Coverage for pygeodesy/formy.py: 98%

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1 

2# -*- coding: utf-8 -*- 

3 

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

5''' 

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

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

8 

9# from pygeodesy.cartesianBase import CartesianBase # _MODS 

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

11 _0_0s, float0_, isnon0, remainder, _umod_PI2, \ 

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

13 _32_0, _90_0, _180_0, _360_0 

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

15 _mean_radius, _spherical_datum, _WGS84, _EWGS84 

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

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

18 _TypeError, _ValueError, _xattr, _xError, \ 

19 _xcallable,_xkwds, _xkwds_pop2 

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

21from pygeodesy.fsums import fsumf_, Fmt, unstr 

22# from pygeodesy.internals import _dunder_nameof # from .named 

23from pygeodesy.interns import _delta_, _distant_, _inside_, _SPACE_, _too_ 

24from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS 

25from pygeodesy.named import _name__, _name2__, _NamedTuple, _xnamed, \ 

26 _dunder_nameof 

27from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, LatLon2Tuple, \ 

28 Intersection3Tuple, PhiLam2Tuple, Vector3Tuple 

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

30# from pygeodesy.triaxials import _hartzell3 # _MODS 

31from pygeodesy.units import _isHeight, _isRadius, Bearing, Degrees_, Distance, \ 

32 Distance_, Height, Lamd, Lat, Lon, Meter_, Phid, \ 

33 Radians, Radians_, Radius, Radius_, Scalar, _100km 

34from pygeodesy.utily import acos1, atan2b, atan2d, degrees2m, _loneg, m2degrees, \ 

35 tan_2, sincos2, sincos2_, sincos2d_, _Wrap 

36# from pygeodesy.vector3d import _otherV3d # _MODS 

37# from pygeodesy.vector3dBase import _xyz_y_z3 # _MODS 

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

39# sphericalNvector, sphericalTrigonometry # _MODS 

40 

41from contextlib import contextmanager 

42from math import asin, atan, atan2, cos, degrees, fabs, radians, sin, sqrt # pow 

43 

44__all__ = _ALL_LAZY.formy 

45__version__ = '24.07.29' 

46 

47_RADIANS2 = (PI / _180_0)**2 # degrees- to radians-squared 

48_ratio_ = 'ratio' 

49_xline_ = 'xline' 

50 

51 

52def _anti2(a, b, n_2, n, n2): 

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

54 ''' 

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

56 if r == a: 

57 r = -r 

58 b += n 

59 if fabs(b) > n: 

60 b = remainder(b, n2) 

61 return float0_(r, b) 

62 

63 

64def antipode(lat, lon, **name): 

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

66 point in C{degrees}. 

67 

68 @arg lat: Latitude (C{degrees}). 

69 @arg lon: Longitude (C{degrees}). 

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

71 

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

73 

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

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

76 ''' 

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

78 

79 

80def antipode_(phi, lam, **name): 

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

82 point in C{radians}. 

83 

84 @arg phi: Latitude (C{radians}). 

85 @arg lam: Longitude (C{radians}). 

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

87 

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

89 

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

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

92 ''' 

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

94 

95 

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

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

98 between two (spherical) points. 

99 

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

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

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

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

104 @kwarg final_wrap: Optional keyword arguments for function 

105 L{pygeodesy.bearing_}. 

106 

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

108 both points coincide. 

109 ''' 

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

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

112 return degrees(r) 

113 

114 

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

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

117 two (spherical) points. 

118 

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

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

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

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

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

124 (C{bool}). 

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

126 B{C{lam2}} (C{bool}). 

127 

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

129 are coincident. 

130 

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

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

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

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

135 ''' 

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

137 if final: # swap plus PI 

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

139 r = PI3 

140 else: 

141 r = PI2 

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

143 

144 x = ca1 * sa2 - sa1 * ca2 * cdb 

145 y = sdb * ca2 

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

147 

148 

149def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf 

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

151 ''' 

152 try: # for LatLon_ and ellipsoidal LatLon 

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

154 except AttributeError: 

155 pass 

156 # XXX spherical version, OK for ellipsoidal ispolar? 

