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 atan, atan2, cos, degrees, fabs, radians, sin, sqrt # pow 

37 

38__all__ = _ALL_LAZY.formy 

39__version__ = '23.06.08' 

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 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 excessGirard_(A, B, C): 

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

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

589 formula. 

590 

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

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

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

594 

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

596 

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

598 

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

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

601 ''' 

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

603 Radians_(B=B), 

604 Radians_(C=C), -PI)) 

605 

606 

607def excessLHuilier_(a, b, c): 

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

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

610 Theorem. 

611 

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

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

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

615 

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

617 

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

619 

620 @see: Function L{excessGirard_} and U{Spherical trigonometry 

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

622 ''' 

623 a = Radians_(a=a) 

624 b = Radians_(b=b) 

625 c = Radians_(c=c) 

626 

627 s = fsumf_(a, b, c) * _0_5 

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

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

630 return Radians(LHuilier=r * _4_0) 

631 

632 

633def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

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

637 method. 

638 

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

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

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

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

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

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

645 L{a_f2Tuple}) or C{None}. 

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

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

648 

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

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

651 if C{B{radius}=0} or C{None}. 

652 

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

654 

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

656 

657 @raise ValueError: Semi-circular longitudinal delta. 

658 

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

660 ''' 

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

662 

663 

664def excessKarney_(phi2, phi1, lam21): 

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

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

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

668 method. 

669 

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

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

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

673 

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

675 

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

677 

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

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

680 ''' 

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

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

683 # 

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

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

686 # 1 + tan(φ1 / 2) · tan(φ2 / 2) 

687 # 

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

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

690 t2 = tan_2(phi2) 

691 t1 = tan_2(phi1) 

692 t = tan_2(lam21, lam21=None) 

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

694 _1_0 + (t1 * t2)) * _2_0) 

695 

696 

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

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

699# 

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

701# 

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

703# 

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

705# 

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

707# 

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

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

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

711# 

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

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

714# 

715# = tan(lambda21 / 2) * tanh((Lambertian(phi1) + 

716# Lambertian(phi2)) / 2) 

717# 

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

719# inverse Gudermannian) function 

720# 

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

722# 

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

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

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

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

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

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

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

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

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

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

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

734# - Charles Karney 

735 

736 

737def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

740 

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

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

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

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

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

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

747 L{a_f2Tuple}) or C{None}. 

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

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

750 

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

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

753 if C{B{radius}=0} or C{None}. 

754 

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

756 

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

758 

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

760 ''' 

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

762 

763 

764def excessQuad_(phi2, phi1, lam21): 

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

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

767 

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

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

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

771 

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

773 

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

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

776 ''' 

777 s = sin((phi2 + phi1) * _0_5) 

778 c = cos((phi2 - phi1) * _0_5) 

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

780 

781 

782def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False): 

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

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

785 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

787 

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

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

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

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

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

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

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

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

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

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

798 

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

800 ellipsoid axes). 

801 

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

803 

804 @note: The meridional and prime_vertical radii of curvature 

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

806 

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

808 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

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

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

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

812 ''' 

813 E = _ellipsoidal(datum, flatLocal) 

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

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

816 

817hubeny = flatLocal # PYCHOK for Karl Hubeny 

818 

819 

820def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True): 

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

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

823 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

825 

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

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

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

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

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

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

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

833 

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

835 

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

837 

838 @note: The meridional and prime_vertical radii of curvature 

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

840 

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

842 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_}, 

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

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

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

846 ''' 

847 E = _ellipsoidal(datum, flatLocal_) 

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

849 

850hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny 

851 

852 

853def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

856 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

857 formula. 

858 

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

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

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

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

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

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

865 L{a_f2Tuple}) to use. 

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

867 and B{C{lon2}} (C{bool}). 

868 

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

870 ellipsoid or datum axes). 

871 

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

873 

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

875 L{cosineForsytheAndoyerLambert},L{cosineLaw}, 

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

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

878 L{vincentys}. 

879 ''' 

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

881 

882 

883def flatPolar_(phi2, phi1, lam21): 

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

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

886 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

887 formula. 

888 

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

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

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

892 

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

894 

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

896 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

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

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

899 ''' 

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

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

902 if a < b: 

903 a, b = b, a 

904 if a < EPS0: 

905 a = _0_0 

906 elif b > 0: 

907 b = b / a # /= chokes PyChecker 

908 c = b * cos(lam21) * _2_0 

909 c = fsumf_(_1_0, b**2, -fabs(c)) 

910 a *= sqrt0(c) 

911 return a 

912 

913 

914def hartzell(pov, los=None, earth=_WGS84, name=NN, **LatLon_and_kwds): 

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

916 from a Point-Of-View in space. 

