1

2# -*- coding: utf-8 -*-

3

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

5'''

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

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

8

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

10 _umod_PI2, float0, isnon0, remainder, _0_0, \

11 _0_125, _0_25, _0_5, _1_0, _2_0, _N_2_0, \

12 _4_0, _32_0, _90_0, _180_0, _360_0

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

14 _mean_radius, _spherical_datum, _WGS84

15# from pygeodesy.ellipsoids import Ellipsoid # from .datums

16 from pygeodesy.errors import _AssertionError, IntersectionError, \

17 LimitError, limiterrors, _ValueError

18 from pygeodesy.fmath import Fdot, euclid, fdot, hypot, hypot2, sqrt0

19 from pygeodesy.fsums import fsum_, unstr

20 from pygeodesy.interns import NN, _distant_, _inside_, _near_, _null_, \

21 _opposite_, _outside_, _too_

22 from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS

23 from pygeodesy.named import _NamedTuple, _xnamed

24 from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, \

25 LatLon2Tuple, PhiLam2Tuple, Vector3Tuple

26# from pygeodesy.streprs import unstr # from .fsums

27 from pygeodesy.units import Degrees_, Distance, Distance_, Height, Lam_, Lat, \

28 Lon, Phi_, Radians, Radians_, Radius, Radius_, \

29 Scalar, _100km

30 from pygeodesy.utily import acos1, atan2b, atan2d, degrees2m, m2degrees, tan_2, \

31 sincos2, sincos2_, sincos2d_, unroll180, unrollPI

32

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

34

35__all__ = _ALL_LAZY.formy

36__version__ = '23.04.05'

37

38_ratio_ = 'ratio'

39_xline_ = 'xline'

40

41

42 def _anti2(a, b, n_2, n, n2):

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

44 '''

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

46 if r == a:

47 r = -r

48 b += n

49 if fabs(b) > n:

50 b = remainder(b, n2)

51 return float0(r, b)

52

53

54 def antipode(lat, lon, name=NN):

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

56 to a given point in C{degrees}.

57

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

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

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

61

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

63

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

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

66 '''

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

68

69

70 def antipode_(phi, lam, name=NN):

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

72 to a given point in C{radians}.

73

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

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

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

77

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

79

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

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

82 '''

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

84

85

86 def _area_or_(excess_, lat1, lat2, radius, d_lon, unused):

87 '''(INTERNAL) Helper for area and spherical excess.

88 '''

89 r = excess_(Phi_(lat2=lat2),

90 Phi_(lat1=lat1), radians(d_lon))

91 if radius:

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

93 return r

94

95

96 def bearing(lat1, lon1, lat2, lon2, **options):

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

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

99

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

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

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

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

104 @kwarg options: Optional keyword arguments for function

105 L{pygeodesy.bearing_}.

106

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

108 zero if start and end point coincide.

109 '''

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

111 Phi_(lat2=lat2), Lam_(lon2=lon2), **options)

112 return degrees(r)

113

114

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

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

117 between a (spherical) start and end point.

118

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

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

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

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

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

124 @kwarg wrap: Wrap and L{pygeodesy.unrollPI} longitudes (C{bool}).

125

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

127 and end point coincide.

128

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

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

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

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

133 '''

134 if final: # swap plus PI

135 phi1, lam1, phi2, lam2 = phi2, lam2, phi1, lam1

136 r = PI3

137 else:

138 r = PI2

139

140 db, _ = unrollPI(lam1, lam2, wrap=wrap)

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

142

143 x = ca1 * sa2 - sa1 * ca2 * cdb

144 y = sdb * ca2

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

146

147

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

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

150 '''

151 try: # for LatLon_ and ellipsoidal LatLon

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

153 except AttributeError:

154 pass

155 # XXX spherical version, OK for ellipsoidal ispolar?

156 a1, b1 = p1.philam

157 a2, b2 = p2.philam

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

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

160 name=_bearingTo2.__name__)

161

162

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

164 '''Return the angle from North for the direction vector

165 M{(lon2 - lon1, lat2 - lat1)} between two points.

166

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

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

169

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

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

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

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

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

175 mean latitude (C{bool}).

176 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).

177

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

179

180 @note: Courtesy of Martin Schultz.

181

182 @see: U{Local, flat earth approximation

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

184 '''

185 d_lon, _ = unroll180(lon1, lon2, wrap=wrap)

186 if adjust: # scale delta lon

187 d_lon *= _scale_deg(lat1, lat2)

188 return atan2b(d_lon, lat2 - lat1)

189

190

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

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

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

194 2013/10/admiralty-manual-of-navigation-vol-1-1964-english501c.pdf>} of the

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

196 fromula.

197

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

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

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

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

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

203 L{Ellipsoid2} or L{a_f2Tuple}).

204 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).

205

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

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

208

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

210

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

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

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

214 L{Ellipsoid.distance2}.

215 '''

216 return _distanceToE(cosineAndoyerLambert_, lat1, lat2, datum,

217 *unroll180(lon1, lon2, wrap=wrap))

218

219

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

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

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

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

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

225 fromula.

