Coverage for pygeodesy/formy.py: 97%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''Formulary of basic geodesy functions and approximations. 

5''' 

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

7from __future__ import division as _; del _ # PYCHOK semicolon 

8 

9# from pygeodesy.cartesianBase import CartesianBase # _MODS 

10from pygeodesy.constants import EPS, EPS0, EPS1, PI, PI2, PI3, PI_2, R_M, \ 

11 _isfinite, float0_, isnon0, remainder, _umod_PI2, \ 

12 _0_0, _0_125, _0_25, _0_5, _1_0, _2_0, _4_0, \ 

13 _32_0, _90_0, _180_0, _360_0 

14from pygeodesy.datums import Datum, Ellipsoid, _ellipsoidal_datum, \ 

15 _mean_radius, _spherical_datum, _WGS84, _EWGS84 

16# from pygeodesy.ellipsoids import Ellipsoid, _EWGS84 # from .datums 

17from pygeodesy.errors import IntersectionError, LimitError, limiterrors, \ 

18 _TypeError, _ValueError, _xattr, _xError, \ 

19 _xkwds, _xkwds_pop 

20from pygeodesy.fmath import euclid, hypot, hypot_, hypot2, sqrt0 

21from pygeodesy.fsums import fsumf_ 

22from pygeodesy.interns import NN, _delta_, _distant_, _inside_, _SPACE_, _too_ 

23from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS 

24from pygeodesy.named import _NamedTuple, _xnamed, Fmt, unstr 

25from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, \ 

26 Intersection3Tuple, LatLon2Tuple, \ 

27 PhiLam2Tuple, Vector3Tuple 

28# from pygeodesy.streprs import Fmt, unstr # from .named 

29# from pygeodesy.triaxials import _hartzell2 # _MODS 

30from pygeodesy.units import _isHeight, _isRadius, Bearing, Degrees, Degrees_, \ 

31 Distance, Distance_, Height, Lam_, Lat, Lon, \ 

32 Meter_, Phi_, Radians, Radians_, Radius, Radius_, \ 

33 Scalar, _100km 

34from pygeodesy.utily import acos1, atan2b, atan2d, degrees2m, _loneg, m2degrees, \ 

35 tan_2, sincos2, sincos2_, sincos2d_, _Wrap 

36# from pygeodesy.vector3d import _otherV3d # _MODS 

37# from pygeodesy.vector3dBase import _xyz_y_z3 # _MODS 

38# from pygeodesy import ellipsoidalExact, ellipsoidalKarney, vector3d, \ 

39# sphericalNvector, sphericalTrigonometry # _MODS 

40 

41from contextlib import contextmanager 

42from math import asin, atan, atan2, cos, degrees, fabs, radians, sin, sqrt # pow 

43 

44__all__ = _ALL_LAZY.formy 

45__version__ = '23.12.03' 

46 

47_D2_R2 = (PI / _180_0)**2 # degrees- to radians-squared 

48_ratio_ = 'ratio' 

49_xline_ = 'xline' 

50 

51 

52def _anti2(a, b, n_2, n, n2): 

53 '''(INTERNAL) Helper for C{antipode} and C{antipode_}. 

54 ''' 

55 r = remainder(a, n) if fabs(a) > n_2 else a 

56 if r == a: 

57 r = -r 

58 b += n 

59 if fabs(b) > n: 

60 b = remainder(b, n2) 

61 return float0_(r, b) 

62 

63 

64def antipode(lat, lon, name=NN): 

65 '''Return the antipode, the point diametrically opposite 

66 to a given point in C{degrees}. 

67 

68 @arg lat: Latitude (C{degrees}). 

69 @arg lon: Longitude (C{degrees}). 

70 @kwarg name: Optional name (C{str}). 

71 

72 @return: A L{LatLon2Tuple}C{(lat, lon)}. 

73 

74 @see: Functions L{antipode_} and L{normal} and U{Geosphere 

75 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}. 

76 ''' 

77 return LatLon2Tuple(*_anti2(lat, lon, _90_0, _180_0, _360_0), name=name) 

78 

79 

80def antipode_(phi, lam, name=NN): 

81 '''Return the antipode, the point diametrically opposite 

82 to a given point in C{radians}. 

83 

84 @arg phi: Latitude (C{radians}). 

85 @arg lam: Longitude (C{radians}). 

86 @kwarg name: Optional name (C{str}). 

87 

88 @return: A L{PhiLam2Tuple}C{(phi, lam)}. 

89 

90 @see: Functions L{antipode} and L{normal_} and U{Geosphere 

91 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}. 

92 ''' 

93 return PhiLam2Tuple(*_anti2(phi, lam, PI_2, PI, PI2), name=name) 

94 

95 

96def bearing(lat1, lon1, lat2, lon2, **final_wrap): 

97 '''Compute the initial or final bearing (forward or reverse 

98 azimuth) between a (spherical) start and end point. 

99 

100 @arg lat1: Start latitude (C{degrees}). 

101 @arg lon1: Start longitude (C{degrees}). 

102 @arg lat2: End latitude (C{degrees}). 

103 @arg lon2: End longitude (C{degrees}). 

104 @kwarg final_wrap: Optional keyword arguments for function 

105 L{pygeodesy.bearing_}. 

106 

107 @return: Initial or final bearing (compass C{degrees360}) or 

108 zero if start and end point coincide. 

109 ''' 

110 r = bearing_(Phi_(lat1=lat1), Lam_(lon1=lon1), 

111 Phi_(lat2=lat2), Lam_(lon2=lon2), **final_wrap) 

112 return degrees(r) 

113 

114 

115def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False): 

116 '''Compute the initial or final bearing (forward or reverse azimuth) 

117 between a (spherical) start and end point. 

118 

119 @arg phi1: Start latitude (C{radians}). 

120 @arg lam1: Start longitude (C{radians}). 

121 @arg phi2: End latitude (C{radians}). 

122 @arg lam2: End longitude (C{radians}). 

123 @kwarg final: Return final bearing if C{True}, initial otherwise (C{bool}). 

124 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{phi2}} and 

125 B{C{lam2}} (C{bool}). 

126 

127 @return: Initial or final bearing (compass C{radiansPI2}) or zero if start 

128 and end point coincide. 

129 

130 @see: U{Bearing<https://www.Movable-Type.co.UK/scripts/latlong.html>}, U{Course 

131 between two points<https://www.EdWilliams.org/avform147.htm#Crs>} and 

132 U{Bearing Between Two Points<https://web.Archive.org/web/20020630205931/ 

133 https://MathForum.org/library/drmath/view/55417.html>}. 

134 ''' 

135 db, phi2, lam2 = _Wrap.philam3(lam1, phi2, lam2, wrap) 

136 if final: # swap plus PI 

137 phi1, lam1, phi2, lam2, db = phi2, lam2, phi1, lam1, -db 

138 r = PI3 

139 else: 

140 r = PI2 

141 sa1, ca1, sa2, ca2, sdb, cdb = sincos2_(phi1, phi2, db) 

142 

143 x = ca1 * sa2 - sa1 * ca2 * cdb 

144 y = sdb * ca2 

145 return _umod_PI2(atan2(y, x) + r) # .utily.wrapPI2 

146 

147 

148def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf 

149 '''(INTERNAL) Compute initial and final bearing. 

150 ''' 

151 try: # for LatLon_ and ellipsoidal LatLon 

152 return p1.bearingTo2(p2, wrap=wrap) 

153 except AttributeError: 

154 pass 

155 # XXX spherical version, OK for ellipsoidal ispolar? 

156 t = p1.philam + p2.philam 

157 return Bearing2Tuple(degrees(bearing_(*t, final=False, wrap=wrap)), 

158 degrees(bearing_(*t, final=True, wrap=wrap)), 

159 name=_bearingTo2.__name__) 

160 

161 

162def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False): 

163 '''Return the angle from North for the direction vector M{(lon2 - lon1, 

164 lat2 - lat1)} between two points. 

165 

166 Suitable only for short, not near-polar vectors up to a few hundred 

167 Km or Miles. Use function L{pygeodesy.bearing} for longer vectors. 

168 

169 @arg lat1: From latitude (C{degrees}). 

170 @arg lon1: From longitude (C{degrees}). 

171 @arg lat2: To latitude (C{degrees}). 

172 @arg lon2: To longitude (C{degrees}). 

173 @kwarg adjust: Adjust the longitudinal delta by the cosine of the 

174 mean latitude (C{bool}). 

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

176 and B{C{lon2}} (C{bool}). 

177 

178 @return: Compass angle from North (C{degrees360}). 

179 

180 @note: Courtesy of Martin Schultz. 

181 

182 @see: U{Local, flat earth approximation 

183 <https://www.EdWilliams.org/avform.htm#flat>}. 

184 ''' 

185 d_lon, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap) 

186 if adjust: # scale delta lon 

187 d_lon *= _scale_deg(lat1, lat2) 

188 return atan2b(d_lon, lat2 - lat1) 

189 

190 

191def cosineAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

192 '''Compute the distance between two (ellipsoidal) points using the U{Andoyer-Lambert 

193 <https://books.google.com/books?id=x2UiAQAAIAAJ>} correction of the U{Law of 

194 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula. 

195 

196 @arg lat1: Start latitude (C{degrees}). 

197 @arg lon1: Start longitude (C{degrees}). 

198 @arg lat2: End latitude (C{degrees}). 

199 @arg lon2: End longitude (C{degrees}). 

200 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

201 L{Ellipsoid2} or L{a_f2Tuple}) to use. 

202 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

203 B{C{lat2}} and B{C{lon2}} (C{bool}). 

204 

205 @return: Distance (C{meter}, same units as the B{C{datum}}'s 

206 ellipsoid axes or C{radians} if B{C{datum}} is C{None}). 

