Coverage for pygeodesy/karney.py: 94%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''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 

7The I{wrapped} class methods return a L{GDict} instance offering access to the C{dict} items 

8either by C{key} or by C{attribute} name. 

9All methods of the I{wrapped} L{Geodesic<geodesicw.Geodesic>} and L{GeodesicLine<geodesicw.GeodesicLine>} 

10classes return a L{GDict} instance offering access to the C{dict} items either by C{key} or by 

11C{attribute} name. 

12 

13With env variable C{PYGEODESY_GEOGRAPHICLIB} left undefined or set to C{"2"}, modules L{geodesicx}, 

14L{geodesicw} and this module will use U{GeographicLib 2.0<https://GeographicLib.SourceForge.io/C++/doc/>} 

15and newer transcoding, otherwise C{1.52} or older. 

16 

17Karney-based functionality 

18========================== 

19 

201. 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{JacobiConformal} -- U{JacobiConformal 

64 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1JacobiConformal.html#details>} 

65 

66 - L{KTransverseMercator} - U{TransverseMercator 

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

68 

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

70 classGeographicLib_1_1LocalCartesian.html>} 

71 

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

73 

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

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

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

77 

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

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

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

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

82 from I{GeographicLib 2.0} 

83 

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

85 

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

87 

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

89 

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

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

92 

93are I{transcoded} from C++ classes in I{Karney}'s U{GeographicLib<https://GeographicLib.SourceForge.io/C++/doc/annotated.html>}. 

94 

952. These C{pygeodesy} modules and classes 

96 

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

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

99 L{GnomonicKarney}, L{HeightIDWkarney} 

100 

101are or use I{wrappers} around I{Karney}'s Python U{geographiclib<https://PyPI.org/project/geographiclib>} 

102C{geodesic}, C++ utility U{GeodSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} or 

103C++ utility U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>}. 

104 

1053. All C{pygeodesy} functions and methods to compute I{ellipsoidal} intersections, nearest points and trilaterations 

106 

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

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

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

110 

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

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

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

114 

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

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

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

118 

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

120 

121are implementations of I{Karney}'s iterative solution posted under U{The B{ellipsoidal} case 

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

123on an ellipsoid of revolution<https://ArXiv.org/pdf/1102.1215.pdf>} (pp 20-21, section B{14. MARITIME BOUNDARIES}). 

124 

1254. The C{pygeodesy} methods to compute I{ellipsoidal} intersections and nearest points 

126 

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

128 

129are I{transcoded} of I{Karney}'s iterative C++ function U{rhumb-intercept 

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

131 

1325. Spherical functions 

133 

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

135 

136in C{pygeodesy} are based on I{Karney}'s post U{Area of a spherical polygon 

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

138''' 

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

140from __future__ import division as _; del _ # PYCHOK semicolon 

141 

142from pygeodesy.basics import _copysign, int1s, isint, itemsorted, neg, unsigned0, \ 

143 _xgeographiclib, _zip, _version_info 

144from pygeodesy.constants import NAN, _isfinite as _math_isfinite, _0_0, \ 

145 _1_16th, _1_0, _2_0, _180_0, _N_180_0, _360_0 

146from pygeodesy.errors import GeodesicError, _ValueError, _xkwds, _xkwds_get1, \ 

147 _xkwds_kwds, _xkwds_not 

148from pygeodesy.fmath import cbrt, fremainder, norm2 

149# from pygeodesy.internals import _version_info # from .basics 

150from pygeodesy.interns import NN, _2_, _a12_, _area_, _azi1_, _azi2_, _azi12_, \ 

151 _composite_, _lat1_, _lat2_, _lon1_, _lon2_, \ 

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

153 _UNDER_, _X_, _BAR_ # PYCHOK used! 