226

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

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

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

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

231 L{Ellipsoid2} or L{a_f2Tuple}).

232

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

234

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

236

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

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

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

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

241 Distance/AndoyerLambert.php>}.

242 '''

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

244 if isnon0(c1) and isnon0(c2):

245 E = _ellipsoidal(datum, cosineAndoyerLambert_)

246 if E.f: # ellipsoidal

247 r2 = atan2(E.b_a * s2, c2)

248 r1 = atan2(E.b_a * s1, c1)

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

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

251 if r:

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

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

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

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

256 r += (c - s) * E.f * _0_125

257 return r

258

259

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

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

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

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

264 formula.

265

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

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

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

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

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

271 L{Ellipsoid2} or L{a_f2Tuple}).

272 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).

273

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

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

276

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

278

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

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

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

282 L{Ellipsoid.distance2}.

283 '''

284 return _distanceToE(cosineForsytheAndoyerLambert_, lat1, lat2, datum,

285 *unroll180(lon1, lon2, wrap=wrap))

286

287

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

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

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

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

292 formula.

293

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

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

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

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

298 L{Ellipsoid2} or L{a_f2Tuple}).

299

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

301

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

303

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

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

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

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

308 Distance/ForsytheCorrection.php>}.

309 '''

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

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

312 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_)

313 if E.f: # ellipsoidal

314 sr, cr, s2r, _ = sincos2_(r, r * _2_0)

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

316 s = (s1 + s2)**2 / (1 + cr)

317 t = (s1 - s2)**2 / (1 - cr)

318 x = s + t

319 y = s - t

320

321 s = 8 * r**2 / sr

322 a = 64 * r + _2_0 * s * cr # 16 * r**2 / tan(r)

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

324 b = -2 * d

325 e = 30 * s2r

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

327

328 t = fsum_( a * x, b * y, -c * x**2, d * x * y, e * y**2)

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

330 return r

331

332

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

334 '''Compute the distance between two points using the

335 U{spherical Law of Cosines

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

337 formula.

338

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

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

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

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

343 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},

344 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}).

345 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).

346

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

348 ellipsoid or datum axes).

349

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

351

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

353 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean},

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

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

356

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

358 '''

359 return _distanceToS(cosineLaw_, lat1, lat2, radius,

360 *unroll180(lon1, lon2, wrap=wrap))

361

362

363 def cosineLaw_(phi2, phi1, lam21):

364 '''Compute the I{angular} distance between two points using the

365 U{spherical Law of Cosines

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

367 formula.

368

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

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

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

372

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

374

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

376 L{cosineForsytheAndoyerLambert_}, L{equirectangular_},

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

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

379

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

381 '''

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

383

384

385 def _distanceToE(func_, lat1, lat2, earth, d_lon, unused):

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

387 '''

388 E = _ellipsoidal(earth, func_)

389 r = func_(Phi_(lat2=lat2),

390 Phi_(lat1=lat1), radians(d_lon), datum=E)

391 return r * E.a

392

393

394 def _distanceToS(func_, lat1, lat2, earth, d_lon, unused, **adjust):

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

396 '''

397 r = func_(Phi_(lat2=lat2),

398 Phi_(lat1=lat1), radians(d_lon), **adjust)

399 return r * _mean_radius(earth, lat1, lat2)

400

401

402 def _ellipsoidal(earth, where):

403 '''(INTERNAL) Helper for distances.

404 '''

405 return earth if isinstance(earth, Ellipsoid) else (

406 earth if isinstance(earth, Datum) else

407 _ellipsoidal_datum(earth, name=where.__name__)).ellipsoid # PYCHOK indent

408

409

410 def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **options):

411 '''Compute the distance between two points using

412 the U{Equirectangular Approximation / Projection

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

414

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

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

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

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

419 @kwarg radius: Mean earth radius, ellipsoid or datum

420 (C{meter}, L{Ellipsoid}, L{Ellipsoid2},

421 L{Datum} or L{a_f2Tuple}).

422 @kwarg options: Optional keyword arguments for function

423 L{equirectangular_}.

424

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

426 the ellipsoid or datum axes).

427

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

429

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

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

432 approximate or accurate distance functions.

433 '''

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

435 Lat(lat2=lat2), Lon(lon2=lon2),

436 **options).distance2) # PYCHOK 4 vs 2-3

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

438

439

440 def equirectangular_(lat1, lon1, lat2, lon2,

441 adjust=True, limit=45, wrap=False):

442 '''Compute the distance between two points using

443 the U{Equirectangular Approximation / Projection

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

445

446 This approximation is valid for short distance of several

447 hundred Km or Miles, see the B{C{limit}} keyword argument and

448 the L{LimitError}.

449

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

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

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

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

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

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

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

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

458 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).

459

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

461 unroll_lon2)}.

462

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

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

465 set to C{True}.

466

467 @see: U{Local, flat earth approximation

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

469 L{equirectangular}, L{cosineAndoyerLambert},

470 L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean},

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

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

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

474 '''

475 d_lat = lat2 - lat1

476 d_lon, ulon2 = unroll180(lon1, lon2, wrap=wrap)

477

478 if limit and limit > 0 and limiterrors() and (fabs(d_lat) > limit or

479 fabs(d_lon) > limit):

480 t = unstr(equirectangular_, lat1, lon1, lat2, lon2, limit=limit)

481 raise LimitError('delta exceeds limit', txt=t)

482

483 if adjust: # scale delta lon

484 d_lon *= _scale_deg(lat1, lat2)

485

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

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

488

489

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

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

492

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

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

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

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

497 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},

498 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}).

