CAP_1p = 1 << 1 # for goedesici/-w

224 _CAP_2 = 1 << 2

225 _CAP_3 = 1 << 3 # for goedesici/-w

226 _CAP_4 = 1 << 4 # for goedesicw

227 _CAP_ALL = 0x1F

228# _CAP_MASK = _CAP_ALL

229 LATITUDE = 1 << 7 # compute latitude C{lat2}

230 LONGITUDE = 1 << 8 | _CAP_3 # compute longitude C{lon2}

231 AZIMUTH = 1 << 9 # azimuths C{azi1} and C{azi2}

232 DISTANCE = 1 << 10 | _CAP_1 # compute distance C{s12}

233 DISTANCE_IN = 1 << 11 | _CAP_1 | _CAP_1p # allow distance C{s12} in Direct

234 REDUCEDLENGTH = 1 << 12 | _CAP_1 | _CAP_2 # compute reduced length C{m12}

235 GEODESICSCALE = 1 << 13 | _CAP_1 | _CAP_2 # compute geodesic scales C{M12} and C{M21}

236 AREA = 1 << 14 | _CAP_4 # compute area C{S12}

237

238 STANDARD = AZIMUTH | DISTANCE | LATITUDE | LONGITUDE

239

240 _DIRECT3 = AZIMUTH | LATITUDE | LONGITUDE # for goedesicw only

241 _INVERSE3 = AZIMUTH | DISTANCE # for goedesicw only

242 _STD = STANDARD # for goedesicw only

243 _STD_LINE = _STD | DISTANCE_IN # for goedesici/-w

244

245 LINE_CAPS = _STD_LINE | REDUCEDLENGTH | GEODESICSCALE # .geodesici only

246 LONG_UNROLL = 1 << 15 # unroll C{lon2} in .Direct and .Position

247# = 1 << 16 # unused

248 LINE_OFF = 1 << 17 # Line without updates from parent geodesic or rhumb

249 REVERSE2 = 1 << 18 # reverse C{azi2}

250 ALL = 0x7F80 | _CAP_ALL # without LONG_UNROLL, LINE_OFF, REVERSE2 and _DEBUG_*

251

252 LATITUDE_LONGITUDE = LATITUDE | LONGITUDE

253 LATITUDE_LONGITUDE_AREA = LATITUDE | LONGITUDE | AREA

254

255 AZIMUTH_DISTANCE = AZIMUTH | DISTANCE

256 AZIMUTH_DISTANCE_AREA = AZIMUTH | DISTANCE | AREA

257

258 _SALP_CALPs_ = 1 << 19 # (INTERNAL) GeodesicExact._GenInverse

259

260 _DEBUG_AREA = 1 << 20 # (INTERNAL) include Line details

261 _DEBUG_DIRECT = 1 << 21 # (INTERNAL) include Direct details

262 _DEBUG_INVERSE = 1 << 22 # (INTERNAL) include Inverse details

263 _DEBUG_LINE = 1 << 23 # (INTERNAL) include Line details

264 _DEBUG_ALL = _DEBUG_AREA | _DEBUG_DIRECT | _DEBUG_INVERSE | \

265 _DEBUG_LINE | _SALP_CALPs_

266

267 _OUT_ALL = ALL # see geographiclib.geodesiccapabilities.py

268 _OUT_MASK = ALL | LONG_UNROLL | REVERSE2 | _DEBUG_ALL

269

270 _AZIMUTH_LATITUDE_LONGITUDE = AZIMUTH | LATITUDE | LONGITUDE

271 _AZIMUTH_LATITUDE_LONG_UNROLL = AZIMUTH | LATITUDE | LONG_UNROLL

272 _DEBUG_DIRECT_LINE = _DEBUG_DIRECT | _DEBUG_LINE

273# _DISTANCE_IN_OUT = DISTANCE_IN & _OUT_MASK # == DISTANCE_IN in .gx, .gxline

274 _REDUCEDLENGTH_GEODESICSCALE = REDUCEDLENGTH | GEODESICSCALE

275# _REDUCEDLENGTH_GEODESICSCALE_DISTANCE = REDUCEDLENGTH | GEODESICSCALE | DISTANCE

276

277 def items(self):

278 '''Yield all I{public} C{Caps} as 2-tuple C{(NAME, mask)}.

279 '''

280 for n, C in Caps.__class__.__dict__.items():

281 if isint(C) and not n.startswith(_UNDER_) \

282 and n.replace(_UNDER_, NN).isupper():

283 yield n, C

284

285 def toStr(self, Cask, sep=_BAR_):

286 '''Return C{Caps} or an C{outmask} as C{str} or tuple of C{str}s.

