Coverage for pygeodesy/formy.py: 98%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''Formulary of basic geodesy functions and approximations. 

5''' 

6# make sure int/int division yields float quotient, see .basics 

7from __future__ import division as _; del _ # PYCHOK semicolon 

8 

9# from pygeodesy.cartesianBase import CartesianBase # _MODS 

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

11 _umod_PI2, float0_, isnon0, remainder, \ 

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

13 _32_0, _90_0, _180_0, _360_0 

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

15 _mean_radius, _spherical_datum, _WGS84, _EWGS84 

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

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

18 _TypeError, _ValueError, _xattr, _xError, \ 

19 _xkwds, _xkwds_pop2 

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

21from pygeodesy.fsums import fsumf_ 

22from pygeodesy.interns import NN, _delta_, _distant_, _inside_, _SPACE_, _too_ 

23from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS 

24from pygeodesy.named import _NamedTuple, _xnamed, Fmt, unstr 

25from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, Intersection3Tuple, \ 

26 LatLon2Tuple, PhiLam2Tuple, Vector3Tuple 

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

28# from pygeodesy.triaxials import _hartzell2 # _MODS 

29from pygeodesy.units import _isHeight, _isRadius, Bearing, Degrees_, Distance, \ 

30 Distance_, Height, Lam_, Lat, Lon, Meter_, Phi_, \ 

31 Radians, Radians_, Radius, Radius_, Scalar, _100km 

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

33 tan_2, sincos2, sincos2_, sincos2d_, _Wrap 

34# from pygeodesy.vector3d import _otherV3d # _MODS 

35# from pygeodesy.vector3dBase import _xyz_y_z3 # _MODS 

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

37# sphericalNvector, sphericalTrigonometry # _MODS 

38 

39from contextlib import contextmanager 

40from math import asin, atan, atan2, cos, degrees, fabs, radians, sin, sqrt # pow 

41 

42__all__ = _ALL_LAZY.formy 

43__version__ = '24.02.18' 

44 

45_RADIANS2 = (PI / _180_0)**2 # degrees- to radians-squared 

46_ratio_ = 'ratio' 

47_xline_ = 'xline' 

48 

49 

50def _anti2(a, b, n_2, n, n2): 

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

52 ''' 

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

54 if r == a: 

55 r = -r 

56 b += n 

57 if fabs(b) > n: 

58 b = remainder(b, n2) 

59 return float0_(r, b) 

60 

61 

62def antipode(lat, lon, name=NN): 

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

64 to a given point in C{degrees}. 

65 

66 @arg lat: Latitude (C{degrees}). 

67 @arg lon: Longitude (C{degrees}). 

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

69 

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

71 

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

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

74 ''' 

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

76 

77 

78def antipode_(phi, lam, name=NN): 

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

80 to a given point in C{radians}. 

81 

82 @arg phi: Latitude (C{radians}). 

83 @arg lam: Longitude (C{radians}). 

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

85 

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

87 

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

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

90 ''' 

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

92 

93 

94def bearing(lat1, lon1, lat2, lon2, **final_wrap): 

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

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

97 

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

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

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

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

102 @kwarg final_wrap: Optional keyword arguments for function 

103 L{pygeodesy.bearing_}. 

104 

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

106 zero if start and end point coincide. 

107 ''' 

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

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

110 return degrees(r) 

111 

112 

113def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False): 

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

115 between a (spherical) start and end point. 

116 

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

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

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

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

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

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

123 B{C{lam2}} (C{bool}). 

124 

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

126 and end point coincide. 

127 

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

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

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

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

132 ''' 

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

134 if final: # swap plus PI 

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

136 r = PI3 

137 else: 

138 r = PI2 

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

140 

141 x = ca1 * sa2 - sa1 * ca2 * cdb 

142 y = sdb * ca2 

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

144 

145 

146def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf 

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

148 ''' 

149 try: # for LatLon_ and ellipsoidal LatLon 

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

151 except AttributeError: 

152 pass 

153 # XXX spherical version, OK for ellipsoidal ispolar? 

154 t = p1.philam + p2.philam 

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

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

157 name=_bearingTo2.__name__) 

158 

159 

160def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False): 

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

162 lat2 - lat1)} between two points. 

163 

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

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

166 

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

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

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

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

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

172 mean latitude (C{bool}). 

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

174 and B{C{lon2}} (C{bool}). 

175 

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

177 

178 @note: Courtesy of Martin Schultz. 

179 

180 @see: U{Local, flat earth approximation 

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

182 ''' 

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

184 if adjust: # scale delta lon 

185 d_lon *= _scale_deg(lat1, lat2) 

186 return atan2b(d_lon, lat2 - lat1) 

187 

188 

189def cosineAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

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

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

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

193 

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

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

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

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

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

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

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

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

202 

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

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

205 

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

207 

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

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

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

211 L{Ellipsoid.distance2}. 

212 ''' 

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

214 

215 

216def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): 

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

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

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

220 

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

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

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

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

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

226 

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

228 

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

230 

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

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

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

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

235 AndoyerLambert.php>}. 

236 ''' 

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

238 if isnon0(c1) and isnon0(c2): 

239 E = _ellipsoidal(datum, cosineAndoyerLambert_) 

240 if E.f: # ellipsoidal 

241 r2 = atan2(E.b_a * s2, c2) 

242 r1 = atan2(E.b_a * s1, c1) 

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

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

245 if r: 

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

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

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

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

250 r += (c - s) * E.f * _0_125 

251 return r 

252 

253 

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

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

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

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

258 

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

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

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

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

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

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

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

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

267 

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

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

270 

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

272 

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

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

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

276 L{Ellipsoid.distance2}. 

277 ''' 

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

279 

280 

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

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

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

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

285 formula. 

