Coverage for pygeodesy/formy.py: 98%

448 statements  

« prev     ^ index     » next       coverage.py v7.6.1, created at 2025-04-25 13:15 -0400

1 

2# -*- coding: utf-8 -*- 

3 

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

5''' 

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

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

8 

9from pygeodesy.basics import _copysign, _isin # _args_kwds_count2 

10# from pygeodesy.cartesianBase import CartesianBase # _MODS 

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

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

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

14 _90_0, _180_0, _360_0 

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

16 _mean_radius, _spherical_datum, _WGS84, _EWGS84 

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

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

19 _TypeError, _ValueError, _xattr, _xError, \ 

20 _xcallable, _xkwds, _xkwds_pop2 

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

22from pygeodesy.fsums import fsumf_, Fmt, unstr 

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

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

25from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS 

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

27from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, LatLon2Tuple, \ 

28 Intersection3Tuple, PhiLam2Tuple 

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

30# from pygeodesy.triaxials import _hartzell3 # _MODS 

31from pygeodesy.units import _isDegrees, _isHeight, _isRadius, Bearing, Degrees_, \ 

32 Distance, Distance_, Height, Lamd, Lat, Lon, Meter_, \ 

33 Phid, Radians, Radians_, Radius, Radius_, Scalar, _100km 

34from pygeodesy.utily import acos1, asin1, atan2, atan2b, degrees2m, hav, _loneg, \ 

35 m2degrees, tan_2, sincos2, sincos2_, _Wrap 

36# from pygeodesy.vector3d import _otherV3d # _MODS 

37# from pygeodesy.vector3dBase import _xyz_y_z3 # _MODS 

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

39# sphericalNvector, sphericalTrigonometry # _MODS 

40 

41from contextlib import contextmanager 

42from math import atan, cos, degrees, fabs, radians, sin, sqrt # pow 

43 

44__all__ = _ALL_LAZY.formy 

45__version__ = '25.04.14' 

46 

47_RADIANS2 = radians(_1_0)**2 # degree to radians-squared 

48_ratio_ = 'ratio' 

49_xline_ = 'xline' 

50 

51 

52def angle2chord(rad, radius=R_M): 

53 '''Get the chord length of a (central) angle or I{angular} distance. 

54 

55 @arg rad: Central angle (C{radians}). 

56 @kwarg radius: Mean earth radius (C{meter}, conventionally), datum (L{Datum}) or ellipsoid 

57 (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) to use or C{None}. 

58 

59 @return: Chord length (C{meter}, same units as B{C{radius}} or if C{B{radius} is None}, C{radians}). 

60 

61 @see: Function L{chord2angle}, method L{intermediateChordTo<sphericalNvector.LatLon.intermediateChordTo>} and 

62 U{great-circle-distance<https://WikiPedia.org/wiki/Great-circle_distance#Relation_between_central_angle_and_chord_length>}. 

63 ''' 

64 d = _isDegrees(rad, iscalar=False) 

65 r = sin((radians(rad) if d else rad) / _2_0) * _2_0 

66 return (degrees(r) if d else r) if radius is None else (_mean_radius(radius) * r) 

67 

68 

69def _anti2(a, b, n_2, n, n2): 

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

71 ''' 

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

73 if r == a: 

74 r = -r 

75 b += n 

76 if fabs(b) > n: 

77 b = remainder(b, n2) 

78 return float0_(r, b) 

79 

80 

81def antipode(lat, lon, **name): 

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

83 point in C{degrees}. 

84 

85 @arg lat: Latitude (C{degrees}). 

86 @arg lon: Longitude (C{degrees}). 

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

88 

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

90 

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

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

93 ''' 

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

95 

96 

97def antipode_(phi, lam, **name): 

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

99 point in C{radians}. 

100 

101 @arg phi: Latitude (C{radians}). 

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

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

104 

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

106 

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

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

109 ''' 

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

111 

112 

113def bearing(lat1, lon1, lat2, lon2, **final_wrap): 

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

115 (spherical) points. 

116 

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

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

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

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

121 @kwarg final_wrap: Optional keyword arguments for function L{pygeodesy.bearing_}. 

122 

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

124 coincide. 

125 ''' 

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

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

128 return degrees(r) 

129 

130 

131def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False): 

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

133 (spherical) points. 

134 

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

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

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

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

139 @kwarg final: If C{True}, return the final, otherwise the initial bearing (C{bool}). 

140 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{phi2}} and B{C{lam2}} 

141 (C{bool}). 

142 

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

144 coincide. 

145 

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

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

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

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

150 ''' 

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

152 if final: # swap plus PI 

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

154 r = PI3 

155 else: 

156 r = PI2 

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

158 

159 x = ca1 * sa2 - sa1 * ca2 * cdb 

160 y = sdb * ca2 

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

162 

163 

164def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf 

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

166 ''' 

167 try: # for LatLon_ and ellipsoidal LatLon 

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

169 except AttributeError: 

170 pass 

171 # XXX spherical version, OK for ellipsoidal ispolar? 

172 t = p1.philam + p2.philam 

173 i = bearing_(*t, final=False, wrap=wrap) 

174 f = bearing_(*t, final=True, wrap=wrap) 

175 return Bearing2Tuple(degrees(i), degrees(f), 

176 name__=_bearingTo2) 

177 

178 

179def chord2angle(chord, radius=R_M): 

180 '''Get the (central) angle from a chord length or distance. 

181 

182 @arg chord: Length or distance (C{meter}, same units as B{C{radius}}). 

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

184 ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) to use. 

185 

186 @return: Angle (C{radians} with sign of B{C{chord}}) or C{0} if C{B{radius}=0}. 

187 

188 @note: The angle will exceed C{PI} if C{B{chord} > B{radius} * 2}. 

189 

190 @see: Function L{angle2chord}. 

191 ''' 

192 m = _mean_radius(radius) 

193 r = fabs(chord / (m * _2_0)) if m > 0 else _0_0 

194 if r: 

195 i = int(r) 

196 if i > 0: 

197 r -= i 

198 i *= PI 

199 r = (asin1(r) + i) * _2_0 

200 return _copysign(r, chord) 

201 

202 

203def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False): 

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

205 lat2 - lat1)} between two points. 

206 

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

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

209 

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

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

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

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

214 @kwarg adjust: Adjust the longitudinal delta by the cosine of the mean 

215 latitude (C{bool}). 

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

217 B{C{lon2}} (C{bool}). 

218 

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

220 

221 @note: Courtesy of Martin Schultz. 

222 

223 @see: U{Local, flat earth approximation 

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

225 ''' 

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

227 if adjust: # scale delta lon 

228 d_lon *= _scale_deg(lat1, lat2) 

229 return atan2b(d_lon, lat2 - lat1) 

230 

231 

232def cosineLaw(lat1, lon1, lat2, lon2, corr=0, earth=None, wrap=False, 

233 datum=_WGS84, radius=R_M): 

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

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

236 formula, optionally corrected. 