287 '''

288 s = (n for n, C in sorted(Caps.items())

289 if C and (Cask & C) == C) # and int1s(C) <= 3

290 return sep.join(s) if sep else tuple(s)

291

292 Caps = Caps() # PYCHOK singleton

293'''I{Enum}-style masks to be bit-C{or}'ed to specify geodesic or

294 rhumb capabilities (C{caps}) and results (C{outmask}).

295

296 C{AREA} - compute area C{S12},

297

298 C{AZIMUTH} - include azimuths C{azi1} and C{azi2},

299

300 C{DISTANCE} - compute distance C{s12} and angular distance C{a12},

301

302 C{DISTANCE_IN} - allow distance C{s12} in C{.Direct},

303

304 C{EMPTY} - nothing, formerly aka C{NONE},

305

306 C{GEODESICSCALE} - compute geodesic scales C{M12} and C{M21},

307

308 C{LATITUDE} - compute latitude C{lat2},

309

310 C{LINE_OFF} - Line without updates from parent geodesic or rhumb.

311

312 C{LONGITUDE} - compute longitude C{lon2},

313

314 C{LONG_UNROLL} - unroll C{lon2} in C{.Direct},

315

316 C{REDUCEDLENGTH} - compute reduced length C{m12},

317

318 C{REVERSE2} - reverse C{azi2},

319

320 and C{ALL} - all of the above.

321

322 C{STANDARD} = C{AZIMUTH | DISTANCE | DISTANCE_IN | LATITUDE | LONGITUDE}'''

323

324_key2Caps = dict(a12 =Caps.DISTANCE, # in GDict._unCaps

325 azi2=Caps.AZIMUTH,

326 lat2=Caps.LATITUDE,

327 lon2=Caps.LONGITUDE,

328 m12 =Caps.REDUCEDLENGTH,

329 M12 =Caps.GEODESICSCALE,

330 M21 =Caps.GEODESICSCALE,

331 s12 =Caps.DISTANCE,

332 S12 =Caps.AREA)

333

334

335 class _CapsBase(_NamedBase): # in .auxilats, .geodesicx.gxbases

336 '''(INTERNAL) Base class for C{Geodesic*}, C{Geodesic*Exact}, C{Intersectool} and C{Rhumb*Base}.

337 '''

338 ALL = Caps.ALL

339 AREA = Caps.AREA

340 AZIMUTH = Caps.AZIMUTH

341 DISTANCE = Caps.DISTANCE

342 DISTANCE_IN = Caps.DISTANCE_IN

343 EMPTY = Caps.EMPTY # aka NONE

344 GEODESICSCALE = Caps.GEODESICSCALE

345 LATITUDE = Caps.LATITUDE

346 LINE_CAPS = Caps.LINE_CAPS

347 LINE_OFF = Caps.LINE_OFF

348 LONGITUDE = Caps.LONGITUDE

349 LONG_UNROLL = Caps.LONG_UNROLL

350 REDUCEDLENGTH = Caps.REDUCEDLENGTH

351 STANDARD = Caps.STANDARD

352 _STD_LINE = Caps._STD_LINE # for geodesici

353

354 _caps = 0 # None

355 _debug = 0 # or Caps._DEBUG_...

356

357 @Property_RO

358 def caps(self):

359 '''Get the capabilities (bit-or'ed C{Caps}).

360 '''

361 return self._caps

362

363 def caps_(self, caps):

364 '''Check the available capabilities.

365

366 @arg caps: Bit-or'ed combination of L{Caps} values

367 for all capabilities to be checked.

368

369 @return: C{True} if I{all} B{C{caps}} are available,

370 C{False} otherwise (C{bool}).

371 '''

372 caps &= Caps._OUT_ALL

373 return (self.caps & caps) == caps

374

375 @property

376 def debug(self):

377 '''Get the C{debug} option (C{bool}).

378 '''

379 return bool(self._debug)

380

381 @debug.setter # PYCHOK setter!

382 def debug(self, debug):

383 '''Set the C{debug} option (C{bool}) to include

384 more details in L{GDict} results.

385 '''

386 self._debug = Caps._DEBUG_ALL if debug else 0

387

388 def _iter2tion(self, r, iter=None, **unused):

389 '''(INTERNAL) Copy C{C{s}.iter} into C{B{r}._iteration}.

