Coverage for pygeodesy/formy.py: 99%

421 statements  

« prev     ^ index     » next       coverage.py v7.2.2, created at 2023-09-20 13:43 -0400

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 

10# from pygeodesy.cartesianBase import CartesianBase # _MODS 

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

12 _umod_PI2, float0_, isnon0, remainder, \ 

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

14 _4_0, _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 _xkwds, _xkwds_pop 

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

22from pygeodesy.fsums import fsumf_, isscalar 

23from pygeodesy.interns import NN, _delta_, _distant_, _SPACE_, _too_ 

24from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS 

25from pygeodesy.named import _NamedTuple, _xnamed, Fmt, unstr 

26from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, \ 

27 Intersection3Tuple, LatLon2Tuple, \ 

28 PhiLam2Tuple, Vector3Tuple 

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

30# from pygeodesy.triaxials import _hartzell3d2 # _MODS 

31from pygeodesy.units import Bearing, Degrees_, Distance, Distance_, Height, \ 

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

33 Radius, Radius_, Scalar, _100km 

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

35 tan_2, sincos2, sincos2_, sincos2d_, _Wrap 

36# from pygeodesy.vector3d import _otherV3d # _MODS 

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

38# sphericalNvector, sphericalTrigonometry # _MODS 

39 

40from contextlib import contextmanager 

41from math import asin, atan, atan2, cos, degrees, fabs, radians, sin, sqrt # pow 

42 

43__all__ = _ALL_LAZY.formy 

44__version__ = '23.09.06' 

45 

46_D2_R2 = (PI / _180_0)**2 # degrees- to radians-squared 

47_ratio_ = 'ratio' 

48_xline_ = 'xline' 

49 

50 

51def _anti2(a, b, n_2, n, n2): 

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

53 ''' 

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

55 if r == a: 

56 r = -r 

57 b += n 

58 if fabs(b) > n: 

59 b = remainder(b, n2) 

60 return float0_(r, b) 

61 

62 

63def antipode(lat, lon, name=NN): 

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

65 to a given point in C{degrees}. 

66 

67 @arg lat: Latitude (C{degrees}). 

68 @arg lon: Longitude (C{degrees}). 

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

70 

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

72 

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

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

75 ''' 

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

77 

78 

79def antipode_(phi, lam, name=NN): 

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

81 to a given point in C{radians}. 

82 

83 @arg phi: Latitude (C{radians}). 

84 @arg lam: Longitude (C{radians}). 

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

86 

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

88 

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

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

91 ''' 

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

93 

94 

95def bearing(lat1, lon1, lat2, lon2, **final_wrap): 

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

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

98 

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

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

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

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

103 @kwarg final_wrap: Optional keyword arguments for function 

104 L{pygeodesy.bearing_}. 

105 

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

107 zero if start and end point coincide. 

108 ''' 

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

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

111 return degrees(r) 

112 

113 

114def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False): 

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

116 between a (spherical) start and end point. 

117 

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

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

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

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

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

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

124 B{C{lam2}} (C{bool}). 

125 

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

127 and end point coincide. 

128 

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

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

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

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

133 ''' 

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

135 if final: # swap plus PI 

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

137 r = PI3 

138 else: 

139 r = PI2 

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

141 

142 x = ca1 * sa2 - sa1 * ca2 * cdb 

143 y = sdb * ca2 

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

145 

146 

147def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf 

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

149 ''' 

150 try: # for LatLon_ and ellipsoidal LatLon 

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

152 except AttributeError: 

153 pass 

154 # XXX spherical version, OK for ellipsoidal ispolar? 

155 a1, b1 = p1.philam 

156 a2, b2 = p2.philam 

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

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

159 name=_bearingTo2.__name__) 

160 

161 

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

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

164 lat2 - lat1)} between two points. 

165 

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

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

168 

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

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

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

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

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

174 mean latitude (C{bool}). 

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

176 and B{C{lon2}} (C{bool}). 

177 

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

179 

180 @note: Courtesy of Martin Schultz. 

181 

182 @see: U{Local, flat earth approximation 

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

184 ''' 

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

186 if adjust: # scale delta lon 

187 d_lon *= _scale_deg(lat1, lat2) 

188 return atan2b(d_lon, lat2 - lat1) 

189 

190 

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

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

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

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

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

196 

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

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

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

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

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

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

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

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

205 

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

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

208 

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

210 

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

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

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

214 L{Ellipsoid.distance2}. 

215 ''' 

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

217 

218 

219def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): 

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

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

222 admiralty-manual-of-navigation-vol-1-1964-english501c.pdf>} 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}, 

229 L{Ellipsoid2} or 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{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

237 L{flatPolar_}, 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 

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

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

262 formula. 

263 

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

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

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

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

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

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

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

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

272 

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

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

275 

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

277 

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

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

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

281 L{Ellipsoid.distance2}. 

282 ''' 

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

284 

285 

286def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): 

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

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

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

290 formula. 

291 

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

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

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

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

296 L{Ellipsoid2} or L{a_f2Tuple}). 

297 

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

299 

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

301 

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

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

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

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

306 Distance/ForsytheCorrection.php>}. 

307 ''' 

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

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

310 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_) 

311 if E.f: # ellipsoidal 

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

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

314 s = (s1 + s2)**2 / (1 + cr) 

315 t = (s1 - s2)**2 / (1 - cr) 

316 x = s + t 

317 y = s - t 

318 

319 s = 8 * r**2 / sr 

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

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

322 b = -2 * d 

323 e = 30 * s2r 

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

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

326 

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

328 return r 

329 

330 

331def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

334 formula. 

335 

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

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

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

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

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

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

342 L{a_f2Tuple}) to use. 

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

344 and B{C{lon2}} (C{bool}). 

345 

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

347 ellipsoid or datum axes). 

348 

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

350 

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

352 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean}, 

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

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

355 

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

357 ''' 

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

359 

360 

361def cosineLaw_(phi2, phi1, lam21): 

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

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

364 formula. 

