Coverage for pygeodesy/formy.py: 98%

439 statements  

« prev     ^ index     » next       coverage.py v7.6.1, created at 2024-11-12 16:17 -0500

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 W_args_kwds_count2 

10# from pygeodesy.cartesianBase import CartesianBase # _MODS 

11from pygeodesy.constants import EPS, EPS0, EPS1, PI, PI2, PI3, PI_2, R_M, \ 

12 _0_0s, float0_, isnon0, remainder, _umod_PI2, \ 

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

14 _32_0, _90_0, _180_0, _360_0 

15from pygeodesy.datums import Datum, Ellipsoid, _ellipsoidal_datum, \ 

16 _mean_radius, _spherical_datum, _WGS84, _EWGS84 

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

18from pygeodesy.errors import IntersectionError, LimitError, limiterrors, \ 

19 _TypeError, _ValueError, _xattr, _xError, \ 

20 _xcallable,_xkwds, _xkwds_pop2 

21from pygeodesy.fmath import euclid, hypot, hypot2, sqrt0 

22from pygeodesy.fsums import fsumf_, Fmt, unstr 

23# from pygeodesy.internals import _DUNDER_nameof # from .named 

24from pygeodesy.interns import _delta_, _distant_, _inside_, _SPACE_, _too_ 

25from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS 

26from pygeodesy.named import _name__, _name2__, _NamedTuple, _xnamed, \ 

27 _DUNDER_nameof 

28from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, LatLon2Tuple, \ 

29 Intersection3Tuple, PhiLam2Tuple, Vector3Tuple 

30# from pygeodesy.streprs import Fmt, unstr # from .fsums 

31# from pygeodesy.triaxials import _hartzell3 # _MODS 

32from pygeodesy.units import _isHeight, _isRadius, Bearing, Degrees_, Distance, \ 

33 Distance_, Height, Lamd, Lat, Lon, Meter_, Phid, \ 

34 Radians, Radians_, Radius, Radius_, Scalar, _100km 

35from pygeodesy.utily import acos1, atan2b, atan2d, degrees2m, _loneg, m2degrees, \ 

36 tan_2, sincos2, sincos2_, sincos2d_, _Wrap 

37# from pygeodesy.vector3d import _otherV3d # _MODS 

38# from pygeodesy.vector3dBase import _xyz_y_z3 # _MODS 

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

40# sphericalNvector, sphericalTrigonometry # _MODS 

41 

42from contextlib import contextmanager 

43from math import asin, atan, atan2, cos, degrees, fabs, radians, sin, sqrt # pow 

44 

45__all__ = _ALL_LAZY.formy 

46__version__ = '24.10.14' 

47 

48_RADIANS2 = (PI / _180_0)**2 # degrees- to radians-squared 

49_ratio_ = 'ratio' 

50_xline_ = 'xline' 

51 

52 

53def _anti2(a, b, n_2, n, n2): 

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

55 ''' 

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

57 if r == a: 

58 r = -r 

59 b += n 

60 if fabs(b) > n: 

61 b = remainder(b, n2) 

62 return float0_(r, b) 

63 

64 

65def antipode(lat, lon, **name): 

66 '''Return the antipode, the point diametrically opposite to a given 

67 point in C{degrees}. 

68 

69 @arg lat: Latitude (C{degrees}). 

70 @arg lon: Longitude (C{degrees}). 

71 @kwarg name: Optional C{B{name}=NN} (C{str}). 

72 

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

74 

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

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

77 ''' 

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

79 

80 

81def antipode_(phi, lam, **name): 

82 '''Return the antipode, the point diametrically opposite to a given 

83 point in C{radians}. 

84 

85 @arg phi: Latitude (C{radians}). 

86 @arg lam: Longitude (C{radians}). 

87 @kwarg name: Optional C{B{name}=NN} (C{str}). 

88 

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

90 

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

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

93 ''' 

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

95 

96 

97def bearing(lat1, lon1, lat2, lon2, **final_wrap): 

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

99 between two (spherical) points. 

100 

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

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

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

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

105 @kwarg final_wrap: Optional keyword arguments for function 

106 L{pygeodesy.bearing_}. 

107 

108 @return: Initial or final bearing (compass C{degrees360}) or zero if 

109 both points coincide. 

110 ''' 

111 r = bearing_(Phid(lat1=lat1), Lamd(lon1=lon1), 

112 Phid(lat2=lat2), Lamd(lon2=lon2), **final_wrap) 

113 return degrees(r) 

114 

115 

116def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False): 

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

118 two (spherical) points. 

119 

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

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

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

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

124 @kwarg final: If C{True}, return the final, otherwise the initial bearing 

125 (C{bool}). 

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

127 B{C{lam2}} (C{bool}). 

128 

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

130 are coincident. 

131 

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

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

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

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

136 ''' 

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

138 if final: # swap plus PI 

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

140 r = PI3 

141 else: 

142 r = PI2 

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

144 

145 x = ca1 * sa2 - sa1 * ca2 * cdb 

146 y = sdb * ca2 

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

148 

149 

150def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf 

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

152 ''' 

153 try: # for LatLon_ and ellipsoidal LatLon 

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

155 except AttributeError: 

156 pass 

157 # XXX spherical version, OK for ellipsoidal ispolar? 

158 t = p1.philam + p2.philam 

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

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

161 name__=_bearingTo2) 

162 

163 

164def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False): 

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

166 lat2 - lat1)} between two points. 

167 

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

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

170 

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

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

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

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

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

176 mean latitude (C{bool}). 

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

178 and B{C{lon2}} (C{bool}). 

179 

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

181 

182 @note: Courtesy of Martin Schultz. 

183 

184 @see: U{Local, flat earth approximation 

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

186 ''' 

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

188 if adjust: # scale delta lon 

189 d_lon *= _scale_deg(lat1, lat2) 

190 return atan2b(d_lon, lat2 - lat1) 

191 

192 

193def cosineAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

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

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

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

197 

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

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

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

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

202 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

203 L{a_f2Tuple}) to use. 

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

205 B{C{lon2}} (C{bool}). 

206 

207 @return: Distance (C{meter}, same units as the B{C{datum}}'s ellipsoid axes or 

208 C{radians} if C{B{datum} is None}). 

209 

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

211 

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

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

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

215 L{Ellipsoid.distance2}. 

216 ''' 

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

218 

219 

220def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): 

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

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

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

224 

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

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

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

228 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

229 L{a_f2Tuple}) to use. 

230 

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

232 

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

234 

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

236 L{cosineLaw_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_}, 

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

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

239 AndoyerLambert.php>}. 

240 ''' 

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

242 if isnon0(c1) and isnon0(c2): 

243 E = _ellipsoidal(datum, cosineAndoyerLambert_) 

244 if E.f: # ellipsoidal 

245 r2 = atan2(E.b_a * s2, c2) 

246 r1 = atan2(E.b_a * s1, c1) 

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

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

249 if r: 

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

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

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

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

254 r += (c - s) * E.f * _0_125 

255 return r 

256 

257 

258def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

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

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

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

262 

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

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

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

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

267 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

268 L{a_f2Tuple}) to use. 

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

270 B{C{lon2}} (C{bool}). 

271 

272 @return: Distance (C{meter}, same units as the B{C{datum}}'s ellipsoid axes or 

273 C{radians} if C{B{datum} is None}). 

274 

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

276 

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

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

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

280 L{Ellipsoid.distance2}. 

281 ''' 

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

283 

284 

285def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): 

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

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

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

289 formula. 