157 t = p1.philam + p2.philam 

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

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

160 name__=_bearingTo2) 

161 

162 

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

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

165 lat2 - lat1)} between two points. 

166 

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

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

169 

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

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

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

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

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

175 mean latitude (C{bool}). 

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

177 and B{C{lon2}} (C{bool}). 

178 

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

180 

181 @note: Courtesy of Martin Schultz. 

182 

183 @see: U{Local, flat earth approximation 

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

185 ''' 

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

187 if adjust: # scale delta lon 

188 d_lon *= _scale_deg(lat1, lat2) 

189 return atan2b(d_lon, lat2 - lat1) 

190 

191 

192def cosineAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

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

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

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

196 

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

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

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

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

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

202 L{a_f2Tuple}) to use. 

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

204 B{C{lon2}} (C{bool}). 

205 

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

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

208 

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

210 

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

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

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

214 L{Ellipsoid.distance2}. 

215 ''' 

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

217 

218 

219def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): 

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

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

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

223 

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

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

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

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

228 L{a_f2Tuple}) to use. 

229 

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

231 

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

233 

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

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

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

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

238 AndoyerLambert.php>}. 

239 ''' 

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

241 if isnon0(c1) and isnon0(c2): 

242 E = _ellipsoidal(datum, cosineAndoyerLambert_) 

243 if E.f: # ellipsoidal 

244 r2 = atan2(E.b_a * s2, c2) 

245 r1 = atan2(E.b_a * s1, c1) 

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

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

248 if r: 

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

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

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

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

253 r += (c - s) * E.f * _0_125 

254 return r 

255 

256 

257def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

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

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

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

261 

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

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

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

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

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

267 L{a_f2Tuple}) to use. 

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

269 B{C{lon2}} (C{bool}). 

270 

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

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

273 

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

275 

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

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

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

279 L{Ellipsoid.distance2}. 

280 ''' 

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

282 

283 

284def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): 

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

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

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

288 formula. 

289 

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

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

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

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

294 L{Ellipsoid2} or L{a_f2Tuple}). 

295 

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

297 

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

299 

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

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

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

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

304 Distance/ForsytheCorrection.php>}. 

305 ''' 

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

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

308 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_) 

309 if E.f: # ellipsoidal 

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

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

312 s = (s1 + s2)**2 / (1 + cr) 

313 t = (s1 - s2)**2 / (1 - cr) 

314 x = s + t 

315 y = s - t 

316 

317 s = 8 * r**2 / sr 

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

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

320 b = -2 * d 

321 e = 30 * s2r 

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

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

324 

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

326 return r 

327 

328 

329def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

358def cosineLaw_(phi2, phi1, lam21): 

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

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

361 

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

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

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

365 

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

367 

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

369 L{cosineForsytheAndoyerLambert_}, L{euclidean_}, 

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

371 L{thomas_} and L{vincentys_}. 

372 

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

374 ''' 

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

376 

377 

378def _d3(wrap, lat1, lon1, lat2, lon2): 

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

380 ''' 

381 if wrap: 

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

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

384 else: # for backward compaibility 

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

386 

387 

388def _dE(func_, earth, *wrap_lls): 

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

390 ''' 

391 E = _ellipsoidal(earth, func_) 

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

393 return r * E.a 

394 

395 

396def _dS(func_, radius, *wrap_lls, **adjust): 

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

398 ''' 

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

400 if radius is not R_M: 

401 _, lat1, _, lat2, _ = wrap_lls 

402 radius = _mean_radius(radius, lat1, lat2) 

403 return r * radius 

404 

405 

406def _eA(excess_, radius, *wrap_lls): 

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

408 ''' 

409 r = excess_(*_d3(*wrap_lls)) 

410 if radius: 

411 _, lat1, _, lat2, _ = wrap_lls 

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

413 return r 

414 

415 

416def _ellipsoidal(earth, where): 

417 '''(INTERNAL) Helper for distances. 

