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.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_, _inside_, _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__ = '23.10.04' 

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 U{Andoyer-Lambert 

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

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

195 

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

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

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

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

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

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

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

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

204 

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

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

207 

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

209 

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

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

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

213 L{Ellipsoid.distance2}. 

214 ''' 

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

216 

217 

218def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): 

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

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

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

222 

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

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

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

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

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

228 

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

230 

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

232 

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

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

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

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

237 AndoyerLambert.php>}. 

238 ''' 

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

240 if isnon0(c1) and isnon0(c2): 

241 E = _ellipsoidal(datum, cosineAndoyerLambert_) 

242 if E.f: # ellipsoidal 

243 r2 = atan2(E.b_a * s2, c2) 

244 r1 = atan2(E.b_a * s1, c1) 

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

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

247 if r: 

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

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

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

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

252 r += (c - s) * E.f * _0_125 

253 return r 

254 

255 

256def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

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

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

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

260 

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

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

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

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

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

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

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

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

269 

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

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

272 

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

274 

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

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

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

278 L{Ellipsoid.distance2}. 

279 ''' 

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

281 

282 

283def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): 

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

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

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

287 formula. 

288 

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

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

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

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

293 L{Ellipsoid2} or L{a_f2Tuple}). 

294 

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

296 

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

298 

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

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

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

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

303 Distance/ForsytheCorrection.php>}. 

304 ''' 

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

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

307 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_) 

308 if E.f: # ellipsoidal 

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

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

311 s = (s1 + s2)**2 / (1 + cr) 

312 t = (s1 - s2)**2 / (1 - cr) 

313 x = s + t 

314 y = s - t 

315 

316 s = 8 * r**2 / sr 

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

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

319 b = -2 * d 

320 e = 30 * s2r 

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

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

323 

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

325 return r 

326 

327 

328def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

331 

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

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

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

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

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

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

338 L{a_f2Tuple}) to use. 

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

340 and B{C{lon2}} (C{bool}). 

341 

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

343 ellipsoid or datum axes). 

344 

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

346 

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

348 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean}, 

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

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

351 

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

353 ''' 

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

355 

356 

357def cosineLaw_(phi2, phi1, lam21): 

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

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

360 

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

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

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

364 

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

366 

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

368 L{cosineForsytheAndoyerLambert_}, L{equirectangular_}, 

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

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

371 

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

373 ''' 

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

375 

376 

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

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

379 ''' 

380 if wrap: 

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

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

383 else: # for backward compaibility 

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

385 

386 

387def _dE(func_, earth, *wrap_lls): 

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

389 ''' 

390 E = _ellipsoidal(earth, func_) 

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

392 return r * E.a 

393 

394 

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

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

397 ''' 

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

399 if radius is not R_M: 

400 _, lat1, _, lat2, _ = wrap_lls 

401 radius = _mean_radius(radius, lat1, lat2) 

402 return r * radius 

403 

404 

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

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

407 ''' 

408 r = excess_(*_d3(*wrap_lls)) 

409 if radius: 

410 _, lat1, _, lat2, _ = wrap_lls 

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

412 return r 

413 

414 

415def _ellipsoidal(earth, where): 

416 '''(INTERNAL) Helper for distances. 

417 ''' 

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

419 earth if isinstance(earth, Ellipsoid) else 

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

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

422 

423 

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

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

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

427 

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

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

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

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

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

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

434 L{a_f2Tuple}). 

435 @kwarg adjust_limit_wrap: Optional keyword arguments for 

436 function L{equirectangular_}. 

437 

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

439 the ellipsoid or datum axes). 

440 

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

442 

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

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

445 approximate or accurate distance functions. 

446 ''' 

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

448 Lat(lat2=lat2), Lon(lon2=lon2), 

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

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

451 

452 

453def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap): 

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

455 and L{hausdorff._HausdorffMeterRedians} classes. 

456 ''' 

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

458 

459 

460def equirectangular_(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False): 

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

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

463 

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

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

466 

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

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

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

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

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

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

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

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

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

476 and B{C{lon2}} (C{bool}). 

477 

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

479 unroll_lon2)} in C{degrees squared}. 

480 

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

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

483 set to C{True}. 

484 

485 @see: U{Local, flat earth approximation 

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

487 L{equirectangular}, L{cosineAndoyerLambert}, 

488 L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean}, 

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

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

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

492 ''' 

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

494 d_lat = lat2 - lat1 

495 

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

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

498 if d > limit: 

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

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

501 limit=limit, wrap=wrap) 

502 raise LimitError(s, txt=t) 

503 

504 if adjust: # scale delta lon 

505 d_lon *= _scale_deg(lat1, lat2) 

506 

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

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

509 

510 

511def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False): 

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

513 

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

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

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

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

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

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

520 L{a_f2Tuple}) to use. 

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

522 the mean latitude (C{bool}). 

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

524 and B{C{lon2}} (C{bool}). 

525 

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

527 ellipsoid or datum axes). 

528 

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

530 

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

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

533 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

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

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

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

537 ''' 

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

539 

540 

541def euclidean_(phi2, phi1, lam21, adjust=True): 

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

543 

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

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

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

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

548 of the mean latitude (C{bool}). 

549 

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

551 

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

553 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_}, 

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

555 and L{vincentys_}. 