917 

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

919 or L{Vector3d}). 

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

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

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

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

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

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

926 point plus C{LatLon} keyword arguments, include 

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

928 

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

930 B{C{pov}} or B{C{LatLon}}). 

931 

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

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

934 the earth or points in an opposite direction. 

935 

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

937 

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

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

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

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

942 ''' 

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

944 _spherical_datum(earth, name=hartzell.__name__) 

945 try: 

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

947 except Exception as x: 

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

949 

950# else: 

951# E = D.ellipsoid 

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

953# 

954# def _Error(txt): 

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

956# 

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

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

959# q2 = E.b2_a2 # == c2 / a2 

960# bc = E.a * E.b # == b * c 

961# 

962# V3 = _MODS.vector3d._otherV3d 

963# p3 = V3(pov=pov) 

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

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

966# 

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

968# ux, vy, wz = u3.times_(p3).xyz 

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

970# 

971# t = c2, c2, b2 

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

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

974# raise _Error(_near_(_null_)) 

975# 

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

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

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

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

980# if r > 0: 

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

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

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

984# 

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

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

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

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

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

990# raise _Error(_too_(_distant_)) 

991# 

992# r = p3.minus(u3.times(d)) 

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

994 

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

996 if LatLon_and_kwds: 

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

998 r = c.toLatLon(**LatLon_and_kwds) 

999 return r 

1000 

1001 

1002def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1003 '''Compute the distance between two (spherical) points using the 

1004 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} 

1005 formula. 

1006 

1007 @arg lat1: Start latitude (C{degrees}). 

1008 @arg lon1: Start longitude (C{degrees}). 

1009 @arg lat2: End latitude (C{degrees}). 

1010 @arg lon2: End longitude (C{degrees}). 

1011 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1012 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1013 L{a_f2Tuple}) to use. 

1014 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1015 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1016 

1017 @return: Distance (C{meter}, same units as B{C{radius}}). 

1018 

1019 @raise TypeError: Invalid B{C{radius}}. 

1020 

1021 @see: U{Distance between two (spherical) points 

1022 <https://www.EdWilliams.org/avform.htm#Dist>}, functions 

1023 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1024 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1025 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, 

1026 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1027 

1028 @note: See note at function L{vincentys_}. 

1029 ''' 

1030 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2) 

1031 

1032 

1033def haversine_(phi2, phi1, lam21): 

1034 '''Compute the I{angular} distance between two (spherical) points 

1035 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} 

1036 formula. 

1037 

1038 @arg phi2: End latitude (C{radians}). 

1039 @arg phi1: Start latitude (C{radians}). 

1040 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1041 

1042 @return: Angular distance (C{radians}). 

1043 

1044 @see: Functions L{haversine}, L{cosineAndoyerLambert_}, 

1045 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1046 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1047 L{flatPolar_}, L{thomas_} and L{vincentys_}. 

1048 

1049 @note: See note at function L{vincentys_}. 

1050 ''' 

1051 def _hsin(rad): 

1052 return sin(rad * _0_5)**2 

1053 

1054 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine 

1055 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2 

1056 

1057 

1058def heightOf(angle, distance, radius=R_M): 

1059 '''Determine the height above the (spherical) earth' surface after 

1060 traveling along a straight line at a given tilt. 

1061 

1062 @arg angle: Tilt angle above horizontal (C{degrees}). 

1063 @arg distance: Distance along the line (C{meter} or same units as 

1064 B{C{radius}}). 

1065 @kwarg radius: Optional mean earth radius (C{meter}). 

1066 

1067 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}). 

1068 

1069 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}. 

1070 

1071 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>} 

1072 (U{Shapiro et al. 2009, JTECH 

1073 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>} 

1074 and U{Potvin et al. 2012, JTECH 

1075 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}). 

1076 ''' 

1077 r = h = Radius(radius) 

1078 d = fabs(Distance(distance)) 

1079 if d > h: 

1080 d, h = h, d 

1081 

1082 if d > EPS0: # and h > EPS0 

1083 d = d / h # /= h chokes PyChecker 

1084 s = sin(Phi_(angle=angle, clip=_180_0)) 

1085 s = fsumf_(_1_0, _2_0 * s * d, d**2) 

1086 if s > 0: 

1087 return h * sqrt(s) - r 

1088 

1089 raise _ValueError(angle=angle, distance=distance, radius=radius) 

1090 

1091 

1092def horizon(height, radius=R_M, refraction=False): 

1093 '''Determine the distance to the horizon from a given altitude 

1094 above the (spherical) earth. 