286 

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

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

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

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

291 L{Ellipsoid2} or L{a_f2Tuple}). 

292 

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

294 

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

296 

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

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

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

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

301 Distance/ForsytheCorrection.php>}. 

302 ''' 

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

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

305 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_) 

306 if E.f: # ellipsoidal 

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

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

309 s = (s1 + s2)**2 / (1 + cr) 

310 t = (s1 - s2)**2 / (1 - cr) 

311 x = s + t 

312 y = s - t 

313 

314 s = 8 * r**2 / sr 

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

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

317 b = -2 * d 

318 e = 30 * s2r 

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

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

321 

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

323 return r 

324 

325 

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

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

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

329 

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

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

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

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

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

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

336 L{a_f2Tuple}) to use. 

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

338 and B{C{lon2}} (C{bool}). 

339 

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

341 ellipsoid or datum axes). 

342 

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

344 

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

346 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean}, 

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

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

349 

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

351 ''' 

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

353 

354 

355def cosineLaw_(phi2, phi1, lam21): 

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

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

358 

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

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

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

362 

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

364 

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

366 L{cosineForsytheAndoyerLambert_}, L{equirectangular_}, 

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

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

369 

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

371 ''' 

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

373 

374 

375def _d3(wrap, lat1, lon1, lat2, lon2): 

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

377 ''' 

378 if wrap: 

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

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

381 else: # for backward compaibility 

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

383 

384 

385def _dE(func_, earth, *wrap_lls): 

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

387 ''' 

388 E = _ellipsoidal(earth, func_) 

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

390 return r * E.a 

391 

392 

393def _dS(func_, radius, *wrap_lls, **adjust): 

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

395 ''' 

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

397 if radius is not R_M: 

398 _, lat1, _, lat2, _ = wrap_lls 

399 radius = _mean_radius(radius, lat1, lat2) 

400 return r * radius 

401 

402 

403def _eA(excess_, radius, *wrap_lls): 

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

405 ''' 

406 r = excess_(*_d3(*wrap_lls)) 

407 if radius: 

408 _, lat1, _, lat2, _ = wrap_lls 

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

410 return r 

411 

412 

413def _ellipsoidal(earth, where): 

414 '''(INTERNAL) Helper for distances. 

415 ''' 

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

417 earth if isinstance(earth, Ellipsoid) else 

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

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

420 

421 

422def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **adjust_limit_wrap): 

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

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

425 

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

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

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

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

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

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

432 L{a_f2Tuple}). 

433 @kwarg adjust_limit_wrap: Optional keyword arguments for 

434 function L{equirectangular_}. 

435 

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

437 the ellipsoid or datum axes). 

438 

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

440 

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

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

443 approximate or accurate distance functions. 

444 ''' 

445 d = sqrt(equirectangular_(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._FrecherMeterRadians} 

453 and L{hausdorff._HausdorffMeterRedians} classes. 

454 ''' 

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

456 

457 

458def equirectangular_(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 

463 or Miles, see 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 

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

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

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

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

474 and B{C{lon2}} (C{bool}). 

475 

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

477 unroll_lon2)} in C{degrees squared}. 

478 

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

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

481 set to C{True}. 

482 

483 @see: U{Local, flat earth approximation 

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

485 L{equirectangular}, L{cosineAndoyerLambert}, 

486 L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean}, 

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

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

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

490 ''' 

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

492 d_lat = lat2 - lat1 

493 

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

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

496 if d > limit: 

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

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

499 limit=limit, wrap=wrap) 

500 raise LimitError(s, txt=t) 

501 

502 if adjust: # scale delta lon 

503 d_lon *= _scale_deg(lat1, lat2) 

504 

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

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

507 

508 

509def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False): 

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

511 

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

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

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

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

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

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

518 L{a_f2Tuple}) to use. 

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

520 the mean latitude (C{bool}). 

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

522 and B{C{lon2}} (C{bool}). 

523 

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

525 ellipsoid or datum axes). 

526 

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

528 

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

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

531 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

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

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

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

535 ''' 

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

537 

538 

539def euclidean_(phi2, phi1, lam21, adjust=True): 

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

541 

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

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

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

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

546 of the mean latitude (C{bool}). 

547 

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

549 

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

551 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_}, 

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

553 and L{vincentys_}. 

554 ''' 

555 if adjust: 

556 lam21 *= _scale_rad(phi2, phi1) 

557 return euclid(phi2 - phi1, lam21) 

558 

559 

560def excessAbc_(A, b, c): 

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

562 and the included (small) angle. 

563 

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

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

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

567 

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

569 

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

571 

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

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

574 ''' 

575 A = Radians_(A=A) 

576 b = Radians_(b=b) * _0_5 

577 c = Radians_(c=c) * _0_5 

578 

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

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

581 

582 

583def excessCagnoli_(a, b, c): 

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

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

586 

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

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

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

590 

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

592 

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

594 

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

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

597 ''' 

598 a = Radians_(a=a) 

599 b = Radians_(b=b) 

600 c = Radians_(c=c) 

601 

602 s = fsumf_(a, b, c) * _0_5 

603 _s = sin 

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

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

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

607 return Radians(Cagnoli=r * _2_0) 

608 

609 

610def excessGirard_(A, B, C): 

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

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

613 

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

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

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

617 

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

619 

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

621 

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

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

624 ''' 

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

626 Radians_(B=B), 

627 Radians_(C=C), -PI)) 

628 

629 

630def excessLHuilier_(a, b, c): 

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

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

633 

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

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

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

637 

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

639 

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

641 

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

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

644 ''' 

645 a = Radians_(a=a) 

646 b = Radians_(b=b) 

647 c = Radians_(c=c) 

648 

649 s = fsumf_(a, b, c) * _0_5 

650 _t = tan_2 

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

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

653 return Radians(LHuilier=r * _4_0) 

654 

655 

656def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

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

660 method. 

661 

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

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

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

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

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

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

668 L{a_f2Tuple}) or C{None}. 

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

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

671 

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

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

674 if C{B{radius}=0} or C{None}. 