556 ''' 

557 if adjust: 

558 lam21 *= _scale_rad(phi2, phi1) 

559 return euclid(phi2 - phi1, lam21) 

560 

561 

562def excessAbc_(A, b, c): 

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

564 and the included (small) angle. 

565 

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

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

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

569 

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

571 

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

573 

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

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

576 ''' 

577 A = Radians_(A=A) 

578 b = Radians_(b=b) * _0_5 

579 c = Radians_(c=c) * _0_5 

580 

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

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

583 

584 

585def excessCagnoli_(a, b, c): 

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

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

588 

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

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

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

592 

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

594 

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

596 

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

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

599 ''' 

600 a = Radians_(a=a) 

601 b = Radians_(b=b) 

602 c = Radians_(c=c) 

603 

604 s = fsumf_(a, b, c) * _0_5 

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

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

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

608 return Radians(Cagnoli=r * _2_0) 

609 

610 

611def excessGirard_(A, B, C): 

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

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

614 

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

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

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

618 

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

620 

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

622 

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

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

625 ''' 

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

627 Radians_(B=B), 

628 Radians_(C=C), -PI)) 

629 

630 

631def excessLHuilier_(a, b, c): 

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

633 <https://MathWorld.Wolfram.com/LHuiliersTheorem.html>} Theorem. 

634 

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

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

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

638 

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

640 

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

642 

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

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

645 ''' 

646 a = Radians_(a=a) 

647 b = Radians_(b=b) 

648 c = Radians_(c=c) 

649 

650 s = fsumf_(a, b, c) * _0_5 

651 r = tan_2(s) * tan_2(s - a) * tan_2(s - b) * tan_2(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 t2 = tan_2(phi2) 

714 t1 = tan_2(phi1) 

715 t = tan_2(lam21, lam21=None) 

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

717 _1_0 + (t1 * t2)) * _2_0) 

718 

719 

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

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

722# 

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

724# 

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

726# 

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

728# 

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

730# 

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

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

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

734# 

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

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

737# 

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

739# 

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

741# inverse Gudermannian) function 

742# 

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

744# 

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

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

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

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

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

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

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

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

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

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

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

756# - Charles Karney 

757 

758 

759def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

762 

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

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

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

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

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

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

769 L{a_f2Tuple}) or C{None}. 

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

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

772 

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

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

775 if C{B{radius}=0} or C{None}. 

776 

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

778 

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

780 

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

782 ''' 

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

784 

785 

786def excessQuad_(phi2, phi1, lam21): 

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

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

789 

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

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

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

793 

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

795 

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

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

798 ''' 

799 s = sin((phi2 + phi1) * _0_5) 

800 c = cos((phi2 - phi1) * _0_5) 

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

802 

803 

804def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False): 

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

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

807 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

809 

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

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

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

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

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

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

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

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

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

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

820 

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

822 ellipsoid axes). 

823 

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

825 

826 @note: The meridional and prime_vertical radii of curvature 

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

828 

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

830 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

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

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

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

834 ''' 

835 E = _ellipsoidal(datum, flatLocal) 

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

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

838 

839hubeny = flatLocal # PYCHOK for Karl Hubeny 

840 

841 

842def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True): 

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

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

845 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

847 

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

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

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

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

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

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

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

855 

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

857 

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

859 

860 @note: The meridional and prime_vertical radii of curvature 

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

862 

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

864 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_}, 

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

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

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

868 ''' 

869 E = _ellipsoidal(datum, flatLocal_) 

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

871 

872hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny 

873 

874 

875def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

878 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

879 formula. 

880 

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

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

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

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

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

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

887 L{a_f2Tuple}) to use. 

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

889 and B{C{lon2}} (C{bool}). 

890 

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

892 ellipsoid or datum axes). 

893 

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

895 

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

897 L{cosineForsytheAndoyerLambert},L{cosineLaw}, 

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

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

900 L{vincentys}. 

901 ''' 

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

903 

904 

905def flatPolar_(phi2, phi1, lam21): 

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

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

908 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

909 formula. 

910 

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

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

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

914 

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

916 

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

918 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

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

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

921 ''' 

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

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

924 if a < b: 

925 a, b = b, a 

926 if a < EPS0: 

927 a = _0_0 

928 elif b > 0: 

929 b = b / a # /= chokes PyChecker 

930 c = b * cos(lam21) * _2_0 

931 c = fsumf_(_1_0, b**2, -fabs(c)) 

932 a *= sqrt0(c) 

933 return a 

934 

935 

936def _hartzell(inst, los, earth, **kwds): 

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

938 ''' 

939 if earth is not None: 

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

941 inst = inst.toDatum(earth) 

942 h = inst.height 

943 if h > 0: # EPS0 

944 r = hartzell(inst, los=los, earth=earth or inst.datum, **kwds) 

945 elif h < 0: # EPS0 

946 raise IntersectionError(pov=inst, los=los, height=h, txt=_inside_) 

947 else: 

948 r = inst 

949 return r 

950 

951 

952def hartzell(pov, los=None, earth=_WGS84, name=NN, **LatLon_and_kwds): 

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

954 from a Point-Of-View in space. 

955 

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

957 C{LatLon} or L{Vector3d}). 

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

959 or C{None} to point to the earth' center. 

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

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

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

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

964 point plus C{LatLon} keyword arguments, include 

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

966 

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

968 B{C{pov}} or the given C{B{LatLon}_and_kwds}) with C{.heigth} 

969 set to the distance to the B{C{pov}}. 