237 

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

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

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

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

242 @kwarg corr: Use C{B{corr}=2} to apply the U{Forsythe-Andoyer-Lambert 

243 <https://www2.UNB.CA/gge/Pubs/TR77.pdf>}, C{B{corr}=1} for the 

244 U{Andoyer-Lambert<https://Books.Google.com/books?id=x2UiAQAAIAAJ>} 

245 corrected (ellipsoidal) or keep C{B{corr}=0} for the uncorrected 

246 (spherical) C{Law of Cosines} formula (C{int}). 

247 @kwarg earth: Mean earth radius (C{meter}) or datum (L{Datum}) or ellipsoid 

248 (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) to use. 

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

250 (C{bool}). 

251 @kwarg datum: Default ellipsiodal B{C{earth}} (and for backward compatibility). 

252 @kwarg radius: Default spherical B{C{earth}} (and for backward compatibility). 

253 

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

255 ellipsoid axes). 

256 

257 @raise TypeError: Invalid B{C{earth}}, B{C{datum}} or B{C{radius}}. 

258 

259 @raise ValueError: Invalid B{C{corr}}. 

260 

261 @see: Functions L{cosineLaw_}, L{equirectangular}, L{euclidean}, L{flatLocal} / 

262 L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and 

263 method L{Ellipsoid.distance2}. 

264 

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

266 ''' 

267 return _dE(cosineLaw_, earth or datum, wrap, lat1, lon1, lat2, lon2, corr=corr) if corr else \ 

268 _dS(cosineLaw_, earth or radius, wrap, lat1, lon1, lat2, lon2) 

269 

270 

271def cosineLaw_(phi2, phi1, lam21, corr=0, earth=None, datum=_WGS84): 

272 '''Compute the I{angular} distance between two points using the U{Law of Cosines 

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

274 optionally corrected. 

275 

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

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

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

279 @kwarg corr: Use C{B{corr}=2} to apply the U{Forsythe-Andoyer-Lambert 

280 <https://www2.UNB.CA/gge/Pubs/TR77.pdf>}, C{B{corr}=1} for the 

281 U{Andoyer-Lambert<https://Books.Google.com/books?id=x2UiAQAAIAAJ>} 

282 corrected (ellipsoidal) or keep C{B{corr}=0} for the uncorrected 

283 (spherical) C{Law of Cosines} formula (C{int}). 

284 @kwarg earth: Mean earth radius (C{meter}) or datum (L{Datum}) or ellipsoid 

285 (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) to use. 

286 @kwarg datum: Default ellipsoidal B{C{earth}} (and for backward compatibility). 

287 

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

289 

290 @raise TypeError: Invalid B{C{earth}} or B{C{datum}}. 

291 

292 @raise ValueError: Invalid B{C{corr}}. 

293 

294 @see: Functions L{cosineLaw}, L{euclidean_}, L{flatLocal_} / L{hubeny_}, 

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

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

297 AndoyerLambert.php>}. 

298 ''' 

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

300 if corr and isnon0(c1) and isnon0(c2): 

301 E = _ellipsoidal(earth or datum, cosineLaw_) 

302 f = _0_25 * E.f 

303 if f: # ellipsoidal 

304 if corr == 1: # Andoyer-Lambert 

305 r2 = atan2(E.b_a * s2, c2) 

306 r1 = atan2(E.b_a * s1, c1) 

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

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

309 if r: 

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

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

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

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

314 r += (c - s) * _0_5 * f 

315 

316 elif corr == 2: # Forsythe-Andoyer-Lambert 

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

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

319 s = (s1 + s2)**2 / (_1_0 + cr) 

320 t = (s1 - s2)**2 / (_1_0 - cr) 

321 x = s + t 

322 y = s - t 

323 

324 s = 8 * r**2 / sr 

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

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

327 b = -2 * d 

328 e = 30 * s2r 

329 

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

331 t = fdot_( a, x, b, y, e, y**2, -c, x**2, d, x * y) * _0_125 

332 r += fdot_(-r, x, sr, y * 3, t, f) * f 

333 else: 

334 raise _ValueError(corr=corr) 

335 return r 

336 

337 

338def _d3(wrap, lat1, lon1, lat2, lon2): 

339 '''(INTERNAL) Helper for _dE, _dS, .... 

340 ''' 

341 if wrap: 

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

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

344 else: # for backward compaibility 

345 return Phid(lat2=lat2), Phid(lat1=lat1), radians(lon2 - lon1) 

346 

347 

348def _dE(fun_, earth, wrap, *lls, **corr): 

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

350 ''' 

351 E = _ellipsoidal(earth, fun_) 

352 r = fun_(*_d3(wrap, *lls), datum=E, **corr) 

353 return r * E.a 

354 

355 

356def _dS(fun_, radius, wrap, *lls, **adjust): 

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

358 ''' 

359 r = fun_(*_d3(wrap, *lls), **adjust) 

360 if radius is not R_M: 

361 try: # datum? 

362 radius = radius.ellipsoid.R1 

363 except AttributeError: 

364 pass # scalar? 

365 lat1, _, lat2, _ = lls 

366 radius = _mean_radius(radius, lat1, lat2) 

367 return r * radius 

368 

369 

370def _ellipsoidal(earth, where): 

371 '''(INTERNAL) Helper for distances. 

372 ''' 

373 return _EWGS84 if _isin(earth, _EWGS84, _WGS84) else ( 

374 earth if isinstance(earth, Ellipsoid) else 

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

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

377 

378 

379def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **adjust_limit_wrap): 

380 '''Approximate the distance between two points using the U{Equirectangular Approximation 

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

382 

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

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

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

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

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

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

389 @kwarg adjust_limit_wrap: Optionally, keyword arguments for function L{equirectangular4}. 

390 

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

392 ellipsoid axes). 

393 

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

395 

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

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

398 ''' 

399 r = _mean_radius(radius, lat1, lat2) 

400 t = equirectangular4(Lat(lat1=lat1), Lon(lon1=lon1), 

401 Lat(lat2=lat2), Lon(lon2=lon2), 

402 **adjust_limit_wrap) # PYCHOK 4 vs 2-3 

403 return degrees2m(sqrt(t.distance2), radius=r) 

404 

405 

406def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap): 

407 '''(INTERNAL) Helper for classes L{frechet._FrechetMeterRadians} and 

408 L{hausdorff._HausdorffMeterRedians}. 

409 ''' 

410 t = equirectangular4(lat1, lon1, lat2, lon2, **adjust_limit_wrap) 

411 return t.distance2 * _RADIANS2 

412 

413 

414def equirectangular4(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False): 

415 '''Approximate the distance between two points using the U{Equirectangular Approximation 

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

417 

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

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

420 

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

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

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

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

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

426 mean latitude (C{bool}). 