675 

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

677 

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

679 

680 @raise ValueError: Semi-circular longitudinal delta. 

681 

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

683 ''' 

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

685 

686 

687def excessKarney_(phi2, phi1, lam21): 

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

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

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

691 method. 

692 

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

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

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

696 

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

698 

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

700 

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

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

703 ''' 

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

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

706 # 

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

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

709 # 1 + tan(φ1 / 2) · tan(φ2 / 2) 

710 # 

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

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

713 _t = tan_2 

714 t2 = _t(phi2) 

715 t1 = _t(phi1) 

716 t = _t(lam21, lam21=None) 

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

718 _1_0 + (t1 * t2)) * _2_0) 

719 

720 

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

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

723# 

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

725# 

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

727# 

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

729# 

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

731# 

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

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

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

735# 

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

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

738# 

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

740# 

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

742# inverse Gudermannian) function 

743# 

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

745# 

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

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

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

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

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

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

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

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

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

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

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

757# - Charles Karney 

758 

759 

760def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

763 

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

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

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

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

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

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

770 L{a_f2Tuple}) or C{None}. 

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

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

773 

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

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

776 if C{B{radius}=0} or C{None}. 

777 

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

779 

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

781 

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

783 ''' 

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

785 

786 

787def excessQuad_(phi2, phi1, lam21): 

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

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

790 

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

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

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

794 

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

796 

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

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

799 ''' 

800 s = sin((phi2 + phi1) * _0_5) 

801 c = cos((phi2 - phi1) * _0_5) 

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

803 

804 

805def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False): 

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

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

808 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

810 

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

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

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

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

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

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

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

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

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

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

821 

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

823 ellipsoid axes). 

824 

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

826 

827 @note: The meridional and prime_vertical radii of curvature 

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

829 

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

831 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

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

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

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

835 ''' 

836 E = _ellipsoidal(datum, flatLocal) 

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

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

839 

840hubeny = flatLocal # PYCHOK for Karl Hubeny 

841 

842 

843def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True): 

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

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

846 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

848 

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

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

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

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

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

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

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

856 

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

858 

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

860 

861 @note: The meridional and prime_vertical radii of curvature 

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

863 

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

865 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_}, 

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

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

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

869 ''' 

870 E = _ellipsoidal(datum, flatLocal_) 

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

872 

873hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny 

874 

875 

876def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

879 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

880 formula. 

881 

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

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

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

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

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

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

888 L{a_f2Tuple}) to use. 

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

890 and B{C{lon2}} (C{bool}). 

891 

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

893 ellipsoid or datum axes). 

894 

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

896 

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

898 L{cosineForsytheAndoyerLambert},L{cosineLaw}, 

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

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

901 L{vincentys}. 

902 ''' 

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

904 

905 

906def flatPolar_(phi2, phi1, lam21): 

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

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

909 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

910 formula. 

911 

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

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

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

915 

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

917 

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

919 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

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

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

922 ''' 

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

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

925 if a < b: 

926 a, b = b, a 

927 if a < EPS0: 

928 a = _0_0 

929 elif b > 0: 

930 b = b / a # /= chokes PyChecker 

931 c = b * cos(lam21) * _2_0 

932 c = fsumf_(_1_0, b**2, -fabs(c)) 

933 a *= sqrt0(c) 

934 return a 

935 

936 

937def _hartzell(pov, los, earth, **kwds): 

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

939 ''' 

940 if earth is None: 

941 earth = pov.datum 

942 else: 

943 earth = _spherical_datum(earth, name=hartzell.__name__) 

944 pov = pov.toDatum(earth) 

945 h = pov.height 

946 if h < 0: # EPS0 

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

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

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

950 

951 

952def hartzell(pov, los=False, earth=_WGS84, name=NN, **LatLon_and_kwds): 

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

954 a Point-Of-View in space. 

955 

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

957 L{Ecef9Tuple} or L{Vector3d}). 

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

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

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

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

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

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

964 @kwarg LatLon_and_kwds: Optional C{B{LatLon}=None} class to return the 

965 intersection plus additional C{LatLon} keyword 

966 arguments, include B{C{datum}} if different 

967 from B{C{earth}}. 

968 

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

970 the given B{C{LatLon}} instance) with attribute C{heigth} set 

971 to the distance to the B{C{pov}}. 

972 

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

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

975 away from the earth. 

976 

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

978 

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

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

981 ''' 

982 n = hartzell.__name__ 

983 D = earth if isinstance(earth, Datum) else _spherical_datum(earth, name=n) 

984 try: 

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

986 

987 r = _xnamed(r, name or n) 

988 C = _MODS.cartesianBase.CartesianBase 

989 if LatLon_and_kwds: 

990 c = C(r, datum=D, name=r.name) 

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

992 elif isinstance(r, C): 

993 r.height = h 

994 if i: 

995 r._iteration = i 

996 except Exception as x: 

997 raise IntersectionError(pov=pov, los=los, earth=earth, cause=x, **LatLon_and_kwds) 

998 return r 

999 

1000 

1001def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1002 '''Compute the distance between two (spherical) points using the 

1003 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} 

1004 formula. 

1005 

1006 @arg lat1: Start latitude (C{degrees}). 

1007 @arg lon1: Start longitude (C{degrees}). 

1008 @arg lat2: End latitude (C{degrees}). 

1009 @arg lon2: End longitude (C{degrees}). 

1010 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1011 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1012 L{a_f2Tuple}) to use. 

1013 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1014 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1015 

1016 @return: Distance (C{meter}, same units as B{C{radius}}). 

1017 

1018 @raise TypeError: Invalid B{C{radius}}. 

1019 

1020 @see: U{Distance between two (spherical) points 

1021 <https://www.EdWilliams.org/avform.htm#Dist>}, functions 

1022 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1023 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1024 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, 

1025 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1026 

1027 @note: See note at function L{vincentys_}. 

