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 _args_kwds_count2, _copysign # from .constants 

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, _copysign 

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 _DUNDER_nameof # from .named 

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

25from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS 

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

27 _DUNDER_nameof 

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

29 Intersection3Tuple, PhiLam2Tuple 

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

31# from pygeodesy.triaxials import _hartzell3 # _MODS 

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

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

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

35from pygeodesy.utily import acos1, asin1, atan2, atan2b, degrees2m, hav, _loneg, \ 

36 m2degrees, tan_2, sincos2, sincos2_, _Wrap 

37# from pygeodesy.vector3d import _otherV3d # _MODS 

38# from pygeodesy.vector3dBase import _xyz_y_z3 # _MODS 

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

40# sphericalNvector, sphericalTrigonometry # _MODS 

41 

42from contextlib import contextmanager 

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

44 

45__all__ = _ALL_LAZY.formy 

46__version__ = '25.01.05' 

47 

48_RADIANS2 = radians(_1_0)**2 # degree to radians-squared 

49_ratio_ = 'ratio' 

50_xline_ = 'xline' 

51 

52 

53def angle2chord(rad, radius=R_M): 

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

55 

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

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

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

59 

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

61 

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

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

64 ''' 

65 d = _isDegrees(rad, iscalar=False) 

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

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

68 

69 

70def _anti2(a, b, n_2, n, n2): 

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

72 ''' 

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

74 if r == a: 

75 r = -r 

76 b += n 

77 if fabs(b) > n: 

78 b = remainder(b, n2) 

79 return float0_(r, b) 

80 

81 

82def antipode(lat, lon, **name): 

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

84 point in C{degrees}. 

85 

86 @arg lat: Latitude (C{degrees}). 

87 @arg lon: Longitude (C{degrees}). 

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

89 

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

91 

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

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

94 ''' 

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

96 

97 

98def antipode_(phi, lam, **name): 

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

100 point in C{radians}. 

101 

102 @arg phi: Latitude (C{radians}). 

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

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

105 

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

107 

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

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

110 ''' 

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

112 

113 

114def bearing(lat1, lon1, lat2, lon2, **final_wrap): 

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

116 (spherical) points. 

117 

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

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

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

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

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

123 

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

125 coincide. 

126 ''' 

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

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

129 return degrees(r) 

130 

131 

132def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False): 

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

134 (spherical) points. 

135 

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

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

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

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

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

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

142 (C{bool}). 

143 

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

145 coincide. 

146 

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

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

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

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

151 ''' 

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

153 if final: # swap plus PI 

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

155 r = PI3 

156 else: 

157 r = PI2 

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

159 

160 x = ca1 * sa2 - sa1 * ca2 * cdb 

161 y = sdb * ca2 

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

163 

164 

165def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf 

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

167 ''' 

168 try: # for LatLon_ and ellipsoidal LatLon 

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

170 except AttributeError: 

171 pass 

172 # XXX spherical version, OK for ellipsoidal ispolar? 

173 t = p1.philam + p2.philam 

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

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

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

177 name__=_bearingTo2) 

178 

179 

180def chord2angle(chord, radius=R_M): 

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

182 

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

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

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

186 

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

188 

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

190 

191 @see: Function L{angle2chord}. 

192 ''' 

193 m = _mean_radius(radius) 

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

195 if r: 

196 i = int(r) 

197 if i > 0: 

198 r -= i 

199 i *= PI 

200 r = (asin1(r) + i) * _2_0 

201 return _copysign(r, chord) 

202 

203 

204def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False): 

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

206 lat2 - lat1)} between two points. 

207 

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

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

210 

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

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

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

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

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

216 latitude (C{bool}). 

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

218 B{C{lon2}} (C{bool}). 

219 

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

221 

222 @note: Courtesy of Martin Schultz. 

223 

224 @see: U{Local, flat earth approximation 

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

226 ''' 

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

228 if adjust: # scale delta lon 

229 d_lon *= _scale_deg(lat1, lat2) 

230 return atan2b(d_lon, lat2 - lat1) 

231 

232 

233def cosineLaw(lat1, lon1, lat2, lon2, corr=0, earth=None, wrap=False, 

234 datum=_WGS84, radius=R_M): 

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

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

237 formula, optionally corrected. 

238 

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

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

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

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

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

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

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

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

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

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

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

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

251 (C{bool}). 

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

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

254 

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

256 ellipsoid axes). 

257 

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

259 

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

261 

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

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

264 method L{Ellipsoid.distance2}. 

265 

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

267 ''' 

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

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

270 

271 

272def cosineLaw_(phi2, phi1, lam21, corr=0, earth=None, datum=_WGS84): 

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

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

275 optionally corrected. 

276 

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

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

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

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

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

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

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

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

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

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

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

288 

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

290 

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

292 

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

294 

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

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

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

298 AndoyerLambert.php>}. 

299 ''' 

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

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

302 E = _ellipsoidal(earth or datum, cosineLaw_) 

303 f = _0_25 * E.f 

304 if f: # ellipsoidal 

305 if corr == 1: # Andoyer-Lambert 

306 r2 = atan2(E.b_a * s2, c2) 

307 r1 = atan2(E.b_a * s1, c1) 

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

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

310 if r: 

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

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

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

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

315 r += (c - s) * _0_5 * f 

316 

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

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

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

320 s = (s1 + s2)**2 / (_1_0 + cr) 

321 t = (s1 - s2)**2 / (_1_0 - cr) 

322 x = s + t 

323 y = s - t 

324 

325 s = 8 * r**2 / sr 

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

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

328 b = -2 * d 

329 e = 30 * s2r 

330 

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

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

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

334 else: 

335 raise _ValueError(corr=corr) 

336 return r 

337 

338 

339def _d3(wrap, lat1, lon1, lat2, lon2): 

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

341 ''' 

342 if wrap: 

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

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

345 else: # for backward compaibility 

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

347 

348 

349def _dE(fun_, earth, wrap, *lls, **corr): 

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

351 ''' 

352 E = _ellipsoidal(earth, fun_) 

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

354 return r * E.a 

355 

356 

357def _dS(fun_, radius, wrap, *lls, **adjust): 

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

359 ''' 

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

361 if radius is not R_M: 

362 try: # datum? 