390 '''

391 if iter is not None:

392 self._iteration = r._iteration = iter

393 return r

394

395

396 class Direct9Tuple(_GTuple):

397 '''9-Tuple C{(a12, lat2, lon2, azi2, s12, m12, M12, M21, S12)} with arc

398 length C{a12}, angles C{lat2}, C{lon2} and azimuth C{azi2} in C{degrees},

399 distance C{s12} and reduced length C{m12} in C{meter} and area C{S12} in

400 C{meter} I{squared}.

401 '''

402 _Names_ = (_a12_, _lat2_, _lon2_, _azi2_, _s12_, _m12_, _M12_, _M21_, _S12_)

403 _Units_ = (_Azi, _Lat, _Lon, _Azi, _M, _Pass, _Pass, _Pass, _M2)

404

405

406 class GDict(ADict): # XXX _NamedDict

407 '''A C{dict} with both C{key} I{and} C{attribute} access to the C{dict} items.

408

409 Results of all C{geodesic} and C{rhumb} methods (with capitalized named) are

410 returned as L{GDict} instances, see for example L{GeodesicExact} and L{RhumbAux}.

411 '''

412 def toDirect9Tuple(self, dflt=NAN):

413 '''Convert this L{GDict} result to a 9-tuple, like I{Karney}'s method

414 C{geographiclib.geodesic.Geodesic._GenDirect}.

415

416 @kwarg dflt: Default value for missing items (C{any}).

417

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

419 s12, m12, M12, M21, S12)}

420 '''

421 return self._toTuple(Direct9Tuple, dflt)

422

423 def toGeodSolve12Tuple(self, dflt=NAN): # PYCHOK 12 args

424 '''Convert this L{GDict} result to a 12-Tuple, compatible with I{Karney}'s

425 U{GeodSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>}

426 result.

427

428 @kwarg dflt: Default value for missing items (C{any}).

429

430 @return: L{GeodSolve12Tuple}C{(lat1, lon1, azi1, lat2, lon2, azi2,

431 s12, a12, m12, M12, M21, S12)}.

432 '''

433 return self._toTuple(_MODS.geodsolve.GeodSolve12Tuple, dflt)

434

435 def toInverse10Tuple(self, dflt=NAN):

436 '''Convert this L{GDict} result to a 10-tuple, like I{Karney}'s

437 method C{geographiclib.geodesic.Geodesic._GenInverse}.

438

439 @kwarg dflt: Default value for missing items (C{any}).

440

441 @return: L{Inverse10Tuple}C{(a12, s12, salp1, calp1,

442 salp2, calp2, m12, M12, M21, S12)}.

443 '''

444 return self._toTuple(Inverse10Tuple, dflt)

445

446 def _toNAN(self, outmask): # .GeodesicLineExact._GenPosition

447 '''(INTERNAL) Convert this C{GDict} to all C{NAN}s.

448 '''

449 d = dict((n, NAN) for n in GeodSolve12Tuple._Names_)

450 return self.set_(**d)._unCaps(outmask)

451

452 @deprecated_method

453 def toRhumb7Tuple(self, dflt=NAN): # PYCHOK no cover

454 '''DEPRECATED on 23.12.07, use method C{toRhumb8Tuple}.

455

456 @return: A I{DEPRECATED} L{Rhumb7Tuple}.

457 '''

458 return self._toTuple(_MODS.deprecated.classes.Rhumb7Tuple, dflt)

459

460 def toRhumb8Tuple(self, dflt=NAN):

461 '''Convert this L{GDict} result to a 8-tuple.

462

463 @kwarg dflt: Default value for missing items (C{any}).

464

465 @return: L{Rhumb8Tuple}C{(lat1, lon1, lat2, lon2,

466 azi12, s12, S12, a12)}.

467 '''

468 return self._toTuple(Rhumb8Tuple, dflt)

469

470 def toRhumbSolve7Tuple(self, dflt=NAN):

471 '''Convert this L{GDict} result to a 8-tuple.

472

473 @kwarg dflt: Default value for missing items (C{any}).

474

475 @return: L{RhumbSolve7Tuple}C{(lat1, lon1, lat2, lon2,

476 azi12, s12, S12)}.

477 '''

478 return self._toTuple(_MODS.rhumb.solve.RhumbSolve7Tuple, dflt)

479

480 def _toTuple(self, nTuple, dflt):

481 '''(INTERNAL) Convert this C{GDict} to an B{C{nTuple}}.

482 '''

483 _g = getattr

484 t = tuple(_g(self, n, dflt) for n in nTuple._Names_)

485 return nTuple(t, iteration=self._iteration)

486

487 def _2X(self, gl, _2X=_X_): # .Intersectool, .Intersector

488 '''(INTERNAL) Rename C{-2} attr to C{-X} or C{-M}.

