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.basics import isscalar # from .fsums

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

11 _umod_PI2, float0_, isnon0, remainder, \

12 _0_0, _0_125, _0_25, _0_5, _1_0, _2_0, \

13 _4_0, _32_0, _90_0, _180_0, _360_0

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

15 _mean_radius, _spherical_datum, _WGS84

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

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

18 _TypeError, _ValueError, \

19 _xError, _xkwds, _xkwds_pop

20 from pygeodesy.fmath import euclid, hypot, hypot2, sqrt0

21 from pygeodesy.fsums import fsumf_, isscalar

22 from pygeodesy.interns import NN, _distant_, _SPACE_, _too_

23 from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS

24 from pygeodesy.named import _NamedTuple, _xnamed, Fmt, unstr

25 from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, \

26 Intersection3Tuple, LatLon2Tuple, \

27 PhiLam2Tuple, Vector3Tuple

28# from pygeodesy.streprs import Fmt, unstr # from .named

29 from pygeodesy.units import Bearing, Degrees_, Distance, Distance_, Height, \

30 Lam_, Lat, Lon, Meter_, Phi_, Radians, Radians_, \

31 Radius, Radius_, Scalar, _100km

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

33 tan_2, sincos2, sincos2_, sincos2d_, _Wrap

34

35 from contextlib import contextmanager

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

37

38__all__ = _ALL_LAZY.formy

39__version__ = '23.05.15'

40

41_delta_ = 'delta'

42_D2_R2 = (PI / _180_0)**2 # degrees- to radians-squared

43_EWGS84 = _WGS84.ellipsoid

44_ratio_ = 'ratio'

45_xline_ = 'xline'

46

47

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

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

50 '''

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

52 if r == a:

53 r = -r

54 b += n

55 if fabs(b) > n:

56 b = remainder(b, n2)

57 return float0_(r, b)

58

59

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

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

62 to a given point in C{degrees}.

63

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

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

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

67

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

69

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

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

72 '''

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

74

75

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

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

78 to a given point in C{radians}.

79

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

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

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

83

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

85

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

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

88 '''

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

90

91

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

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

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

95

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

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

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

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

100 @kwarg final_wrap: Optional keyword arguments for function

101 L{pygeodesy.bearing_}.

102

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

104 zero if start and end point coincide.

105 '''

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

107 Phi_(lat2=lat2), Lam_(lon2=lon2), **final_wrap)

108 return degrees(r)

109

110

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

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

113 between a (spherical) start and end point.

114

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

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

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

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

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

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

121 B{C{lam2}} (C{bool}).

122

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

124 and end point coincide.

125

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

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

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

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

130 '''

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

132 if final: # swap plus PI

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

134 r = PI3

135 else:

136 r = PI2

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

138

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

140 y = sdb * ca2

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

142

143

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

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

146 '''

147 try: # for LatLon_ and ellipsoidal LatLon

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

149 except AttributeError:

150 pass

151 # XXX spherical version, OK for ellipsoidal ispolar?

152 a1, b1 = p1.philam

153 a2, b2 = p2.philam

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

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

156 name=_bearingTo2.__name__)

157

158

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

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

161 lat2 - lat1)} between two points.

162

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

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

165

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

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

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

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

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

171 mean latitude (C{bool}).

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

173 and B{C{lon2}} (C{bool}).

174

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

176

177 @note: Courtesy of Martin Schultz.

178

179 @see: U{Local, flat earth approximation

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

181 '''

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

183 if adjust: # scale delta lon

184 d_lon *= _scale_deg(lat1, lat2)

185 return atan2b(d_lon, lat2 - lat1)

186

187

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

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

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

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

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

193

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

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

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

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

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

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

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

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

202

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

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

205

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

207

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

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

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

211 L{Ellipsoid.distance2}.

212 '''

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

214

215

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

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

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

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

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

221

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

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

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

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

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

227

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

229

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

231

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

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

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

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

236 AndoyerLambert.php>}.

237 '''

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

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

240 E = _ellipsoidal(datum, cosineAndoyerLambert_)

241 if E.f: # ellipsoidal

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

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

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

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

246 if r:

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

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

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

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

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

252 return r

253

254

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

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

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

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

259 formula.

260

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

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

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

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

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

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

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

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

269

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

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

272

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

274

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

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

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

278 L{Ellipsoid.distance2}.

279 '''

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

281

282

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

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

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

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

287 formula.

288

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

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

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

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

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

294

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

296

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

298

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

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

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

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

303 Distance/ForsytheCorrection.php>}.

304 '''

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

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

307 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_)

308 if E.f: # ellipsoidal

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

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

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

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

313 x = s + t

314 y = s - t

315

316 s = 8 * r**2 / sr

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

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

319 b = -2 * d

320 e = 30 * s2r

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

322

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

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

325 return r

326

327

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

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

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

331 formula.

332

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

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

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

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

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

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

339 L{a_f2Tuple}) to use.

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

341 and B{C{lon2}} (C{bool}).

342

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

344 ellipsoid or datum axes).

345

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

347

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

349 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean},

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

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

352

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

354 '''

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

356

357

358 def cosineLaw_(phi2, phi1, lam21):

359 '''Compute the I{angular} distance between two points using the U{spherical

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

361 formula.

