Coverage for pygeodesy/formy.py: 99%

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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.basics import isscalar # from .fsums 

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

11 _umod_PI2, float0_, isnon0, remainder, \ 

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

13 _4_0, _32_0, _90_0, _180_0, _360_0 

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

15 _mean_radius, _spherical_datum, _WGS84 

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

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

18 _TypeError, _ValueError, \ 

19 _xError, _xkwds, _xkwds_pop 

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

21from pygeodesy.fsums import fsumf_, isscalar 

22from pygeodesy.interns import NN, _delta_, _distant_, _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 

29from pygeodesy.units import Bearing, Degrees_, Distance, Distance_, Height, \ 

30 Lam_, Lat, Lon, Meter_, Phi_, Radians, Radians_, \ 

31 Radius, Radius_, Scalar, _100km 

32from pygeodesy.utily import acos1, atan2b, atan2d, degrees2m, m2degrees, \ 

33 tan_2, sincos2, sincos2_, sincos2d_, _Wrap 

34 

35from contextlib import contextmanager 

36from math import asin, atan, atan2, cos, degrees, fabs, radians, sin, sqrt # pow 

37 

38__all__ = _ALL_LAZY.formy 

39__version__ = '23.08.11' 

40 

41_D2_R2 = (PI / _180_0)**2 # degrees- to radians-squared 

42_EWGS84 = _WGS84.ellipsoid 

43_ratio_ = 'ratio' 

44_xline_ = 'xline' 

45 

46 

47def _anti2(a, b, n_2, n, n2): 

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

49 ''' 

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

51 if r == a: 

52 r = -r 

53 b += n 

54 if fabs(b) > n: 

55 b = remainder(b, n2) 

56 return float0_(r, b) 

57 

58 

59def antipode(lat, lon, name=NN): 

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

61 to a given point in C{degrees}. 

62 

63 @arg lat: Latitude (C{degrees}). 

64 @arg lon: Longitude (C{degrees}). 

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

66 

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

68 

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

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

71 ''' 

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

73 

74 

75def antipode_(phi, lam, name=NN): 

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

77 to a given point in C{radians}. 

78 

79 @arg phi: Latitude (C{radians}). 

80 @arg lam: Longitude (C{radians}). 

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

82 

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

84 

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

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

87 ''' 

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

89 

90 

91def bearing(lat1, lon1, lat2, lon2, **final_wrap): 

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

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

94 

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

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

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

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

99 @kwarg final_wrap: Optional keyword arguments for function 

100 L{pygeodesy.bearing_}. 

101 

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

103 zero if start and end point coincide. 

104 ''' 

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

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

107 return degrees(r) 

108 

109 

110def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False): 

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

112 between a (spherical) start and end point. 

113 

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

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

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

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

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

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

120 B{C{lam2}} (C{bool}). 

121 

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

123 and end point coincide. 

124 

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

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

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

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

129 ''' 

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

131 if final: # swap plus PI 

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

133 r = PI3 

134 else: 

135 r = PI2 

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

137 

138 x = ca1 * sa2 - sa1 * ca2 * cdb 

139 y = sdb * ca2 

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

141 

142 

143def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf 

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

145 ''' 

146 try: # for LatLon_ and ellipsoidal LatLon 

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

148 except AttributeError: 

149 pass 

150 # XXX spherical version, OK for ellipsoidal ispolar? 

151 a1, b1 = p1.philam 

152 a2, b2 = p2.philam 

153 return Bearing2Tuple(degrees(bearing_(a1, b1, a2, b2, final=False, wrap=wrap)), 

154 degrees(bearing_(a1, b1, a2, b2, final=True, wrap=wrap)), 

155 name=_bearingTo2.__name__) 

156 

157 

158def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False): 

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

160 lat2 - lat1)} between two points. 

161 

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

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

164 

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

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

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

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

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

170 mean latitude (C{bool}). 

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

172 and B{C{lon2}} (C{bool}). 

173 

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

175 

176 @note: Courtesy of Martin Schultz. 

177 

178 @see: U{Local, flat earth approximation 

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

180 ''' 

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

182 if adjust: # scale delta lon 

183 d_lon *= _scale_deg(lat1, lat2) 

184 return atan2b(d_lon, lat2 - lat1) 

185 

186 

187def cosineAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

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

189 U{Andoyer-Lambert correction<https://NavLib.net/wp-content/uploads/2013/10/ 

190 admiralty-manual-of-navigation-vol-1-1964-english501c.pdf>} of the U{Law of 

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

192 

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

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

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

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

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

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

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

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

201 

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

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

204 

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

206 

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

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

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

210 L{Ellipsoid.distance2}. 

211 ''' 

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

213 

214 

215def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): 

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

217 U{Andoyer-Lambert correction<https://NavLib.net/wp-content/uploads/2013/10/ 

218 admiralty-manual-of-navigation-vol-1-1964-english501c.pdf>} of the U{Law of 

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

220 

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

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

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

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

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

226 

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

228 

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

230 

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

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

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

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

235 AndoyerLambert.php>}. 

236 ''' 

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

238 if isnon0(c1) and isnon0(c2): 

239 E = _ellipsoidal(datum, cosineAndoyerLambert_) 

240 if E.f: # ellipsoidal 

241 r2 = atan2(E.b_a * s2, c2) 

242 r1 = atan2(E.b_a * s1, c1) 

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

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

245 if r: 

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

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

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

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

250 r += (c - s) * E.f * _0_125 

251 return r 

252 

253 

254def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

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

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

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

258 formula. 

259 

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

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

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

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

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

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

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

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

268 

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

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

271 

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

273 

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

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

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

277 L{Ellipsoid.distance2}. 

278 ''' 

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

280 

281 

282def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): 

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

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

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

286 formula. 

287 

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

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

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

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

292 L{Ellipsoid2} or L{a_f2Tuple}). 

293 

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

295 

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

297 

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

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

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

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

302 Distance/ForsytheCorrection.php>}. 

303 ''' 

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

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

306 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_) 

307 if E.f: # ellipsoidal 

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

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

310 s = (s1 + s2)**2 / (1 + cr) 

311 t = (s1 - s2)**2 / (1 - cr) 

312 x = s + t 

313 y = s - t 

314 

315 s = 8 * r**2 / sr 

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

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

318 b = -2 * d 

319 e = 30 * s2r 

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

321 

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

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

324 return r 

325 

326 

327def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

330 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 

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

360 formula. 

