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__ = '23.03.19'

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_(func_, lat1, lat2, radius, d_lon, unused):

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

88 '''

89 r = func_(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: Function L{excessGirard}, L{excessLHuilier}, 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}, 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}, 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 units of B{C{radius}}

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

625 C{None} or C{0}.

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: Function 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}, 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((lam(phi1) + lam(phi2)) / 2)

691#

692# where lambda21 = lambda2 - lambda1 and lam(x) is the Lambertian (or inverse

693# Gudermannian) function

694#

695# lam(x) = asinh(tan(x)) = atanh(sin(x)) = 2 * atanh(tan(x / 2))

696#

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

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

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

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

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

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

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

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

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

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

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

708# - Charles Karney

709

710

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

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

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

714

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

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

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

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

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

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

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

722

723 @return: Surface area, I{signed} (I{square} C{meter}, or units of B{C{radius}}

724 I{squared}) or I{spherical excess} (C{radians}) if B{C{radius}} is

725 C{None} or C{0}.

726

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

728

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

730

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

732 '''

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

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

735

736

737 def excessQuad_(phi2, phi1, lam21):

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

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

740

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

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

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

744

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

746

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

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

749 '''

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

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

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

753

754

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

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

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

758 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}

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

760

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

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

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

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

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

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

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

768

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

770 ellipsoid axes).

771

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

773

774 @note: The meridional and prime_vertical radii of curvature

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

776

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

778 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},

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

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

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

782 '''

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

784 return flatLocal_(Phi_(lat2=lat2),

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

786

787 hubeny = flatLocal # PYCHOK for Karl Hubeny

788

789

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

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

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

793 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}

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

795

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

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

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

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

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

801

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

803

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

805

806 @note: The meridional and prime_vertical radii of curvature

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

808

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

810 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_},

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

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

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

814 '''

815 E = _ellipsoidal(datum, flatLocal_)

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

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

818

819 hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny

820

821

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

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

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

825 Geographical_distance#Polar_coordinate_flat-Earth_formula>}

826 formula.

827

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

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

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

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

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

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

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

835

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

837 ellipsoid or datum axes).

838

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

840

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

842 L{cosineForsytheAndoyerLambert},L{cosineLaw},

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

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

845 L{vincentys}.

846 '''

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

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

849

850

851 def flatPolar_(phi2, phi1, lam21):

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

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

854 Geographical_distance#Polar_coordinate_flat-Earth_formula>}

855 formula.

856

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

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

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

860

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

862

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

864 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

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

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

867 '''

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

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

870 if a < b:

871 a, b = b, a

872 if a < EPS0:

873 a = _0_0

874 elif b > 0:

875 b = b / a # /= chokes PyChecker

876 c = b * cos(lam21) * _2_0

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

878 a *= sqrt0(c)

879 return a

880

881

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

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

884 from a Point-Of-View in space.

885

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

887 or L{Vector3d}).

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

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

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

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

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

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

894 point plus C{LatLon} keyword arguments, include

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

896

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

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

899

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

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

902 the earth or points in an opposite direction.

903

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

905

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

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

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

909 '''

910 def _Error(txt):

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

912

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

914 _spherical_datum(earth, name=hartzell.__name__)

915 E = D.ellipsoid

916

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

918 try:

919 t = _MODS.triaxials

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

921 except Exception as x:

922 raise _Error(str(x))

923 else:

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

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

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

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

928

929 V3 = _MODS.vector3d._otherV3d

930 p3 = V3(pov=pov)

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

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

933

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

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

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

937

938 t = c2, c2, b2

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

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

941 raise _Error(_near_(_null_))

942

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

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

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

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

947 if r > 0:

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

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

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

951

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

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

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

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

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

957 raise _Error(_too_(_distant_))

958

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

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

961

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

963 if LatLon_and_kwds:

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

965 r = c.toLatLon(**LatLon_and_kwds)

966 return r

967

968

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

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

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

972 formula.

973

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

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

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

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

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

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

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

981

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

983

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

985

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

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

988 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},

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

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

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

992

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

994 '''

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

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

997

998

999 def haversine_(phi2, phi1, lam21):

1000 '''Compute the I{angular} distance between two (spherical) points

1001 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}

1002 formula.

1003

1004 @arg phi2: End latitude (C{radians}).

1005 @arg phi1: Start latitude (C{radians}).

1006 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).

1007

1008 @return: Angular distance (C{radians}).