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

428 or C{0} for unlimited. 

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

430 (C{bool}). 

431 

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

433 C{distance2} in C{degrees squared}. 

434 

435 @raise LimitError: The lat- or longitudinal delta exceeds the B{C{-limit..limit}} 

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

437 

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

439 functions L{equirectangular}, L{cosineLaw}, L{euclidean}, L{flatLocal} / 

440 L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and methods 

441 L{Ellipsoid.distance2}, C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

442 ''' 

443 if wrap: 

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

445 else: 

446 d_lon, ulon2 = (lon2 - lon1), lon2 

447 d_lat = lat2 - lat1 

448 

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

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

451 if d > limit: 

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

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

454 limit=limit, wrap=wrap) 

455 raise LimitError(s, txt=t) 

456 

457 if adjust: # scale delta lon 

458 d_lon *= _scale_deg(lat1, lat2) 

459 

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

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

462 

463 

464def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False): 

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

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 radius: Mean earth radius (C{meter}), datum (L{Datum}) or ellipsoid 

472 (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) to use. 

473 @kwarg adjust: Adjust the longitudinal delta by the cosine of the mean 

474 latitude (C{bool}). 

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

476 B{C{lon2}} (C{bool}). 

477 

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

479 or datum axes). 

480 

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

482 

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

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

485 L{euclidean_}, L{cosineLaw}, L{equirectangular}, L{flatLocal} / 

486 L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} 

487 and methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*} and 

488 C{LatLon.equirectangularTo}. 

489 ''' 

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

491 

492 

493def euclidean_(phi2, phi1, lam21, adjust=True): 

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

495 

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

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

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

499 @kwarg adjust: Adjust the longitudinal delta by the cosine of the mean 

500 latitude (C{bool}). 

501 

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

503 

504 @see: Functions L{euclid}, L{euclidean}, L{cosineLaw_}, L{flatLocal_} / 

505 L{hubeny_}, L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_}. 

506 ''' 

507 if adjust: 

508 lam21 *= _scale_rad(phi2, phi1) 

509 return euclid(phi2 - phi1, lam21) 

510 

511 

512def excessAbc_(A, b, c): 

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

514 and the included (small) angle. 

515 

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

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

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

519 

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

521 

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

523 

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

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

526 ''' 

527 A = Radians_(A=A) 

528 b = Radians_(b=b) * _0_5 

529 c = Radians_(c=c) * _0_5 

530 

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

532 s = sA * sb * sc 

533 c = cA * sb * sc + cc * cb 

534 return atan2(s, c) * _2_0 

535 

536 

537def excessCagnoli_(a, b, c): 

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

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

540 

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

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

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

544 

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

546 

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

548 

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

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

551 ''' 

552 a = Radians_(a=a) 

553 b = Radians_(b=b) 

554 c = Radians_(c=c) 

555 

556 r = _maprod(cos, a * _0_5, b * _0_5, c * _0_5) 

557 if r: 

558 s = fsumf_(a, b, c) * _0_5 

559 t = _maprod(sin, s, s - a, s - b, s - c) 

560 r = asin1(sqrt(t) * _0_5 / r) if t > 0 else _0_0 

561 return Radians(Cagnoli=r * _2_0) 

562 

563 

564def excessGirard_(A, B, C): 

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

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

567 

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

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

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

571 

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

573 

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

575 

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

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

578 ''' 

579 r = fsumf_(Radians_(A=A), 

580 Radians_(B=B), 

581 Radians_(C=C), -PI) 

582 return Radians(Girard=r) 

583 

584 

585def excessLHuilier_(a, b, c): 

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

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

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{excessCagnoli_}, L{excessGirard_} and U{Spherical 

598 trigonometry<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 = _maprod(tan_2, s, s - a, s - b, s - c) 

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

607 return Radians(LHuilier=r * _4_0) 

608 

609 

610def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

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

614 method. 

615 

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

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

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

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

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

621 (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) or C{None}. 

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

623 B{C{lon2}} (C{bool}). 

624 

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

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

627 if C{B{radius}=0} or C{None}. 

628 

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

630 

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

632 

633 @raise ValueError: Semi-circular longitudinal delta. 

634 

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

636 ''' 

637 r = excessKarney_(*_d3(wrap, lat1, lon1, lat2, lon2)) 

638 if radius: 

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

640 return r 

641 

642 

643def excessKarney_(phi2, phi1, lam21): 

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

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

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

647 method. 

648 

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

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

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

652 

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

654 

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

656 

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

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

659 ''' 

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

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

662 # 

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

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

665 # 1 + tan(φ1 / 2) · tan(φ2 / 2) 

666 # 

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

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

669 t2 = tan_2(phi2) 

670 t1 = tan_2(phi1) 

671 c = (t1 * t2) + _1_0 

672 s = (t1 + t2) * tan_2(lam21, lam21=None) 

673 return Radians(Karney=atan2(s, c) * _2_0) 

674 

675 

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

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

678# 

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

680# 

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

682# 

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

684# 

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

686# 

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

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

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

690# 

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

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

693# 

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

695# 

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

697# inverse Gudermannian) function 

698# 

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

700# 

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

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

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

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

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

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

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

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

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

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

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

712# - Charles Karney 

713 

714 

715def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

718 

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

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

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

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

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

724 (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) or C{None}. 

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

726 B{C{lon2}} (C{bool}). 

727 

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

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

730 if C{B{radius}=0} or C{None}. 

731 

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

733 

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

735 

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

737 ''' 

738 r = excessQuad_(*_d3(wrap, lat1, lon1, lat2, lon2)) 

739 if radius: 

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

741 return r 

742 

743 

744def excessQuad_(phi2, phi1, lam21): 

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

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

747 

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

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

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

751 

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

753 

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

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

756 ''' 

757 c = cos((phi2 - phi1) * _0_5) 

758 s = sin((phi2 + phi1) * _0_5) * tan_2(lam21) 

759 return Radians(Quad=atan2(s, c) * _2_0) 

760 

761 

762def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False): 

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

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

765 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

767 

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

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

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

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

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

773 or L{a_f2Tuple}) to use. 

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

775 method L{pygeodesy.Ellipsoid.roc2_}. 

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

777 B{C{lon2}} (C{bool}). 

778 

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

780 

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

782 

783 @note: The meridional and prime_vertical radii of curvature are taken and 

784 scaled at the mean of both latitude. 