1095 

1096 @arg height: Altitude (C{meter} or same units as B{C{radius}}). 

1097 @kwarg radius: Optional mean earth radius (C{meter}). 

1098 @kwarg refraction: Consider atmospheric refraction (C{bool}). 

1099 

1100 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}). 

1101 

1102 @raise ValueError: Invalid B{C{height}} or B{C{radius}}. 

1103 

1104 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}. 

1105 ''' 

1106 h, r = Height(height), Radius(radius) 

1107 if min(h, r) < 0: 

1108 raise _ValueError(height=height, radius=radius) 

1109 

1110 if refraction: 

1111 d2 = 2.415750694528 * h * r # 2.0 / 0.8279 

1112 else: 

1113 d2 = h * fsumf_(r, r, h) 

1114 return sqrt0(d2) 

1115 

1116 

1117class _idllmn6(object): # see also .geodesicw._wargs, .vector2d._numpy 

1118 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}. 

1119 ''' 

1120 @contextmanager # <https://www.python.org/dev/peps/pep-0343/> Examples 

1121 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds): 

1122 try: 

1123 if wrap: 

1124 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap) 

1125 kwds = _xkwds(kwds, wrap=wrap) # for _xError 

1126 m = small if small is _100km else Meter_(small=small) 

1127 n = (intersections2 if s else intersection2).__name__ 

1128 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m: 

1129 d, m = None, _MODS.vector3d 

1130 _i = m._intersects2 if s else m._intersect3d3 

1131 elif isscalar(datum) and datum < 0 and not s: 

1132 d = _spherical_datum(-datum, name=n) 

1133 m = _MODS.sphericalNvector 

1134 _i = m.intersection 

1135 else: 

1136 d = _spherical_datum(datum, name=n) 

1137 if d.isSpherical: 

1138 m = _MODS.sphericalTrigonometry 

1139 _i = m._intersects2 if s else m._intersect 

1140 elif d.isEllipsoidal: 

1141 try: 

1142 if d.ellipsoid.geodesic: 

1143 pass 

1144 m = _MODS.ellipsoidalKarney 

1145 except ImportError: 

1146 m = _MODS.ellipsoidalExact 

1147 _i = m._intersections2 if s else m._intersection3 # ellispoidalBaseDI 

1148 else: 

1149 raise _TypeError(datum=datum) 

1150 yield _i, d, lat2, lon2, m, n 

1151 

1152 except (TypeError, ValueError) as x: 

1153 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum, 

1154 lat2=lat2, lon2=lon2, small=small, **kwds) 

1155 

1156_idllmn6 = _idllmn6() # PYCHOK singleton 

1157 

1158 

1159def intersection2(lat1, lon1, bearing1, 

1160 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True 

1161 '''I{Conveniently} compute the intersection of two lines each defined 

1162 by a (geodetic) point and a bearing from North, using either ... 

1163 

1164 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km 

1165 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1166 

1167 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}} 

1168 or a C{scalar B{datum}} representing the earth radius, conventionally 

1169 in C{meter} or ... 

1170 

1171 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative} 

1172 C{scalar}, (negative) earth radius, conventionally in C{meter} or ... 

1173 

1174 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}} 

1175 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1176 is installed, otherwise ... 

1177 

1178 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal. 

1179 

1180 @arg lat1: Latitude of the first point (C{degrees}). 

1181 @arg lon1: Longitude of the first point (C{degrees}). 

1182 @arg bearing1: Bearing at the first point (compass C{degrees}). 

1183 @arg lat2: Latitude of the second point (C{degrees}). 

1184 @arg lon2: Longitude of the second point (C{degrees}). 

1185 @arg bearing2: Bearing at the second point (compass C{degrees}). 

1186 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1187 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1188 radius (C{meter}, same units as B{C{radius1}} and 

1189 B{C{radius2}}) or C{None}. 

1190 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1191 and B{C{lon2}} (C{bool}). 

1192 @kwarg small: Upper limit for small distances (C{meter}). 

1193 

1194 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and 

1195 longitude of the intersection point. 

1196 

1197 @raise IntersectionError: Ambiguous or infinite intersection 

1198 or colinear, parallel or otherwise 

1199 non-intersecting lines. 

1200 

1201 @raise TypeError: Invalid B{C{datum}}. 

1202 

1203 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}}, 

1204 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}. 

