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, _EWGS84 

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

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

18 _TypeError, _ValueError, _xattr, _xError, \ 

19 _xkwds, _xkwds_pop 

20from pygeodesy.fmath import euclid, hypot, 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__ = '23.08.20' 

40 

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

42_ratio_ = 'ratio' 

43_xline_ = 'xline' 

44 

45 

46def _anti2(a, b, n_2, n, n2): 

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

48 ''' 

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

50 if r == a: 

51 r = -r 

52 b += n 

53 if fabs(b) > n: 

54 b = remainder(b, n2) 

55 return float0_(r, b) 

56 

57 

58def antipode(lat, lon, name=NN): 

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

60 to a given point in C{degrees}. 

61 

62 @arg lat: Latitude (C{degrees}). 

63 @arg lon: Longitude (C{degrees}). 

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

65 

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

67 

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

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

70 ''' 

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

72 

73 

74def antipode_(phi, lam, name=NN): 

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

76 to a given point in C{radians}. 

77 

78 @arg phi: Latitude (C{radians}). 

79 @arg lam: Longitude (C{radians}). 

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

81 

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

83 

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

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

86 ''' 

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

88 

89 

90def bearing(lat1, lon1, lat2, lon2, **final_wrap): 

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

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

93 

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

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

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

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

98 @kwarg final_wrap: Optional keyword arguments for function 

99 L{pygeodesy.bearing_}. 

100 

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

102 zero if start and end point coincide. 

103 ''' 

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

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

106 return degrees(r) 

107 

108 

109def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False): 

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

111 between a (spherical) start and end point. 

112 

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

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

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

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

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

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

119 B{C{lam2}} (C{bool}). 

120 

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

122 and end point coincide. 

123 

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

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

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

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

128 ''' 

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

130 if final: # swap plus PI 

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

132 r = PI3 

133 else: 

134 r = PI2 

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

136 

137 x = ca1 * sa2 - sa1 * ca2 * cdb 

138 y = sdb * ca2 

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

140 

141 

142def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf 

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

144 ''' 

145 try: # for LatLon_ and ellipsoidal LatLon 

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

147 except AttributeError: 

148 pass 

149 # XXX spherical version, OK for ellipsoidal ispolar? 

150 a1, b1 = p1.philam 

151 a2, b2 = p2.philam 

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

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

154 name=_bearingTo2.__name__) 

155 

156 

157def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False): 

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

159 lat2 - lat1)} between two points. 

160 

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

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

163 

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

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

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

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

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

169 mean latitude (C{bool}). 

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

171 and B{C{lon2}} (C{bool}). 

172 

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

174 

175 @note: Courtesy of Martin Schultz. 

176 

177 @see: U{Local, flat earth approximation 

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

179 ''' 

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

181 if adjust: # scale delta lon 

182 d_lon *= _scale_deg(lat1, lat2) 

183 return atan2b(d_lon, lat2 - lat1) 

184 

185 

186def cosineAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

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

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

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

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

191 

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

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

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

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

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

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

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

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

200 

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

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

203 

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

205 

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

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

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

209 L{Ellipsoid.distance2}. 

210 ''' 

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

212 

213 

214def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): 

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

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

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

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

219 

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

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

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

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

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

225 

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

227 

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

229 

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

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

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

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

234 AndoyerLambert.php>}. 

235 ''' 

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

237 if isnon0(c1) and isnon0(c2): 

238 E = _ellipsoidal(datum, cosineAndoyerLambert_) 

239 if E.f: # ellipsoidal 

240 r2 = atan2(E.b_a * s2, c2) 

241 r1 = atan2(E.b_a * s1, c1) 

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

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

244 if r: 

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

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

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

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

249 r += (c - s) * E.f * _0_125 

250 return r 

251 

252 

253def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

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

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

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

257 formula. 

258 

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

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

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

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

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

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

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

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

267 

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

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

270 

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

272 

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

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

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

276 L{Ellipsoid.distance2}. 

277 ''' 

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

279 

280 

281def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): 

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

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

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

285 formula. 

286 

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

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

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

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

291 L{Ellipsoid2} or L{a_f2Tuple}). 

292 

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

294 

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

296 

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

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

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

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

301 Distance/ForsytheCorrection.php>}. 

302 ''' 

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

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

305 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_) 

306 if E.f: # ellipsoidal 

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

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

309 s = (s1 + s2)**2 / (1 + cr) 

310 t = (s1 - s2)**2 / (1 - cr) 

311 x = s + t 

312 y = s - t 

313 

314 s = 8 * r**2 / sr 

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

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

317 b = -2 * d 

318 e = 30 * s2r 

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

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

321 

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

323 return r 

324 

325 

326def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

329 formula. 

330 

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

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

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

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

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

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

337 L{a_f2Tuple}) to use. 

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

339 and B{C{lon2}} (C{bool}). 

340 

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

342 ellipsoid or datum axes). 

343 

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

345 

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

347 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean}, 

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

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

350 

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

352 ''' 

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

354 

355 

356def cosineLaw_(phi2, phi1, lam21): 

357 '''Compute the I{angular} distance between two points using the U{spherical 

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

359 formula. 

360 

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

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

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

364 

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

366 

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

368 L{cosineForsytheAndoyerLambert_}, L{equirectangular_}, 

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

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

371 

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

373 ''' 

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

375 

376 

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

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

379 ''' 

380 if wrap: 

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

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

383 else: # for backward compaibility 

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

385 

386 

387def _dE(func_, earth, *wrap_lls): 

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

389 ''' 

390 E = _ellipsoidal(earth, func_) 

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

392 return r * E.a 

393 

394 

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

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

397 ''' 

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

399 if radius is not R_M: 

400 _, lat1, _, lat2, _ = wrap_lls 

401 radius = _mean_radius(radius, lat1, lat2) 

402 return r * radius 

403 

404 

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

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

407 ''' 

408 r = excess_(*_d3(*wrap_lls)) 

409 if radius: 

410 _, lat1, _, lat2, _ = wrap_lls 

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

412 return r 

413 

414 

415def _ellipsoidal(earth, where): 

416 '''(INTERNAL) Helper for distances. 