363 radius = radius.ellipsoid.R1 

364 except AttributeError: 

365 pass # scalar? 

366 lat1, _, lat2, _ = lls 

367 radius = _mean_radius(radius, lat1, lat2) 

368 return r * radius 

369 

370 

371def _ellipsoidal(earth, where): 

372 '''(INTERNAL) Helper for distances. 

373 ''' 

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

375 earth if isinstance(earth, Ellipsoid) else 

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

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

378 

379 

380def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **adjust_limit_wrap): 

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

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

383 

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

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

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

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

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

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

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

391 

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

393 ellipsoid axes). 

394 

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

396 

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

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

399 ''' 

400 r = _mean_radius(radius, lat1, lat2) 

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

402 Lat(lat2=lat2), Lon(lon2=lon2), 

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

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

405 

406 

407def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap): 

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

409 L{hausdorff._HausdorffMeterRedians}. 

410 ''' 

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

412 return t.distance2 * _RADIANS2 

413 

414 

415def equirectangular4(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False): 

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

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

418 

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

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

421 

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

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

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

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

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

427 mean latitude (C{bool}). 

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

429 or C{0} for unlimited. 

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

431 (C{bool}). 

432 

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

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

435 

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

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

438 

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

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

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

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

443 ''' 

444 if wrap: 

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

446 else: 

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

448 d_lat = lat2 - lat1 

449 

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

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

452 if d > limit: 

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

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

455 limit=limit, wrap=wrap) 

456 raise LimitError(s, txt=t) 

457 

458 if adjust: # scale delta lon 

459 d_lon *= _scale_deg(lat1, lat2) 

460 

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

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

463 

464 

465def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False): 

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

467 

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

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

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

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

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

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

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

475 latitude (C{bool}). 

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

477 B{C{lon2}} (C{bool}). 

478 

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

480 or datum axes). 

481 

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

483 

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

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

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

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

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

489 C{LatLon.equirectangularTo}. 

490 ''' 

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

492 

493 

494def euclidean_(phi2, phi1, lam21, adjust=True): 

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

496 

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

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

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

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

501 latitude (C{bool}). 

502 

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

504 

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

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

507 ''' 

508 if adjust: 

509 lam21 *= _scale_rad(phi2, phi1) 

510 return euclid(phi2 - phi1, lam21) 

511 

512 

513def excessAbc_(A, b, c): 

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

515 and the included (small) angle. 

516 

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

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

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

520 

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

522 

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

524 

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

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

527 ''' 

528 A = Radians_(A=A) 

529 b = Radians_(b=b) * _0_5 

530 c = Radians_(c=c) * _0_5 

531 

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

533 s = sA * sb * sc 

534 c = cA * sb * sc + cc * cb 

535 return atan2(s, c) * _2_0 

536 

537 

538def excessCagnoli_(a, b, c): 

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

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

541 

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

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

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

545 

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

547 

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

549 

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

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

552 ''' 

553 a = Radians_(a=a) 

554 b = Radians_(b=b) 

555 c = Radians_(c=c) 

556 

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

558 if r: 

559 s = fsumf_(a, b, c) * _0_5 

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

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

562 return Radians(Cagnoli=r * _2_0) 

563 

564 

565def excessGirard_(A, B, C): 

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

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

568 

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

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

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

572 

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

574 

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

576 

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

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

579 ''' 

580 r = fsumf_(Radians_(A=A), 

581 Radians_(B=B), 

582 Radians_(C=C), -PI) 

583 return Radians(Girard=r) 

584 

585 

586def excessLHuilier_(a, b, c): 

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

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

589 

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

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

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

593 

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

595 

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

597 

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

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

600 ''' 

601 a = Radians_(a=a) 

602 b = Radians_(b=b) 

603 c = Radians_(c=c) 

604 

605 s = fsumf_(a, b, c) * _0_5 

606 r = _maprod(tan_2, s, s - a, s - b, s - c) 

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

608 return Radians(LHuilier=r * _4_0) 

609 

610 

611def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

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

615 method. 

616 

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

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

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

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

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

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

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

624 B{C{lon2}} (C{bool}). 

625 

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

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

628 if C{B{radius}=0} or C{None}. 

629 

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

631 

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

633 

634 @raise ValueError: Semi-circular longitudinal delta. 

635 

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

637 ''' 

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

639 if radius: 

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

641 return r 

642 

643 

644def excessKarney_(phi2, phi1, lam21): 

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

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

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

648 method. 