1205 

1206 @see: Method L{RhumbLine.intersection2}. 

1207 

1208 @note: The returned intersections may be near-antipodal. 

1209 ''' 

1210 b1 = Bearing(bearing1=bearing1) 

1211 b2 = Bearing(bearing2=bearing2) 

1212 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1213 small, wrap, False, bearing1=b1, bearing2=b2) as t: 

1214 _i, d, lat2, lon2, m, n = t 

1215 if d is None: 

1216 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1, 

1217 m.Vector3d(lon2, lat2, 0), b2, useZ=False) 

1218 t = LatLon2Tuple(t.y, t.x, name=n) 

1219 

1220 else: 

1221 t = _i(m.LatLon(lat1, lon1, datum=d), b1, 

1222 m.LatLon(lat2, lon2, datum=d), b2, height=0, wrap=False) 

1223 if isinstance(t, Intersection3Tuple): # ellipsoidal 

1224 t, _, _ = t 

1225 t = LatLon2Tuple(t.lat, t.lon, name=n) 

1226 return t 

1227 

1228 

1229def intersections2(lat1, lon1, radius1, 

1230 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True 

1231 '''I{Conveniently} compute the intersections of two circles each defined 

1232 by a (geodetic) center point and a radius, using either ... 

1233 

1234 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km 

1235 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1236 

1237 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}} 

1238 or a C{scalar B{datum}} representing the earth radius, conventionally 

1239 in C{meter} or ... 

1240 

1241 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}} 

1242 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1243 is installed, otherwise ... 

1244 

1245 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal. 

1246 

1247 @arg lat1: Latitude of the first circle center (C{degrees}). 

1248 @arg lon1: Longitude of the first circle center (C{degrees}). 

1249 @arg radius1: Radius of the first circle (C{meter}, conventionally). 

1250 @arg lat2: Latitude of the second circle center (C{degrees}). 

1251 @arg lon2: Longitude of the second circle center (C{degrees}). 

1252 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}). 

1253 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1254 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1255 radius (C{meter}, same units as B{C{radius1}} and 

1256 B{C{radius2}}) or C{None}. 

1257 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1258 and B{C{lon2}} (C{bool}). 

1259 @kwarg small: Upper limit for small distances (C{meter}). 

1260 

1261 @return: 2-Tuple of the intersection points, each a 

1262 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the 

1263 points are the same instance, aka the I{radical center}. 

1264 

1265 @raise IntersectionError: Concentric, antipodal, invalid or 

1266 non-intersecting circles or no 

1267 convergence. 

1268 

1269 @raise TypeError: Invalid B{C{datum}}. 

1270 

1271 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}}, 

1272 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}. 

1273 ''' 

1274 r1 = Radius_(radius1=radius1) 

1275 r2 = Radius_(radius2=radius2) 

1276 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1277 small, wrap, True, radius1=r1, radius2=r2) as t: 

1278 _i, d, lat2, lon2, m, n = t 

1279 if d is None: 

1280 r1 = m2degrees(r1, radius=R_M, lat=lat1) 

1281 r2 = m2degrees(r2, radius=R_M, lat=lat2) 

1282 

1283 def _V2T(x, y, _, **unused): # _ == z unused 

1284 return LatLon2Tuple(y, x, name=n) 

1285 

1286 t = _i(m.Vector3d(lon1, lat1, 0), r1, 

1287 m.Vector3d(lon2, lat2, 0), r2, sphere=False, 

1288 Vector=_V2T) 

1289 else: 

1290 def _LL2T(lat, lon, **unused): 

1291 return LatLon2Tuple(lat, lon, name=n) 

1292 

1293 t = _i(m.LatLon(lat1, lon1, datum=d), r1, 

1294 m.LatLon(lat2, lon2, datum=d), r2, 

1295 LatLon=_LL2T, height=0, wrap=False) 

1296 return t 

1297 

1298 

1299def isantipode(lat1, lon1, lat2, lon2, eps=EPS): 

1300 '''Check whether two points are I{antipodal}, on diametrically 

1301 opposite sides of the earth. 

1302 

1303 @arg lat1: Latitude of one point (C{degrees}). 

1304 @arg lon1: Longitude of one point (C{degrees}). 

1305 @arg lat2: Latitude of the other point (C{degrees}). 

1306 @arg lon2: Longitude of the other point (C{degrees}). 

1307 @kwarg eps: Tolerance for near-equality (C{degrees}). 