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

500 mean latitude (C{bool}).

501 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).

502

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

504 ellipsoid or datum axes).

505

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

507

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

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

510 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},

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

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

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

514 '''

515 return _distanceToS(euclidean_, lat1, lat2, radius,

516 *unroll180(lon1, lon2, wrap=wrap),

517 adjust=adjust)

518

519

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

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

522 (spherical) points.

523

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

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

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

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

528 of the mean latitude (C{bool}).

529

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

531

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

533 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_},

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

535 and L{vincentys_}.

536 '''

537 if adjust:

538 lam21 *= _scale_rad(phi2, phi1)

539 return euclid(phi2 - phi1, lam21)

540

541

542 def excessAbc_(A, b, c):

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

544 from two sides and the included angle.

545

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

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

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

549

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

551

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

553

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

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

556 '''

557 sA, cA, sb, cb, sc, cc = sincos2_(Radians_(A=A), Radians_(b=b) * _0_5,

558 Radians_(c=c) * _0_5)

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

560

561

562 def excessGirard_(A, B, C):

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

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

565 formula.

566

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

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

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

570

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

572

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

574

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

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

577 '''

578 return Radians(Girard=fsum_(Radians_(A=A),

579 Radians_(B=B),

580 Radians_(C=C), -PI))

581

582

583 def excessLHuilier_(a, b, c):

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

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

586 Theorem.

587

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

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

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

591

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

593

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

595

596 @see: Function L{excessGirard_} and U{Spherical trigonometry

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

598 '''

599 a = Radians_(a=a)

600 b = Radians_(b=b)

601 c = Radians_(c=c)

602

603 s = fsum_(a, b, c) * _0_5

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

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

606 return Radians(LHuilier=r * _4_0)

607

608

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

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

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

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

613 method.

614

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

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

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

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

619 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, L{Ellipsoid},

620 L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}) or C{None}.

621 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).

622

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

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

625 if C{B{radius}=0} or C{None}.

626

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

628

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

630

631 @raise ValueError: Semi-circular longitudinal delta.

632

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

634 '''

635 return _area_or_(excessKarney_, lat1, lat2, radius,

636 *unroll180(lon1, lon2, wrap=wrap))

637

638

639 def excessKarney_(phi2, phi1, lam21):

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

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

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

643 method.

644

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

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

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

648

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

650

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

652

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

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

655 '''

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

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

658 #

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

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

661 # 1 + tan(φ1 / 2) · tan(φ2 / 2)

662 #

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

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

665 t2 = tan_2(phi2)

666 t1 = tan_2(phi1)

667 t = tan_2(lam21, lam21=None)

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

669 _1_0 + (t1 * t2)) * _2_0)

670

671

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

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

674#

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

676#

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

678#

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

680#

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

682#

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

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

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

686#

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

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

689#

690# = tan(lambda21 / 2) * tanh((Lambertian(phi1) +

691# Lambertian(phi2)) / 2)

692#

693# where lambda21 = lambda2 - lambda1 and lamb(x) is the Lambertian (or

694# inverse Gudermannian) function

695#

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

697#

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

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

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

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

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

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

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

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

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

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

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

709# - Charles Karney

710

711

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

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

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

715

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

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

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

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

720 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},

721 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}) or C{None}.

722 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).

723

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

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

726 if C{B{radius}=0} or C{None}.

727

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

729

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

731

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

733 '''

734 return _area_or_(excessQuad_, lat1, lat2, radius,

735 *unroll180(lon1, lon2, wrap=wrap))

736

737

738 def excessQuad_(phi2, phi1, lam21):

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

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

741

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

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

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

745

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

747

748 @see: Function L{excessQuad}, U{Spherical trigonometry

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

750 '''

751 s = sin((phi2 + phi1) * _0_5)

752 c = cos((phi2 - phi1) * _0_5)

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

754

755

756 def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):

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

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

759 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}

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

761

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

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

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

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

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

767 L{Ellipsoid2} or L{a_f2Tuple}).

768 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).

769

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

771 ellipsoid axes).

772

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

774

775 @note: The meridional and prime_vertical radii of curvature

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

777

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

779 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},

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

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

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

783 '''

784 d, _ = unroll180(lon1, lon2, wrap=wrap)

785 return flatLocal_(Phi_(lat2=lat2),

786 Phi_(lat1=lat1), radians(d), datum=datum)

787

788 hubeny = flatLocal # PYCHOK for Karl Hubeny

789

790

791 def flatLocal_(phi2, phi1, lam21, datum=_WGS84):

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

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

794 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}

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

796

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

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

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

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

801 L{Ellipsoid2} or L{a_f2Tuple}).

802

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

804

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

806

807 @note: The meridional and prime_vertical radii of curvature

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

809

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

811 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_},

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

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

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

815 '''

816 E = _ellipsoidal(datum, flatLocal_)

817 m, n = E.roc2_((phi2 + phi1) * _0_5, scaled=True)

818 return hypot(m * (phi2 - phi1), n * lam21)

819

820 hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny

821

822

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

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

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

826 Geographical_distance#Polar_coordinate_flat-Earth_formula>}

827 formula.