361 

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

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

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

365 

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

367 

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

369 L{cosineForsytheAndoyerLambert_}, L{equirectangular_}, 

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

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

372 

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

374 ''' 

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

376 

377 

378def _d3(wrap, lat1, lon1, lat2, lon2): 

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

380 ''' 

381 if wrap: 

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

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

384 else: # for backward compaibility 

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

386 

387 

388def _dE(func_, earth, *wrap_lls): 

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

390 ''' 

391 E = _ellipsoidal(earth, func_) 

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

393 return r * E.a 

394 

395 

396def _dS(func_, radius, *wrap_lls, **adjust): 

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

398 ''' 

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

400 if radius is not R_M: 

401 _, lat1, _, lat2, _ = wrap_lls 

402 radius = _mean_radius(radius, lat1, lat2) 

403 return r * radius 

404 

405 

406def _eA(excess_, radius, *wrap_lls): 

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

408 ''' 

409 r = excess_(*_d3(*wrap_lls)) 

410 if radius: 

411 _, lat1, _, lat2, _ = wrap_lls 

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

413 return r 

414 

415 

416def _ellipsoidal(earth, where): 

417 '''(INTERNAL) Helper for distances. 

418 ''' 

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

420 earth if isinstance(earth, Ellipsoid) else 

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

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

423 

424 

425def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **adjust_limit_wrap): 

426 '''Compute the distance between two points using 

427 the U{Equirectangular Approximation / Projection 

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

429 

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

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

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

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

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

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

436 L{a_f2Tuple}). 

437 @kwarg adjust_limit_wrap: Optional keyword arguments for 

438 function L{equirectangular_}. 

439 

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

441 the ellipsoid or datum axes). 

442 

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

444 

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

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

447 approximate or accurate distance functions. 

448 ''' 

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

450 Lat(lat2=lat2), Lon(lon2=lon2), 

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

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

453 

454 

455def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap): 

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

457 and L{hausdorff._HausdorffMeterRedians} classes. 

458 ''' 

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

460 

461 

462def equirectangular_(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False): 

463 '''Compute the distance between two points using the U{Equirectangular 

464 Approximation / Projection 

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

466 

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

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

469 

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

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

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

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

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

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

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

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

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

479 and B{C{lon2}} (C{bool}). 

480 

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

482 unroll_lon2)} in C{degrees squared}. 

483 

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

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

486 set to C{True}. 

487 

488 @see: U{Local, flat earth approximation 

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

490 L{equirectangular}, L{cosineAndoyerLambert}, 

491 L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean}, 

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

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

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

495 ''' 

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

497 d_lat = lat2 - lat1 

498 

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

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

501 if d > limit: 

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

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

504 limit=limit, wrap=wrap) 

505 raise LimitError(s, txt=t) 

506 

507 if adjust: # scale delta lon 

508 d_lon *= _scale_deg(lat1, lat2) 

509 

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

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

512 

513 

514def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False): 

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

516 

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

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

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

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

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

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

523 L{a_f2Tuple}) to use. 

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

525 the mean latitude (C{bool}). 

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

527 and B{C{lon2}} (C{bool}). 

528 

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

530 ellipsoid or datum axes). 

531 

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

533 

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

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

536 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

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

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

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

540 ''' 

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

542 

543 

544def euclidean_(phi2, phi1, lam21, adjust=True): 

545 '''Approximate the I{angular} C{Euclidean} distance between two 

546 (spherical) points. 

547 

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

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

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

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

552 of the mean latitude (C{bool}). 

553 

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

555 

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

557 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_}, 

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

559 and L{vincentys_}. 

560 ''' 

561 if adjust: 

562 lam21 *= _scale_rad(phi2, phi1) 

563 return euclid(phi2 - phi1, lam21) 

564 

565 

566def excessAbc_(A, b, c): 

567 '''Compute the I{spherical excess} C{E} of a (spherical) triangle 

568 from two sides and the included (small) angle. 

569 

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

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

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

573 

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

575 

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

577 

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

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

580 ''' 

581 sA, cA, sb, cb, sc, cc = sincos2_(Radians_(A=A), Radians_(b=b) * _0_5, 

582 Radians_(c=c) * _0_5) 

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

584 

585 

586def excessCagnoli_(a, b, c): 

587 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using 

588 U{Cagnoli's<https://Zenodo.org/record/35392>} (D.34) formula. 

589 

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

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

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

593 

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

595 

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

597 

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

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

600 ''' 

601 a = Radians_(a=a) 

602 b = Radians_(b=b) 

603 c = Radians_(c=c) 

604 

605 s = fsumf_(a, b, c) * _0_5 

606 r = sin(s) * sin(s - a) * sin(s - b) * sin(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 

614 U{Girard's<https://MathWorld.Wolfram.com/GirardsSphericalExcessFormula.html>} 

615 formula. 

616 

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

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

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

620 

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

622 

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

624 

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

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

627 ''' 

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

629 Radians_(B=B), 

630 Radians_(C=C), -PI)) 

631 

632 

633def excessLHuilier_(a, b, c): 

634 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using 

635 U{L'Huilier's<https://MathWorld.Wolfram.com/LHuiliersTheorem.html>} 

636 Theorem. 

637 

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

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

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

641 

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

643 

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

645 

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

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

648 ''' 

649 a = Radians_(a=a) 

650 b = Radians_(b=b) 

651 c = Radians_(c=c) 

652 

653 s = fsumf_(a, b, c) * _0_5 

654 r = tan_2(s) * tan_2(s - a) * tan_2(s - b) * tan_2(s - c) 

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

656 return Radians(LHuilier=r * _4_0) 

657 

658 

659def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

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

663 method. 

664 

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

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

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

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

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

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

671 L{a_f2Tuple}) or C{None}. 

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

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

674 

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

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

677 if C{B{radius}=0} or C{None}. 