207 

208 @raise TypeError: Invalid B{C{datum}}. 

209 

210 @see: Functions L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert}, 

211 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, 

212 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method 

213 L{Ellipsoid.distance2}. 

214 ''' 

215 return _dE(cosineAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2) 

216 

217 

218def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): 

219 '''Compute the I{angular} distance between two (ellipsoidal) points using the U{Andoyer-Lambert 

220 <https://books.google.com/books?id=x2UiAQAAIAAJ>} correction of the U{Law of 

221 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula. 

222 

223 @arg phi2: End latitude (C{radians}). 

224 @arg phi1: Start latitude (C{radians}). 

225 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

226 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

227 L{Ellipsoid2} or L{a_f2Tuple}) to use. 

228 

229 @return: Angular distance (C{radians}). 

230 

231 @raise TypeError: Invalid B{C{datum}}. 

232 

233 @see: Functions L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert_}, 

234 L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

235 L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP 

236 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/Distance/ 

237 AndoyerLambert.php>}. 

238 ''' 

239 s2, c2, s1, c1, r, c21 = _sincosa6(phi2, phi1, lam21) 

240 if isnon0(c1) and isnon0(c2): 

241 E = _ellipsoidal(datum, cosineAndoyerLambert_) 

242 if E.f: # ellipsoidal 

243 r2 = atan2(E.b_a * s2, c2) 

244 r1 = atan2(E.b_a * s1, c1) 

245 s2, c2, s1, c1 = sincos2_(r2, r1) 

246 r = acos1(s1 * s2 + c1 * c2 * c21) 

247 if r: 

248 sr, _, sr_2, cr_2 = sincos2_(r, r * _0_5) 

249 if isnon0(sr_2) and isnon0(cr_2): 

250 s = (sr + r) * ((s1 - s2) / sr_2)**2 

251 c = (sr - r) * ((s1 + s2) / cr_2)**2 

252 r += (c - s) * E.f * _0_125 

253 return r 

254 

255 

256def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

257 '''Compute the distance between two (ellipsoidal) points using the U{Forsythe-Andoyer-Lambert 

258 <https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} correction of the U{Law of Cosines 

259 <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula. 

260 

261 @arg lat1: Start latitude (C{degrees}). 

262 @arg lon1: Start longitude (C{degrees}). 

263 @arg lat2: End latitude (C{degrees}). 

264 @arg lon2: End longitude (C{degrees}). 

265 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

266 L{Ellipsoid2} or L{a_f2Tuple}) to use. 

267 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

268 B{C{lat2}} and B{C{lon2}} (C{bool}). 

269 

270 @return: Distance (C{meter}, same units as the B{C{datum}}'s 

271 ellipsoid axes or C{radians} if B{C{datum}} is C{None}). 

272 

273 @raise TypeError: Invalid B{C{datum}}. 

274 

275 @see: Functions L{cosineForsytheAndoyerLambert_}, L{cosineAndoyerLambert}, 

276 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, 

277 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method 

278 L{Ellipsoid.distance2}. 

279 ''' 

280 return _dE(cosineForsytheAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2) 

281 

282 

283def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): 

284 '''Compute the I{angular} distance between two (ellipsoidal) points using the 

285 U{Forsythe-Andoyer-Lambert<https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} correction of 

286 the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} 

287 formula. 

288 

289 @arg phi2: End latitude (C{radians}). 

290 @arg phi1: Start latitude (C{radians}). 

291 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

292 @kwarg datum: Datum (L{Datum}) or ellipsoid to use (L{Ellipsoid}, 

293 L{Ellipsoid2} or L{a_f2Tuple}). 

294 

295 @return: Angular distance (C{radians}). 

296 

297 @raise TypeError: Invalid B{C{datum}}. 

298 

299 @see: Functions L{cosineForsytheAndoyerLambert}, L{cosineAndoyerLambert_}, 

300 L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

301 L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP 

302 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/ 

303 Distance/ForsytheCorrection.php>}. 

304 ''' 

305 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21) 

306 if r and isnon0(c1) and isnon0(c2): 

307 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_) 

308 if E.f: # ellipsoidal 

309 sr, cr, s2r, _ = sincos2_(r, r * 2) 

310 if isnon0(sr) and fabs(cr) < EPS1: 

311 s = (s1 + s2)**2 / (1 + cr) 

312 t = (s1 - s2)**2 / (1 - cr) 

313 x = s + t 

314 y = s - t 

315 

316 s = 8 * r**2 / sr 

317 a = 64 * r + s * cr * 2 # 16 * r**2 / tan(r) 

318 d = 48 * sr + s # 8 * r**2 / tan(r) 

319 b = -2 * d 

320 e = 30 * s2r 

321 c = fsumf_(30 * r, e * _0_5, s * cr) # 8 * r**2 / tan(r) 

322 t = fsumf_( a * x, e * y**2, b * y, -c * x**2, d * x * y) 

323 

324 r += fsumf_(-r * x, 3 * y * sr, t * E.f / _32_0) * E.f * _0_25 

325 return r 

326 

327 

328def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

329 '''Compute the distance between two points using the U{spherical Law of Cosines 

330 <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula. 

331 

332 @arg lat1: Start latitude (C{degrees}). 

333 @arg lon1: Start longitude (C{degrees}). 

334 @arg lat2: End latitude (C{degrees}). 

335 @arg lon2: End longitude (C{degrees}). 

336 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

337 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

338 L{a_f2Tuple}) to use. 

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

340 and B{C{lon2}} (C{bool}). 

341 

342 @return: Distance (C{meter}, same units as B{C{radius}} or the 

343 ellipsoid or datum axes). 

344 

345 @raise TypeError: Invalid B{C{radius}}. 

346 

347 @see: Functions L{cosineLaw_}, L{cosineAndoyerLambert}, 

348 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean}, 

349 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and 

350 L{vincentys} and method L{Ellipsoid.distance2}. 

351 

352 @note: See note at function L{vincentys_}. 

353 ''' 

354 return _dS(cosineLaw_, radius, wrap, lat1, lon1, lat2, lon2) 

355 

356 

357def cosineLaw_(phi2, phi1, lam21): 

358 '''Compute the I{angular} distance between two points using the U{spherical Law of 

359 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula. 

360 

361 @arg phi2: End latitude (C{radians}). 

362 @arg phi1: Start latitude (C{radians}). 

363 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

364 

365 @return: Angular distance (C{radians}). 

366 

367 @see: Functions L{cosineLaw}, L{cosineAndoyerLambert_}, 

368 L{cosineForsytheAndoyerLambert_}, L{equirectangular_}, 

369 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_}, 

370 L{haversine_}, L{thomas_} and L{vincentys_}. 

371 

372 @note: See note at function L{vincentys_}. 

373 ''' 

374 return _sincosa6(phi2, phi1, lam21)[4] 

375 

376 

377def _d3(wrap, lat1, lon1, lat2, lon2): 

378 '''(INTERNAL) Helper for _dE, _dS and _eA. 

379 ''' 

380 if wrap: 

381 d_lon, lat2, _ = _Wrap.latlon3(lon1, lat2, lon2, wrap) 

382 return radians(lat2), Phi_(lat1=lat1), radians(d_lon) 

383 else: # for backward compaibility 

384 return Phi_(lat2=lat2), Phi_(lat1=lat1), Phi_(d_lon=lon2 - lon1) 

385 

386 

387def _dE(func_, earth, *wrap_lls): 

388 '''(INTERNAL) Helper for ellipsoidal distances. 

389 ''' 

390 E = _ellipsoidal(earth, func_) 

391 r = func_(*_d3(*wrap_lls), datum=E) 

392 return r * E.a 

393 

394 

395def _dS(func_, radius, *wrap_lls, **adjust): 

396 '''(INTERNAL) Helper for spherical distances. 

397 ''' 

398 r = func_(*_d3(*wrap_lls), **adjust) 

399 if radius is not R_M: 

400 _, lat1, _, lat2, _ = wrap_lls 

401 radius = _mean_radius(radius, lat1, lat2) 

402 return r * radius 

403 

404 

405def _eA(excess_, radius, *wrap_lls): 

406 '''(INTERNAL) Helper for spherical excess or area. 

407 ''' 

408 r = excess_(*_d3(*wrap_lls)) 

409 if radius: 

410 _, lat1, _, lat2, _ = wrap_lls 

411 r *= _mean_radius(radius, lat1, lat2)**2 

412 return r 

413 

414 

415def _ellipsoidal(earth, where): 

416 '''(INTERNAL) Helper for distances. 

417 ''' 

418 return _EWGS84 if earth in (_WGS84, _EWGS84) else ( 

419 earth if isinstance(earth, Ellipsoid) else 

420 (earth if isinstance(earth, Datum) else # PYCHOK indent 

421 _ellipsoidal_datum(earth, name=where.__name__)).ellipsoid) 

422 

423 

424def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **adjust_limit_wrap): 

425 '''Compute the distance between two points using the U{Equirectangular Approximation 

426 / Projection<https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}. 

427 

428 @arg lat1: Start latitude (C{degrees}). 

429 @arg lon1: Start longitude (C{degrees}). 

430 @arg lat2: End latitude (C{degrees}). 

431 @arg lon2: End longitude (C{degrees}). 

432 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

433 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

434 L{a_f2Tuple}). 

435 @kwarg adjust_limit_wrap: Optional keyword arguments for 

436 function L{equirectangular_}. 

437 

438 @return: Distance (C{meter}, same units as B{C{radius}} or 

439 the ellipsoid or datum axes). 

440 

441 @raise TypeError: Invalid B{C{radius}}. 

442 

443 @see: Function L{equirectangular_} for more details, the 

444 available B{C{options}}, errors, restrictions and other, 

445 approximate or accurate distance functions. 