1009

1010 @see: Functions L{haversine}, L{cosineAndoyerLambert_},

1011 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

1012 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},

1013 L{flatPolar_}, L{thomas_} and L{vincentys_}.

1014

1015 @note: See note at function L{vincentys_}.

1016 '''

1017 def _hsin(rad):

1018 return sin(rad * _0_5)**2

1019

1020 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine

1021 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2

1022

1023

1024 def heightOf(angle, distance, radius=R_M):

1025 '''Determine the height above the (spherical) earth' surface after

1026 traveling along a straight line at a given tilt.

1027

1028 @arg angle: Tilt angle above horizontal (C{degrees}).

1029 @arg distance: Distance along the line (C{meter} or same units as

1030 B{C{radius}}).

1031 @kwarg radius: Optional mean earth radius (C{meter}).

1032

1033 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}).

1034

1035 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}.

1036

1037 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>}

1038 (U{Shapiro et al. 2009, JTECH

1039 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>}

1040 and U{Potvin et al. 2012, JTECH

1041 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}).

1042 '''

1043 r = h = Radius(radius)

1044 d = fabs(Distance(distance))

1045 if d > h:

1046 d, h = h, d

1047

1048 if d > EPS0: # and h > EPS0

1049 d = d / h # /= h chokes PyChecker

1050 s = sin(Phi_(angle=angle, clip=_180_0))

1051 s = fsum_(_1_0, _2_0 * s * d, d**2)

1052 if s > 0:

1053 return h * sqrt(s) - r

1054

1055 raise _ValueError(angle=angle, distance=distance, radius=radius)

1056

1057

1058 def horizon(height, radius=R_M, refraction=False):

1059 '''Determine the distance to the horizon from a given altitude

1060 above the (spherical) earth.

1061

1062 @arg height: Altitude (C{meter} or same units as B{C{radius}}).

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

1064 @kwarg refraction: Consider atmospheric refraction (C{bool}).

1065

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

1067

1068 @raise ValueError: Invalid B{C{height}} or B{C{radius}}.

1069

1070 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}.

1071 '''

1072 h, r = Height(height), Radius(radius)

1073 if min(h, r) < 0:

1074 raise _ValueError(height=height, radius=radius)

1075

1076 if refraction:

1077 d2 = 2.415750694528 * h * r # 2.0 / 0.8279

1078 else:

1079 d2 = h * fsum_(r, r, h)

1080 return sqrt0(d2)

1081

1082

1083 def intersections2(lat1, lon1, radius1,

1084 lat2, lon2, radius2, datum=None, wrap=True):

1085 '''Conveniently compute the intersections of two circles each defined

1086 by a (geodetic) center point and a radius, using either ...

1087

1088 1) L{vector3d.intersections2} for small distances (below 100 KM or

1089 about 0.9 degrees) or if no B{C{datum}} is specified, or ...

1090

1091 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}}

1092 or if B{C{datum}} is a C{scalar} representing the earth radius,

1093 conventionally in C{meter} or ...

1094

1095 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}}

1096 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}

1097 is installed, otherwise ...

1098

1099 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal.

1100

1101 @arg lat1: Latitude of the first circle center (C{degrees}).

1102 @arg lon1: Longitude of the first circle center (C{degrees}).

1103 @arg radius1: Radius of the first circle (C{meter}, conventionally).

1104 @arg lat2: Latitude of the second circle center (C{degrees}).

1105 @arg lon2: Longitude of the second circle center (C{degrees}).

1106 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}).

1107 @kwarg datum: Optional ellipsoidal or spherical datum (L{Datum},

1108 L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple} or

1109 C{scalar} earth radius in C{meter}, same units as

1110 B{C{radius1}} and B{C{radius2}}) or C{None}.

1111 @kwarg wrap: Wrap and unroll longitudes (C{bool}).

1112

1113 @return: 2-Tuple of the intersection points, each a

1114 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles,

1115 the points are the same instance, aka I{radical center}.

1116

1117 @raise IntersectionError: Concentric, antipodal, invalid or

1118 non-intersecting circles or no

1119 convergence.

1120

1121 @raise TypeError: Invalid B{C{datum}}.

1122

1123 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}},

1124 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}.