970 

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

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

973 the earth or points in an opposite direction. 

974 

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

976 

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

978 methods L{Ellipsoid.hartzell4}, C{Cartesian.hartzell}, C{LatLon.hartzell} 

979 and U{I{Satellite Line-of-Sight Intersection with Earth}<https:// 

980 StephenHartzell.Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}. 

981 ''' 

982 n = hartzell.__name__ 

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

984 _spherical_datum(earth, name=n) 

985 try: 

986 r, h = _MODS.triaxials._hartzell3d2(pov, los, D.ellipsoid._triaxial) 

987 except Exception as x: 

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

989 

990 r = _xnamed(r, name or n) 

991 C = _MODS.cartesianBase.CartesianBase 

992 if LatLon_and_kwds: 

993 c = C(r, datum=D, name=r.name) 

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

995 elif isinstance(r, C): 

996 r.height = h 

997 return r 

998 

999 

1000def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1001 '''Compute the distance between two (spherical) points using the 

1002 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} 

1003 formula. 

1004 

1005 @arg lat1: Start latitude (C{degrees}). 

1006 @arg lon1: Start longitude (C{degrees}). 

1007 @arg lat2: End latitude (C{degrees}). 

1008 @arg lon2: End longitude (C{degrees}). 

1009 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1010 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1011 L{a_f2Tuple}) to use. 

1012 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1013 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1014 

1015 @return: Distance (C{meter}, same units as B{C{radius}}). 

1016 

1017 @raise TypeError: Invalid B{C{radius}}. 

1018 

1019 @see: U{Distance between two (spherical) points 

1020 <https://www.EdWilliams.org/avform.htm#Dist>}, functions 

1021 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1022 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1023 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, 

1024 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1025 

1026 @note: See note at function L{vincentys_}. 

1027 ''' 

1028 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2) 

1029 

1030 

1031def haversine_(phi2, phi1, lam21): 

1032 '''Compute the I{angular} distance between two (spherical) points 

1033 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} 

1034 formula. 

1035 

1036 @arg phi2: End latitude (C{radians}). 

1037 @arg phi1: Start latitude (C{radians}). 

1038 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1039 

1040 @return: Angular distance (C{radians}). 

1041 

1042 @see: Functions L{haversine}, L{cosineAndoyerLambert_}, 

1043 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1044 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1045 L{flatPolar_}, L{thomas_} and L{vincentys_}. 

1046 

1047 @note: See note at function L{vincentys_}. 

1048 ''' 

1049 def _hsin(rad): 

1050 return sin(rad * _0_5)**2 

1051 

1052 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine 

1053 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2 

1054 

1055 

1056def heightOf(angle, distance, radius=R_M): 

1057 '''Determine the height above the (spherical) earth' surface after 

1058 traveling along a straight line at a given tilt. 

1059 

1060 @arg angle: Tilt angle above horizontal (C{degrees}). 

1061 @arg distance: Distance along the line (C{meter} or same units as 

1062 B{C{radius}}). 

1063 @kwarg radius: Optional mean earth radius (C{meter}). 

1064 

1065 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}). 

1066 

1067 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}. 

1068 

1069 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>} 

1070 (U{Shapiro et al. 2009, JTECH 

1071 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>} 

1072 and U{Potvin et al. 2012, JTECH 

1073 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}). 

1074 ''' 

1075 r = h = Radius(radius) 

1076 d = fabs(Distance(distance)) 

1077 if d > h: 

1078 d, h = h, d 

1079 

1080 if d > EPS0: # and h > EPS0 

1081 d = d / h # /= h chokes PyChecker 

1082 s = sin(Phi_(angle=angle, clip=_180_0)) 

1083 s = fsumf_(_1_0, _2_0 * s * d, d**2) 

1084 if s > 0: 

1085 return h * sqrt(s) - r 

1086 

1087 raise _ValueError(angle=angle, distance=distance, radius=radius) 

1088 

1089 

1090def heightOrthometric(h_ll, N): 

1091 '''Get the I{orthometric} height B{H}, the height above the geoid, earth surface. 

1092 

1093 @arg h_ll: The height above the ellipsoid (C{meter}) or an I{ellipsoidal} 

1094 location (C{LatLon} with a C{height} or C{h} attribute). 

1095 @arg N: The I{geoid} height (C{meter}), the height of the geoid above the 

1096 ellipsoid at the same B{C{h_ll}} location. 

1097 

1098 @return: I{Orthometric} height C{B{H} = B{h} - B{N}} (C{meter}, same units 

1099 as B{C{h}} and B{C{N}}). 

1100 

1101 @see: U{Ellipsoid, Geoid, and Othometric Heights<https://www.NGS.NOAA.gov/ 

1102 GEOID/PRESENTATIONS/2007_02_24_CCPS/Roman_A_PLSC2007notes.pdf>}, page 

1103 6 and module L{pygeodesy.geoids}. 

1104 ''' 

1105 h = h_ll if isscalar(h_ll) else _xattr(h_ll, height=_xattr(h_ll, h=0)) 

1106 return Height(H=Height(h=h) - Height(N=N)) 

1107 

1108 

1109def horizon(height, radius=R_M, refraction=False): 

1110 '''Determine the distance to the horizon from a given altitude 

1111 above the (spherical) earth. 