649 

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

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

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

653 

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

655 

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

657 

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

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

660 ''' 

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

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

663 # 

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

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

666 # 1 + tan(φ1 / 2) · tan(φ2 / 2) 

667 # 

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

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

670 t2 = tan_2(phi2) 

671 t1 = tan_2(phi1) 

672 c = (t1 * t2) + _1_0 

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

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

675 

676 

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

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

679# 

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

681# 

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

683# 

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

685# 

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

687# 

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

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

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

691# 

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

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

694# 

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

696# 

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

698# inverse Gudermannian) function 

699# 

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

701# 

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

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

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

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

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

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

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

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

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

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

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

713# - Charles Karney 

714 

715 

716def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

719 

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

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

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

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

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

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

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

727 B{C{lon2}} (C{bool}). 

728 

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

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

731 if C{B{radius}=0} or C{None}. 

732 

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

734 

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

736 

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

738 ''' 

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

740 if radius: 

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

742 return r 

743 

744 

745def excessQuad_(phi2, phi1, lam21): 

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

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

748 

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

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

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

752 

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

754 

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

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

757 ''' 

758 c = cos((phi2 - phi1) * _0_5) 

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

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

761 

762 

763def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False): 

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

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

766 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

768 

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

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

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

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

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

774 or L{a_f2Tuple}) to use. 

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

776 method L{pygeodesy.Ellipsoid.roc2_}. 

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

778 B{C{lon2}} (C{bool}). 

779 

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

781 

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

783 

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

785 scaled at the mean of both latitude. 

786 

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

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

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

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

791 ''' 

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

793 E = _ellipsoidal(datum, flatLocal) 

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

795 

796hubeny = flatLocal # PYCHOK for Karl Hubeny 

797 

798 

799def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True): 

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

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

802 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

804 

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

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

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

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

809 or L{a_f2Tuple}) to use. 

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

811 method L{pygeodesy.Ellipsoid.roc2_}. 

812 

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

814 

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

816 

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

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

819 

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

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

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

823 ''' 

824 E = _ellipsoidal(datum, flatLocal_) 

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

826 

827hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny 

828 

829 

830def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

833 #Polar_coordinate_flat-Earth_formula>} formula. 

834 

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

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

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

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

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

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

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

842 (C{bool}). 

843 

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

845 ellipsoid axes). 

846 

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

848 

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

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

851 ''' 

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

853 

854 

855def flatPolar_(phi2, phi1, lam21): 

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

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

858 #Polar_coordinate_flat-Earth_formula>} formula. 

859 

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

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

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

863 

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

865 

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

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

868 ''' 

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

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

871 if a < b: 

872 a, b = b, a 

873 if a < EPS0: 

874 a = _0_0 

875 elif b > 0: 

876 b = b / a # /= chokes PyChecker 

877 c = b * cos(lam21) * _2_0 

878 c = fsumf_(_1_0, b**2, -fabs(c)) 

879 a *= sqrt0(c) 

880 return a 

881 

882 

883def _hartzell(pov, los, earth, **kwds): 

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

885 ''' 

886 if earth is None: 

887 earth = pov.datum 

888 else: 

889 earth = _spherical_datum(earth, name__=hartzell) 

890 pov = pov.toDatum(earth) 

891 h = pov.height 

892 if h < 0: # EPS0 

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

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

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

896 

897 

898def hartzell(pov, los=False, earth=_WGS84, **name_LatLon_and_kwds): 

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

900 a Point-Of-View in space. 

901 

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

903 L{Ecef9Tuple} or L{Vector3d}). 

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

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

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

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

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

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

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

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

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

913 

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

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

916 the distance to the B{C{pov}}. 

917 

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

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

920 away from the earth. 

921 

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

923 

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

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

926 ''' 

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

928 try: 

929 D = _spherical_datum(earth, name__=hartzell) 

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

931 

932 C = _MODS.cartesianBase.CartesianBase 

933 if kwds: 

934 c = C(r, datum=D) 

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

936 elif isinstance(r, C): 

937 r.height = h 

938 if i: 

939 r._iteration = i 

940 except Exception as x: 

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

942 return _xnamed(r, n) if n else r 

943 

944 

945def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

948 

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

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

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

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

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

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

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

956 B{C{lon2}} (C{bool}). 

957 

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

959 

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

961 

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

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

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

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

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

967 

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

969 ''' 

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

971 

972 

973def haversine_(phi2, phi1, lam21): 

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

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

976 

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

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

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

980 

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

982 

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

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

985 

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

987 ''' 

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

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

990 

991 

992def heightOf(angle, distance, radius=R_M): 

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

994 traveling along a straight line at a given tilt. 

995 

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

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

998 B{C{radius}}). 

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

1000 

1001 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}). 

1002 

1003 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}. 

1004 

1005 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>} 

1006 (U{Shapiro et al. 2009, JTECH 

1007 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>} 

1008 and U{Potvin et al. 2012, JTECH 

1009 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}). 

1010 ''' 

1011 r = h = Radius(radius) 

1012 d = fabs(Distance(distance)) 

1013 if d > h: 

1014 d, h = h, d 

1015 

1016 if d > EPS0: # and h > EPS0 

1017 d = d / h # /= h chokes PyChecker 

1018 s = sin(Phid(angle=angle, clip=_180_0)) 

1019 s = fsumf_(_1_0, s * d * _2_0, d**2) 

1020 if s > 0: 

1021 return h * sqrt(s) - r 

1022 

1023 raise _ValueError(angle=angle, distance=distance, radius=radius) 

1024 

1025 

1026def heightOrthometric(h_loc, N): 

1027 '''Get the I{orthometric} height B{H}, the height above the geoid, earth surface. 