489 '''

490 X = GDict(self)

491 for n in (_lat2_, _lon2_, _azi2_, _s12_, _a12_):

492 if n in X: # X._X = X._2

493 X[n[:-1] + _2X] = X.pop(n)

494 v = getattr(gl, n, X)

495 if v is not X: # X._2 = gl._2

496 X[n] = v

497 return X

498

499 def _unCaps(self, outmask): # in .geodsolve

500 '''(INTERNAL) Remove superfluous items.

501 '''

502 for k, C in _key2Caps.items():

503 if k in self and (outmask & C) != C:

504 self.pop(k) # delattr(self, k)

505 return self

506

507

508 class GeodSolve12Tuple(_GTuple):

509 '''12-Tuple C{(lat1, lon1, azi1, lat2, lon2, azi2, s12, a12, m12, M12, M21, S12)} with

510 angles C{lat1}, C{lon1}, C{azi1}, C{lat2}, C{lon2} and C{azi2} and arc C{a12} all in

511 C{degrees}, initial C{azi1} and final C{azi2} forward azimuths, distance C{s12} and

512 reduced length C{m12} in C{meter}, area C{S12} in C{meter} I{squared} and geodesic

513 scale factors C{M12} and C{M21}, both C{scalar}, see U{GeodSolve

514 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>}.

515 '''

516 # from GeodSolve --help option -f ... lat1 lon1 azi1 lat2 lon2 azi2 s12 a12 m12 M12 M21 S12

517 _Names_ = (_lat1_, _lon1_, _azi1_, _lat2_, _lon2_, _azi2_, _s12_, _a12_, _m12_, _M12_, _M21_, _S12_)

518 _Units_ = (_Lat, _Lon, _Azi, _Lat, _Lon, _Azi, _M, _Deg, _Pass, _Pass, _Pass, _M2)

519

520

521 class Inverse10Tuple(_GTuple):

522 '''10-Tuple C{(a12, s12, salp1, calp1, salp2, calp2, m12, M12, M21, S12)} with arc length

523 C{a12} in C{degrees}, distance C{s12} and reduced length C{m12} in C{meter}, area

524 C{S12} in C{meter} I{squared} and the sines C{salp1}, C{salp2} and cosines C{calp1},

525 C{calp2} of the initial C{1} and final C{2} (forward) azimuths.

526 '''

527 _Names_ = (_a12_, _s12_, 'salp1', 'calp1', 'salp2', 'calp2', _m12_, _M12_, _M21_, _S12_)

528 _Units_ = (_Azi, _M, _Pass, _Pass, _Pass, _Pass, _Pass, _Pass, _Pass, _M2)

529

530 def toGDict(self, **updates):

531 '''Convert this C{Inverse10Tuple} to a L{GDict}.

532

533 @kwarg updates: Optional items to apply (C{nam=value} pairs)

534 '''

535 return _GTuple.toGDict(self, azi1=atan2d(self.salp1, self.calp1), # PYCHOK namedTuple

536 azi2=atan2d(self.salp2, self.calp2), # PYCHOK namedTuple

537 **updates) # PYCHOK indent

538

539

540 class _kWrapped(object): # in .geodesicw

541 ''''(INTERNAL) Wrapper for some of I{Karney}'s U{geographiclib

542 <https://PyPI.org/project/geographiclib>} classes.

543 '''

544

545 @property_ROnce

546 def geographiclib(self):

547 '''Lazily import C{geographiclib}, provided the U{geographiclib

548 <https://PyPI.org/project/geographiclib>} package is installed,

549 otherwise raise a C{LazyImportError}.

550 '''

551 g = _xgeographiclib(self.__class__.__module__, 1, 49)

552 from geographiclib.geodesic import Geodesic

553 g.Geodesic = Geodesic

554 from geographiclib.geodesicline import GeodesicLine

555 g.GeodesicLine = GeodesicLine

556 from geographiclib.geomath import Math

557 g.Math = Math

558 return g

559

560 @property_ROnce

561 def Math(self):

562 '''Wrap the C{geomath.Math} class, provided the U{geographiclib

563 <https://PyPI.org/project/geographiclib>} package is installed,

564 otherwise C{None}.

565 '''

566 try:

567 g = self.geographiclib

568 M = g.Math

569 if _version_info(g) < (2,):

570 if _K_2_0:

571 M = None

572# elif not _K_2_0: # XXX set 2.0?