365 

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

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

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

369 

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

371 

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

373 L{cosineForsytheAndoyerLambert_}, L{equirectangular_}, 

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

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

376 

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

378 ''' 

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

380 

381 

382def _d3(wrap, lat1, lon1, lat2, lon2): 

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

384 ''' 

385 if wrap: 

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

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

388 else: # for backward compaibility 

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

390 

391 

392def _dE(func_, earth, *wrap_lls): 

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

394 ''' 

395 E = _ellipsoidal(earth, func_) 

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

397 return r * E.a 

398 

399 

400def _dS(func_, radius, *wrap_lls, **adjust): 

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

402 ''' 

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

404 if radius is not R_M: 

405 _, lat1, _, lat2, _ = wrap_lls 

406 radius = _mean_radius(radius, lat1, lat2) 

407 return r * radius 

408 

409 

410def _eA(excess_, radius, *wrap_lls): 

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

412 ''' 

413 r = excess_(*_d3(*wrap_lls)) 

414 if radius: 

415 _, lat1, _, lat2, _ = wrap_lls 

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

417 return r 

418 

419 

420def _ellipsoidal(earth, where): 

421 '''(INTERNAL) Helper for distances. 

422 ''' 

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

424 earth if isinstance(earth, Ellipsoid) else 

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

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

427 

428 

429def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **adjust_limit_wrap): 

430 '''Compute the distance between two points using 

431 the U{Equirectangular Approximation / Projection 

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

433 

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

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

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

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

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

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

440 L{a_f2Tuple}). 

441 @kwarg adjust_limit_wrap: Optional keyword arguments for 

442 function L{equirectangular_}. 

443 

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

445 the ellipsoid or datum axes). 

446 

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

448 

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

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

451 approximate or accurate distance functions. 

452 ''' 

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

454 Lat(lat2=lat2), Lon(lon2=lon2), 

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

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

457 

458 

459def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap): 

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

461 and L{hausdorff._HausdorffMeterRedians} classes. 

462 ''' 

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

464 

465 

466def equirectangular_(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False): 

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

468 Approximation / Projection 

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

470 

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

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

473 

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

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

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

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

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

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

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

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

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

483 and B{C{lon2}} (C{bool}). 

484 

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

486 unroll_lon2)} in C{degrees squared}. 

487 

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

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

490 set to C{True}. 

491 

492 @see: U{Local, flat earth approximation 

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

494 L{equirectangular}, L{cosineAndoyerLambert}, 

495 L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean}, 

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

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

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

499 ''' 

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

501 d_lat = lat2 - lat1 

502 

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

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

505 if d > limit: 

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

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

508 limit=limit, wrap=wrap) 

509 raise LimitError(s, txt=t) 

510 

511 if adjust: # scale delta lon 

512 d_lon *= _scale_deg(lat1, lat2) 

513 

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

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

516 

517 

518def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False): 

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

520 

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

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

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

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

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

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

527 L{a_f2Tuple}) to use. 

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

529 the mean latitude (C{bool}). 

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

531 and B{C{lon2}} (C{bool}). 

532 

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

534 ellipsoid or datum axes). 

535 

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

537 

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

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

540 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

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

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

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

544 ''' 

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

546 

547 

548def euclidean_(phi2, phi1, lam21, adjust=True): 

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

550 (spherical) points. 

551 

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

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

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

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

556 of the mean latitude (C{bool}). 

557 

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

559 

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

561 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_}, 

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

563 and L{vincentys_}. 

564 ''' 

565 if adjust: 

566 lam21 *= _scale_rad(phi2, phi1) 

567 return euclid(phi2 - phi1, lam21) 

568 

569 

570def excessAbc_(A, b, c): 

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

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

573 

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

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

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

577 

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

579 

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

581 

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

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

584 ''' 

585 A = Radians_(A=A) 

586 b = Radians_(b=b) * _0_5 

587 c = Radians_(c=c) * _0_5 

588 

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

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

591 

592 

593def excessCagnoli_(a, b, c): 

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

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

596 

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

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

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

600 

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

602 

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

604 

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

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

607 ''' 

608 a = Radians_(a=a) 

609 b = Radians_(b=b) 

610 c = Radians_(c=c) 

611 

612 s = fsumf_(a, b, c) * _0_5 

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

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

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

616 return Radians(Cagnoli=r * _2_0) 

617 

618 

619def excessGirard_(A, B, C): 

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

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

622 formula. 

623 

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

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

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

627 

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

629 

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

631 

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

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

634 ''' 

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

636 Radians_(B=B), 

637 Radians_(C=C), -PI)) 

638 

639 

640def excessLHuilier_(a, b, c): 

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

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

643 Theorem. 

644 

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

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

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

648 

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

650 

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

652 

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

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

655 ''' 

656 a = Radians_(a=a) 

657 b = Radians_(b=b) 

658 c = Radians_(c=c) 

659 

660 s = fsumf_(a, b, c) * _0_5 

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

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

663 return Radians(LHuilier=r * _4_0) 

664 

665 

666def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

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

670 method. 

671 

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

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

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

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

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

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

678 L{a_f2Tuple}) or C{None}. 

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

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

681 

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

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

684 if C{B{radius}=0} or C{None}. 

685 

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

687 

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

689 

690 @raise ValueError: Semi-circular longitudinal delta. 

691 

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

693 ''' 

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

695 

696 

697def excessKarney_(phi2, phi1, lam21): 

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

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

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

701 method. 

702 

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

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

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

706 

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

708 

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

710 

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

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

713 ''' 

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

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

716 # 

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

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

719 # 1 + tan(φ1 / 2) · tan(φ2 / 2) 

720 # 

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

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

723 t2 = tan_2(phi2) 

724 t1 = tan_2(phi1) 

725 t = tan_2(lam21, lam21=None) 

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

727 _1_0 + (t1 * t2)) * _2_0) 

728 

729 

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

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

732# 

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

734# 

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

736# 

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

738# 

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

740# 

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

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

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

744# 

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

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

747# 

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

749# 

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

751# inverse Gudermannian) function 

752# 

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

754# 

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

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

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

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

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

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

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

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

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

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

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

766# - Charles Karney 

767 

768 

769def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

772 

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

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

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

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

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

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

779 L{a_f2Tuple}) or C{None}. 

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

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

782 

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

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

785 if C{B{radius}=0} or C{None}. 