362

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

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

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

366

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

368

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

370 L{cosineForsytheAndoyerLambert_}, L{equirectangular_},

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

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

373

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

375 '''

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

377

378

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

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

381 '''

382 if wrap:

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

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

385 else: # for backward compaibility

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

387

388

389 def _dE(func_, earth, *wrap_lls):

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

391 '''

392 E = _ellipsoidal(earth, func_)

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

394 return r * E.a

395

396

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

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

399 '''

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

401 if radius is not R_M:

402 _, lat1, _, lat2, _ = wrap_lls

403 radius = _mean_radius(radius, lat1, lat2)

404 return r * radius

405

406

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

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

409 '''

410 r = excess_(*_d3(*wrap_lls))

411 if radius:

412 _, lat1, _, lat2, _ = wrap_lls

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

414 return r

415

416

417 def _ellipsoidal(earth, where):

418 '''(INTERNAL) Helper for distances.

419 '''

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

421 earth if isinstance(earth, Ellipsoid) else

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

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

424

425

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

427 '''Compute the distance between two points using

428 the U{Equirectangular Approximation / Projection

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

430

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

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

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

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

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

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

437 L{a_f2Tuple}).

438 @kwarg adjust_limit_wrap: Optional keyword arguments for

439 function L{equirectangular_}.

440

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

442 the ellipsoid or datum axes).

443

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

445

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

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

448 approximate or accurate distance functions.

449 '''

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

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

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

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

454

455

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

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

458 and L{hausdorff._HausdorffMeterRedians} classes.

459 '''

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

461

462

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

464 '''Compute the distance between two points using the U{Equirectangular

465 Approximation / Projection

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

467

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

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

470

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

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

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

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

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

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

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

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

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

480 and B{C{lon2}} (C{bool}).

481

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

483 unroll_lon2)} in C{degrees squared}.

484

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

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

487 set to C{True}.

488

489 @see: U{Local, flat earth approximation

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

491 L{equirectangular}, L{cosineAndoyerLambert},

492 L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean},

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

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

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

496 '''

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

498 d_lat = lat2 - lat1

499

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

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

502 if d > limit:

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

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

505 limit=limit, wrap=wrap)

506 raise LimitError(s, txt=t)

507

508 if adjust: # scale delta lon

509 d_lon *= _scale_deg(lat1, lat2)

510

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

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

513

514

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

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

517

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

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

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

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

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

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

524 L{a_f2Tuple}) to use.

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

526 the mean latitude (C{bool}).

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

528 and B{C{lon2}} (C{bool}).

529

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

531 ellipsoid or datum axes).

532

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

534

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

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

537 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},

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

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

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

541 '''

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

543

544

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

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

547 (spherical) points.

548

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

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

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

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

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

554

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

556

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

558 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_},

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

560 and L{vincentys_}.

561 '''

562 if adjust:

563 lam21 *= _scale_rad(phi2, phi1)

564 return euclid(phi2 - phi1, lam21)

565

566

567 def excessAbc_(A, b, c):

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

569 from two sides and the included angle.

570

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

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

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

574

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

576

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

578

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

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

581 '''

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

583 Radians_(c=c) * _0_5)

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

585

586

587 def excessGirard_(A, B, C):

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

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

590 formula.

591

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

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

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

595

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

597

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

599

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

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

602 '''

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

604 Radians_(B=B),

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

606

607

608 def excessLHuilier_(a, b, c):

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

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

611 Theorem.

612

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

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

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

616

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

618

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

620

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

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

623 '''

624 a = Radians_(a=a)

625 b = Radians_(b=b)

626 c = Radians_(c=c)

627

628 s = fsumf_(a, b, c) * _0_5

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

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

631 return Radians(LHuilier=r * _4_0)

632

633

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

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

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

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

638 method.

639

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

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

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

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

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

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

646 L{a_f2Tuple}) or C{None}.

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

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

649

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

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

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

653

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

655

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

657

658 @raise ValueError: Semi-circular longitudinal delta.

659

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

661 '''

662 return _eA(excessKarney_, radius, wrap, lat1, lon1, lat2, lon2)

663

664

665 def excessKarney_(phi2, phi1, lam21):

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

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

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

669 method.

670

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

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

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

674

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

676

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

678

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

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

681 '''

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

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

684 #

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

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

687 # 1 + tan(φ1 / 2) · tan(φ2 / 2)

688 #

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

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

691 t2 = tan_2(phi2)

692 t1 = tan_2(phi1)

693 t = tan_2(lam21, lam21=None)

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

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

696

697

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

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

700#

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

702#

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

704#

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

706#

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

708#

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

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

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

712#

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

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

715#

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

717# Lambertian(phi2)) / 2)

718#

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

720# inverse Gudermannian) function

721#

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

723#

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

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

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

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

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

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

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

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

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

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

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

735# - Charles Karney

736

737

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

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

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

741

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

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

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

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

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

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

748 L{a_f2Tuple}) or C{None}.

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

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

751

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

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

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

755

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

757

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

759

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

761 '''

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

763

764

765 def excessQuad_(phi2, phi1, lam21):

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

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

768

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

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

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

772

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

774

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

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

777 '''

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

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

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

781

782

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

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

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

786 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}

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

788

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

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

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

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

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

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

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

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

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

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

799

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

801 ellipsoid axes).