1028 ''' 

1029 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2) 

1030 

1031 

1032def haversine_(phi2, phi1, lam21): 

1033 '''Compute the I{angular} distance between two (spherical) points 

1034 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} 

1035 formula. 

1036 

1037 @arg phi2: End latitude (C{radians}). 

1038 @arg phi1: Start latitude (C{radians}). 

1039 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1040 

1041 @return: Angular distance (C{radians}). 

1042 

1043 @see: Functions L{haversine}, L{cosineAndoyerLambert_}, 

1044 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1045 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1046 L{flatPolar_}, L{thomas_} and L{vincentys_}. 

1047 

1048 @note: See note at function L{vincentys_}. 

1049 ''' 

1050 def _hsin(rad): 

1051 return sin(rad * _0_5)**2 

1052 

1053 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine 

1054 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2 

1055 

1056 

1057def heightOf(angle, distance, radius=R_M): 

1058 '''Determine the height above the (spherical) earth' surface after 

1059 traveling along a straight line at a given tilt. 

1060 

1061 @arg angle: Tilt angle above horizontal (C{degrees}). 

1062 @arg distance: Distance along the line (C{meter} or same units as 

1063 B{C{radius}}). 

1064 @kwarg radius: Optional mean earth radius (C{meter}). 

1065 

1066 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}). 

1067 

1068 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}. 

1069 

1070 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>} 

1071 (U{Shapiro et al. 2009, JTECH 

1072 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>} 

1073 and U{Potvin et al. 2012, JTECH 

1074 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}). 

1075 ''' 

1076 r = h = Radius(radius) 

1077 d = fabs(Distance(distance)) 

1078 if d > h: 

1079 d, h = h, d 

1080 

1081 if d > EPS0: # and h > EPS0 

1082 d = d / h # /= h chokes PyChecker 

1083 s = sin(Phi_(angle=angle, clip=_180_0)) 

1084 s = fsumf_(_1_0, s * d * _2_0, d**2) 

1085 if s > 0: 

1086 return h * sqrt(s) - r 

1087 

1088 raise _ValueError(angle=angle, distance=distance, radius=radius) 

1089 

1090 

1091def heightOrthometric(h_ll, N): 

1092 '''Get the I{orthometric} height B{H}, the height above the geoid, earth surface. 

1093 

1094 @arg h_ll: The height above the ellipsoid (C{meter}) or an I{ellipsoidal} 

1095 location (C{LatLon} with a C{height} or C{h} attribute). 

1096 @arg N: The I{geoid} height (C{meter}), the height of the geoid above the 

1097 ellipsoid at the same B{C{h_ll}} location. 

1098 

1099 @return: I{Orthometric} height C{B{H} = B{h} - B{N}} (C{meter}, same units 

1100 as B{C{h}} and B{C{N}}). 

1101 

1102 @see: U{Ellipsoid, Geoid, and Othometric Heights<https://www.NGS.NOAA.gov/ 

1103 GEOID/PRESENTATIONS/2007_02_24_CCPS/Roman_A_PLSC2007notes.pdf>}, page 

1104 6 and module L{pygeodesy.geoids}. 

1105 ''' 

1106 h = h_ll if _isHeight(h_ll) else _xattr(h_ll, height=_xattr(h_ll, h=0)) 

1107 return Height(H=Height(h=h) - Height(N=N)) 

1108 

1109 

1110def horizon(height, radius=R_M, refraction=False): 

1111 '''Determine the distance to the horizon from a given altitude 

1112 above the (spherical) earth. 

1113 

1114 @arg height: Altitude (C{meter} or same units as B{C{radius}}). 

1115 @kwarg radius: Optional mean earth radius (C{meter}). 

1116 @kwarg refraction: Consider atmospheric refraction (C{bool}). 

1117 

1118 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}). 

1119 

1120 @raise ValueError: Invalid B{C{height}} or B{C{radius}}. 

1121 

1122 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}. 

1123 ''' 

1124 h, r = Height(height), Radius(radius) 

1125 if min(h, r) < 0: 

1126 raise _ValueError(height=height, radius=radius) 

1127 

1128 d2 = ((r * 2.415750694528) if refraction else # 2.0 / 0.8279 

1129 fsumf_(r, r, h)) * h 

1130 return sqrt0(d2) 

1131 

1132 

1133class _idllmn6(object): # see also .geodesicw._wargs, .latlonBase._toCartesian3, .vector2d._numpy 

1134 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}. 

1135 ''' 

1136 @contextmanager # <https://www.Python.org/dev/peps/pep-0343/> Examples 

1137 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds): 

1138 try: 

1139 if wrap: 

1140 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap) 

1141 kwds = _xkwds(kwds, wrap=wrap) # for _xError 

1142 m = small if small is _100km else Meter_(small=small) 

1143 n = (intersections2 if s else intersection2).__name__ 

1144 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m: 

1145 d, m = None, _MODS.vector3d 

1146 _i = m._intersects2 if s else m._intersect3d3 

1147 elif _isRadius(datum) and datum < 0 and not s: 

1148 d = _spherical_datum(-datum, name=n) 

1149 m = _MODS.sphericalNvector 

1150 _i = m.intersection 

1151 else: 

1152 d = _spherical_datum(datum, name=n) 

1153 if d.isSpherical: 

1154 m = _MODS.sphericalTrigonometry 

1155 _i = m._intersects2 if s else m._intersect 

1156 elif d.isEllipsoidal: 

1157 try: 

1158 if d.ellipsoid.geodesic: 

1159 pass 

1160 m = _MODS.ellipsoidalKarney 

1161 except ImportError: 

1162 m = _MODS.ellipsoidalExact 

1163 _i = m._intersections2 if s else m._intersection3 # ellispoidalBaseDI 

1164 else: 

1165 raise _TypeError(datum=datum) 

1166 yield _i, d, lat2, lon2, m, n 

1167 

1168 except (TypeError, ValueError) as x: 

1169 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum, 

1170 lat2=lat2, lon2=lon2, small=small, **kwds) 

1171 

1172_idllmn6 = _idllmn6() # PYCHOK singleton 

1173 

1174 

1175def intersection2(lat1, lon1, bearing1, 

1176 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True 

1177 '''I{Conveniently} compute the intersection of two lines each defined 

1178 by a (geodetic) point and a bearing from North, using either ... 