1028 

1029 @arg h_loc: The height above the ellipsoid (C{meter}) or an I{ellipsoidal} 

1030 location (C{LatLon} or C{Cartesian} with a C{height} or C{h} 

1031 attribute), otherwise C{0 meter}. 

1032 @arg N: The I{geoid} height (C{meter}), the height of the geoid above the 

1033 ellipsoid at the same B{C{h_loc}} location. 

1034 

1035 @return: I{Orthometric} height C{B{H} = B{h} - B{N}} (C{meter}, same units 

1036 as B{C{h}} and B{C{N}}). 

1037 

1038 @see: U{Ellipsoid, Geoid, and Orthometric Heights<https://www.NGS.NOAA.gov/ 

1039 GEOID/PRESENTATIONS/2007_02_24_CCPS/Roman_A_PLSC2007notes.pdf>}, page 

1040 6 and module L{pygeodesy.geoids}. 

1041 ''' 

1042 h = h_loc if _isHeight(h_loc) else _xattr(h_loc, height=_xattr(h_loc, h=0)) 

1043 return Height(H=Height(h=h) - Height(N=N)) 

1044 

1045 

1046def horizon(height, radius=R_M, refraction=False): 

1047 '''Determine the distance to the horizon from a given altitude above the 

1048 (spherical) earth. 

1049 

1050 @arg height: Altitude (C{meter} or same units as B{C{radius}}). 

1051 @kwarg radius: Optional mean earth radius (C{meter}). 

1052 @kwarg refraction: Consider atmospheric refraction (C{bool}). 

1053 

1054 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}). 

1055 

1056 @raise ValueError: Invalid B{C{height}} or B{C{radius}}. 

1057 

1058 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}. 

1059 ''' 

1060 h, r = Height(height), Radius(radius) 

1061 if min(h, r) < 0: 

1062 raise _ValueError(height=height, radius=radius) 

1063 

1064 if refraction: 

1065 r *= 2.415750694528 # 2.0 / 0.8279 

1066 else: 

1067 r += r + h 

1068 return sqrt0(r * h) 

1069 

1070 

1071class _idllmn6(object): # see also .geodesicw._wargs, .latlonBase._toCartesian3, .vector2d._numpy 

1072 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}. 

1073 ''' 

1074 @contextmanager # <https://www.Python.org/dev/peps/pep-0343/> Examples 

1075 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds): 

1076 try: 

1077 if wrap: 

1078 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap) 

1079 kwds = _xkwds(kwds, wrap=wrap) # for _xError 

1080 m = small if small is _100km else Meter_(small=small) 

1081 n = _DUNDER_nameof(intersections2 if s else intersection2) 

1082 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m: 

1083 d, m = None, _MODS.vector3d 

1084 _i = m._intersects2 if s else m._intersect3d3 

1085 elif _isRadius(datum) and datum < 0 and not s: 

1086 d = _spherical_datum(-datum, name=n) 

1087 m = _MODS.sphericalNvector 

1088 _i = m.intersection 

1089 else: 

1090 d = _spherical_datum(datum, name=n) 

1091 if d.isSpherical: 

1092 m = _MODS.sphericalTrigonometry 

1093 _i = m._intersects2 if s else m._intersect 

1094 elif d.isEllipsoidal: 

1095 try: 

1096 if d.ellipsoid.geodesic: 

1097 pass 

1098 m = _MODS.ellipsoidalKarney 

1099 except ImportError: 

1100 m = _MODS.ellipsoidalExact 

1101 _i = m._intersections2 if s else m._intersection3 # ellipsoidalBaseDI 

1102 else: 

1103 raise _TypeError(datum=datum) 

1104 yield _i, d, lat2, lon2, m, n 

1105 

1106 except (TypeError, ValueError) as x: 

1107 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum, 

1108 lat2=lat2, lon2=lon2, small=small, **kwds) 

1109 

1110_idllmn6 = _idllmn6() # PYCHOK singleton 

1111 

1112 

1113def intersection2(lat1, lon1, bearing1, 

1114 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True 

1115 '''I{Conveniently} compute the intersection of two lines each defined by 

1116 a (geodetic) point and a bearing from North, using either ... 

1117 

1118 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km 

1119 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1120 

1121 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}} 

1122 or a C{scalar B{datum}} representing the earth radius, conventionally 

1123 in C{meter} or ... 

1124 

1125 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative} 

1126 C{scalar}, (negative) earth radius, conventionally in C{meter} or ... 

1127 

1128 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}} 

1129 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1130 is installed, otherwise ... 

1131 

1132 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal. 

1133 

1134 @arg lat1: Latitude of the first point (C{degrees}). 

1135 @arg lon1: Longitude of the first point (C{degrees}). 

1136 @arg bearing1: Bearing at the first point (compass C{degrees360}). 

1137 @arg lat2: Latitude of the second point (C{degrees}). 

1138 @arg lon2: Longitude of the second point (C{degrees}). 

1139 @arg bearing2: Bearing at the second point (compass C{degrees360}). 

1140 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1141 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth radius 

1142 (C{meter}, same units as B{C{radius1}} and B{C{radius2}}) 

1143 or C{None}. 

1144 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and 

1145 B{C{lon2}} (C{bool}). 

1146 @kwarg small: Upper limit for small distances (C{meter}). 

1147 

1148 @return: Intersection point (L{LatLon2Tuple}C{(lat, lon)}). 

1149 

1150 @raise IntersectionError: No or an ambiguous intersection or colinear, 

1151 parallel or otherwise non-intersecting lines. 

1152 

1153 @raise TypeError: Invalid B{C{datum}}. 