154from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS, _getenv 

155from pygeodesy.named import ADict, _NamedBase, _NamedTuple, notImplemented, _Pass 

156from pygeodesy.props import deprecated_method, Property_RO 

157from pygeodesy.units import Bearing as _Azi, Degrees as _Deg, Lat, Lon, \ 

158 Meter as _M, Meter2 as _M2, Number_ 

159from pygeodesy.utily import atan2d, sincos2d, tand, _unrollon, fabs 

160 

161# from math import fabs # from .utily 

162 

163__all__ = _ALL_LAZY.karney 

164__version__ = '24.07.09' 

165 

166_K_2_0 = _getenv('PYGEODESY_GEOGRAPHICLIB', _2_) == _2_ 

167_perimeter_ = 'perimeter' 

168 

169 

170class _GTuple(_NamedTuple): # in .testNamedTuples 

171 '''(INTERNAL) Helper. 

172 ''' 

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

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

175 

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

177 ''' 

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

179 if updates: 

180 r.update(updates) 

181 if self._iteration is not None: 

182 r._iteration = self._iteration 

183 return r 

184 

185 

186class _Lat(Lat): 

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

188 ''' 

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

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

191 Lat.__new__(_Lat, *lat, **kwds) 

192 

193 

194class _Lon(Lon): 

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

196 ''' 

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

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

199 Lon.__new__(_Lon, *lon, **kwds) 

200 

201 

202class Area3Tuple(_NamedTuple): # in .geodesicx.gxarea 

203 '''3-Tuple C{(number, perimeter, area)} with the C{number} 

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

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

206 ''' 

207 _Names_ = (_number_, _perimeter_, _area_) 

208 _Units_ = ( Number_, _M, _M2) 

209 

210 

211class Caps(object): # PYCHOK 

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

213 ''' 