417 ''' 

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

419 earth if isinstance(earth, Ellipsoid) else 

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

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

422 

423 

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

425 '''Compute the distance between two points using 

426 the U{Equirectangular Approximation / Projection 

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

428 

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

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

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

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

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

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

435 L{a_f2Tuple}). 

436 @kwarg adjust_limit_wrap: Optional keyword arguments for 

437 function L{equirectangular_}. 

438 

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

440 the ellipsoid or datum axes). 

441 

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

443 

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

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

446 approximate or accurate distance functions. 

447 ''' 

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

449 Lat(lat2=lat2), Lon(lon2=lon2), 

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

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

452 

453 

454def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap): 

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

456 and L{hausdorff._HausdorffMeterRedians} classes. 

457 ''' 

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

459 

460 

461def equirectangular_(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False): 

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

463 Approximation / Projection 

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

465 

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

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

468 

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

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

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

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

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

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

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

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

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

478 and B{C{lon2}} (C{bool}). 

479 

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

481 unroll_lon2)} in C{degrees squared}. 

482 

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

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

485 set to C{True}. 

486 

487 @see: U{Local, flat earth approximation 

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

489 L{equirectangular}, L{cosineAndoyerLambert}, 

490 L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean}, 

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

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

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

494 ''' 

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

496 d_lat = lat2 - lat1 

497 

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

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

500 if d > limit: 

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

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

503 limit=limit, wrap=wrap) 

504 raise LimitError(s, txt=t) 

505 

506 if adjust: # scale delta lon 

507 d_lon *= _scale_deg(lat1, lat2) 

508 

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

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

511 

512 

513def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False): 

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

515 

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

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

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

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

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

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

522 L{a_f2Tuple}) to use. 

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

524 the mean latitude (C{bool}). 

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

526 and B{C{lon2}} (C{bool}). 

527 

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

529 ellipsoid or datum axes). 

530 

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

532 

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

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

535 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

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

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

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

539 ''' 

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

541 

542 

543def euclidean_(phi2, phi1, lam21, adjust=True): 

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

545 (spherical) points. 

546 

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

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

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

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

551 of the mean latitude (C{bool}). 

552 

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

554 

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

556 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_}, 

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

558 and L{vincentys_}. 

559 ''' 

560 if adjust: 

561 lam21 *= _scale_rad(phi2, phi1) 

562 return euclid(phi2 - phi1, lam21) 

563 

564 

565def excessAbc_(A, b, c): 

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

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

568 

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

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

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

572 

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

574 

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

576 

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

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

579 ''' 

580 A = Radians_(A=A) 

581 b = Radians_(b=b) * _0_5 

582 c = Radians_(c=c) * _0_5 

583 

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

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

586 

587 

588def excessCagnoli_(a, b, c): 

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

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

591 

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

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

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

595 

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

597 

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

599 

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

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

602 ''' 

603 a = Radians_(a=a) 

604 b = Radians_(b=b) 

605 c = Radians_(c=c) 

606 

607 s = fsumf_(a, b, c) * _0_5 

608 r = sin(s) * sin(s - a) * sin(s - b) * sin(s - c) 

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

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

611 return Radians(Cagnoli=r * _2_0) 

612 

613 

614def excessGirard_(A, B, C): 

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

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

617 formula. 

618 

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

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

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

622 

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

624 

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

626 

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

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

629 ''' 

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

631 Radians_(B=B), 

632 Radians_(C=C), -PI)) 

633 

634 

635def excessLHuilier_(a, b, c): 

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

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

638 Theorem. 

639 

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

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

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

643 

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

645 

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

647 

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

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

650 ''' 

651 a = Radians_(a=a) 

652 b = Radians_(b=b) 

653 c = Radians_(c=c) 

654 

655 s = fsumf_(a, b, c) * _0_5 

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

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

658 return Radians(LHuilier=r * _4_0) 

659 

660 

661def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

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

665 method. 

666 

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

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

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

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

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

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

673 L{a_f2Tuple}) or C{None}. 

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

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

676 

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

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

679 if C{B{radius}=0} or C{None}. 

680 

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

682 

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

684 

685 @raise ValueError: Semi-circular longitudinal delta. 

686 

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

688 ''' 

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

690 

691 

692def excessKarney_(phi2, phi1, lam21): 

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

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

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

696 method. 

697 

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

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

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

701 

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

703 

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

705 

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

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

708 ''' 

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

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

711 # 

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

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

714 # 1 + tan(φ1 / 2) · tan(φ2 / 2) 

715 # 

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

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

718 t2 = tan_2(phi2) 

719 t1 = tan_2(phi1) 

720 t = tan_2(lam21, lam21=None) 

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

722 _1_0 + (t1 * t2)) * _2_0) 

723 

724 

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

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

727# 

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

729# 

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

731# 

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

733# 

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

735# 

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

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

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

739# 

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

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

742# 

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

744# 

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

746# inverse Gudermannian) function 

747# 

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

749# 

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

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

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

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

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

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

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

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

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

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

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

761# - Charles Karney 

762 

763 

764def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

767 

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

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

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

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

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

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

774 L{a_f2Tuple}) or C{None}. 

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

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

777 

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

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

780 if C{B{radius}=0} or C{None}. 

781 

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

783 

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

785 

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

787 ''' 

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

789 

790 

791def excessQuad_(phi2, phi1, lam21): 

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

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

794 

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

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

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

798 

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

800 

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

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

803 ''' 

804 s = sin((phi2 + phi1) * _0_5) 

805 c = cos((phi2 - phi1) * _0_5) 

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

807 

808 

809def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False): 

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

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

812 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

814 

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

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

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

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

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

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

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

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

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

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

825 

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

827 ellipsoid axes). 