786 

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

788 

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

790 

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

792 ''' 

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

794 

795 

796def excessQuad_(phi2, phi1, lam21): 

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

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

799 

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

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

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

803 

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

805 

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

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

808 ''' 

809 s = sin((phi2 + phi1) * _0_5) 

810 c = cos((phi2 - phi1) * _0_5) 

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

812 

813 

814def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False): 

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

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

817 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

819 

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

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

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

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

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

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

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

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

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

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

830 

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

832 ellipsoid axes). 

833 

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

835 

836 @note: The meridional and prime_vertical radii of curvature 

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

838 

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

840 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

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

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

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

844 ''' 

845 E = _ellipsoidal(datum, flatLocal) 

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

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

848 

849hubeny = flatLocal # PYCHOK for Karl Hubeny 

850 

851 

852def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True): 

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

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

855 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

857 

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

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

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

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

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

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

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

865 

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

867 

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

869 

870 @note: The meridional and prime_vertical radii of curvature 

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

872 

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

874 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_}, 

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

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

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

878 ''' 

879 E = _ellipsoidal(datum, flatLocal_) 

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

881 

882hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny 

883 

884 

885def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

888 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

889 formula. 

890 

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

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

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

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

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

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

897 L{a_f2Tuple}) to use. 

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

899 and B{C{lon2}} (C{bool}). 

900 

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

902 ellipsoid or datum axes). 

903 

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

905 

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

907 L{cosineForsytheAndoyerLambert},L{cosineLaw}, 

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

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

910 L{vincentys}. 

911 ''' 

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

913 

914 

915def flatPolar_(phi2, phi1, lam21): 

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

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

918 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

919 formula. 

920 

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

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

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

924 

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

926 

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

928 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

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

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

931 ''' 

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

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

934 if a < b: 

935 a, b = b, a 

936 if a < EPS0: 

937 a = _0_0 

938 elif b > 0: 

939 b = b / a # /= chokes PyChecker 

940 c = b * cos(lam21) * _2_0 

941 c = fsumf_(_1_0, b**2, -fabs(c)) 

942 a *= sqrt0(c) 

943 return a 

944 

945 

946def hartzell(pov, los=None, earth=_WGS84, name=NN, **LatLon_and_kwds): 

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

948 from a Point-Of-View in space. 

949 

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

951 or L{Vector3d}). 

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

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

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

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

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

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

958 point plus C{LatLon} keyword arguments, include 

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

960 

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

962 B{C{pov}} or B{C{LatLon}}). 

963 

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

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

966 the earth or points in an opposite direction. 

967 

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

969 

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

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

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

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

974 ''' 

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

976 _spherical_datum(earth, name=hartzell.__name__) 

977 try: 

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

979 except Exception as x: 

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

981 

982# else: 

983# E = D.ellipsoid 

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

985# 

986# def _Error(txt): 

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

988# 

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

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

991# q2 = E.b2_a2 # == c2 / a2 

992# bc = E.a * E.b # == b * c 

993# 

994# V3 = _MODS.vector3d._otherV3d 

995# p3 = V3(pov=pov) 

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

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

998# 

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

1000# ux, vy, wz = u3.times_(p3).xyz 

1001# u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz 

1002# 

1003# t = c2, c2, b2 

1004# m = fdot(t, u2, v2, w2) # a2 factored out 

1005# if m < EPS0: # zero or near-null LOS vector 

1006# raise _Error(_near_(_null_)) 

1007# 

1008# # a2 and b2 factored out, b2 == a2 and b2 / a2 == 1 

1009# r = fsumf_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2, 

1010# c2 * u2, -u2 * z2, -w2 * x2, ux * wz * 2, 

1011# -w2 * y2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2) 

1012# if r > 0: 

1013# r = sqrt(r) * bc # == a * a * b * c / a2 

1014# elif r < 0: # LOS pointing away from or missing the earth 

1015# raise _Error(_opposite_ if max(ux, vy, wz) > 0 else _outside_) 

1016# 

1017# d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out 

1018# if d > 0: # POV inside or LOS missing, outside the earth 

1019# s = fsumf_(_1_0, x2 / a2, y2 / b2, z2 / c2, _N_2_0) # like _sideOf 

1020# raise _Error(_outside_ if s > 0 else _inside_) 

1021# elif fsumf_(x2, y2, z2) < d**2: # d past earth center 

1022# raise _Error(_too_(_distant_)) 

1023# 

1024# r = p3.minus(u3.times(d)) 

1025# # h = p3.minus(r).length # distance to ellipsoid 

1026 

1027 r = _xnamed(r, name or hartzell.__name__) 

1028 if LatLon_and_kwds: 

1029 c = _MODS.cartesianBase.CartesianBase(r, datum=D, name=r.name) 

1030 r = c.toLatLon(**LatLon_and_kwds) 

1031 return r 

1032 

1033 

1034def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1035 '''Compute the distance between two (spherical) points using the 

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

1037 formula. 

1038 

1039 @arg lat1: Start latitude (C{degrees}). 

1040 @arg lon1: Start longitude (C{degrees}). 

1041 @arg lat2: End latitude (C{degrees}). 

1042 @arg lon2: End longitude (C{degrees}). 

1043 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1044 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1045 L{a_f2Tuple}) to use. 

1046 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1047 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1048 

1049 @return: Distance (C{meter}, same units as B{C{radius}}). 

1050 

1051 @raise TypeError: Invalid B{C{radius}}. 

1052 

1053 @see: U{Distance between two (spherical) points 

1054 <https://www.EdWilliams.org/avform.htm#Dist>}, functions 

1055 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1056 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1057 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, 

1058 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1059 

1060 @note: See note at function L{vincentys_}. 