573# _K_2_0 = True

574 except (AttributeError, ImportError):

575 M = None

576 return M

577

578 @property_RO

579 def Math_K_2(self):

580 return (_2_4_ if _K_2_4 else

581 (_2_ if _K_2_0 else _1_)) if self.Math else NN

582

583_wrapped = _kWrapped() # PYCHOK singleton, .datum, .test/base.py

584

585

586 class Rhumb8Tuple(_GTuple):

587 '''8-Tuple C{(lat1, lon1, lat2, lon2, azi12, s12, S12, a12)} with lat- C{lat1},

588 C{lat2} and longitudes C{lon1}, C{lon2} of both points, the azimuth of the

589 rhumb line C{azi12}, the distance C{s12}, the area C{S12} under the rhumb

590 line and the angular distance C{a12} between both points.

591 '''

592 _Names_ = (_lat1_, _lon1_, _lat2_, _lon2_, _azi12_, _s12_, _S12_, _a12_)

593 _Units_ = ( Lat, Lon, Lat, Lon, _Azi, _M, _M2, _Deg)

594

595 def toDirect9Tuple(self, dflt=NAN, **a12_azi1_azi2_m12_M12_M21):

596 '''Convert this L{Rhumb8Tuple} result to a 9-tuple, like I{Karney}'s

597 method C{geographiclib.geodesic.Geodesic._GenDirect}.

598

599 @kwarg dflt: Default value for missing items (C{any}).

600 @kwarg a12_azi1_azi2_m12_M12_M21: Optional keyword arguments

601 to specify or override L{Inverse10Tuple} items.

602

603 @return: L{Direct9Tuple}C{(a12, lat2, lon2, azi2, s12,

604 m12, M12, M21, S12)}

605 '''

606 d = dict(azi1=self.azi12, M12=_1_0, m12=self.s12, # PYCHOK attr

607 azi2=self.azi12, M21=_1_0) # PYCHOK attr

608 if a12_azi1_azi2_m12_M12_M21:

609 d.update(a12_azi1_azi2_m12_M12_M21)

610 return self._toTuple(Direct9Tuple, dflt, d)

611

612 def toInverse10Tuple(self, dflt=NAN, **a12_m12_M12_M21_salp1_calp1_salp2_calp2):

613 '''Convert this L{Rhumb8Tuple} to a 10-tuple, like I{Karney}'s

614 method C{geographiclib.geodesic.Geodesic._GenInverse}.

615

616 @kwarg dflt: Default value for missing items (C{any}).

617 @kwarg a12_m12_M12_M21_salp1_calp1_salp2_calp2: Optional keyword

618 arguments to specify or override L{Inverse10Tuple} items.

619

620 @return: L{Inverse10Tuple}C{(a12, s12, salp1, calp1, salp2, calp2,

621 m12, M12, M21, S12)}.

622 '''

623 s, c = sincos2d(self.azi12) # PYCHOK attr

624 d = dict(salp1=s, calp1=c, M12=_1_0, m12=self.s12, # PYCHOK attr

625 salp2=s, calp2=c, M21=_1_0)

626 if a12_m12_M12_M21_salp1_calp1_salp2_calp2:

627 d.update(a12_m12_M12_M21_salp1_calp1_salp2_calp2)

628 return self._toTuple(Inverse10Tuple, dflt, d)

629

630 def _toTuple(self, nTuple, dflt, updates={}):

631 '''(INTERNAL) Convert this C{Rhumb8Tuple} to an B{C{nTuple}}.

632 '''

633 _g = self.toGDict(**updates).get

634 t = tuple(_g(n, dflt) for n in nTuple._Names_)

635 return nTuple(t, name=self.name)

636

637 @deprecated_method

638 def _to7Tuple(self):

639 '''DEPRECATED, do not use!'''

640 return _MODS.deprecated.classes.Rhumb7Tuple(self[:-1])

641

642

643 class _Xables(object):

644 '''(INTERNAL) Get I{Karney}'s executable paths from/and env vars.