785 

786 @see: Functions L{flatLocal_} or L{hubeny_}, L{cosineLaw}, L{equirectangular}, 

787 L{euclidean}, L{flatPolar}, L{haversine}, L{thomas} and L{vincentys}, method 

788 L{Ellipsoid.distance2} and U{local, flat earth approximation 

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

790 ''' 

791 t = _d3(wrap, lat1, lon1, lat2, lon2) 

792 E = _ellipsoidal(datum, flatLocal) 

793 return E._hubeny_2(*t, scaled=scaled, squared=False) * E.a 

794 

795hubeny = flatLocal # PYCHOK for Karl Hubeny 

796 

797 

798def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True): 

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

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

801 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

803 

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

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

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

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

808 or L{a_f2Tuple}) to use. 

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

810 method L{pygeodesy.Ellipsoid.roc2_}. 

811 

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

813 

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

815 

816 @note: The meridional and prime_vertical radii of curvature are taken and 

817 scaled I{at the mean of both latitude}. 

818 

819 @see: Functions L{flatLocal} or L{hubeny}, L{cosineLaw_}, L{flatPolar_}, 

820 L{euclidean_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{local, 

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

822 ''' 

823 E = _ellipsoidal(datum, flatLocal_) 

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

825 

826hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny 

827 

828 

829def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

830 '''Compute the distance between two (spherical) points using the U{polar 

831 coordinate flat-Earth<https://WikiPedia.org/wiki/Geographical_distance 

832 #Polar_coordinate_flat-Earth_formula>} formula. 

833 

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

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

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

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

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

839 (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) to use. 

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

841 (C{bool}). 

842 

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

844 ellipsoid axes). 

845 

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

847 

848 @see: Functions L{flatPolar_}, L{cosineLaw}, L{flatLocal} / L{hubeny}, 

849 L{equirectangular}, L{euclidean}, L{haversine}, L{thomas} and L{vincentys}. 

850 ''' 

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

852 

853 

854def flatPolar_(phi2, phi1, lam21): 

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

856 U{polar coordinate flat-Earth<https://WikiPedia.org/wiki/Geographical_distance 

857 #Polar_coordinate_flat-Earth_formula>} formula. 

858 

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

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

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

862 

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

864 

865 @see: Functions L{flatPolar}, L{cosineLaw_}, L{euclidean_}, L{flatLocal_} / 

866 L{hubeny_}, L{haversine_}, L{thomas_} and L{vincentys_}. 

867 ''' 

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

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

870 if a < b: 

871 a, b = b, a 

872 if a < EPS0: 

873 a = _0_0 

874 elif b > 0: 

875 b = b / a # /= chokes PyChecker 

876 c = b * cos(lam21) * _2_0 

877 c = fsumf_(_1_0, b**2, -fabs(c)) 

878 a *= sqrt0(c) 

879 return a 

880 

881 

882def _hartzell(pov, los, earth, **kwds): 

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

884 ''' 

885 if earth is None: 

886 earth = pov.datum 

887 else: 

888 earth = _spherical_datum(earth, name__=hartzell) 

889 pov = pov.toDatum(earth) 

890 h = pov.height 

891 if h < 0: # EPS0 

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

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

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

895 

896 

897def hartzell(pov, los=False, earth=_WGS84, **name_LatLon_and_kwds): 

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

899 a Point-Of-View in space. 

900 

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

902 L{Ecef9Tuple} or L{Vector3d}). 

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

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

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

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

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

908 @kwarg name_LatLon_and_kwds: Optional C{B{name}="hartzell"} (C{str}), class 

909 C{B{LatLon}=None} to return the intersection and optionally, 

910 additional C{LatLon} keyword arguments, include the B{C{datum}} 

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

912 

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

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

915 the distance to the B{C{pov}}. 

916 

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

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

919 away from the earth. 

920 

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

922 

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

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

925 ''' 

926 n, kwds = _name2__(name_LatLon_and_kwds, name__=hartzell) 

927 try: 

928 D = _spherical_datum(earth, name__=hartzell) 

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

930 

931 C = _MODS.cartesianBase.CartesianBase 

932 if kwds: 

933 c = C(r, datum=D) 

934 r = c.toLatLon(**_xkwds(kwds, height=h)) 

935 elif isinstance(r, C): 

936 r.height = h 

937 if i: 

938 r._iteration = i 

939 except Exception as x: 

940 raise IntersectionError(pov=pov, los=los, earth=earth, cause=x, **kwds) 

941 return _xnamed(r, n) if n else r 

942 

943 

944def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

945 '''Compute the distance between two (spherical) points using the U{Haversine 

946 <https://www.Movable-Type.co.UK/scripts/latlong.html>} formula. 

947 

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

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

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

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

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

953 (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) to use. 

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

955 B{C{lon2}} (C{bool}). 

956 

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

958 

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

960 

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

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

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

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

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

966 

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

968 ''' 

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

970 

971 

972def haversine_(phi2, phi1, lam21): 

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

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

975 

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

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

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

979 

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

981 

982 @see: Functions L{haversine}, L{cosineLaw_}, L{euclidean_}, L{flatLocal_} / 

983 L{hubeny_}, L{flatPolar_}, L{thomas_} and L{vincentys_}. 

984 

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

986 ''' 

987 h = hav(phi2 - phi1) + cos(phi1) * cos(phi2) * hav(lam21) # haversine 

988 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin1(sqrt(h)) * 2 

989 

990 

991def heightOf(angle, distance, radius=R_M): 

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

993 traveling along a straight line at a given tilt. 

994 

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

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

997 B{C{radius}}). 

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

999 

1000 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}). 

1001 

1002 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}. 

1003 

1004 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>} 

1005 (U{Shapiro et al. 2009, JTECH 

1006 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>} 

1007 and U{Potvin et al. 2012, JTECH 

1008 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}). 

1009 ''' 

1010 r = h = Radius(radius) 

1011 d = fabs(Distance(distance)) 

1012 if d > h: 

1013 d, h = h, d 

1014 

1015 if d > EPS0: # and h > EPS0 

1016 d = d / h # /= h chokes PyChecker 

1017 s = sin(Phid(angle=angle, clip=_180_0)) 

1018 s = fsumf_(_1_0, s * d * _2_0, d**2) 

1019 if s > 0: 

1020 return h * sqrt(s) - r 

1021 

1022 raise _ValueError(angle=angle, distance=distance, radius=radius) 

1023 

1024 

1025def heightOrthometric(h_loc, N): 

1026 '''Get the I{orthometric} height B{H}, the height above the geoid, earth surface. 

1027 

1028 @arg h_loc: The height above the ellipsoid (C{meter}) or an I{ellipsoidal} 

1029 location (C{LatLon} or C{Cartesian} with a C{height} or C{h} 

1030 attribute), otherwise C{0 meter}. 