1125 '''

1126 if datum is None or euclidean(lat1, lon1, lat2, lon2, radius=R_M,

1127 adjust=True, wrap=wrap) < _100km:

1128 def _V2T(x, y, _, **unused): # _ == z unused

1129 return LatLon2Tuple(y, x, name=intersections2.__name__)

1130

1131 r1 = m2degrees(Radius_(radius1=radius1), radius=R_M, lat=lat1)

1132 r2 = m2degrees(Radius_(radius2=radius2), radius=R_M, lat=lat2)

1133

1134 _, lon2 = unroll180(lon1, lon2, wrap=wrap)

1135 m = _MODS.vector3d

1136 t = m.intersections2(m.Vector3d(lon1, lat1, 0), r1,

1137 m.Vector3d(lon2, lat2, 0), r2, sphere=False,

1138 Vector=_V2T)

1139 else:

1140 def _LL2T(lat, lon, **unused):

1141 return LatLon2Tuple(lat, lon, name=intersections2.__name__)

1142

1143 d = _spherical_datum(datum, name=intersections2.__name__)

1144 if d.isSpherical:

1145 m = _MODS.sphericalTrigonometry

1146 elif d.isEllipsoidal:

1147 try:

1148 if d.ellipsoid.geodesic:

1149 pass

1150 m = _MODS.ellipsoidalKarney

1151 except ImportError:

1152 m = _MODS.ellipsoidalExact

1153 else:

1154 raise _AssertionError(datum=d)

1155

1156 t = m.intersections2(m.LatLon(lat1, lon1, datum=d), radius1,

1157 m.LatLon(lat2, lon2, datum=d), radius2,

1158 LatLon=_LL2T, height=0, wrap=wrap)

1159 return t

1160

1161

1162 def isantipode(lat1, lon1, lat2, lon2, eps=EPS):

1163 '''Check whether two points are antipodal, on diametrically

1164 opposite sides of the earth.

1165

1166 @arg lat1: Latitude of one point (C{degrees}).

1167 @arg lon1: Longitude of one point (C{degrees}).

1168 @arg lat2: Latitude of the other point (C{degrees}).

1169 @arg lon2: Longitude of the other point (C{degrees}).

1170 @kwarg eps: Tolerance for near-equality (C{degrees}).

1171

1172 @return: C{True} if points are antipodal within the

1173 B{C{eps}} tolerance, C{False} otherwise.

1174

1175 @see: Functions L{isantipode_} and L{antipode}.

1176 '''

1177 return True if (fabs(lat1 + lat2) <= eps and

1178 fabs(lon1 + lon2) <= eps) else \

1179 _MODS.latlonBase._isequalTo(antipode(lat1, lon1),

1180 normal(lat2, lon2), eps=eps)

1181

1182

1183 def isantipode_(phi1, lam1, phi2, lam2, eps=EPS):

1184 '''Check whether two points are antipodal, on diametrically

1185 opposite sides of the earth.

1186

1187 @arg phi1: Latitude of one point (C{radians}).

1188 @arg lam1: Longitude of one point (C{radians}).

1189 @arg phi2: Latitude of the other point (C{radians}).

1190 @arg lam2: Longitude of the other point (C{radians}).

1191 @kwarg eps: Tolerance for near-equality (C{radians}).

1192

1193 @return: C{True} if points are antipodal within the

1194 B{C{eps}} tolerance, C{False} otherwise.

1195

1196 @see: Functions L{isantipode} and L{antipode_}.

1197 '''

1198 return True if (fabs(phi1 + phi2) <= eps and

1199 fabs(lam1 + lam2) <= eps) else \

1200 _MODS.latlonBase._isequalTo_(antipode_(phi1, lam1),

1201 normal_(phi2, lam2), eps=eps)

1202

1203

1204 def isnormal(lat, lon, eps=0):

1205 '''Check whether B{C{lat}} I{and} B{C{lon}} are within the I{normal}

1206 range in C{degrees}.

1207

1208 @arg lat: Latitude (C{degrees}).

1209 @arg lon: Longitude (C{degrees}).

1210 @kwarg eps: Optional tolerance below C{90} and C{180 degrees}.

1211

1212 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and

1213 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise.

1214

1215 @see: Functions L{isnormal_} and L{normal}.

1216 '''

1217 return (_90_0 - fabs(lat)) >= eps and (_180_0 - fabs(lon)) >= eps

1218

1219

1220 def isnormal_(phi, lam, eps=0):

1221 '''Check whether B{C{phi}} I{and} B{C{lam}} are within the I{normal}

1222 range in C{radians}.

1223

1224 @arg phi: Latitude (C{radians}).

1225 @arg lam: Longitude (C{radians}).

1226 @kwarg eps: Optional tolerance below C{PI/2} and C{PI radians}.