1308 

1309 @return: C{True} if points are antipodal within the 

1310 B{C{eps}} tolerance, C{False} otherwise. 

1311 

1312 @see: Functions L{isantipode_} and L{antipode}. 

1313 ''' 

1314 return (fabs(lat1 + lat2) <= eps and 

1315 fabs(lon1 + lon2) <= eps) or _isequalTo( 

1316 normal(lat1, lon1), antipode(lat2, lon2), eps) 

1317 

1318 

1319def isantipode_(phi1, lam1, phi2, lam2, eps=EPS): 

1320 '''Check whether two points are I{antipodal}, on diametrically 

1321 opposite sides of the earth. 

1322 

1323 @arg phi1: Latitude of one point (C{radians}). 

1324 @arg lam1: Longitude of one point (C{radians}). 

1325 @arg phi2: Latitude of the other point (C{radians}). 

1326 @arg lam2: Longitude of the other point (C{radians}). 

1327 @kwarg eps: Tolerance for near-equality (C{radians}). 

1328 

1329 @return: C{True} if points are antipodal within the 

1330 B{C{eps}} tolerance, C{False} otherwise. 

1331 

1332 @see: Functions L{isantipode} and L{antipode_}. 

1333 ''' 

1334 return (fabs(phi1 + phi2) <= eps and 

1335 fabs(lam1 + lam2) <= eps) or _isequalTo_( 

1336 normal_(phi1, lam1), antipode_(phi2, lam2), eps) 

1337 

1338 

1339def _isequalTo(p1, p2, eps=EPS): 

1340 '''Compare 2 point lat-/lons ignoring C{class}. 

1341 ''' 

1342 return (fabs(p1.lat - p2.lat) <= eps and 

1343 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon) 

1344 

1345 

1346def _isequalTo_(p1, p2, eps=EPS): 

1347 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}. 

1348 ''' 

1349 return (fabs(p1.phi - p2.phi) <= eps and 

1350 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam) 

1351 

1352 

1353def isnormal(lat, lon, eps=0): 

1354 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their 

1355 respective I{normal} range in C{degrees}. 

1356 

1357 @arg lat: Latitude (C{degrees}). 

1358 @arg lon: Longitude (C{degrees}). 

1359 @kwarg eps: Optional tolerance C{degrees}). 

1360 

1361 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and 

1362 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise. 

1363 

1364 @see: Functions L{isnormal_} and L{normal}. 

1365 ''' 

1366 return (_90_0 - fabs(lat)) >= eps and (_180_0 - fabs(lon)) >= eps 

1367 

1368 

1369def isnormal_(phi, lam, eps=0): 

1370 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their 

1371 respective I{normal} range in C{radians}. 

1372 

1373 @arg phi: Latitude (C{radians}). 

1374 @arg lam: Longitude (C{radians}). 

1375 @kwarg eps: Optional tolerance C{radians}). 

1376 

1377 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and 

1378 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise. 

1379 

1380 @see: Functions L{isnormal} and L{normal_}. 

1381 ''' 

1382 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps 

1383 

1384 

1385def latlon2n_xyz(lat, lon, name=NN): 

1386 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1387 earth's surface) X, Y and Z components. 

1388 

1389 @arg lat: Latitude (C{degrees}). 

1390 @arg lon: Longitude (C{degrees}). 

1391 @kwarg name: Optional name (C{str}). 

1392 

1393 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1394 

1395 @see: Function L{philam2n_xyz}. 

1396 

1397 @note: These are C{n-vector} x, y and z components, 

1398 I{NOT} geocentric ECEF x, y and z coordinates! 

1399 ''' 

1400 return _2n_xyz(name, *sincos2d_(lat, lon)) 

1401 

1402 

1403def _normal2(a, b, n_2, n, n2): 

1404 '''(INTERNAL) Helper for C{normal} and C{normal_}. 

1405 ''' 

1406 if fabs(b) > n: 

1407 b = remainder(b, n2) 

1408 if fabs(a) > n_2: 

1409 r = remainder(a, n) 

1410 if r != a: 

1411 a = -r 

1412 b -= n if b > 0 else -n 

1413 return float0_(a, b) 

1414 

1415 

1416def normal(lat, lon, name=NN): 

1417 '''Normalize a lat- I{and} longitude pair in C{degrees}. 

1418 

1419 @arg lat: Latitude (C{degrees}). 

1420 @arg lon: Longitude (C{degrees}). 

1421 @kwarg name: Optional name (C{str}). 

1422 

1423 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90} 

1424 and C{abs(lon) <= 180}. 

1425 

1426 @see: Functions L{normal_} and L{isnormal}. 