1031 @arg N: The I{geoid} height (C{meter}), the height of the geoid above the 

1032 ellipsoid at the same B{C{h_loc}} location. 

1033 

1034 @return: I{Orthometric} height C{B{H} = B{h} - B{N}} (C{meter}, same units 

1035 as B{C{h}} and B{C{N}}). 

1036 

1037 @see: U{Ellipsoid, Geoid, and Orthometric Heights<https://www.NGS.NOAA.gov/ 

1038 GEOID/PRESENTATIONS/2007_02_24_CCPS/Roman_A_PLSC2007notes.pdf>}, page 

1039 6 and module L{pygeodesy.geoids}. 

1040 ''' 

1041 h = h_loc if _isHeight(h_loc) else _xattr(h_loc, height=_xattr(h_loc, h=0)) 

1042 return Height(H=Height(h=h) - Height(N=N)) 

1043 

1044 

1045def horizon(height, radius=R_M, refraction=False): 

1046 '''Determine the distance to the horizon from a given altitude above the 

1047 (spherical) earth. 

1048 

1049 @arg height: Altitude (C{meter} or same units as B{C{radius}}). 

1050 @kwarg radius: Optional mean earth radius (C{meter}). 

1051 @kwarg refraction: Consider atmospheric refraction (C{bool}). 

1052 

1053 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}). 

1054 

1055 @raise ValueError: Invalid B{C{height}} or B{C{radius}}. 

1056 

1057 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}. 

1058 ''' 

1059 h, r = Height(height), Radius(radius) 

1060 if min(h, r) < 0: 

1061 raise _ValueError(height=height, radius=radius) 

1062 

1063 if refraction: 

1064 r *= 2.415750694528 # 2.0 / 0.8279 

1065 else: 

1066 r += r + h 

1067 return sqrt0(r * h) 

1068 

1069 

1070class _idllmn6(object): # see also .geodesicw._wargs, .latlonBase._toCartesian3, .vector2d._numpy 

1071 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}. 

1072 ''' 

1073 @contextmanager # <https://www.Python.org/dev/peps/pep-0343/> Examples 

1074 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds): 

1075 try: 

1076 if wrap: 

1077 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap) 

1078 kwds = _xkwds(kwds, wrap=wrap) # for _xError 

1079 m = small if small is _100km else Meter_(small=small) 

1080 n = typename(intersections2 if s else intersection2) 

1081 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m: 

1082 d, m = None, _MODS.vector3d 

1083 _i = m._intersects2 if s else m._intersect3d3 

1084 elif _isRadius(datum) and datum < 0 and not s: 

1085 d = _spherical_datum(-datum, name=n) 

1086 m = _MODS.sphericalNvector 

1087 _i = m.intersection 

1088 else: 

1089 d = _spherical_datum(datum, name=n) 

1090 if d.isSpherical: 

1091 m = _MODS.sphericalTrigonometry 

1092 _i = m._intersects2 if s else m._intersect 

1093 elif d.isEllipsoidal: 

1094 try: 

1095 if d.ellipsoid.geodesic: 

1096 pass 

1097 m = _MODS.ellipsoidalKarney 

1098 except ImportError: 

1099 m = _MODS.ellipsoidalExact 

1100 _i = m._intersections2 if s else m._intersection3 # ellipsoidalBaseDI 

1101 else: 

1102 raise _TypeError(datum=datum) 

1103 yield _i, d, lat2, lon2, m, n 

1104 

1105 except (TypeError, ValueError) as x: 

1106 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum, 

1107 lat2=lat2, lon2=lon2, small=small, **kwds) 

1108 

1109_idllmn6 = _idllmn6() # PYCHOK singleton 

1110 

1111 

1112def intersection2(lat1, lon1, bearing1, 

1113 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True 

1114 '''I{Conveniently} compute the intersection of two lines each defined by 

1115 a (geodetic) point and a bearing from North, using either ... 

1116 

1117 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km 

1118 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1119 

1120 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}} 

1121 or a C{scalar B{datum}} representing the earth radius, conventionally 

1122 in C{meter} or ... 

1123 

1124 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative} 

1125 C{scalar}, (negative) earth radius, conventionally in C{meter} or ... 

1126 

1127 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}} 

1128 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1129 is installed, otherwise ... 

1130 

1131 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal. 

1132 

1133 @arg lat1: Latitude of the first point (C{degrees}). 

1134 @arg lon1: Longitude of the first point (C{degrees}). 

1135 @arg bearing1: Bearing at the first point (compass C{degrees360}). 

1136 @arg lat2: Latitude of the second point (C{degrees}). 

1137 @arg lon2: Longitude of the second point (C{degrees}). 

1138 @arg bearing2: Bearing at the second point (compass C{degrees360}). 

1139 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1140 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth radius 

1141 (C{meter}, same units as B{C{radius1}} and B{C{radius2}}) 

1142 or C{None}. 

1143 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and 

1144 B{C{lon2}} (C{bool}). 

1145 @kwarg small: Upper limit for small distances (C{meter}). 

1146 

1147 @return: Intersection point (L{LatLon2Tuple}C{(lat, lon)}). 

1148 

1149 @raise IntersectionError: No or an ambiguous intersection or colinear, 

1150 parallel or otherwise non-intersecting lines. 

1151 

1152 @raise TypeError: Invalid B{C{datum}}. 

1153 

1154 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}}, B{C{lat2}}, 

1155 B{C{lon2}} or B{C{bearing2}}. 

1156 

1157 @see: Method L{RhumbLine.intersection2}. 

1158 ''' 

1159 b1 = Bearing(bearing1=bearing1) 

1160 b2 = Bearing(bearing2=bearing2) 

1161 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1162 small, wrap, False, bearing1=b1, bearing2=b2) as t: 

1163 _i, d, lat2, lon2, m, n = t 

1164 if d is None: 

1165 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1, 

1166 m.Vector3d(lon2, lat2, 0), b2, useZ=False) 

1167 t = LatLon2Tuple(t.y, t.x, name=n) 

1168 

1169 else: 

1170 t = _i(m.LatLon(lat1, lon1, datum=d), b1, 

1171 m.LatLon(lat2, lon2, datum=d), b2, 

1172 LatLon=None, height=0, wrap=False) 

1173 if isinstance(t, Intersection3Tuple): # ellipsoidal 

1174 t, _, _ = t 

1175 t = LatLon2Tuple(t.lat, t.lon, name=n) 

1176 return t 

1177 

1178 

1179def intersections2(lat1, lon1, radius1, 

1180 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True 

1181 '''I{Conveniently} compute the intersections of two circles each defined 

1182 by a (geodetic) center point and a radius, using either ... 