1112 

1113 @arg height: Altitude (C{meter} or same units as B{C{radius}}). 

1114 @kwarg radius: Optional mean earth radius (C{meter}). 

1115 @kwarg refraction: Consider atmospheric refraction (C{bool}). 

1116 

1117 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}). 

1118 

1119 @raise ValueError: Invalid B{C{height}} or B{C{radius}}. 

1120 

1121 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}. 

1122 ''' 

1123 h, r = Height(height), Radius(radius) 

1124 if min(h, r) < 0: 

1125 raise _ValueError(height=height, radius=radius) 

1126 

1127 if refraction: 

1128 d2 = 2.415750694528 * h * r # 2.0 / 0.8279 

1129 else: 

1130 d2 = h * fsumf_(r, r, h) 

1131 return sqrt0(d2) 

1132 

1133 

1134class _idllmn6(object): # see also .geodesicw._wargs, .vector2d._numpy 

1135 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}. 

1136 ''' 

1137 @contextmanager # <https://www.python.org/dev/peps/pep-0343/> Examples 

1138 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds): 

1139 try: 

1140 if wrap: 

1141 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap) 

1142 kwds = _xkwds(kwds, wrap=wrap) # for _xError 

1143 m = small if small is _100km else Meter_(small=small) 

1144 n = (intersections2 if s else intersection2).__name__ 

1145 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m: 

1146 d, m = None, _MODS.vector3d 

1147 _i = m._intersects2 if s else m._intersect3d3 

1148 elif isscalar(datum) and datum < 0 and not s: 

1149 d = _spherical_datum(-datum, name=n) 

1150 m = _MODS.sphericalNvector 

1151 _i = m.intersection 

1152 else: 

1153 d = _spherical_datum(datum, name=n) 

1154 if d.isSpherical: 

1155 m = _MODS.sphericalTrigonometry 

1156 _i = m._intersects2 if s else m._intersect 

1157 elif d.isEllipsoidal: 

1158 try: 

1159 if d.ellipsoid.geodesic: 

1160 pass 

1161 m = _MODS.ellipsoidalKarney 

1162 except ImportError: 

1163 m = _MODS.ellipsoidalExact 

1164 _i = m._intersections2 if s else m._intersection3 # ellispoidalBaseDI 

1165 else: 

1166 raise _TypeError(datum=datum) 

1167 yield _i, d, lat2, lon2, m, n 

1168 

1169 except (TypeError, ValueError) as x: 

1170 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum, 

1171 lat2=lat2, lon2=lon2, small=small, **kwds) 

1172 

1173_idllmn6 = _idllmn6() # PYCHOK singleton 

1174 

1175 

1176def intersection2(lat1, lon1, bearing1, 

1177 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True 

1178 '''I{Conveniently} compute the intersection of two lines each defined 

1179 by a (geodetic) point and a bearing from North, using either ... 

1180 

1181 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km 

1182 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1183 

1184 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}} 

1185 or a C{scalar B{datum}} representing the earth radius, conventionally 

1186 in C{meter} or ... 

1187 

1188 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative} 

1189 C{scalar}, (negative) earth radius, conventionally in C{meter} or ... 

1190 

1191 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}} 

1192 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1193 is installed, otherwise ... 

1194 

1195 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal. 

1196 

1197 @arg lat1: Latitude of the first point (C{degrees}). 

1198 @arg lon1: Longitude of the first point (C{degrees}). 

1199 @arg bearing1: Bearing at the first point (compass C{degrees}). 

1200 @arg lat2: Latitude of the second point (C{degrees}). 

1201 @arg lon2: Longitude of the second point (C{degrees}). 

1202 @arg bearing2: Bearing at the second point (compass C{degrees}). 

1203 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1204 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1205 radius (C{meter}, same units as B{C{radius1}} and 

1206 B{C{radius2}}) or C{None}. 

1207 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1208 and B{C{lon2}} (C{bool}). 

1209 @kwarg small: Upper limit for small distances (C{meter}). 

1210 

1211 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and 

1212 longitude of the intersection point. 

1213 

1214 @raise IntersectionError: Ambiguous or infinite intersection 

1215 or colinear, parallel or otherwise 

1216 non-intersecting lines. 

1217 

1218 @raise TypeError: Invalid B{C{datum}}. 

1219 

1220 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}}, 

1221 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}. 

1222 

1223 @see: Method L{RhumbLine.intersection2}. 

1224 

1225 @note: The returned intersections may be near-antipodal. 

1226 ''' 

1227 b1 = Bearing(bearing1=bearing1) 

1228 b2 = Bearing(bearing2=bearing2) 

1229 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1230 small, wrap, False, bearing1=b1, bearing2=b2) as t: 

1231 _i, d, lat2, lon2, m, n = t 

1232 if d is None: 

1233 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1, 

1234 m.Vector3d(lon2, lat2, 0), b2, useZ=False) 

1235 t = LatLon2Tuple(t.y, t.x, name=n) 

1236 

1237 else: 

1238 t = _i(m.LatLon(lat1, lon1, datum=d), b1, 

1239 m.LatLon(lat2, lon2, datum=d), b2, height=0, wrap=False) 

1240 if isinstance(t, Intersection3Tuple): # ellipsoidal 

1241 t, _, _ = t 

1242 t = LatLon2Tuple(t.lat, t.lon, name=n) 

1243 return t 

1244 

1245 

1246def intersections2(lat1, lon1, radius1, 

1247 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True 

1248 '''I{Conveniently} compute the intersections of two circles each defined 

1249 by a (geodetic) center point and a radius, using either ... 