214 EMPTY = 0 # formerly aka NONE 

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

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

217 _CAP_2 = 1 << 2 

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

219# _CAP_4 = 1 << 4 

220# _CAP_ALL = 0x1F 

221# _CAP_MASK = _CAP_ALL 

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

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

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

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

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

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

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

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

230 

231 STANDARD = AZIMUTH | DISTANCE | DISTANCE_IN | LATITUDE | LONGITUDE 

232 ALL = 0x7F80 # without LONG_UNROLL, LINE_OFF, REVERSE2 and _DEBUG_* 

233 

234 _DIRECT3 = AZIMUTH | LATITUDE | LONGITUDE | _CAP_3 # for goedesicw only 

235 _INVERSE3 = AZIMUTH | DISTANCE | _CAP_1 # for goedesicw only 

236 _STD = STANDARD | _CAP_3 | _CAP_1 # for goedesicw only 

237 _STD_LINE = _STD | _CAP_2 | _CAP_1p # for goedesici/-w 

238 

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

240 

241 LINE_OFF = 1 << 15 # Line without updates from parent geodesic or rhumb 

242 LONG_UNROLL = 1 << 16 # unroll C{lon2} in .Direct and .Position 

243 REVERSE2 = 1 << 17 # reverse C{azi2} 

244 

245 LATITUDE_LONGITUDE = LATITUDE | LONGITUDE 

246 LATITUDE_LONGITUDE_AREA = LATITUDE | LONGITUDE | AREA 

247 

248 AZIMUTH_DISTANCE = AZIMUTH | DISTANCE 

249 AZIMUTH_DISTANCE_AREA = AZIMUTH | DISTANCE | AREA 

250 

251 _ANGLE_ONLY = 1 << 18 # angular distance C{a12} only 

252 _S_CALPs_ = 1 << 19 # (INTERNAL) GeodesicExact._GenInverse 

253 

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

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

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

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

258 _DEBUG_ALL = _DEBUG_AREA | _DEBUG_DIRECT | _DEBUG_INVERSE | \ 

259 _DEBUG_LINE | _ANGLE_ONLY | _S_CALPs_ 

260 

261 _OUT_ALL = ALL 

262 _OUT_MASK = ALL | LONG_UNROLL | REVERSE2 | _DEBUG_ALL 

263 

264 _AZIMUTH_LATITUDE_LONGITUDE = AZIMUTH | LATITUDE | LONGITUDE 

265 _DEBUG_DIRECT_LINE = _DEBUG_DIRECT | _DEBUG_LINE 

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

267 _LINE = AZIMUTH | LATITUDE | LONG_UNROLL 

268 _REDUCEDLENGTH_GEODESICSCALE = REDUCEDLENGTH | GEODESICSCALE 

269# _REDUCEDLENGTH_GEODESICSCALE_DISTANCE = REDUCEDLENGTH | GEODESICSCALE | DISTANCE 

270 

271 def toStr(self, Csk, sep=_BAR_): 

272 '''Return a C{Caps} or C{outmask} as C{str} or tuple of C{str}s. 

273 ''' 

274 s = [] 

275 for c, C in itemsorted(self.__class__.__dict__): 

276 if isint(C) and (Csk & C) and int1s(C) == 1 \ 

277 and (C in (Caps.REVERSE2, Caps._S_CALPs_) 

278 or c.replace(_UNDER_, NN).isupper()): 

279 s.append(c) 

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

281 

282Caps = Caps() # PYCHOK singleton 

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

284rhumb capabilities (C{caps}) and expected results (C{outmask}). 

285 

286C{AREA} - compute area C{S12}, 

287 

288C{AZIMUTH} - include azimuths C{azi1} and C{azi2}, 

289 

290C{DISTANCE} - compute distance C{s12}, 

291 

292C{DISTANCE_IN} - allow distance C{s12} in C{.Direct}, 

293 

294C{EMPTY} - nothing, formerly aka C{NONE}, 

295 

296C{GEODESICSCALE} - compute geodesic scales C{M12} and C{M21}, 

297 

298C{LATITUDE} - compute latitude C{lat2}, 

299 

300C{LINE_OFF} - Line without updates from parent geodesic or rhumb. 

301 

302C{LONGITUDE} - compute longitude C{lon2}, 

303 

304C{LONG_UNROLL} - unroll C{lon2} in C{.Direct}, 

305 

306C{REDUCEDLENGTH} - compute reduced length C{m12}, 

307 

308C{REVERSE2} - reverse C{azi2}, 

309 

310and C{ALL} - all of the above. 

311 

312C{STANDARD} = C{AZIMUTH | DISTANCE | DISTANCE_IN | LATITUDE | LONGITUDE}''' 

313 

314_KEY2Caps = dict(azi2=Caps.AZIMUTH, # see GDict._unCaps 

315 lat2=Caps.LATITUDE, 

316 lon2=Caps.LONGITUDE, 

317 m12 =Caps.REDUCEDLENGTH, 

318 M12 =Caps.GEODESICSCALE, 

319 M21 =Caps.GEODESICSCALE, 

320 s12 =Caps.DISTANCE, 

321 S12 =Caps.AREA) 

322 

323 

324class _CapsBase(_NamedBase): # in .auxilats, .geodesicx.gxbases 

325 '''(INTERNAL) Base class for C{[_]Geodesic*Exact}. 

326 ''' 

327 ALL = Caps.ALL 

328 AREA = Caps.AREA 

329 AZIMUTH = Caps.AZIMUTH 

330 DISTANCE = Caps.DISTANCE 

331 DISTANCE_IN = Caps.DISTANCE_IN 

332 EMPTY = Caps.EMPTY # aka NONE 

333 GEODESICSCALE = Caps.GEODESICSCALE 

334 LATITUDE = Caps.LATITUDE 

335 LINE_CAPS = Caps.LINE_CAPS 

336 LINE_OFF = Caps.LINE_OFF 

337 LONGITUDE = Caps.LONGITUDE 

338 LONG_UNROLL = Caps.LONG_UNROLL 

339 REDUCEDLENGTH = Caps.REDUCEDLENGTH 

340 STANDARD = Caps.STANDARD 

341 _STD_LINE = Caps._STD_LINE # for geodesici 

342 

343 _caps = 0 # None 

344 _debug = 0 # or Caps._DEBUG_... 