1183 

1184 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km 

1185 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1186 

1187 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}} 

1188 or a C{scalar B{datum}} representing the earth radius, conventionally 

1189 in C{meter} or ... 

1190 

1191 3) L{ellipsoidalKarney.intersections2} 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 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal. 

1196 

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

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

1199 @arg radius1: Radius of the first circle (C{meter}, conventionally). 

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

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

1202 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}). 

1203 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} 

1204 or L{a_f2Tuple}) or C{scalar} earth radius (C{meter}, same units as 

1205 B{C{radius1}} and B{C{radius2}}) or C{None}. 

1206 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and B{C{lon2}} 

1207 (C{bool}). 

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

1209 

1210 @return: 2-Tuple of the intersection points, each a L{LatLon2Tuple}C{(lat, lon)}. 

1211 Both points are the same instance, aka the I{radical center} if the 

1212 circles are abutting 

1213 

1214 @raise IntersectionError: Concentric, antipodal, invalid or non-intersecting 

1215 circles or no convergence. 

1216 

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

1218 

1219 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}}, B{C{lat2}}, 

1220 B{C{lon2}} or B{C{radius2}}. 

1221 ''' 

1222 r1 = Radius_(radius1=radius1) 

1223 r2 = Radius_(radius2=radius2) 

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

1225 small, wrap, True, radius1=r1, radius2=r2) as t: 

1226 _i, d, lat2, lon2, m, n = t 

1227 if d is None: 

1228 r1 = m2degrees(r1, radius=R_M, lat=lat1) 

1229 r2 = m2degrees(r2, radius=R_M, lat=lat2) 

1230 

1231 def _V2T(x, y, _, **unused): # _ == z unused 

1232 return LatLon2Tuple(y, x, name=n) 

1233 

1234 t = _i(m.Vector3d(lon1, lat1, 0), r1, 

1235 m.Vector3d(lon2, lat2, 0), r2, sphere=False, 

1236 Vector=_V2T) 

1237 else: 

1238 def _LL2T(lat, lon, **unused): 

1239 return LatLon2Tuple(lat, lon, name=n) 

1240 

1241 t = _i(m.LatLon(lat1, lon1, datum=d), r1, 

1242 m.LatLon(lat2, lon2, datum=d), r2, 

1243 LatLon=_LL2T, height=0, wrap=False) 

1244 return t 

1245 

1246 

1247def isantipode(lat1, lon1, lat2, lon2, eps=EPS): 

1248 '''Check whether two points are I{antipodal}, on diametrically 

1249 opposite sides of the earth. 

1250 

1251 @arg lat1: Latitude of one point (C{degrees}). 

1252 @arg lon1: Longitude of one point (C{degrees}). 

1253 @arg lat2: Latitude of the other point (C{degrees}). 

1254 @arg lon2: Longitude of the other point (C{degrees}). 

1255 @kwarg eps: Tolerance for near-equality (C{degrees}). 

1256 

1257 @return: C{True} if points are antipodal within the 

1258 B{C{eps}} tolerance, C{False} otherwise. 

1259 

1260 @see: Functions L{isantipode_} and L{antipode}. 

1261 ''' 

1262 return (fabs(lat1 + lat2) <= eps and 

1263 fabs(lon1 + lon2) <= eps) or _isequalTo( 

1264 normal(lat1, lon1), antipode(lat2, lon2), eps) 

1265 

1266 

1267def isantipode_(phi1, lam1, phi2, lam2, eps=EPS): 

1268 '''Check whether two points are I{antipodal}, on diametrically 

1269 opposite sides of the earth. 

1270 

1271 @arg phi1: Latitude of one point (C{radians}). 

1272 @arg lam1: Longitude of one point (C{radians}). 

1273 @arg phi2: Latitude of the other point (C{radians}). 

1274 @arg lam2: Longitude of the other point (C{radians}). 

1275 @kwarg eps: Tolerance for near-equality (C{radians}). 

1276 

1277 @return: C{True} if points are antipodal within the 

1278 B{C{eps}} tolerance, C{False} otherwise. 

1279 

1280 @see: Functions L{isantipode} and L{antipode_}. 

1281 ''' 

1282 return (fabs(phi1 + phi2) <= eps and 

1283 fabs(lam1 + lam2) <= eps) or _isequalTo_( 

1284 normal_(phi1, lam1), antipode_(phi2, lam2), eps) 

1285 

1286 

1287def _isequalTo(p1, p2, eps=EPS): 

1288 '''Compare 2 point lat-/lons ignoring C{class}. 

1289 ''' 

1290 return (fabs(p1.lat - p2.lat) <= eps and 

1291 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon) 

1292 

1293 

1294def _isequalTo_(p1, p2, eps=EPS): # underscore_! 

1295 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}. 

1296 ''' 

1297 return (fabs(p1.phi - p2.phi) <= eps and 

1298 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam) 

1299 

1300 

1301def isnormal(lat, lon, eps=0): 

1302 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their 

1303 respective I{normal} range in C{degrees}. 

1304 

1305 @arg lat: Latitude (C{degrees}). 

1306 @arg lon: Longitude (C{degrees}). 

1307 @kwarg eps: Optional tolerance C{degrees}). 

1308 

1309 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and 

1310 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} otherwise. 

1311 

1312 @see: Functions L{isnormal_} and L{normal}. 

1313 ''' 

1314 return _loneg(fabs(lon)) >= eps and (_90_0 - fabs(lat)) >= eps # co-latitude 

1315 

1316 

1317def isnormal_(phi, lam, eps=0): 

1318 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their 

1319 respective I{normal} range in C{radians}. 

1320 

1321 @arg phi: Latitude (C{radians}). 

1322 @arg lam: Longitude (C{radians}). 

1323 @kwarg eps: Optional tolerance C{radians}). 

1324 

1325 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and 

1326 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise. 

1327 

1328 @see: Functions L{isnormal} and L{normal_}. 

1329 ''' 

1330 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps 

1331 

1332 

1333def _maprod(fun_, *ts): 

1334 '''(INTERNAL) Helper for C{excessCagnoli_} and C{excessLHuilier_}. 

1335 ''' 

1336 return fprod(map(fun_, ts)) 

1337 

1338 

1339def _normal2(a, b, n_2, n, n2): 

1340 '''(INTERNAL) Helper for C{normal} and C{normal_}. 

1341 ''' 

1342 if fabs(b) > n: 

1343 b = remainder(b, n2) 

1344 if fabs(a) > n_2: 

1345 r = remainder(a, n) 

1346 if r != a: 

1347 a = -r 

1348 b -= n if b > 0 else -n 

1349 return float0_(a, b) 

1350 

1351 

1352def normal(lat, lon, **name): 

1353 '''Normalize a lat- I{and} longitude pair in C{degrees}. 

1354 

1355 @arg lat: Latitude (C{degrees}). 