1427 ''' 

1428 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0), 

1429 name=name or normal.__name__) 

1430 

1431 

1432def normal_(phi, lam, name=NN): 

1433 '''Normalize a lat- I{and} longitude pair in C{radians}. 

1434 

1435 @arg phi: Latitude (C{radians}). 

1436 @arg lam: Longitude (C{radians}). 

1437 @kwarg name: Optional name (C{str}). 

1438 

1439 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2} 

1440 and C{abs(lam) <= PI}. 

1441 

1442 @see: Functions L{normal} and L{isnormal_}. 

1443 ''' 

1444 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2), 

1445 name=name or normal_.__name__) 

1446 

1447 

1448def _2n_xyz(name, sa, ca, sb, cb): 

1449 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}. 

1450 ''' 

1451 # Kenneth Gade eqn 3, but using right-handed 

1452 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N 

1453 return Vector3Tuple(ca * cb, ca * sb, sa, name=name) 

1454 

1455 

1456def n_xyz2latlon(x, y, z, name=NN): 

1457 '''Convert C{n-vector} components to lat- and longitude in C{degrees}. 

1458 

1459 @arg x: X component (C{scalar}). 

1460 @arg y: Y component (C{scalar}). 

1461 @arg z: Z component (C{scalar}). 

1462 @kwarg name: Optional name (C{str}). 

1463 

1464 @return: A L{LatLon2Tuple}C{(lat, lon)}. 

1465 

1466 @see: Function L{n_xyz2philam}. 

1467 ''' 

1468 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name) 

1469 

1470 

1471def n_xyz2philam(x, y, z, name=NN): 

1472 '''Convert C{n-vector} components to lat- and longitude in C{radians}. 

1473 

1474 @arg x: X component (C{scalar}). 

1475 @arg y: Y component (C{scalar}). 

1476 @arg z: Z component (C{scalar}). 

1477 @kwarg name: Optional name (C{str}). 

1478 

1479 @return: A L{PhiLam2Tuple}C{(phi, lam)}. 

1480 

1481 @see: Function L{n_xyz2latlon}. 

1482 ''' 

1483 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name) 

1484 

1485 

1486def _opposes(d, m, n, n2): 

1487 '''(INETNAL) Helper for C{opposing} and C{opposing_}. 

1488 ''' 

1489 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1 

1490 return False if d < m or d > (n2 - m) else ( 

1491 True if (n - m) < d < (n + m) else None) 

1492 

1493 

1494def opposing(bearing1, bearing2, margin=_90_0): 

1495 '''Compare the direction of two bearings given in C{degrees}. 

1496 

1497 @arg bearing1: First bearing (compass C{degrees}). 

1498 @arg bearing2: Second bearing (compass C{degrees}). 

1499 @kwarg margin: Optional, interior angle bracket (C{degrees}). 

1500 

1501 @return: C{True} if both bearings point in opposite, C{False} if 

1502 in similar or C{None} if in perpendicular directions. 

1503 

1504 @see: Function L{opposing_}. 

1505 ''' 

1506 m = Degrees_(margin=margin, low=EPS0, high=_90_0) 

1507 return _opposes(bearing2 - bearing1, m,_180_0, _360_0) 

1508 

1509 

1510def opposing_(radians1, radians2, margin=PI_2): 

1511 '''Compare the direction of two bearings given in C{radians}. 

1512 

1513 @arg radians1: First bearing (C{radians}). 

1514 @arg radians2: Second bearing (C{radians}). 

1515 @kwarg margin: Optional, interior angle bracket (C{radians}). 

1516 

1517 @return: C{True} if both bearings point in opposite, C{False} if 

1518 in similar or C{None} if in perpendicular directions. 

1519 

1520 @see: Function L{opposing}. 

1521 ''' 

1522 m = Radians_(margin=margin, low=EPS0, high=PI_2) 

1523 return _opposes(radians2 - radians1, m, PI, PI2) 

1524 

1525 

1526def philam2n_xyz(phi, lam, name=NN): 

1527 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1528 earth's surface) X, Y and Z components. 

1529 

1530 @arg phi: Latitude (C{radians}). 

1531 @arg lam: Longitude (C{radians}). 

1532 @kwarg name: Optional name (C{str}). 

1533 

1534 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1535 

1536 @see: Function L{latlon2n_xyz}. 