1154 

1155 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}}, B{C{lat2}}, 

1156 B{C{lon2}} or B{C{bearing2}}. 

1157 

1158 @see: Method L{RhumbLine.intersection2}. 

1159 ''' 

1160 b1 = Bearing(bearing1=bearing1) 

1161 b2 = Bearing(bearing2=bearing2) 

1162 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1163 small, wrap, False, bearing1=b1, bearing2=b2) as t: 

1164 _i, d, lat2, lon2, m, n = t 

1165 if d is None: 

1166 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1, 

1167 m.Vector3d(lon2, lat2, 0), b2, useZ=False) 

1168 t = LatLon2Tuple(t.y, t.x, name=n) 

1169 

1170 else: 

1171 t = _i(m.LatLon(lat1, lon1, datum=d), b1, 

1172 m.LatLon(lat2, lon2, datum=d), b2, 

1173 LatLon=None, height=0, wrap=False) 

1174 if isinstance(t, Intersection3Tuple): # ellipsoidal 

1175 t, _, _ = t 

1176 t = LatLon2Tuple(t.lat, t.lon, name=n) 

1177 return t 

1178 

1179 

1180def intersections2(lat1, lon1, radius1, 

1181 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True 

1182 '''I{Conveniently} compute the intersections of two circles each defined 

1183 by a (geodetic) center point and a radius, using either ... 

1184 

1185 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km 

1186 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1187 

1188 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}} 

1189 or a C{scalar B{datum}} representing the earth radius, conventionally 

1190 in C{meter} or ... 

1191 

1192 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}} 

1193 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1194 is installed, otherwise ... 

1195 

1196 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal. 

1197 

1198 @arg lat1: Latitude of the first circle center (C{degrees}). 

1199 @arg lon1: Longitude of the first circle center (C{degrees}). 

1200 @arg radius1: Radius of the first circle (C{meter}, conventionally). 

1201 @arg lat2: Latitude of the second circle center (C{degrees}). 

1202 @arg lon2: Longitude of the second circle center (C{degrees}). 

1203 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}). 

1204 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} 

1205 or L{a_f2Tuple}) or C{scalar} earth radius (C{meter}, same units as 

1206 B{C{radius1}} and B{C{radius2}}) or C{None}. 

1207 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and B{C{lon2}} 

1208 (C{bool}). 

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

1210 

1211 @return: 2-Tuple of the intersection points, each a L{LatLon2Tuple}C{(lat, lon)}. 

1212 Both points are the same instance, aka the I{radical center} if the 

1213 circles are abutting 

1214 

1215 @raise IntersectionError: Concentric, antipodal, invalid or non-intersecting 

1216 circles or no convergence. 

1217 

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

1219 

1220 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}}, B{C{lat2}}, 

1221 B{C{lon2}} or B{C{radius2}}. 

1222 ''' 

1223 r1 = Radius_(radius1=radius1) 

1224 r2 = Radius_(radius2=radius2) 

1225 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1226 small, wrap, True, radius1=r1, radius2=r2) as t: 

1227 _i, d, lat2, lon2, m, n = t 

1228 if d is None: 

1229 r1 = m2degrees(r1, radius=R_M, lat=lat1) 

1230 r2 = m2degrees(r2, radius=R_M, lat=lat2) 

1231 

1232 def _V2T(x, y, _, **unused): # _ == z unused 

1233 return LatLon2Tuple(y, x, name=n) 

1234 

1235 t = _i(m.Vector3d(lon1, lat1, 0), r1, 

1236 m.Vector3d(lon2, lat2, 0), r2, sphere=False, 

1237 Vector=_V2T) 

1238 else: 

1239 def _LL2T(lat, lon, **unused): 

1240 return LatLon2Tuple(lat, lon, name=n) 

1241 

1242 t = _i(m.LatLon(lat1, lon1, datum=d), r1, 

1243 m.LatLon(lat2, lon2, datum=d), r2, 

1244 LatLon=_LL2T, height=0, wrap=False) 

1245 return t 

1246 

1247 

1248def isantipode(lat1, lon1, lat2, lon2, eps=EPS): 

1249 '''Check whether two points are I{antipodal}, on diametrically 

1250 opposite sides of the earth. 

1251 

1252 @arg lat1: Latitude of one point (C{degrees}). 

1253 @arg lon1: Longitude of one point (C{degrees}). 

1254 @arg lat2: Latitude of the other point (C{degrees}). 

1255 @arg lon2: Longitude of the other point (C{degrees}). 

1256 @kwarg eps: Tolerance for near-equality (C{degrees}). 

1257 

1258 @return: C{True} if points are antipodal within the 

1259 B{C{eps}} tolerance, C{False} otherwise. 

1260 

1261 @see: Functions L{isantipode_} and L{antipode}. 

1262 ''' 

1263 return (fabs(lat1 + lat2) <= eps and 

1264 fabs(lon1 + lon2) <= eps) or _isequalTo( 

1265 normal(lat1, lon1), antipode(lat2, lon2), eps) 

1266 

1267 

1268def isantipode_(phi1, lam1, phi2, lam2, eps=EPS): 

1269 '''Check whether two points are I{antipodal}, on diametrically 

1270 opposite sides of the earth. 

1271 

1272 @arg phi1: Latitude of one point (C{radians}). 

1273 @arg lam1: Longitude of one point (C{radians}). 

1274 @arg phi2: Latitude of the other point (C{radians}). 

1275 @arg lam2: Longitude of the other point (C{radians}). 