290 

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

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

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

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

295 L{Ellipsoid2} or L{a_f2Tuple}). 

296 

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

298 

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

300 

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

302 L{cosineLaw_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_}, 

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

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

305 Distance/ForsytheCorrection.php>}. 

306 ''' 

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

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

309 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_) 

310 if E.f: # ellipsoidal 

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

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

313 s = (s1 + s2)**2 / (1 + cr) 

314 t = (s1 - s2)**2 / (1 - cr) 

315 x = s + t 

316 y = s - t 

317 

318 s = 8 * r**2 / sr 

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

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

321 b = -2 * d 

322 e = 30 * s2r 

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

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

325 

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

327 return r 

328 

329 

330def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

333 

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

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

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

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

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

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

340 L{a_f2Tuple}) to use. 

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

342 and B{C{lon2}} (C{bool}). 

343 

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

345 ellipsoid or datum axes). 

346 

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

348 

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

350 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean}, 

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

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

353 

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

355 ''' 

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

357 

358 

359def cosineLaw_(phi2, phi1, lam21): 

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

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

362 

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

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

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

366 

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

368 

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

370 L{cosineForsytheAndoyerLambert_}, L{euclidean_}, 

371 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_}, 

372 L{thomas_} and L{vincentys_}. 

373 

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

375 ''' 

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

377 

378 

379def _d3(wrap, lat1, lon1, lat2, lon2): 

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

381 ''' 

382 if wrap: 

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

384 return radians(lat2), Phid(lat1=lat1), radians(d_lon) 

385 else: # for backward compaibility 

386 return Phid(lat2=lat2), Phid(lat1=lat1), Phid(d_lon=lon2 - lon1) 

387 

388 

389def _dE(func_, earth, *wrap_lls): 

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

391 ''' 

392 E = _ellipsoidal(earth, func_) 

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

394 return r * E.a 

395 

396 

397def _dS(func_, radius, *wrap_lls, **adjust): 

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

399 ''' 

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

401 if radius is not R_M: 

402 _, lat1, _, lat2, _ = wrap_lls 

403 radius = _mean_radius(radius, lat1, lat2) 

404 return r * radius 

405 

406 

407def _eA(excess_, radius, *wrap_lls): 

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

409 ''' 

410 r = excess_(*_d3(*wrap_lls)) 

411 if radius: 

412 _, lat1, _, lat2, _ = wrap_lls 

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

414 return r 

415 

416 

417def _ellipsoidal(earth, where): 

418 '''(INTERNAL) Helper for distances. 

419 ''' 

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

421 earth if isinstance(earth, Ellipsoid) else 

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

423 _ellipsoidal_datum(earth, name__=where)).ellipsoid) 

424 

425 

426def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **adjust_limit_wrap): 

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

428 / Projection<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}) or ellipsoid 

435 (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). 

436 @kwarg adjust_limit_wrap: Optional keyword arguments for function L{equirectangular4}. 

437 

438 @return: Distance (C{meter}, same units as B{C{radius}} or the ellipsoid or datum axes). 

439 

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

441 

442 @see: Function L{equirectangular4} for more details, the available B{C{options}}, 

443 errors, restrictions and other, approximate or accurate distance functions. 

444 ''' 

445 d = sqrt(equirectangular4(Lat(lat1=lat1), Lon(lon1=lon1), 

446 Lat(lat2=lat2), Lon(lon2=lon2), 

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

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

449 

450 

451def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap): 

452 '''(INTERNAL) Helper for the L{frechet._FrechetMeterRadians} 

453 and L{hausdorff._HausdorffMeterRedians} classes. 

454 ''' 

455 return equirectangular4(lat1, lon1, lat2, lon2, **adjust_limit_wrap).distance2 * _RADIANS2 

456 

457 

458def equirectangular4(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False): 

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

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

461 

462 This approximation is valid for short distance of several hundred Km or Miles, see 

463 the B{C{limit}} keyword argument and L{LimitError}. 

464 

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

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

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

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

469 @kwarg adjust: Adjust the wrapped, unrolled longitudinal delta by the cosine of the mean 

470 latitude (C{bool}). 

471 @kwarg limit: Optional limit for lat- and longitudinal deltas (C{degrees}) or C{None} 

472 or C{0} for unlimited. 

473 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and B{C{lon2}} 

474 (C{bool}). 

475 

476 @return: A L{Distance4Tuple}C{(distance2, delta_lat, delta_lon, unroll_lon2)} 

477 in C{degrees squared}. 

478 

479 @raise LimitError: If the lat- and/or longitudinal delta exceeds the B{C{-limit..limit}} 

480 range and L{limiterrors<pygeodesy.limiterrors>} is C{True}. 

481 

482 @see: U{Local, flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>}, 

483 functions L{equirectangular}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

484 L{cosineLaw}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, 

485 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*} 

486 and C{LatLon.equirectangularTo}. 

487 ''' 

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

489 d_lat = lat2 - lat1 

490 

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

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

493 if d > limit: 

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

495 s = unstr(equirectangular4, lat1, lon1, lat2, lon2, 

496 limit=limit, wrap=wrap) 

497 raise LimitError(s, txt=t) 

498 

499 if adjust: # scale delta lon 

500 d_lon *= _scale_deg(lat1, lat2) 

501 

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

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

504 

505 

506def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False): 

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

508 

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

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

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

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

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

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

515 L{a_f2Tuple}) to use. 

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

517 the mean latitude (C{bool}). 

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

519 and B{C{lon2}} (C{bool}). 

520 

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

522 ellipsoid or datum axes). 

523 

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

525 

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

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

528 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

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

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

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

532 ''' 

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

534 

535 

536def euclidean_(phi2, phi1, lam21, adjust=True): 

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

538 

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

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

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

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

543 of the mean latitude (C{bool}). 