1179 

1180 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km 

1181 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1182 

1183 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}} 

1184 or a C{scalar B{datum}} representing the earth radius, conventionally 

1185 in C{meter} or ... 

1186 

1187 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative} 

1188 C{scalar}, (negative) earth radius, conventionally in C{meter} or ... 

1189 

1190 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}} 

1191 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1192 is installed, otherwise ... 

1193 

1194 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal. 

1195 

1196 @arg lat1: Latitude of the first point (C{degrees}). 

1197 @arg lon1: Longitude of the first point (C{degrees}). 

1198 @arg bearing1: Bearing at the first point (compass C{degrees}). 

1199 @arg lat2: Latitude of the second point (C{degrees}). 

1200 @arg lon2: Longitude of the second point (C{degrees}). 

1201 @arg bearing2: Bearing at the second point (compass C{degrees}). 

1202 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1203 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1204 radius (C{meter}, same units as B{C{radius1}} and 

1205 B{C{radius2}}) or C{None}. 

1206 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1207 and B{C{lon2}} (C{bool}). 

1208 @kwarg small: Upper limit for small distances (C{meter}). 

1209 

1210 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and 

1211 longitude of the intersection point. 

1212 

1213 @raise IntersectionError: Ambiguous or infinite intersection 

1214 or colinear, parallel or otherwise 

1215 non-intersecting lines. 

1216 

1217 @raise TypeError: Invalid B{C{datum}}. 

1218 

1219 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}}, 

1220 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}. 

1221 

1222 @see: Method L{RhumbLine.intersection2}. 

1223 

1224 @note: The returned intersections may be near-antipodal. 

1225 ''' 

1226 b1 = Bearing(bearing1=bearing1) 

1227 b2 = Bearing(bearing2=bearing2) 

1228 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1229 small, wrap, False, bearing1=b1, bearing2=b2) as t: 

1230 _i, d, lat2, lon2, m, n = t 

1231 if d is None: 

1232 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1, 

1233 m.Vector3d(lon2, lat2, 0), b2, useZ=False) 

1234 t = LatLon2Tuple(t.y, t.x, name=n) 

1235 

1236 else: 

1237 t = _i(m.LatLon(lat1, lon1, datum=d), b1, 

1238 m.LatLon(lat2, lon2, datum=d), b2, height=0, wrap=False) 

1239 if isinstance(t, Intersection3Tuple): # ellipsoidal 

1240 t, _, _ = t 

1241 t = LatLon2Tuple(t.lat, t.lon, name=n) 

1242 return t 

1243 

1244 

1245def intersections2(lat1, lon1, radius1, 

1246 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True 

1247 '''I{Conveniently} compute the intersections of two circles each defined 

1248 by a (geodetic) center point and a radius, using either ... 

1249 

1250 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km 

1251 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1252 

1253 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}} 

1254 or a C{scalar B{datum}} representing the earth radius, conventionally 

1255 in C{meter} or ... 

1256 

1257 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}} 

1258 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1259 is installed, otherwise ... 

1260 

1261 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal. 

1262 

1263 @arg lat1: Latitude of the first circle center (C{degrees}). 

1264 @arg lon1: Longitude of the first circle center (C{degrees}). 

1265 @arg radius1: Radius of the first circle (C{meter}, conventionally). 

1266 @arg lat2: Latitude of the second circle center (C{degrees}). 

1267 @arg lon2: Longitude of the second circle center (C{degrees}). 

1268 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}). 

1269 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1270 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1271 radius (C{meter}, same units as B{C{radius1}} and 

1272 B{C{radius2}}) or C{None}. 

1273 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1274 and B{C{lon2}} (C{bool}). 

1275 @kwarg small: Upper limit for small distances (C{meter}). 

1276 

1277 @return: 2-Tuple of the intersection points, each a 

1278 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the 

1279 points are the same instance, aka the I{radical center}. 

1280 

1281 @raise IntersectionError: Concentric, antipodal, invalid or 

1282 non-intersecting circles or no 

1283 convergence. 

1284 

1285 @raise TypeError: Invalid B{C{datum}}. 

1286 

1287 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}}, 

1288 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}. 

1289 ''' 

1290 r1 = Radius_(radius1=radius1) 

1291 r2 = Radius_(radius2=radius2) 

1292 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1293 small, wrap, True, radius1=r1, radius2=r2) as t: 

1294 _i, d, lat2, lon2, m, n = t 

1295 if d is None: 

1296 r1 = m2degrees(r1, radius=R_M, lat=lat1) 

1297 r2 = m2degrees(r2, radius=R_M, lat=lat2) 

1298 

1299 def _V2T(x, y, _, **unused): # _ == z unused 

1300 return LatLon2Tuple(y, x, name=n) 

1301 

1302 t = _i(m.Vector3d(lon1, lat1, 0), r1, 

1303 m.Vector3d(lon2, lat2, 0), r2, sphere=False, 

1304 Vector=_V2T) 

1305 else: 

1306 def _LL2T(lat, lon, **unused): 

1307 return LatLon2Tuple(lat, lon, name=n) 

1308 

1309 t = _i(m.LatLon(lat1, lon1, datum=d), r1, 

1310 m.LatLon(lat2, lon2, datum=d), r2, 

1311 LatLon=_LL2T, height=0, wrap=False) 

1312 return t 

1313 

1314 

1315def isantipode(lat1, lon1, lat2, lon2, eps=EPS): 

1316 '''Check whether two points are I{antipodal}, on diametrically 

1317 opposite sides of the earth. 