1356 @arg lon: Longitude (C{degrees}). 

1357 @kwarg name: Optional C{B{name}="normal"} (C{str}). 

1358 

1359 @return: L{LatLon2Tuple}C{(lat, lon)} with C{-90 <= lat <= 90} 

1360 and C{-180 <= lon <= 180}. 

1361 

1362 @see: Functions L{normal_} and L{isnormal}. 

1363 ''' 

1364 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0), 

1365 name=_name__(name, name__=normal)) 

1366 

1367 

1368def normal_(phi, lam, **name): 

1369 '''Normalize a lat- I{and} longitude pair in C{radians}. 

1370 

1371 @arg phi: Latitude (C{radians}). 

1372 @arg lam: Longitude (C{radians}). 

1373 @kwarg name: Optional C{B{name}="normal_"} (C{str}). 

1374 

1375 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2} 

1376 and C{abs(lam) <= PI}. 

1377 

1378 @see: Functions L{normal} and L{isnormal_}. 

1379 ''' 

1380 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2), 

1381 name=_name__(name, name__=normal_)) 

1382 

1383 

1384def _opposes(d, m, n, n2): 

1385 '''(INTERNAL) Helper for C{opposing} and C{opposing_}. 

1386 ''' 

1387 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1 

1388 return False if d < m or d > (n2 - m) else ( 

1389 True if (n - m) < d < (n + m) else None) 

1390 

1391 

1392def opposing(bearing1, bearing2, margin=_90_0): 

1393 '''Compare the direction of two bearings given in C{degrees}. 

1394 

1395 @arg bearing1: First bearing (compass C{degrees}). 

1396 @arg bearing2: Second bearing (compass C{degrees}). 

1397 @kwarg margin: Optional, interior angle bracket (C{degrees}). 

1398 

1399 @return: C{True} if both bearings point in opposite, C{False} if 

1400 in similar or C{None} if in I{perpendicular} directions. 

1401 

1402 @see: Function L{opposing_}. 

1403 ''' 

1404 m = Degrees_(margin=margin, low=EPS0, high=_90_0) 

1405 return _opposes(bearing2 - bearing1, m, _180_0, _360_0) 

1406 

1407 

1408def opposing_(radians1, radians2, margin=PI_2): 

1409 '''Compare the direction of two bearings given in C{radians}. 

1410 

1411 @arg radians1: First bearing (C{radians}). 

1412 @arg radians2: Second bearing (C{radians}). 

1413 @kwarg margin: Optional, interior angle bracket (C{radians}). 

1414 

1415 @return: C{True} if both bearings point in opposite, C{False} if 

1416 in similar or C{None} if in perpendicular directions. 

1417 

1418 @see: Function L{opposing}. 

1419 ''' 

1420 m = Radians_(margin=margin, low=EPS0, high=PI_2) 

1421 return _opposes(radians2 - radians1, m, PI, PI2) 

1422 

1423 

1424def _Propy(func, nargs, kwds): 

1425 '''(INTERNAL) Helper for the C{frechet.[-]Frechet**} and 

1426 C{hausdorff.[-]Hausdorff*} classes. 

1427 ''' 

1428 try: 

1429 _xcallable(distance=func) 

1430 # assert _args_kwds_count2(func)[0] == nargs + int(ismethod(func)) 

1431 _ = func(*_0_0s(nargs), **kwds) 

1432 except Exception as x: 

1433 t = unstr(func, **kwds) 

1434 raise _TypeError(t, cause=x) 

1435 return func 

1436 

1437 

1438def _radical2(d, r1, r2, **name): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d 

1439 # (INTERNAL) See C{radical2} below 

1440 # assert d > EPS0 

1441 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5 

1442 n = _name__(name, name__=radical2) 

1443 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d, name=n) 

1444 

1445 

1446def radical2(distance, radius1, radius2, **name): 

1447 '''Compute the I{radical ratio} and I{radical line} of two U{intersecting 

1448 circles<https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}. 

1449 

1450 The I{radical line} is perpendicular to the axis thru the centers of 

1451 the circles at C{(0, 0)} and C{(B{distance}, 0)}. 

1452 

1453 @arg distance: Distance between the circle centers (C{scalar}). 

1454 @arg radius1: Radius of the first circle (C{scalar}). 

1455 @arg radius2: Radius of the second circle (C{scalar}). 

1456 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1457 

1458 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <= ratio <= 

1459 1.0} and C{xline} is along the B{C{distance}}. 

1460 

1461 @raise IntersectionError: The B{C{distance}} exceeds the sum of 

1462 B{C{radius1}} and B{C{radius2}}. 

1463 

1464 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or B{C{radius2}}. 

1465 ''' 

1466 d = Distance_(distance, low=_0_0) 

1467 r1 = Radius_(radius1=radius1) 

1468 r2 = Radius_(radius2=radius2) 

1469 if d > (r1 + r2): 

1470 raise IntersectionError(distance=d, radius1=r1, radius2=r2, 

1471 txt=_too_(_distant_)) 

1472 return _radical2(d, r1, r2, **name) if d > EPS0 else \ 

1473 Radical2Tuple(_0_5, _0_0, **name) 

1474 

1475 

1476class Radical2Tuple(_NamedTuple): 

1477 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and 

1478 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0} 

1479 ''' 

1480 _Names_ = (_ratio_, _xline_) 

1481 _Units_ = ( Scalar, Scalar) 

1482 

1483 

1484def _radistance(inst): 

1485 '''(INTERNAL) Helper for the L{frechet._FrechetMeterRadians} 

1486 and L{hausdorff._HausdorffMeterRedians} classes. 

1487 ''' 

1488 wrap_, kwds_ = _xkwds_pop2(inst._kwds, wrap=False) 

1489 func_ = inst._func_ 

1490 try: # calling lower-overhead C{func_} 

1491 func_(0, _0_25, _0_5, **kwds_) 

1492 wrap_ = _Wrap._philamop(wrap_) 

1493 except TypeError: 

1494 return inst.distance 

1495 

1496 def _philam(p): 

1497 try: 

1498 return p.phi, p.lam 

1499 except AttributeError: # no .phi or .lam 

1500 return radians(p.lat), radians(p.lon) 

1501 

1502 def _func_wrap(point1, point2): 

1503 phi1, lam1 = wrap_(*_philam(point1)) 

1504 phi2, lam2 = wrap_(*_philam(point2)) 

1505 return func_(phi2, phi1, lam2 - lam1, **kwds_) 

1506 

1507 inst._units = inst._units_ 

1508 return _func_wrap 

1509 

1510 

1511def _scale_deg(lat1, lat2): # degrees 

1512 # scale factor cos(mean of lats) for delta lon 

1513 m = fabs(lat1 + lat2) * _0_5 

1514 return cos(radians(m)) if m < _90_0 else _0_0 

1515 

1516 

1517def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights 

1518 # scale factor cos(mean of phis) for delta lam 

1519 m = fabs(phi1 + phi2) * _0_5 

1520 return cos(m) if m < PI_2 else _0_0 

1521 

1522 

1523def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw 

1524 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine. 