446 ''' 

447 d = sqrt(equirectangular_(Lat(lat1=lat1), Lon(lon1=lon1), 

448 Lat(lat2=lat2), Lon(lon2=lon2), 

449 **adjust_limit_wrap).distance2) # PYCHOK 4 vs 2-3 

450 return degrees2m(d, radius=_mean_radius(radius, lat1, lat2)) 

451 

452 

453def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap): 

454 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians} 

455 and L{hausdorff._HausdorffMeterRedians} classes. 

456 ''' 

457 return equirectangular_(lat1, lon1, lat2, lon2, **adjust_limit_wrap).distance2 * _D2_R2 

458 

459 

460def equirectangular_(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False): 

461 '''Compute the distance between two points using the U{Equirectangular Approximation 

462 / Projection<https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}. 

463 

464 This approximation is valid for short distance of several hundred Km 

465 or Miles, see the B{C{limit}} keyword argument and L{LimitError}. 

466 

467 @arg lat1: Start latitude (C{degrees}). 

468 @arg lon1: Start longitude (C{degrees}). 

469 @arg lat2: End latitude (C{degrees}). 

470 @arg lon2: End longitude (C{degrees}). 

471 @kwarg adjust: Adjust the wrapped, unrolled longitudinal delta 

472 by the cosine of the mean latitude (C{bool}). 

473 @kwarg limit: Optional limit for lat- and longitudinal deltas 

474 (C{degrees}) or C{None} or C{0} for unlimited. 

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

476 and B{C{lon2}} (C{bool}). 

477 

478 @return: A L{Distance4Tuple}C{(distance2, delta_lat, delta_lon, 

479 unroll_lon2)} in C{degrees squared}. 

480 

481 @raise LimitError: If the lat- and/or longitudinal delta exceeds the 

482 B{C{-limit..limit}} range and L{pygeodesy.limiterrors} 

483 set to C{True}. 

484 

485 @see: U{Local, flat earth approximation 

486 <https://www.EdWilliams.org/avform.htm#flat>}, functions 

487 L{equirectangular}, L{cosineAndoyerLambert}, 

488 L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean}, 

489 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} 

490 and L{vincentys} and methods L{Ellipsoid.distance2}, 

491 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

492 ''' 

493 d_lon, lat2, ulon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap) 

494 d_lat = lat2 - lat1 

495 

496 if limit and limit > 0 and limiterrors(): 

497 d = max(fabs(d_lat), fabs(d_lon)) 

498 if d > limit: 

499 t = _SPACE_(_delta_, Fmt.PAREN_g(d), Fmt.exceeds_limit(limit)) 

500 s = unstr(equirectangular_, lat1, lon1, lat2, lon2, 

501 limit=limit, wrap=wrap) 

502 raise LimitError(s, txt=t) 

503 

504 if adjust: # scale delta lon 

505 d_lon *= _scale_deg(lat1, lat2) 

506 

507 d2 = hypot2(d_lat, d_lon) # degrees squared! 

508 return Distance4Tuple(d2, d_lat, d_lon, ulon2 - lon2) 

509 

510 

511def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False): 

512 '''Approximate the C{Euclidean} distance between two (spherical) points. 

513 

514 @arg lat1: Start latitude (C{degrees}). 

515 @arg lon1: Start longitude (C{degrees}). 

516 @arg lat2: End latitude (C{degrees}). 

517 @arg lon2: End longitude (C{degrees}). 

518 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

519 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

520 L{a_f2Tuple}) to use. 

521 @kwarg adjust: Adjust the longitudinal delta by the cosine of 

522 the mean latitude (C{bool}). 

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

524 and B{C{lon2}} (C{bool}). 

525 

526 @return: Distance (C{meter}, same units as B{C{radius}} or the 

527 ellipsoid or datum axes). 

528 

529 @raise TypeError: Invalid B{C{radius}}. 

530 

531 @see: U{Distance between two (spherical) points 

532 <https://www.EdWilliams.org/avform.htm#Dist>}, functions L{euclid}, 

533 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

534 L{cosineLaw}, L{equirectangular}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

535 L{haversine}, L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, 

536 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

537 ''' 

538 return _dS(euclidean_, radius, wrap, lat1, lon1, lat2, lon2, adjust=adjust) 

539 

540 

541def euclidean_(phi2, phi1, lam21, adjust=True): 

542 '''Approximate the I{angular} C{Euclidean} distance between two (spherical) points. 

543 

544 @arg phi2: End latitude (C{radians}). 

545 @arg phi1: Start latitude (C{radians}). 

546 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

547 @kwarg adjust: Adjust the longitudinal delta by the cosine 

548 of the mean latitude (C{bool}). 

549 

550 @return: Angular distance (C{radians}). 

551 

552 @see: Functions L{euclid}, L{euclidean}, L{cosineAndoyerLambert_}, 

553 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_}, 

554 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_}, L{thomas_} 

555 and L{vincentys_}. 

556 ''' 

557 if adjust: 

558 lam21 *= _scale_rad(phi2, phi1) 

559 return euclid(phi2 - phi1, lam21) 

560 

561 

562def excessAbc_(A, b, c): 

563 '''Compute the I{spherical excess} C{E} of a (spherical) triangle from two sides 

564 and the included (small) angle. 

565 

566 @arg A: An interior triangle angle (C{radians}). 

567 @arg b: Frist adjacent triangle side (C{radians}). 

568 @arg c: Second adjacent triangle side (C{radians}). 

569 

570 @return: Spherical excess (C{radians}). 

571 

572 @raise UnitError: Invalid B{C{A}}, B{C{b}} or B{C{c}}. 

573 

574 @see: Functions L{excessGirard_}, L{excessLHuilier_} and U{Spherical 

575 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}. 

576 ''' 

577 A = Radians_(A=A) 

578 b = Radians_(b=b) * _0_5 

579 c = Radians_(c=c) * _0_5 

580 

581 sA, cA, sb, cb, sc, cc = sincos2_(A, b, c) 

582 return atan2(sA * sb * sc, cb * cc + cA * sb * sc) * _2_0 

583 

584 

585def excessCagnoli_(a, b, c): 

586 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{Cagnoli's 

587 <https://Zenodo.org/record/35392>} (D.34) formula. 

588 

589 @arg a: First triangle side (C{radians}). 

590 @arg b: Second triangle side (C{radians}). 

591 @arg c: Third triangle side (C{radians}). 

592 

593 @return: Spherical excess (C{radians}). 

594 

595 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}. 

596 

597 @see: Function L{excessLHuilier_} and U{Spherical trigonometry 

598 <https://WikiPedia.org/wiki/Spherical_trigonometry>}. 

599 ''' 

600 a = Radians_(a=a) 

601 b = Radians_(b=b) 

602 c = Radians_(c=c) 

603 

604 s = fsumf_(a, b, c) * _0_5 

605 _s = sin 

606 r = _s(s) * _s(s - a) * _s(s - b) * _s(s - c) 

607 c = cos(a * _0_5) * cos(b * _0_5) * cos(c * _0_5) 

608 r = asin(sqrt(r) * _0_5 / c) if c and r > 0 else _0_0 

609 return Radians(Cagnoli=r * _2_0) 

610 

611 

612def excessGirard_(A, B, C): 

613 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{Girard's 

614 <https://MathWorld.Wolfram.com/GirardsSphericalExcessFormula.html>} formula. 

615 

616 @arg A: First interior triangle angle (C{radians}). 

617 @arg B: Second interior triangle angle (C{radians}). 

618 @arg C: Third interior triangle angle (C{radians}). 

619 

620 @return: Spherical excess (C{radians}). 

621 

622 @raise UnitError: Invalid B{C{A}}, B{C{B}} or B{C{C}}. 

623 

624 @see: Function L{excessLHuilier_} and U{Spherical trigonometry 

625 <https://WikiPedia.org/wiki/Spherical_trigonometry>}. 

626 ''' 

627 return Radians(Girard=fsumf_(Radians_(A=A), 

628 Radians_(B=B), 

629 Radians_(C=C), -PI)) 

630 

631 

632def excessLHuilier_(a, b, c): 

633 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{L'Huilier's 

634 <https://MathWorld.Wolfram.com/LHuiliersTheorem.html>}'s Theorem. 

635 

636 @arg a: First triangle side (C{radians}). 

637 @arg b: Second triangle side (C{radians}). 

638 @arg c: Third triangle side (C{radians}). 

639 

640 @return: Spherical excess (C{radians}). 

641 

642 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}. 

643 

644 @see: Function L{excessCagnoli_}, L{excessGirard_} and U{Spherical 

645 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}. 

646 ''' 

647 a = Radians_(a=a) 

648 b = Radians_(b=b) 

649 c = Radians_(c=c) 

650 

651 s = fsumf_(a, b, c) * _0_5 

652 _t = tan_2 

653 r = _t(s) * _t(s - a) * _t(s - b) * _t(s - c) 

654 r = atan(sqrt(r)) if r > 0 else _0_0 

655 return Radians(LHuilier=r * _4_0) 

656 

657 

658def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

659 '''Compute the surface area of a (spherical) quadrilateral bounded by a 

660 segment of a great circle, two meridians and the equator using U{Karney's 

661 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>} 

662 method. 

663 

664 @arg lat1: Start latitude (C{degrees}). 

665 @arg lon1: Start longitude (C{degrees}). 

666 @arg lat2: End latitude (C{degrees}). 

667 @arg lon2: End longitude (C{degrees}). 

668 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

669 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

670 L{a_f2Tuple}) or C{None}. 

671 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

672 B{C{lat2}} and B{C{lon2}} (C{bool}). 

673 

674 @return: Surface area, I{signed} (I{square} C{meter} or the same units as 

675 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians}) 

676 if C{B{radius}=0} or C{None}. 

