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 _args_kwds_count2, _copysign # from .constants

10# from pygeodesy.cartesianBase import CartesianBase # _MODS

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

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

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

14 _32_0, _90_0, _180_0, _360_0, _copysign

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

16 _mean_radius, _spherical_datum, _WGS84, _EWGS84

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

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

19 _TypeError, _ValueError, _xattr, _xError, \

20 _xcallable,_xkwds, _xkwds_pop2

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

22 from pygeodesy.fsums import fsumf_, Fmt, unstr

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

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

25 from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS

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

27 _DUNDER_nameof

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

29 Intersection3Tuple, PhiLam2Tuple

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

31# from pygeodesy.triaxials import _hartzell3 # _MODS

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

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

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

35 from pygeodesy.utily import acos1, atan2, atan2b, degrees2m, _loneg, m2degrees, \

36 tan_2, sincos2, sincos2_, _Wrap

37# from pygeodesy.vector3d import _otherV3d # _MODS

38# from pygeodesy.vector3dBase import _xyz_y_z3 # _MODS

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

40# sphericalNvector, sphericalTrigonometry # _MODS

41

42 from contextlib import contextmanager

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

44

45__all__ = _ALL_LAZY.formy

46__version__ = '24.12.06'

47

48_RADIANS2 = (PI / _180_0)**2 # degrees- to radians-squared

49_ratio_ = 'ratio'

50_xline_ = 'xline'

51

52

53 def angle2chord(rad, radius=R_M):

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

55

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

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

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

59

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

61

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

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

64 '''

65 d = _isDegrees(rad, iscalar=False)

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

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

68

69

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

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

72 '''

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

74 if r == a:

75 r = -r

76 b += n

77 if fabs(b) > n:

78 b = remainder(b, n2)

79 return float0_(r, b)

80

81

82 def antipode(lat, lon, **name):

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

84 point in C{degrees}.

85

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

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

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

89

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

91

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

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

94 '''

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

96

97

98 def antipode_(phi, lam, **name):

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

100 point in C{radians}.

101

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

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

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

105

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

107

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

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

110 '''

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

112

113

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

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

116 between two (spherical) points.

117

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

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

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

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

122 @kwarg final_wrap: Optional keyword arguments for function

123 L{pygeodesy.bearing_}.

124

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

126 both points coincide.

127 '''

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

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

130 return degrees(r)

131

132

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

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

135 two (spherical) points.

136

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

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

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

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

141 @kwarg final: If C{True}, return the final, otherwise the initial bearing

142 (C{bool}).

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

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

145

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

147 are coincident.

148

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

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

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

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

153 '''

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

155 if final: # swap plus PI

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

157 r = PI3

158 else:

159 r = PI2

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

161

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

163 y = sdb * ca2

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

165

166

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

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

169 '''

170 try: # for LatLon_ and ellipsoidal LatLon

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

172 except AttributeError:

173 pass

174 # XXX spherical version, OK for ellipsoidal ispolar?

175 t = p1.philam + p2.philam

176 return Bearing2Tuple(degrees(bearing_(*t, final=False, wrap=wrap)),

177 degrees(bearing_(*t, final=True, wrap=wrap)),

178 name__=_bearingTo2)

179

180

181 def chord2angle(chord, radius=R_M):

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

183

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

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

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

187

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

189

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

191

192 @see: Function L{angle2chord}.

193 '''

194 m = _mean_radius(radius)

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

196 if r:

197 i = int(r)

198 if i > 0:

199 r -= i

200 i *= PI

201 r = asin(r) + i

202 return _copysign(r * _2_0, chord)

203

204

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

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

207 lat2 - lat1)} between two points.

208

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

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

211

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

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

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

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

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

217 mean latitude (C{bool}).

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

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

220

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

222

223 @note: Courtesy of Martin Schultz.

224

225 @see: U{Local, flat earth approximation

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

227 '''

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

229 if adjust: # scale delta lon

230 d_lon *= _scale_deg(lat1, lat2)

231 return atan2b(d_lon, lat2 - lat1)

232

233

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

235 '''Compute the distance between two (ellipsoidal) points using the U{Andoyer-Lambert

236 <https://books.google.com/books?id=x2UiAQAAIAAJ>} correction of the U{Law of

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

238

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

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

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

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

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

244 L{a_f2Tuple}) to use.

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

246 B{C{lon2}} (C{bool}).

247

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

249 C{radians} if C{B{datum} is None}).

250

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

252

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

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

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

256 L{Ellipsoid.distance2}.

257 '''

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

259

260

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

262 '''Compute the I{angular} distance between two (ellipsoidal) points using the U{Andoyer-Lambert

263 <https://books.google.com/books?id=x2UiAQAAIAAJ>} correction of the U{Law of

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

265

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

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

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

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

270 L{a_f2Tuple}) to use.

271

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

273

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

275

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

277 L{cosineLaw_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},

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

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

280 AndoyerLambert.php>}.

281 '''

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

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

284 E = _ellipsoidal(datum, cosineAndoyerLambert_)

285 if E.f: # ellipsoidal

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

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

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

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

290 if r:

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

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

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

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

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

296 return r

297

298

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

300 '''Compute the distance between two (ellipsoidal) points using the U{Forsythe-Andoyer-Lambert

301 <https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} correction of the U{Law of Cosines

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

303

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

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

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

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

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

309 L{a_f2Tuple}) to use.

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

311 B{C{lon2}} (C{bool}).

312

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

314 C{radians} if C{B{datum} is None}).

315

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

317

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

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

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

321 L{Ellipsoid.distance2}.

322 '''

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

324

325

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

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

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

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

330 formula.