678 

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

680 

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

682 

683 @raise ValueError: Semi-circular longitudinal delta. 

684 

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

686 ''' 

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

688 

689 

690def excessKarney_(phi2, phi1, lam21): 

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

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

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

694 method. 

695 

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

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

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

699 

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

701 

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

703 

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

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

706 ''' 

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

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

709 # 

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

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

712 # 1 + tan(φ1 / 2) · tan(φ2 / 2) 

713 # 

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

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

716 t2 = tan_2(phi2) 

717 t1 = tan_2(phi1) 

718 t = tan_2(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((Lambertian(phi1) + 

742# Lambertian(phi2)) / 2) 

743# 

744# where lambda21 = lambda2 - lambda1 and lamb(x) is the Lambertian (or 

745# inverse Gudermannian) function 

746# 

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

748# 

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

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

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

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

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

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

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

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

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

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

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

760# - Charles Karney 

761 

762 

763def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

766 

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

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

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

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

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

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

773 L{a_f2Tuple}) or C{None}. 

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

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

776 

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

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

779 if C{B{radius}=0} or C{None}. 

780 

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

782 

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

784 

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

786 ''' 

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

788 

789 

790def excessQuad_(phi2, phi1, lam21): 

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

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

793 

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

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

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

797 

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

799 

800 @see: Function L{excessQuad}, U{Spherical trigonometry 

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

802 ''' 

803 s = sin((phi2 + phi1) * _0_5) 

804 c = cos((phi2 - phi1) * _0_5) 

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

806 

807 

808def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False): 

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

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

811 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

813 

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

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

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

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

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

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

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

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

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

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

824 

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

826 ellipsoid axes). 

827 

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

829 

830 @note: The meridional and prime_vertical radii of curvature 

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

832 

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

834 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

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

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

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

838 ''' 

839 E = _ellipsoidal(datum, flatLocal) 

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

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

842 

843hubeny = flatLocal # PYCHOK for Karl Hubeny 

844 

845 

846def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True): 

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

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

849 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

851 

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

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

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

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

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

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

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

859 

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

861 

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

863 

864 @note: The meridional and prime_vertical radii of curvature 

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

866 

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

868 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_}, 

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

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

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

872 ''' 

873 E = _ellipsoidal(datum, flatLocal_) 

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

875 

876hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny 

877 

878 

879def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

882 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

883 formula. 

884 

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

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

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

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

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

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

891 L{a_f2Tuple}) to use. 

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

893 and B{C{lon2}} (C{bool}). 

894 

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

896 ellipsoid or datum axes). 

897 

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

899 

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

901 L{cosineForsytheAndoyerLambert},L{cosineLaw}, 

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

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

904 L{vincentys}. 

905 ''' 

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

907 

908 

909def flatPolar_(phi2, phi1, lam21): 

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

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

912 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

913 formula. 

914 

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

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

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

918 

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

920 

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

922 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

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

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

925 ''' 

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

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

928 if a < b: 

929 a, b = b, a 

930 if a < EPS0: 

931 a = _0_0 

932 elif b > 0: 

933 b = b / a # /= chokes PyChecker 

934 c = b * cos(lam21) * _2_0 

935 c = fsumf_(_1_0, b**2, -fabs(c)) 

936 a *= sqrt0(c) 

937 return a 

938 

939 

940def hartzell(pov, los=None, earth=_WGS84, name=NN, **LatLon_and_kwds): 

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

942 from a Point-Of-View in space. 

943 

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

945 or L{Vector3d}). 

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

947 C{None} to point to the earth' center. 

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

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

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

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

952 point plus C{LatLon} keyword arguments, include 

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

954 

955 @return: The earth intersection (L{Vector3d}, C{Cartesian type} of 

956 B{C{pov}} or B{C{LatLon}}). 

957 

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

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

960 the earth or points in an opposite direction. 

961 

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

963 

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

965 method L{Ellipsoid.hartzell4} and U{I{Satellite Line-of-Sight 

966 Intersection with Earth}<https://StephenHartzell.Medium.com/ 

967 satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}. 

968 ''' 

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

970 _spherical_datum(earth, name=hartzell.__name__) 

971 try: 

972 r, _ = _MODS.triaxials._hartzell3d2(pov, los, D.ellipsoid._triaxial) 

973 except Exception as x: 

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

975 

976# else: 

977# E = D.ellipsoid 

978# # Triaxial(a, b, c) == (E.a, E.a, E.b) 

979# 

980# def _Error(txt): 

981# return IntersectionError(pov=pov, los=los, earth=earth, txt=txt) 

982# 

983# a2 = b2 = E.a2 # earth' x, y, ... 

984# c2 = E.b2 # ... z semi-axis squared 

985# q2 = E.b2_a2 # == c2 / a2 

986# bc = E.a * E.b # == b * c 

987# 

988# V3 = _MODS.vector3d._otherV3d 

989# p3 = V3(pov=pov) 

990# u3 = V3(los=los) if los else p3.negate() 

991# u3 = u3.unit() # unit vector, opposing signs 

992# 

993# x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz 

994# ux, vy, wz = u3.times_(p3).xyz 

995# u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz 

996# 

997# t = c2, c2, b2 

998# m = fdot(t, u2, v2, w2) # a2 factored out 

999# if m < EPS0: # zero or near-null LOS vector 

1000# raise _Error(_near_(_null_)) 

1001# 

1002# # a2 and b2 factored out, b2 == a2 and b2 / a2 == 1 

1003# r = fsumf_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2, 

1004# c2 * u2, -u2 * z2, -w2 * x2, ux * wz * 2, 

1005# -w2 * y2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2) 

1006# if r > 0: 

1007# r = sqrt(r) * bc # == a * a * b * c / a2 

1008# elif r < 0: # LOS pointing away from or missing the earth 

1009# raise _Error(_opposite_ if max(ux, vy, wz) > 0 else _outside_) 

1010# 

1011# d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out 

1012# if d > 0: # POV inside or LOS missing, outside the earth 

1013# s = fsumf_(_1_0, x2 / a2, y2 / b2, z2 / c2, _N_2_0) # like _sideOf 

1014# raise _Error(_outside_ if s > 0 else _inside_) 

1015# elif fsumf_(x2, y2, z2) < d**2: # d past earth center 

1016# raise _Error(_too_(_distant_)) 

1017# 

1018# r = p3.minus(u3.times(d)) 

1019# # h = p3.minus(r).length # distance to ellipsoid 

1020 

1021 r = _xnamed(r, name or hartzell.__name__) 

1022 if LatLon_and_kwds: 

1023 c = _MODS.cartesianBase.CartesianBase(r, datum=D, name=r.name) 

1024 r = c.toLatLon(**LatLon_and_kwds) 

1025 return r 

1026 

1027 

1028def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1029 '''Compute the distance between two (spherical) points using the 

1030 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} 

1031 formula. 