1061 ''' 

1062 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2) 

1063 

1064 

1065def haversine_(phi2, phi1, lam21): 

1066 '''Compute the I{angular} distance between two (spherical) points 

1067 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} 

1068 formula. 

1069 

1070 @arg phi2: End latitude (C{radians}). 

1071 @arg phi1: Start latitude (C{radians}). 

1072 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1073 

1074 @return: Angular distance (C{radians}). 

1075 

1076 @see: Functions L{haversine}, L{cosineAndoyerLambert_}, 

1077 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1078 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1079 L{flatPolar_}, L{thomas_} and L{vincentys_}. 

1080 

1081 @note: See note at function L{vincentys_}. 

1082 ''' 

1083 def _hsin(rad): 

1084 return sin(rad * _0_5)**2 

1085 

1086 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine 

1087 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2 

1088 

1089 

1090def heightOf(angle, distance, radius=R_M): 

1091 '''Determine the height above the (spherical) earth' surface after 

1092 traveling along a straight line at a given tilt. 

1093 

1094 @arg angle: Tilt angle above horizontal (C{degrees}). 

1095 @arg distance: Distance along the line (C{meter} or same units as 

1096 B{C{radius}}). 

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

1098 

1099 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}). 

1100 

1101 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}. 

1102 

1103 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>} 

1104 (U{Shapiro et al. 2009, JTECH 

1105 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>} 

1106 and U{Potvin et al. 2012, JTECH 

1107 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}). 

1108 ''' 

1109 r = h = Radius(radius) 

1110 d = fabs(Distance(distance)) 

1111 if d > h: 

1112 d, h = h, d 

1113 

1114 if d > EPS0: # and h > EPS0 

1115 d = d / h # /= h chokes PyChecker 

1116 s = sin(Phi_(angle=angle, clip=_180_0)) 

1117 s = fsumf_(_1_0, _2_0 * s * d, d**2) 

1118 if s > 0: 

1119 return h * sqrt(s) - r 

1120 

1121 raise _ValueError(angle=angle, distance=distance, radius=radius) 

1122 

1123 

1124def heightOrthometric(h_ll, N): 

1125 '''Get the I{orthometric} height B{H}, the height above the geoid, earth surface. 

1126 

1127 @arg h_ll: The height above the ellipsoid (C{meter}) or an I{ellipsoidal} 

1128 location (C{LatLon} with a C{height} or C{h} attribute). 

1129 @arg N: The I{geoid} height (C{meter}), the height of the geoid above the 

1130 ellipsoid at the same B{C{h_ll}} location. 

1131 

1132 @return: I{Orthometric} height C{B{H} = B{h} - B{N}} (C{meter}, same units 

1133 as B{C{h}} and B{C{N}}). 

1134 

1135 @see: U{Ellipsoid, Geoid, and Othometric Heights<https://www.NGS.NOAA.gov/ 

1136 GEOID/PRESENTATIONS/2007_02_24_CCPS/Roman_A_PLSC2007notes.pdf>}, page 

1137 6 and module L{pygeodesy.geoids}. 

1138 ''' 

1139 h = h_ll if isscalar(h_ll) else _xattr(h_ll, height=_xattr(h_ll, h=0)) 

1140 return Height(H=Height(h=h) - Height(N=N)) 

1141 

1142 

1143def horizon(height, radius=R_M, refraction=False): 

1144 '''Determine the distance to the horizon from a given altitude 

1145 above the (spherical) earth. 

1146 

1147 @arg height: Altitude (C{meter} or same units as B{C{radius}}). 

1148 @kwarg radius: Optional mean earth radius (C{meter}). 

1149 @kwarg refraction: Consider atmospheric refraction (C{bool}). 

1150 

1151 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}). 

1152 

1153 @raise ValueError: Invalid B{C{height}} or B{C{radius}}. 

1154 

1155 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}. 

1156 ''' 

1157 h, r = Height(height), Radius(radius) 

1158 if min(h, r) < 0: 

1159 raise _ValueError(height=height, radius=radius) 

1160 

1161 if refraction: 

1162 d2 = 2.415750694528 * h * r # 2.0 / 0.8279 

1163 else: 

1164 d2 = h * fsumf_(r, r, h) 

1165 return sqrt0(d2) 

1166 

1167 

1168class _idllmn6(object): # see also .geodesicw._wargs, .vector2d._numpy 

1169 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}. 

1170 ''' 

1171 @contextmanager # <https://www.python.org/dev/peps/pep-0343/> Examples 

1172 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds): 

1173 try: 

1174 if wrap: 

1175 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap) 

1176 kwds = _xkwds(kwds, wrap=wrap) # for _xError 

1177 m = small if small is _100km else Meter_(small=small) 

1178 n = (intersections2 if s else intersection2).__name__ 

1179 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m: 

1180 d, m = None, _MODS.vector3d 

1181 _i = m._intersects2 if s else m._intersect3d3 

1182 elif isscalar(datum) and datum < 0 and not s: 

1183 d = _spherical_datum(-datum, name=n) 

1184 m = _MODS.sphericalNvector 

1185 _i = m.intersection 

1186 else: 

1187 d = _spherical_datum(datum, name=n) 

1188 if d.isSpherical: 

1189 m = _MODS.sphericalTrigonometry 

1190 _i = m._intersects2 if s else m._intersect 

1191 elif d.isEllipsoidal: 

1192 try: 

1193 if d.ellipsoid.geodesic: 

1194 pass 

1195 m = _MODS.ellipsoidalKarney 

1196 except ImportError: 

1197 m = _MODS.ellipsoidalExact 

1198 _i = m._intersections2 if s else m._intersection3 # ellispoidalBaseDI 

1199 else: 

1200 raise _TypeError(datum=datum) 

1201 yield _i, d, lat2, lon2, m, n 

1202 

1203 except (TypeError, ValueError) as x: 

1204 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum, 

1205 lat2=lat2, lon2=lon2, small=small, **kwds) 

1206 

1207_idllmn6 = _idllmn6() # PYCHOK singleton 

1208 

1209 

1210def intersection2(lat1, lon1, bearing1, 

1211 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True 

1212 '''I{Conveniently} compute the intersection of two lines each defined 

1213 by a (geodetic) point and a bearing from North, using either ... 