828 

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

830 

831 @note: The meridional and prime_vertical radii of curvature 

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

833 

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

835 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

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

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

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

839 ''' 

840 E = _ellipsoidal(datum, flatLocal) 

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

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

843 

844hubeny = flatLocal # PYCHOK for Karl Hubeny 

845 

846 

847def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True): 

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

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

850 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

852 

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

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

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

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

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

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

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

860 

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

862 

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

864 

865 @note: The meridional and prime_vertical radii of curvature 

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

867 

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

869 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_}, 

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

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

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

873 ''' 

874 E = _ellipsoidal(datum, flatLocal_) 

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

876 

877hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny 

878 

879 

880def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

883 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

884 formula. 

885 

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

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

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

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

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

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

892 L{a_f2Tuple}) to use. 

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

894 and B{C{lon2}} (C{bool}). 

895 

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

897 ellipsoid or datum axes). 

898 

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

900 

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

902 L{cosineForsytheAndoyerLambert},L{cosineLaw}, 

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

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

905 L{vincentys}. 

906 ''' 

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

908 

909 

910def flatPolar_(phi2, phi1, lam21): 

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

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

913 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

914 formula. 

915 

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

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

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

919 

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

921 

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

923 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

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

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

926 ''' 

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

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

929 if a < b: 

930 a, b = b, a 

931 if a < EPS0: 

932 a = _0_0 

933 elif b > 0: 

934 b = b / a # /= chokes PyChecker 

935 c = b * cos(lam21) * _2_0 

936 c = fsumf_(_1_0, b**2, -fabs(c)) 

937 a *= sqrt0(c) 

938 return a 

939 

940 

941def hartzell(pov, los=None, earth=_WGS84, name=NN, **LatLon_and_kwds): 

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

943 from a Point-Of-View in space. 

944 

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

946 or L{Vector3d}). 

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

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

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

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

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

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

953 point plus C{LatLon} keyword arguments, include 

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

955 

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

957 B{C{pov}} or B{C{LatLon}}). 

958 

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

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

961 the earth or points in an opposite direction. 

962 

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

964 

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

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

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

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

969 ''' 

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

971 _spherical_datum(earth, name=hartzell.__name__) 

972 try: 

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

974 except Exception as x: 

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

976 

977# else: 

978# E = D.ellipsoid 

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

980# 

981# def _Error(txt): 

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

983# 

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

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

986# q2 = E.b2_a2 # == c2 / a2 

987# bc = E.a * E.b # == b * c 

988# 

989# V3 = _MODS.vector3d._otherV3d 

990# p3 = V3(pov=pov) 

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

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

993# 

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

995# ux, vy, wz = u3.times_(p3).xyz 

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

997# 

998# t = c2, c2, b2 

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

1000# if m < EPS0: # zero or near-null LOS vector 

1001# raise _Error(_near_(_null_)) 

1002# 

1003# # a2 and b2 factored out, b2 == a2 and b2 / a2 == 1 

1004# r = fsumf_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2, 

1005# c2 * u2, -u2 * z2, -w2 * x2, ux * wz * 2, 

1006# -w2 * y2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2) 

1007# if r > 0: 

1008# r = sqrt(r) * bc # == a * a * b * c / a2 

1009# elif r < 0: # LOS pointing away from or missing the earth 

1010# raise _Error(_opposite_ if max(ux, vy, wz) > 0 else _outside_) 

1011# 

1012# d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out 

1013# if d > 0: # POV inside or LOS missing, outside the earth 

1014# s = fsumf_(_1_0, x2 / a2, y2 / b2, z2 / c2, _N_2_0) # like _sideOf 

1015# raise _Error(_outside_ if s > 0 else _inside_) 

1016# elif fsumf_(x2, y2, z2) < d**2: # d past earth center 

1017# raise _Error(_too_(_distant_)) 

1018# 

1019# r = p3.minus(u3.times(d)) 

1020# # h = p3.minus(r).length # distance to ellipsoid 

1021 

1022 r = _xnamed(r, name or hartzell.__name__) 

1023 if LatLon_and_kwds: 

1024 c = _MODS.cartesianBase.CartesianBase(r, datum=D, name=r.name) 

1025 r = c.toLatLon(**LatLon_and_kwds) 

1026 return r 

1027 

1028 

1029def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1030 '''Compute the distance between two (spherical) points using the 

1031 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} 

1032 formula. 

1033 

1034 @arg lat1: Start latitude (C{degrees}). 

1035 @arg lon1: Start longitude (C{degrees}). 

1036 @arg lat2: End latitude (C{degrees}). 

1037 @arg lon2: End longitude (C{degrees}). 

1038 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1039 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1040 L{a_f2Tuple}) to use. 

1041 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1042 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1043 

1044 @return: Distance (C{meter}, same units as B{C{radius}}). 

1045 

1046 @raise TypeError: Invalid B{C{radius}}. 

1047 

1048 @see: U{Distance between two (spherical) points 

1049 <https://www.EdWilliams.org/avform.htm#Dist>}, functions 

1050 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1051 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1052 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, 

1053 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1054 

1055 @note: See note at function L{vincentys_}. 

1056 ''' 

1057 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2) 

1058 

1059 

1060def haversine_(phi2, phi1, lam21): 

1061 '''Compute the I{angular} distance between two (spherical) points 

1062 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} 

1063 formula. 

1064 

1065 @arg phi2: End latitude (C{radians}). 

1066 @arg phi1: Start latitude (C{radians}). 

1067 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1068 

1069 @return: Angular distance (C{radians}). 

1070 

1071 @see: Functions L{haversine}, L{cosineAndoyerLambert_}, 

1072 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1073 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1074 L{flatPolar_}, L{thomas_} and L{vincentys_}. 