1227

1228 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and

1229 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise.

1230

1231 @see: Functions L{isnormal} and L{normal_}.

1232 '''

1233 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps

1234

1235

1236 def latlon2n_xyz(lat, lon, name=NN):

1237 '''Convert lat-, longitude to C{n-vector} (I{normal} to the

1238 earth's surface) X, Y and Z components.

1239

1240 @arg lat: Latitude (C{degrees}).

1241 @arg lon: Longitude (C{degrees}).

1242 @kwarg name: Optional name (C{str}).

1243

1244 @return: A L{Vector3Tuple}C{(x, y, z)}.

1245

1246 @see: Function L{philam2n_xyz}.

1247

1248 @note: These are C{n-vector} x, y and z components,

1249 I{NOT} geocentric ECEF x, y and z coordinates!

1250 '''

1251 return _2n_xyz(name, *sincos2d_(lat, lon))

1252

1253

1254 def _normal2(a, b, n_2, n, n2):

1255 '''(INTERNAL) Helper for C{normal} and C{normal_}.

1256 '''

1257 if fabs(b) > n:

1258 b = remainder(b, n2)

1259 r = remainder(a, n) if fabs(a) > n_2 else a

1260 if r != a:

1261 r = -r

1262 b -= n if b > 0 else -n

1263 return float0(r, b)

1264

1265

1266 def normal(lat, lon, name=NN):

1267 '''Normalize a lat- I{and} longitude pair in C{degrees}.

1268

1269 @arg lat: Latitude (C{degrees}).

1270 @arg lon: Longitude (C{degrees}).

1271 @kwarg name: Optional name (C{str}).

1272

1273 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90}

1274 and C{abs(lon) <= 180}.

1275

1276 @see: Functions L{normal_} and L{isnormal}.

1277 '''

1278 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0), name=name)

1279

1280

1281 def normal_(phi, lam, name=NN):

1282 '''Normalize a lat- I{and} longitude pair in C{radians}.

1283

1284 @arg phi: Latitude (C{radians}).

1285 @arg lam: Longitude (C{radians}).

1286 @kwarg name: Optional name (C{str}).

1287

1288 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2}

1289 and C{abs(lam) <= PI}.

1290

1291 @see: Functions L{normal} and L{isnormal_}.

1292 '''

1293 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2), name=name)

1294

1295

1296 def _2n_xyz(name, sa, ca, sb, cb):

1297 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}.

1298 '''

1299 # Kenneth Gade eqn 3, but using right-handed

1300 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N

1301 return Vector3Tuple(ca * cb, ca * sb, sa, name=name)

1302

1303

1304 def n_xyz2latlon(x, y, z, name=NN):

1305 '''Convert C{n-vector} components to lat- and longitude in C{degrees}.

1306

1307 @arg x: X component (C{scalar}).

1308 @arg y: Y component (C{scalar}).

1309 @arg z: Z component (C{scalar}).

1310 @kwarg name: Optional name (C{str}).

1311

1312 @return: A L{LatLon2Tuple}C{(lat, lon)}.

1313

1314 @see: Function L{n_xyz2philam}.

1315 '''

1316 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name)

1317

1318

1319 def n_xyz2philam(x, y, z, name=NN):

1320 '''Convert C{n-vector} components to lat- and longitude in C{radians}.

1321

1322 @arg x: X component (C{scalar}).

1323 @arg y: Y component (C{scalar}).

1324 @arg z: Z component (C{scalar}).

1325 @kwarg name: Optional name (C{str}).

1326

1327 @return: A L{PhiLam2Tuple}C{(phi, lam)}.

1328

1329 @see: Function L{n_xyz2latlon}.

1330 '''

1331 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name)

1332

1333

1334 def _opposes(d, m, n, n2):

1335 '''(INETNAL) Helper for C{opposing} and C{opposing_}.

1336 '''

1337 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1

1338 return False if d < m or d > (n2 - m) else (

1339 True if (n - m) < d < (n + m) else None)

1340

1341

1342 def opposing(bearing1, bearing2, margin=_90_0):

1343 '''Compare the direction of two bearings given in C{degrees}.

1344

1345 @arg bearing1: First bearing (compass C{degrees}).

1346 @arg bearing2: Second bearing (compass C{degrees}).

1347 @kwarg margin: Optional, interior angle bracket (C{degrees}).

1348

1349 @return: C{True} if both bearings point in opposite, C{False} if

1350 in similar or C{None} if in perpendicular directions.