331

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

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

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

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

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

337

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

339

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

341

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

343 L{cosineLaw_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},

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

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

346 Distance/ForsytheCorrection.php>}.

347 '''

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

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

350 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_)

351 if E.f: # ellipsoidal

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

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

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

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

356 x = s + t

357 y = s - t

358

359 s = 8 * r**2 / sr

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

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

362 b = -2 * d

363 e = 30 * s2r

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

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

366

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

368 return r

369

370

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

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

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

374

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

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

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

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

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

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

381 L{a_f2Tuple}) to use.

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

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

384

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

386 ellipsoid or datum axes).

387

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

389

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

391 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean},

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

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

394

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

396 '''

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

398

399

400 def cosineLaw_(phi2, phi1, lam21):

401 '''Compute the I{angular} distance between two points using the U{spherical Law of

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

403

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

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

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

407

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

409

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

411 L{cosineForsytheAndoyerLambert_}, L{euclidean_},

412 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_},

413 L{thomas_} and L{vincentys_}.

414

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

416 '''

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

418

419

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

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

422 '''

423 if wrap:

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

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

426 else: # for backward compaibility

427 return Phid(lat2=lat2), Phid(lat1=lat1), Phid(d_lon=lon2 - lon1)

428

429

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

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

432 '''

433 E = _ellipsoidal(earth, func_)

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

435 return r * E.a

436

437

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

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

440 '''

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

442 if radius is not R_M:

443 _, lat1, _, lat2, _ = wrap_lls

444 radius = _mean_radius(radius, lat1, lat2)

445 return r * radius

446

447

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

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

450 '''

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

452 if radius:

453 _, lat1, _, lat2, _ = wrap_lls

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

455 return r

456

457

458 def _ellipsoidal(earth, where):

459 '''(INTERNAL) Helper for distances.

460 '''

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

462 earth if isinstance(earth, Ellipsoid) else

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

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

465

466

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

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

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

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

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

477 @kwarg adjust_limit_wrap: Optional keyword arguments for function L{equirectangular4}.

478

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

480

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

482

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

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

485 '''

486 d = sqrt(equirectangular4(Lat(lat1=lat1), Lon(lon1=lon1),

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

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

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

490

491

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

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

494 L{hausdorff._HausdorffMeterRedians}.

495 '''

496 return equirectangular4(lat1, lon1, lat2, lon2, **adjust_limit_wrap).distance2 * _RADIANS2

497

498

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

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

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

502

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

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

505

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

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

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

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

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

511 latitude (C{bool}).

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

513 or C{0} for unlimited.

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

515 (C{bool}).

516

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

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

519

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

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

522

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

524 functions L{equirectangular}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},

525 L{cosineLaw}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine},

526 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*}

527 and C{LatLon.equirectangularTo}.

528 '''

529 if wrap:

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

531 else:

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

533 d_lat = lat2 - lat1

534

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

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

537 if d > limit:

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

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

540 limit=limit, wrap=wrap)

541 raise LimitError(s, txt=t)

542

543 if adjust: # scale delta lon

544 d_lon *= _scale_deg(lat1, lat2)

545

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

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

548

549

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

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

552

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

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

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

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

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

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

559 L{a_f2Tuple}) to use.

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

561 the mean latitude (C{bool}).

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

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

564

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

566 ellipsoid or datum axes).

567

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

569

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

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

572 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},

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

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

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

576 '''

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

578

579

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

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

582

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

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

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

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

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

588

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

590

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

592 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

593 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_},

594 L{thomas_} and L{vincentys_}.

595 '''

596 if adjust:

597 lam21 *= _scale_rad(phi2, phi1)

598 return euclid(phi2 - phi1, lam21)

599

600

601 def excessAbc_(A, b, c):

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

603 and the included (small) angle.

604

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

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

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

608

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

610

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

612

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

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

615 '''

616 A = Radians_(A=A)

617 b = Radians_(b=b) * _0_5

618 c = Radians_(c=c) * _0_5

619

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

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

622

623

624 def excessCagnoli_(a, b, c):

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

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

627

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

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

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

631

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

633

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

635

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

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

638 '''

639 a = Radians_(a=a)

640 b = Radians_(b=b)

641 c = Radians_(c=c)

642

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

644 _s = sin

645 r = _s(s) * _s(s - a) * _s(s - b) * _s(s - c)

646 c = cos(a * _0_5) * cos(b * _0_5) * cos(c * _0_5)

647 r = asin(sqrt(r) * _0_5 / c) if c and r > 0 else _0_0

648 return Radians(Cagnoli=r * _2_0)

649

650

651 def excessGirard_(A, B, C):

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

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

654

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

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

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

658

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

660

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

662

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

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

665 '''

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

667 Radians_(B=B),

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

669

670

671 def excessLHuilier_(a, b, c):

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

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

674

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

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

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

678

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

680

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

682

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

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

685 '''

686 a = Radians_(a=a)

687 b = Radians_(b=b)

688 c = Radians_(c=c)

689

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

691 _t = tan_2

692 r = _t(s) * _t(s - a) * _t(s - b) * _t(s - c)

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

694 return Radians(LHuilier=r * _4_0)

695

696

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

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

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

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

701 method.

702

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

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

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

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

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

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

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

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

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

712

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

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

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

716

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

718

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

720

721 @raise ValueError: Semi-circular longitudinal delta.

722

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

724 '''

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

726

727

728 def excessKarney_(phi2, phi1, lam21):

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

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

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

732 method.