1075 

1076 @note: See note at function L{vincentys_}. 

1077 ''' 

1078 def _hsin(rad): 

1079 return sin(rad * _0_5)**2 

1080 

1081 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine 

1082 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2 

1083 

1084 

1085def heightOf(angle, distance, radius=R_M): 

1086 '''Determine the height above the (spherical) earth' surface after 

1087 traveling along a straight line at a given tilt. 

1088 

1089 @arg angle: Tilt angle above horizontal (C{degrees}). 

1090 @arg distance: Distance along the line (C{meter} or same units as 

1091 B{C{radius}}). 

1092 @kwarg radius: Optional mean earth radius (C{meter}). 

1093 

1094 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}). 

1095 

1096 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}. 

1097 

1098 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>} 

1099 (U{Shapiro et al. 2009, JTECH 

1100 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>} 

1101 and U{Potvin et al. 2012, JTECH 

1102 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}). 

1103 ''' 

1104 r = h = Radius(radius) 

1105 d = fabs(Distance(distance)) 

1106 if d > h: 

1107 d, h = h, d 

1108 

1109 if d > EPS0: # and h > EPS0 

1110 d = d / h # /= h chokes PyChecker 

1111 s = sin(Phi_(angle=angle, clip=_180_0)) 

1112 s = fsumf_(_1_0, _2_0 * s * d, d**2) 

1113 if s > 0: 

1114 return h * sqrt(s) - r 

1115 

1116 raise _ValueError(angle=angle, distance=distance, radius=radius) 

1117 

1118 

1119def heightOrthometric(h_ll, N): 

1120 '''Get the I{orthometric} height B{H}, the height above the geoid, earth surface. 

1121 

1122 @arg h_ll: The height above the ellipsoid (C{meter}) or an I{ellipsoidal} 

1123 location (C{LatLon} with a C{height} or C{h} attribute). 

1124 @arg N: The I{geoid} height (C{meter}), the height of the geoid above the 

1125 ellipsoid at the same B{C{h_ll}} location. 

1126 

1127 @return: I{Orthometric} height C{B{H} = B{h} - B{N}} (C{meter}, same units 

1128 as B{C{h}} and B{C{N}}). 

1129 

1130 @see: U{Ellipsoid, Geoid, and Othometric Heights<https://www.NGS.NOAA.gov/ 

1131 GEOID/PRESENTATIONS/2007_02_24_CCPS/Roman_A_PLSC2007notes.pdf>}, page 

1132 6 and module L{pygeodesy.geoids}. 

1133 ''' 

1134 h = h_ll if isscalar(h_ll) else _xattr(h_ll, height=_xattr(h_ll, h=0)) 

1135 return Height(H=Height(h=h) - Height(N=N)) 

1136 

1137 

1138def horizon(height, radius=R_M, refraction=False): 

1139 '''Determine the distance to the horizon from a given altitude 

1140 above the (spherical) earth. 

1141 

1142 @arg height: Altitude (C{meter} or same units as B{C{radius}}). 

1143 @kwarg radius: Optional mean earth radius (C{meter}). 

1144 @kwarg refraction: Consider atmospheric refraction (C{bool}). 

1145 

1146 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}). 

1147 

1148 @raise ValueError: Invalid B{C{height}} or B{C{radius}}. 

1149 

1150 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}. 

1151 ''' 

1152 h, r = Height(height), Radius(radius) 

1153 if min(h, r) < 0: 

1154 raise _ValueError(height=height, radius=radius) 

1155 

1156 if refraction: 

1157 d2 = 2.415750694528 * h * r # 2.0 / 0.8279 

1158 else: 

1159 d2 = h * fsumf_(r, r, h) 

1160 return sqrt0(d2) 

1161 

1162 

1163class _idllmn6(object): # see also .geodesicw._wargs, .vector2d._numpy 

1164 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}. 

1165 ''' 

1166 @contextmanager # <https://www.python.org/dev/peps/pep-0343/> Examples 

1167 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds): 

1168 try: 

1169 if wrap: 

1170 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap) 

1171 kwds = _xkwds(kwds, wrap=wrap) # for _xError 

1172 m = small if small is _100km else Meter_(small=small) 

1173 n = (intersections2 if s else intersection2).__name__ 

1174 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m: 

1175 d, m = None, _MODS.vector3d 

1176 _i = m._intersects2 if s else m._intersect3d3 

1177 elif isscalar(datum) and datum < 0 and not s: 

1178 d = _spherical_datum(-datum, name=n) 

1179 m = _MODS.sphericalNvector 

1180 _i = m.intersection 

1181 else: 

1182 d = _spherical_datum(datum, name=n) 

1183 if d.isSpherical: 

1184 m = _MODS.sphericalTrigonometry 

1185 _i = m._intersects2 if s else m._intersect 

1186 elif d.isEllipsoidal: 

1187 try: 

1188 if d.ellipsoid.geodesic: 

1189 pass 

1190 m = _MODS.ellipsoidalKarney 

1191 except ImportError: 

1192 m = _MODS.ellipsoidalExact 

1193 _i = m._intersections2 if s else m._intersection3 # ellispoidalBaseDI 

1194 else: 

1195 raise _TypeError(datum=datum) 

1196 yield _i, d, lat2, lon2, m, n 

1197 

1198 except (TypeError, ValueError) as x: 

1199 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum, 

1200 lat2=lat2, lon2=lon2, small=small, **kwds) 

1201 

1202_idllmn6 = _idllmn6() # PYCHOK singleton 

1203 

1204 

1205def intersection2(lat1, lon1, bearing1, 

1206 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True 

1207 '''I{Conveniently} compute the intersection of two lines each defined 

1208 by a (geodetic) point and a bearing from North, using either ... 