1318 

1319 @arg lat1: Latitude of one point (C{degrees}). 

1320 @arg lon1: Longitude of one point (C{degrees}). 

1321 @arg lat2: Latitude of the other point (C{degrees}). 

1322 @arg lon2: Longitude of the other point (C{degrees}). 

1323 @kwarg eps: Tolerance for near-equality (C{degrees}). 

1324 

1325 @return: C{True} if points are antipodal within the 

1326 B{C{eps}} tolerance, C{False} otherwise. 

1327 

1328 @see: Functions L{isantipode_} and L{antipode}. 

1329 ''' 

1330 return (fabs(lat1 + lat2) <= eps and 

1331 fabs(lon1 + lon2) <= eps) or _isequalTo( 

1332 normal(lat1, lon1), antipode(lat2, lon2), eps) 

1333 

1334 

1335def isantipode_(phi1, lam1, phi2, lam2, eps=EPS): 

1336 '''Check whether two points are I{antipodal}, on diametrically 

1337 opposite sides of the earth. 

1338 

1339 @arg phi1: Latitude of one point (C{radians}). 

1340 @arg lam1: Longitude of one point (C{radians}). 

1341 @arg phi2: Latitude of the other point (C{radians}). 

1342 @arg lam2: Longitude of the other point (C{radians}). 

1343 @kwarg eps: Tolerance for near-equality (C{radians}). 

1344 

1345 @return: C{True} if points are antipodal within the 

1346 B{C{eps}} tolerance, C{False} otherwise. 

1347 

1348 @see: Functions L{isantipode} and L{antipode_}. 

1349 ''' 

1350 return (fabs(phi1 + phi2) <= eps and 

1351 fabs(lam1 + lam2) <= eps) or _isequalTo_( 

1352 normal_(phi1, lam1), antipode_(phi2, lam2), eps) 

1353 

1354 

1355def _isequalTo(p1, p2, eps=EPS): 

1356 '''Compare 2 point lat-/lons ignoring C{class}. 

1357 ''' 

1358 return (fabs(p1.lat - p2.lat) <= eps and 

1359 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon) 

1360 

1361 

1362def _isequalTo_(p1, p2, eps=EPS): 

1363 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}. 

1364 ''' 

1365 return (fabs(p1.phi - p2.phi) <= eps and 

1366 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam) 

1367 

1368 

1369def isnormal(lat, lon, eps=0): 

1370 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their 

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

1372 

1373 @arg lat: Latitude (C{degrees}). 

1374 @arg lon: Longitude (C{degrees}). 

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

1376 

1377 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and 

1378 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise. 

1379 

1380 @see: Functions L{isnormal_} and L{normal}. 

1381 ''' 

1382 return (_90_0 - fabs(lat)) >= eps and _loneg(fabs(lon)) >= eps 

1383 

1384 

1385def isnormal_(phi, lam, eps=0): 

1386 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their 

1387 respective I{normal} range in C{radians}. 

1388 

1389 @arg phi: Latitude (C{radians}). 

1390 @arg lam: Longitude (C{radians}). 

1391 @kwarg eps: Optional tolerance C{radians}). 

1392 

1393 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and 

1394 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise. 

1395 

1396 @see: Functions L{isnormal} and L{normal_}. 

1397 ''' 

1398 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps 

1399 

1400 

1401def latlon2n_xyz(lat, lon, name=NN): 

1402 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1403 earth's surface) X, Y and Z components. 

1404 

1405 @arg lat: Latitude (C{degrees}). 

1406 @arg lon: Longitude (C{degrees}). 

1407 @kwarg name: Optional name (C{str}). 

1408 

1409 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1410 

1411 @see: Function L{philam2n_xyz}. 

1412 

1413 @note: These are C{n-vector} x, y and z components, 

1414 I{NOT} geocentric ECEF x, y and z coordinates! 

1415 ''' 

1416 return _2n_xyz(name, *sincos2d_(lat, lon)) 

1417 

1418 

1419def _normal2(a, b, n_2, n, n2): 

1420 '''(INTERNAL) Helper for C{normal} and C{normal_}. 

1421 ''' 

1422 if fabs(b) > n: 

1423 b = remainder(b, n2) 

1424 if fabs(a) > n_2: 

1425 r = remainder(a, n) 

1426 if r != a: 

1427 a = -r 

1428 b -= n if b > 0 else -n 

1429 return float0_(a, b) 

1430 

1431 

1432def normal(lat, lon, name=NN): 

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

1434 

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

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

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

1438 

1439 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90} 

1440 and C{abs(lon) <= 180}. 

1441 

1442 @see: Functions L{normal_} and L{isnormal}. 

1443 ''' 

1444 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0), 

1445 name=name or normal.__name__) 

1446 

1447 

1448def normal_(phi, lam, name=NN): 

1449 '''Normalize a lat- I{and} longitude pair in C{radians}. 

1450 

1451 @arg phi: Latitude (C{radians}). 

1452 @arg lam: Longitude (C{radians}). 

1453 @kwarg name: Optional name (C{str}). 

1454 

1455 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2} 

1456 and C{abs(lam) <= PI}. 

1457 

1458 @see: Functions L{normal} and L{isnormal_}. 

1459 ''' 

1460 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2), 

1461 name=name or normal_.__name__) 

1462 

1463 

1464def _2n_xyz(name, sa, ca, sb, cb): 

1465 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}. 

1466 ''' 

1467 # Kenneth Gade eqn 3, but using right-handed 

1468 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N 

1469 return Vector3Tuple(ca * cb, ca * sb, sa, name=name) 

1470 

1471 

1472def n_xyz2latlon(x, y, z, name=NN): 

1473 '''Convert C{n-vector} components to lat- and longitude in C{degrees}. 

1474 

1475 @arg x: X component (C{scalar}). 

1476 @arg y: Y component (C{scalar}). 

1477 @arg z: Z component (C{scalar}). 