1214 

1215 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km 

1216 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1217 

1218 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}} 

1219 or a C{scalar B{datum}} representing the earth radius, conventionally 

1220 in C{meter} or ... 

1221 

1222 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative} 

1223 C{scalar}, (negative) earth radius, conventionally in C{meter} or ... 

1224 

1225 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}} 

1226 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1227 is installed, otherwise ... 

1228 

1229 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal. 

1230 

1231 @arg lat1: Latitude of the first point (C{degrees}). 

1232 @arg lon1: Longitude of the first point (C{degrees}). 

1233 @arg bearing1: Bearing at the first point (compass C{degrees}). 

1234 @arg lat2: Latitude of the second point (C{degrees}). 

1235 @arg lon2: Longitude of the second point (C{degrees}). 

1236 @arg bearing2: Bearing at the second point (compass C{degrees}). 

1237 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1238 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1239 radius (C{meter}, same units as B{C{radius1}} and 

1240 B{C{radius2}}) or C{None}. 

1241 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1242 and B{C{lon2}} (C{bool}). 

1243 @kwarg small: Upper limit for small distances (C{meter}). 

1244 

1245 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and 

1246 longitude of the intersection point. 

1247 

1248 @raise IntersectionError: Ambiguous or infinite intersection 

1249 or colinear, parallel or otherwise 

1250 non-intersecting lines. 

1251 

1252 @raise TypeError: Invalid B{C{datum}}. 

1253 

1254 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}}, 

1255 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}. 

1256 

1257 @see: Method L{RhumbLine.intersection2}. 

1258 

1259 @note: The returned intersections may be near-antipodal. 

1260 ''' 

1261 b1 = Bearing(bearing1=bearing1) 

1262 b2 = Bearing(bearing2=bearing2) 

1263 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1264 small, wrap, False, bearing1=b1, bearing2=b2) as t: 

1265 _i, d, lat2, lon2, m, n = t 

1266 if d is None: 

1267 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1, 

1268 m.Vector3d(lon2, lat2, 0), b2, useZ=False) 

1269 t = LatLon2Tuple(t.y, t.x, name=n) 

1270 

1271 else: 

1272 t = _i(m.LatLon(lat1, lon1, datum=d), b1, 

1273 m.LatLon(lat2, lon2, datum=d), b2, height=0, wrap=False) 

1274 if isinstance(t, Intersection3Tuple): # ellipsoidal 

1275 t, _, _ = t 

1276 t = LatLon2Tuple(t.lat, t.lon, name=n) 

1277 return t 

1278 

1279 

1280def intersections2(lat1, lon1, radius1, 

1281 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True 

1282 '''I{Conveniently} compute the intersections of two circles each defined 

1283 by a (geodetic) center point and a radius, using either ... 

1284 

1285 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km 

1286 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1287 

1288 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}} 

1289 or a C{scalar B{datum}} representing the earth radius, conventionally 

1290 in C{meter} or ... 

1291 

1292 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}} 

1293 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1294 is installed, otherwise ... 

1295 

1296 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal. 

1297 

1298 @arg lat1: Latitude of the first circle center (C{degrees}). 

1299 @arg lon1: Longitude of the first circle center (C{degrees}). 

1300 @arg radius1: Radius of the first circle (C{meter}, conventionally). 

1301 @arg lat2: Latitude of the second circle center (C{degrees}). 

1302 @arg lon2: Longitude of the second circle center (C{degrees}). 

1303 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}). 

1304 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1305 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1306 radius (C{meter}, same units as B{C{radius1}} and 

1307 B{C{radius2}}) or C{None}. 

1308 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1309 and B{C{lon2}} (C{bool}). 

1310 @kwarg small: Upper limit for small distances (C{meter}). 

1311 

1312 @return: 2-Tuple of the intersection points, each a 

1313 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the 

1314 points are the same instance, aka the I{radical center}. 

1315 

1316 @raise IntersectionError: Concentric, antipodal, invalid or 

1317 non-intersecting circles or no 

1318 convergence. 

1319 

1320 @raise TypeError: Invalid B{C{datum}}. 

1321 

1322 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}}, 

1323 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}. 

1324 ''' 

1325 r1 = Radius_(radius1=radius1) 

1326 r2 = Radius_(radius2=radius2) 

1327 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1328 small, wrap, True, radius1=r1, radius2=r2) as t: 

1329 _i, d, lat2, lon2, m, n = t 

1330 if d is None: 

1331 r1 = m2degrees(r1, radius=R_M, lat=lat1) 

1332 r2 = m2degrees(r2, radius=R_M, lat=lat2) 

1333 

1334 def _V2T(x, y, _, **unused): # _ == z unused 

1335 return LatLon2Tuple(y, x, name=n) 

1336 

1337 t = _i(m.Vector3d(lon1, lat1, 0), r1, 

1338 m.Vector3d(lon2, lat2, 0), r2, sphere=False, 

1339 Vector=_V2T) 

1340 else: 

1341 def _LL2T(lat, lon, **unused): 

1342 return LatLon2Tuple(lat, lon, name=n) 

1343 

1344 t = _i(m.LatLon(lat1, lon1, datum=d), r1, 

1345 m.LatLon(lat2, lon2, datum=d), r2, 

1346 LatLon=_LL2T, height=0, wrap=False) 

1347 return t 

1348 

1349 

1350def isantipode(lat1, lon1, lat2, lon2, eps=EPS): 

1351 '''Check whether two points are I{antipodal}, on diametrically 

1352 opposite sides of the earth. 

1353 

1354 @arg lat1: Latitude of one point (C{degrees}). 

1355 @arg lon1: Longitude of one point (C{degrees}). 

1356 @arg lat2: Latitude of the other point (C{degrees}). 

1357 @arg lon2: Longitude of the other point (C{degrees}). 

1358 @kwarg eps: Tolerance for near-equality (C{degrees}). 