345 

346 @Property_RO 

347 def caps(self): 

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

349 ''' 

350 return self._caps 

351 

352 def caps_(self, caps): 

353 '''Check the available capabilities. 

354 

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

356 for all capabilities to be checked. 

357 

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

359 C{False} otherwise (C{bool}). 

360 ''' 

361 caps &= Caps._OUT_ALL 

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

363 

364 @property 

365 def debug(self): 

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

367 ''' 

368 return bool(self._debug) 

369 

370 @debug.setter # PYCHOK setter! 

371 def debug(self, debug): 

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

373 more details in L{GDict} results. 

374 ''' 

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

376 

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

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

379 ''' 

380 if iter is not None: 

381 self._iteration = r._iteration = iter 

382 return r 

383 

384 

385class Direct9Tuple(_GTuple): 

386 '''9-Tuple C{(a12, lat2, lon2, azi2, s12, m12, M12, M21, S12)} with arc 

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

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

389 C{meter} I{squared}. 

390 ''' 

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

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

393 

394 

395class GDict(ADict): # XXX _NamedDict 

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

397 

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

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

400 ''' 

401 def toDirect9Tuple(self, dflt=NAN): 

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

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

404 

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

406 

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

408 s12, m12, M12, M21, S12)} 

409 ''' 

410 return self._toTuple(Direct9Tuple, dflt) 

411 

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

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

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

415 result. 

416 

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

418 

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

420 s12, a12, m12, M12, M21, S12)}. 

421 ''' 

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

423 

424 def toInverse10Tuple(self, dflt=NAN): 

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

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

427 

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

429 

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

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

432 ''' 

433 return self._toTuple(Inverse10Tuple, dflt) 

434 

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

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

437 ''' 

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

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

440 

441 @deprecated_method 

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

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

444 

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

446 ''' 

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

448 

449 def toRhumb8Tuple(self, dflt=NAN): 

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

451 

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

453 

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

455 azi12, s12, S12, a12)}. 

456 ''' 

457 return self._toTuple(Rhumb8Tuple, dflt) 

458 

459 def toRhumbSolve7Tuple(self, dflt=NAN): 

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

461 

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

463 

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

465 azi12, s12, S12)}. 

466 ''' 

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

468 

469 def _toTuple(self, nTuple, dflt): 

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

471 ''' 

472 _g = getattr 

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

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

475 

476 def _2X(self, gl, _2X=_X_): # .Intesectool, .Intersector 

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

478 ''' 

479 X = GDict(self) 

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

481 if n in X: # X._X = X._2 

482 X[n[:-1] + _2X] = X.pop(n) 

483 v = getattr(gl, n, X) 

484 if v is not X: # X._2 = gl._2 

485 X[n] = v 

486 return X 

487 

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

489 '''(INTERNAL) Remove superfluous items. 

490 ''' 

491 for k, m in _KEY2Caps.items(): 

492 if k in self and not (outmask & m): 

493 self.pop(k) # delattr(self, k) 

494 return self 

495 

496 

497class GeodSolve12Tuple(_GTuple): 

498 '''12-Tuple C{(lat1, lon1, azi1, lat2, lon2, azi2, s12, a12, m12, M12, M21, S12)} with 

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

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

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

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

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

504 ''' 

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

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

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

508 

509 

510class Inverse10Tuple(_GTuple): 

511 '''10-Tuple C{(a12, s12, salp1, calp1, salp2, calp2, m12, M12, M21, S12)} with arc length 

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

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

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

515 ''' 

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

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

518 

519 def toGDict(self, **updates): 

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

521 

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

523 ''' 

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

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

526 **updates) # PYCHOK indent 

527 

528 

529class _kWrapped(object): # in .geodesicw 

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

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

532 ''' 

533 

534 @Property_RO 

535 def geographiclib(self): 

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

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

538 otherwise raise a C{LazyImportError}. 