1478 @kwarg name: Optional name (C{str}). 

1479 

1480 @return: A L{LatLon2Tuple}C{(lat, lon)}. 

1481 

1482 @see: Function L{n_xyz2philam}. 

1483 ''' 

1484 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name) 

1485 

1486 

1487def n_xyz2philam(x, y, z, name=NN): 

1488 '''Convert C{n-vector} components to lat- and longitude in C{radians}. 

1489 

1490 @arg x: X component (C{scalar}). 

1491 @arg y: Y component (C{scalar}). 

1492 @arg z: Z component (C{scalar}). 

1493 @kwarg name: Optional name (C{str}). 

1494 

1495 @return: A L{PhiLam2Tuple}C{(phi, lam)}. 

1496 

1497 @see: Function L{n_xyz2latlon}. 

1498 ''' 

1499 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name) 

1500 

1501 

1502def _opposes(d, m, n, n2): 

1503 '''(INTERNAL) Helper for C{opposing} and C{opposing_}. 

1504 ''' 

1505 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1 

1506 return False if d < m or d > (n2 - m) else ( 

1507 True if (n - m) < d < (n + m) else None) 

1508 

1509 

1510def opposing(bearing1, bearing2, margin=_90_0): 

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

1512 

1513 @arg bearing1: First bearing (compass C{degrees}). 

1514 @arg bearing2: Second bearing (compass C{degrees}). 

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

1516 

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

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

1519 

1520 @see: Function L{opposing_}. 

1521 ''' 

1522 m = Degrees_(margin=margin, low=EPS0, high=_90_0) 

1523 return _opposes(bearing2 - bearing1, m, _180_0, _360_0) 

1524 

1525 

1526def opposing_(radians1, radians2, margin=PI_2): 

1527 '''Compare the direction of two bearings given in C{radians}. 

1528 

1529 @arg radians1: First bearing (C{radians}). 

1530 @arg radians2: Second bearing (C{radians}). 

1531 @kwarg margin: Optional, interior angle bracket (C{radians}). 

1532 

1533 @return: C{True} if both bearings point in opposite, C{False} if 

1534 in similar or C{None} if in perpendicular directions. 

1535 

1536 @see: Function L{opposing}. 

1537 ''' 

1538 m = Radians_(margin=margin, low=EPS0, high=PI_2) 

1539 return _opposes(radians2 - radians1, m, PI, PI2) 

1540 

1541 

1542def philam2n_xyz(phi, lam, name=NN): 

1543 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1544 earth's surface) X, Y and Z components. 

1545 

1546 @arg phi: Latitude (C{radians}). 

1547 @arg lam: Longitude (C{radians}). 

1548 @kwarg name: Optional name (C{str}). 

1549 

1550 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1551 

1552 @see: Function L{latlon2n_xyz}. 

1553 

1554 @note: These are C{n-vector} x, y and z components, 

1555 I{NOT} geocentric ECEF x, y and z coordinates! 

1556 ''' 

1557 return _2n_xyz(name, *sincos2_(phi, lam)) 

1558 

1559 

1560def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d 

1561 # (INTERNAL) See C{radical2} below 

1562 # assert d > EPS0 

1563 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5 

1564 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d) 

1565 

1566 

1567def radical2(distance, radius1, radius2): 

1568 '''Compute the I{radical ratio} and I{radical line} of two 

1569 U{intersecting circles<https://MathWorld.Wolfram.com/ 

1570 Circle-CircleIntersection.html>}. 

1571 

1572 The I{radical line} is perpendicular to the axis thru the 

1573 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}. 

1574 

1575 @arg distance: Distance between the circle centers (C{scalar}). 

1576 @arg radius1: Radius of the first circle (C{scalar}). 

1577 @arg radius2: Radius of the second circle (C{scalar}). 

1578 

1579 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <= 

1580 ratio <= 1.0} and C{xline} is along the B{C{distance}}. 

1581 

1582 @raise IntersectionError: The B{C{distance}} exceeds the sum 

1583 of B{C{radius1}} and B{C{radius2}}. 

1584 

1585 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or 

1586 B{C{radius2}}. 

1587 

1588 @see: U{Circle-Circle Intersection 

1589 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}. 

1590 ''' 

1591 d = Distance_(distance, low=_0_0) 

1592 r1 = Radius_(radius1=radius1) 

1593 r2 = Radius_(radius2=radius2) 

1594 if d > (r1 + r2): 

1595 raise IntersectionError(distance=d, radius1=r1, radius2=r2, 

1596 txt=_too_(_distant_)) 

1597 return _radical2(d, r1, r2) if d > EPS0 else \ 

1598 Radical2Tuple(_0_5, _0_0) 

1599 

1600 

1601class Radical2Tuple(_NamedTuple): 

1602 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and 

1603 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0} 

1604 ''' 

1605 _Names_ = (_ratio_, _xline_) 

1606 _Units_ = ( Scalar, Scalar) 

1607 

1608 

1609def _radistance(inst): 

1610 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians} 

1611 and L{hausdorff._HausdorffMeterRedians} classes. 

1612 ''' 

1613 wrap_, kwds_ = _xkwds_pop2(inst._kwds, wrap=False) 

1614 func_ = inst._func_ 

1615 try: # calling lower-overhead C{func_} 

1616 func_(0, _0_25, _0_5, **kwds_) 

1617 wrap_ = _Wrap._philamop(wrap_) 

1618 except TypeError: 

1619 return inst.distance 

1620 

1621 def _philam(p): 

1622 try: 

1623 return p.phi, p.lam 

1624 except AttributeError: # no .phi or .lam 

1625 return radians(p.lat), radians(p.lon) 

1626 

1627 def _func_wrap(point1, point2): 

1628 phi1, lam1 = wrap_(*_philam(point1)) 

1629 phi2, lam2 = wrap_(*_philam(point2)) 

1630 return func_(phi2, phi1, lam2 - lam1, **kwds_) 

1631 

1632 inst._units = inst._units_ 

1633 return _func_wrap 

1634 

1635 

1636def _scale_deg(lat1, lat2): # degrees 

1637 # scale factor cos(mean of lats) for delta lon 

1638 m = fabs(lat1 + lat2) * _0_5 

1639 return cos(radians(m)) if m < 90 else _0_0 

1640 

1641 

1642def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights 

1643 # scale factor cos(mean of phis) for delta lam 

1644 m = fabs(phi1 + phi2) * _0_5 

1645 return cos(m) if m < PI_2 else _0_0 

1646 

1647 

1648def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw 

1649 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine. 