1359 

1360 @return: C{True} if points are antipodal within the 

1361 B{C{eps}} tolerance, C{False} otherwise. 

1362 

1363 @see: Functions L{isantipode_} and L{antipode}. 

1364 ''' 

1365 return (fabs(lat1 + lat2) <= eps and 

1366 fabs(lon1 + lon2) <= eps) or _isequalTo( 

1367 normal(lat1, lon1), antipode(lat2, lon2), eps) 

1368 

1369 

1370def isantipode_(phi1, lam1, phi2, lam2, eps=EPS): 

1371 '''Check whether two points are I{antipodal}, on diametrically 

1372 opposite sides of the earth. 

1373 

1374 @arg phi1: Latitude of one point (C{radians}). 

1375 @arg lam1: Longitude of one point (C{radians}). 

1376 @arg phi2: Latitude of the other point (C{radians}). 

1377 @arg lam2: Longitude of the other point (C{radians}). 

1378 @kwarg eps: Tolerance for near-equality (C{radians}). 

1379 

1380 @return: C{True} if points are antipodal within the 

1381 B{C{eps}} tolerance, C{False} otherwise. 

1382 

1383 @see: Functions L{isantipode} and L{antipode_}. 

1384 ''' 

1385 return (fabs(phi1 + phi2) <= eps and 

1386 fabs(lam1 + lam2) <= eps) or _isequalTo_( 

1387 normal_(phi1, lam1), antipode_(phi2, lam2), eps) 

1388 

1389 

1390def _isequalTo(p1, p2, eps=EPS): 

1391 '''Compare 2 point lat-/lons ignoring C{class}. 

1392 ''' 

1393 return (fabs(p1.lat - p2.lat) <= eps and 

1394 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon) 

1395 

1396 

1397def _isequalTo_(p1, p2, eps=EPS): 

1398 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}. 

1399 ''' 

1400 return (fabs(p1.phi - p2.phi) <= eps and 

1401 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam) 

1402 

1403 

1404def isnormal(lat, lon, eps=0): 

1405 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their 

1406 respective I{normal} range in C{degrees}. 

1407 

1408 @arg lat: Latitude (C{degrees}). 

1409 @arg lon: Longitude (C{degrees}). 

1410 @kwarg eps: Optional tolerance C{degrees}). 

1411 

1412 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and 

1413 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise. 

1414 

1415 @see: Functions L{isnormal_} and L{normal}. 

1416 ''' 

1417 return (_90_0 - fabs(lat)) >= eps and _loneg(fabs(lon)) >= eps 

1418 

1419 

1420def isnormal_(phi, lam, eps=0): 

1421 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their 

1422 respective I{normal} range in C{radians}. 

1423 

1424 @arg phi: Latitude (C{radians}). 

1425 @arg lam: Longitude (C{radians}). 

1426 @kwarg eps: Optional tolerance C{radians}). 

1427 

1428 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and 

1429 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise. 

1430 

1431 @see: Functions L{isnormal} and L{normal_}. 

1432 ''' 

1433 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps 

1434 

1435 

1436def latlon2n_xyz(lat, lon, name=NN): 

1437 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1438 earth's surface) X, Y and Z components. 

1439 

1440 @arg lat: Latitude (C{degrees}). 

1441 @arg lon: Longitude (C{degrees}). 

1442 @kwarg name: Optional name (C{str}). 

1443 

1444 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1445 

1446 @see: Function L{philam2n_xyz}. 

1447 

1448 @note: These are C{n-vector} x, y and z components, 

1449 I{NOT} geocentric ECEF x, y and z coordinates! 

1450 ''' 

1451 return _2n_xyz(name, *sincos2d_(lat, lon)) 

1452 

1453 

1454def _normal2(a, b, n_2, n, n2): 

1455 '''(INTERNAL) Helper for C{normal} and C{normal_}. 

1456 ''' 

1457 if fabs(b) > n: 

1458 b = remainder(b, n2) 

1459 if fabs(a) > n_2: 

1460 r = remainder(a, n) 

1461 if r != a: 

1462 a = -r 

1463 b -= n if b > 0 else -n 

1464 return float0_(a, b) 

1465 

1466 

1467def normal(lat, lon, name=NN): 

1468 '''Normalize a lat- I{and} longitude pair in C{degrees}. 

1469 

1470 @arg lat: Latitude (C{degrees}). 

1471 @arg lon: Longitude (C{degrees}). 

1472 @kwarg name: Optional name (C{str}). 

1473 

1474 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90} 

1475 and C{abs(lon) <= 180}. 

1476 

1477 @see: Functions L{normal_} and L{isnormal}. 

1478 ''' 

1479 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0), 

1480 name=name or normal.__name__) 

1481 

1482 

1483def normal_(phi, lam, name=NN): 

1484 '''Normalize a lat- I{and} longitude pair in C{radians}. 

1485 

1486 @arg phi: Latitude (C{radians}). 

1487 @arg lam: Longitude (C{radians}). 

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

1489 

1490 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2} 

1491 and C{abs(lam) <= PI}. 

1492 

1493 @see: Functions L{normal} and L{isnormal_}. 

1494 ''' 

1495 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2), 

1496 name=name or normal_.__name__) 

1497 

1498 

1499def _2n_xyz(name, sa, ca, sb, cb): 

1500 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}. 

1501 ''' 

1502 # Kenneth Gade eqn 3, but using right-handed 

1503 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N 

1504 return Vector3Tuple(ca * cb, ca * sb, sa, name=name) 

1505 

1506 

1507def n_xyz2latlon(x, y, z, name=NN): 

1508 '''Convert C{n-vector} components to lat- and longitude in C{degrees}. 

1509 

1510 @arg x: X component (C{scalar}). 

1511 @arg y: Y component (C{scalar}). 

1512 @arg z: Z component (C{scalar}). 

1513 @kwarg name: Optional name (C{str}). 

1514 

1515 @return: A L{LatLon2Tuple}C{(lat, lon)}. 