1032 

1033 @arg lat1: Start latitude (C{degrees}). 

1034 @arg lon1: Start longitude (C{degrees}). 

1035 @arg lat2: End latitude (C{degrees}). 

1036 @arg lon2: End longitude (C{degrees}). 

1037 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1038 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1039 L{a_f2Tuple}) to use. 

1040 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1041 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1042 

1043 @return: Distance (C{meter}, same units as B{C{radius}}). 

1044 

1045 @raise TypeError: Invalid B{C{radius}}. 

1046 

1047 @see: U{Distance between two (spherical) points 

1048 <https://www.EdWilliams.org/avform.htm#Dist>}, functions 

1049 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1050 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1051 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, 

1052 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1053 

1054 @note: See note at function L{vincentys_}. 

1055 ''' 

1056 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2) 

1057 

1058 

1059def haversine_(phi2, phi1, lam21): 

1060 '''Compute the I{angular} distance between two (spherical) points 

1061 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} 

1062 formula. 

1063 

1064 @arg phi2: End latitude (C{radians}). 

1065 @arg phi1: Start latitude (C{radians}). 

1066 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1067 

1068 @return: Angular distance (C{radians}). 

1069 

1070 @see: Functions L{haversine}, L{cosineAndoyerLambert_}, 

1071 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1072 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1073 L{flatPolar_}, L{thomas_} and L{vincentys_}. 

1074 

1075 @note: See note at function L{vincentys_}. 

1076 ''' 

1077 def _hsin(rad): 

1078 return sin(rad * _0_5)**2 

1079 

1080 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine 

1081 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2 

1082 

1083 

1084def heightOf(angle, distance, radius=R_M): 

1085 '''Determine the height above the (spherical) earth' surface after 

1086 traveling along a straight line at a given tilt. 

1087 

1088 @arg angle: Tilt angle above horizontal (C{degrees}). 

1089 @arg distance: Distance along the line (C{meter} or same units as 

1090 B{C{radius}}). 

1091 @kwarg radius: Optional mean earth radius (C{meter}). 

1092 

1093 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}). 

1094 

1095 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}. 

1096 

1097 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>} 

1098 (U{Shapiro et al. 2009, JTECH 

1099 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>} 

1100 and U{Potvin et al. 2012, JTECH 

1101 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}). 

1102 ''' 

1103 r = h = Radius(radius) 

1104 d = fabs(Distance(distance)) 

1105 if d > h: 

1106 d, h = h, d 

1107 

1108 if d > EPS0: # and h > EPS0 

1109 d = d / h # /= h chokes PyChecker 

1110 s = sin(Phi_(angle=angle, clip=_180_0)) 

1111 s = fsumf_(_1_0, _2_0 * s * d, d**2) 

1112 if s > 0: 

1113 return h * sqrt(s) - r 

1114 

1115 raise _ValueError(angle=angle, distance=distance, radius=radius) 

1116 

1117 

1118def horizon(height, radius=R_M, refraction=False): 

1119 '''Determine the distance to the horizon from a given altitude 

1120 above the (spherical) earth. 

1121 

1122 @arg height: Altitude (C{meter} or same units as B{C{radius}}). 

1123 @kwarg radius: Optional mean earth radius (C{meter}). 

1124 @kwarg refraction: Consider atmospheric refraction (C{bool}). 

1125 

1126 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}). 

1127 

1128 @raise ValueError: Invalid B{C{height}} or B{C{radius}}. 

1129 

1130 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}. 

1131 ''' 

1132 h, r = Height(height), Radius(radius) 

1133 if min(h, r) < 0: 

1134 raise _ValueError(height=height, radius=radius) 

1135 

1136 if refraction: 

1137 d2 = 2.415750694528 * h * r # 2.0 / 0.8279 

1138 else: 

1139 d2 = h * fsumf_(r, r, h) 

1140 return sqrt0(d2) 

1141 

1142 

1143class _idllmn6(object): # see also .geodesicw._wargs, .vector2d._numpy 

1144 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}. 

1145 ''' 

1146 @contextmanager # <https://www.python.org/dev/peps/pep-0343/> Examples 

1147 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds): 

1148 try: 

1149 if wrap: 

1150 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap) 

1151 kwds = _xkwds(kwds, wrap=wrap) # for _xError 

1152 m = small if small is _100km else Meter_(small=small) 

1153 n = (intersections2 if s else intersection2).__name__ 

1154 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m: 

1155 d, m = None, _MODS.vector3d 

1156 _i = m._intersects2 if s else m._intersect3d3 

1157 elif isscalar(datum) and datum < 0 and not s: 

1158 d = _spherical_datum(-datum, name=n) 

1159 m = _MODS.sphericalNvector 

1160 _i = m.intersection 

1161 else: 

1162 d = _spherical_datum(datum, name=n) 

1163 if d.isSpherical: 

1164 m = _MODS.sphericalTrigonometry 

1165 _i = m._intersects2 if s else m._intersect 

1166 elif d.isEllipsoidal: 

1167 try: 

1168 if d.ellipsoid.geodesic: 

1169 pass 

1170 m = _MODS.ellipsoidalKarney 

1171 except ImportError: 

1172 m = _MODS.ellipsoidalExact 

1173 _i = m._intersections2 if s else m._intersection3 # ellispoidalBaseDI 

1174 else: 

1175 raise _TypeError(datum=datum) 

1176 yield _i, d, lat2, lon2, m, n 

1177 

1178 except (TypeError, ValueError) as x: 

1179 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum, 

1180 lat2=lat2, lon2=lon2, small=small, **kwds) 

1181 

1182_idllmn6 = _idllmn6() # PYCHOK singleton 

1183 

1184 

1185def intersection2(lat1, lon1, bearing1, 

1186 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True 

1187 '''I{Conveniently} compute the intersection of two lines each defined 

1188 by a (geodetic) point and a bearing from North, using either ... 