1351

1352 @see: Function L{opposing_}.

1353 '''

1354 m = Degrees_(margin=margin, low=EPS0, high=_90_0)

1355 return _opposes(bearing2 - bearing1, m,_180_0, _360_0)

1356

1357

1358 def opposing_(radians1, radians2, margin=PI_2):

1359 '''Compare the direction of two bearings given in C{radians}.

1360

1361 @arg radians1: First bearing (C{radians}).

1362 @arg radians2: Second bearing (C{radians}).

1363 @kwarg margin: Optional, interior angle bracket (C{radians}).

1364

1365 @return: C{True} if both bearings point in opposite, C{False} if

1366 in similar or C{None} if in perpendicular directions.

1367

1368 @see: Function L{opposing}.

1369 '''

1370 m = Radians_(margin=margin, low=EPS0, high=PI_2)

1371 return _opposes(radians2 - radians1, m, PI, PI2)

1372

1373

1374 def philam2n_xyz(phi, lam, name=NN):

1375 '''Convert lat-, longitude to C{n-vector} (I{normal} to the

1376 earth's surface) X, Y and Z components.

1377

1378 @arg phi: Latitude (C{radians}).

1379 @arg lam: Longitude (C{radians}).

1380 @kwarg name: Optional name (C{str}).

1381

1382 @return: A L{Vector3Tuple}C{(x, y, z)}.

1383

1384 @see: Function L{latlon2n_xyz}.

1385

1386 @note: These are C{n-vector} x, y and z components,

1387 I{NOT} geocentric ECEF x, y and z coordinates!

1388 '''

1389 return _2n_xyz(name, *sincos2_(phi, lam))

1390

1391

1392 def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d

1393 # (INTERNAL) See C{radical2} below

1394 # assert d > EPS0

1395 r = fsum_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5

1396 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d)

1397

1398

1399 def radical2(distance, radius1, radius2):

1400 '''Compute the I{radical ratio} and I{radical line} of two

1401 U{intersecting circles<https://MathWorld.Wolfram.com/

1402 Circle-CircleIntersection.html>}.

1403

1404 The I{radical line} is perpendicular to the axis thru the

1405 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}.

1406

1407 @arg distance: Distance between the circle centers (C{scalar}).

1408 @arg radius1: Radius of the first circle (C{scalar}).

1409 @arg radius2: Radius of the second circle (C{scalar}).

1410

1411 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <=

1412 ratio <= 1.0} and C{xline} is along the B{C{distance}}.

1413

1414 @raise IntersectionError: The B{C{distance}} exceeds the sum

1415 of B{C{radius1}} and B{C{radius2}}.

1416

1417 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or

1418 B{C{radius2}}.

1419

1420 @see: U{Circle-Circle Intersection

1421 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}.

1422 '''

1423 d = Distance_(distance, low=_0_0)

1424 r1 = Radius_(radius1=radius1)

1425 r2 = Radius_(radius2=radius2)

1426 if d > (r1 + r2):

1427 raise IntersectionError(distance=d, radius1=r1, radius2=r2,

1428 txt=_too_(_distant_))

1429 return _radical2(d, r1, r2) if d > EPS0 else \

1430 Radical2Tuple(_0_5, _0_0)

1431

1432

1433 class Radical2Tuple(_NamedTuple):

1434 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and

1435 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0}

1436 '''

1437 _Names_ = (_ratio_, _xline_)

1438 _Units_ = ( Scalar, Scalar)

1439

1440

1441 def _scale_deg(lat1, lat2): # degrees

1442 # scale factor cos(mean of lats) for delta lon

1443 m = fabs(lat1 + lat2) * _0_5

1444 return cos(radians(m)) if m < 90 else _0_0

1445

1446

1447 def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights

1448 # scale factor cos(mean of phis) for delta lam

1449 m = fabs(phi1 + phi2) * _0_5

1450 return cos(m) if m < PI_2 else _0_0

1451

1452

1453 def _sincosa6(phi2, phi1, lam21):

1454 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine.

1455 '''

1456 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21)

1457 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21

1458

1459

1460 def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):

1461 '''Compute the distance between two (ellipsoidal) points using

1462 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}

1463 formula.

1464

1465 @arg lat1: Start latitude (C{degrees}).

1466 @arg lon1: Start longitude (C{degrees}).

1467 @arg lat2: End latitude (C{degrees}).

1468 @arg lon2: End longitude (C{degrees}).

1469 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},

1470 L{Ellipsoid2} or L{a_f2Tuple}).

1471 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).