544 

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

546 

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

548 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

549 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_}, 

550 L{thomas_} and L{vincentys_}. 

551 ''' 

552 if adjust: 

553 lam21 *= _scale_rad(phi2, phi1) 

554 return euclid(phi2 - phi1, lam21) 

555 

556 

557def excessAbc_(A, b, c): 

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

559 and the included (small) angle. 

560 

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

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

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

564 

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

566 

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

568 

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

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

571 ''' 

572 A = Radians_(A=A) 

573 b = Radians_(b=b) * _0_5 

574 c = Radians_(c=c) * _0_5 

575 

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

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

578 

579 

580def excessCagnoli_(a, b, c): 

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

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

583 

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

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

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

587 

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

589 

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

591 

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

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

594 ''' 

595 a = Radians_(a=a) 

596 b = Radians_(b=b) 

597 c = Radians_(c=c) 

598 

599 s = fsumf_(a, b, c) * _0_5 

600 _s = sin 

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

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

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

604 return Radians(Cagnoli=r * _2_0) 

605 

606 

607def excessGirard_(A, B, C): 

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

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

610 

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

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

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

614 

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

616 

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

618 

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

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

621 ''' 

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

623 Radians_(B=B), 

624 Radians_(C=C), -PI)) 

625 

626 

627def excessLHuilier_(a, b, c): 

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

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

630 

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

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

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

634 

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

636 

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

638 

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

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

641 ''' 

642 a = Radians_(a=a) 

643 b = Radians_(b=b) 

644 c = Radians_(c=c) 

645 

646 s = fsumf_(a, b, c) * _0_5 

647 _t = tan_2 

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

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

650 return Radians(LHuilier=r * _4_0) 

651 

652 

653def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

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

657 method. 

658 

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

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

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

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

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

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

665 L{a_f2Tuple}) or C{None}. 

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

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

668 

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

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

671 if C{B{radius}=0} or C{None}. 

672 

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

674 

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

676 

677 @raise ValueError: Semi-circular longitudinal delta. 

678 

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

680 ''' 

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

682 

683 

684def excessKarney_(phi2, phi1, lam21): 

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

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

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

688 method. 

689 

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

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

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

693 

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

695 

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

697 

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

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

700 ''' 

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

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

703 # 

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

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

706 # 1 + tan(φ1 / 2) · tan(φ2 / 2) 

707 # 

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

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

710 _t = tan_2 

711 t2 = _t(phi2) 

712 t1 = _t(phi1) 

713 t = _t(lam21, lam21=None) 

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

715 _1_0 + (t1 * t2)) * _2_0) 

716 

717 

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

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

720# 

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

722# 

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

724# 

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

726# 

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

728# 

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

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

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

732# 

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

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

735# 

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

737# 

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

739# inverse Gudermannian) function 

740# 

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

742# 

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

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

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

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

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

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

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

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

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

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

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

754# - Charles Karney 

755 

756 

757def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

760 

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

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

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

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

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

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

767 L{a_f2Tuple}) or C{None}. 

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

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

770 

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

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

773 if C{B{radius}=0} or C{None}. 

774 

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

776 

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

778 

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

780 ''' 

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

782 

783 

784def excessQuad_(phi2, phi1, lam21): 

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

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

787 

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

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

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

791 

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

793 

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

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

796 ''' 

797 s = sin((phi2 + phi1) * _0_5) 

798 c = cos((phi2 - phi1) * _0_5) 

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

800 

801 

802def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False): 

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

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

805 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

807 

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

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

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

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

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

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

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

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

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

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

818 

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

820 ellipsoid axes). 

821 

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

823 

824 @note: The meridional and prime_vertical radii of curvature 

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

826 

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

828 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

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

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

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

832 ''' 

833 E = _ellipsoidal(datum, flatLocal) 

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

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

836 

837hubeny = flatLocal # PYCHOK for Karl Hubeny 

838 

839 

840def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True): 

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

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

843 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

845 

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

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

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

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

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

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

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

853 

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

855 

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

857 

858 @note: The meridional and prime_vertical radii of curvature 

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

860 

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

862 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_}, 

863 L{euclidean_}, L{haversine_}, L{thomas_} and L{vincentys_} and 

864 U{local, flat earth approximation 

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

866 ''' 

867 E = _ellipsoidal(datum, flatLocal_) 

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

869 

870hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny 

871 

872 

873def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

876 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

877 formula. 

878 

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

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

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

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

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

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

885 L{a_f2Tuple}) to use. 

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

887 and B{C{lon2}} (C{bool}). 

888 

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

890 ellipsoid or datum axes). 

891 

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

893 

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

895 L{cosineForsytheAndoyerLambert},L{cosineLaw}, 

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

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

898 L{vincentys}. 

899 ''' 

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

901 

902 

903def flatPolar_(phi2, phi1, lam21): 

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

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

906 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

907 formula. 

908 

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

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

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

912 

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

914 

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

916 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

917 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{haversine_}, 

918 L{thomas_} and L{vincentys_}. 

919 ''' 

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

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

922 if a < b: 

923 a, b = b, a 

924 if a < EPS0: 

925 a = _0_0 

926 elif b > 0: 

927 b = b / a # /= chokes PyChecker 

928 c = b * cos(lam21) * _2_0 

929 c = fsumf_(_1_0, b**2, -fabs(c)) 

930 a *= sqrt0(c) 

931 return a 

932 

933 

934def _hartzell(pov, los, earth, **kwds): 

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

936 ''' 

937 if earth is None: 

938 earth = pov.datum 

939 else: 

940 earth = _spherical_datum(earth, name__=hartzell) 

941 pov = pov.toDatum(earth) 

942 h = pov.height 

943 if h < 0: # EPS0 

944 t = _SPACE_(Fmt.PARENSPACED(height=h), _inside_) 

945 raise IntersectionError(pov=pov, earth=earth, txt=t) 

946 return hartzell(pov, los=los, earth=earth, **kwds) if h > 0 else pov # EPS0 

947 

948 

949def hartzell(pov, los=False, earth=_WGS84, **name_LatLon_and_kwds): 

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

951 a Point-Of-View in space. 

952 

953 @arg pov: Point-Of-View outside the earth (C{LatLon}, C{Cartesian}, 

954 L{Ecef9Tuple} or L{Vector3d}). 

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

956 C{True} for the I{normal, plumb} onto the surface or C{False} 

957 or C{None} to point to the center of the earth. 

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

959 L{a_f2Tuple} or a C{scalar} earth radius in C{meter}). 

960 @kwarg name_LatLon_and_kwds: Optional, overriding C{B{name}="hartzell"} 

961 (C{str}), class C{B{LatLon}=None} to return the intersection 

962 plus additional C{LatLon} keyword arguments, include the 

963 B{C{datum}} if different and to convert from B{C{earth}}. 

964 

965 @return: The intersection (L{Vector3d}, B{C{pov}}'s C{cartesian type} or the 

966 given B{C{LatLon}} instance) with attribute C{height} set to the 

967 distance to the B{C{pov}}. 