677 

678 @raise TypeError: Invalid B{C{radius}}. 

679 

680 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}. 

681 

682 @raise ValueError: Semi-circular longitudinal delta. 

683 

684 @see: Functions L{excessKarney_} and L{excessQuad}. 

685 ''' 

686 return _eA(excessKarney_, radius, wrap, lat1, lon1, lat2, lon2) 

687 

688 

689def excessKarney_(phi2, phi1, lam21): 

690 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded by 

691 a segment of a great circle, two meridians and the equator using U{Karney's 

692 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>} 

693 method. 

694 

695 @arg phi2: End latitude (C{radians}). 

696 @arg phi1: Start latitude (C{radians}). 

697 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

698 

699 @return: Spherical excess, I{signed} (C{radians}). 

700 

701 @raise ValueError: Semi-circular longitudinal delta B{C{lam21}}. 

702 

703 @see: Function L{excessKarney} and U{Area of a spherical polygon 

704 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}. 

705 ''' 

706 # from: Veness <https://www.Movable-Type.co.UK/scripts/latlong.html> Area 

707 # method due to Karney: for each edge of the polygon, 

708 # 

709 # tan(Δλ / 2) · (tan(φ1 / 2) + tan(φ2 / 2)) 

710 # tan(E / 2) = ----------------------------------------- 

711 # 1 + tan(φ1 / 2) · tan(φ2 / 2) 

712 # 

713 # where E is the spherical excess of the trapezium obtained by extending 

714 # the edge to the equator-circle vector for each edge (see also ***). 

715 _t = tan_2 

716 t2 = _t(phi2) 

717 t1 = _t(phi1) 

718 t = _t(lam21, lam21=None) 

719 return Radians(Karney=atan2(t * (t1 + t2), 

720 _1_0 + (t1 * t2)) * _2_0) 

721 

722 

723# ***) Original post no longer available, following is a copy of the main part 

724# <http://OSGeo-org.1560.x6.Nabble.com/Area-of-a-spherical-polygon-td3841625.html> 

725# 

726# The area of a polygon on a (unit) sphere is given by the spherical excess 

727# 

728# A = 2 * pi - sum(exterior angles) 

729# 

730# However this is badly conditioned if the polygon is small. In this case, use 

731# 

732# A = sum(S12{i, i+1}) over the edges of the polygon 

733# 

734# where S12 is the area of the quadrilateral bounded by an edge of the polygon, 

735# two meridians and the equator, i.e. with vertices (phi1, lambda1), (phi2, 

736# lambda2), (0, lambda1) and (0, lambda2). S12 is given by 

737# 

738# tan(S12 / 2) = tan(lambda21 / 2) * (tan(phi1 / 2) + tan(phi2 / 2)) / 

739# (tan(phi1 / 2) * tan(phi2 / 2) + 1) 

740# 

741# = tan(lambda21 / 2) * tanh((Lamb(phi1) + Lamb(phi2)) / 2) 

742# 

743# where lambda21 = lambda2 - lambda1 and Lamb(x) is the Lambertian (or the 

744# inverse Gudermannian) function 

745# 

746# Lambertian(x) = asinh(tan(x)) = atanh(sin(x)) = 2 * atanh(tan(x / 2)) 

747# 

748# Notes: The formula for S12 is exact, except that... 

749# - it is indeterminate if an edge is a semi-circle 

750# - the formula for A applies only if the polygon does not include a pole 

751# (if it does, then add +/- 2 * pi to the result) 

752# - in the limit of small phi and lambda, S12 reduces to the trapezoidal 

753# formula, S12 = (lambda2 - lambda1) * (phi1 + phi2) / 2 

754# - I derived this result from the equation for the area of a spherical 

755# triangle in terms of two edges and the included angle given by, e.g. 

756# U{Todhunter, I. - Spherical Trigonometry (1871), Sec. 103, Eq. (2) 

757# <http://Books.Google.com/books?id=3uBHAAAAIAAJ&pg=PA71>} 

758# - I would be interested to know if this formula for S12 is already known 

759# - Charles Karney 

760 

761 

762def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

763 '''Compute the surface area of a (spherical) quadrilateral bounded by a segment 

764 of a great circle, two meridians and the equator. 

765 

766 @arg lat1: Start latitude (C{degrees}). 

767 @arg lon1: Start longitude (C{degrees}). 

768 @arg lat2: End latitude (C{degrees}). 

769 @arg lon2: End longitude (C{degrees}). 

770 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

771 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

772 L{a_f2Tuple}) or C{None}. 

773 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

774 B{C{lat2}} and B{C{lon2}} (C{bool}). 

775 

776 @return: Surface area, I{signed} (I{square} C{meter} or the same units as 

777 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians}) 

778 if C{B{radius}=0} or C{None}. 

779 

780 @raise TypeError: Invalid B{C{radius}}. 

781 

782 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}. 

783 

784 @see: Function L{excessQuad_} and L{excessKarney}. 

785 ''' 

786 return _eA(excessQuad_, radius, wrap, lat1, lon1, lat2, lon2) 

787 

788 

789def excessQuad_(phi2, phi1, lam21): 

790 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded 

791 by a segment of a great circle, two meridians and the equator. 

792 

793 @arg phi2: End latitude (C{radians}). 

794 @arg phi1: Start latitude (C{radians}). 

795 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

796 

797 @return: Spherical excess, I{signed} (C{radians}). 

798 

799 @see: Function L{excessQuad} and U{Spherical trigonometry 

800 <https://WikiPedia.org/wiki/Spherical_trigonometry>}. 

801 ''' 

802 s = sin((phi2 + phi1) * _0_5) 

803 c = cos((phi2 - phi1) * _0_5) 

804 return Radians(Quad=atan2(tan_2(lam21) * s, c) * _2_0) 

805 

806 

807def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False): 

808 '''Compute the distance between two (ellipsoidal) points using 

809 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/ 

810 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

811 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula. 

812 

813 @arg lat1: Start latitude (C{degrees}). 

814 @arg lon1: Start longitude (C{degrees}). 

815 @arg lat2: End latitude (C{degrees}). 

816 @arg lon2: End longitude (C{degrees}). 

817 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

818 L{Ellipsoid2} or L{a_f2Tuple}) to use. 

819 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}), 

820 see method L{pygeodesy.Ellipsoid.roc2_}. 

821 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

822 B{C{lat2}} and B{C{lon2}} (C{bool}). 

823 

824 @return: Distance (C{meter}, same units as the B{C{datum}}'s 

825 ellipsoid axes). 

826 

827 @raise TypeError: Invalid B{C{datum}}. 

828 

829 @note: The meridional and prime_vertical radii of curvature 

830 are taken and scaled at the mean of both latitude. 

831 

832 @see: Functions L{flatLocal_} or L{hubeny_}, L{cosineLaw}, L{flatPolar}, 

833 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

834 L{equirectangular}, L{euclidean}, L{haversine}, L{thomas}, 

835 L{vincentys}, method L{Ellipsoid.distance2} and U{local, flat 

836 earth approximation<https://www.EdWilliams.org/avform.htm#flat>}. 

837 ''' 

838 E = _ellipsoidal(datum, flatLocal) 

839 return E._hubeny_2(*_d3(wrap, lat1, lon1, lat2, lon2), 

840 scaled=scaled, squared=False) * E.a 

841 

842hubeny = flatLocal # PYCHOK for Karl Hubeny 

843 

844 

845def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True): 

846 '''Compute the I{angular} distance between two (ellipsoidal) points using 

847 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/ 

848 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

849 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula. 

850 

851 @arg phi2: End latitude (C{radians}). 

852 @arg phi1: Start latitude (C{radians}). 

853 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

854 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

855 L{Ellipsoid2} or L{a_f2Tuple}) to use. 

856 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}), 

857 see method L{pygeodesy.Ellipsoid.roc2_}. 

858 

859 @return: Angular distance (C{radians}). 

860 

861 @raise TypeError: Invalid B{C{datum}}. 

862 

863 @note: The meridional and prime_vertical radii of curvature 

864 are taken and scaled I{at the mean of both latitude}. 

865 

866 @see: Functions L{flatLocal} or L{hubeny}, L{cosineAndoyerLambert_}, 

867 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_}, 

868 L{equirectangular_}, L{euclidean_}, L{haversine_}, L{thomas_} 

869 and L{vincentys_} and U{local, flat earth approximation 

870 <https://www.EdWilliams.org/avform.htm#flat>}. 

871 ''' 

872 E = _ellipsoidal(datum, flatLocal_) 

873 return E._hubeny_2(phi2, phi1, lam21, scaled=scaled, squared=False) 

874 

875hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny 

876 

877 

878def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

879 '''Compute the distance between two (spherical) points using 

880 the U{polar coordinate flat-Earth <https://WikiPedia.org/wiki/ 

881 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

882 formula. 

883 

884 @arg lat1: Start latitude (C{degrees}). 

885 @arg lon1: Start longitude (C{degrees}). 

886 @arg lat2: End latitude (C{degrees}). 

887 @arg lon2: End longitude (C{degrees}). 

888 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

889 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

890 L{a_f2Tuple}) to use. 

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

892 and B{C{lon2}} (C{bool}). 

893 

894 @return: Distance (C{meter}, same units as B{C{radius}} or the 

895 ellipsoid or datum axes). 

896 

897 @raise TypeError: Invalid B{C{radius}}. 

898 

899 @see: Functions L{flatPolar_}, L{cosineAndoyerLambert}, 

900 L{cosineForsytheAndoyerLambert},L{cosineLaw}, 

901 L{flatLocal}/L{hubeny}, L{equirectangular}, 

902 L{euclidean}, L{haversine}, L{thomas} and 

903 L{vincentys}. 

904 ''' 

905 return _dS(flatPolar_, radius, wrap, lat1, lon1, lat2, lon2) 

906 

907 

908def flatPolar_(phi2, phi1, lam21): 

909 '''Compute the I{angular} distance between two (spherical) points 

910 using the U{polar coordinate flat-Earth<https://WikiPedia.org/wiki/ 

911 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

912 formula. 