802

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

804

805 @note: The meridional and prime_vertical radii of curvature

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

807

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

809 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},

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

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

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

813 '''

814 E = _ellipsoidal(datum, flatLocal)

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

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

817

818 hubeny = flatLocal # PYCHOK for Karl Hubeny

819

820

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

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

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

824 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}

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

826

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

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

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

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

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

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

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

834

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

836

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

838

839 @note: The meridional and prime_vertical radii of curvature

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

841

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

843 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_},

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

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

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

847 '''

848 E = _ellipsoidal(datum, flatLocal_)

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

850

851 hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny

852

853

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

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

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

857 Geographical_distance#Polar_coordinate_flat-Earth_formula>}

858 formula.

859

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

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

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

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

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

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

866 L{a_f2Tuple}) to use.

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

868 and B{C{lon2}} (C{bool}).

869

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

871 ellipsoid or datum axes).

872

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

874

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

876 L{cosineForsytheAndoyerLambert},L{cosineLaw},

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

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

879 L{vincentys}.

880 '''

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

882

883

884 def flatPolar_(phi2, phi1, lam21):

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

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

887 Geographical_distance#Polar_coordinate_flat-Earth_formula>}

888 formula.

889

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

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

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

893

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

895

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

897 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

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

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

900 '''

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

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

903 if a < b:

904 a, b = b, a

905 if a < EPS0:

906 a = _0_0

907 elif b > 0:

908 b = b / a # /= chokes PyChecker

909 c = b * cos(lam21) * _2_0

910 c = fsumf_(_1_0, b**2, -fabs(c))

911 a *= sqrt0(c)

912 return a

913

914

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

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

917 from a Point-Of-View in space.

918

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

920 or L{Vector3d}).

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

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

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

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

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

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

927 point plus C{LatLon} keyword arguments, include

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

929

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

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

932

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

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

935 the earth or points in an opposite direction.

936

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

938

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

940 method L{Ellipsoid.hartzell4} and U{I{Satellite Line-of-Sight

941 Intersection with Earth}<https://StephenHartzell.Medium.com/

942 satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}.

943 '''

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

945 _spherical_datum(earth, name=hartzell.__name__)

946 try:

947 r, _ = _MODS.triaxials._hartzell3d2(pov, los, D.ellipsoid._triaxial)

948 except Exception as x:

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

950

951# else:

952# E = D.ellipsoid

953# # Triaxial(a, b, c) == (E.a, E.a, E.b)

954#

955# def _Error(txt):

956# return IntersectionError(pov=pov, los=los, earth=earth, txt=txt)

957#

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

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

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

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

962#

963# V3 = _MODS.vector3d._otherV3d

964# p3 = V3(pov=pov)

965# u3 = V3(los=los) if los else p3.negate()

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

967#

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

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

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

971#

972# t = c2, c2, b2

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

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

975# raise _Error(_near_(_null_))

976#

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

978# r = fsumf_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2,

979# c2 * u2, -u2 * z2, -w2 * x2, ux * wz * 2,

980# -w2 * y2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2)

981# if r > 0:

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

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

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

985#

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

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

988# s = fsumf_(_1_0, x2 / a2, y2 / b2, z2 / c2, _N_2_0) # like _sideOf

989# raise _Error(_outside_ if s > 0 else _inside_)

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

991# raise _Error(_too_(_distant_))

992#

993# r = p3.minus(u3.times(d))

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

995

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

997 if LatLon_and_kwds:

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

999 r = c.toLatLon(**LatLon_and_kwds)

1000 return r

1001

1002

1003 def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):

1004 '''Compute the distance between two (spherical) points using the

1005 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}

1006 formula.

1007

1008 @arg lat1: Start latitude (C{degrees}).

1009 @arg lon1: Start longitude (C{degrees}).

1010 @arg lat2: End latitude (C{degrees}).

1011 @arg lon2: End longitude (C{degrees}).

1012 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})

1013 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or

1014 L{a_f2Tuple}) to use.

1015 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll

1016 B{C{lat2}} and B{C{lon2}} (C{bool}).

1017

1018 @return: Distance (C{meter}, same units as B{C{radius}}).

1019

1020 @raise TypeError: Invalid B{C{radius}}.

1021

1022 @see: U{Distance between two (spherical) points

1023 <https://www.EdWilliams.org/avform.htm#Dist>}, functions

1024 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},

1025 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},

1026 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},

1027 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.

1028

1029 @note: See note at function L{vincentys_}.

1030 '''

1031 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2)

1032

1033

1034 def haversine_(phi2, phi1, lam21):

1035 '''Compute the I{angular} distance between two (spherical) points

1036 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}

1037 formula.

1038

1039 @arg phi2: End latitude (C{radians}).

1040 @arg phi1: Start latitude (C{radians}).

1041 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).

1042

1043 @return: Angular distance (C{radians}).

1044

1045 @see: Functions L{haversine}, L{cosineAndoyerLambert_},

1046 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

1047 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},

1048 L{flatPolar_}, L{thomas_} and L{vincentys_}.

1049

1050 @note: See note at function L{vincentys_}.

1051 '''

1052 def _hsin(rad):

1053 return sin(rad * _0_5)**2

1054

1055 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine

1056 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2

1057

1058

1059 def heightOf(angle, distance, radius=R_M):

1060 '''Determine the height above the (spherical) earth' surface after

1061 traveling along a straight line at a given tilt.