1525 ''' 

1526 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21) 

1527 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21 

1528 

1529 

1530def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

1531 '''Compute the distance between two (ellipsoidal) points using U{Thomas' 

1532 <https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} formula. 

1533 

1534 @arg lat1: Start latitude (C{degrees}). 

1535 @arg lon1: Start longitude (C{degrees}). 

1536 @arg lat2: End latitude (C{degrees}). 

1537 @arg lon2: End longitude (C{degrees}). 

1538 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} 

1539 or L{a_f2Tuple}) to use. 

1540 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and 

1541 B{C{lon2}} (C{bool}). 

1542 

1543 @return: Distance (C{meter}, same units as the B{C{datum}}'s or ellipsoid axes). 

1544 

1545 @raise TypeError: Invalid B{C{datum}}. 

1546 

1547 @see: Functions L{thomas_}, L{cosineLaw}, L{equirectangular}, L{euclidean}, 

1548 L{flatLocal} / L{hubeny}, L{flatPolar}, L{haversine}, L{vincentys} and 

1549 method L{Ellipsoid.distance2}. 

1550 ''' 

1551 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2) 

1552 

1553 

1554def thomas_(phi2, phi1, lam21, datum=_WGS84): 

1555 '''Compute the I{angular} distance between two (ellipsoidal) points using 

1556 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} formula. 

1557 

1558 @arg phi2: End latitude (C{radians}). 

1559 @arg phi1: Start latitude (C{radians}). 

1560 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1561 @kwarg datum: Datum (L{Datum}) ?or ellipsoid to use (L{Ellipsoid}, 

1562 L{Ellipsoid2} or L{a_f2Tuple}). 

1563 

1564 @return: Angular distance (C{radians}). 

1565 

1566 @raise TypeError: Invalid B{C{datum}}. 

1567 

1568 @see: Functions L{thomas}, L{cosineLaw_}, L{euclidean_}, L{flatLocal_} / 

1569 L{hubeny_}, L{flatPolar_}, L{haversine_} and L{vincentys_} and 

1570 U{Geodesy-PHP<https://GitHub.com/jtejido/geodesy-php/blob/master/ 

1571 src/Geodesy/Distance/ThomasFormula.php>}. 

1572 ''' 

1573 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21) 

1574 if r and isnon0(c1) and isnon0(c2): 

1575 E = _ellipsoidal(datum, thomas_) 

1576 f = E.f * _0_25 

1577 if f: # ellipsoidal 

1578 r1 = atan2(E.b_a * s1, c1) 

1579 r2 = atan2(E.b_a * s2, c2) 

1580 

1581 j = (r2 + r1) * _0_5 

1582 k = (r2 - r1) * _0_5 

1583 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5) 

1584 

1585 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2) 

1586 u = _1_0 - h 

1587 if isnon0(u) and isnon0(h): 

1588 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h) 

1589 sr, cr = sincos2(r) 

1590 if isnon0(sr): 

1591 u = (sj * ck)**2 * 2 / u 

1592 h = (sk * cj)**2 * 2 / h 

1593 x = u + h 

1594 y = u - h 

1595 

1596 b = r * 2 

1597 s = r / sr 

1598 e = 4 * s**2 

1599 d = 2 * cr 

1600 a = e * d 

1601 c = s - (a - d) * _0_5 

1602 

1603 t = fdot_(a, x, -b, y, -d, y**2, c, x**2, e, x * y) * _0_25 

1604 r -= fdot_(s, x, -1, y, -t, f) * f * sr 

1605 return r 

1606 

1607 

1608def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1609 '''Compute the distance between two (spherical) points using 

1610 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1611 spherical formula. 

1612 

1613 @arg lat1: Start latitude (C{degrees}). 

1614 @arg lon1: Start longitude (C{degrees}). 

1615 @arg lat2: End latitude (C{degrees}). 

1616 @arg lon2: End longitude (C{degrees}). 

1617 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) or 

1618 ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) 

1619 to use. 

1620 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and 

1621 B{C{lon2}} (C{bool}). 

1622 

1623 @return: Distance (C{meter}, same units as B{C{radius}}). 

1624 

1625 @raise UnitError: Invalid B{C{radius}}. 

1626 

1627 @see: Functions L{vincentys_}, L{cosineLaw}, L{equirectangular}, L{euclidean}, 

1628 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine} and L{thomas} and 

1629 methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*} and 

1630 C{LatLon.equirectangularTo}. 

1631 

1632 @note: See note at function L{vincentys_}. 

1633 ''' 

1634 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2) 

1635 

1636 

1637def vincentys_(phi2, phi1, lam21): 

1638 '''Compute the I{angular} distance between two (spherical) points using 

1639 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1640 spherical formula. 

1641 

1642 @arg phi2: End latitude (C{radians}). 

1643 @arg phi1: Start latitude (C{radians}). 

1644 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1645 

1646 @return: Angular distance (C{radians}). 

1647 

1648 @see: Functions L{vincentys}, L{cosineLaw_}, L{euclidean_}, L{flatLocal_} / 

1649 L{hubeny_}, L{flatPolar_}, L{haversine_} and L{thomas_}. 

1650 

1651 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_} produce 

1652 equivalent results, but L{vincentys_} is suitable for antipodal 

1653 points and slightly more expensive (M{3 cos, 3 sin, 1 hypot, 1 atan2, 

1654 6 mul, 2 add}) than L{haversine_} (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 

1655 mul, 1 add}) and L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}). 

1656 ''' 

1657 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21) 

1658 

1659 c = c2 * c21 

1660 x = s1 * s2 + c1 * c 

1661 y = c1 * s2 - s1 * c 

1662 return atan2(hypot(c2 * s21, y), x) 

1663 

1664# **) MIT License 

1665# 

1666# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved. 

1667# 

1668# Permission is hereby granted, free of charge, to any person obtaining a 

1669# copy of this software and associated documentation files (the "Software"), 

1670# to deal in the Software without restriction, including without limitation 

1671# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1672# and/or sell copies of the Software, and to permit persons to whom the 

1673# Software is furnished to do so, subject to the following conditions: 

1674# 

1675# The above copyright notice and this permission notice shall be included 

1676# in all copies or substantial portions of the Software. 

1677# 

1678# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1679# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1680# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1681# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1682# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1683# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1684# OTHER DEALINGS IN THE SOFTWARE.