1250 

1251 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km 

1252 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1253 

1254 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}} 

1255 or a C{scalar B{datum}} representing the earth radius, conventionally 

1256 in C{meter} or ... 

1257 

1258 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}} 

1259 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1260 is installed, otherwise ... 

1261 

1262 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal. 

1263 

1264 @arg lat1: Latitude of the first circle center (C{degrees}). 

1265 @arg lon1: Longitude of the first circle center (C{degrees}). 

1266 @arg radius1: Radius of the first circle (C{meter}, conventionally). 

1267 @arg lat2: Latitude of the second circle center (C{degrees}). 

1268 @arg lon2: Longitude of the second circle center (C{degrees}). 

1269 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}). 

1270 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1271 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1272 radius (C{meter}, same units as B{C{radius1}} and 

1273 B{C{radius2}}) or C{None}. 

1274 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1275 and B{C{lon2}} (C{bool}). 

1276 @kwarg small: Upper limit for small distances (C{meter}). 

1277 

1278 @return: 2-Tuple of the intersection points, each a 

1279 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the 

1280 points are the same instance, aka the I{radical center}. 

1281 

1282 @raise IntersectionError: Concentric, antipodal, invalid or 

1283 non-intersecting circles or no 

1284 convergence. 

1285 

1286 @raise TypeError: Invalid B{C{datum}}. 

1287 

1288 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}}, 

1289 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}. 

1290 ''' 

1291 r1 = Radius_(radius1=radius1) 

1292 r2 = Radius_(radius2=radius2) 

1293 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1294 small, wrap, True, radius1=r1, radius2=r2) as t: 

1295 _i, d, lat2, lon2, m, n = t 

1296 if d is None: 

1297 r1 = m2degrees(r1, radius=R_M, lat=lat1) 

1298 r2 = m2degrees(r2, radius=R_M, lat=lat2) 

1299 

1300 def _V2T(x, y, _, **unused): # _ == z unused 

1301 return LatLon2Tuple(y, x, name=n) 

1302 

1303 t = _i(m.Vector3d(lon1, lat1, 0), r1, 

1304 m.Vector3d(lon2, lat2, 0), r2, sphere=False, 

1305 Vector=_V2T) 

1306 else: 

1307 def _LL2T(lat, lon, **unused): 

1308 return LatLon2Tuple(lat, lon, name=n) 

1309 

1310 t = _i(m.LatLon(lat1, lon1, datum=d), r1, 

1311 m.LatLon(lat2, lon2, datum=d), r2, 

1312 LatLon=_LL2T, height=0, wrap=False) 

1313 return t 

1314 

1315 

1316def isantipode(lat1, lon1, lat2, lon2, eps=EPS): 

1317 '''Check whether two points are I{antipodal}, on diametrically 

1318 opposite sides of the earth. 

1319 

1320 @arg lat1: Latitude of one point (C{degrees}). 

1321 @arg lon1: Longitude of one point (C{degrees}). 

1322 @arg lat2: Latitude of the other point (C{degrees}). 

1323 @arg lon2: Longitude of the other point (C{degrees}). 

1324 @kwarg eps: Tolerance for near-equality (C{degrees}). 

1325 

1326 @return: C{True} if points are antipodal within the 

1327 B{C{eps}} tolerance, C{False} otherwise. 

1328 

1329 @see: Functions L{isantipode_} and L{antipode}. 

1330 ''' 

1331 return (fabs(lat1 + lat2) <= eps and 

1332 fabs(lon1 + lon2) <= eps) or _isequalTo( 

1333 normal(lat1, lon1), antipode(lat2, lon2), eps) 

1334 

1335 

1336def isantipode_(phi1, lam1, phi2, lam2, eps=EPS): 

1337 '''Check whether two points are I{antipodal}, on diametrically 

1338 opposite sides of the earth. 

1339 

1340 @arg phi1: Latitude of one point (C{radians}). 

1341 @arg lam1: Longitude of one point (C{radians}). 

1342 @arg phi2: Latitude of the other point (C{radians}). 

1343 @arg lam2: Longitude of the other point (C{radians}). 

1344 @kwarg eps: Tolerance for near-equality (C{radians}). 

1345 

1346 @return: C{True} if points are antipodal within the 

1347 B{C{eps}} tolerance, C{False} otherwise. 

1348 

1349 @see: Functions L{isantipode} and L{antipode_}. 

1350 ''' 

1351 return (fabs(phi1 + phi2) <= eps and 

1352 fabs(lam1 + lam2) <= eps) or _isequalTo_( 

1353 normal_(phi1, lam1), antipode_(phi2, lam2), eps) 

1354 

1355 

1356def _isequalTo(p1, p2, eps=EPS): 

1357 '''Compare 2 point lat-/lons ignoring C{class}. 

1358 ''' 

1359 return (fabs(p1.lat - p2.lat) <= eps and 

1360 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon) 

1361 

1362 

1363def _isequalTo_(p1, p2, eps=EPS): 

1364 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}. 

1365 ''' 

1366 return (fabs(p1.phi - p2.phi) <= eps and 

1367 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam) 

1368 

1369 

1370def isnormal(lat, lon, eps=0): 

1371 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their 

1372 respective I{normal} range in C{degrees}. 