828

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

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

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

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

833 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},

834 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}).

835 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).

836

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

838 ellipsoid or datum axes).

839

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

841

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

843 L{cosineForsytheAndoyerLambert},L{cosineLaw},

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

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

846 L{vincentys}.

847 '''

848 return _distanceToS(flatPolar_, lat1, lat2, radius,

849 *unroll180(lon1, lon2, wrap=wrap))

850

851

852 def flatPolar_(phi2, phi1, lam21):

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

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

855 Geographical_distance#Polar_coordinate_flat-Earth_formula>}

856 formula.

857

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

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

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

861

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

863

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

865 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

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

867 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 = fsum_(_1_0, b**2, -fabs(c))

879 a *= sqrt0(c)

880 return a

881

882

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

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

885 from a Point-Of-View in space.

886

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

888 or L{Vector3d}).

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

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

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

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

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

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

895 point plus C{LatLon} keyword arguments, include

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

897

898 @return: The earth intersection (L{Vector3d}, C{Cartesian type} of

899 B{C{pov}} or B{C{LatLon}}).

900

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

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

903 the earth or points in an opposite direction.

904

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

906

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

908 U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell.

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

910 '''

911 def _Error(txt):

912 return IntersectionError(pov=pov, los=los, earth=earth, txt=txt)

913

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

915 _spherical_datum(earth, name=hartzell.__name__)

916 E = D.ellipsoid

917

918 if E.b > E.a: # PYCHOK no cover

919 try:

920 t = _MODS.triaxials

921 r, _ = t._hartzell3d2(pov, los, t.Triaxial_(E.a, E.a, E.b))

922 except Exception as x:

923 raise _Error(str(x))

924 else:

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

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

927 q2 = E.b2_a2 # == c2 / a2

928 bc = E.a * E.b # == b * c

929

930 V3 = _MODS.vector3d._otherV3d

931 p3 = V3(pov=pov)

932 u3 = V3(los=los) if los else p3.negate()

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

934

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

936 ux, vy, wz = u3.times_(p3).xyz

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

938

939 t = c2, c2, b2

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

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

942 raise _Error(_near_(_null_))

943

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

945 r = fsum_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2,

946 c2 * u2, -u2 * z2, -w2 * x2, ux * wz * 2,

947 -w2 * y2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2, floats=True)

948 if r > 0:

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

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

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

952

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

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

955 s = fsum_(_1_0, x2 / a2, y2 / b2, z2 / c2, _N_2_0, floats=True) # like _sideOf

956 raise _Error(_outside_ if s > 0 else _inside_)

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

958 raise _Error(_too_(_distant_))

959

960 r = p3.minus(u3.times(d))

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

962

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

964 if LatLon_and_kwds:

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

966 r = c.toLatLon(**LatLon_and_kwds)

967 return r

968

969

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

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

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

973 formula.

974

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

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

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

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

979 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},

980 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}).

981 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).

982

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

984

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

986

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

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

989 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},

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

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

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

993

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

995 '''

996 return _distanceToS(haversine_, lat1, lat2, radius,

997 *unroll180(lon1, lon2, wrap=wrap))

998

999

1000 def haversine_(phi2, phi1, lam21):

1001 '''Compute the I{angular} distance between two (spherical) points

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

1003 formula.

1004

1005 @arg phi2: End latitude (C{radians}).

1006 @arg phi1: Start latitude (C{radians}).

1007 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).

1008

1009 @return: Angular distance (C{radians}).

1010

1011 @see: Functions L{haversine}, L{cosineAndoyerLambert_},

1012 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

1013 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},

1014 L{flatPolar_}, L{thomas_} and L{vincentys_}.

1015

1016 @note: See note at function L{vincentys_}.

1017 '''

1018 def _hsin(rad):

1019 return sin(rad * _0_5)**2

1020

1021 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine

1022 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2

1023

1024

1025 def heightOf(angle, distance, radius=R_M):

1026 '''Determine the height above the (spherical) earth' surface after

1027 traveling along a straight line at a given tilt.

1028

1029 @arg angle: Tilt angle above horizontal (C{degrees}).

1030 @arg distance: Distance along the line (C{meter} or same units as

1031 B{C{radius}}).

1032 @kwarg radius: Optional mean earth radius (C{meter}).

1033

1034 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}).

1035

1036 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}.

1037

1038 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>}

1039 (U{Shapiro et al. 2009, JTECH

1040 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>}

1041 and U{Potvin et al. 2012, JTECH

1042 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}).

1043 '''

1044 r = h = Radius(radius)

1045 d = fabs(Distance(distance))

1046 if d > h:

1047 d, h = h, d

1048

1049 if d > EPS0: # and h > EPS0

1050 d = d / h # /= h chokes PyChecker

1051 s = sin(Phi_(angle=angle, clip=_180_0))

1052 s = fsum_(_1_0, _2_0 * s * d, d**2)

1053 if s > 0:

1054 return h * sqrt(s) - r

1055

1056 raise _ValueError(angle=angle, distance=distance, radius=radius)

1057

1058

1059 def horizon(height, radius=R_M, refraction=False):

1060 '''Determine the distance to the horizon from a given altitude

1061 above the (spherical) earth.