1650 ''' 

1651 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21) 

1652 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21 

1653 

1654 

1655def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

1656 '''Compute the distance between two (ellipsoidal) points using 

1657 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1658 formula. 

1659 

1660 @arg lat1: Start latitude (C{degrees}). 

1661 @arg lon1: Start longitude (C{degrees}). 

1662 @arg lat2: End latitude (C{degrees}). 

1663 @arg lon2: End longitude (C{degrees}). 

1664 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1665 L{Ellipsoid2} or L{a_f2Tuple}) to use. 

1666 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1667 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1668 

1669 @return: Distance (C{meter}, same units as the B{C{datum}}'s 

1670 ellipsoid axes). 

1671 

1672 @raise TypeError: Invalid B{C{datum}}. 

1673 

1674 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1675 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, 

1676 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}. 

1677 ''' 

1678 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2) 

1679 

1680 

1681def thomas_(phi2, phi1, lam21, datum=_WGS84): 

1682 '''Compute the I{angular} distance between two (ellipsoidal) points using 

1683 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1684 formula. 

1685 

1686 @arg phi2: End latitude (C{radians}). 

1687 @arg phi1: Start latitude (C{radians}). 

1688 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1689 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, 

1690 L{Ellipsoid2} or L{a_f2Tuple}). 

1691 

1692 @return: Angular distance (C{radians}). 

1693 

1694 @raise TypeError: Invalid B{C{datum}}. 

1695 

1696 @see: Functions L{thomas}, L{cosineAndoyerLambert_}, 

1697 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1698 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1699 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP 

1700 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/ 

1701 Distance/ThomasFormula.php>}. 

1702 ''' 

1703 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21) 

1704 if r and isnon0(c1) and isnon0(c2): 

1705 E = _ellipsoidal(datum, thomas_) 

1706 if E.f: 

1707 r1 = atan2(E.b_a * s1, c1) 

1708 r2 = atan2(E.b_a * s2, c2) 

1709 

1710 j = (r2 + r1) * _0_5 

1711 k = (r2 - r1) * _0_5 

1712 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5) 

1713 

1714 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2) 

1715 u = _1_0 - h 

1716 if isnon0(u) and isnon0(h): 

1717 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h) 

1718 sr, cr = sincos2(r) 

1719 if isnon0(sr): 

1720 u = 2 * (sj * ck)**2 / u 

1721 h = 2 * (sk * cj)**2 / h 

1722 x = u + h 

1723 y = u - h 

1724 

1725 s = r / sr 

1726 e = 4 * s**2 

1727 d = 2 * cr 

1728 a = e * d 

1729 b = 2 * r 

1730 c = s - (a - d) * _0_5 

1731 f = E.f * _0_25 

1732 

1733 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y) 

1734 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr 

1735 return r 

1736 

1737 

1738def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1739 '''Compute the distance between two (spherical) points using 

1740 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1741 spherical formula. 

1742 

1743 @arg lat1: Start latitude (C{degrees}). 

1744 @arg lon1: Start longitude (C{degrees}). 

1745 @arg lat2: End latitude (C{degrees}). 

1746 @arg lon2: End longitude (C{degrees}). 

1747 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1748 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1749 L{a_f2Tuple}) to use. 

1750 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1751 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1752 

1753 @return: Distance (C{meter}, same units as B{C{radius}}). 

1754 

1755 @raise UnitError: Invalid B{C{radius}}. 

1756 

1757 @see: Functions L{vincentys_}, L{cosineAndoyerLambert}, 

1758 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular}, 

1759 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1760 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2}, 

1761 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1762 

1763 @note: See note at function L{vincentys_}. 

1764 ''' 

1765 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2) 

1766 

1767 

1768def vincentys_(phi2, phi1, lam21): 

1769 '''Compute the I{angular} distance between two (spherical) points using 

1770 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1771 spherical formula. 

1772 

1773 @arg phi2: End latitude (C{radians}). 

1774 @arg phi1: Start latitude (C{radians}). 

1775 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1776 

1777 @return: Angular distance (C{radians}). 

1778 

1779 @see: Functions L{vincentys}, L{cosineAndoyerLambert_}, 

1780 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1781 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1782 L{flatPolar_}, L{haversine_} and L{thomas_}. 

1783 

1784 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_} 

1785 produce equivalent results, but L{vincentys_} is suitable 

1786 for antipodal points and slightly more expensive (M{3 cos, 

1787 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_} 

1788 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and 

1789 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}). 

1790 ''' 

1791 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21) 

1792 

1793 c = c2 * c21 

1794 x = s1 * s2 + c1 * c 

1795 y = c1 * s2 - s1 * c 

1796 return atan2(hypot(c2 * s21, y), x) 

1797 

1798# **) MIT License 

1799# 

1800# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

1801# 

1802# Permission is hereby granted, free of charge, to any person obtaining a 

1803# copy of this software and associated documentation files (the "Software"), 

1804# to deal in the Software without restriction, including without limitation 

1805# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1806# and/or sell copies of the Software, and to permit persons to whom the 

1807# Software is furnished to do so, subject to the following conditions: 

1808# 

1809# The above copyright notice and this permission notice shall be included 

1810# in all copies or substantial portions of the Software. 

1811# 

1812# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1813# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1814# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1815# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1816# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

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