418 ''' 

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

420 earth if isinstance(earth, Ellipsoid) else 

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

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

423 

424 

425def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **adjust_limit_wrap): 

426 '''Compute the distance between two points using the U{Equirectangular Approximation 

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

428 

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

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

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

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

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

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

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

436 

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

438 

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

440 

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

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

443 ''' 

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

445 Lat(lat2=lat2), Lon(lon2=lon2), 

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

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

448 

449 

450def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap): 

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

452 and L{hausdorff._HausdorffMeterRedians} classes. 

453 ''' 

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

455 

456 

457def equirectangular4(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False): 

458 '''Compute the distance between two points using the U{Equirectangular Approximation 

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

460 

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

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

463 

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

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

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

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

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

469 latitude (C{bool}). 

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

471 or C{0} for unlimited. 

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

473 (C{bool}). 

474 

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

476 in C{degrees squared}. 

477 

478 @raise LimitError: If the lat- and/or longitudinal delta exceeds the B{C{-limit..limit}} 

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

480 

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

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

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

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

485 and C{LatLon.equirectangularTo}. 

486 ''' 

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

488 d_lat = lat2 - lat1 

489 

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

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

492 if d > limit: 

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

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

495 limit=limit, wrap=wrap) 

496 raise LimitError(s, txt=t) 

497 

498 if adjust: # scale delta lon 

499 d_lon *= _scale_deg(lat1, lat2) 

500 

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

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

503 

504 

505def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False): 

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

507 

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

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

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

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

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

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

514 L{a_f2Tuple}) to use. 

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

516 the mean latitude (C{bool}). 

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

518 and B{C{lon2}} (C{bool}). 

519 

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

521 ellipsoid or datum axes). 

522 

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

524 

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

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

527 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

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

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

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

531 ''' 

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

533 

534 

535def euclidean_(phi2, phi1, lam21, adjust=True): 

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

537 

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

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

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

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

542 of the mean latitude (C{bool}). 

543 

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

545 

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

547 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

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

549 L{thomas_} and L{vincentys_}. 

550 ''' 

551 if adjust: 

552 lam21 *= _scale_rad(phi2, phi1) 

553 return euclid(phi2 - phi1, lam21) 

554 

555 

556def excessAbc_(A, b, c): 

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

558 and the included (small) angle. 

559 

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

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

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

563 

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

565 

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

567 

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

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

570 ''' 

571 A = Radians_(A=A) 

572 b = Radians_(b=b) * _0_5 

573 c = Radians_(c=c) * _0_5 

574 

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

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

577 

578 

579def excessCagnoli_(a, b, c): 

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

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

582 

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

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

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

586 

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

588 

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

590 

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

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

593 ''' 

594 a = Radians_(a=a) 

595 b = Radians_(b=b) 

596 c = Radians_(c=c) 

597 

598 s = fsumf_(a, b, c) * _0_5 

599 _s = sin 

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

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

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

603 return Radians(Cagnoli=r * _2_0) 

604 

605 

606def excessGirard_(A, B, C): 

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

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

609 

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

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

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

613 

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

615 

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

617 

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

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

620 ''' 

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

622 Radians_(B=B), 

623 Radians_(C=C), -PI)) 

624 

625 

626def excessLHuilier_(a, b, c): 

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

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

629 

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

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

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

633 

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

635 

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

637 

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

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

640 ''' 

641 a = Radians_(a=a) 

642 b = Radians_(b=b) 

643 c = Radians_(c=c) 

644 

645 s = fsumf_(a, b, c) * _0_5 

646 _t = tan_2 

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

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

649 return Radians(LHuilier=r * _4_0) 

650 

651 

652def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

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

656 method. 

657 

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

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

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

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

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

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

664 L{a_f2Tuple}) or C{None}. 

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

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

667 

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

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

670 if C{B{radius}=0} or C{None}. 

671 

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

673 

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

675 

676 @raise ValueError: Semi-circular longitudinal delta. 

677 

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

679 ''' 

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

681 

682 

683def excessKarney_(phi2, phi1, lam21): 

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

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

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

687 method. 