1062

1063 @arg height: Altitude (C{meter} or same units as B{C{radius}}).

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

1065 @kwarg refraction: Consider atmospheric refraction (C{bool}).

1066

1067 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}).

1068

1069 @raise ValueError: Invalid B{C{height}} or B{C{radius}}.

1070

1071 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}.

1072 '''

1073 h, r = Height(height), Radius(radius)

1074 if min(h, r) < 0:

1075 raise _ValueError(height=height, radius=radius)

1076

1077 if refraction:

1078 d2 = 2.415750694528 * h * r # 2.0 / 0.8279

1079 else:

1080 d2 = h * fsum_(r, r, h)

1081 return sqrt0(d2)

1082

1083

1084 def intersections2(lat1, lon1, radius1,

1085 lat2, lon2, radius2, datum=None, wrap=True):

1086 '''Conveniently compute the intersections of two circles each defined

1087 by a (geodetic) center point and a radius, using either ...

1088

1089 1) L{vector3d.intersections2} for small distances (below 100 KM or

1090 about 0.9 degrees) or if no B{C{datum}} is specified, or ...

1091

1092 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}}

1093 or if B{C{datum}} is a C{scalar} representing the earth radius,

1094 conventionally in C{meter} or ...

1095

1096 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}}

1097 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}

1098 is installed, otherwise ...

1099

1100 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal.

1101

1102 @arg lat1: Latitude of the first circle center (C{degrees}).

1103 @arg lon1: Longitude of the first circle center (C{degrees}).

1104 @arg radius1: Radius of the first circle (C{meter}, conventionally).

1105 @arg lat2: Latitude of the second circle center (C{degrees}).

1106 @arg lon2: Longitude of the second circle center (C{degrees}).

1107 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}).

1108 @kwarg datum: Optional ellipsoidal or spherical datum (L{Datum},

1109 L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple} or

1110 C{scalar} earth radius in C{meter}, same units as

1111 B{C{radius1}} and B{C{radius2}}) or C{None}.

1112 @kwarg wrap: Wrap and unroll longitudes (C{bool}).

1113

1114 @return: 2-Tuple of the intersection points, each a

1115 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles,

1116 the points are the same instance, aka I{radical center}.

1117

1118 @raise IntersectionError: Concentric, antipodal, invalid or

1119 non-intersecting circles or no

1120 convergence.

1121

1122 @raise TypeError: Invalid B{C{datum}}.

1123

1124 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}},

1125 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}.

1126 '''

1127 if datum is None or euclidean(lat1, lon1, lat2, lon2, radius=R_M,

1128 adjust=True, wrap=wrap) < _100km:

1129 def _V2T(x, y, _, **unused): # _ == z unused

1130 return LatLon2Tuple(y, x, name=intersections2.__name__)

1131

1132 r1 = m2degrees(Radius_(radius1=radius1), radius=R_M, lat=lat1)

1133 r2 = m2degrees(Radius_(radius2=radius2), radius=R_M, lat=lat2)

1134

1135 _, lon2 = unroll180(lon1, lon2, wrap=wrap)

1136 m = _MODS.vector3d

1137 t = m.intersections2(m.Vector3d(lon1, lat1, 0), r1,

1138 m.Vector3d(lon2, lat2, 0), r2, sphere=False,

1139 Vector=_V2T)

1140 else:

1141 def _LL2T(lat, lon, **unused):

1142 return LatLon2Tuple(lat, lon, name=intersections2.__name__)

1143

1144 d = _spherical_datum(datum, name=intersections2.__name__)

1145 if d.isSpherical:

1146 m = _MODS.sphericalTrigonometry

1147 elif d.isEllipsoidal:

1148 try:

1149 if d.ellipsoid.geodesic:

1150 pass

1151 m = _MODS.ellipsoidalKarney

1152 except ImportError:

1153 m = _MODS.ellipsoidalExact

1154 else:

1155 raise _AssertionError(datum=d)

1156

1157 t = m.intersections2(m.LatLon(lat1, lon1, datum=d), radius1,

1158 m.LatLon(lat2, lon2, datum=d), radius2,

1159 LatLon=_LL2T, height=0, wrap=wrap)

1160 return t

1161

1162

1163 def isantipode(lat1, lon1, lat2, lon2, eps=EPS):

1164 '''Check whether two points are antipodal, on diametrically

1165 opposite sides of the earth.

1166

1167 @arg lat1: Latitude of one point (C{degrees}).

1168 @arg lon1: Longitude of one point (C{degrees}).

1169 @arg lat2: Latitude of the other point (C{degrees}).

1170 @arg lon2: Longitude of the other point (C{degrees}).

1171 @kwarg eps: Tolerance for near-equality (C{degrees}).

1172

1173 @return: C{True} if points are antipodal within the

1174 B{C{eps}} tolerance, C{False} otherwise.

1175

1176 @see: Functions L{isantipode_} and L{antipode}.