1189 

1190 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km 

1191 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1192 

1193 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}} 

1194 or a C{scalar B{datum}} representing the earth radius, conventionally 

1195 in C{meter} or ... 

1196 

1197 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative} 

1198 C{scalar}, (negative) earth radius, conventionally in C{meter} or ... 

1199 

1200 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}} 

1201 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1202 is installed, otherwise ... 

1203 

1204 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal. 

1205 

1206 @arg lat1: Latitude of the first point (C{degrees}). 

1207 @arg lon1: Longitude of the first point (C{degrees}). 

1208 @arg bearing1: Bearing at the first point (compass C{degrees}). 

1209 @arg lat2: Latitude of the second point (C{degrees}). 

1210 @arg lon2: Longitude of the second point (C{degrees}). 

1211 @arg bearing2: Bearing at the second point (compass C{degrees}). 

1212 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1213 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1214 radius (C{meter}, same units as B{C{radius1}} and 

1215 B{C{radius2}}) or C{None}. 

1216 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1217 and B{C{lon2}} (C{bool}). 

1218 @kwarg small: Upper limit for small distances (C{meter}). 

1219 

1220 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and 

1221 longitude of the intersection point. 

1222 

1223 @raise IntersectionError: Ambiguous or infinite intersection 

1224 or colinear, parallel or otherwise 

1225 non-intersecting lines. 

1226 

1227 @raise TypeError: Invalid B{C{datum}}. 

1228 

1229 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}}, 

1230 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}. 

1231 

1232 @see: Method L{RhumbLine.intersection2}. 

1233 

1234 @note: The returned intersections may be near-antipodal. 

1235 ''' 

1236 b1 = Bearing(bearing1=bearing1) 

1237 b2 = Bearing(bearing2=bearing2) 

1238 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1239 small, wrap, False, bearing1=b1, bearing2=b2) as t: 

1240 _i, d, lat2, lon2, m, n = t 

1241 if d is None: 

1242 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1, 

1243 m.Vector3d(lon2, lat2, 0), b2, useZ=False) 

1244 t = LatLon2Tuple(t.y, t.x, name=n) 

1245 

1246 else: 

1247 t = _i(m.LatLon(lat1, lon1, datum=d), b1, 

1248 m.LatLon(lat2, lon2, datum=d), b2, height=0, wrap=False) 

1249 if isinstance(t, Intersection3Tuple): # ellipsoidal 

1250 t, _, _ = t 

1251 t = LatLon2Tuple(t.lat, t.lon, name=n) 

1252 return t 

1253 

1254 

1255def intersections2(lat1, lon1, radius1, 

1256 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True 

1257 '''I{Conveniently} compute the intersections of two circles each defined 

1258 by a (geodetic) center point and a radius, using either ... 

1259 

1260 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km 

1261 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1262 

1263 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}} 

1264 or a C{scalar B{datum}} representing the earth radius, conventionally 

1265 in C{meter} or ... 

1266 

1267 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}} 

1268 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1269 is installed, otherwise ... 

1270 

1271 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal. 

1272 

1273 @arg lat1: Latitude of the first circle center (C{degrees}). 

1274 @arg lon1: Longitude of the first circle center (C{degrees}). 

1275 @arg radius1: Radius of the first circle (C{meter}, conventionally). 

1276 @arg lat2: Latitude of the second circle center (C{degrees}). 

1277 @arg lon2: Longitude of the second circle center (C{degrees}). 

1278 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}). 

1279 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1280 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1281 radius (C{meter}, same units as B{C{radius1}} and 

1282 B{C{radius2}}) or C{None}. 

1283 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1284 and B{C{lon2}} (C{bool}). 

1285 @kwarg small: Upper limit for small distances (C{meter}). 

1286 

1287 @return: 2-Tuple of the intersection points, each a 

1288 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the 

1289 points are the same instance, aka the I{radical center}. 

1290 

1291 @raise IntersectionError: Concentric, antipodal, invalid or 

1292 non-intersecting circles or no 

1293 convergence. 

1294 

1295 @raise TypeError: Invalid B{C{datum}}. 

1296 

1297 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}}, 

1298 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}. 

1299 ''' 

1300 r1 = Radius_(radius1=radius1) 

1301 r2 = Radius_(radius2=radius2) 

1302 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1303 small, wrap, True, radius1=r1, radius2=r2) as t: 

1304 _i, d, lat2, lon2, m, n = t 

1305 if d is None: 

1306 r1 = m2degrees(r1, radius=R_M, lat=lat1) 

1307 r2 = m2degrees(r2, radius=R_M, lat=lat2) 

1308 

1309 def _V2T(x, y, _, **unused): # _ == z unused 

1310 return LatLon2Tuple(y, x, name=n) 

1311 

1312 t = _i(m.Vector3d(lon1, lat1, 0), r1, 

1313 m.Vector3d(lon2, lat2, 0), r2, sphere=False, 

1314 Vector=_V2T) 

1315 else: 

1316 def _LL2T(lat, lon, **unused): 

1317 return LatLon2Tuple(lat, lon, name=n) 

1318 

1319 t = _i(m.LatLon(lat1, lon1, datum=d), r1, 

1320 m.LatLon(lat2, lon2, datum=d), r2, 

1321 LatLon=_LL2T, height=0, wrap=False) 

1322 return t 

1323 

1324 

1325def isantipode(lat1, lon1, lat2, lon2, eps=EPS): 

1326 '''Check whether two points are I{antipodal}, on diametrically 

1327 opposite sides of the earth. 