968 

969 @raise IntersectionError: Invalid B{C{pov}} or B{C{pov}} inside the earth or 

970 invalid B{C{los}} or B{C{los}} points outside or 

971 away from the earth. 

972 

973 @raise TypeError: Invalid B{C{earth}}, C{ellipsoid} or C{datum}. 

974 

975 @see: Class L{Los}, functions L{tyr3d} and L{hartzell4} and methods 

976 L{Ellipsoid.hartzell4} and any C{Cartesian.hartzell} and C{LatLon.hartzell}. 

977 ''' 

978 n, LatLon_and_kwds = _name2__(name_LatLon_and_kwds, name__=hartzell) 

979 try: 

980 D = _spherical_datum(earth, name__=hartzell) 

981 r, h, i = _MODS.triaxials._hartzell3(pov, los, D.ellipsoid._triaxial) 

982 

983 C = _MODS.cartesianBase.CartesianBase 

984 if LatLon_and_kwds: 

985 c = C(r, datum=D) 

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

987 elif isinstance(r, C): 

988 r.height = h 

989 if i: 

990 r._iteration = i 

991 except Exception as x: 

992 raise IntersectionError(pov=pov, los=los, earth=earth, cause=x, 

993 **LatLon_and_kwds) 

994 return _xnamed(r, n) if n else r 

995 

996 

997def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

1000 formula. 

1001 

1002 @arg lat1: Start latitude (C{degrees}). 

1003 @arg lon1: Start longitude (C{degrees}). 

1004 @arg lat2: End latitude (C{degrees}). 

1005 @arg lon2: End longitude (C{degrees}). 

1006 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1007 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1008 L{a_f2Tuple}) to use. 

1009 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1010 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1011 

1012 @return: Distance (C{meter}, same units as B{C{radius}}). 

1013 

1014 @raise TypeError: Invalid B{C{radius}}. 

1015 

1016 @see: U{Distance between two (spherical) points 

1017 <https://www.EdWilliams.org/avform.htm#Dist>}, functions 

1018 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1019 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1020 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, 

1021 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1022 

1023 @note: See note at function L{vincentys_}. 

1024 ''' 

1025 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2) 

1026 

1027 

1028def haversine_(phi2, phi1, lam21): 

1029 '''Compute the I{angular} distance between two (spherical) points 

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

1031 formula. 

1032 

1033 @arg phi2: End latitude (C{radians}). 

1034 @arg phi1: Start latitude (C{radians}). 

1035 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1036 

1037 @return: Angular distance (C{radians}). 

1038 

1039 @see: Functions L{haversine}, L{cosineAndoyerLambert_}, 

1040 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1041 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_}, 

1042 L{thomas_} and L{vincentys_}. 

1043 

1044 @note: See note at function L{vincentys_}. 

1045 ''' 

1046 def _hsin(rad): 

1047 return sin(rad * _0_5)**2 

1048 

1049 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine 

1050 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2 

1051 

1052 

1053def heightOf(angle, distance, radius=R_M): 

1054 '''Determine the height above the (spherical) earth' surface after 

1055 traveling along a straight line at a given tilt. 

1056 

1057 @arg angle: Tilt angle above horizontal (C{degrees}). 

1058 @arg distance: Distance along the line (C{meter} or same units as 

1059 B{C{radius}}). 

1060 @kwarg radius: Optional mean earth radius (C{meter}). 

1061 

1062 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}). 

1063 

1064 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}. 

1065 

1066 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>} 

1067 (U{Shapiro et al. 2009, JTECH 

1068 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>} 

1069 and U{Potvin et al. 2012, JTECH 

1070 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}). 

1071 ''' 

1072 r = h = Radius(radius) 

1073 d = fabs(Distance(distance)) 

1074 if d > h: 

1075 d, h = h, d 

1076 

1077 if d > EPS0: # and h > EPS0 

1078 d = d / h # /= h chokes PyChecker 

1079 s = sin(Phid(angle=angle, clip=_180_0)) 

1080 s = fsumf_(_1_0, s * d * _2_0, d**2) 

1081 if s > 0: 

1082 return h * sqrt(s) - r 

1083 

1084 raise _ValueError(angle=angle, distance=distance, radius=radius) 

1085 

1086 

1087def heightOrthometric(h_ll, N): 

1088 '''Get the I{orthometric} height B{H}, the height above the geoid, earth surface. 

1089 

1090 @arg h_ll: The height above the ellipsoid (C{meter}) or an I{ellipsoidal} 

1091 location (C{LatLon} with a C{height} or C{h} attribute). 

1092 @arg N: The I{geoid} height (C{meter}), the height of the geoid above the 

1093 ellipsoid at the same B{C{h_ll}} location. 

1094 

1095 @return: I{Orthometric} height C{B{H} = B{h} - B{N}} (C{meter}, same units 

1096 as B{C{h}} and B{C{N}}). 

1097 

1098 @see: U{Ellipsoid, Geoid, and Othometric Heights<https://www.NGS.NOAA.gov/ 

1099 GEOID/PRESENTATIONS/2007_02_24_CCPS/Roman_A_PLSC2007notes.pdf>}, page 

1100 6 and module L{pygeodesy.geoids}. 

1101 ''' 

1102 h = h_ll if _isHeight(h_ll) else _xattr(h_ll, height=_xattr(h_ll, h=0)) 

1103 return Height(H=Height(h=h) - Height(N=N)) 

1104 

1105 

1106def horizon(height, radius=R_M, refraction=False): 

1107 '''Determine the distance to the horizon from a given altitude above the 

1108 (spherical) earth. 

1109 

1110 @arg height: Altitude (C{meter} or same units as B{C{radius}}). 

1111 @kwarg radius: Optional mean earth radius (C{meter}). 

1112 @kwarg refraction: Consider atmospheric refraction (C{bool}). 

1113 

1114 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}). 

1115 

1116 @raise ValueError: Invalid B{C{height}} or B{C{radius}}. 

1117 

1118 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}. 

1119 ''' 

1120 h, r = Height(height), Radius(radius) 

1121 if min(h, r) < 0: 

1122 raise _ValueError(height=height, radius=radius) 

1123 

1124 d2 = ((r * 2.415750694528) if refraction else # 2.0 / 0.8279 

1125 fsumf_(r, r, h)) * h 

1126 return sqrt0(d2) 

1127 

1128 

1129class _idllmn6(object): # see also .geodesicw._wargs, .latlonBase._toCartesian3, .vector2d._numpy 

1130 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}. 