1177 '''

1178 return True if (fabs(lat1 + lat2) <= eps and

1179 fabs(lon1 + lon2) <= eps) else \

1180 _MODS.latlonBase._isequalTo(antipode(lat1, lon1),

1181 normal(lat2, lon2), eps=eps)

1182

1183

1184 def isantipode_(phi1, lam1, phi2, lam2, eps=EPS):

1185 '''Check whether two points are antipodal, on diametrically

1186 opposite sides of the earth.

1187

1188 @arg phi1: Latitude of one point (C{radians}).

1189 @arg lam1: Longitude of one point (C{radians}).

1190 @arg phi2: Latitude of the other point (C{radians}).

1191 @arg lam2: Longitude of the other point (C{radians}).

1192 @kwarg eps: Tolerance for near-equality (C{radians}).

1193

1194 @return: C{True} if points are antipodal within the

1195 B{C{eps}} tolerance, C{False} otherwise.

1196

1197 @see: Functions L{isantipode} and L{antipode_}.

1198 '''

1199 return True if (fabs(phi1 + phi2) <= eps and

1200 fabs(lam1 + lam2) <= eps) else \

1201 _MODS.latlonBase._isequalTo_(antipode_(phi1, lam1),

1202 normal_(phi2, lam2), eps=eps)

1203

1204

1205 def isnormal(lat, lon, eps=0):

1206 '''Check whether B{C{lat}} I{and} B{C{lon}} are within the I{normal}

1207 range in C{degrees}.

1208

1209 @arg lat: Latitude (C{degrees}).

1210 @arg lon: Longitude (C{degrees}).

1211 @kwarg eps: Optional tolerance below C{90} and C{180 degrees}.

1212

1213 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and

1214 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise.

1215

1216 @see: Functions L{isnormal_} and L{normal}.

1217 '''

1218 return (_90_0 - fabs(lat)) >= eps and (_180_0 - fabs(lon)) >= eps

1219

1220

1221 def isnormal_(phi, lam, eps=0):

1222 '''Check whether B{C{phi}} I{and} B{C{lam}} are within the I{normal}

1223 range in C{radians}.

1224

1225 @arg phi: Latitude (C{radians}).

1226 @arg lam: Longitude (C{radians}).

1227 @kwarg eps: Optional tolerance below C{PI/2} and C{PI radians}.

1228

1229 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and

1230 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise.

1231

1232 @see: Functions L{isnormal} and L{normal_}.

1233 '''

1234 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps

1235

1236

1237 def latlon2n_xyz(lat, lon, name=NN):

1238 '''Convert lat-, longitude to C{n-vector} (I{normal} to the

1239 earth's surface) X, Y and Z components.

1240

1241 @arg lat: Latitude (C{degrees}).

1242 @arg lon: Longitude (C{degrees}).

1243 @kwarg name: Optional name (C{str}).

1244

1245 @return: A L{Vector3Tuple}C{(x, y, z)}.

1246

1247 @see: Function L{philam2n_xyz}.

1248

1249 @note: These are C{n-vector} x, y and z components,

1250 I{NOT} geocentric ECEF x, y and z coordinates!

1251 '''

1252 return _2n_xyz(name, *sincos2d_(lat, lon))

1253

1254

1255 def _normal2(a, b, n_2, n, n2):

1256 '''(INTERNAL) Helper for C{normal} and C{normal_}.

1257 '''

1258 if fabs(b) > n:

1259 b = remainder(b, n2)

1260 r = remainder(a, n) if fabs(a) > n_2 else a

1261 if r != a:

1262 r = -r

1263 b -= n if b > 0 else -n

1264 return float0(r, b)

1265

1266

1267 def normal(lat, lon, name=NN):

1268 '''Normalize a lat- I{and} longitude pair in C{degrees}.

1269

1270 @arg lat: Latitude (C{degrees}).

1271 @arg lon: Longitude (C{degrees}).

1272 @kwarg name: Optional name (C{str}).

1273

1274 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90}

1275 and C{abs(lon) <= 180}.

1276

1277 @see: Functions L{normal_} and L{isnormal}.

1278 '''

1279 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0), name=name)

1280

1281

1282 def normal_(phi, lam, name=NN):

1283 '''Normalize a lat- I{and} longitude pair in C{radians}.

1284

1285 @arg phi: Latitude (C{radians}).

1286 @arg lam: Longitude (C{radians}).

1287 @kwarg name: Optional name (C{str}).

1288

1289 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2}

1290 and C{abs(lam) <= PI}.

1291

1292 @see: Functions L{normal} and L{isnormal_}.

1293 '''

1294 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2), name=name)

1295

1296

1297 def _2n_xyz(name, sa, ca, sb, cb):

1298 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}.

1299 '''

1300 # Kenneth Gade eqn 3, but using right-handed

1301 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N

1302 return Vector3Tuple(ca * cb, ca * sb, sa, name=name)

1303

1304

1305 def n_xyz2latlon(x, y, z, name=NN):

1306 '''Convert C{n-vector} components to lat- and longitude in C{degrees}.

1307

1308 @arg x: X component (C{scalar}).

1309 @arg y: Y component (C{scalar}).

1310 @arg z: Z component (C{scalar}).

1311 @kwarg name: Optional name (C{str}).

1312

1313 @return: A L{LatLon2Tuple}C{(lat, lon)}.