733

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

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

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

737

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

739

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

741

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

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

744 '''

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

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

747 #

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

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

750 # 1 + tan(φ1 / 2) · tan(φ2 / 2)

751 #

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

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

754 _t = tan_2

755 t2 = _t(phi2)

756 t1 = _t(phi1)

757 t = _t(lam21, lam21=None)

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

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

760

761

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

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

764#

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

766#

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

768#

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

770#

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

772#

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

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

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

776#

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

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

779#

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

781#

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

783# inverse Gudermannian) function

784#

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

786#

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

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

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

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

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

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

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

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

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

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

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

798# - Charles Karney

799

800

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

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

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

804

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

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

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

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

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

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

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

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

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

814

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

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

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

818

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

820

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

822

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

824 '''

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

826

827

828 def excessQuad_(phi2, phi1, lam21):

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

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

831

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

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

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

835

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

837

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

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

840 '''

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

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

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

844

845

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

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

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

849 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}

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

851

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

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

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

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

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

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

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

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

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

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

862

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

864 ellipsoid axes).

865

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

867

868 @note: The meridional and prime_vertical radii of curvature

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

870

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

872 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},

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

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

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

876 '''

877 E = _ellipsoidal(datum, flatLocal)

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

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

880

881 hubeny = flatLocal # PYCHOK for Karl Hubeny

882

883

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

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

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

887 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}

888 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} 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 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},

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

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

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

897

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

899

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

901

902 @note: The meridional and prime_vertical radii of curvature

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

904

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

906 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_},

907 L{euclidean_}, L{haversine_}, L{thomas_} and L{vincentys_} and

908 U{local, flat earth approximation

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

910 '''

911 E = _ellipsoidal(datum, flatLocal_)

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

913

914 hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny

915

916

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

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

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

920 Geographical_distance#Polar_coordinate_flat-Earth_formula>}

921 formula.

922

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

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

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

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

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

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

929 L{a_f2Tuple}) to use.

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

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

932

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

934 ellipsoid or datum axes).

935

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

937

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

939 L{cosineForsytheAndoyerLambert},L{cosineLaw},

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

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

942 L{vincentys}.

943 '''

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

945

946

947 def flatPolar_(phi2, phi1, lam21):

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

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

950 Geographical_distance#Polar_coordinate_flat-Earth_formula>}

951 formula.

952

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

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

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

956

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

958

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

960 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

961 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{haversine_},

962 L{thomas_} and L{vincentys_}.

963 '''

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

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

966 if a < b:

967 a, b = b, a

968 if a < EPS0:

969 a = _0_0

970 elif b > 0:

971 b = b / a # /= chokes PyChecker

972 c = b * cos(lam21) * _2_0

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

974 a *= sqrt0(c)

975 return a

976

977

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

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

980 '''

981 if earth is None:

982 earth = pov.datum

983 else:

984 earth = _spherical_datum(earth, name__=hartzell)

985 pov = pov.toDatum(earth)

986 h = pov.height

987 if h < 0: # EPS0

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

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

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

991

992

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

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

995 a Point-Of-View in space.

996

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

998 L{Ecef9Tuple} or L{Vector3d}).

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

1000 C{True} for the I{normal, plumb} onto the surface or C{False}

1001 or C{None} to point to the center of the earth.

1002 @kwarg earth: The earth model (L{Datum}, L{Ellipsoid}, L{Ellipsoid2},

1003 L{a_f2Tuple} or a C{scalar} earth radius in C{meter}).

1004 @kwarg name_LatLon_and_kwds: Optional, overriding C{B{name}="hartzell"}

1005 (C{str}), class C{B{LatLon}=None} to return the intersection

1006 plus additional C{LatLon} keyword arguments, include the

1007 B{C{datum}} if different and to convert from B{C{earth}}.

1008

1009 @return: The intersection (L{Vector3d}, B{C{pov}}'s C{cartesian type} or the

1010 given B{C{LatLon}} instance) with attribute C{height} set to the

1011 distance to the B{C{pov}}.

1012

1013 @raise IntersectionError: Invalid B{C{pov}} or B{C{pov}} inside the earth or

1014 invalid B{C{los}} or B{C{los}} points outside or

1015 away from the earth.

1016

1017 @raise TypeError: Invalid B{C{earth}}, C{ellipsoid} or C{datum}.

1018

1019 @see: Class L{Los}, functions L{tyr3d} and L{hartzell4} and methods

1020 L{Ellipsoid.hartzell4} and any C{Cartesian.hartzell} and C{LatLon.hartzell}.

1021 '''

1022 n, LatLon_and_kwds = _name2__(name_LatLon_and_kwds, name__=hartzell)

1023 try:

1024 D = _spherical_datum(earth, name__=hartzell)

1025 r, h, i = _MODS.triaxials._hartzell3(pov, los, D.ellipsoid._triaxial)

1026

1027 C = _MODS.cartesianBase.CartesianBase

1028 if LatLon_and_kwds:

1029 c = C(r, datum=D)

1030 r = c.toLatLon(**_xkwds(LatLon_and_kwds, height=h))

1031 elif isinstance(r, C):

1032 r.height = h

1033 if i:

1034 r._iteration = i

1035 except Exception as x:

1036 raise IntersectionError(pov=pov, los=los, earth=earth, cause=x,

1037 **LatLon_and_kwds)

1038 return _xnamed(r, n) if n else r

1039

1040

1041 def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):

1042 '''Compute the distance between two (spherical) points using the

1043 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} formula.

1044

1045 @arg lat1: Start latitude (C{degrees}).