1373 

1374 @arg lat: Latitude (C{degrees}). 

1375 @arg lon: Longitude (C{degrees}). 

1376 @kwarg eps: Optional tolerance C{degrees}). 

1377 

1378 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and 

1379 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise. 

1380 

1381 @see: Functions L{isnormal_} and L{normal}. 

1382 ''' 

1383 return (_90_0 - fabs(lat)) >= eps and _loneg(fabs(lon)) >= eps 

1384 

1385 

1386def isnormal_(phi, lam, eps=0): 

1387 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their 

1388 respective I{normal} range in C{radians}. 

1389 

1390 @arg phi: Latitude (C{radians}). 

1391 @arg lam: Longitude (C{radians}). 

1392 @kwarg eps: Optional tolerance C{radians}). 

1393 

1394 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and 

1395 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise. 

1396 

1397 @see: Functions L{isnormal} and L{normal_}. 

1398 ''' 

1399 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps 

1400 

1401 

1402def latlon2n_xyz(lat, lon, name=NN): 

1403 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1404 earth's surface) X, Y and Z components. 

1405 

1406 @arg lat: Latitude (C{degrees}). 

1407 @arg lon: Longitude (C{degrees}). 

1408 @kwarg name: Optional name (C{str}). 

1409 

1410 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1411 

1412 @see: Function L{philam2n_xyz}. 

1413 

1414 @note: These are C{n-vector} x, y and z components, 

1415 I{NOT} geocentric ECEF x, y and z coordinates! 

1416 ''' 

1417 return _2n_xyz(name, *sincos2d_(lat, lon)) 

1418 

1419 

1420def _normal2(a, b, n_2, n, n2): 

1421 '''(INTERNAL) Helper for C{normal} and C{normal_}. 

1422 ''' 

1423 if fabs(b) > n: 

1424 b = remainder(b, n2) 

1425 if fabs(a) > n_2: 

1426 r = remainder(a, n) 

1427 if r != a: 

1428 a = -r 

1429 b -= n if b > 0 else -n 

1430 return float0_(a, b) 

1431 

1432 

1433def normal(lat, lon, name=NN): 

1434 '''Normalize a lat- I{and} longitude pair in C{degrees}. 

1435 

1436 @arg lat: Latitude (C{degrees}). 

1437 @arg lon: Longitude (C{degrees}). 

1438 @kwarg name: Optional name (C{str}). 

1439 

1440 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90} 

1441 and C{abs(lon) <= 180}. 

1442 

1443 @see: Functions L{normal_} and L{isnormal}. 

1444 ''' 

1445 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0), 

1446 name=name or normal.__name__) 

1447 

1448 

1449def normal_(phi, lam, name=NN): 

1450 '''Normalize a lat- I{and} longitude pair in C{radians}. 

1451 

1452 @arg phi: Latitude (C{radians}). 

1453 @arg lam: Longitude (C{radians}). 

1454 @kwarg name: Optional name (C{str}). 

1455 

1456 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2} 

1457 and C{abs(lam) <= PI}. 

1458 

1459 @see: Functions L{normal} and L{isnormal_}. 

1460 ''' 

1461 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2), 

1462 name=name or normal_.__name__) 

1463 

1464 

1465def _2n_xyz(name, sa, ca, sb, cb): 

1466 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}. 

1467 ''' 

1468 # Kenneth Gade eqn 3, but using right-handed 

1469 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N 

1470 return Vector3Tuple(ca * cb, ca * sb, sa, name=name) 

1471 

1472 

1473def n_xyz2latlon(x, y, z, name=NN): 

1474 '''Convert C{n-vector} components to lat- and longitude in C{degrees}. 

1475 

1476 @arg x: X component (C{scalar}). 

1477 @arg y: Y component (C{scalar}). 

1478 @arg z: Z component (C{scalar}). 

1479 @kwarg name: Optional name (C{str}). 

1480 

1481 @return: A L{LatLon2Tuple}C{(lat, lon)}. 

1482 

1483 @see: Function L{n_xyz2philam}. 

1484 ''' 

1485 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name) 

1486 

1487 

1488def n_xyz2philam(x, y, z, name=NN): 

1489 '''Convert C{n-vector} components to lat- and longitude in C{radians}. 

1490 

1491 @arg x: X component (C{scalar}). 

1492 @arg y: Y component (C{scalar}). 

1493 @arg z: Z component (C{scalar}). 

1494 @kwarg name: Optional name (C{str}). 

1495 

1496 @return: A L{PhiLam2Tuple}C{(phi, lam)}. 

1497 

1498 @see: Function L{n_xyz2latlon}. 

1499 ''' 

1500 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name) 

1501 

1502 

1503def _opposes(d, m, n, n2): 

1504 '''(INTERNAL) Helper for C{opposing} and C{opposing_}. 

1505 ''' 

1506 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1 

1507 return False if d < m or d > (n2 - m) else ( 

1508 True if (n - m) < d < (n + m) else None) 

1509 

1510 

1511def opposing(bearing1, bearing2, margin=_90_0): 

1512 '''Compare the direction of two bearings given in C{degrees}. 

1513 

1514 @arg bearing1: First bearing (compass C{degrees}). 

1515 @arg bearing2: Second bearing (compass C{degrees}). 