1314

1315 @see: Function L{n_xyz2philam}.

1316 '''

1317 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name)

1318

1319

1320 def n_xyz2philam(x, y, z, name=NN):

1321 '''Convert C{n-vector} components to lat- and longitude in C{radians}.

1322

1323 @arg x: X component (C{scalar}).

1324 @arg y: Y component (C{scalar}).

1325 @arg z: Z component (C{scalar}).

1326 @kwarg name: Optional name (C{str}).

1327

1328 @return: A L{PhiLam2Tuple}C{(phi, lam)}.

1329

1330 @see: Function L{n_xyz2latlon}.

1331 '''

1332 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name)

1333

1334

1335 def _opposes(d, m, n, n2):

1336 '''(INETNAL) Helper for C{opposing} and C{opposing_}.

1337 '''

1338 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1

1339 return False if d < m or d > (n2 - m) else (

1340 True if (n - m) < d < (n + m) else None)

1341

1342

1343 def opposing(bearing1, bearing2, margin=_90_0):

1344 '''Compare the direction of two bearings given in C{degrees}.

1345

1346 @arg bearing1: First bearing (compass C{degrees}).

1347 @arg bearing2: Second bearing (compass C{degrees}).

1348 @kwarg margin: Optional, interior angle bracket (C{degrees}).

1349

1350 @return: C{True} if both bearings point in opposite, C{False} if

1351 in similar or C{None} if in perpendicular directions.

1352

1353 @see: Function L{opposing_}.

1354 '''

1355 m = Degrees_(margin=margin, low=EPS0, high=_90_0)

1356 return _opposes(bearing2 - bearing1, m,_180_0, _360_0)

1357

1358

1359 def opposing_(radians1, radians2, margin=PI_2):

1360 '''Compare the direction of two bearings given in C{radians}.

1361

1362 @arg radians1: First bearing (C{radians}).

1363 @arg radians2: Second bearing (C{radians}).

1364 @kwarg margin: Optional, interior angle bracket (C{radians}).

1365

1366 @return: C{True} if both bearings point in opposite, C{False} if

1367 in similar or C{None} if in perpendicular directions.

1368

1369 @see: Function L{opposing}.

1370 '''

1371 m = Radians_(margin=margin, low=EPS0, high=PI_2)

1372 return _opposes(radians2 - radians1, m, PI, PI2)

1373

1374

1375 def philam2n_xyz(phi, lam, name=NN):

1376 '''Convert lat-, longitude to C{n-vector} (I{normal} to the

1377 earth's surface) X, Y and Z components.

1378

1379 @arg phi: Latitude (C{radians}).

1380 @arg lam: Longitude (C{radians}).

1381 @kwarg name: Optional name (C{str}).

1382

1383 @return: A L{Vector3Tuple}C{(x, y, z)}.

1384

1385 @see: Function L{latlon2n_xyz}.

1386

1387 @note: These are C{n-vector} x, y and z components,

1388 I{NOT} geocentric ECEF x, y and z coordinates!

1389 '''

1390 return _2n_xyz(name, *sincos2_(phi, lam))

1391

1392

1393 def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d

1394 # (INTERNAL) See C{radical2} below

1395 # assert d > EPS0

1396 r = fsum_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5

1397 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d)

1398

1399

1400 def radical2(distance, radius1, radius2):

1401 '''Compute the I{radical ratio} and I{radical line} of two

1402 U{intersecting circles<https://MathWorld.Wolfram.com/

1403 Circle-CircleIntersection.html>}.

1404

1405 The I{radical line} is perpendicular to the axis thru the

1406 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}.

1407

1408 @arg distance: Distance between the circle centers (C{scalar}).

1409 @arg radius1: Radius of the first circle (C{scalar}).

1410 @arg radius2: Radius of the second circle (C{scalar}).

1411

1412 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <=

1413 ratio <= 1.0} and C{xline} is along the B{C{distance}}.

1414

1415 @raise IntersectionError: The B{C{distance}} exceeds the sum

1416 of B{C{radius1}} and B{C{radius2}}.

1417

1418 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or

1419 B{C{radius2}}.

1420

1421 @see: U{Circle-Circle Intersection

1422 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}.

1423 '''

1424 d = Distance_(distance, low=_0_0)

1425 r1 = Radius_(radius1=radius1)

1426 r2 = Radius_(radius2=radius2)

1427 if d > (r1 + r2):

1428 raise IntersectionError(distance=d, radius1=r1, radius2=r2,

1429 txt=_too_(_distant_))

1430 return _radical2(d, r1, r2) if d > EPS0 else \

1431 Radical2Tuple(_0_5, _0_0)

1432

1433

1434 class Radical2Tuple(_NamedTuple):

1435 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and

1436 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0}

1437 '''

1438 _Names_ = (_ratio_, _xline_)

1439 _Units_ = ( Scalar, Scalar)

1440

1441

1442 def _scale_deg(lat1, lat2): # degrees

1443 # scale factor cos(mean of lats) for delta lon

1444 m = fabs(lat1 + lat2) * _0_5

1445 return cos(radians(m)) if m < 90 else _0_0

1446

1447

1448 def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights

1449 # scale factor cos(mean of phis) for delta lam

1450 m = fabs(phi1 + phi2) * _0_5

1451 return cos(m) if m < PI_2 else _0_0

1452

1453

1454 def _sincosa6(phi2, phi1, lam21):

1455 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine.