1046 @arg lon1: Start longitude (C{degrees}).

1047 @arg lat2: End latitude (C{degrees}).

1048 @arg lon2: End longitude (C{degrees}).

1049 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) or ellipsoid

1050 (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) to use.

1051 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and

1052 B{C{lon2}} (C{bool}).

1053

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

1055

1056 @raise TypeError: Invalid B{C{radius}}.

1057

1058 @see: U{Distance between two (spherical) points

1059 <https://www.EdWilliams.org/avform.htm#Dist>}, functions

1060 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},

1061 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},

1062 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},

1063 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.

1064

1065 @note: See note at function L{vincentys_}.

1066 '''

1067 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2)

1068

1069

1070 def haversine_(phi2, phi1, lam21):

1071 '''Compute the I{angular} distance between two (spherical) points using the

1072 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} formula.

1073

1074 @arg phi2: End latitude (C{radians}).

1075 @arg phi1: Start latitude (C{radians}).

1076 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).

1077

1078 @return: Angular distance (C{radians}).

1079

1080 @see: Functions L{haversine}, L{cosineAndoyerLambert_},

1081 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

1082 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},

1083 L{thomas_} and L{vincentys_}.

1084

1085 @note: See note at function L{vincentys_}.

1086 '''

1087 def _hsin(rad):

1088 return sin(rad * _0_5)**2

1089

1090 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine

1091 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2

1092

1093

1094 def heightOf(angle, distance, radius=R_M):

1095 '''Determine the height above the (spherical) earth' surface after

1096 traveling along a straight line at a given tilt.

1097

1098 @arg angle: Tilt angle above horizontal (C{degrees}).

1099 @arg distance: Distance along the line (C{meter} or same units as

1100 B{C{radius}}).

1101 @kwarg radius: Optional mean earth radius (C{meter}).

1102

1103 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}).

1104

1105 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}.

1106

1107 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>}

1108 (U{Shapiro et al. 2009, JTECH

1109 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>}

1110 and U{Potvin et al. 2012, JTECH

1111 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}).

1112 '''

1113 r = h = Radius(radius)

1114 d = fabs(Distance(distance))

1115 if d > h:

1116 d, h = h, d

1117

1118 if d > EPS0: # and h > EPS0

1119 d = d / h # /= h chokes PyChecker

1120 s = sin(Phid(angle=angle, clip=_180_0))

1121 s = fsumf_(_1_0, s * d * _2_0, d**2)

1122 if s > 0:

1123 return h * sqrt(s) - r

1124

1125 raise _ValueError(angle=angle, distance=distance, radius=radius)

1126

1127

1128 def heightOrthometric(h_ll, N):

1129 '''Get the I{orthometric} height B{H}, the height above the geoid, earth surface.

1130

1131 @arg h_ll: The height above the ellipsoid (C{meter}) or an I{ellipsoidal}

1132 location (C{LatLon} with a C{height} or C{h} attribute).

1133 @arg N: The I{geoid} height (C{meter}), the height of the geoid above the

1134 ellipsoid at the same B{C{h_ll}} location.

1135

1136 @return: I{Orthometric} height C{B{H} = B{h} - B{N}} (C{meter}, same units

1137 as B{C{h}} and B{C{N}}).

1138

1139 @see: U{Ellipsoid, Geoid, and Othometric Heights<https://www.NGS.NOAA.gov/

1140 GEOID/PRESENTATIONS/2007_02_24_CCPS/Roman_A_PLSC2007notes.pdf>}, page

1141 6 and module L{pygeodesy.geoids}.

1142 '''

1143 h = h_ll if _isHeight(h_ll) else _xattr(h_ll, height=_xattr(h_ll, h=0))

1144 return Height(H=Height(h=h) - Height(N=N))

1145

1146

1147 def horizon(height, radius=R_M, refraction=False):

1148 '''Determine the distance to the horizon from a given altitude above the

1149 (spherical) earth.

1150

1151 @arg height: Altitude (C{meter} or same units as B{C{radius}}).

1152 @kwarg radius: Optional mean earth radius (C{meter}).

1153 @kwarg refraction: Consider atmospheric refraction (C{bool}).

1154

1155 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}).

1156

1157 @raise ValueError: Invalid B{C{height}} or B{C{radius}}.

1158

1159 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}.

1160 '''

1161 h, r = Height(height), Radius(radius)

1162 if min(h, r) < 0:

1163 raise _ValueError(height=height, radius=radius)

1164

1165 d2 = ((r * 2.415750694528) if refraction else # 2.0 / 0.8279

1166 fsumf_(r, r, h)) * h

1167 return sqrt0(d2)

1168

1169

1170 class _idllmn6(object): # see also .geodesicw._wargs, .latlonBase._toCartesian3, .vector2d._numpy

1171 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}.