913 

914 @arg phi2: End latitude (C{radians}). 

915 @arg phi1: Start latitude (C{radians}). 

916 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

917 

918 @return: Angular distance (C{radians}). 

919 

920 @see: Functions L{flatPolar}, L{cosineAndoyerLambert_}, 

921 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

922 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

923 L{haversine_}, L{thomas_} and L{vincentys_}. 

924 ''' 

925 a = fabs(PI_2 - phi1) # co-latitude 

926 b = fabs(PI_2 - phi2) # co-latitude 

927 if a < b: 

928 a, b = b, a 

929 if a < EPS0: 

930 a = _0_0 

931 elif b > 0: 

932 b = b / a # /= chokes PyChecker 

933 c = b * cos(lam21) * _2_0 

934 c = fsumf_(_1_0, b**2, -fabs(c)) 

935 a *= sqrt0(c) 

936 return a 

937 

938 

939def _hartzell(inst, los, earth, **kwds): 

940 '''(INTERNAL) Helper for C{CartesianBase.hartzell} and C{LatLonBase.hartzell}. 

941 ''' 

942 if earth is not None: 

943 earth = _spherical_datum(earth, name=hartzell.__name__) 

944 inst = inst.toDatum(earth) 

945 h = inst.height 

946 if h > 0: # EPS0 

947 r = hartzell(inst, los=los, earth=earth or inst.datum, **kwds) 

948 elif h < 0: # EPS0 

949 raise IntersectionError(pov=inst, los=los, height=h, txt=_inside_) 

950 else: 

951 r = inst 

952 return r 

953 

954 

955def hartzell(pov, los=None, earth=_WGS84, name=NN, **LatLon_and_kwds): 

956 '''Compute the intersection of the earth's surface and a Line-Of-Sight 

957 from a Point-Of-View in space. 

958 

959 @arg pov: Point-Of-View outside the earth (C{Cartesian}, L{Ecef9Tuple} 

960 C{LatLon} or L{Vector3d}). 

961 @kwarg los: Line-Of-Sight, I{direction} to earth (L{Los}, L{Vector3d}) 

962 or C{None} to point to the earth' center. 

963 @kwarg earth: The earth model (L{Datum}, L{Ellipsoid}, L{Ellipsoid2}, 

964 L{a_f2Tuple} or C{scalar} radius in C{meter}). 

965 @kwarg name: Optional name (C{str}). 

966 @kwarg LatLon_and_kwds: Optional C{LatLon} class for the intersection 

967 point plus C{LatLon} keyword arguments, include 

968 B{C{datum}} if different from B{C{earth}}. 

969 

970 @return: The intersection point (L{Vector3d}, the C{Cartesian type} of 

971 B{C{pov}} or the given C{B{LatLon}_and_kwds}) with C{.heigth} 

972 set to the distance to the B{C{pov}}. 

973 

974 @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, B{C{pov}} 

975 is inside the earth or B{C{los}} points outside 

976 the earth or points in an opposite direction. 

977 

978 @raise TypeError: Invalid B{C{pov}}, B{C{los}} or B{C{earth}}. 

979 

980 @see: Function L{pygeodesy.hartzell4}, L{pygeodesy.tyr3d} for B{C{los}}, 

981 methods L{Ellipsoid.hartzell4}, C{Cartesian.hartzell}, C{LatLon.hartzell} 

982 and U{I{Satellite Line-of-Sight Intersection with Earth}<https:// 

983 StephenHartzell.Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}. 

984 ''' 

985 n = hartzell.__name__ 

986 D = earth if isinstance(earth, Datum) else \ 

987 _spherical_datum(earth, name=n) 

988 try: 

989 r, h = _MODS.triaxials._hartzell2(pov, los, D.ellipsoid._triaxial) 

990 except Exception as x: 

991 raise IntersectionError(pov=pov, los=los, earth=earth, cause=x) 

992 

993 r = _xnamed(r, name or n) 

994 C = _MODS.cartesianBase.CartesianBase 

995 if LatLon_and_kwds: 

996 c = C(r, datum=D, name=r.name) 

997 r = c.toLatLon(**_xkwds(LatLon_and_kwds, height=h)) 

998 elif isinstance(r, C): 

999 r.height = h 

1000 return r 

1001 

1002 

1003def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1004 '''Compute the distance between two (spherical) points using the 

1005 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} 

1006 formula. 

1007 

1008 @arg lat1: Start latitude (C{degrees}). 

1009 @arg lon1: Start longitude (C{degrees}). 

1010 @arg lat2: End latitude (C{degrees}). 

1011 @arg lon2: End longitude (C{degrees}). 

1012 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1013 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1014 L{a_f2Tuple}) to use. 

1015 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1016 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1017 

1018 @return: Distance (C{meter}, same units as B{C{radius}}). 

1019 

1020 @raise TypeError: Invalid B{C{radius}}. 

1021 

1022 @see: U{Distance between two (spherical) points 

1023 <https://www.EdWilliams.org/avform.htm#Dist>}, functions 

1024 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1025 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1026 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, 

1027 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1028 

1029 @note: See note at function L{vincentys_}. 

1030 ''' 

1031 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2) 

1032 

1033 

1034def haversine_(phi2, phi1, lam21): 

1035 '''Compute the I{angular} distance between two (spherical) points 

1036 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} 

1037 formula. 

1038 

1039 @arg phi2: End latitude (C{radians}). 

1040 @arg phi1: Start latitude (C{radians}). 

1041 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1042 

1043 @return: Angular distance (C{radians}). 

1044 

1045 @see: Functions L{haversine}, L{cosineAndoyerLambert_}, 

1046 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1047 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1048 L{flatPolar_}, L{thomas_} and L{vincentys_}. 

1049 

1050 @note: See note at function L{vincentys_}. 

1051 ''' 

1052 def _hsin(rad): 

1053 return sin(rad * _0_5)**2 

1054 

1055 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine 

1056 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2 

1057 

1058 

1059def heightOf(angle, distance, radius=R_M): 

1060 '''Determine the height above the (spherical) earth' surface after 

1061 traveling along a straight line at a given tilt. 

1062 

1063 @arg angle: Tilt angle above horizontal (C{degrees}). 

1064 @arg distance: Distance along the line (C{meter} or same units as 

1065 B{C{radius}}). 

1066 @kwarg radius: Optional mean earth radius (C{meter}). 

1067 

1068 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}). 

1069 

1070 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}. 

1071 

1072 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>} 

1073 (U{Shapiro et al. 2009, JTECH 

1074 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>} 

1075 and U{Potvin et al. 2012, JTECH 

1076 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}). 

1077 ''' 

1078 r = h = Radius(radius) 

1079 d = fabs(Distance(distance)) 

1080 if d > h: 

1081 d, h = h, d 

1082 

1083 if d > EPS0: # and h > EPS0 

1084 d = d / h # /= h chokes PyChecker 

1085 s = sin(Phi_(angle=angle, clip=_180_0)) 

1086 s = fsumf_(_1_0, s * d * _2_0, d**2) 

1087 if s > 0: 

1088 return h * sqrt(s) - r 

1089 

1090 raise _ValueError(angle=angle, distance=distance, radius=radius) 

1091 

1092 

1093def heightOrthometric(h_ll, N): 

1094 '''Get the I{orthometric} height B{H}, the height above the geoid, earth surface. 

1095 

1096 @arg h_ll: The height above the ellipsoid (C{meter}) or an I{ellipsoidal} 

1097 location (C{LatLon} with a C{height} or C{h} attribute). 

1098 @arg N: The I{geoid} height (C{meter}), the height of the geoid above the 

1099 ellipsoid at the same B{C{h_ll}} location. 

1100 

1101 @return: I{Orthometric} height C{B{H} = B{h} - B{N}} (C{meter}, same units 

1102 as B{C{h}} and B{C{N}}). 

1103 

1104 @see: U{Ellipsoid, Geoid, and Othometric Heights<https://www.NGS.NOAA.gov/ 

1105 GEOID/PRESENTATIONS/2007_02_24_CCPS/Roman_A_PLSC2007notes.pdf>}, page 

1106 6 and module L{pygeodesy.geoids}. 

1107 ''' 

1108 h = h_ll if _isHeight(h_ll) else _xattr(h_ll, height=_xattr(h_ll, h=0)) 

1109 return Height(H=Height(h=h) - Height(N=N)) 

1110 

1111 

1112def horizon(height, radius=R_M, refraction=False): 

1113 '''Determine the distance to the horizon from a given altitude 

1114 above the (spherical) earth. 

1115 

1116 @arg height: Altitude (C{meter} or same units as B{C{radius}}). 

1117 @kwarg radius: Optional mean earth radius (C{meter}). 

1118 @kwarg refraction: Consider atmospheric refraction (C{bool}). 

1119 

1120 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}). 

1121 

1122 @raise ValueError: Invalid B{C{height}} or B{C{radius}}. 

1123 

1124 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}. 

1125 ''' 

1126 h, r = Height(height), Radius(radius) 

1127 if min(h, r) < 0: 

1128 raise _ValueError(height=height, radius=radius) 

1129 

1130 d2 = ((r * 2.415750694528) if refraction else # 2.0 / 0.8279 

1131 fsumf_(r, r, h)) * h 

1132 return sqrt0(d2) 

1133 

1134 

1135class _idllmn6(object): # see also .geodesicw._wargs, .latlonBase._toCartesian3, .vector2d._numpy 

1136 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}. 