1328 

1329 @arg lat1: Latitude of one point (C{degrees}). 

1330 @arg lon1: Longitude of one point (C{degrees}). 

1331 @arg lat2: Latitude of the other point (C{degrees}). 

1332 @arg lon2: Longitude of the other point (C{degrees}). 

1333 @kwarg eps: Tolerance for near-equality (C{degrees}). 

1334 

1335 @return: C{True} if points are antipodal within the 

1336 B{C{eps}} tolerance, C{False} otherwise. 

1337 

1338 @see: Functions L{isantipode_} and L{antipode}. 

1339 ''' 

1340 return (fabs(lat1 + lat2) <= eps and 

1341 fabs(lon1 + lon2) <= eps) or _isequalTo( 

1342 normal(lat1, lon1), antipode(lat2, lon2), eps) 

1343 

1344 

1345def isantipode_(phi1, lam1, phi2, lam2, eps=EPS): 

1346 '''Check whether two points are I{antipodal}, on diametrically 

1347 opposite sides of the earth. 

1348 

1349 @arg phi1: Latitude of one point (C{radians}). 

1350 @arg lam1: Longitude of one point (C{radians}). 

1351 @arg phi2: Latitude of the other point (C{radians}). 

1352 @arg lam2: Longitude of the other point (C{radians}). 

1353 @kwarg eps: Tolerance for near-equality (C{radians}). 

1354 

1355 @return: C{True} if points are antipodal within the 

1356 B{C{eps}} tolerance, C{False} otherwise. 

1357 

1358 @see: Functions L{isantipode} and L{antipode_}. 

1359 ''' 

1360 return (fabs(phi1 + phi2) <= eps and 

1361 fabs(lam1 + lam2) <= eps) or _isequalTo_( 

1362 normal_(phi1, lam1), antipode_(phi2, lam2), eps) 

1363 

1364 

1365def _isequalTo(p1, p2, eps=EPS): 

1366 '''Compare 2 point lat-/lons ignoring C{class}. 

1367 ''' 

1368 return (fabs(p1.lat - p2.lat) <= eps and 

1369 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon) 

1370 

1371 

1372def _isequalTo_(p1, p2, eps=EPS): 

1373 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}. 

1374 ''' 

1375 return (fabs(p1.phi - p2.phi) <= eps and 

1376 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam) 

1377 

1378 

1379def isnormal(lat, lon, eps=0): 

1380 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their 

1381 respective I{normal} range in C{degrees}. 

1382 

1383 @arg lat: Latitude (C{degrees}). 

1384 @arg lon: Longitude (C{degrees}). 

1385 @kwarg eps: Optional tolerance C{degrees}). 

1386 

1387 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and 

1388 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise. 

1389 

1390 @see: Functions L{isnormal_} and L{normal}. 

1391 ''' 

1392 return (_90_0 - fabs(lat)) >= eps and (_180_0 - fabs(lon)) >= eps 

1393 

1394 

1395def isnormal_(phi, lam, eps=0): 

1396 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their 

1397 respective I{normal} range in C{radians}. 

1398 

1399 @arg phi: Latitude (C{radians}). 

1400 @arg lam: Longitude (C{radians}). 

1401 @kwarg eps: Optional tolerance C{radians}). 

1402 

1403 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and 

1404 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise. 

1405 

1406 @see: Functions L{isnormal} and L{normal_}. 

1407 ''' 

1408 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps 

1409 

1410 

1411def latlon2n_xyz(lat, lon, name=NN): 

1412 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1413 earth's surface) X, Y and Z components. 

1414 

1415 @arg lat: Latitude (C{degrees}). 

1416 @arg lon: Longitude (C{degrees}). 

1417 @kwarg name: Optional name (C{str}). 

1418 

1419 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1420 

1421 @see: Function L{philam2n_xyz}. 

1422 

1423 @note: These are C{n-vector} x, y and z components, 

1424 I{NOT} geocentric ECEF x, y and z coordinates! 

1425 ''' 

1426 return _2n_xyz(name, *sincos2d_(lat, lon)) 

1427 

1428 

1429def _normal2(a, b, n_2, n, n2): 

1430 '''(INTERNAL) Helper for C{normal} and C{normal_}. 

1431 ''' 

1432 if fabs(b) > n: 

1433 b = remainder(b, n2) 

1434 if fabs(a) > n_2: 

1435 r = remainder(a, n) 

1436 if r != a: 

1437 a = -r 

1438 b -= n if b > 0 else -n 

1439 return float0_(a, b) 

1440 

1441 

1442def normal(lat, lon, name=NN): 

1443 '''Normalize a lat- I{and} longitude pair in C{degrees}. 

1444 

1445 @arg lat: Latitude (C{degrees}). 

1446 @arg lon: Longitude (C{degrees}). 

1447 @kwarg name: Optional name (C{str}). 

1448 

1449 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90} 

1450 and C{abs(lon) <= 180}. 

1451 

1452 @see: Functions L{normal_} and L{isnormal}. 

1453 ''' 

1454 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0), 

1455 name=name or normal.__name__) 

1456 

1457 

1458def normal_(phi, lam, name=NN): 

1459 '''Normalize a lat- I{and} longitude pair in C{radians}. 

1460 

1461 @arg phi: Latitude (C{radians}). 

1462 @arg lam: Longitude (C{radians}). 

1463 @kwarg name: Optional name (C{str}). 

1464 

1465 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2} 

1466 and C{abs(lam) <= PI}. 

1467 

1468 @see: Functions L{normal} and L{isnormal_}. 

1469 ''' 

1470 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2), 

1471 name=name or normal_.__name__) 

1472 

1473 

1474def _2n_xyz(name, sa, ca, sb, cb): 

1475 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}. 

1476 ''' 

1477 # Kenneth Gade eqn 3, but using right-handed 

1478 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N 

1479 return Vector3Tuple(ca * cb, ca * sb, sa, name=name) 

1480 

1481 

1482def n_xyz2latlon(x, y, z, name=NN): 

1483 '''Convert C{n-vector} components to lat- and longitude in C{degrees}. 

1484 

1485 @arg x: X component (C{scalar}). 