1172 '''

1173 @contextmanager # <https://www.Python.org/dev/peps/pep-0343/> Examples

1174 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds):

1175 try:

1176 if wrap:

1177 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)

1178 kwds = _xkwds(kwds, wrap=wrap) # for _xError

1179 m = small if small is _100km else Meter_(small=small)

1180 n = _DUNDER_nameof(intersections2 if s else intersection2)

1181 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m:

1182 d, m = None, _MODS.vector3d

1183 _i = m._intersects2 if s else m._intersect3d3

1184 elif _isRadius(datum) and datum < 0 and not s:

1185 d = _spherical_datum(-datum, name=n)

1186 m = _MODS.sphericalNvector

1187 _i = m.intersection

1188 else:

1189 d = _spherical_datum(datum, name=n)

1190 if d.isSpherical:

1191 m = _MODS.sphericalTrigonometry

1192 _i = m._intersects2 if s else m._intersect

1193 elif d.isEllipsoidal:

1194 try:

1195 if d.ellipsoid.geodesic:

1196 pass

1197 m = _MODS.ellipsoidalKarney

1198 except ImportError:

1199 m = _MODS.ellipsoidalExact

1200 _i = m._intersections2 if s else m._intersection3 # ellipsoidalBaseDI

1201 else:

1202 raise _TypeError(datum=datum)

1203 yield _i, d, lat2, lon2, m, n

1204

1205 except (TypeError, ValueError) as x:

1206 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum,

1207 lat2=lat2, lon2=lon2, small=small, **kwds)

1208

1209_idllmn6 = _idllmn6() # PYCHOK singleton

1210

1211

1212 def intersection2(lat1, lon1, bearing1,

1213 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True

1214 '''I{Conveniently} compute the intersection of two lines each defined

1215 by a (geodetic) point and a bearing from North, using either ...

1216

1217 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km

1218 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...

1219

1220 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}}

1221 or a C{scalar B{datum}} representing the earth radius, conventionally

1222 in C{meter} or ...

1223

1224 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative}

1225 C{scalar}, (negative) earth radius, conventionally in C{meter} or ...

1226

1227 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}}

1228 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}

1229 is installed, otherwise ...

1230

1231 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal.

1232

1233 @arg lat1: Latitude of the first point (C{degrees}).

1234 @arg lon1: Longitude of the first point (C{degrees}).

1235 @arg bearing1: Bearing at the first point (compass C{degrees}).

1236 @arg lat2: Latitude of the second point (C{degrees}).

1237 @arg lon2: Longitude of the second point (C{degrees}).

1238 @arg bearing2: Bearing at the second point (compass C{degrees}).

1239 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},

1240 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth

1241 radius (C{meter}, same units as B{C{radius1}} and

1242 B{C{radius2}}) or C{None}.

1243 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}

1244 and B{C{lon2}} (C{bool}).

1245 @kwarg small: Upper limit for small distances (C{meter}).

1246

1247 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and

1248 longitude of the intersection point.

1249

1250 @raise IntersectionError: Ambiguous or infinite intersection

1251 or colinear, parallel or otherwise

1252 non-intersecting lines.

1253

1254 @raise TypeError: Invalid B{C{datum}}.

1255

1256 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}},

1257 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}.

1258

1259 @see: Method L{RhumbLine.intersection2}.

1260

1261 @note: The returned intersections may be near-antipodal.

1262 '''

1263 b1 = Bearing(bearing1=bearing1)

1264 b2 = Bearing(bearing2=bearing2)

1265 with _idllmn6(datum, lat1, lon1, lat2, lon2,

1266 small, wrap, False, bearing1=b1, bearing2=b2) as t:

1267 _i, d, lat2, lon2, m, n = t

1268 if d is None:

1269 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1,

1270 m.Vector3d(lon2, lat2, 0), b2, useZ=False)

1271 t = LatLon2Tuple(t.y, t.x, name=n)

1272

1273 else:

1274 t = _i(m.LatLon(lat1, lon1, datum=d), b1,

1275 m.LatLon(lat2, lon2, datum=d), b2,

1276 LatLon=None, height=0, wrap=False)

1277 if isinstance(t, Intersection3Tuple): # ellipsoidal

1278 t, _, _ = t

1279 t = LatLon2Tuple(t.lat, t.lon, name=n)

1280 return t

1281

1282

1283 def intersections2(lat1, lon1, radius1,

1284 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True

1285 '''I{Conveniently} compute the intersections of two circles each defined

1286 by a (geodetic) center point and a radius, using either ...

1287

1288 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km

1289 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...

1290

1291 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}}

1292 or a C{scalar B{datum}} representing the earth radius, conventionally

1293 in C{meter} or ...

1294

1295 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}}

1296 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}

1297 is installed, otherwise ...

1298

1299 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal.

1300

1301 @arg lat1: Latitude of the first circle center (C{degrees}).

1302 @arg lon1: Longitude of the first circle center (C{degrees}).

1303 @arg radius1: Radius of the first circle (C{meter}, conventionally).

1304 @arg lat2: Latitude of the second circle center (C{degrees}).

1305 @arg lon2: Longitude of the second circle center (C{degrees}).

1306 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}).

1307 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},

1308 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth

1309 radius (C{meter}, same units as B{C{radius1}} and

1310 B{C{radius2}}) or C{None}.

1311 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}

1312 and B{C{lon2}} (C{bool}).

1313 @kwarg small: Upper limit for small distances (C{meter}).

1314

1315 @return: 2-Tuple of the intersection points, each a

1316 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the

1317 points are the same instance, aka the I{radical center}.

1318

1319 @raise IntersectionError: Concentric, antipodal, invalid or

1320 non-intersecting circles or no

1321 convergence.

1322

1323 @raise TypeError: Invalid B{C{datum}}.

1324

1325 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}},

1326 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}.