539 ''' 

540 g = _xgeographiclib(self.__class__, 1, 49) 

541 from geographiclib.geodesic import Geodesic 

542 g.Geodesic = Geodesic 

543 from geographiclib.geodesicline import GeodesicLine 

544 g.GeodesicLine = GeodesicLine 

545 from geographiclib.geomath import Math 

546 g.Math = Math 

547 return g 

548 

549 @Property_RO 

550 def Math(self): 

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

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

553 otherwise C{None}. 

554 ''' 

555 try: 

556 g = self.geographiclib 

557 M = g.Math 

558 if _version_info(g) < (2,): 

559 if _K_2_0: 

560 M = None 

561# elif not _K_2_0: # XXX set 2.0? 

562# _K_2_0 = True 

563 except (AttributeError, ImportError): 

564 M = None 

565 return M 

566 

567_wrapped = _kWrapped() # PYCHOK singleton, .datum, .test/base.py 

568 

569 

570class Rhumb8Tuple(_GTuple): 

571 '''8-Tuple C{(lat1, lon1, lat2, lon2, azi12, s12, S12, a12)} with lat- C{lat1}, 

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

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

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

575 ''' 

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

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

578 

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

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

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

582 

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

584 @kwarg a12_azi1_azi2_m12_M12_M21: Optional keyword arguments 

585 to specify or override L{Inverse10Tuple} items. 

586 

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

588 m12, M12, M21, S12)} 

589 ''' 

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

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

592 if a12_azi1_azi2_m12_M12_M21: 

593 d.update(a12_azi1_azi2_m12_M12_M21) 

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

595 

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

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

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

599 

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

601 @kwarg a12_m12_M12_M21_salp1_calp1_salp2_calp2: Optional keyword 

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

603 

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

605 m12, M12, M21, S12)}. 

606 ''' 

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

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

609 salp2=s, calp2=c, M21=_1_0) 

610 if a12_m12_M12_M21_salp1_calp1_salp2_calp2: 

611 d.update(a12_m12_M12_M21_salp1_calp1_salp2_calp2) 

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

613 

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

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

616 ''' 

617 _g = self.toGDict(**updates).get 

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

619 return nTuple(t, name=self.name) 

620 

621 @deprecated_method 

622 def _to7Tuple(self): 

623 '''DEPRECATED, do not use!''' 

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

625 

626 

627def _around(x): # in .utily.sincos2d 

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

629 

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

631 

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

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

634 ''' 

635 try: 

636 return _wrapped.Math.AngRound(x) 

637 except AttributeError: 

638 if x: 

639 y = _1_16th - fabs(x) 

640 if y > 0: # fabs(x) < _1_16th 

641 x = _copysign(_1_16th - y, x) 

642 else: 

643 x = _0_0 # -0 to 0 

644 return x 

645 

646 

647def _atan2d(y, x): 

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

649 ''' 

650 try: 

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

652 except AttributeError: 

653 return atan2d(y, x) 

654 

655 

656def _cbrt(x): 

657 '''Return C{cubic root(B{x})}. 

658 ''' 

659 try: 

660 return _wrapped.Math.cbrt(x) 

661 except AttributeError: 

662 return cbrt(x) 

663 

664 

665def _copyBit(x, y): 

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

667 ''' 

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

669 

670 

671def _2cos2x(cx, sx): # in .auxDST, .auxLat, .gxbases 

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

673 ''' 

674 r = cx - sx 

675 if r: 

676 r *= (cx + sx) * _2_0 

677 return r 

678 

679 

680def _diff182(deg0, deg, K_2_0=False): 

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

682 

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

684 ''' 

685 try: 

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

687 except AttributeError: 