1486 @arg y: Y component (C{scalar}). 

1487 @arg z: Z component (C{scalar}). 

1488 @kwarg name: Optional name (C{str}). 

1489 

1490 @return: A L{LatLon2Tuple}C{(lat, lon)}. 

1491 

1492 @see: Function L{n_xyz2philam}. 

1493 ''' 

1494 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name) 

1495 

1496 

1497def n_xyz2philam(x, y, z, name=NN): 

1498 '''Convert C{n-vector} components to lat- and longitude in C{radians}. 

1499 

1500 @arg x: X component (C{scalar}). 

1501 @arg y: Y component (C{scalar}). 

1502 @arg z: Z component (C{scalar}). 

1503 @kwarg name: Optional name (C{str}). 

1504 

1505 @return: A L{PhiLam2Tuple}C{(phi, lam)}. 

1506 

1507 @see: Function L{n_xyz2latlon}. 

1508 ''' 

1509 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name) 

1510 

1511 

1512def _opposes(d, m, n, n2): 

1513 '''(INTERNAL) Helper for C{opposing} and C{opposing_}. 

1514 ''' 

1515 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1 

1516 return False if d < m or d > (n2 - m) else ( 

1517 True if (n - m) < d < (n + m) else None) 

1518 

1519 

1520def opposing(bearing1, bearing2, margin=_90_0): 

1521 '''Compare the direction of two bearings given in C{degrees}. 

1522 

1523 @arg bearing1: First bearing (compass C{degrees}). 

1524 @arg bearing2: Second bearing (compass C{degrees}). 

1525 @kwarg margin: Optional, interior angle bracket (C{degrees}). 

1526 

1527 @return: C{True} if both bearings point in opposite, C{False} if 

1528 in similar or C{None} if in perpendicular directions. 

1529 

1530 @see: Function L{opposing_}. 

1531 ''' 

1532 m = Degrees_(margin=margin, low=EPS0, high=_90_0) 

1533 return _opposes(bearing2 - bearing1, m, _180_0, _360_0) 

1534 

1535 

1536def opposing_(radians1, radians2, margin=PI_2): 

1537 '''Compare the direction of two bearings given in C{radians}. 

1538 

1539 @arg radians1: First bearing (C{radians}). 

1540 @arg radians2: Second bearing (C{radians}). 

1541 @kwarg margin: Optional, interior angle bracket (C{radians}). 

1542 

1543 @return: C{True} if both bearings point in opposite, C{False} if 

1544 in similar or C{None} if in perpendicular directions. 

1545 

1546 @see: Function L{opposing}. 

1547 ''' 

1548 m = Radians_(margin=margin, low=EPS0, high=PI_2) 

1549 return _opposes(radians2 - radians1, m, PI, PI2) 

1550 

1551 

1552def philam2n_xyz(phi, lam, name=NN): 

1553 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1554 earth's surface) X, Y and Z components. 

1555 

1556 @arg phi: Latitude (C{radians}). 

1557 @arg lam: Longitude (C{radians}). 

1558 @kwarg name: Optional name (C{str}). 

1559 

1560 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1561 

1562 @see: Function L{latlon2n_xyz}. 

1563 

1564 @note: These are C{n-vector} x, y and z components, 

1565 I{NOT} geocentric ECEF x, y and z coordinates! 

1566 ''' 

1567 return _2n_xyz(name, *sincos2_(phi, lam)) 

1568 

1569 

1570def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d 

1571 # (INTERNAL) See C{radical2} below 

1572 # assert d > EPS0 

1573 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5 

1574 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d) 

1575 

1576 

1577def radical2(distance, radius1, radius2): 

1578 '''Compute the I{radical ratio} and I{radical line} of two 

1579 U{intersecting circles<https://MathWorld.Wolfram.com/ 

1580 Circle-CircleIntersection.html>}. 

1581 

1582 The I{radical line} is perpendicular to the axis thru the 

1583 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}. 

1584 

1585 @arg distance: Distance between the circle centers (C{scalar}). 

1586 @arg radius1: Radius of the first circle (C{scalar}). 

1587 @arg radius2: Radius of the second circle (C{scalar}). 

1588 

1589 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <= 

1590 ratio <= 1.0} and C{xline} is along the B{C{distance}}. 

1591 

1592 @raise IntersectionError: The B{C{distance}} exceeds the sum 

1593 of B{C{radius1}} and B{C{radius2}}. 

1594 

1595 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or 

1596 B{C{radius2}}. 

1597 

1598 @see: U{Circle-Circle Intersection 

1599 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}. 

1600 ''' 

1601 d = Distance_(distance, low=_0_0) 

1602 r1 = Radius_(radius1=radius1) 

1603 r2 = Radius_(radius2=radius2) 

1604 if d > (r1 + r2): 

1605 raise IntersectionError(distance=d, radius1=r1, radius2=r2, 

1606 txt=_too_(_distant_)) 

1607 return _radical2(d, r1, r2) if d > EPS0 else \ 

1608 Radical2Tuple(_0_5, _0_0) 

1609 

1610 

1611class Radical2Tuple(_NamedTuple): 

1612 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and 

1613 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0} 

1614 ''' 

1615 _Names_ = (_ratio_, _xline_) 

1616 _Units_ = ( Scalar, Scalar) 

1617 

1618 

1619def _radistance(inst): 

1620 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians} 

1621 and L{hausdorff._HausdorffMeterRedians} classes. 