1327 '''

1328 r1 = Radius_(radius1=radius1)

1329 r2 = Radius_(radius2=radius2)

1330 with _idllmn6(datum, lat1, lon1, lat2, lon2,

1331 small, wrap, True, radius1=r1, radius2=r2) as t:

1332 _i, d, lat2, lon2, m, n = t

1333 if d is None:

1334 r1 = m2degrees(r1, radius=R_M, lat=lat1)

1335 r2 = m2degrees(r2, radius=R_M, lat=lat2)

1336

1337 def _V2T(x, y, _, **unused): # _ == z unused

1338 return LatLon2Tuple(y, x, name=n)

1339

1340 t = _i(m.Vector3d(lon1, lat1, 0), r1,

1341 m.Vector3d(lon2, lat2, 0), r2, sphere=False,

1342 Vector=_V2T)

1343 else:

1344 def _LL2T(lat, lon, **unused):

1345 return LatLon2Tuple(lat, lon, name=n)

1346

1347 t = _i(m.LatLon(lat1, lon1, datum=d), r1,

1348 m.LatLon(lat2, lon2, datum=d), r2,

1349 LatLon=_LL2T, height=0, wrap=False)

1350 return t

1351

1352

1353 def isantipode(lat1, lon1, lat2, lon2, eps=EPS):

1354 '''Check whether two points are I{antipodal}, on diametrically

1355 opposite sides of the earth.

1356

1357 @arg lat1: Latitude of one point (C{degrees}).

1358 @arg lon1: Longitude of one point (C{degrees}).

1359 @arg lat2: Latitude of the other point (C{degrees}).

1360 @arg lon2: Longitude of the other point (C{degrees}).

1361 @kwarg eps: Tolerance for near-equality (C{degrees}).

1362

1363 @return: C{True} if points are antipodal within the

1364 B{C{eps}} tolerance, C{False} otherwise.

1365

1366 @see: Functions L{isantipode_} and L{antipode}.

1367 '''

1368 return (fabs(lat1 + lat2) <= eps and

1369 fabs(lon1 + lon2) <= eps) or _isequalTo(

1370 normal(lat1, lon1), antipode(lat2, lon2), eps)

1371

1372

1373 def isantipode_(phi1, lam1, phi2, lam2, eps=EPS):

1374 '''Check whether two points are I{antipodal}, on diametrically

1375 opposite sides of the earth.

1376

1377 @arg phi1: Latitude of one point (C{radians}).

1378 @arg lam1: Longitude of one point (C{radians}).

1379 @arg phi2: Latitude of the other point (C{radians}).

1380 @arg lam2: Longitude of the other point (C{radians}).

1381 @kwarg eps: Tolerance for near-equality (C{radians}).

1382

1383 @return: C{True} if points are antipodal within the

1384 B{C{eps}} tolerance, C{False} otherwise.

1385

1386 @see: Functions L{isantipode} and L{antipode_}.

1387 '''

1388 return (fabs(phi1 + phi2) <= eps and

1389 fabs(lam1 + lam2) <= eps) or _isequalTo_(

1390 normal_(phi1, lam1), antipode_(phi2, lam2), eps)

1391

1392

1393 def _isequalTo(p1, p2, eps=EPS):

1394 '''Compare 2 point lat-/lons ignoring C{class}.

1395 '''

1396 return (fabs(p1.lat - p2.lat) <= eps and

1397 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon)

1398

1399

1400 def _isequalTo_(p1, p2, eps=EPS):

1401 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}.

1402 '''

1403 return (fabs(p1.phi - p2.phi) <= eps and

1404 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam)

1405

1406

1407 def isnormal(lat, lon, eps=0):

1408 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their

1409 respective I{normal} range in C{degrees}.

1410

1411 @arg lat: Latitude (C{degrees}).

1412 @arg lon: Longitude (C{degrees}).

1413 @kwarg eps: Optional tolerance C{degrees}).

1414

1415 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and

1416 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise.

1417

1418 @see: Functions L{isnormal_} and L{normal}.

1419 '''

1420 return (_90_0 - fabs(lat)) >= eps and _loneg(fabs(lon)) >= eps

1421

1422

1423 def isnormal_(phi, lam, eps=0):

1424 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their

1425 respective I{normal} range in C{radians}.

1426

1427 @arg phi: Latitude (C{radians}).

1428 @arg lam: Longitude (C{radians}).

1429 @kwarg eps: Optional tolerance C{radians}).

1430

1431 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and

1432 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise.

1433

1434 @see: Functions L{isnormal} and L{normal_}.

1435 '''

1436 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps

1437

1438

1439 def _normal2(a, b, n_2, n, n2):

1440 '''(INTERNAL) Helper for C{normal} and C{normal_}.

1441 '''

1442 if fabs(b) > n:

1443 b = remainder(b, n2)

1444 if fabs(a) > n_2:

1445 r = remainder(a, n)

1446 if r != a:

1447 a = -r

1448 b -= n if b > 0 else -n

1449 return float0_(a, b)

1450

1451

1452 def normal(lat, lon, **name):

1453 '''Normalize a lat- I{and} longitude pair in C{degrees}.

1454

1455 @arg lat: Latitude (C{degrees}).

1456 @arg lon: Longitude (C{degrees}).

1457 @kwarg name: Optional, overriding C{B{name}="normal"} (C{str}).

1458

1459 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90}

1460 and C{abs(lon) <= 180}.

1461

1462 @see: Functions L{normal_} and L{isnormal}.

1463 '''

1464 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0),

1465 name=_name__(name, name__=normal))

1466

1467

1468 def normal_(phi, lam, **name):

1469 '''Normalize a lat- I{and} longitude pair in C{radians}.

1470

1471 @arg phi: Latitude (C{radians}).

1472 @arg lam: Longitude (C{radians}).

1473 @kwarg name: Optional, overriding C{B{name}="normal_"} (C{str}).

1474

1475 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2}

1476 and C{abs(lam) <= PI}.

1477

1478 @see: Functions L{normal} and L{isnormal_}.