1131 ''' 

1132 @contextmanager # <https://www.Python.org/dev/peps/pep-0343/> Examples 

1133 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds): 

1134 try: 

1135 if wrap: 

1136 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap) 

1137 kwds = _xkwds(kwds, wrap=wrap) # for _xError 

1138 m = small if small is _100km else Meter_(small=small) 

1139 n = _DUNDER_nameof(intersections2 if s else intersection2) 

1140 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m: 

1141 d, m = None, _MODS.vector3d 

1142 _i = m._intersects2 if s else m._intersect3d3 

1143 elif _isRadius(datum) and datum < 0 and not s: 

1144 d = _spherical_datum(-datum, name=n) 

1145 m = _MODS.sphericalNvector 

1146 _i = m.intersection 

1147 else: 

1148 d = _spherical_datum(datum, name=n) 

1149 if d.isSpherical: 

1150 m = _MODS.sphericalTrigonometry 

1151 _i = m._intersects2 if s else m._intersect 

1152 elif d.isEllipsoidal: 

1153 try: 

1154 if d.ellipsoid.geodesic: 

1155 pass 

1156 m = _MODS.ellipsoidalKarney 

1157 except ImportError: 

1158 m = _MODS.ellipsoidalExact 

1159 _i = m._intersections2 if s else m._intersection3 # ellipsoidalBaseDI 

1160 else: 

1161 raise _TypeError(datum=datum) 

1162 yield _i, d, lat2, lon2, m, n 

1163 

1164 except (TypeError, ValueError) as x: 

1165 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum, 

1166 lat2=lat2, lon2=lon2, small=small, **kwds) 

1167 

1168_idllmn6 = _idllmn6() # PYCHOK singleton 

1169 

1170 

1171def intersection2(lat1, lon1, bearing1, 

1172 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True 

1173 '''I{Conveniently} compute the intersection of two lines each defined 

1174 by a (geodetic) point and a bearing from North, using either ... 

1175 

1176 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km 

1177 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1178 

1179 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}} 

1180 or a C{scalar B{datum}} representing the earth radius, conventionally 

1181 in C{meter} or ... 

1182 

1183 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative} 

1184 C{scalar}, (negative) earth radius, conventionally in C{meter} or ... 

1185 

1186 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}} 

1187 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1188 is installed, otherwise ... 

1189 

1190 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal. 

1191 

1192 @arg lat1: Latitude of the first point (C{degrees}). 

1193 @arg lon1: Longitude of the first point (C{degrees}). 

1194 @arg bearing1: Bearing at the first point (compass C{degrees}). 

1195 @arg lat2: Latitude of the second point (C{degrees}). 

1196 @arg lon2: Longitude of the second point (C{degrees}). 

1197 @arg bearing2: Bearing at the second point (compass C{degrees}). 

1198 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1199 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1200 radius (C{meter}, same units as B{C{radius1}} and 

1201 B{C{radius2}}) or C{None}. 

1202 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1203 and B{C{lon2}} (C{bool}). 

1204 @kwarg small: Upper limit for small distances (C{meter}). 

1205 

1206 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and 

1207 longitude of the intersection point. 

1208 

1209 @raise IntersectionError: Ambiguous or infinite intersection 

1210 or colinear, parallel or otherwise 

1211 non-intersecting lines. 

1212 

1213 @raise TypeError: Invalid B{C{datum}}. 

1214 

1215 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}}, 

1216 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}. 

1217 

1218 @see: Method L{RhumbLine.intersection2}. 

1219 

1220 @note: The returned intersections may be near-antipodal. 

1221 ''' 

1222 b1 = Bearing(bearing1=bearing1) 

1223 b2 = Bearing(bearing2=bearing2) 

1224 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1225 small, wrap, False, bearing1=b1, bearing2=b2) as t: 

1226 _i, d, lat2, lon2, m, n = t 

1227 if d is None: 

1228 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1, 

1229 m.Vector3d(lon2, lat2, 0), b2, useZ=False) 

1230 t = LatLon2Tuple(t.y, t.x, name=n) 

1231 

1232 else: 

1233 t = _i(m.LatLon(lat1, lon1, datum=d), b1, 

1234 m.LatLon(lat2, lon2, datum=d), b2, 

1235 LatLon=None, height=0, wrap=False) 

1236 if isinstance(t, Intersection3Tuple): # ellipsoidal 

1237 t, _, _ = t 

1238 t = LatLon2Tuple(t.lat, t.lon, name=n) 

1239 return t 

1240 

1241 

1242def intersections2(lat1, lon1, radius1, 

1243 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True 

1244 '''I{Conveniently} compute the intersections of two circles each defined 

1245 by a (geodetic) center point and a radius, using either ... 

1246 

1247 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km 

1248 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1249 

1250 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}} 

1251 or a C{scalar B{datum}} representing the earth radius, conventionally 

1252 in C{meter} or ... 

1253 

1254 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}} 

1255 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1256 is installed, otherwise ... 

1257 

1258 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal. 

1259 

1260 @arg lat1: Latitude of the first circle center (C{degrees}). 

1261 @arg lon1: Longitude of the first circle center (C{degrees}). 

1262 @arg radius1: Radius of the first circle (C{meter}, conventionally). 

1263 @arg lat2: Latitude of the second circle center (C{degrees}). 

1264 @arg lon2: Longitude of the second circle center (C{degrees}). 

1265 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}). 

1266 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1267 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1268 radius (C{meter}, same units as B{C{radius1}} and 

1269 B{C{radius2}}) or C{None}. 

1270 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1271 and B{C{lon2}} (C{bool}). 

1272 @kwarg small: Upper limit for small distances (C{meter}). 

1273 

1274 @return: 2-Tuple of the intersection points, each a 

1275 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the 

1276 points are the same instance, aka the I{radical center}. 

1277 

1278 @raise IntersectionError: Concentric, antipodal, invalid or 

1279 non-intersecting circles or no 

1280 convergence. 

1281 

1282 @raise TypeError: Invalid B{C{datum}}. 

1283 

1284 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}}, 

1285 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}. 