1062

1063 @arg angle: Tilt angle above horizontal (C{degrees}).

1064 @arg distance: Distance along the line (C{meter} or same units as

1065 B{C{radius}}).

1066 @kwarg radius: Optional mean earth radius (C{meter}).

1067

1068 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}).

1069

1070 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}.

1071

1072 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>}

1073 (U{Shapiro et al. 2009, JTECH

1074 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>}

1075 and U{Potvin et al. 2012, JTECH

1076 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}).

1077 '''

1078 r = h = Radius(radius)

1079 d = fabs(Distance(distance))

1080 if d > h:

1081 d, h = h, d

1082

1083 if d > EPS0: # and h > EPS0

1084 d = d / h # /= h chokes PyChecker

1085 s = sin(Phi_(angle=angle, clip=_180_0))

1086 s = fsumf_(_1_0, _2_0 * s * d, d**2)

1087 if s > 0:

1088 return h * sqrt(s) - r

1089

1090 raise _ValueError(angle=angle, distance=distance, radius=radius)

1091

1092

1093 def horizon(height, radius=R_M, refraction=False):

1094 '''Determine the distance to the horizon from a given altitude

1095 above the (spherical) earth.

1096

1097 @arg height: Altitude (C{meter} or same units as B{C{radius}}).

1098 @kwarg radius: Optional mean earth radius (C{meter}).

1099 @kwarg refraction: Consider atmospheric refraction (C{bool}).

1100

1101 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}).

1102

1103 @raise ValueError: Invalid B{C{height}} or B{C{radius}}.

1104

1105 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}.

1106 '''

1107 h, r = Height(height), Radius(radius)

1108 if min(h, r) < 0:

1109 raise _ValueError(height=height, radius=radius)

1110

1111 if refraction:

1112 d2 = 2.415750694528 * h * r # 2.0 / 0.8279

1113 else:

1114 d2 = h * fsumf_(r, r, h)

1115 return sqrt0(d2)

1116

1117

1118 class _idllmn6(object): # see also .geodesicw._wargs, .vector2d._numpy

1119 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}.

1120 '''

1121 @contextmanager # <https://www.python.org/dev/peps/pep-0343/> Examples

1122 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds):

1123 try:

1124 if wrap:

1125 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)

1126 kwds = _xkwds(kwds, wrap=wrap) # for _xError

1127 m = small if small is _100km else Meter_(small=small)

1128 n = (intersections2 if s else intersection2).__name__

1129 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m:

1130 d, m = None, _MODS.vector3d

1131 _i = m._intersects2 if s else m._intersect3d3

1132 elif isscalar(datum) and datum < 0 and not s:

1133 d = _spherical_datum(-datum, name=n)

1134 m = _MODS.sphericalNvector

1135 _i = m.intersection

1136 else:

1137 d = _spherical_datum(datum, name=n)

1138 if d.isSpherical:

1139 m = _MODS.sphericalTrigonometry

1140 _i = m._intersects2 if s else m._intersect

1141 elif d.isEllipsoidal:

1142 try:

1143 if d.ellipsoid.geodesic:

1144 pass

1145 m = _MODS.ellipsoidalKarney

1146 except ImportError:

1147 m = _MODS.ellipsoidalExact

1148 _i = m._intersections2 if s else m._intersection3 # ellispoidalBaseDI

1149 else:

1150 raise _TypeError(datum=datum)

1151 yield _i, d, lat2, lon2, m, n

1152

1153 except (TypeError, ValueError) as x:

1154 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum,

1155 lat2=lat2, lon2=lon2, small=small, **kwds)

1156

1157_idllmn6 = _idllmn6() # PYCHOK singleton

1158

1159

1160 def intersection2(lat1, lon1, bearing1,

1161 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True

1162 '''I{Conveniently} compute the intersection of two lines each defined

1163 by a (geodetic) point and a bearing from North, using either ...

1164

1165 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km

1166 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...

1167

1168 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}}

1169 or a C{scalar B{datum}} representing the earth radius, conventionally

1170 in C{meter} or ...

1171

1172 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative}

1173 C{scalar}, (negative) earth radius, conventionally in C{meter} or ...

1174

1175 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}}

1176 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}

1177 is installed, otherwise ...

1178

1179 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal.

1180

1181 @arg lat1: Latitude of the first point (C{degrees}).

1182 @arg lon1: Longitude of the first point (C{degrees}).

1183 @arg bearing1: Bearing at the first point (compass C{degrees}).

1184 @arg lat2: Latitude of the second point (C{degrees}).

1185 @arg lon2: Longitude of the second point (C{degrees}).

1186 @arg bearing2: Bearing at the second point (compass C{degrees}).

1187 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},

1188 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth

1189 radius (C{meter}, same units as B{C{radius1}} and

1190 B{C{radius2}}) or C{None}.

1191 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}

1192 and B{C{lon2}} (C{bool}).

1193 @kwarg small: Upper limit for small distances (C{meter}).

1194

1195 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and

1196 longitude of the intersection point.

1197

1198 @raise IntersectionError: Ambiguous or infinite intersection

1199 or colinear, parallel or otherwise

1200 non-intersecting lines.

1201

1202 @raise TypeError: Invalid B{C{datum}}.

1203

1204 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}},

1205 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}.

1206

1207 @see: Method L{RhumbLine.intersection2}.