688 if K_2_0 or _K_2_0: # geographiclib 2.0 

689 _r, _360 = fremainder, _360_0 

690 d, t = _sum2(_r(-deg0, _360), 

691 _r( deg, _360)) 

692 d, t = _sum2(_r( d, _360), t) 

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

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

695 else: 

696 _n = _norm180 

697 d, t = _sum2(_n(-deg0), _n(deg)) 

698 d = _n(d) 

699 if t > 0 and d == _180_0: 

700 d = _N_180_0 

701 d, t = _sum2(d, t) 

702 return d, t 

703 

704 

705def _fix90(deg): # mimick Math.LatFix 

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

707 

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

709 ''' 

710 try: 

711 return _wrapped.Math.LatFix(deg) 

712 except AttributeError: 

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

714 

715 

716def _isfinite(x): # mimick geomath.Math.isfinite 

717 '''Check finiteness of C{x}. 

718 

719 @return: C{True} if finite. 

720 ''' 

721 try: 

722 return _wrapped.Math.isfinite(x) 

723 except AttributeError: 

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

725 

726 

727def _llz2line(line, **llz2): 

728 '''(INTERNAL) Set C{line.lat2, .lon2, .azi2} from C{llz2}. 

729 ''' 

730 if llz2: 

731 llz2 = _xkwds_not(None, **_xkwds_kwds(llz2, lat2=None, lon2=None, azi2=None)) 

732 if llz2: 

733 line.__dict__.update(llz2) 

734 return line 

735 

736 

737def _norm2(x, y): # mimick geomath.Math.norm 

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

739 

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

741 ''' 

742 try: 

743 return _wrapped.Math.norm(x, y) 

744 except AttributeError: 

745 return norm2(x, y) 

746 

747 

748def _norm180(deg): # mimick geomath.Math.AngNormalize 

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

750 

751 @return: Reduced angle C{degrees}. 

752 ''' 

753 try: 

754 return _wrapped.Math.AngNormalize(deg) 

755 except AttributeError: 

756 d = fremainder(deg, _360_0) 

757 if d in (_180_0, _N_180_0): 

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

759 return d 

760 

761 

762def _polygon(geodesic, points, closed, line, wrap): 

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

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

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

766 or C{geodesicw.Geodesic} instance. 

767 ''' 

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

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

770 

771 if _MODS.booleans.isBoolean(points): 

772 # recursive call for each boolean clip 

773 

774 def _a_p(clip, *args, **unused): 

775 return _polygon(geodesic, clip, *args) 

776 

777 if not closed: # closed only 

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

779 

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

781 

782 gP = geodesic.Polygon(line) 

783 _A = gP.AddPoint 

784 

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

786 p1 = p0 = Ps[0] 

787 

788 # note, lon deltas are unrolled, by default 

789 _A(p1.lat, p1.lon) 

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

791 if wrap and not Ps.looped: 

792 p2 = _unrollon(p1, p2) 

793 _A(p2.lat, p2.lon) 

794 p1 = p2 

795 if closed and line and p1 != p0: 

796 _A(p0.lat, p0.lon) 

797 

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

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

800 

801 

802def _polynomial(x, cs, i, j): # PYCHOK shared 

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

804 different signature and cascaded summation as C{karney._sum2_}. 

805 

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

807 ''' 

808 # assert 0 <= i <= j <= len(cs) 

809# try: 

810# return _wrapped.Math.polyval(j - i - 1, cs, i, x) 

811# except AttributeError: 

812# s, t = cs[i], _0_0 

813# for c in cs[i+1:j]: 

814# s, t = _sum2_(s * x, t * x, c) 

815# return s # + t 

816 s = cs[i] 

817 i += 1 

818 if x and i < j: 

819 s, _ = _sum2_(s, _0_0, x=x, *cs[i:j]) 

820 return s # + t 

821 

822 

823def _remainder(x, y): 

824 '''Remainder of C{x / y}. 