1286 ''' 

1287 r1 = Radius_(radius1=radius1) 

1288 r2 = Radius_(radius2=radius2) 

1289 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1290 small, wrap, True, radius1=r1, radius2=r2) as t: 

1291 _i, d, lat2, lon2, m, n = t 

1292 if d is None: 

1293 r1 = m2degrees(r1, radius=R_M, lat=lat1) 

1294 r2 = m2degrees(r2, radius=R_M, lat=lat2) 

1295 

1296 def _V2T(x, y, _, **unused): # _ == z unused 

1297 return LatLon2Tuple(y, x, name=n) 

1298 

1299 t = _i(m.Vector3d(lon1, lat1, 0), r1, 

1300 m.Vector3d(lon2, lat2, 0), r2, sphere=False, 

1301 Vector=_V2T) 

1302 else: 

1303 def _LL2T(lat, lon, **unused): 

1304 return LatLon2Tuple(lat, lon, name=n) 

1305 

1306 t = _i(m.LatLon(lat1, lon1, datum=d), r1, 

1307 m.LatLon(lat2, lon2, datum=d), r2, 

1308 LatLon=_LL2T, height=0, wrap=False) 

1309 return t 

1310 

1311 

1312def isantipode(lat1, lon1, lat2, lon2, eps=EPS): 

1313 '''Check whether two points are I{antipodal}, on diametrically 

1314 opposite sides of the earth. 

1315 

1316 @arg lat1: Latitude of one point (C{degrees}). 

1317 @arg lon1: Longitude of one point (C{degrees}). 

1318 @arg lat2: Latitude of the other point (C{degrees}). 

1319 @arg lon2: Longitude of the other point (C{degrees}). 

1320 @kwarg eps: Tolerance for near-equality (C{degrees}). 

1321 

1322 @return: C{True} if points are antipodal within the 

1323 B{C{eps}} tolerance, C{False} otherwise. 

1324 

1325 @see: Functions L{isantipode_} and L{antipode}. 

1326 ''' 

1327 return (fabs(lat1 + lat2) <= eps and 

1328 fabs(lon1 + lon2) <= eps) or _isequalTo( 

1329 normal(lat1, lon1), antipode(lat2, lon2), eps) 

1330 

1331 

1332def isantipode_(phi1, lam1, phi2, lam2, eps=EPS): 

1333 '''Check whether two points are I{antipodal}, on diametrically 

1334 opposite sides of the earth. 

1335 

1336 @arg phi1: Latitude of one point (C{radians}). 

1337 @arg lam1: Longitude of one point (C{radians}). 

1338 @arg phi2: Latitude of the other point (C{radians}). 

1339 @arg lam2: Longitude of the other point (C{radians}). 

1340 @kwarg eps: Tolerance for near-equality (C{radians}). 

1341 

1342 @return: C{True} if points are antipodal within the 

1343 B{C{eps}} tolerance, C{False} otherwise. 

1344 

1345 @see: Functions L{isantipode} and L{antipode_}. 

1346 ''' 

1347 return (fabs(phi1 + phi2) <= eps and 

1348 fabs(lam1 + lam2) <= eps) or _isequalTo_( 

1349 normal_(phi1, lam1), antipode_(phi2, lam2), eps) 

1350 

1351 

1352def _isequalTo(p1, p2, eps=EPS): 

1353 '''Compare 2 point lat-/lons ignoring C{class}. 

1354 ''' 

1355 return (fabs(p1.lat - p2.lat) <= eps and 

1356 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon) 

1357 

1358 

1359def _isequalTo_(p1, p2, eps=EPS): 

1360 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}. 

1361 ''' 

1362 return (fabs(p1.phi - p2.phi) <= eps and 

1363 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam) 

1364 

1365 

1366def isnormal(lat, lon, eps=0): 

1367 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their 

1368 respective I{normal} range in C{degrees}. 

1369 

1370 @arg lat: Latitude (C{degrees}). 

1371 @arg lon: Longitude (C{degrees}). 

1372 @kwarg eps: Optional tolerance C{degrees}). 

1373 

1374 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and 

1375 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise. 

1376 

1377 @see: Functions L{isnormal_} and L{normal}. 

1378 ''' 

1379 return (_90_0 - fabs(lat)) >= eps and _loneg(fabs(lon)) >= eps 

1380 

1381 

1382def isnormal_(phi, lam, eps=0): 

1383 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their 

1384 respective I{normal} range in C{radians}. 

1385 

1386 @arg phi: Latitude (C{radians}). 

1387 @arg lam: Longitude (C{radians}). 

1388 @kwarg eps: Optional tolerance C{radians}). 

1389 

1390 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and 

1391 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise. 

1392 

1393 @see: Functions L{isnormal} and L{normal_}. 

1394 ''' 

1395 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps 

1396 

1397 

1398def latlon2n_xyz(lat, lon, **name): 

1399 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1400 earth's surface) X, Y and Z components. 

1401 

1402 @arg lat: Latitude (C{degrees}). 

1403 @arg lon: Longitude (C{degrees}). 

1404 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1405 

1406 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1407 

1408 @see: Function L{philam2n_xyz}. 

1409 

1410 @note: These are C{n-vector} x, y and z components, 

1411 I{NOT} geocentric ECEF x, y and z coordinates! 

1412 ''' 

1413 return _2n_xyz(name, *sincos2d_(lat, lon)) 

1414 

1415 

1416def _normal2(a, b, n_2, n, n2): 

1417 '''(INTERNAL) Helper for C{normal} and C{normal_}. 

1418 ''' 

1419 if fabs(b) > n: 

1420 b = remainder(b, n2) 

1421 if fabs(a) > n_2: 

1422 r = remainder(a, n) 

1423 if r != a: 

1424 a = -r 

1425 b -= n if b > 0 else -n 

1426 return float0_(a, b) 

1427 

1428 

1429def normal(lat, lon, **name): 

1430 '''Normalize a lat- I{and} longitude pair in C{degrees}. 

1431 

1432 @arg lat: Latitude (C{degrees}). 

1433 @arg lon: Longitude (C{degrees}). 

1434 @kwarg name: Optional, overriding C{B{name}="normal"} (C{str}). 

1435 

1436 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90} 

1437 and C{abs(lon) <= 180}. 

1438 

1439 @see: Functions L{normal_} and L{isnormal}. 

1440 ''' 

1441 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0), 

1442 name=_name__(name, name__=normal)) 

1443 

1444 

1445def normal_(phi, lam, **name): 

1446 '''Normalize a lat- I{and} longitude pair in C{radians}. 

1447 

1448 @arg phi: Latitude (C{radians}). 

1449 @arg lam: Longitude (C{radians}). 

1450 @kwarg name: Optional, overriding C{B{name}="normal_"} (C{str}). 

1451 

1452 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2} 

1453 and C{abs(lam) <= PI}. 

1454 

1455 @see: Functions L{normal} and L{isnormal_}. 

1456 ''' 

1457 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2), 

1458 name=_name__(name, name__=normal_)) 

1459 

1460 

1461def _2n_xyz(name, sa, ca, sb, cb): # name always **name 

1462 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}. 