645 '''

646 bin_ = '/opt/local/bin/' # '/opt/local/Cellar/geographiclib/2.3/bin/' # HomeBrew on macOS

647 ENV = NN

648

649 def GeoConvert(self, *dir_):

650 return self._path(self.GeoConvert, *dir_)

651

652 def GeodSolve(self, *dir_):

653 return self._path(self.GeodSolve, *dir_)

654

655 def IntersectTool(self, *dir_):

656 return self._path(self.IntersectTool, *dir_)

657

658 def RhumbSolve(self, *dir_):

659 return self._path(self.RhumbSolve, *dir_)

660

661 def name_version(self, path, base=True):

662 # return C{(path + ' ' + version)} of an executable

663 if path:

664 try:

665 r, s = _popen2((path, '--version'))

666 if base:

667 path = _MODS.os.path.basename(path)

668 r = _SPACE_(path, r.split()[-1])

669 else:

670 r = _MODS.streprs.Fmt.PARENSPACED(r, s)

671 return r

672 except (IndexError, IOError, OSError):

673 pass

674 return NN

675

676 def _path(self, which, *dir_):

677 self.ENV = E = _PYGEODESY(which)

678 return _getenv(E, NN) or \

679 (NN(dir_[0], which.__name__) if dir_ else E)

680

681 def X_not(self, path):

682 return 'env %s=%r not executable' % (self.ENV, path)

683

684 def X_OK(self, path): # is C{path} an executable?

685 os = _MODS.os # import os

686 return os.access(path, os.X_OK) if path else False

687

688_Xables = _Xables() # PYCHOK singleton

689

690

691 def _around(x): # in .utily.sincos2d

692 '''I{Coarsen} a scalar by rounding small values to underflow to C{0.0}.

693

694 @return: Coarsened value (C{float}).

695

696 @see: I{Karney}'s U{geomath.Math.AngRound<https://SourceForge.net/p/

697 geographiclib/code/ci/release/tree/python/geographiclib/geomath.py>}

698 '''

699 try:

700 return _wrapped.Math.AngRound(x)

701 except AttributeError:

702 if x:

703 y = _1_16th - fabs(x)

704 if y > 0: # fabs(x) < _1_16th

705 x = _copysign(_1_16th - y, x)

706 else:

707 x = _0_0 # -0 to 0

708 return x

709

710

711 def _atan2d(y, x):

712 '''Return C{atan2(B{y}, B{x})} in C{degrees}.

713 '''

714 try:

715 return _wrapped.Math.atan2d(y, x)

716 except AttributeError:

717 return atan2d(y, x)

718

719

720 def _cbrt(x):

721 '''Return C{cubic root(B{x})}.

722 '''

723 try:

724 return _wrapped.Math.cbrt(x)

725 except AttributeError:

726 return cbrt(x)

727

728

729 def _copyBit(x, y):

730 '''Like C{copysign0(B{x}, B{y})}, with C{B{x} > 0}.

731 '''

732 return (-x) if _signBit(y) else x

733

734

735 def _2cos2x(cx, sx): # in .auxDST, .auxLat, .gxbases

736 '''Return M{2 * cos(2 * x)} from cos(x) and sin(x).

737 '''

738 r = cx - sx

739 if r:

740 r *= (cx + sx) * _2_0

741 return r

742

743

744 def _diff182(deg0, deg, K_2_0=False):

745 '''Compute C{deg - deg0}, reduced to C{[-180,180]} accurately.

746

747 @return: 2-Tuple C{(delta_angle, residual)} in C{degrees}.

748 '''

749 try:

750 return _wrapped.Math.AngDiff(deg0, deg)

751 except AttributeError:

752 if K_2_0 or _K_2_0: # geographiclib 2.0

753 _r, _360 = fremainder, _360_0

754 d, t = _sum2(_r(-deg0, _360),

755 _r( deg, _360))

756 d, t = _sum2(_r( d, _360), t)

757 if d in (_0_0, _180_0, _N_180_0):

758 d = _copysign(d, -t if t else (deg - deg0))

759 else:

760 _n = _norm180

761 d, t = _sum2(_n(-deg0), _n(deg))

762 d = _n(d)

763 if t > 0 and d == _180_0:

764 d = _N_180_0

765 d, t = _sum2(d, t)

766 return d, t

767

768

769 def _fix90(deg): # mimick Math.LatFix

770 '''Replace angle in C{degrees} outside [-90,90] by NAN.

771

772 @return: Angle C{degrees} or NAN.

773 '''

774 try:

775 return _wrapped.Math.LatFix(deg)

776 except AttributeError:

777 return NAN if fabs(deg) > 90 else deg

778

779

780 def _isfinite(x): # mimick geomath.Math.isfinite

781 '''Check finiteness of C{x}.

782

783 @return: C{True} if finite.

784 '''

785 try:

786 return _wrapped.Math.isfinite(x)

787 except AttributeError:

788 return _math_isfinite(x) # and fabs(x) <= _MAX

789

790

791 def _llz2gl(gl, **llz2): # see .geodesici._llz2G

792 '''(INTERNAL) Set C{gl.lat2, .lon2, .azi2} from C{llz2}.