1208

1209 @note: The returned intersections may be near-antipodal.

1210 '''

1211 b1 = Bearing(bearing1=bearing1)

1212 b2 = Bearing(bearing2=bearing2)

1213 with _idllmn6(datum, lat1, lon1, lat2, lon2,

1214 small, wrap, False, bearing1=b1, bearing2=b2) as t:

1215 _i, d, lat2, lon2, m, n = t

1216 if d is None:

1217 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1,

1218 m.Vector3d(lon2, lat2, 0), b2, useZ=False)

1219 t = LatLon2Tuple(t.y, t.x, name=n)

1220

1221 else:

1222 t = _i(m.LatLon(lat1, lon1, datum=d), b1,

1223 m.LatLon(lat2, lon2, datum=d), b2, height=0, wrap=False)

1224 if isinstance(t, Intersection3Tuple): # ellipsoidal

1225 t, _, _ = t

1226 t = LatLon2Tuple(t.lat, t.lon, name=n)

1227 return t

1228

1229

1230 def intersections2(lat1, lon1, radius1,

1231 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True

1232 '''I{Conveniently} compute the intersections of two circles each defined

1233 by a (geodetic) center point and a radius, using either ...

1234

1235 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km

1236 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...

1237

1238 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}}

1239 or a C{scalar B{datum}} representing the earth radius, conventionally

1240 in C{meter} or ...

1241

1242 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}}

1243 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}

1244 is installed, otherwise ...

1245

1246 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal.

1247

1248 @arg lat1: Latitude of the first circle center (C{degrees}).

1249 @arg lon1: Longitude of the first circle center (C{degrees}).

1250 @arg radius1: Radius of the first circle (C{meter}, conventionally).

1251 @arg lat2: Latitude of the second circle center (C{degrees}).

1252 @arg lon2: Longitude of the second circle center (C{degrees}).

1253 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}).

1254 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},

1255 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth

1256 radius (C{meter}, same units as B{C{radius1}} and

1257 B{C{radius2}}) or C{None}.

1258 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}

1259 and B{C{lon2}} (C{bool}).

1260 @kwarg small: Upper limit for small distances (C{meter}).

1261

1262 @return: 2-Tuple of the intersection points, each a

1263 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the

1264 points are the same instance, aka the I{radical center}.

1265

1266 @raise IntersectionError: Concentric, antipodal, invalid or

1267 non-intersecting circles or no

1268 convergence.

1269

1270 @raise TypeError: Invalid B{C{datum}}.

1271

1272 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}},

1273 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}.

1274 '''

1275 r1 = Radius_(radius1=radius1)

1276 r2 = Radius_(radius2=radius2)

1277 with _idllmn6(datum, lat1, lon1, lat2, lon2,

1278 small, wrap, True, radius1=r1, radius2=r2) as t:

1279 _i, d, lat2, lon2, m, n = t

1280 if d is None:

1281 r1 = m2degrees(r1, radius=R_M, lat=lat1)

1282 r2 = m2degrees(r2, radius=R_M, lat=lat2)

1283

1284 def _V2T(x, y, _, **unused): # _ == z unused

1285 return LatLon2Tuple(y, x, name=n)

1286

1287 t = _i(m.Vector3d(lon1, lat1, 0), r1,

1288 m.Vector3d(lon2, lat2, 0), r2, sphere=False,

1289 Vector=_V2T)

1290 else:

1291 def _LL2T(lat, lon, **unused):

1292 return LatLon2Tuple(lat, lon, name=n)

1293

1294 t = _i(m.LatLon(lat1, lon1, datum=d), r1,

1295 m.LatLon(lat2, lon2, datum=d), r2,

1296 LatLon=_LL2T, height=0, wrap=False)

1297 return t

1298

1299

1300 def isantipode(lat1, lon1, lat2, lon2, eps=EPS):

1301 '''Check whether two points are I{antipodal}, on diametrically

1302 opposite sides of the earth.

1303

1304 @arg lat1: Latitude of one point (C{degrees}).

1305 @arg lon1: Longitude of one point (C{degrees}).

1306 @arg lat2: Latitude of the other point (C{degrees}).

1307 @arg lon2: Longitude of the other point (C{degrees}).

1308 @kwarg eps: Tolerance for near-equality (C{degrees}).

1309

1310 @return: C{True} if points are antipodal within the

1311 B{C{eps}} tolerance, C{False} otherwise.

1312

1313 @see: Functions L{isantipode_} and L{antipode}.

1314 '''

1315 return (fabs(lat1 + lat2) <= eps and

1316 fabs(lon1 + lon2) <= eps) or _isequalTo(

1317 normal(lat1, lon1), antipode(lat2, lon2), eps)

1318

1319

1320 def isantipode_(phi1, lam1, phi2, lam2, eps=EPS):

1321 '''Check whether two points are I{antipodal}, on diametrically

1322 opposite sides of the earth.

1323

1324 @arg phi1: Latitude of one point (C{radians}).

1325 @arg lam1: Longitude of one point (C{radians}).

1326 @arg phi2: Latitude of the other point (C{radians}).

1327 @arg lam2: Longitude of the other point (C{radians}).

1328 @kwarg eps: Tolerance for near-equality (C{radians}).

1329

1330 @return: C{True} if points are antipodal within the

1331 B{C{eps}} tolerance, C{False} otherwise.

1332

1333 @see: Functions L{isantipode} and L{antipode_}.