1463 ''' 

1464 # Kenneth Gade eqn 3, but using right-handed 

1465 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N 

1466 return Vector3Tuple(ca * cb, ca * sb, sa, **name) 

1467 

1468 

1469def n_xyz2latlon(x, y, z, **name): 

1470 '''Convert C{n-vector} components to lat- and longitude in C{degrees}. 

1471 

1472 @arg x: X component (C{scalar}). 

1473 @arg y: Y component (C{scalar}). 

1474 @arg z: Z component (C{scalar}). 

1475 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1476 

1477 @return: A L{LatLon2Tuple}C{(lat, lon)}. 

1478 

1479 @see: Function L{n_xyz2philam}. 

1480 ''' 

1481 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), **name) 

1482 

1483 

1484def n_xyz2philam(x, y, z, **name): 

1485 '''Convert C{n-vector} components to lat- and longitude in C{radians}. 

1486 

1487 @arg x: X component (C{scalar}). 

1488 @arg y: Y component (C{scalar}). 

1489 @arg z: Z component (C{scalar}). 

1490 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1491 

1492 @return: A L{PhiLam2Tuple}C{(phi, lam)}. 

1493 

1494 @see: Function L{n_xyz2latlon}. 

1495 ''' 

1496 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), **name) 

1497 

1498 

1499def _opposes(d, m, n, n2): 

1500 '''(INTERNAL) Helper for C{opposing} and C{opposing_}. 

1501 ''' 

1502 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1 

1503 return False if d < m or d > (n2 - m) else ( 

1504 True if (n - m) < d < (n + m) else None) 

1505 

1506 

1507def opposing(bearing1, bearing2, margin=_90_0): 

1508 '''Compare the direction of two bearings given in C{degrees}. 

1509 

1510 @arg bearing1: First bearing (compass C{degrees}). 

1511 @arg bearing2: Second bearing (compass C{degrees}). 

1512 @kwarg margin: Optional, interior angle bracket (C{degrees}). 

1513 

1514 @return: C{True} if both bearings point in opposite, C{False} if 

1515 in similar or C{None} if in I{perpendicular} directions. 

1516 

1517 @see: Function L{opposing_}. 

1518 ''' 

1519 m = Degrees_(margin=margin, low=EPS0, high=_90_0) 

1520 return _opposes(bearing2 - bearing1, m, _180_0, _360_0) 

1521 

1522 

1523def opposing_(radians1, radians2, margin=PI_2): 

1524 '''Compare the direction of two bearings given in C{radians}. 

1525 

1526 @arg radians1: First bearing (C{radians}). 

1527 @arg radians2: Second bearing (C{radians}). 

1528 @kwarg margin: Optional, interior angle bracket (C{radians}). 

1529 

1530 @return: C{True} if both bearings point in opposite, C{False} if 

1531 in similar or C{None} if in perpendicular directions. 

1532 

1533 @see: Function L{opposing}. 

1534 ''' 

1535 m = Radians_(margin=margin, low=EPS0, high=PI_2) 

1536 return _opposes(radians2 - radians1, m, PI, PI2) 

1537 

1538 

1539def philam2n_xyz(phi, lam, **name): 

1540 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1541 earth's surface) X, Y and Z components. 

1542 

1543 @arg phi: Latitude (C{radians}). 

1544 @arg lam: Longitude (C{radians}). 

1545 @kwarg name: Optional name (C{str}). 

1546 

1547 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1548 

1549 @see: Function L{latlon2n_xyz}. 

1550 

1551 @note: These are C{n-vector} x, y and z components, 

1552 I{NOT} geocentric ECEF x, y and z coordinates! 

1553 ''' 

1554 return _2n_xyz(name, *sincos2_(phi, lam)) 

1555 

1556 

1557def _Propy(func, nargs, kwds): 

1558 '''(INTERNAL) Helper for the C{frechet.[-]Frechet**} and 

1559 C{hausdorff.[-]Hausdorff*} classes. 

1560 ''' 

1561 try: 

1562 _xcallable(distance=func) 

1563 # assert _args_kwds_count2(func)[0] == nargs + int(ismethod(func)) 

1564 _ = func(*_0_0s(nargs), **kwds) 

1565 except Exception as x: 

1566 t = unstr(func, **kwds) 

1567 raise _TypeError(t, cause=x) 

1568 return func 

1569 

1570 

1571def _radical2(d, r1, r2, **name): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d 

1572 # (INTERNAL) See C{radical2} below 

1573 # assert d > EPS0 

1574 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5 

1575 n = _name__(name, name__=radical2) 

1576 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d, name=n) 

1577 

1578 

1579def radical2(distance, radius1, radius2, **name): 

1580 '''Compute the I{radical ratio} and I{radical line} of two 

1581 U{intersecting circles<https://MathWorld.Wolfram.com/ 

1582 Circle-CircleIntersection.html>}. 

1583 

1584 The I{radical line} is perpendicular to the axis thru the 

1585 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}. 

1586 

1587 @arg distance: Distance between the circle centers (C{scalar}). 

1588 @arg radius1: Radius of the first circle (C{scalar}). 

1589 @arg radius2: Radius of the second circle (C{scalar}). 

1590 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1591 

1592 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <= 

1593 ratio <= 1.0} and C{xline} is along the B{C{distance}}. 

1594 

1595 @raise IntersectionError: The B{C{distance}} exceeds the sum 

1596 of B{C{radius1}} and B{C{radius2}}. 

1597 

1598 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or 

1599 B{C{radius2}}. 

1600 

1601 @see: U{Circle-Circle Intersection 

1602 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}. 

1603 ''' 

1604 d = Distance_(distance, low=_0_0) 

1605 r1 = Radius_(radius1=radius1) 

1606 r2 = Radius_(radius2=radius2) 

1607 if d > (r1 + r2): 

1608 raise IntersectionError(distance=d, radius1=r1, radius2=r2, 

1609 txt=_too_(_distant_)) 

1610 return _radical2(d, r1, r2, **name) if d > EPS0 else \ 

1611 Radical2Tuple(_0_5, _0_0, **name) 

1612 

1613 

1614class Radical2Tuple(_NamedTuple): 

1615 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and 

1616 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0} 

1617 ''' 

1618 _Names_ = (_ratio_, _xline_) 

1619 _Units_ = ( Scalar, Scalar) 

1620 

1621 

1622def _radistance(inst): 

1623 '''(INTERNAL) Helper for the L{frechet._FrechetMeterRadians} 

1624 and L{hausdorff._HausdorffMeterRedians} classes. 