1209 

1210 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km 

1211 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1212 

1213 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}} 

1214 or a C{scalar B{datum}} representing the earth radius, conventionally 

1215 in C{meter} or ... 

1216 

1217 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative} 

1218 C{scalar}, (negative) earth radius, conventionally in C{meter} or ... 

1219 

1220 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}} 

1221 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1222 is installed, otherwise ... 

1223 

1224 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal. 

1225 

1226 @arg lat1: Latitude of the first point (C{degrees}). 

1227 @arg lon1: Longitude of the first point (C{degrees}). 

1228 @arg bearing1: Bearing at the first point (compass C{degrees}). 

1229 @arg lat2: Latitude of the second point (C{degrees}). 

1230 @arg lon2: Longitude of the second point (C{degrees}). 

1231 @arg bearing2: Bearing at the second point (compass C{degrees}). 

1232 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1233 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1234 radius (C{meter}, same units as B{C{radius1}} and 

1235 B{C{radius2}}) or C{None}. 

1236 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1237 and B{C{lon2}} (C{bool}). 

1238 @kwarg small: Upper limit for small distances (C{meter}). 

1239 

1240 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and 

1241 longitude of the intersection point. 

1242 

1243 @raise IntersectionError: Ambiguous or infinite intersection 

1244 or colinear, parallel or otherwise 

1245 non-intersecting lines. 

1246 

1247 @raise TypeError: Invalid B{C{datum}}. 

1248 

1249 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}}, 

1250 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}. 

1251 

1252 @see: Method L{RhumbLine.intersection2}. 

1253 

1254 @note: The returned intersections may be near-antipodal. 

1255 ''' 

1256 b1 = Bearing(bearing1=bearing1) 

1257 b2 = Bearing(bearing2=bearing2) 

1258 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1259 small, wrap, False, bearing1=b1, bearing2=b2) as t: 

1260 _i, d, lat2, lon2, m, n = t 

1261 if d is None: 

1262 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1, 

1263 m.Vector3d(lon2, lat2, 0), b2, useZ=False) 

1264 t = LatLon2Tuple(t.y, t.x, name=n) 

1265 

1266 else: 

1267 t = _i(m.LatLon(lat1, lon1, datum=d), b1, 

1268 m.LatLon(lat2, lon2, datum=d), b2, height=0, wrap=False) 

1269 if isinstance(t, Intersection3Tuple): # ellipsoidal 

1270 t, _, _ = t 

1271 t = LatLon2Tuple(t.lat, t.lon, name=n) 

1272 return t 

1273 

1274 

1275def intersections2(lat1, lon1, radius1, 

1276 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True 

1277 '''I{Conveniently} compute the intersections of two circles each defined 

1278 by a (geodetic) center point and a radius, using either ... 

1279 

1280 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km 

1281 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1282 

1283 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}} 

1284 or a C{scalar B{datum}} representing the earth radius, conventionally 

1285 in C{meter} or ... 

1286 

1287 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}} 

1288 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1289 is installed, otherwise ... 

1290 

1291 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal. 

1292 

1293 @arg lat1: Latitude of the first circle center (C{degrees}). 

1294 @arg lon1: Longitude of the first circle center (C{degrees}). 

1295 @arg radius1: Radius of the first circle (C{meter}, conventionally). 

1296 @arg lat2: Latitude of the second circle center (C{degrees}). 

1297 @arg lon2: Longitude of the second circle center (C{degrees}). 

1298 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}). 

1299 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1300 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1301 radius (C{meter}, same units as B{C{radius1}} and 

1302 B{C{radius2}}) or C{None}. 

1303 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1304 and B{C{lon2}} (C{bool}). 

1305 @kwarg small: Upper limit for small distances (C{meter}). 

1306 

1307 @return: 2-Tuple of the intersection points, each a 

1308 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the 

1309 points are the same instance, aka the I{radical center}. 

1310 

1311 @raise IntersectionError: Concentric, antipodal, invalid or 

1312 non-intersecting circles or no 

1313 convergence. 

1314 

1315 @raise TypeError: Invalid B{C{datum}}. 

1316 

1317 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}}, 

1318 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}. 

1319 ''' 

1320 r1 = Radius_(radius1=radius1) 

1321 r2 = Radius_(radius2=radius2) 

1322 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1323 small, wrap, True, radius1=r1, radius2=r2) as t: 

1324 _i, d, lat2, lon2, m, n = t 

1325 if d is None: 

1326 r1 = m2degrees(r1, radius=R_M, lat=lat1) 

1327 r2 = m2degrees(r2, radius=R_M, lat=lat2) 

1328 

1329 def _V2T(x, y, _, **unused): # _ == z unused 

1330 return LatLon2Tuple(y, x, name=n) 

1331 

1332 t = _i(m.Vector3d(lon1, lat1, 0), r1, 

1333 m.Vector3d(lon2, lat2, 0), r2, sphere=False, 

1334 Vector=_V2T) 

1335 else: 

1336 def _LL2T(lat, lon, **unused): 

1337 return LatLon2Tuple(lat, lon, name=n) 

1338 

1339 t = _i(m.LatLon(lat1, lon1, datum=d), r1, 

1340 m.LatLon(lat2, lon2, datum=d), r2, 

1341 LatLon=_LL2T, height=0, wrap=False) 

1342 return t 

1343 

1344 

1345def isantipode(lat1, lon1, lat2, lon2, eps=EPS): 

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

1347 opposite sides of the earth. 

1348 

1349 @arg lat1: Latitude of one point (C{degrees}). 

1350 @arg lon1: Longitude of one point (C{degrees}). 

1351 @arg lat2: Latitude of the other point (C{degrees}). 

1352 @arg lon2: Longitude of the other point (C{degrees}). 