1537 

1538 @note: These are C{n-vector} x, y and z components, 

1539 I{NOT} geocentric ECEF x, y and z coordinates! 

1540 ''' 

1541 return _2n_xyz(name, *sincos2_(phi, lam)) 

1542 

1543 

1544def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d 

1545 # (INTERNAL) See C{radical2} below 

1546 # assert d > EPS0 

1547 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5 

1548 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d) 

1549 

1550 

1551def radical2(distance, radius1, radius2): 

1552 '''Compute the I{radical ratio} and I{radical line} of two 

1553 U{intersecting circles<https://MathWorld.Wolfram.com/ 

1554 Circle-CircleIntersection.html>}. 

1555 

1556 The I{radical line} is perpendicular to the axis thru the 

1557 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}. 

1558 

1559 @arg distance: Distance between the circle centers (C{scalar}). 

1560 @arg radius1: Radius of the first circle (C{scalar}). 

1561 @arg radius2: Radius of the second circle (C{scalar}). 

1562 

1563 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <= 

1564 ratio <= 1.0} and C{xline} is along the B{C{distance}}. 

1565 

1566 @raise IntersectionError: The B{C{distance}} exceeds the sum 

1567 of B{C{radius1}} and B{C{radius2}}. 

1568 

1569 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or 

1570 B{C{radius2}}. 

1571 

1572 @see: U{Circle-Circle Intersection 

1573 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}. 

1574 ''' 

1575 d = Distance_(distance, low=_0_0) 

1576 r1 = Radius_(radius1=radius1) 

1577 r2 = Radius_(radius2=radius2) 

1578 if d > (r1 + r2): 

1579 raise IntersectionError(distance=d, radius1=r1, radius2=r2, 

1580 txt=_too_(_distant_)) 

1581 return _radical2(d, r1, r2) if d > EPS0 else \ 

1582 Radical2Tuple(_0_5, _0_0) 

1583 

1584 

1585class Radical2Tuple(_NamedTuple): 

1586 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and 

1587 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0} 

1588 ''' 

1589 _Names_ = (_ratio_, _xline_) 

1590 _Units_ = ( Scalar, Scalar) 

1591 

1592 

1593def _radistance(inst): 

1594 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians} 

1595 and L{hausdorff._HausdorffMeterRedians} classes. 

1596 ''' 

1597 kwds_ = _xkwds(inst._kwds, wrap=False) 

1598 wrap_ = _xkwds_pop(kwds_, wrap=False) 

1599 func_ = inst._func_ 

1600 try: # calling lower-overhead C{func_} 

1601 func_(0, _0_25, _0_5, **kwds_) 

1602 wrap_ = _Wrap._philamop(wrap_) 

1603 except TypeError: 

1604 return inst.distance 

1605 

1606 def _philam(p): 

1607 try: 

1608 return p.phi, p.lam 

1609 except AttributeError: # no .phi or .lam 

1610 return radians(p.lat), radians(p.lon) 

1611 

1612 def _func_wrap(point1, point2): 

1613 phi1, lam1 = wrap_(*_philam(point1)) 

1614 phi2, lam2 = wrap_(*_philam(point2)) 

1615 return func_(phi2, phi1, lam2 - lam1, **kwds_) 

1616 

1617 inst._units = inst._units_ 

1618 return _func_wrap 

1619 

1620 

1621def _scale_deg(lat1, lat2): # degrees 

1622 # scale factor cos(mean of lats) for delta lon 

1623 m = fabs(lat1 + lat2) * _0_5 

1624 return cos(radians(m)) if m < 90 else _0_0 

1625 

1626 

1627def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights 

1628 # scale factor cos(mean of phis) for delta lam 

1629 m = fabs(phi1 + phi2) * _0_5 

1630 return cos(m) if m < PI_2 else _0_0 

1631 

1632 

1633def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw 

1634 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine. 

1635 ''' 

1636 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21) 

1637 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21 

1638 

1639 

1640def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

1641 '''Compute the distance between two (ellipsoidal) points using 

1642 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1643 formula. 

1644 

1645 @arg lat1: Start latitude (C{degrees}). 

1646 @arg lon1: Start longitude (C{degrees}). 

1647 @arg lat2: End latitude (C{degrees}). 

1648 @arg lon2: End longitude (C{degrees}). 

1649 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1650 L{Ellipsoid2} or L{a_f2Tuple}) to use. 

1651 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1652 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1653 

1654 @return: Distance (C{meter}, same units as the B{C{datum}}'s 

1655 ellipsoid axes). 

1656 

1657 @raise TypeError: Invalid B{C{datum}}. 

1658 

1659 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1660 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, 

1661 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}. 