1334 '''

1335 return (fabs(phi1 + phi2) <= eps and

1336 fabs(lam1 + lam2) <= eps) or _isequalTo_(

1337 normal_(phi1, lam1), antipode_(phi2, lam2), eps)

1338

1339

1340 def _isequalTo(p1, p2, eps=EPS):

1341 '''Compare 2 point lat-/lons ignoring C{class}.

1342 '''

1343 return (fabs(p1.lat - p2.lat) <= eps and

1344 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon)

1345

1346

1347 def _isequalTo_(p1, p2, eps=EPS):

1348 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}.

1349 '''

1350 return (fabs(p1.phi - p2.phi) <= eps and

1351 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam)

1352

1353

1354 def isnormal(lat, lon, eps=0):

1355 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their

1356 respective I{normal} range in C{degrees}.

1357

1358 @arg lat: Latitude (C{degrees}).

1359 @arg lon: Longitude (C{degrees}).

1360 @kwarg eps: Optional tolerance C{degrees}).

1361

1362 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and

1363 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise.

1364

1365 @see: Functions L{isnormal_} and L{normal}.

1366 '''

1367 return (_90_0 - fabs(lat)) >= eps and (_180_0 - fabs(lon)) >= eps

1368

1369

1370 def isnormal_(phi, lam, eps=0):

1371 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their

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

1373

1374 @arg phi: Latitude (C{radians}).

1375 @arg lam: Longitude (C{radians}).

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

1377

1378 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and

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

1380

1381 @see: Functions L{isnormal} and L{normal_}.

1382 '''

1383 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps

1384

1385

1386 def latlon2n_xyz(lat, lon, name=NN):

1387 '''Convert lat-, longitude to C{n-vector} (I{normal} to the

1388 earth's surface) X, Y and Z components.

1389

1390 @arg lat: Latitude (C{degrees}).

1391 @arg lon: Longitude (C{degrees}).

1392 @kwarg name: Optional name (C{str}).

1393

1394 @return: A L{Vector3Tuple}C{(x, y, z)}.

1395

1396 @see: Function L{philam2n_xyz}.

1397

1398 @note: These are C{n-vector} x, y and z components,

1399 I{NOT} geocentric ECEF x, y and z coordinates!

1400 '''

1401 return _2n_xyz(name, *sincos2d_(lat, lon))

1402

1403

1404 def _normal2(a, b, n_2, n, n2):

1405 '''(INTERNAL) Helper for C{normal} and C{normal_}.

1406 '''

1407 if fabs(b) > n:

1408 b = remainder(b, n2)

1409 if fabs(a) > n_2:

1410 r = remainder(a, n)

1411 if r != a:

1412 a = -r

1413 b -= n if b > 0 else -n

1414 return float0_(a, b)

1415

1416

1417 def normal(lat, lon, name=NN):

1418 '''Normalize a lat- I{and} longitude pair in C{degrees}.

1419

1420 @arg lat: Latitude (C{degrees}).

1421 @arg lon: Longitude (C{degrees}).

1422 @kwarg name: Optional name (C{str}).

1423

1424 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90}

1425 and C{abs(lon) <= 180}.

1426

1427 @see: Functions L{normal_} and L{isnormal}.

1428 '''

1429 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0),

1430 name=name or normal.__name__)

1431

1432

1433 def normal_(phi, lam, name=NN):

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

1435

1436 @arg phi: Latitude (C{radians}).

1437 @arg lam: Longitude (C{radians}).

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

1439

1440 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2}

1441 and C{abs(lam) <= PI}.

1442

1443 @see: Functions L{normal} and L{isnormal_}.

1444 '''

1445 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2),

1446 name=name or normal_.__name__)

1447

1448

1449 def _2n_xyz(name, sa, ca, sb, cb):

1450 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}.

1451 '''

1452 # Kenneth Gade eqn 3, but using right-handed

1453 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N

1454 return Vector3Tuple(ca * cb, ca * sb, sa, name=name)

1455

1456

1457 def n_xyz2latlon(x, y, z, name=NN):

1458 '''Convert C{n-vector} components to lat- and longitude in C{degrees}.

1459

1460 @arg x: X component (C{scalar}).

1461 @arg y: Y component (C{scalar}).

1462 @arg z: Z component (C{scalar}).

1463 @kwarg name: Optional name (C{str}).

1464

1465 @return: A L{LatLon2Tuple}C{(lat, lon)}.

1466

1467 @see: Function L{n_xyz2philam}.

1468 '''

1469 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name)

1470

1471

1472 def n_xyz2philam(x, y, z, name=NN):

1473 '''Convert C{n-vector} components to lat- and longitude in C{radians}.

1474

1475 @arg x: X component (C{scalar}).

1476 @arg y: Y component (C{scalar}).

1477 @arg z: Z component (C{scalar}).

1478 @kwarg name: Optional name (C{str}).

1479

1480 @return: A L{PhiLam2Tuple}C{(phi, lam)}.

1481

1482 @see: Function L{n_xyz2latlon}.

1483 '''

1484 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name)

1485

1486

1487 def _opposes(d, m, n, n2):

1488 '''(INETNAL) Helper for C{opposing} and C{opposing_}.