1479 '''

1480 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2),

1481 name=_name__(name, name__=normal_))

1482

1483

1484 def _opposes(d, m, n, n2):

1485 '''(INTERNAL) Helper for C{opposing} and C{opposing_}.

1486 '''

1487 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1

1488 return False if d < m or d > (n2 - m) else (

1489 True if (n - m) < d < (n + m) else None)

1490

1491

1492 def opposing(bearing1, bearing2, margin=_90_0):

1493 '''Compare the direction of two bearings given in C{degrees}.

1494

1495 @arg bearing1: First bearing (compass C{degrees}).

1496 @arg bearing2: Second bearing (compass C{degrees}).

1497 @kwarg margin: Optional, interior angle bracket (C{degrees}).

1498

1499 @return: C{True} if both bearings point in opposite, C{False} if

1500 in similar or C{None} if in I{perpendicular} directions.

1501

1502 @see: Function L{opposing_}.

1503 '''

1504 m = Degrees_(margin=margin, low=EPS0, high=_90_0)

1505 return _opposes(bearing2 - bearing1, m, _180_0, _360_0)

1506

1507

1508 def opposing_(radians1, radians2, margin=PI_2):

1509 '''Compare the direction of two bearings given in C{radians}.

1510

1511 @arg radians1: First bearing (C{radians}).

1512 @arg radians2: Second bearing (C{radians}).

1513 @kwarg margin: Optional, interior angle bracket (C{radians}).

1514

1515 @return: C{True} if both bearings point in opposite, C{False} if

1516 in similar or C{None} if in perpendicular directions.

1517

1518 @see: Function L{opposing}.

1519 '''

1520 m = Radians_(margin=margin, low=EPS0, high=PI_2)

1521 return _opposes(radians2 - radians1, m, PI, PI2)

1522

1523

1524 def _Propy(func, nargs, kwds):

1525 '''(INTERNAL) Helper for the C{frechet.[-]Frechet**} and

1526 C{hausdorff.[-]Hausdorff*} classes.

1527 '''

1528 try:

1529 _xcallable(distance=func)

1530 # assert _args_kwds_count2(func)[0] == nargs + int(ismethod(func))

1531 _ = func(*_0_0s(nargs), **kwds)

1532 except Exception as x:

1533 t = unstr(func, **kwds)

1534 raise _TypeError(t, cause=x)

1535 return func

1536

1537

1538 def _radical2(d, r1, r2, **name): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d

1539 # (INTERNAL) See C{radical2} below

1540 # assert d > EPS0

1541 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5

1542 n = _name__(name, name__=radical2)

1543 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d, name=n)

1544

1545

1546 def radical2(distance, radius1, radius2, **name):

1547 '''Compute the I{radical ratio} and I{radical line} of two

1548 U{intersecting circles<https://MathWorld.Wolfram.com/

1549 Circle-CircleIntersection.html>}.

1550

1551 The I{radical line} is perpendicular to the axis thru the

1552 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}.

1553

1554 @arg distance: Distance between the circle centers (C{scalar}).

1555 @arg radius1: Radius of the first circle (C{scalar}).

1556 @arg radius2: Radius of the second circle (C{scalar}).

1557 @kwarg name: Optional C{B{name}=NN} (C{str}).

1558

1559 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <=

1560 ratio <= 1.0} and C{xline} is along the B{C{distance}}.

1561

1562 @raise IntersectionError: The B{C{distance}} exceeds the sum

1563 of B{C{radius1}} and B{C{radius2}}.

1564

1565 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or

1566 B{C{radius2}}.

1567

1568 @see: U{Circle-Circle Intersection

1569 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}.

1570 '''

1571 d = Distance_(distance, low=_0_0)

1572 r1 = Radius_(radius1=radius1)

1573 r2 = Radius_(radius2=radius2)

1574 if d > (r1 + r2):

1575 raise IntersectionError(distance=d, radius1=r1, radius2=r2,

1576 txt=_too_(_distant_))

1577 return _radical2(d, r1, r2, **name) if d > EPS0 else \

1578 Radical2Tuple(_0_5, _0_0, **name)

1579

1580

1581 class Radical2Tuple(_NamedTuple):

1582 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and

1583 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0}

1584 '''

1585 _Names_ = (_ratio_, _xline_)

1586 _Units_ = ( Scalar, Scalar)

1587

1588

1589 def _radistance(inst):

1590 '''(INTERNAL) Helper for the L{frechet._FrechetMeterRadians}

1591 and L{hausdorff._HausdorffMeterRedians} classes.

1592 '''

1593 wrap_, kwds_ = _xkwds_pop2(inst._kwds, wrap=False)

1594 func_ = inst._func_

1595 try: # calling lower-overhead C{func_}

1596 func_(0, _0_25, _0_5, **kwds_)

1597 wrap_ = _Wrap._philamop(wrap_)

1598 except TypeError:

1599 return inst.distance

1600

1601 def _philam(p):

1602 try:

1603 return p.phi, p.lam

1604 except AttributeError: # no .phi or .lam

1605 return radians(p.lat), radians(p.lon)

1606

1607 def _func_wrap(point1, point2):

1608 phi1, lam1 = wrap_(*_philam(point1))

1609 phi2, lam2 = wrap_(*_philam(point2))

1610 return func_(phi2, phi1, lam2 - lam1, **kwds_)

1611

1612 inst._units = inst._units_

1613 return _func_wrap

1614

1615

1616 def _scale_deg(lat1, lat2): # degrees

1617 # scale factor cos(mean of lats) for delta lon

1618 m = fabs(lat1 + lat2) * _0_5

1619 return cos(radians(m)) if m < _90_0 else _0_0

1620

1621

1622 def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights

1623 # scale factor cos(mean of phis) for delta lam

1624 m = fabs(phi1 + phi2) * _0_5

1625 return cos(m) if m < PI_2 else _0_0

1626

1627

1628 def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw

1629 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine.