1276 @kwarg eps: Tolerance for near-equality (C{radians}). 

1277 

1278 @return: C{True} if points are antipodal within the 

1279 B{C{eps}} tolerance, C{False} otherwise. 

1280 

1281 @see: Functions L{isantipode} and L{antipode_}. 

1282 ''' 

1283 return (fabs(phi1 + phi2) <= eps and 

1284 fabs(lam1 + lam2) <= eps) or _isequalTo_( 

1285 normal_(phi1, lam1), antipode_(phi2, lam2), eps) 

1286 

1287 

1288def _isequalTo(p1, p2, eps=EPS): 

1289 '''Compare 2 point lat-/lons ignoring C{class}. 

1290 ''' 

1291 return (fabs(p1.lat - p2.lat) <= eps and 

1292 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon) 

1293 

1294 

1295def _isequalTo_(p1, p2, eps=EPS): # underscore_! 

1296 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}. 

1297 ''' 

1298 return (fabs(p1.phi - p2.phi) <= eps and 

1299 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam) 

1300 

1301 

1302def isnormal(lat, lon, eps=0): 

1303 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their 

1304 respective I{normal} range in C{degrees}. 

1305 

1306 @arg lat: Latitude (C{degrees}). 

1307 @arg lon: Longitude (C{degrees}). 

1308 @kwarg eps: Optional tolerance C{degrees}). 

1309 

1310 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and 

1311 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} otherwise. 

1312 

1313 @see: Functions L{isnormal_} and L{normal}. 

1314 ''' 

1315 return _loneg(fabs(lon)) >= eps and (_90_0 - fabs(lat)) >= eps # co-latitude 

1316 

1317 

1318def isnormal_(phi, lam, eps=0): 

1319 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their 

1320 respective I{normal} range in C{radians}. 

1321 

1322 @arg phi: Latitude (C{radians}). 

1323 @arg lam: Longitude (C{radians}). 

1324 @kwarg eps: Optional tolerance C{radians}). 

1325 

1326 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and 

1327 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise. 

1328 

1329 @see: Functions L{isnormal} and L{normal_}. 

1330 ''' 

1331 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps 

1332 

1333 

1334def _maprod(fun_, *ts): 

1335 '''(INTERNAL) Helper for C{excessCagnoli_} and C{excessLHuilier_}. 

1336 ''' 

1337 return fprod(map(fun_, ts)) 

1338 

1339 

1340def _normal2(a, b, n_2, n, n2): 

1341 '''(INTERNAL) Helper for C{normal} and C{normal_}. 

1342 ''' 

1343 if fabs(b) > n: 

1344 b = remainder(b, n2) 

1345 if fabs(a) > n_2: 

1346 r = remainder(a, n) 

1347 if r != a: 

1348 a = -r 

1349 b -= n if b > 0 else -n 

1350 return float0_(a, b) 

1351 

1352 

1353def normal(lat, lon, **name): 

1354 '''Normalize a lat- I{and} longitude pair in C{degrees}. 

1355 

1356 @arg lat: Latitude (C{degrees}). 

1357 @arg lon: Longitude (C{degrees}). 

1358 @kwarg name: Optional C{B{name}="normal"} (C{str}). 

1359 

1360 @return: L{LatLon2Tuple}C{(lat, lon)} with C{-90 <= lat <= 90} 

1361 and C{-180 <= lon <= 180}. 

1362 

1363 @see: Functions L{normal_} and L{isnormal}. 

1364 ''' 

1365 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0), 

1366 name=_name__(name, name__=normal)) 

1367 

1368 

1369def normal_(phi, lam, **name): 

1370 '''Normalize a lat- I{and} longitude pair in C{radians}. 

1371 

1372 @arg phi: Latitude (C{radians}). 

1373 @arg lam: Longitude (C{radians}). 

1374 @kwarg name: Optional C{B{name}="normal_"} (C{str}). 

1375 

1376 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2} 

1377 and C{abs(lam) <= PI}. 

1378 

1379 @see: Functions L{normal} and L{isnormal_}. 

1380 ''' 

1381 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2), 

1382 name=_name__(name, name__=normal_)) 

1383 

1384 

1385def _opposes(d, m, n, n2): 

1386 '''(INTERNAL) Helper for C{opposing} and C{opposing_}. 

1387 ''' 

1388 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1 

1389 return False if d < m or d > (n2 - m) else ( 

1390 True if (n - m) < d < (n + m) else None) 

1391 

1392 

1393def opposing(bearing1, bearing2, margin=_90_0): 

1394 '''Compare the direction of two bearings given in C{degrees}. 

1395 

1396 @arg bearing1: First bearing (compass C{degrees}). 

1397 @arg bearing2: Second bearing (compass C{degrees}). 

1398 @kwarg margin: Optional, interior angle bracket (C{degrees}). 

1399 

1400 @return: C{True} if both bearings point in opposite, C{False} if 

1401 in similar or C{None} if in I{perpendicular} directions. 

1402 

1403 @see: Function L{opposing_}. 

1404 ''' 

1405 m = Degrees_(margin=margin, low=EPS0, high=_90_0) 

1406 return _opposes(bearing2 - bearing1, m, _180_0, _360_0) 

1407 

1408 

1409def opposing_(radians1, radians2, margin=PI_2): 

1410 '''Compare the direction of two bearings given in C{radians}. 

1411 

1412 @arg radians1: First bearing (C{radians}). 

1413 @arg radians2: Second bearing (C{radians}). 

1414 @kwarg margin: Optional, interior angle bracket (C{radians}). 