1137 ''' 

1138 @contextmanager # <https://www.Python.org/dev/peps/pep-0343/> Examples 

1139 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds): 

1140 try: 

1141 if wrap: 

1142 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap) 

1143 kwds = _xkwds(kwds, wrap=wrap) # for _xError 

1144 m = small if small is _100km else Meter_(small=small) 

1145 n = (intersections2 if s else intersection2).__name__ 

1146 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m: 

1147 d, m = None, _MODS.vector3d 

1148 _i = m._intersects2 if s else m._intersect3d3 

1149 elif _isRadius(datum) and datum < 0 and not s: 

1150 d = _spherical_datum(-datum, name=n) 

1151 m = _MODS.sphericalNvector 

1152 _i = m.intersection 

1153 else: 

1154 d = _spherical_datum(datum, name=n) 

1155 if d.isSpherical: 

1156 m = _MODS.sphericalTrigonometry 

1157 _i = m._intersects2 if s else m._intersect 

1158 elif d.isEllipsoidal: 

1159 try: 

1160 if d.ellipsoid.geodesic: 

1161 pass 

1162 m = _MODS.ellipsoidalKarney 

1163 except ImportError: 

1164 m = _MODS.ellipsoidalExact 

1165 _i = m._intersections2 if s else m._intersection3 # ellispoidalBaseDI 

1166 else: 

1167 raise _TypeError(datum=datum) 

1168 yield _i, d, lat2, lon2, m, n 

1169 

1170 except (TypeError, ValueError) as x: 

1171 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum, 

1172 lat2=lat2, lon2=lon2, small=small, **kwds) 

1173 

1174_idllmn6 = _idllmn6() # PYCHOK singleton 

1175 

1176 

1177def intersection2(lat1, lon1, bearing1, 

1178 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True 

1179 '''I{Conveniently} compute the intersection of two lines each defined 

1180 by a (geodetic) point and a bearing from North, using either ... 

1181 

1182 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km 

1183 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1184 

1185 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}} 

1186 or a C{scalar B{datum}} representing the earth radius, conventionally 

1187 in C{meter} or ... 

1188 

1189 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative} 

1190 C{scalar}, (negative) earth radius, conventionally in C{meter} or ... 

1191 

1192 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}} 

1193 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1194 is installed, otherwise ... 

1195 

1196 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal. 

1197 

1198 @arg lat1: Latitude of the first point (C{degrees}). 

1199 @arg lon1: Longitude of the first point (C{degrees}). 

1200 @arg bearing1: Bearing at the first point (compass C{degrees}). 

1201 @arg lat2: Latitude of the second point (C{degrees}). 

1202 @arg lon2: Longitude of the second point (C{degrees}). 

1203 @arg bearing2: Bearing at the second point (compass C{degrees}). 

1204 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1205 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1206 radius (C{meter}, same units as B{C{radius1}} and 

1207 B{C{radius2}}) or C{None}. 

1208 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1209 and B{C{lon2}} (C{bool}). 

1210 @kwarg small: Upper limit for small distances (C{meter}). 

1211 

1212 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and 

1213 longitude of the intersection point. 

1214 

1215 @raise IntersectionError: Ambiguous or infinite intersection 

1216 or colinear, parallel or otherwise 

1217 non-intersecting lines. 

1218 

1219 @raise TypeError: Invalid B{C{datum}}. 

1220 

1221 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}}, 

1222 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}. 

1223 

1224 @see: Method L{RhumbLine.intersection2}. 

1225 

1226 @note: The returned intersections may be near-antipodal. 

1227 ''' 

1228 b1 = Bearing(bearing1=bearing1) 

1229 b2 = Bearing(bearing2=bearing2) 

1230 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1231 small, wrap, False, bearing1=b1, bearing2=b2) as t: 

1232 _i, d, lat2, lon2, m, n = t 

1233 if d is None: 

1234 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1, 

1235 m.Vector3d(lon2, lat2, 0), b2, useZ=False) 

1236 t = LatLon2Tuple(t.y, t.x, name=n) 

1237 

1238 else: 

1239 t = _i(m.LatLon(lat1, lon1, datum=d), b1, 

1240 m.LatLon(lat2, lon2, datum=d), b2, height=0, wrap=False) 

1241 if isinstance(t, Intersection3Tuple): # ellipsoidal 

1242 t, _, _ = t 

1243 t = LatLon2Tuple(t.lat, t.lon, name=n) 

1244 return t 

1245 

1246 

1247def intersections2(lat1, lon1, radius1, 

1248 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True 

1249 '''I{Conveniently} compute the intersections of two circles each defined 

1250 by a (geodetic) center point and a radius, using either ... 

1251 

1252 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km 

1253 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1254 

1255 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}} 

1256 or a C{scalar B{datum}} representing the earth radius, conventionally 

1257 in C{meter} or ... 

1258 

1259 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}} 

1260 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1261 is installed, otherwise ... 

1262 

1263 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal. 

1264 

1265 @arg lat1: Latitude of the first circle center (C{degrees}). 

1266 @arg lon1: Longitude of the first circle center (C{degrees}). 

1267 @arg radius1: Radius of the first circle (C{meter}, conventionally). 

1268 @arg lat2: Latitude of the second circle center (C{degrees}). 

1269 @arg lon2: Longitude of the second circle center (C{degrees}). 

1270 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}). 

1271 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1272 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1273 radius (C{meter}, same units as B{C{radius1}} and 

1274 B{C{radius2}}) or C{None}. 

1275 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1276 and B{C{lon2}} (C{bool}). 

1277 @kwarg small: Upper limit for small distances (C{meter}). 

1278 

1279 @return: 2-Tuple of the intersection points, each a 

1280 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the 

1281 points are the same instance, aka the I{radical center}. 

1282 

1283 @raise IntersectionError: Concentric, antipodal, invalid or 

1284 non-intersecting circles or no 

1285 convergence. 

1286 

1287 @raise TypeError: Invalid B{C{datum}}. 

1288 

1289 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}}, 

1290 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}. 

1291 ''' 

1292 r1 = Radius_(radius1=radius1) 

1293 r2 = Radius_(radius2=radius2) 

1294 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1295 small, wrap, True, radius1=r1, radius2=r2) as t: 

1296 _i, d, lat2, lon2, m, n = t 

1297 if d is None: 

1298 r1 = m2degrees(r1, radius=R_M, lat=lat1) 

1299 r2 = m2degrees(r2, radius=R_M, lat=lat2) 

1300 

1301 def _V2T(x, y, _, **unused): # _ == z unused 

1302 return LatLon2Tuple(y, x, name=n) 

1303 

1304 t = _i(m.Vector3d(lon1, lat1, 0), r1, 

1305 m.Vector3d(lon2, lat2, 0), r2, sphere=False, 

1306 Vector=_V2T) 

1307 else: 

1308 def _LL2T(lat, lon, **unused): 

1309 return LatLon2Tuple(lat, lon, name=n) 

1310 

1311 t = _i(m.LatLon(lat1, lon1, datum=d), r1, 

1312 m.LatLon(lat2, lon2, datum=d), r2, 

1313 LatLon=_LL2T, height=0, wrap=False) 

1314 return t 

1315 

1316 

1317def isantipode(lat1, lon1, lat2, lon2, eps=EPS): 

1318 '''Check whether two points are I{antipodal}, on diametrically 

1319 opposite sides of the earth. 

1320 

1321 @arg lat1: Latitude of one point (C{degrees}). 

1322 @arg lon1: Longitude of one point (C{degrees}). 

1323 @arg lat2: Latitude of the other point (C{degrees}). 

1324 @arg lon2: Longitude of the other point (C{degrees}). 

1325 @kwarg eps: Tolerance for near-equality (C{degrees}). 

1326 

1327 @return: C{True} if points are antipodal within the 

1328 B{C{eps}} tolerance, C{False} otherwise. 

1329 

1330 @see: Functions L{isantipode_} and L{antipode}. 

1331 ''' 

1332 return (fabs(lat1 + lat2) <= eps and 

1333 fabs(lon1 + lon2) <= eps) or _isequalTo( 

1334 normal(lat1, lon1), antipode(lat2, lon2), eps) 

1335 

1336 

1337def isantipode_(phi1, lam1, phi2, lam2, eps=EPS): 

1338 '''Check whether two points are I{antipodal}, on diametrically 

1339 opposite sides of the earth. 

1340 

1341 @arg phi1: Latitude of one point (C{radians}). 

1342 @arg lam1: Longitude of one point (C{radians}). 

1343 @arg phi2: Latitude of the other point (C{radians}). 

1344 @arg lam2: Longitude of the other point (C{radians}). 

1345 @kwarg eps: Tolerance for near-equality (C{radians}). 

1346 

1347 @return: C{True} if points are antipodal within the 

1348 B{C{eps}} tolerance, C{False} otherwise. 

1349 

1350 @see: Functions L{isantipode} and L{antipode_}. 

1351 ''' 

1352 return (fabs(phi1 + phi2) <= eps and 

1353 fabs(lam1 + lam2) <= eps) or _isequalTo_( 

1354 normal_(phi1, lam1), antipode_(phi2, lam2), eps) 

1355 

1356 

1357def _isequalTo(p1, p2, eps=EPS): 

1358 '''Compare 2 point lat-/lons ignoring C{class}. 

1359 ''' 

1360 return (fabs(p1.lat - p2.lat) <= eps and 

1361 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon) 

1362 

1363 

1364def _isequalTo_(p1, p2, eps=EPS): 

1365 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}. 

1366 ''' 

1367 return (fabs(p1.phi - p2.phi) <= eps and 

1368 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam) 

1369 

1370 

1371def isnormal(lat, lon, eps=0): 

1372 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their 

1373 respective I{normal} range in C{degrees}. 