1622 ''' 

1623 kwds_ = _xkwds(inst._kwds, wrap=False) 

1624 wrap_ = _xkwds_pop(kwds_, wrap=False) 

1625 func_ = inst._func_ 

1626 try: # calling lower-overhead C{func_} 

1627 func_(0, _0_25, _0_5, **kwds_) 

1628 wrap_ = _Wrap._philamop(wrap_) 

1629 except TypeError: 

1630 return inst.distance 

1631 

1632 def _philam(p): 

1633 try: 

1634 return p.phi, p.lam 

1635 except AttributeError: # no .phi or .lam 

1636 return radians(p.lat), radians(p.lon) 

1637 

1638 def _func_wrap(point1, point2): 

1639 phi1, lam1 = wrap_(*_philam(point1)) 

1640 phi2, lam2 = wrap_(*_philam(point2)) 

1641 return func_(phi2, phi1, lam2 - lam1, **kwds_) 

1642 

1643 inst._units = inst._units_ 

1644 return _func_wrap 

1645 

1646 

1647def _scale_deg(lat1, lat2): # degrees 

1648 # scale factor cos(mean of lats) for delta lon 

1649 m = fabs(lat1 + lat2) * _0_5 

1650 return cos(radians(m)) if m < 90 else _0_0 

1651 

1652 

1653def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights 

1654 # scale factor cos(mean of phis) for delta lam 

1655 m = fabs(phi1 + phi2) * _0_5 

1656 return cos(m) if m < PI_2 else _0_0 

1657 

1658 

1659def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw 

1660 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine. 

1661 ''' 

1662 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21) 

1663 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21 

1664 

1665 

1666def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

1667 '''Compute the distance between two (ellipsoidal) points using 

1668 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1669 formula. 

1670 

1671 @arg lat1: Start latitude (C{degrees}). 

1672 @arg lon1: Start longitude (C{degrees}). 

1673 @arg lat2: End latitude (C{degrees}). 

1674 @arg lon2: End longitude (C{degrees}). 

1675 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1676 L{Ellipsoid2} or L{a_f2Tuple}) to use. 

1677 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1678 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1679 

1680 @return: Distance (C{meter}, same units as the B{C{datum}}'s 

1681 ellipsoid axes). 

1682 

1683 @raise TypeError: Invalid B{C{datum}}. 

1684 

1685 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1686 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, 

1687 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}. 

1688 ''' 

1689 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2) 

1690 

1691 

1692def thomas_(phi2, phi1, lam21, datum=_WGS84): 

1693 '''Compute the I{angular} distance between two (ellipsoidal) points using 

1694 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1695 formula. 

1696 

1697 @arg phi2: End latitude (C{radians}). 

1698 @arg phi1: Start latitude (C{radians}). 

1699 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1700 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, 

1701 L{Ellipsoid2} or L{a_f2Tuple}). 

1702 

1703 @return: Angular distance (C{radians}). 

1704 

1705 @raise TypeError: Invalid B{C{datum}}. 

1706 

1707 @see: Functions L{thomas}, L{cosineAndoyerLambert_}, 

1708 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1709 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1710 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP 

1711 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/ 

1712 Distance/ThomasFormula.php>}. 

1713 ''' 

1714 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21) 

1715 if r and isnon0(c1) and isnon0(c2): 

1716 E = _ellipsoidal(datum, thomas_) 

1717 if E.f: 

1718 r1 = atan2(E.b_a * s1, c1) 

1719 r2 = atan2(E.b_a * s2, c2) 

1720 

1721 j = (r2 + r1) * _0_5 

1722 k = (r2 - r1) * _0_5 

1723 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5) 

1724 

1725 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2) 

1726 u = _1_0 - h 

1727 if isnon0(u) and isnon0(h): 

1728 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h) 

1729 sr, cr = sincos2(r) 

1730 if isnon0(sr): 

1731 u = 2 * (sj * ck)**2 / u 

1732 h = 2 * (sk * cj)**2 / h 

1733 x = u + h 

1734 y = u - h 

1735 

1736 s = r / sr 

1737 e = 4 * s**2 

1738 d = 2 * cr 

1739 a = e * d 

1740 b = 2 * r 

1741 c = s - (a - d) * _0_5 

1742 f = E.f * _0_25 

1743 

1744 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y) 

1745 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr 

1746 return r 

1747 

1748 

1749def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1750 '''Compute the distance between two (spherical) points using 

1751 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1752 spherical formula. 

1753 

1754 @arg lat1: Start latitude (C{degrees}). 

1755 @arg lon1: Start longitude (C{degrees}). 

1756 @arg lat2: End latitude (C{degrees}). 

1757 @arg lon2: End longitude (C{degrees}). 

1758 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1759 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1760 L{a_f2Tuple}) to use. 

1761 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1762 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1763 

1764 @return: Distance (C{meter}, same units as B{C{radius}}). 

1765 

1766 @raise UnitError: Invalid B{C{radius}}. 

1767 

1768 @see: Functions L{vincentys_}, L{cosineAndoyerLambert}, 

1769 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular}, 

1770 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1771 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2}, 

1772 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1773 

1774 @note: See note at function L{vincentys_}. 

1775 ''' 

1776 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2) 

1777 

1778 

1779def vincentys_(phi2, phi1, lam21): 

1780 '''Compute the I{angular} distance between two (spherical) points using 

1781 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1782 spherical formula. 

1783 

1784 @arg phi2: End latitude (C{radians}). 

1785 @arg phi1: Start latitude (C{radians}). 

1786 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1787 

1788 @return: Angular distance (C{radians}). 

1789 

1790 @see: Functions L{vincentys}, L{cosineAndoyerLambert_}, 

1791 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1792 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1793 L{flatPolar_}, L{haversine_} and L{thomas_}. 

1794 

1795 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_} 

1796 produce equivalent results, but L{vincentys_} is suitable 

1797 for antipodal points and slightly more expensive (M{3 cos, 

1798 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_} 

1799 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and 

1800 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}). 

1801 ''' 

1802 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21) 

1803 

1804 c = c2 * c21 

1805 x = s1 * s2 + c1 * c 

1806 y = c1 * s2 - s1 * c 

1807 return atan2(hypot(c2 * s21, y), x) 

1808 

1809# **) MIT License 

1810# 

1811# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved. 

1812# 

1813# Permission is hereby granted, free of charge, to any person obtaining a 

1814# copy of this software and associated documentation files (the "Software"), 

1815# to deal in the Software without restriction, including without limitation 

1816# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1817# and/or sell copies of the Software, and to permit persons to whom the 

1818# Software is furnished to do so, subject to the following conditions: 

1819# 

1820# The above copyright notice and this permission notice shall be included 

1821# in all copies or substantial portions of the Software. 

1822# 

1823# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1824# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1825# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1826# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1827# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

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