793 '''

794 if llz2:

795 for n in (_lat2_, _lon2_, _azi2_): # _lat1_, _lon1_, _azi1_

796 v = llz2.get(n, None)

797 if v is not None:

798 setattr(gl, n, v)

799 return gl

800

801

802 def _norm2(x, y): # mimick geomath.Math.norm

803 '''Normalize C{B{x}} and C{B{y}}.

804

805 @return: 2-Tuple of C{(B{x}, B{y})}, normalized.

806 '''

807 try:

808 return _wrapped.Math.norm(x, y)

809 except AttributeError:

810 return norm2(x, y)

811

812

813 def _norm180(deg): # mimick geomath.Math.AngNormalize

814 '''Reduce angle in C{degrees} to (-180,180].

815

816 @return: Reduced angle C{degrees}.

817 '''

818 try:

819 return _wrapped.Math.AngNormalize(deg)

820 except AttributeError:

821 d = fremainder(deg, _360_0)

822 if d in (_180_0, _N_180_0):

823 d = _copysign(_180_0, deg) if _K_2_0 else _180_0

824 return d

825

826

827 def _polygon(geodesic, points, closed, line, wrap):

828 '''(INTERNAL) Compute the area or perimeter of a polygon,

829 using a L{GeodesicExact}, L{GeodesicSolve} or (if the

830 C{geographiclib} package is installed) a C{Geodesic}

831 or C{geodesicw.Geodesic} instance.

832 '''

833 if not wrap: # capability LONG_UNROLL can't be off

834 notImplemented(None, wrap=wrap, up=3)

835

836 if _MODS.booleans.isBoolean(points):

837 # recursive call for each boolean clip

838

839 def _a_p(clip, *args, **unused):

840 return _polygon(geodesic, clip, *args)

841

842 if not closed: # closed only

843 raise _ValueError(closed=closed, points=_composite_)

844

845 return points._sum1(_a_p, closed, line, wrap)

846

847 gP = geodesic.Polygon(line)

848 _A = gP.AddPoint

849

850 Ps = _MODS.iters.PointsIter(points, loop=1, wrap=wrap) # base=LatLonEllipsoidalBase(0, 0)

851 p1 = p0 = Ps[0]

852

853 # note, lon deltas are unrolled, by default

854 _A(p1.lat, p1.lon)

855 for p2 in Ps.iterate(closed=closed):

856 if wrap and not Ps.looped:

857 p2 = _unrollon(p1, p2)

858 _A(p2.lat, p2.lon)

859 p1 = p2

860 if closed and line and p1 != p0:

861 _A(p0.lat, p0.lon)

862

863 # gP.Compute returns (number_of_points, perimeter, signed area)

864 return gP.Compute(False, True)[1 if line else 2]

865

866

867 try:

868 from math import fma as _fma # since 3.13

869

870 def _poly_fma(x, s, *cs):

871 for c in cs:

872 s = _fma(s, x, c)

873 return s

874

875 except ImportError: # Python 3.12-

876

877 def _poly_fma(x, s, *cs): # PYCHOK redef

878 t = _0_0

879 for c in cs:

880 s, t, _ = _sum3(s * x, t * x, c)

881 return s + t

882

883# def _poly_fma(x, *cs):

884# S = Fhorner(x, *cs, incx=False)

885# return float(S)

886

887 def _polynomial(x, cs, i, j): # PYCHOK shared

888 '''(INTERNAL) Like C++ C{GeographicLib.Math.hpp.polyval} but with a

889 signature and cascaded summation different from C{karney._sum3}.

890

891 @return: M{sum(x**(j - k - 1) * cs[k] for k in range(i, j)}

892 '''

893 if (i + 1) < j <= len(cs): # load _Rtuple._tuple

894 try:

895 r = _wrapped.Math.polyval(j - i - 1, cs, i, x)

896 except AttributeError:

897 r = _poly_fma(x, *cs[i:j])

898 else:

899 r = cs[i]

900 return float(r)

901

902

903 def _remainder(x, y):

904 '''Remainder of C{x / y}.

905

906 @return: Remainder in the range M{[-y / 2, y / 2]}, preserving signed 0.0.

907 '''

908 try:

909 return _wrapped.Math.remainder(x, y)

910 except AttributeError:

911 return fremainder(x, y)

912

913

914 if _K_2_0:

915 from math import cos as _cos, sin as _sin

916

917 def _sincos2(rad):

918 return _sin(rad), _cos(rad)

919

920 _signBit = _MODS.basics.signBit

921 else:

922 _sincos2 = _MODS.utily.sincos2 # PYCHOK shared

923

924 def _signBit(x):

925 '''(INTERNAL) GeographicLib 1.52-.