1456 '''

1457 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21)

1458 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21

1459

1460

1461 def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):

1462 '''Compute the distance between two (ellipsoidal) points using

1463 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}

1464 formula.

1465

1466 @arg lat1: Start latitude (C{degrees}).

1467 @arg lon1: Start longitude (C{degrees}).

1468 @arg lat2: End latitude (C{degrees}).

1469 @arg lon2: End longitude (C{degrees}).

1470 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},

1471 L{Ellipsoid2} or L{a_f2Tuple}).

1472 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).

1473

1474 @return: Distance (C{meter}, same units as the B{C{datum}}'s

1475 ellipsoid axes).

1476

1477 @raise TypeError: Invalid B{C{datum}}.

1478

1479 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},

1480 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},

1481 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}.

1482 '''

1483 return _distanceToE(thomas_, lat1, lat2, datum,

1484 *unroll180(lon1, lon2, wrap=wrap))

1485

1486

1487 def thomas_(phi2, phi1, lam21, datum=_WGS84):

1488 '''Compute the I{angular} distance between two (ellipsoidal) points using

1489 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}

1490 formula.

1491

1492 @arg phi2: End latitude (C{radians}).

1493 @arg phi1: Start latitude (C{radians}).

1494 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).

1495 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},

1496 L{Ellipsoid2} or L{a_f2Tuple}).

1497

1498 @return: Angular distance (C{radians}).

1499

1500 @raise TypeError: Invalid B{C{datum}}.

1501

1502 @see: Functions L{thomas}, L{cosineAndoyerLambert_},

1503 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

1504 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},

1505 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP

1506 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/

1507 Distance/ThomasFormula.php>}.

1508 '''

1509 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)

1510 if r and isnon0(c1) and isnon0(c2):

1511 E = _ellipsoidal(datum, thomas_)

1512 if E.f:

1513 r1 = atan2(E.b_a * s1, c1)

1514 r2 = atan2(E.b_a * s2, c2)

1515

1516 j = (r2 + r1) * _0_5

1517 k = (r2 - r1) * _0_5

1518 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5)

1519

1520 h = fsum_(sk**2, (ck * h)**2, -(sj * h)**2)

1521 u = _1_0 - h

1522 if isnon0(u) and isnon0(h):

1523 r = atan(sqrt0(h / u)) * _2_0 # == acos(1 - 2 * h)

1524 sr, cr = sincos2(r)

1525 if isnon0(sr):

1526 u = 2 * (sj * ck)**2 / u

1527 h = 2 * (sk * cj)**2 / h

1528 x = u + h

1529 y = u - h

1530

1531 s = r / sr

1532 e = 4 * s**2

1533 d = 2 * cr

1534 a = e * d

1535 b = 2 * r

1536 c = s - (a - d) * _0_5

1537 f = E.f * _0_25

1538

1539 t = fsum_(a * x, -b * y, c * x**2, -d * y**2, e * x * y)

1540 r -= fsum_(s * x, -y, -t * f * _0_25) * f * sr

1541 return r

1542

1543

1544 def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):

1545 '''Compute the distance between two (spherical) points using

1546 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}

1547 spherical formula.

1548

1549 @arg lat1: Start latitude (C{degrees}).

1550 @arg lon1: Start longitude (C{degrees}).

1551 @arg lat2: End latitude (C{degrees}).

1552 @arg lon2: End longitude (C{degrees}).

1553 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},

1554 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}).

1555 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).

1556

1557 @return: Distance (C{meter}, same units as B{C{radius}}).

1558

1559 @raise UnitError: Invalid B{C{radius}}.

1560

1561 @see: Functions L{vincentys_}, L{cosineAndoyerLambert},

1562 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular},

1563 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},

1564 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2},

1565 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.

1566

1567 @note: See note at function L{vincentys_}.

1568 '''

1569 return _distanceToS(vincentys_, lat1, lat2, radius,

1570 *unroll180(lon1, lon2, wrap=wrap))

1571

1572

1573 def vincentys_(phi2, phi1, lam21):

1574 '''Compute the I{angular} distance between two (spherical) points using

1575 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}

1576 spherical formula.

1577

1578 @arg phi2: End latitude (C{radians}).

1579 @arg phi1: Start latitude (C{radians}).

1580 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).

1581

1582 @return: Angular distance (C{radians}).

1583

1584 @see: Functions L{vincentys}, L{cosineAndoyerLambert_},

1585 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

1586 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},

1587 L{flatPolar_}, L{haversine_} and L{thomas_}.

1588

1589 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_}

1590 produce equivalent results, but L{vincentys_} is suitable

1591 for antipodal points and slightly more expensive (M{3 cos,

1592 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_}

1593 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and

1594 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}).

1595 '''

1596 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21)

1597

1598 c = c2 * c21

1599 x = s1 * s2 + c1 * c

1600 y = c1 * s2 - s1 * c

1601 return atan2(hypot(c2 * s21, y), x)

1602

1603# **) MIT License

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