1374 

1375 @arg lat: Latitude (C{degrees}). 

1376 @arg lon: Longitude (C{degrees}). 

1377 @kwarg eps: Optional tolerance C{degrees}). 

1378 

1379 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and 

1380 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise. 

1381 

1382 @see: Functions L{isnormal_} and L{normal}. 

1383 ''' 

1384 return (_90_0 - fabs(lat)) >= eps and _loneg(fabs(lon)) >= eps 

1385 

1386 

1387def isnormal_(phi, lam, eps=0): 

1388 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their 

1389 respective I{normal} range in C{radians}. 

1390 

1391 @arg phi: Latitude (C{radians}). 

1392 @arg lam: Longitude (C{radians}). 

1393 @kwarg eps: Optional tolerance C{radians}). 

1394 

1395 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and 

1396 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise. 

1397 

1398 @see: Functions L{isnormal} and L{normal_}. 

1399 ''' 

1400 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps 

1401 

1402 

1403def latlon2n_xyz(lat, lon, name=NN): 

1404 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1405 earth's surface) X, Y and Z components. 

1406 

1407 @arg lat: Latitude (C{degrees}). 

1408 @arg lon: Longitude (C{degrees}). 

1409 @kwarg name: Optional name (C{str}). 

1410 

1411 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1412 

1413 @see: Function L{philam2n_xyz}. 

1414 

1415 @note: These are C{n-vector} x, y and z components, 

1416 I{NOT} geocentric ECEF x, y and z coordinates! 

1417 ''' 

1418 return _2n_xyz(name, *sincos2d_(lat, lon)) 

1419 

1420 

1421def _normal2(a, b, n_2, n, n2): 

1422 '''(INTERNAL) Helper for C{normal} and C{normal_}. 

1423 ''' 

1424 if fabs(b) > n: 

1425 b = remainder(b, n2) 

1426 if fabs(a) > n_2: 

1427 r = remainder(a, n) 

1428 if r != a: 

1429 a = -r 

1430 b -= n if b > 0 else -n 

1431 return float0_(a, b) 

1432 

1433 

1434def normal(lat, lon, name=NN): 

1435 '''Normalize a lat- I{and} longitude pair in C{degrees}. 

1436 

1437 @arg lat: Latitude (C{degrees}). 

1438 @arg lon: Longitude (C{degrees}). 

1439 @kwarg name: Optional name (C{str}). 

1440 

1441 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90} 

1442 and C{abs(lon) <= 180}. 

1443 

1444 @see: Functions L{normal_} and L{isnormal}. 

1445 ''' 

1446 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0), 

1447 name=name or normal.__name__) 

1448 

1449 

1450def normal_(phi, lam, name=NN): 

1451 '''Normalize a lat- I{and} longitude pair in C{radians}. 

1452 

1453 @arg phi: Latitude (C{radians}). 

1454 @arg lam: Longitude (C{radians}). 

1455 @kwarg name: Optional name (C{str}). 

1456 

1457 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2} 

1458 and C{abs(lam) <= PI}. 

1459 

1460 @see: Functions L{normal} and L{isnormal_}. 

1461 ''' 

1462 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2), 

1463 name=name or normal_.__name__) 

1464 

1465 

1466def _2n_xyz(name, sa, ca, sb, cb): 

1467 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}. 

1468 ''' 

1469 # Kenneth Gade eqn 3, but using right-handed 

1470 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N 

1471 return Vector3Tuple(ca * cb, ca * sb, sa, name=name) 

1472 

1473 

1474def n_xyz2latlon(x, y, z, name=NN): 

1475 '''Convert C{n-vector} components to lat- and longitude in C{degrees}. 

1476 

1477 @arg x: X component (C{scalar}). 

1478 @arg y: Y component (C{scalar}). 

1479 @arg z: Z component (C{scalar}). 

1480 @kwarg name: Optional name (C{str}). 

1481 

1482 @return: A L{LatLon2Tuple}C{(lat, lon)}. 

1483 

1484 @see: Function L{n_xyz2philam}. 

1485 ''' 

1486 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name) 

1487 

1488 

1489def n_xyz2philam(x, y, z, name=NN): 

1490 '''Convert C{n-vector} components to lat- and longitude in C{radians}. 

1491 

1492 @arg x: X component (C{scalar}). 

1493 @arg y: Y component (C{scalar}). 

1494 @arg z: Z component (C{scalar}). 

1495 @kwarg name: Optional name (C{str}). 

1496 

1497 @return: A L{PhiLam2Tuple}C{(phi, lam)}. 

1498 

1499 @see: Function L{n_xyz2latlon}. 

1500 ''' 

1501 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name) 

1502 

1503 

1504def _opposes(d, m, n, n2): 

1505 '''(INTERNAL) Helper for C{opposing} and C{opposing_}. 

1506 ''' 

1507 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1 

1508 return False if d < m or d > (n2 - m) else ( 

1509 True if (n - m) < d < (n + m) else None) 

1510 

1511 

1512def opposing(bearing1, bearing2, margin=_90_0): 

1513 '''Compare the direction of two bearings given in C{degrees}. 

1514 

1515 @arg bearing1: First bearing (compass C{degrees}). 

1516 @arg bearing2: Second bearing (compass C{degrees}). 

1517 @kwarg margin: Optional, interior angle bracket (C{degrees}). 

1518 

1519 @return: C{True} if both bearings point in opposite, C{False} if 

1520 in similar or C{None} if in I{perpendicular} directions. 

1521 

1522 @see: Function L{opposing_}. 

1523 ''' 

1524 m = Degrees_(margin=margin, low=EPS0, high=_90_0) 

1525 return _opposes(bearing2 - bearing1, m, _180_0, _360_0) 

1526 

1527 

1528def opposing_(radians1, radians2, margin=PI_2): 

1529 '''Compare the direction of two bearings given in C{radians}. 

1530 

1531 @arg radians1: First bearing (C{radians}). 

1532 @arg radians2: Second bearing (C{radians}). 

1533 @kwarg margin: Optional, interior angle bracket (C{radians}). 

1534 

1535 @return: C{True} if both bearings point in opposite, C{False} if 

1536 in similar or C{None} if in perpendicular directions. 

1537 

1538 @see: Function L{opposing}. 

1539 ''' 

1540 m = Radians_(margin=margin, low=EPS0, high=PI_2) 

1541 return _opposes(radians2 - radians1, m, PI, PI2) 

1542 

1543 

1544def philam2n_xyz(phi, lam, name=NN): 

1545 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1546 earth's surface) X, Y and Z components. 

1547 

1548 @arg phi: Latitude (C{radians}). 

1549 @arg lam: Longitude (C{radians}). 

1550 @kwarg name: Optional name (C{str}). 

1551 

1552 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1553 

1554 @see: Function L{latlon2n_xyz}. 

1555 

1556 @note: These are C{n-vector} x, y and z components, 

1557 I{NOT} geocentric ECEF x, y and z coordinates! 

1558 ''' 

1559 return _2n_xyz(name, *sincos2_(phi, lam)) 

1560 

1561 

1562def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d 

1563 # (INTERNAL) See C{radical2} below 

1564 # assert d > EPS0 

1565 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5 

1566 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d) 

1567 

1568 

1569def radical2(distance, radius1, radius2): 

1570 '''Compute the I{radical ratio} and I{radical line} of two 

1571 U{intersecting circles<https://MathWorld.Wolfram.com/ 

1572 Circle-CircleIntersection.html>}. 

1573 

1574 The I{radical line} is perpendicular to the axis thru the 

1575 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}. 

1576 

1577 @arg distance: Distance between the circle centers (C{scalar}). 

1578 @arg radius1: Radius of the first circle (C{scalar}). 

1579 @arg radius2: Radius of the second circle (C{scalar}). 

1580 

1581 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <= 

1582 ratio <= 1.0} and C{xline} is along the B{C{distance}}. 

1583 

1584 @raise IntersectionError: The B{C{distance}} exceeds the sum 

1585 of B{C{radius1}} and B{C{radius2}}. 

1586 

1587 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or 

1588 B{C{radius2}}. 

1589 

1590 @see: U{Circle-Circle Intersection 

1591 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}. 

1592 ''' 

1593 d = Distance_(distance, low=_0_0) 

1594 r1 = Radius_(radius1=radius1) 

1595 r2 = Radius_(radius2=radius2) 

1596 if d > (r1 + r2): 

1597 raise IntersectionError(distance=d, radius1=r1, radius2=r2, 

1598 txt=_too_(_distant_)) 

1599 return _radical2(d, r1, r2) if d > EPS0 else \ 

1600 Radical2Tuple(_0_5, _0_0) 

1601 

1602 

1603class Radical2Tuple(_NamedTuple): 

1604 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and 

1605 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0} 

1606 ''' 

1607 _Names_ = (_ratio_, _xline_) 

1608 _Units_ = ( Scalar, Scalar) 

1609 

1610 

1611def _radistance(inst): 

1612 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians} 

1613 and L{hausdorff._HausdorffMeterRedians} classes. 

1614 ''' 

1615 kwds_ = _xkwds(inst._kwds, wrap=False) 

1616 wrap_ = _xkwds_pop(kwds_, wrap=False) 

1617 func_ = inst._func_ 

1618 try: # calling lower-overhead C{func_} 

1619 func_(0, _0_25, _0_5, **kwds_) 

1620 wrap_ = _Wrap._philamop(wrap_) 

1621 except TypeError: 

1622 return inst.distance 

1623 

1624 def _philam(p): 

1625 try: 

1626 return p.phi, p.lam 

1627 except AttributeError: # no .phi or .lam 

1628 return radians(p.lat), radians(p.lon) 

1629 

1630 def _func_wrap(point1, point2): 

1631 phi1, lam1 = wrap_(*_philam(point1)) 

1632 phi2, lam2 = wrap_(*_philam(point2)) 

1633 return func_(phi2, phi1, lam2 - lam1, **kwds_) 

1634 

1635 inst._units = inst._units_ 

1636 return _func_wrap 

1637 

1638 

1639def rtp2xyz(r, theta, phi): 

1640 '''Convert spherical C{(r, theta, phi)} to cartesian C{(x, y, z)} coordinates. 