1516 

1517 @see: Function L{n_xyz2philam}. 

1518 ''' 

1519 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name) 

1520 

1521 

1522def n_xyz2philam(x, y, z, name=NN): 

1523 '''Convert C{n-vector} components to lat- and longitude in C{radians}. 

1524 

1525 @arg x: X component (C{scalar}). 

1526 @arg y: Y component (C{scalar}). 

1527 @arg z: Z component (C{scalar}). 

1528 @kwarg name: Optional name (C{str}). 

1529 

1530 @return: A L{PhiLam2Tuple}C{(phi, lam)}. 

1531 

1532 @see: Function L{n_xyz2latlon}. 

1533 ''' 

1534 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name) 

1535 

1536 

1537def _opposes(d, m, n, n2): 

1538 '''(INTERNAL) Helper for C{opposing} and C{opposing_}. 

1539 ''' 

1540 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1 

1541 return False if d < m or d > (n2 - m) else ( 

1542 True if (n - m) < d < (n + m) else None) 

1543 

1544 

1545def opposing(bearing1, bearing2, margin=_90_0): 

1546 '''Compare the direction of two bearings given in C{degrees}. 

1547 

1548 @arg bearing1: First bearing (compass C{degrees}). 

1549 @arg bearing2: Second bearing (compass C{degrees}). 

1550 @kwarg margin: Optional, interior angle bracket (C{degrees}). 

1551 

1552 @return: C{True} if both bearings point in opposite, C{False} if 

1553 in similar or C{None} if in I{perpendicular} directions. 

1554 

1555 @see: Function L{opposing_}. 

1556 ''' 

1557 m = Degrees_(margin=margin, low=EPS0, high=_90_0) 

1558 return _opposes(bearing2 - bearing1, m, _180_0, _360_0) 

1559 

1560 

1561def opposing_(radians1, radians2, margin=PI_2): 

1562 '''Compare the direction of two bearings given in C{radians}. 

1563 

1564 @arg radians1: First bearing (C{radians}). 

1565 @arg radians2: Second bearing (C{radians}). 

1566 @kwarg margin: Optional, interior angle bracket (C{radians}). 

1567 

1568 @return: C{True} if both bearings point in opposite, C{False} if 

1569 in similar or C{None} if in perpendicular directions. 

1570 

1571 @see: Function L{opposing}. 

1572 ''' 

1573 m = Radians_(margin=margin, low=EPS0, high=PI_2) 

1574 return _opposes(radians2 - radians1, m, PI, PI2) 

1575 

1576 

1577def philam2n_xyz(phi, lam, name=NN): 

1578 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1579 earth's surface) X, Y and Z components. 

1580 

1581 @arg phi: Latitude (C{radians}). 

1582 @arg lam: Longitude (C{radians}). 

1583 @kwarg name: Optional name (C{str}). 

1584 

1585 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1586 

1587 @see: Function L{latlon2n_xyz}. 

1588 

1589 @note: These are C{n-vector} x, y and z components, 

1590 I{NOT} geocentric ECEF x, y and z coordinates! 

1591 ''' 

1592 return _2n_xyz(name, *sincos2_(phi, lam)) 

1593 

1594 

1595def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d 

1596 # (INTERNAL) See C{radical2} below 

1597 # assert d > EPS0 

1598 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5 

1599 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d) 

1600 

1601 

1602def radical2(distance, radius1, radius2): 

1603 '''Compute the I{radical ratio} and I{radical line} of two 

1604 U{intersecting circles<https://MathWorld.Wolfram.com/ 

1605 Circle-CircleIntersection.html>}. 

1606 

1607 The I{radical line} is perpendicular to the axis thru the 

1608 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}. 

1609 

1610 @arg distance: Distance between the circle centers (C{scalar}). 

1611 @arg radius1: Radius of the first circle (C{scalar}). 

1612 @arg radius2: Radius of the second circle (C{scalar}). 

1613 

1614 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <= 

1615 ratio <= 1.0} and C{xline} is along the B{C{distance}}. 

1616 

1617 @raise IntersectionError: The B{C{distance}} exceeds the sum 

1618 of B{C{radius1}} and B{C{radius2}}. 

1619 

1620 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or 

1621 B{C{radius2}}. 

1622 

1623 @see: U{Circle-Circle Intersection 

1624 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}. 

1625 ''' 

1626 d = Distance_(distance, low=_0_0) 

1627 r1 = Radius_(radius1=radius1) 

1628 r2 = Radius_(radius2=radius2) 

1629 if d > (r1 + r2): 

1630 raise IntersectionError(distance=d, radius1=r1, radius2=r2, 

1631 txt=_too_(_distant_)) 

1632 return _radical2(d, r1, r2) if d > EPS0 else \ 

1633 Radical2Tuple(_0_5, _0_0) 

1634 

1635 

1636class Radical2Tuple(_NamedTuple): 

1637 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and 

1638 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0} 

1639 ''' 

1640 _Names_ = (_ratio_, _xline_) 

1641 _Units_ = ( Scalar, Scalar) 

1642 

1643 

1644def _radistance(inst): 

1645 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians} 

1646 and L{hausdorff._HausdorffMeterRedians} classes. 

1647 ''' 

1648 kwds_ = _xkwds(inst._kwds, wrap=False) 

1649 wrap_ = _xkwds_pop(kwds_, wrap=False) 

1650 func_ = inst._func_ 

1651 try: # calling lower-overhead C{func_} 

1652 func_(0, _0_25, _0_5, **kwds_) 

1653 wrap_ = _Wrap._philamop(wrap_) 

1654 except TypeError: 

1655 return inst.distance 

1656 

1657 def _philam(p): 

1658 try: 

1659 return p.phi, p.lam 

1660 except AttributeError: # no .phi or .lam 

1661 return radians(p.lat), radians(p.lon) 

1662 

1663 def _func_wrap(point1, point2): 

1664 phi1, lam1 = wrap_(*_philam(point1)) 

1665 phi2, lam2 = wrap_(*_philam(point2)) 

1666 return func_(phi2, phi1, lam2 - lam1, **kwds_) 

1667 

1668 inst._units = inst._units_ 

1669 return _func_wrap 

1670 

1671 

1672def _scale_deg(lat1, lat2): # degrees 

1673 # scale factor cos(mean of lats) for delta lon 

1674 m = fabs(lat1 + lat2) * _0_5 

1675 return cos(radians(m)) if m < 90 else _0_0 

1676 

1677 

1678def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights 

1679 # scale factor cos(mean of phis) for delta lam 

1680 m = fabs(phi1 + phi2) * _0_5 

1681 return cos(m) if m < PI_2 else _0_0 

1682 

1683 

1684def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw 

1685 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine. 