1415 

1416 @return: C{True} if both bearings point in opposite, C{False} if 

1417 in similar or C{None} if in perpendicular directions. 

1418 

1419 @see: Function L{opposing}. 

1420 ''' 

1421 m = Radians_(margin=margin, low=EPS0, high=PI_2) 

1422 return _opposes(radians2 - radians1, m, PI, PI2) 

1423 

1424 

1425def _Propy(func, nargs, kwds): 

1426 '''(INTERNAL) Helper for the C{frechet.[-]Frechet**} and 

1427 C{hausdorff.[-]Hausdorff*} classes. 

1428 ''' 

1429 try: 

1430 _xcallable(distance=func) 

1431 # assert _args_kwds_count2(func)[0] == nargs + int(ismethod(func)) 

1432 _ = func(*_0_0s(nargs), **kwds) 

1433 except Exception as x: 

1434 t = unstr(func, **kwds) 

1435 raise _TypeError(t, cause=x) 

1436 return func 

1437 

1438 

1439def _radical2(d, r1, r2, **name): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d 

1440 # (INTERNAL) See C{radical2} below 

1441 # assert d > EPS0 

1442 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5 

1443 n = _name__(name, name__=radical2) 

1444 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d, name=n) 

1445 

1446 

1447def radical2(distance, radius1, radius2, **name): 

1448 '''Compute the I{radical ratio} and I{radical line} of two U{intersecting 

1449 circles<https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}. 

1450 

1451 The I{radical line} is perpendicular to the axis thru the centers of 

1452 the circles at C{(0, 0)} and C{(B{distance}, 0)}. 

1453 

1454 @arg distance: Distance between the circle centers (C{scalar}). 

1455 @arg radius1: Radius of the first circle (C{scalar}). 

1456 @arg radius2: Radius of the second circle (C{scalar}). 

1457 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1458 

1459 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <= ratio <= 

1460 1.0} and C{xline} is along the B{C{distance}}. 

1461 

1462 @raise IntersectionError: The B{C{distance}} exceeds the sum of 

1463 B{C{radius1}} and B{C{radius2}}. 

1464 

1465 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or B{C{radius2}}. 

1466 ''' 

1467 d = Distance_(distance, low=_0_0) 

1468 r1 = Radius_(radius1=radius1) 

1469 r2 = Radius_(radius2=radius2) 

1470 if d > (r1 + r2): 

1471 raise IntersectionError(distance=d, radius1=r1, radius2=r2, 

1472 txt=_too_(_distant_)) 

1473 return _radical2(d, r1, r2, **name) if d > EPS0 else \ 

1474 Radical2Tuple(_0_5, _0_0, **name) 

1475 

1476 

1477class Radical2Tuple(_NamedTuple): 

1478 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and 

1479 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0} 

1480 ''' 

1481 _Names_ = (_ratio_, _xline_) 

1482 _Units_ = ( Scalar, Scalar) 

1483 

1484 

1485def _radistance(inst): 

1486 '''(INTERNAL) Helper for the L{frechet._FrechetMeterRadians} 

1487 and L{hausdorff._HausdorffMeterRedians} classes. 

1488 ''' 

1489 wrap_, kwds_ = _xkwds_pop2(inst._kwds, wrap=False) 

1490 func_ = inst._func_ 

1491 try: # calling lower-overhead C{func_} 

1492 func_(0, _0_25, _0_5, **kwds_) 

1493 wrap_ = _Wrap._philamop(wrap_) 

1494 except TypeError: 

1495 return inst.distance 

1496 

1497 def _philam(p): 

1498 try: 

1499 return p.phi, p.lam 

1500 except AttributeError: # no .phi or .lam 

1501 return radians(p.lat), radians(p.lon) 

1502 

1503 def _func_wrap(point1, point2): 

1504 phi1, lam1 = wrap_(*_philam(point1)) 

1505 phi2, lam2 = wrap_(*_philam(point2)) 

1506 return func_(phi2, phi1, lam2 - lam1, **kwds_) 

1507 

1508 inst._units = inst._units_ 

1509 return _func_wrap 

1510 

1511 

1512def _scale_deg(lat1, lat2): # degrees 

1513 # scale factor cos(mean of lats) for delta lon 

1514 m = fabs(lat1 + lat2) * _0_5 

1515 return cos(radians(m)) if m < _90_0 else _0_0 

1516 

1517 

1518def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights 

1519 # scale factor cos(mean of phis) for delta lam 

1520 m = fabs(phi1 + phi2) * _0_5 

1521 return cos(m) if m < PI_2 else _0_0 

1522 

1523 

1524def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw 

1525 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine. 

1526 ''' 

1527 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21) 

1528 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21 

1529 

1530 

1531def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

1532 '''Compute the distance between two (ellipsoidal) points using U{Thomas' 

1533 <https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} formula. 

1534 

1535 @arg lat1: Start latitude (C{degrees}). 

1536 @arg lon1: Start longitude (C{degrees}). 

1537 @arg lat2: End latitude (C{degrees}). 

1538 @arg lon2: End longitude (C{degrees}). 

1539 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} 

1540 or L{a_f2Tuple}) to use. 

1541 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and 

1542 B{C{lon2}} (C{bool}). 

1543 

1544 @return: Distance (C{meter}, same units as the B{C{datum}}'s or ellipsoid axes). 

1545 

1546 @raise TypeError: Invalid B{C{datum}}. 