1641 

1642 @arg r: Radial distance (C{scalar}, conventially C{meter}). 

1643 @arg theta: Inclination (C{degrees} with respect to the positive z-axis). 

1644 @arg phi: Azimuthal angle (C{degrees}). 

1645 

1646 @return: L{Vector3Tuple}C{(x, y, z)} in C{meter}, same units as C{r}. 

1647 

1648 @see: Functions L{rtp2xyz_} and L{xyz2rtp}. 

1649 ''' 

1650 return rtp2xyz_(r, radians(theta), radians(phi)) 

1651 

1652 

1653def rtp2xyz_(r, theta, phi): 

1654 '''Convert spherical C{(r, theta, phi)} to cartesian C{(x, y, z)} coordinates. 

1655 

1656 @arg r: Radial distance (C{scalar}, conventially C{meter}). 

1657 @arg theta: Inclination (C{radians} with respect to the positive z-axis). 

1658 @arg phi: Azimuthal angle (C{radians}). 

1659 

1660 @return: L{Vector3Tuple}C{(x, y, z)} in C{meter}, same units as C{r}. 

1661 

1662 @see: U{Physics<https://WikiPedia.org/wiki/Spherical_coordinate_system>} 

1663 convention (ISO 80000-2:2019). 

1664 ''' 

1665 if r and _isfinite(r): 

1666 s, z, y, x = sincos2_(theta, phi) 

1667 s *= r 

1668 x *= s 

1669 y *= s 

1670 z *= r 

1671 else: 

1672 x = y = z = r 

1673 return Vector3Tuple(x, y, z) 

1674 

1675 

1676def _scale_deg(lat1, lat2): # degrees 

1677 # scale factor cos(mean of lats) for delta lon 

1678 m = fabs(lat1 + lat2) * _0_5 

1679 return cos(radians(m)) if m < 90 else _0_0 

1680 

1681 

1682def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights 

1683 # scale factor cos(mean of phis) for delta lam 

1684 m = fabs(phi1 + phi2) * _0_5 

1685 return cos(m) if m < PI_2 else _0_0 

1686 

1687 

1688def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw 

1689 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine. 

1690 ''' 

1691 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21) 

1692 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21 

1693 

1694 

1695def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

1696 '''Compute the distance between two (ellipsoidal) points using 

1697 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1698 formula. 

1699 

1700 @arg lat1: Start latitude (C{degrees}). 

1701 @arg lon1: Start longitude (C{degrees}). 

1702 @arg lat2: End latitude (C{degrees}). 

1703 @arg lon2: End longitude (C{degrees}). 

1704 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1705 L{Ellipsoid2} or L{a_f2Tuple}) to use. 

1706 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1707 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1708 

1709 @return: Distance (C{meter}, same units as the B{C{datum}}'s 

1710 ellipsoid axes). 

1711 

1712 @raise TypeError: Invalid B{C{datum}}. 

1713 

1714 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1715 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, 

1716 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}. 

1717 ''' 

1718 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2) 

1719 

1720 

1721def thomas_(phi2, phi1, lam21, datum=_WGS84): 

1722 '''Compute the I{angular} distance between two (ellipsoidal) points using 

1723 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1724 formula. 

1725 

1726 @arg phi2: End latitude (C{radians}). 

1727 @arg phi1: Start latitude (C{radians}). 

1728 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1729 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, 

1730 L{Ellipsoid2} or L{a_f2Tuple}). 

1731 

1732 @return: Angular distance (C{radians}). 

1733 

1734 @raise TypeError: Invalid B{C{datum}}. 

1735 

1736 @see: Functions L{thomas}, L{cosineAndoyerLambert_}, 

1737 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1738 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1739 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP 

1740 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/ 

1741 Distance/ThomasFormula.php>}. 

1742 ''' 

1743 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21) 

1744 if r and isnon0(c1) and isnon0(c2): 

1745 E = _ellipsoidal(datum, thomas_) 

1746 if E.f: 

1747 r1 = atan2(E.b_a * s1, c1) 

1748 r2 = atan2(E.b_a * s2, c2) 

1749 

1750 j = (r2 + r1) * _0_5 

1751 k = (r2 - r1) * _0_5 

1752 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5) 

1753 

1754 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2) 

1755 u = _1_0 - h 

1756 if isnon0(u) and isnon0(h): 

1757 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h) 

1758 sr, cr = sincos2(r) 

1759 if isnon0(sr): 

1760 u = 2 * (sj * ck)**2 / u 

1761 h = 2 * (sk * cj)**2 / h 

1762 x = u + h 

1763 y = u - h 

1764 

1765 s = r / sr 

1766 e = 4 * s**2 

1767 d = 2 * cr 

1768 a = e * d 

1769 b = 2 * r 

1770 c = s - (a - d) * _0_5 

1771 f = E.f * _0_25 

1772 

1773 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y) 

1774 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr 

1775 return r 

1776 

1777 

1778def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1779 '''Compute the distance between two (spherical) points using 

1780 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1781 spherical formula. 

1782 

1783 @arg lat1: Start latitude (C{degrees}). 

1784 @arg lon1: Start longitude (C{degrees}). 

1785 @arg lat2: End latitude (C{degrees}). 

1786 @arg lon2: End longitude (C{degrees}). 

1787 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1788 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1789 L{a_f2Tuple}) to use. 

1790 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1791 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1792 

1793 @return: Distance (C{meter}, same units as B{C{radius}}). 

1794 

1795 @raise UnitError: Invalid B{C{radius}}. 

1796 

1797 @see: Functions L{vincentys_}, L{cosineAndoyerLambert}, 

1798 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular}, 

1799 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1800 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2}, 

1801 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1802 

1803 @note: See note at function L{vincentys_}. 

1804 ''' 

1805 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2) 

1806 

1807 

1808def vincentys_(phi2, phi1, lam21): 

1809 '''Compute the I{angular} distance between two (spherical) points using 

1810 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1811 spherical formula. 

1812 

1813 @arg phi2: End latitude (C{radians}). 

1814 @arg phi1: Start latitude (C{radians}). 

1815 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1816 

1817 @return: Angular distance (C{radians}). 

1818 

1819 @see: Functions L{vincentys}, L{cosineAndoyerLambert_}, 

1820 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1821 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1822 L{flatPolar_}, L{haversine_} and L{thomas_}. 

1823 

1824 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_} 

1825 produce equivalent results, but L{vincentys_} is suitable 

1826 for antipodal points and slightly more expensive (M{3 cos, 

1827 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_} 

1828 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and 

1829 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}). 

1830 ''' 

1831 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21) 

1832 

1833 c = c2 * c21 

1834 x = s1 * s2 + c1 * c 

1835 y = c1 * s2 - s1 * c 

1836 return atan2(hypot(c2 * s21, y), x) 

1837 

1838 

1839def xyz2rtp(x_xyz, *y_z): 

1840 '''Convert cartesian C{(x, y, z)} to spherical C{(r, theta, phi)} coordinates. 

1841 

1842 @return: 3-Tuple C{(r, theta, phi)} in C{degrees}. 

1843 

1844 @see: Function L{xyz2rtp_}. 

1845 ''' 

1846 r, t, p = xyz2rtp_(x_xyz, *y_z) 

1847 return r, Degrees(theta=degrees(t)), Degrees(phi=degrees(p)) 

1848 

1849 

1850def xyz2rtp_(x_xyz, *y_z): 

1851 '''Convert cartesian C{(x, y, z)} to spherical C{(r, theta, phi)} coordinates. 

1852 

1853 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian}, L{Ecef9Tuple}, 

1854 C{Nvector}, L{Vector3d}, L{Vector3Tuple}, L{Vector4Tuple} or a 

1855 C{tuple} or C{list} of 3+ C{scalar} items) if no C{y_z} specified. 

1856 @arg y_z: Y and Z component (C{scalar}s), ignored if C{x_xyz} is not a C{scalar}. 

1857 

1858 @return: 3-Tuple C{(r, theta, phi)} with radial distance C{r} (C{meter}, same 

1859 units as C{x}, C{y} and C{z}), inclination C{theta} (with respect to 

1860 the positive z-axis) and azimuthal angle C{phi} in C{radians}. 

1861 

1862 @see: U{Physics<https://WikiPedia.org/wiki/Spherical_coordinate_system>} 

1863 convention (ISO 80000-2:2019). 

1864 ''' 

1865 x, y, z = _MODS.vector3dBase._xyz3(xyz2rtp, x_xyz, *y_z) 

1866 r = hypot_(x, y, z) 

1867 if r: 

1868 t = acos1(z / r) 

1869 p = atan2(y, x) 

1870 if p < 0: 

1871 p += PI2 

1872 else: 

1873 t = p = _0_0 

1874 return r, Radians(theta=t), Radians(phi=p) 

1875 

1876# **) MIT License 

1877# 

1878# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

1879# 

1880# Permission is hereby granted, free of charge, to any person obtaining a 

1881# copy of this software and associated documentation files (the "Software"), 

1882# to deal in the Software without restriction, including without limitation 

1883# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1884# and/or sell copies of the Software, and to permit persons to whom the 

1885# Software is furnished to do so, subject to the following conditions: 

1886# 

1887# The above copyright notice and this permission notice shall be included 

1888# in all copies or substantial portions of the Software. 

1889# 

1890# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1891# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1892# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1893# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1894# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1895# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

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