1516 @kwarg margin: Optional, interior angle bracket (C{degrees}). 

1517 

1518 @return: C{True} if both bearings point in opposite, C{False} if 

1519 in similar or C{None} if in I{perpendicular} directions. 

1520 

1521 @see: Function L{opposing_}. 

1522 ''' 

1523 m = Degrees_(margin=margin, low=EPS0, high=_90_0) 

1524 return _opposes(bearing2 - bearing1, m, _180_0, _360_0) 

1525 

1526 

1527def opposing_(radians1, radians2, margin=PI_2): 

1528 '''Compare the direction of two bearings given in C{radians}. 

1529 

1530 @arg radians1: First bearing (C{radians}). 

1531 @arg radians2: Second bearing (C{radians}). 

1532 @kwarg margin: Optional, interior angle bracket (C{radians}). 

1533 

1534 @return: C{True} if both bearings point in opposite, C{False} if 

1535 in similar or C{None} if in perpendicular directions. 

1536 

1537 @see: Function L{opposing}. 

1538 ''' 

1539 m = Radians_(margin=margin, low=EPS0, high=PI_2) 

1540 return _opposes(radians2 - radians1, m, PI, PI2) 

1541 

1542 

1543def philam2n_xyz(phi, lam, name=NN): 

1544 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1545 earth's surface) X, Y and Z components. 

1546 

1547 @arg phi: Latitude (C{radians}). 

1548 @arg lam: Longitude (C{radians}). 

1549 @kwarg name: Optional name (C{str}). 

1550 

1551 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1552 

1553 @see: Function L{latlon2n_xyz}. 

1554 

1555 @note: These are C{n-vector} x, y and z components, 

1556 I{NOT} geocentric ECEF x, y and z coordinates! 

1557 ''' 

1558 return _2n_xyz(name, *sincos2_(phi, lam)) 

1559 

1560 

1561def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d 

1562 # (INTERNAL) See C{radical2} below 

1563 # assert d > EPS0 

1564 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5 

1565 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d) 

1566 

1567 

1568def radical2(distance, radius1, radius2): 

1569 '''Compute the I{radical ratio} and I{radical line} of two 

1570 U{intersecting circles<https://MathWorld.Wolfram.com/ 

1571 Circle-CircleIntersection.html>}. 

1572 

1573 The I{radical line} is perpendicular to the axis thru the 

1574 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}. 

1575 

1576 @arg distance: Distance between the circle centers (C{scalar}). 

1577 @arg radius1: Radius of the first circle (C{scalar}). 

1578 @arg radius2: Radius of the second circle (C{scalar}). 

1579 

1580 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <= 

1581 ratio <= 1.0} and C{xline} is along the B{C{distance}}. 

1582 

1583 @raise IntersectionError: The B{C{distance}} exceeds the sum 

1584 of B{C{radius1}} and B{C{radius2}}. 

1585 

1586 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or 

1587 B{C{radius2}}. 

1588 

1589 @see: U{Circle-Circle Intersection 

1590 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}. 

1591 ''' 

1592 d = Distance_(distance, low=_0_0) 

1593 r1 = Radius_(radius1=radius1) 

1594 r2 = Radius_(radius2=radius2) 

1595 if d > (r1 + r2): 

1596 raise IntersectionError(distance=d, radius1=r1, radius2=r2, 

1597 txt=_too_(_distant_)) 

1598 return _radical2(d, r1, r2) if d > EPS0 else \ 

1599 Radical2Tuple(_0_5, _0_0) 

1600 

1601 

1602class Radical2Tuple(_NamedTuple): 

1603 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and 

1604 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0} 

1605 ''' 

1606 _Names_ = (_ratio_, _xline_) 

1607 _Units_ = ( Scalar, Scalar) 

1608 

1609 

1610def _radistance(inst): 

1611 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians} 

1612 and L{hausdorff._HausdorffMeterRedians} classes. 

1613 ''' 

1614 kwds_ = _xkwds(inst._kwds, wrap=False) 

1615 wrap_ = _xkwds_pop(kwds_, wrap=False) 

1616 func_ = inst._func_ 

1617 try: # calling lower-overhead C{func_} 

1618 func_(0, _0_25, _0_5, **kwds_) 

1619 wrap_ = _Wrap._philamop(wrap_) 

1620 except TypeError: 

1621 return inst.distance 

1622 

1623 def _philam(p): 

1624 try: 

1625 return p.phi, p.lam 

1626 except AttributeError: # no .phi or .lam 

1627 return radians(p.lat), radians(p.lon) 

1628 

1629 def _func_wrap(point1, point2): 

1630 phi1, lam1 = wrap_(*_philam(point1)) 

1631 phi2, lam2 = wrap_(*_philam(point2)) 

1632 return func_(phi2, phi1, lam2 - lam1, **kwds_) 

1633 

1634 inst._units = inst._units_ 

1635 return _func_wrap 

1636 

1637 

1638def _scale_deg(lat1, lat2): # degrees 

1639 # scale factor cos(mean of lats) for delta lon 

1640 m = fabs(lat1 + lat2) * _0_5 

1641 return cos(radians(m)) if m < 90 else _0_0 

1642 

1643 

1644def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights 

1645 # scale factor cos(mean of phis) for delta lam 

1646 m = fabs(phi1 + phi2) * _0_5 

1647 return cos(m) if m < PI_2 else _0_0 

1648 

1649 

1650def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw 

1651 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine. 