1547 

1548 @see: Functions L{thomas_}, L{cosineLaw}, L{equirectangular}, L{euclidean}, 

1549 L{flatLocal} / L{hubeny}, L{flatPolar}, L{haversine}, L{vincentys} and 

1550 method L{Ellipsoid.distance2}. 

1551 ''' 

1552 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2) 

1553 

1554 

1555def thomas_(phi2, phi1, lam21, datum=_WGS84): 

1556 '''Compute the I{angular} distance between two (ellipsoidal) points using 

1557 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} formula. 

1558 

1559 @arg phi2: End latitude (C{radians}). 

1560 @arg phi1: Start latitude (C{radians}). 

1561 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1562 @kwarg datum: Datum (L{Datum}) ?or ellipsoid to use (L{Ellipsoid}, 

1563 L{Ellipsoid2} or L{a_f2Tuple}). 

1564 

1565 @return: Angular distance (C{radians}). 

1566 

1567 @raise TypeError: Invalid B{C{datum}}. 

1568 

1569 @see: Functions L{thomas}, L{cosineLaw_}, L{euclidean_}, L{flatLocal_} / 

1570 L{hubeny_}, L{flatPolar_}, L{haversine_} and L{vincentys_} and 

1571 U{Geodesy-PHP<https://GitHub.com/jtejido/geodesy-php/blob/master/ 

1572 src/Geodesy/Distance/ThomasFormula.php>}. 

1573 ''' 

1574 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21) 

1575 if r and isnon0(c1) and isnon0(c2): 

1576 E = _ellipsoidal(datum, thomas_) 

1577 f = E.f * _0_25 

1578 if f: # ellipsoidal 

1579 r1 = atan2(E.b_a * s1, c1) 

1580 r2 = atan2(E.b_a * s2, c2) 

1581 

1582 j = (r2 + r1) * _0_5 

1583 k = (r2 - r1) * _0_5 

1584 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5) 

1585 

1586 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2) 

1587 u = _1_0 - h 

1588 if isnon0(u) and isnon0(h): 

1589 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h) 

1590 sr, cr = sincos2(r) 

1591 if isnon0(sr): 

1592 u = (sj * ck)**2 * 2 / u 

1593 h = (sk * cj)**2 * 2 / h 

1594 x = u + h 

1595 y = u - h 

1596 

1597 b = r * 2 

1598 s = r / sr 

1599 e = 4 * s**2 

1600 d = 2 * cr 

1601 a = e * d 

1602 c = s - (a - d) * _0_5 

1603 

1604 t = fdot_(a, x, -b, y, -d, y**2, c, x**2, e, x * y) * _0_25 

1605 r -= fdot_(s, x, -1, y, -t, f) * f * sr 

1606 return r 

1607 

1608 

1609def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1610 '''Compute the distance between two (spherical) points using 

1611 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1612 spherical formula. 

1613 

1614 @arg lat1: Start latitude (C{degrees}). 

1615 @arg lon1: Start longitude (C{degrees}). 

1616 @arg lat2: End latitude (C{degrees}). 

1617 @arg lon2: End longitude (C{degrees}). 

1618 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) or 

1619 ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) 

1620 to use. 

1621 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and 

1622 B{C{lon2}} (C{bool}). 

1623 

1624 @return: Distance (C{meter}, same units as B{C{radius}}). 

1625 

1626 @raise UnitError: Invalid B{C{radius}}. 

1627 

1628 @see: Functions L{vincentys_}, L{cosineLaw}, L{equirectangular}, L{euclidean}, 

1629 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine} and L{thomas} and 

1630 methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*} and 

1631 C{LatLon.equirectangularTo}. 

1632 

1633 @note: See note at function L{vincentys_}. 

1634 ''' 

1635 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2) 

1636 

1637 

1638def vincentys_(phi2, phi1, lam21): 

1639 '''Compute the I{angular} distance between two (spherical) points using 

1640 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1641 spherical formula. 

1642 

1643 @arg phi2: End latitude (C{radians}). 

1644 @arg phi1: Start latitude (C{radians}). 

1645 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1646 

1647 @return: Angular distance (C{radians}). 

1648 

1649 @see: Functions L{vincentys}, L{cosineLaw_}, L{euclidean_}, L{flatLocal_} / 

1650 L{hubeny_}, L{flatPolar_}, L{haversine_} and L{thomas_}. 

1651 

1652 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_} produce 

1653 equivalent results, but L{vincentys_} is suitable for antipodal 

1654 points and slightly more expensive (M{3 cos, 3 sin, 1 hypot, 1 atan2, 

1655 6 mul, 2 add}) than L{haversine_} (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 

1656 mul, 1 add}) and L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}). 

1657 ''' 

1658 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21) 

1659 

1660 c = c2 * c21 

1661 x = s1 * s2 + c1 * c 

1662 y = c1 * s2 - s1 * c 

1663 return atan2(hypot(c2 * s21, y), x) 

1664 

1665# **) MIT License 

1666# 

1667# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved. 

1668# 

1669# Permission is hereby granted, free of charge, to any person obtaining a 

1670# copy of this software and associated documentation files (the "Software"), 

1671# to deal in the Software without restriction, including without limitation 

1672# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1673# and/or sell copies of the Software, and to permit persons to whom the 

1674# Software is furnished to do so, subject to the following conditions: 

1675# 

1676# The above copyright notice and this permission notice shall be included 

1677# in all copies or substantial portions of the Software. 

1678# 

1679# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1680# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1681# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1682# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1683# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1684# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1685# OTHER DEALINGS IN THE SOFTWARE.