1625 ''' 

1626 wrap_, kwds_ = _xkwds_pop2(inst._kwds, wrap=False) 

1627 func_ = inst._func_ 

1628 try: # calling lower-overhead C{func_} 

1629 func_(0, _0_25, _0_5, **kwds_) 

1630 wrap_ = _Wrap._philamop(wrap_) 

1631 except TypeError: 

1632 return inst.distance 

1633 

1634 def _philam(p): 

1635 try: 

1636 return p.phi, p.lam 

1637 except AttributeError: # no .phi or .lam 

1638 return radians(p.lat), radians(p.lon) 

1639 

1640 def _func_wrap(point1, point2): 

1641 phi1, lam1 = wrap_(*_philam(point1)) 

1642 phi2, lam2 = wrap_(*_philam(point2)) 

1643 return func_(phi2, phi1, lam2 - lam1, **kwds_) 

1644 

1645 inst._units = inst._units_ 

1646 return _func_wrap 

1647 

1648 

1649def _scale_deg(lat1, lat2): # degrees 

1650 # scale factor cos(mean of lats) for delta lon 

1651 m = fabs(lat1 + lat2) * _0_5 

1652 return cos(radians(m)) if m < 90 else _0_0 

1653 

1654 

1655def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights 

1656 # scale factor cos(mean of phis) for delta lam 

1657 m = fabs(phi1 + phi2) * _0_5 

1658 return cos(m) if m < PI_2 else _0_0 

1659 

1660 

1661def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw 

1662 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine. 

1663 ''' 

1664 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21) 

1665 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21 

1666 

1667 

1668def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

1669 '''Compute the distance between two (ellipsoidal) points using 

1670 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1671 formula. 

1672 

1673 @arg lat1: Start latitude (C{degrees}). 

1674 @arg lon1: Start longitude (C{degrees}). 

1675 @arg lat2: End latitude (C{degrees}). 

1676 @arg lon2: End longitude (C{degrees}). 

1677 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1678 L{Ellipsoid2} or L{a_f2Tuple}) to use. 

1679 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1680 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1681 

1682 @return: Distance (C{meter}, same units as the B{C{datum}}'s 

1683 ellipsoid axes). 

1684 

1685 @raise TypeError: Invalid B{C{datum}}. 

1686 

1687 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1688 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, 

1689 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}. 

1690 ''' 

1691 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2) 

1692 

1693 

1694def thomas_(phi2, phi1, lam21, datum=_WGS84): 

1695 '''Compute the I{angular} distance between two (ellipsoidal) points using 

1696 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1697 formula. 

1698 

1699 @arg phi2: End latitude (C{radians}). 

1700 @arg phi1: Start latitude (C{radians}). 

1701 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1702 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, 

1703 L{Ellipsoid2} or L{a_f2Tuple}). 

1704 

1705 @return: Angular distance (C{radians}). 

1706 

1707 @raise TypeError: Invalid B{C{datum}}. 

1708 

1709 @see: Functions L{thomas}, L{cosineAndoyerLambert_}, 

1710 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1711 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_}, 

1712 L{haversine_} and L{vincentys_} and U{Geodesy-PHP 

1713 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/ 

1714 Distance/ThomasFormula.php>}. 

1715 ''' 

1716 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21) 

1717 if r and isnon0(c1) and isnon0(c2): 

1718 E = _ellipsoidal(datum, thomas_) 

1719 if E.f: 

1720 r1 = atan2(E.b_a * s1, c1) 

1721 r2 = atan2(E.b_a * s2, c2) 

1722 

1723 j = (r2 + r1) * _0_5 

1724 k = (r2 - r1) * _0_5 

1725 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5) 

1726 

1727 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2) 

1728 u = _1_0 - h 

1729 if isnon0(u) and isnon0(h): 

1730 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h) 

1731 sr, cr = sincos2(r) 

1732 if isnon0(sr): 

1733 u = 2 * (sj * ck)**2 / u 

1734 h = 2 * (sk * cj)**2 / h 

1735 x = u + h 

1736 y = u - h 

1737 

1738 s = r / sr 

1739 e = 4 * s**2 

1740 d = 2 * cr 

1741 a = e * d 

1742 b = 2 * r 

1743 c = s - (a - d) * _0_5 

1744 f = E.f * _0_25 

1745 

1746 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y) 

1747 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr 

1748 return r 

1749 

1750 

1751def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1752 '''Compute the distance between two (spherical) points using 

1753 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1754 spherical formula. 

1755 

1756 @arg lat1: Start latitude (C{degrees}). 

1757 @arg lon1: Start longitude (C{degrees}). 

1758 @arg lat2: End latitude (C{degrees}). 

1759 @arg lon2: End longitude (C{degrees}). 

1760 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1761 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1762 L{a_f2Tuple}) to use. 

1763 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1764 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1765 

1766 @return: Distance (C{meter}, same units as B{C{radius}}). 

1767 

1768 @raise UnitError: Invalid B{C{radius}}. 

1769 

1770 @see: Functions L{vincentys_}, L{cosineAndoyerLambert}, 

1771 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular}, 

1772 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1773 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2}, 

1774 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1775 

1776 @note: See note at function L{vincentys_}. 

1777 ''' 

1778 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2) 

1779 

1780 

1781def vincentys_(phi2, phi1, lam21): 

1782 '''Compute the I{angular} distance between two (spherical) points using 

1783 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1784 spherical formula. 

1785 

1786 @arg phi2: End latitude (C{radians}). 

1787 @arg phi1: Start latitude (C{radians}). 

1788 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1789 

1790 @return: Angular distance (C{radians}). 

1791 

1792 @see: Functions L{vincentys}, L{cosineAndoyerLambert_}, 

1793 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1794 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_}, 

1795 L{haversine_} and L{thomas_}. 

1796 

1797 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_} 

1798 produce equivalent results, but L{vincentys_} is suitable 

1799 for antipodal points and slightly more expensive (M{3 cos, 

1800 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_} 

1801 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and 

1802 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}). 

1803 ''' 

1804 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21) 

1805 

1806 c = c2 * c21 

1807 x = s1 * s2 + c1 * c 

1808 y = c1 * s2 - s1 * c 

1809 return atan2(hypot(c2 * s21, y), x) 

1810 

1811# **) MIT License 

1812# 

1813# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

1814# 

1815# Permission is hereby granted, free of charge, to any person obtaining a 

1816# copy of this software and associated documentation files (the "Software"), 

1817# to deal in the Software without restriction, including without limitation 

1818# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1819# and/or sell copies of the Software, and to permit persons to whom the 

1820# Software is furnished to do so, subject to the following conditions: 

1821# 

1822# The above copyright notice and this permission notice shall be included 

1823# in all copies or substantial portions of the Software. 

1824# 

1825# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1826# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1827# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1828# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1829# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1830# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1831# OTHER DEALINGS IN THE SOFTWARE.