1630 '''

1631 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21)

1632 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21

1633

1634

1635 def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):

1636 '''Compute the distance between two (ellipsoidal) points using

1637 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}

1638 formula.

1639

1640 @arg lat1: Start latitude (C{degrees}).

1641 @arg lon1: Start longitude (C{degrees}).

1642 @arg lat2: End latitude (C{degrees}).

1643 @arg lon2: End longitude (C{degrees}).

1644 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},

1645 L{Ellipsoid2} or L{a_f2Tuple}) to use.

1646 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll

1647 B{C{lat2}} and B{C{lon2}} (C{bool}).

1648

1649 @return: Distance (C{meter}, same units as the B{C{datum}}'s

1650 ellipsoid axes).

1651

1652 @raise TypeError: Invalid B{C{datum}}.

1653

1654 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},

1655 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},

1656 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}.

1657 '''

1658 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2)

1659

1660

1661 def thomas_(phi2, phi1, lam21, datum=_WGS84):

1662 '''Compute the I{angular} distance between two (ellipsoidal) points using

1663 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}

1664 formula.

1665

1666 @arg phi2: End latitude (C{radians}).

1667 @arg phi1: Start latitude (C{radians}).

1668 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).

1669 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},

1670 L{Ellipsoid2} or L{a_f2Tuple}).

1671

1672 @return: Angular distance (C{radians}).

1673

1674 @raise TypeError: Invalid B{C{datum}}.

1675

1676 @see: Functions L{thomas}, L{cosineAndoyerLambert_},

1677 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

1678 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},

1679 L{haversine_} and L{vincentys_} and U{Geodesy-PHP

1680 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/

1681 Distance/ThomasFormula.php>}.

1682 '''

1683 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)

1684 if r and isnon0(c1) and isnon0(c2):

1685 E = _ellipsoidal(datum, thomas_)

1686 if E.f:

1687 r1 = atan2(E.b_a * s1, c1)

1688 r2 = atan2(E.b_a * s2, c2)

1689

1690 j = (r2 + r1) * _0_5

1691 k = (r2 - r1) * _0_5

1692 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5)

1693

1694 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2)

1695 u = _1_0 - h

1696 if isnon0(u) and isnon0(h):

1697 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h)

1698 sr, cr = sincos2(r)

1699 if isnon0(sr):

1700 u = 2 * (sj * ck)**2 / u

1701 h = 2 * (sk * cj)**2 / h

1702 x = u + h

1703 y = u - h

1704

1705 s = r / sr

1706 e = 4 * s**2

1707 d = 2 * cr

1708 a = e * d

1709 b = 2 * r

1710 c = s - (a - d) * _0_5

1711 f = E.f * _0_25

1712

1713 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y)

1714 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr

1715 return r

1716

1717

1718 def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):

1719 '''Compute the distance between two (spherical) points using

1720 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}

1721 spherical formula.

1722

1723 @arg lat1: Start latitude (C{degrees}).

1724 @arg lon1: Start longitude (C{degrees}).

1725 @arg lat2: End latitude (C{degrees}).

1726 @arg lon2: End longitude (C{degrees}).

1727 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})

1728 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or

1729 L{a_f2Tuple}) to use.

1730 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll

1731 B{C{lat2}} and B{C{lon2}} (C{bool}).

1732

1733 @return: Distance (C{meter}, same units as B{C{radius}}).

1734

1735 @raise UnitError: Invalid B{C{radius}}.

1736

1737 @see: Functions L{vincentys_}, L{cosineAndoyerLambert},

1738 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular},

1739 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},

1740 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2},

1741 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.

1742

1743 @note: See note at function L{vincentys_}.

1744 '''

1745 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2)

1746

1747

1748 def vincentys_(phi2, phi1, lam21):

1749 '''Compute the I{angular} distance between two (spherical) points using

1750 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}

1751 spherical formula.

1752

1753 @arg phi2: End latitude (C{radians}).

1754 @arg phi1: Start latitude (C{radians}).

1755 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).

1756

1757 @return: Angular distance (C{radians}).

1758

1759 @see: Functions L{vincentys}, L{cosineAndoyerLambert_},

1760 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},

1761 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},

1762 L{haversine_} and L{thomas_}.

1763

1764 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_}

1765 produce equivalent results, but L{vincentys_} is suitable

1766 for antipodal points and slightly more expensive (M{3 cos,

1767 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_}

1768 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and

1769 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}).

1770 '''

1771 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21)

1772

1773 c = c2 * c21

1774 x = s1 * s2 + c1 * c

1775 y = c1 * s2 - s1 * c

1776 return atan2(hypot(c2 * s21, y), x)

1777

1778# **) MIT License

1779#

1780# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.

1781#

1782# Permission is hereby granted, free of charge, to any person obtaining a

1783# copy of this software and associated documentation files (the "Software"),

1784# to deal in the Software without restriction, including without limitation

1785# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1786# and/or sell copies of the Software, and to permit persons to whom the

1787# Software is furnished to do so, subject to the following conditions:

1788#

1789# The above copyright notice and this permission notice shall be included

1790# in all copies or substantial portions of the Software.

1791#

1792# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1793# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1794# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1795# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1796# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1797# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1798# OTHER DEALINGS IN THE SOFTWARE.