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

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(lat1 + lat2) <= eps and 

1361 fabs(lon1 + lon2) <= eps) or _isequalTo( 

1362 normal(lat1, lon1), antipode(lat2, lon2), eps) 

1363 

1364 

1365def isantipode_(phi1, lam1, phi2, lam2, eps=EPS): 

1366 '''Check whether two points are I{antipodal}, on diametrically 

1367 opposite sides of the earth. 

1368 

1369 @arg phi1: Latitude of one point (C{radians}). 

1370 @arg lam1: Longitude of one point (C{radians}). 

1371 @arg phi2: Latitude of the other point (C{radians}). 

1372 @arg lam2: Longitude of the other point (C{radians}). 

1373 @kwarg eps: Tolerance for near-equality (C{radians}). 

1374 

1375 @return: C{True} if points are antipodal within the 

1376 B{C{eps}} tolerance, C{False} otherwise. 

1377 

1378 @see: Functions L{isantipode} and L{antipode_}. 

1379 ''' 

1380 return (fabs(phi1 + phi2) <= eps and 

1381 fabs(lam1 + lam2) <= eps) or _isequalTo_( 

1382 normal_(phi1, lam1), antipode_(phi2, lam2), eps) 

1383 

1384 

1385def _isequalTo(p1, p2, eps=EPS): 

1386 '''Compare 2 point lat-/lons ignoring C{class}. 

1387 ''' 

1388 return (fabs(p1.lat - p2.lat) <= eps and 

1389 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon) 

1390 

1391 

1392def _isequalTo_(p1, p2, eps=EPS): 

1393 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}. 

1394 ''' 

1395 return (fabs(p1.phi - p2.phi) <= eps and 

1396 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam) 

1397 

1398 

1399def isnormal(lat, lon, eps=0): 

1400 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their 

1401 respective I{normal} range in C{degrees}. 

1402 

1403 @arg lat: Latitude (C{degrees}). 

1404 @arg lon: Longitude (C{degrees}). 

1405 @kwarg eps: Optional tolerance C{degrees}). 

1406 

1407 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and 

1408 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise. 

1409 

1410 @see: Functions L{isnormal_} and L{normal}. 

1411 ''' 

1412 return (_90_0 - fabs(lat)) >= eps and (_180_0 - fabs(lon)) >= eps 

1413 

1414 

1415def isnormal_(phi, lam, eps=0): 

1416 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their 

1417 respective I{normal} range in C{radians}. 

1418 

1419 @arg phi: Latitude (C{radians}). 

1420 @arg lam: Longitude (C{radians}). 

1421 @kwarg eps: Optional tolerance C{radians}). 

1422 

1423 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and 

1424 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise. 

1425 

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

1427 ''' 

1428 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps 

1429 

1430 

1431def latlon2n_xyz(lat, lon, name=NN): 

1432 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1433 earth's surface) X, Y and Z components. 

1434 

1435 @arg lat: Latitude (C{degrees}). 

1436 @arg lon: Longitude (C{degrees}). 

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

1438 

1439 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1440 

1441 @see: Function L{philam2n_xyz}. 

1442 

1443 @note: These are C{n-vector} x, y and z components, 

1444 I{NOT} geocentric ECEF x, y and z coordinates! 

1445 ''' 

1446 return _2n_xyz(name, *sincos2d_(lat, lon)) 

1447 

1448 

1449def _normal2(a, b, n_2, n, n2): 

1450 '''(INTERNAL) Helper for C{normal} and C{normal_}. 

1451 ''' 

1452 if fabs(b) > n: 

1453 b = remainder(b, n2) 

1454 if fabs(a) > n_2: 

1455 r = remainder(a, n) 

1456 if r != a: 

1457 a = -r 

1458 b -= n if b > 0 else -n 

1459 return float0_(a, b) 

1460 

1461 

1462def normal(lat, lon, name=NN): 

1463 '''Normalize a lat- I{and} longitude pair in C{degrees}. 

1464 

1465 @arg lat: Latitude (C{degrees}). 

1466 @arg lon: Longitude (C{degrees}). 

1467 @kwarg name: Optional name (C{str}). 

1468 

1469 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90} 

1470 and C{abs(lon) <= 180}. 

1471 

1472 @see: Functions L{normal_} and L{isnormal}. 

1473 ''' 

1474 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0), 

1475 name=name or normal.__name__) 

1476 

1477 

1478def normal_(phi, lam, name=NN): 

1479 '''Normalize a lat- I{and} longitude pair in C{radians}. 

1480 

1481 @arg phi: Latitude (C{radians}). 

1482 @arg lam: Longitude (C{radians}). 

1483 @kwarg name: Optional name (C{str}). 

1484 

1485 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2} 

1486 and C{abs(lam) <= PI}. 

1487 

1488 @see: Functions L{normal} and L{isnormal_}. 

1489 ''' 

1490 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2), 

1491 name=name or normal_.__name__) 

1492 

1493 

1494def _2n_xyz(name, sa, ca, sb, cb): 

1495 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}. 

1496 ''' 

1497 # Kenneth Gade eqn 3, but using right-handed 

1498 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N 

1499 return Vector3Tuple(ca * cb, ca * sb, sa, name=name) 

1500 

1501 

1502def n_xyz2latlon(x, y, z, name=NN): 

1503 '''Convert C{n-vector} components to lat- and longitude in C{degrees}. 

1504 

1505 @arg x: X component (C{scalar}). 

1506 @arg y: Y component (C{scalar}). 

1507 @arg z: Z component (C{scalar}). 

1508 @kwarg name: Optional name (C{str}). 

1509 

1510 @return: A L{LatLon2Tuple}C{(lat, lon)}. 

1511 

1512 @see: Function L{n_xyz2philam}. 

1513 ''' 

1514 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name) 

1515 

1516 

1517def n_xyz2philam(x, y, z, name=NN): 

1518 '''Convert C{n-vector} components to lat- and longitude in C{radians}. 

1519 

1520 @arg x: X component (C{scalar}). 

1521 @arg y: Y component (C{scalar}). 