1686 ''' 

1687 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21) 

1688 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21 

1689 

1690 

1691def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

1692 '''Compute the distance between two (ellipsoidal) points using 

1693 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1694 formula. 

1695 

1696 @arg lat1: Start latitude (C{degrees}). 

1697 @arg lon1: Start longitude (C{degrees}). 

1698 @arg lat2: End latitude (C{degrees}). 

1699 @arg lon2: End longitude (C{degrees}). 

1700 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1701 L{Ellipsoid2} or L{a_f2Tuple}) to use. 

1702 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1703 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1704 

1705 @return: Distance (C{meter}, same units as the B{C{datum}}'s 

1706 ellipsoid axes). 

1707 

1708 @raise TypeError: Invalid B{C{datum}}. 

1709 

1710 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1711 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, 

1712 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}. 

1713 ''' 

1714 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2) 

1715 

1716 

1717def thomas_(phi2, phi1, lam21, datum=_WGS84): 

1718 '''Compute the I{angular} distance between two (ellipsoidal) points using 

1719 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1720 formula. 

1721 

1722 @arg phi2: End latitude (C{radians}). 

1723 @arg phi1: Start latitude (C{radians}). 

1724 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1725 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, 

1726 L{Ellipsoid2} or L{a_f2Tuple}). 

1727 

1728 @return: Angular distance (C{radians}). 

1729 

1730 @raise TypeError: Invalid B{C{datum}}. 

1731 

1732 @see: Functions L{thomas}, L{cosineAndoyerLambert_}, 

1733 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1734 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1735 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP 

1736 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/ 

1737 Distance/ThomasFormula.php>}. 

1738 ''' 

1739 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21) 

1740 if r and isnon0(c1) and isnon0(c2): 

1741 E = _ellipsoidal(datum, thomas_) 

1742 if E.f: 

1743 r1 = atan2(E.b_a * s1, c1) 

1744 r2 = atan2(E.b_a * s2, c2) 

1745 

1746 j = (r2 + r1) * _0_5 

1747 k = (r2 - r1) * _0_5 

1748 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5) 

1749 

1750 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2) 

1751 u = _1_0 - h 

1752 if isnon0(u) and isnon0(h): 

1753 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h) 

1754 sr, cr = sincos2(r) 

1755 if isnon0(sr): 

1756 u = 2 * (sj * ck)**2 / u 

1757 h = 2 * (sk * cj)**2 / h 

1758 x = u + h 

1759 y = u - h 

1760 

1761 s = r / sr 

1762 e = 4 * s**2 

1763 d = 2 * cr 

1764 a = e * d 

1765 b = 2 * r 

1766 c = s - (a - d) * _0_5 

1767 f = E.f * _0_25 

1768 

1769 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y) 

1770 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr 

1771 return r 

1772 

1773 

1774def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1775 '''Compute the distance between two (spherical) points using 

1776 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1777 spherical formula. 

1778 

1779 @arg lat1: Start latitude (C{degrees}). 

1780 @arg lon1: Start longitude (C{degrees}). 

1781 @arg lat2: End latitude (C{degrees}). 

1782 @arg lon2: End longitude (C{degrees}). 

1783 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1784 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1785 L{a_f2Tuple}) to use. 

1786 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1787 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1788 

1789 @return: Distance (C{meter}, same units as B{C{radius}}). 

1790 

1791 @raise UnitError: Invalid B{C{radius}}. 

1792 

1793 @see: Functions L{vincentys_}, L{cosineAndoyerLambert}, 

1794 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular}, 

1795 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1796 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2}, 

1797 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1798 

1799 @note: See note at function L{vincentys_}. 

1800 ''' 

1801 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2) 

1802 

1803 

1804def vincentys_(phi2, phi1, lam21): 

1805 '''Compute the I{angular} distance between two (spherical) points using 

1806 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1807 spherical formula. 

1808 

1809 @arg phi2: End latitude (C{radians}). 

1810 @arg phi1: Start latitude (C{radians}). 

1811 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1812 

1813 @return: Angular distance (C{radians}). 

1814 

1815 @see: Functions L{vincentys}, L{cosineAndoyerLambert_}, 

1816 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1817 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1818 L{flatPolar_}, L{haversine_} and L{thomas_}. 

1819 

1820 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_} 

1821 produce equivalent results, but L{vincentys_} is suitable 

1822 for antipodal points and slightly more expensive (M{3 cos, 

1823 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_} 

1824 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and 

1825 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}). 

1826 ''' 

1827 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21) 

1828 

1829 c = c2 * c21 

1830 x = s1 * s2 + c1 * c 

1831 y = c1 * s2 - s1 * c 

1832 return atan2(hypot(c2 * s21, y), x) 

1833 

1834# **) MIT License 

1835# 

1836# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved. 

1837# 

1838# Permission is hereby granted, free of charge, to any person obtaining a 

1839# copy of this software and associated documentation files (the "Software"), 

1840# to deal in the Software without restriction, including without limitation 

1841# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1842# and/or sell copies of the Software, and to permit persons to whom the 

1843# Software is furnished to do so, subject to the following conditions: 

1844# 

1845# The above copyright notice and this permission notice shall be included 

1846# in all copies or substantial portions of the Software. 

1847# 

1848# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1849# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1850# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1851# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1852# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1853# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1854# OTHER DEALINGS IN THE SOFTWARE.