1489 '''

1490 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1

1491 return False if d < m or d > (n2 - m) else (

1492 True if (n - m) < d < (n + m) else None)

1493

1494

1495 def opposing(bearing1, bearing2, margin=_90_0):

1496 '''Compare the direction of two bearings given in C{degrees}.

1497

1498 @arg bearing1: First bearing (compass C{degrees}).

1499 @arg bearing2: Second bearing (compass C{degrees}).

1500 @kwarg margin: Optional, interior angle bracket (C{degrees}).

1501

1502 @return: C{True} if both bearings point in opposite, C{False} if

1503 in similar or C{None} if in perpendicular directions.

1504

1505 @see: Function L{opposing_}.

1506 '''

1507 m = Degrees_(margin=margin, low=EPS0, high=_90_0)

1508 return _opposes(bearing2 - bearing1, m,_180_0, _360_0)

1509

1510

1511 def opposing_(radians1, radians2, margin=PI_2):

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

1513

1514 @arg radians1: First bearing (C{radians}).

1515 @arg radians2: Second bearing (C{radians}).

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

1517

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

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

1520

1521 @see: Function L{opposing}.

1522 '''

1523 m = Radians_(margin=margin, low=EPS0, high=PI_2)

1524 return _opposes(radians2 - radians1, m, PI, PI2)

1525

1526

1527 def philam2n_xyz(phi, lam, name=NN):

1528 '''Convert lat-, longitude to C{n-vector} (I{normal} to the

1529 earth's surface) X, Y and Z components.

1530

1531 @arg phi: Latitude (C{radians}).

1532 @arg lam: Longitude (C{radians}).

1533 @kwarg name: Optional name (C{str}).

1534

1535 @return: A L{Vector3Tuple}C{(x, y, z)}.

1536

1537 @see: Function L{latlon2n_xyz}.

1538

1539 @note: These are C{n-vector} x, y and z components,

1540 I{NOT} geocentric ECEF x, y and z coordinates!

1541 '''

1542 return _2n_xyz(name, *sincos2_(phi, lam))

1543

1544

1545 def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d

1546 # (INTERNAL) See C{radical2} below

1547 # assert d > EPS0

1548 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5

1549 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d)

1550

1551

1552 def radical2(distance, radius1, radius2):

1553 '''Compute the I{radical ratio} and I{radical line} of two

1554 U{intersecting circles<https://MathWorld.Wolfram.com/

1555 Circle-CircleIntersection.html>}.

1556

1557 The I{radical line} is perpendicular to the axis thru the

1558 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}.

1559

1560 @arg distance: Distance between the circle centers (C{scalar}).

1561 @arg radius1: Radius of the first circle (C{scalar}).

1562 @arg radius2: Radius of the second circle (C{scalar}).

1563

1564 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <=

1565 ratio <= 1.0} and C{xline} is along the B{C{distance}}.

1566

1567 @raise IntersectionError: The B{C{distance}} exceeds the sum

1568 of B{C{radius1}} and B{C{radius2}}.

1569

1570 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or

1571 B{C{radius2}}.

1572

1573 @see: U{Circle-Circle Intersection

1574 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}.

1575 '''

1576 d = Distance_(distance, low=_0_0)

1577 r1 = Radius_(radius1=radius1)

1578 r2 = Radius_(radius2=radius2)

1579 if d > (r1 + r2):

1580 raise IntersectionError(distance=d, radius1=r1, radius2=r2,

1581 txt=_too_(_distant_))

1582 return _radical2(d, r1, r2) if d > EPS0 else \

1583 Radical2Tuple(_0_5, _0_0)

1584

1585

1586 class Radical2Tuple(_NamedTuple):

1587 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and

1588 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0}

1589 '''

1590 _Names_ = (_ratio_, _xline_)

1591 _Units_ = ( Scalar, Scalar)

1592

1593

1594 def _radistance(inst):

1595 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians}

1596 and L{hausdorff._HausdorffMeterRedians} classes.

1597 '''

1598 kwds_ = _xkwds(inst._kwds, wrap=False)

1599 wrap_ = _xkwds_pop(kwds_, wrap=False)

1600 func_ = inst._func_

1601 try: # calling lower-overhead C{func_}

1602 func_(0, _0_25, _0_5, **kwds_)

1603 wrap_ = _Wrap._philamop(wrap_)

1604 except TypeError:

1605 return inst.distance

1606

1607 _rad = radians

1608

1609 def _philam(p):

1610 return _rad(p.lat), _rad(p.lon)

1611

1612 def _radistance(point1, point2):

1613 try:

1614 phi1, lam1 = wrap_(point1.phi, point1.lam)

1615 phi2, lam2 = wrap_(point2.phi, point2.lam)

1616 except AttributeError: # no 'phi or .lam

1617 phi1, lam1 = wrap_(*_philam(point1))

1618 phi2, lam2 = wrap_(*_philam(point2))

1619 return func_(phi2, phi1, lam2 - lam1, **kwds_)

1620

1621 inst._units = inst._units_

1622 return _radistance

1623

1624

1625 def _scale_deg(lat1, lat2): # degrees

1626 # scale factor cos(mean of lats) for delta lon

1627 m = fabs(lat1 + lat2) * _0_5

1628 return cos(radians(m)) if m < 90 else _0_0

1629

1630

1631 def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights

1632 # scale factor cos(mean of phis) for delta lam

1633 m = fabs(phi1 + phi2) * _0_5

1634 return cos(m) if m < PI_2 else _0_0

1635

1636

1637 def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw

1638 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine.