1522 @arg z: Z component (C{scalar}). 

1523 @kwarg name: Optional name (C{str}). 

1524 

1525 @return: A L{PhiLam2Tuple}C{(phi, lam)}. 

1526 

1527 @see: Function L{n_xyz2latlon}. 

1528 ''' 

1529 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name) 

1530 

1531 

1532def _opposes(d, m, n, n2): 

1533 '''(INTERNAL) Helper for C{opposing} and C{opposing_}. 

1534 ''' 

1535 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1 

1536 return False if d < m or d > (n2 - m) else ( 

1537 True if (n - m) < d < (n + m) else None) 

1538 

1539 

1540def opposing(bearing1, bearing2, margin=_90_0): 

1541 '''Compare the direction of two bearings given in C{degrees}. 

1542 

1543 @arg bearing1: First bearing (compass C{degrees}). 

1544 @arg bearing2: Second bearing (compass C{degrees}). 

1545 @kwarg margin: Optional, interior angle bracket (C{degrees}). 

1546 

1547 @return: C{True} if both bearings point in opposite, C{False} if 

1548 in similar or C{None} if in perpendicular directions. 

1549 

1550 @see: Function L{opposing_}. 

1551 ''' 

1552 m = Degrees_(margin=margin, low=EPS0, high=_90_0) 

1553 return _opposes(bearing2 - bearing1, m, _180_0, _360_0) 

1554 

1555 

1556def opposing_(radians1, radians2, margin=PI_2): 

1557 '''Compare the direction of two bearings given in C{radians}. 

1558 

1559 @arg radians1: First bearing (C{radians}). 

1560 @arg radians2: Second bearing (C{radians}). 

1561 @kwarg margin: Optional, interior angle bracket (C{radians}). 

1562 

1563 @return: C{True} if both bearings point in opposite, C{False} if 

1564 in similar or C{None} if in perpendicular directions. 

1565 

1566 @see: Function L{opposing}. 

1567 ''' 

1568 m = Radians_(margin=margin, low=EPS0, high=PI_2) 

1569 return _opposes(radians2 - radians1, m, PI, PI2) 

1570 

1571 

1572def philam2n_xyz(phi, lam, name=NN): 

1573 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1574 earth's surface) X, Y and Z components. 

1575 

1576 @arg phi: Latitude (C{radians}). 

1577 @arg lam: Longitude (C{radians}). 

1578 @kwarg name: Optional name (C{str}). 

1579 

1580 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1581 

1582 @see: Function L{latlon2n_xyz}. 

1583 

1584 @note: These are C{n-vector} x, y and z components, 

1585 I{NOT} geocentric ECEF x, y and z coordinates! 

1586 ''' 

1587 return _2n_xyz(name, *sincos2_(phi, lam)) 

1588 

1589 

1590def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d 

1591 # (INTERNAL) See C{radical2} below 

1592 # assert d > EPS0 

1593 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5 

1594 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d) 

1595 

1596 

1597def radical2(distance, radius1, radius2): 

1598 '''Compute the I{radical ratio} and I{radical line} of two 

1599 U{intersecting circles<https://MathWorld.Wolfram.com/ 

1600 Circle-CircleIntersection.html>}. 

1601 

1602 The I{radical line} is perpendicular to the axis thru the 

1603 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}. 

1604 

1605 @arg distance: Distance between the circle centers (C{scalar}). 

1606 @arg radius1: Radius of the first circle (C{scalar}). 

1607 @arg radius2: Radius of the second circle (C{scalar}). 

1608 

1609 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <= 

1610 ratio <= 1.0} and C{xline} is along the B{C{distance}}. 

1611 

1612 @raise IntersectionError: The B{C{distance}} exceeds the sum 

1613 of B{C{radius1}} and B{C{radius2}}. 

1614 

1615 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or 

1616 B{C{radius2}}. 

1617 

1618 @see: U{Circle-Circle Intersection 

1619 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}. 

1620 ''' 

1621 d = Distance_(distance, low=_0_0) 

1622 r1 = Radius_(radius1=radius1) 

1623 r2 = Radius_(radius2=radius2) 

1624 if d > (r1 + r2): 

1625 raise IntersectionError(distance=d, radius1=r1, radius2=r2, 

1626 txt=_too_(_distant_)) 

1627 return _radical2(d, r1, r2) if d > EPS0 else \ 

1628 Radical2Tuple(_0_5, _0_0) 

1629 

1630 

1631class Radical2Tuple(_NamedTuple): 

1632 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and 

1633 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0} 

1634 ''' 

1635 _Names_ = (_ratio_, _xline_) 

1636 _Units_ = ( Scalar, Scalar) 

1637 

1638 

1639def _radistance(inst): 

1640 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians} 

1641 and L{hausdorff._HausdorffMeterRedians} classes. 

1642 ''' 

1643 kwds_ = _xkwds(inst._kwds, wrap=False) 

1644 wrap_ = _xkwds_pop(kwds_, wrap=False) 

1645 func_ = inst._func_ 

1646 try: # calling lower-overhead C{func_} 

1647 func_(0, _0_25, _0_5, **kwds_) 

1648 wrap_ = _Wrap._philamop(wrap_) 

1649 except TypeError: 

1650 return inst.distance 

1651 

1652 def _philam(p): 

1653 try: 

1654 return p.phi, p.lam 

1655 except AttributeError: # no .phi or .lam 

1656 return radians(p.lat), radians(p.lon) 

1657 

1658 def _func_wrap(point1, point2): 

1659 phi1, lam1 = wrap_(*_philam(point1)) 

1660 phi2, lam2 = wrap_(*_philam(point2)) 

1661 return func_(phi2, phi1, lam2 - lam1, **kwds_) 

1662 

1663 inst._units = inst._units_ 

1664 return _func_wrap 

1665 

1666 

1667def _scale_deg(lat1, lat2): # degrees 

1668 # scale factor cos(mean of lats) for delta lon 

1669 m = fabs(lat1 + lat2) * _0_5 

1670 return cos(radians(m)) if m < 90 else _0_0 

1671 

1672 

1673def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights 

1674 # scale factor cos(mean of phis) for delta lam 

1675 m = fabs(phi1 + phi2) * _0_5 

1676 return cos(m) if m < PI_2 else _0_0 

1677 

1678 

1679def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw 

1680 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine. 