825 

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

827 ''' 

828 try: 

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

830 except AttributeError: 

831 return fremainder(x, y) 

832 

833 

834if _K_2_0: 

835 from math import cos as _cos, sin as _sin 

836 

837 def _sincos2(rad): 

838 return _sin(rad), _cos(rad) 

839 

840 _signBit = _MODS.basics.signBit 

841else: 

842 _sincos2 = _MODS.utily.sincos2 # PYCHOK shared 

843 

844 def _signBit(x): 

845 '''(INTERNAL) GeographicLib 1.52-. 

846 ''' 

847 return x < 0 

848 

849 

850def _sincos2d(deg): 

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

852 

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

854 ''' 

855 try: 

856 return _wrapped.Math.sincosd(deg) 

857 except AttributeError: 

858 return sincos2d(deg) 

859 

860 

861def _sincos2de(deg, t): 

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

863 

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

865 ''' 

866 try: 

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

868 except AttributeError: 

869 return sincos2d(deg, adeg=t) 

870 

871 

872def _sum2(u, v): # mimick geomath.Math.sum, actually sum2 

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

874 

875 @return: 2-Tuple C{(B{u} + B{v}, residual)}. 

876 

877 @note: The C{residual} can be the same as B{C{u}} or B{C{v}}. 

878 

879 @see: U{Algorithm 3.1<https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>}. 

880 ''' 

881 try: 

882 return _wrapped.Math.sum(u, v) 

883 except AttributeError: 

884 s = u + v 

885 r = s - v 

886 t = s - r 

887 # if Algorithm_3_1: 

888 # t = (u - t) + (v + r) 

889 # elif C_CPP: # Math::sum C/C++ 

890 # r -= u 

891 # t -= v 

892 # t += r 

893 # t = -t 

894 # else: 

895 t = (u - r) + (v - t) 

896 return s, t 

897 

898 

899def _sum2_(s, t, *vs, **x): 

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

901 

902 @kwarg x: Optional polynomial C{B{x}=1} (C{scalar}). 

903 

904 @return: 2-Tuple C{(B{s} + B{t} + sum(B{vs}), residual)}. 

905 

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

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

908 function C{_sum2} above. 

909 

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

911 ''' 

912 x = _xkwds_get1(x, x=_1_0) 

913 p = x != _1_0 

914 

915 _s2, _u0 = _sum2, unsigned0 

916 for v in vs: 

917 if p: 

918 s *= x 

919 t *= x 

920 if v: 

921 t, u = _s2(t, v) # start at the least- 

922 if s: 

923 s, t = _s2(s, t) # significant end 

924 if s: 

925 t += u # accumulate u into t 

926# elif t: # s == 0 implies t == 0 

927# raise _AssertionError(t=t, txt_not_=_0_) 

928 else: 

929 s = _u0(u) # result is u, t = 0 

930 else: 

931 s, t = _u0(t), u 

932 return s, t 

933 

934 

935def _tand(x): 

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

937 ''' 

938 try: 

939 return _wrapped.Math.tand(x) 

940 except AttributeError: 

941 return tand(x) 

942 

943 

944def _unroll2(lon1, lon2, wrap=False): # see .ellipsoidalBaseDI._intersects2 

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

946 

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

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

949 ''' 

950 if wrap: 

951 d, t = _diff182(lon1, lon2) 

952 lon2, _ = _sum2_(d, t, lon1) # (lon1 + d) + t 

953 else: 

954 lon2 = _norm180(lon2) 

955 return (lon2 - lon1), lon2 

956 

957 

958def _unsigned2(x): 

959 '''(INTERNAL) Unsign B{C{x}}. 

960 ''' 

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

962 

963 

964__all__ += _ALL_DOCS(_CapsBase) 

965 

966# **) MIT License 

967# 

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

969# 

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

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

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

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

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

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

976# 

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