1472

1473 @return: Distance (C{meter}, same units as the B{C{datum}}'s

1474 ellipsoid axes).

1475

1476 @raise TypeError: Invalid B{C{datum}}.

1477

1478 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},

1479 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},

1480 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}.

1481 '''

1482 return _distanceToE(thomas_, lat1, lat2, datum,

1483 *unroll180(lon1, lon2, wrap=wrap))

1484

1485

1486 def thomas_(phi2, phi1, lam21, datum=_WGS84):

1487 '''Compute the I{angular} distance between two (ellipsoidal) points using

1488 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}

1489 formula.

1490

1491 @arg phi2: End latitude (C{radians}).

1492 @arg phi1: Start latitude (C{radians}).

1493 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).

1494 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},

1495 L{Ellipsoid2} or L{a_f2Tuple}).

1496

1497 @return: Angular distance (C{radians}).

1498

1499 @raise TypeError: Invalid B{C{datum}}.

1500

1501 @see: Functions L{thomas}, L{cosineAndoyerLambert_},

1502 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

1503 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},

1504 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP

1505 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/

1506 Distance/ThomasFormula.php>}.

1507 '''

1508 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)

1509 if r and isnon0(c1) and isnon0(c2):

1510 E = _ellipsoidal(datum, thomas_)

1511 if E.f:

1512 r1 = atan2(E.b_a * s1, c1)

1513 r2 = atan2(E.b_a * s2, c2)

1514

1515 j = (r2 + r1) * _0_5

1516 k = (r2 - r1) * _0_5

1517 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5)

1518

1519 h = fsum_(sk**2, (ck * h)**2, -(sj * h)**2)

1520 u = _1_0 - h

1521 if isnon0(u) and isnon0(h):

1522 r = atan(sqrt0(h / u)) * _2_0 # == acos(1 - 2 * h)

1523 sr, cr = sincos2(r)

1524 if isnon0(sr):

1525 u = 2 * (sj * ck)**2 / u

1526 h = 2 * (sk * cj)**2 / h

1527 x = u + h

1528 y = u - h

1529

1530 s = r / sr

1531 e = 4 * s**2

1532 d = 2 * cr

1533 a = e * d

1534 b = 2 * r

1535 c = s - (a - d) * _0_5

1536 f = E.f * _0_25

1537

1538 t = fsum_(a * x, -b * y, c * x**2, -d * y**2, e * x * y)

1539 r -= fsum_(s * x, -y, -t * f * _0_25) * f * sr

1540 return r

1541

1542

1543 def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):

1544 '''Compute the distance between two (spherical) points using

1545 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}

1546 spherical formula.

1547

1548 @arg lat1: Start latitude (C{degrees}).

1549 @arg lon1: Start longitude (C{degrees}).

1550 @arg lat2: End latitude (C{degrees}).

1551 @arg lon2: End longitude (C{degrees}).

1552 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},

1553 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}).

1554 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).

1555

1556 @return: Distance (C{meter}, same units as B{C{radius}}).

1557

1558 @raise UnitError: Invalid B{C{radius}}.

1559

1560 @see: Functions L{vincentys_}, L{cosineAndoyerLambert},

1561 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular},

1562 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},

1563 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2},

1564 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.

1565

1566 @note: See note at function L{vincentys_}.

1567 '''

1568 return _distanceToS(vincentys_, lat1, lat2, radius,

1569 *unroll180(lon1, lon2, wrap=wrap))

1570

1571

1572 def vincentys_(phi2, phi1, lam21):

1573 '''Compute the I{angular} distance between two (spherical) points using

1574 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}

1575 spherical formula.

1576

1577 @arg phi2: End latitude (C{radians}).

1578 @arg phi1: Start latitude (C{radians}).

1579 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).

1580

1581 @return: Angular distance (C{radians}).

1582

1583 @see: Functions L{vincentys}, L{cosineAndoyerLambert_},

1584 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

1585 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},

1586 L{flatPolar_}, L{haversine_} and L{thomas_}.

1587

1588 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_}

1589 produce equivalent results, but L{vincentys_} is suitable

1590 for antipodal points and slightly more expensive (M{3 cos,

1591 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_}

1592 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and

1593 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}).

1594 '''

1595 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21)

1596

1597 c = c2 * c21

1598 x = s1 * s2 + c1 * c

1599 y = c1 * s2 - s1 * c

1600 return atan2(hypot(c2 * s21, y), x)

1601

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