1681 ''' 

1682 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21) 

1683 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21 

1684 

1685 

1686def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

1687 '''Compute the distance between two (ellipsoidal) points using 

1688 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1689 formula. 

1690 

1691 @arg lat1: Start latitude (C{degrees}). 

1692 @arg lon1: Start longitude (C{degrees}). 

1693 @arg lat2: End latitude (C{degrees}). 

1694 @arg lon2: End longitude (C{degrees}). 

1695 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1696 L{Ellipsoid2} or L{a_f2Tuple}) to use. 

1697 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1698 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1699 

1700 @return: Distance (C{meter}, same units as the B{C{datum}}'s 

1701 ellipsoid axes). 

1702 

1703 @raise TypeError: Invalid B{C{datum}}. 

1704 

1705 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1706 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, 

1707 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}. 

1708 ''' 

1709 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2) 

1710 

1711 

1712def thomas_(phi2, phi1, lam21, datum=_WGS84): 

1713 '''Compute the I{angular} distance between two (ellipsoidal) points using 

1714 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1715 formula. 

1716 

1717 @arg phi2: End latitude (C{radians}). 

1718 @arg phi1: Start latitude (C{radians}). 

1719 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1720 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, 

1721 L{Ellipsoid2} or L{a_f2Tuple}). 

1722 

1723 @return: Angular distance (C{radians}). 

1724 

1725 @raise TypeError: Invalid B{C{datum}}. 

1726 

1727 @see: Functions L{thomas}, L{cosineAndoyerLambert_}, 

1728 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1729 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1730 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP 

1731 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/ 

1732 Distance/ThomasFormula.php>}. 

1733 ''' 

1734 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21) 

1735 if r and isnon0(c1) and isnon0(c2): 

1736 E = _ellipsoidal(datum, thomas_) 

1737 if E.f: 

1738 r1 = atan2(E.b_a * s1, c1) 

1739 r2 = atan2(E.b_a * s2, c2) 

1740 

1741 j = (r2 + r1) * _0_5 

1742 k = (r2 - r1) * _0_5 

1743 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5) 

1744 

1745 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2) 

1746 u = _1_0 - h 

1747 if isnon0(u) and isnon0(h): 

1748 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h) 

1749 sr, cr = sincos2(r) 

1750 if isnon0(sr): 

1751 u = 2 * (sj * ck)**2 / u 

1752 h = 2 * (sk * cj)**2 / h 

1753 x = u + h 

1754 y = u - h 

1755 

1756 s = r / sr 

1757 e = 4 * s**2 

1758 d = 2 * cr 

1759 a = e * d 

1760 b = 2 * r 

1761 c = s - (a - d) * _0_5 

1762 f = E.f * _0_25 

1763 

1764 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y) 

1765 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr 

1766 return r 

1767 

1768 

1769def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1770 '''Compute the distance between two (spherical) points using 

1771 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1772 spherical formula. 

1773 

1774 @arg lat1: Start latitude (C{degrees}). 

1775 @arg lon1: Start longitude (C{degrees}). 

1776 @arg lat2: End latitude (C{degrees}). 

1777 @arg lon2: End longitude (C{degrees}). 

1778 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1779 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1780 L{a_f2Tuple}) to use. 

1781 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1782 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1783 

1784 @return: Distance (C{meter}, same units as B{C{radius}}). 

1785 

1786 @raise UnitError: Invalid B{C{radius}}. 

1787 

1788 @see: Functions L{vincentys_}, L{cosineAndoyerLambert}, 

1789 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular}, 

1790 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1791 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2}, 

1792 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1793 

1794 @note: See note at function L{vincentys_}. 

1795 ''' 

1796 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2) 

1797 

1798 

1799def vincentys_(phi2, phi1, lam21): 

1800 '''Compute the I{angular} distance between two (spherical) points using 

1801 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1802 spherical formula. 

1803 

1804 @arg phi2: End latitude (C{radians}). 

1805 @arg phi1: Start latitude (C{radians}). 

1806 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1807 

1808 @return: Angular distance (C{radians}). 

1809 

1810 @see: Functions L{vincentys}, L{cosineAndoyerLambert_}, 

1811 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1812 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1813 L{flatPolar_}, L{haversine_} and L{thomas_}. 

1814 

1815 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_} 

1816 produce equivalent results, but L{vincentys_} is suitable 

1817 for antipodal points and slightly more expensive (M{3 cos, 

1818 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_} 

1819 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and 

1820 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}). 

1821 ''' 

1822 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21) 

1823 

1824 c = c2 * c21 

1825 x = s1 * s2 + c1 * c 

1826 y = c1 * s2 - s1 * c 

1827 return atan2(hypot(c2 * s21, y), x) 

1828 

1829# **) MIT License 

1830# 

1831# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved. 

1832# 

1833# Permission is hereby granted, free of charge, to any person obtaining a 

1834# copy of this software and associated documentation files (the "Software"), 

1835# to deal in the Software without restriction, including without limitation 

1836# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1837# and/or sell copies of the Software, and to permit persons to whom the 

1838# Software is furnished to do so, subject to the following conditions: 

1839# 

1840# The above copyright notice and this permission notice shall be included 

1841# in all copies or substantial portions of the Software. 

1842# 

1843# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1844# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1845# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1846# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1847# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1848# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1849# OTHER DEALINGS IN THE SOFTWARE.