1652 ''' 

1653 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21) 

1654 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21 

1655 

1656 

1657def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

1658 '''Compute the distance between two (ellipsoidal) points using 

1659 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1660 formula. 

1661 

1662 @arg lat1: Start latitude (C{degrees}). 

1663 @arg lon1: Start longitude (C{degrees}). 

1664 @arg lat2: End latitude (C{degrees}). 

1665 @arg lon2: End longitude (C{degrees}). 

1666 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1667 L{Ellipsoid2} or L{a_f2Tuple}) to use. 

1668 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1669 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1670 

1671 @return: Distance (C{meter}, same units as the B{C{datum}}'s 

1672 ellipsoid axes). 

1673 

1674 @raise TypeError: Invalid B{C{datum}}. 

1675 

1676 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1677 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, 

1678 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}. 

1679 ''' 

1680 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2) 

1681 

1682 

1683def thomas_(phi2, phi1, lam21, datum=_WGS84): 

1684 '''Compute the I{angular} distance between two (ellipsoidal) points using 

1685 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1686 formula. 

1687 

1688 @arg phi2: End latitude (C{radians}). 

1689 @arg phi1: Start latitude (C{radians}). 

1690 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1691 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, 

1692 L{Ellipsoid2} or L{a_f2Tuple}). 

1693 

1694 @return: Angular distance (C{radians}). 

1695 

1696 @raise TypeError: Invalid B{C{datum}}. 

1697 

1698 @see: Functions L{thomas}, L{cosineAndoyerLambert_}, 

1699 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1700 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1701 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP 

1702 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/ 

1703 Distance/ThomasFormula.php>}. 

1704 ''' 

1705 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21) 

1706 if r and isnon0(c1) and isnon0(c2): 

1707 E = _ellipsoidal(datum, thomas_) 

1708 if E.f: 

1709 r1 = atan2(E.b_a * s1, c1) 

1710 r2 = atan2(E.b_a * s2, c2) 

1711 

1712 j = (r2 + r1) * _0_5 

1713 k = (r2 - r1) * _0_5 

1714 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5) 

1715 

1716 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2) 

1717 u = _1_0 - h 

1718 if isnon0(u) and isnon0(h): 

1719 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h) 

1720 sr, cr = sincos2(r) 

1721 if isnon0(sr): 

1722 u = 2 * (sj * ck)**2 / u 

1723 h = 2 * (sk * cj)**2 / h 

1724 x = u + h 

1725 y = u - h 

1726 

1727 s = r / sr 

1728 e = 4 * s**2 

1729 d = 2 * cr 

1730 a = e * d 

1731 b = 2 * r 

1732 c = s - (a - d) * _0_5 

1733 f = E.f * _0_25 

1734 

1735 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y) 

1736 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr 

1737 return r 

1738 

1739 

1740def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1741 '''Compute the distance between two (spherical) points using 

1742 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1743 spherical formula. 

1744 

1745 @arg lat1: Start latitude (C{degrees}). 

1746 @arg lon1: Start longitude (C{degrees}). 

1747 @arg lat2: End latitude (C{degrees}). 

1748 @arg lon2: End longitude (C{degrees}). 

1749 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1750 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1751 L{a_f2Tuple}) to use. 

1752 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1753 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1754 

1755 @return: Distance (C{meter}, same units as B{C{radius}}). 

1756 

1757 @raise UnitError: Invalid B{C{radius}}. 

1758 

1759 @see: Functions L{vincentys_}, L{cosineAndoyerLambert}, 

1760 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular}, 

1761 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1762 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2}, 

1763 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1764 

1765 @note: See note at function L{vincentys_}. 

1766 ''' 

1767 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2) 

1768 

1769 

1770def vincentys_(phi2, phi1, lam21): 

1771 '''Compute the I{angular} distance between two (spherical) points using 

1772 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1773 spherical formula. 

1774 

1775 @arg phi2: End latitude (C{radians}). 

1776 @arg phi1: Start latitude (C{radians}). 

1777 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1778 

1779 @return: Angular distance (C{radians}). 

1780 

1781 @see: Functions L{vincentys}, L{cosineAndoyerLambert_}, 

1782 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1783 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, 

1784 L{flatPolar_}, L{haversine_} and L{thomas_}. 

1785 

1786 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_} 

1787 produce equivalent results, but L{vincentys_} is suitable 

1788 for antipodal points and slightly more expensive (M{3 cos, 

1789 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_} 

1790 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and 

1791 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}). 

1792 ''' 

1793 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21) 

1794 

1795 c = c2 * c21 

1796 x = s1 * s2 + c1 * c 

1797 y = c1 * s2 - s1 * c 

1798 return atan2(hypot(c2 * s21, y), x) 

1799 

1800# **) MIT License 

1801# 

1802# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved. 

1803# 

1804# Permission is hereby granted, free of charge, to any person obtaining a 

1805# copy of this software and associated documentation files (the "Software"), 

1806# to deal in the Software without restriction, including without limitation 

1807# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1808# and/or sell copies of the Software, and to permit persons to whom the 

1809# Software is furnished to do so, subject to the following conditions: 

1810# 

1811# The above copyright notice and this permission notice shall be included 

1812# in all copies or substantial portions of the Software. 

1813# 

1814# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1815# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1816# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

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