1662 ''' 

1663 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2) 

1664 

1665 

1666def thomas_(phi2, phi1, lam21, datum=_WGS84): 

1667 '''Compute the I{angular} distance between two (ellipsoidal) points using 

1668 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1669 formula. 

1670 

1671 @arg phi2: End latitude (C{radians}). 

1672 @arg phi1: Start latitude (C{radians}). 

1673 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1674 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, 

1675 L{Ellipsoid2} or L{a_f2Tuple}). 

1676 

1677 @return: Angular distance (C{radians}). 

1678 

1679 @raise TypeError: Invalid B{C{datum}}. 

1680 

1681 @see: Functions L{thomas}, L{cosineAndoyerLambert_}, 

1682 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1683 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1684 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP 

1685 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/ 

1686 Distance/ThomasFormula.php>}. 

1687 ''' 

1688 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21) 

1689 if r and isnon0(c1) and isnon0(c2): 

1690 E = _ellipsoidal(datum, thomas_) 

1691 if E.f: 

1692 r1 = atan2(E.b_a * s1, c1) 

1693 r2 = atan2(E.b_a * s2, c2) 

1694 

1695 j = (r2 + r1) * _0_5 

1696 k = (r2 - r1) * _0_5 

1697 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5) 

1698 

1699 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2) 

1700 u = _1_0 - h 

1701 if isnon0(u) and isnon0(h): 

1702 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h) 

1703 sr, cr = sincos2(r) 

1704 if isnon0(sr): 

1705 u = 2 * (sj * ck)**2 / u 

1706 h = 2 * (sk * cj)**2 / h 

1707 x = u + h 

1708 y = u - h 

1709 

1710 s = r / sr 

1711 e = 4 * s**2 

1712 d = 2 * cr 

1713 a = e * d 

1714 b = 2 * r 

1715 c = s - (a - d) * _0_5 

1716 f = E.f * _0_25 

1717 

1718 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y) 

1719 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr 

1720 return r 

1721 

1722 

1723def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1724 '''Compute the distance between two (spherical) points using 

1725 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1726 spherical formula. 

1727 

1728 @arg lat1: Start latitude (C{degrees}). 

1729 @arg lon1: Start longitude (C{degrees}). 

1730 @arg lat2: End latitude (C{degrees}). 

1731 @arg lon2: End longitude (C{degrees}). 

1732 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1733 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1734 L{a_f2Tuple}) to use. 

1735 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1736 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1737 

1738 @return: Distance (C{meter}, same units as B{C{radius}}). 

1739 

1740 @raise UnitError: Invalid B{C{radius}}. 

1741 

1742 @see: Functions L{vincentys_}, L{cosineAndoyerLambert}, 

1743 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular}, 

1744 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1745 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2}, 

1746 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1747 

1748 @note: See note at function L{vincentys_}. 

1749 ''' 

1750 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2) 

1751 

1752 

1753def vincentys_(phi2, phi1, lam21): 

1754 '''Compute the I{angular} distance between two (spherical) points using 

1755 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1756 spherical formula. 

1757 

1758 @arg phi2: End latitude (C{radians}). 

1759 @arg phi1: Start latitude (C{radians}). 

1760 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1761 

1762 @return: Angular distance (C{radians}). 

1763 

1764 @see: Functions L{vincentys}, L{cosineAndoyerLambert_}, 

1765 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1766 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1767 L{flatPolar_}, L{haversine_} and L{thomas_}. 

1768 

1769 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_} 

1770 produce equivalent results, but L{vincentys_} is suitable 

1771 for antipodal points and slightly more expensive (M{3 cos, 

1772 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_} 

1773 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and 

1774 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}). 

1775 ''' 

1776 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21) 

1777 

1778 c = c2 * c21 

1779 x = s1 * s2 + c1 * c 

1780 y = c1 * s2 - s1 * c 

1781 return atan2(hypot(c2 * s21, y), x) 

1782 

1783# **) MIT License 

1784# 

1785# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved. 

1786# 

1787# Permission is hereby granted, free of charge, to any person obtaining a 

1788# copy of this software and associated documentation files (the "Software"), 

1789# to deal in the Software without restriction, including without limitation 

1790# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1791# and/or sell copies of the Software, and to permit persons to whom the 

1792# Software is furnished to do so, subject to the following conditions: 

1793# 

1794# The above copyright notice and this permission notice shall be included 

1795# in all copies or substantial portions of the Software. 

1796# 

1797# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1798# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1799# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1800# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1801# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1802# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1803# OTHER DEALINGS IN THE SOFTWARE.