926 '''

927 return x < 0

928

929

930 def _sincos2d(deg):

931 '''Return sine and cosine of an angle in C{degrees}.

932

933 @return: 2-Tuple C{(sin(B{deg}), cos(B{deg}))}.

934 '''

935 try:

936 return _wrapped.Math.sincosd(deg)

937 except AttributeError:

938 return sincos2d(deg)

939

940

941 def _sincos2de(deg, t):

942 '''Return sine and cosine of a corrected angle in C{degrees}.

943

944 @return: 2-Tuple C{(sin(B{deg}), cos(B{deg}))}.

945 '''

946 try:

947 return _wrapped.Math.sincosde(deg, t)

948 except AttributeError:

949 return sincos2d(deg, adeg=t)

950

951

952 def _sum2(a, b): # mimick geomath.Math.sum, actually sum2

953 '''Error-free summation like C{geomath.Math.sum}.

954

955 @return: 2-Tuple C{(B{a} + B{b}, residual)}.

956

957 @note: The C{residual} can be the same as B{C{a}} or B{C{b}}.

958

959 @see: U{TwoSum<https://accurate-algorithms.readthedocs.io/en/latest/ch04summation.html>}

960 and I{Knuth}'s U{Algorithm 3.1<https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>}.

961 '''

962 try:

963 return _wrapped.Math.sum(a, b)

964 except AttributeError:

965 # if Algorithm_3_1:

966 s = a + b

967 r = s - b

968 t = s - r

969 # elif C_CPP: # Math::sum C/C++

970 # r -= a; t -= b; t += r; t = -t

971 # else:

972 t = (a - r) + (b - t)

973 # assert fabs(s) >= fabs(t)

974 return s, t

975

976

977 def _sum3(s, t, *xs):

978 '''Accumulate any B{C{xs}} into a previous C{_sum2(s, t)}.

979

980 @return: 3-Tuple C{(s, t, n)} where C{s} is the sum of B{s}, B{t} and all

981 B{xs}, C{t} the residual and C{n} the number of zero C{xs}.

982

983 @see: I{Karney's} C++ U{Accumulator<https://GeographicLib.SourceForge.io/

984 C++/doc/Accumulator_8hpp_source.html>} comments for more details and

985 function C{_sum2} above.

986

987 @note: Not "error-free", see C{pygeodesy.test/testKarney.py}.

988 '''

989 z = 0

990 for x in xs:

991 if x:

992 t, r = _sum2(t, x) # start at the least-

993 if s:

994 s, t = _sum2(s, t) # -significant end

995 if s:

996 t += r # accumulate r into t

997 else:

998 # assert t == 0 # s == 0 implies t == 0

999 s = unsigned0(r) # result is r, t = 0

1000 else:

1001 s, t = unsigned0(t), r

1002 else:

1003 z += 1

1004 # assert fabs(s) >= fabs(t)

1005 return s, t, z

1006

1007

1008 def _tand(x):

1009 '''Return C{tan(B{x})} in C{degrees}.

1010 '''

1011 try:

1012 return _wrapped.Math.tand(x)

1013 except AttributeError:

1014 return tand(x)

1015

1016

1017 def _unroll2(lon1, lon2, wrap=False): # see .ellipsoidalBaseDI._intersects2

1018 '''Unroll B{C{lon2 - lon1}} like C{geodesic.Geodesic.Inverse}.

1019

1020 @return: 2-Tuple C{(B{lon2} - B{lon1}, B{lon2})} with B{C{lon2}}

1021 unrolled if C{B{wrap} is True}, normalized otherwise.

1022 '''

1023 if wrap:

1024 d, t = _diff182(lon1, lon2)

1025 lon2, t, _ = _sum3(d, t, lon1) # (lon1 + d) + t

1026 lon2 += t

1027 else:

1028 lon2 = _norm180(lon2)

1029 return (lon2 - lon1), lon2

1030

1031

1032 def _unsigned2(x):

1033 '''(INTERNAL) Unsign B{C{x}}.

1034 '''

1035 return (neg(x), True) if _signBit(x) else (x, False)

1036

1037

1038__all__ += _ALL_DOCS(_CapsBase)

1039

1040# **) MIT License

1041#

1042# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.

1043#

1044# Permission is hereby granted, free of charge, to any person obtaining a

1045# copy of this software and associated documentation files (the "Software"),

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1050#

1051# The above copyright notice and this permission notice shall be included

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1054# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

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