1639 '''

1640 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21)

1641 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21

1642

1643

1644 def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):

1645 '''Compute the distance between two (ellipsoidal) points using

1646 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}

1647 formula.

1648

1649 @arg lat1: Start latitude (C{degrees}).

1650 @arg lon1: Start longitude (C{degrees}).

1651 @arg lat2: End latitude (C{degrees}).

1652 @arg lon2: End longitude (C{degrees}).

1653 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},

1654 L{Ellipsoid2} or L{a_f2Tuple}) to use.

1655 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll

1656 B{C{lat2}} and B{C{lon2}} (C{bool}).

1657

1658 @return: Distance (C{meter}, same units as the B{C{datum}}'s

1659 ellipsoid axes).

1660

1661 @raise TypeError: Invalid B{C{datum}}.

1662

1663 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},

1664 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},

1665 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}.

1666 '''

1667 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2)

1668

1669

1670 def thomas_(phi2, phi1, lam21, datum=_WGS84):

1671 '''Compute the I{angular} distance between two (ellipsoidal) points using

1672 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}

1673 formula.

1674

1675 @arg phi2: End latitude (C{radians}).

1676 @arg phi1: Start latitude (C{radians}).

1677 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).

1678 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},

1679 L{Ellipsoid2} or L{a_f2Tuple}).

1680

1681 @return: Angular distance (C{radians}).

1682

1683 @raise TypeError: Invalid B{C{datum}}.

1684

1685 @see: Functions L{thomas}, L{cosineAndoyerLambert_},

1686 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

1687 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},

1688 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP

1689 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/

1690 Distance/ThomasFormula.php>}.

1691 '''

1692 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)

1693 if r and isnon0(c1) and isnon0(c2):

1694 E = _ellipsoidal(datum, thomas_)

1695 if E.f:

1696 r1 = atan2(E.b_a * s1, c1)

1697 r2 = atan2(E.b_a * s2, c2)

1698

1699 j = (r2 + r1) * _0_5

1700 k = (r2 - r1) * _0_5

1701 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5)

1702

1703 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2)

1704 u = _1_0 - h

1705 if isnon0(u) and isnon0(h):

1706 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h)

1707 sr, cr = sincos2(r)

1708 if isnon0(sr):

1709 u = 2 * (sj * ck)**2 / u

1710 h = 2 * (sk * cj)**2 / h

1711 x = u + h

1712 y = u - h

1713

1714 s = r / sr

1715 e = 4 * s**2

1716 d = 2 * cr

1717 a = e * d

1718 b = 2 * r

1719 c = s - (a - d) * _0_5

1720 f = E.f * _0_25

1721

1722 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y)

1723 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr

1724 return r

1725

1726

1727 def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):

1728 '''Compute the distance between two (spherical) points using

1729 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}

1730 spherical formula.

1731

1732 @arg lat1: Start latitude (C{degrees}).

1733 @arg lon1: Start longitude (C{degrees}).

1734 @arg lat2: End latitude (C{degrees}).

1735 @arg lon2: End longitude (C{degrees}).

1736 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})

1737 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or

1738 L{a_f2Tuple}) to use.

1739 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll

1740 B{C{lat2}} and B{C{lon2}} (C{bool}).

1741

1742 @return: Distance (C{meter}, same units as B{C{radius}}).

1743

1744 @raise UnitError: Invalid B{C{radius}}.

1745

1746 @see: Functions L{vincentys_}, L{cosineAndoyerLambert},

1747 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular},

1748 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},

1749 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2},

1750 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.

1751

1752 @note: See note at function L{vincentys_}.

1753 '''

1754 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2)

1755

1756

1757 def vincentys_(phi2, phi1, lam21):

1758 '''Compute the I{angular} distance between two (spherical) points using

1759 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}

1760 spherical formula.

1761

1762 @arg phi2: End latitude (C{radians}).

1763 @arg phi1: Start latitude (C{radians}).

1764 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).

1765

1766 @return: Angular distance (C{radians}).

1767

1768 @see: Functions L{vincentys}, L{cosineAndoyerLambert_},

1769 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

1770 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},

1771 L{flatPolar_}, L{haversine_} and L{thomas_}.

1772

1773 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_}

1774 produce equivalent results, but L{vincentys_} is suitable

1775 for antipodal points and slightly more expensive (M{3 cos,

1776 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_}

1777 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and

1778 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}).

1779 '''

1780 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21)

1781

1782 c = c2 * c21

1783 x = s1 * s2 + c1 * c

1784 y = c1 * s2 - s1 * c

1785 return atan2(hypot(c2 * s21, y), x)

1786

1787# **) MIT License

1788#

1789# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.

1790#

1791# Permission is hereby granted, free of charge, to any person obtaining a

1792# copy of this software and associated documentation files (the "Software"),

1793# to deal in the Software without restriction, including without limitation

1794# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1795# and/or sell copies of the Software, and to permit persons to whom the

1796# Software is furnished to do so, subject to the following conditions:

1797#

1798# The above copyright notice and this permission notice shall be included

1799# in all copies or substantial portions of the Software.

1800#

1801# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1802# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1803# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1804# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1805# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1806# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1807# OTHER DEALINGS IN THE SOFTWARE.