688 

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

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

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

692 

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

694 

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

696 

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

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

699 ''' 

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

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

702 # 

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

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

705 # 1 + tan(φ1 / 2) · tan(φ2 / 2) 

706 # 

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

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

709 _t = tan_2 

710 t2 = _t(phi2) 

711 t1 = _t(phi1) 

712 t = _t(lam21, lam21=None) 

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

714 _1_0 + (t1 * t2)) * _2_0) 

715 

716 

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

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

719# 

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

721# 

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

723# 

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

725# 

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

727# 

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

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

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

731# 

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

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

734# 

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

736# 

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

738# inverse Gudermannian) function 

739# 

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

741# 

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

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

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

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

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

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

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

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

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

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

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

753# - Charles Karney 

754 

755 

756def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

759 

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

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

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

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

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

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

766 L{a_f2Tuple}) or C{None}. 

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

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

769 

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

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

772 if C{B{radius}=0} or C{None}. 

773 

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

775 

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

777 

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

779 ''' 

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

781 

782 

783def excessQuad_(phi2, phi1, lam21): 

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

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

786 

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

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

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

790 

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

792 

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

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

795 ''' 

796 s = sin((phi2 + phi1) * _0_5) 

797 c = cos((phi2 - phi1) * _0_5) 

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

799 

800 

801def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False): 

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

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

804 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

806 

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

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

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

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

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

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

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

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

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

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

817 

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

819 ellipsoid axes). 

820 

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

822 

823 @note: The meridional and prime_vertical radii of curvature 

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

825 

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

827 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

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

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

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

831 ''' 

832 E = _ellipsoidal(datum, flatLocal) 

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

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

835 

836hubeny = flatLocal # PYCHOK for Karl Hubeny 

837 

838 

839def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True): 

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

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

842 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

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

844 

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

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

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

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

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

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

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

852 

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

854 

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

856 

857 @note: The meridional and prime_vertical radii of curvature 

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

859 

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

861 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_}, 

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

863 U{local, flat earth approximation 

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

865 ''' 

866 E = _ellipsoidal(datum, flatLocal_) 

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

868 

869hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny 

870 

871 

872def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

875 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

876 formula. 

877 

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

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

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

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

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

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

884 L{a_f2Tuple}) to use. 

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

886 and B{C{lon2}} (C{bool}). 

887 

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

889 ellipsoid or datum axes). 

890 

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

892 

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

894 L{cosineForsytheAndoyerLambert},L{cosineLaw}, 

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

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

897 L{vincentys}. 

898 ''' 

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

900 

901 

902def flatPolar_(phi2, phi1, lam21): 

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

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

905 Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

906 formula. 

907 

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

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

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

911 

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

913 

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

915 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

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

917 L{thomas_} and L{vincentys_}. 

918 ''' 

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

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

921 if a < b: 

922 a, b = b, a 

923 if a < EPS0: 

924 a = _0_0 

925 elif b > 0: 

926 b = b / a # /= chokes PyChecker 

927 c = b * cos(lam21) * _2_0 

928 c = fsumf_(_1_0, b**2, -fabs(c)) 

929 a *= sqrt0(c) 

930 return a 

931 

932 

933def _hartzell(pov, los, earth, **kwds): 

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

935 ''' 

936 if earth is None: 

937 earth = pov.datum 

938 else: 

939 earth = _spherical_datum(earth, name__=hartzell) 

940 pov = pov.toDatum(earth) 

941 h = pov.height 

942 if h < 0: # EPS0 

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

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

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

946 

947 

948def hartzell(pov, los=False, earth=_WGS84, **name_LatLon_and_kwds): 

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

950 a Point-Of-View in space. 

951 

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

953 L{Ecef9Tuple} or L{Vector3d}). 

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

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

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

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

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

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

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

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

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

963 

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

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

966 distance to the B{C{pov}}. 

967 

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

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

970 away from the earth. 

971 

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

973 

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

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

976 ''' 

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

978 try: 

979 D = _spherical_datum(earth, name__=hartzell) 

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

981 

982 C = _MODS.cartesianBase.CartesianBase 

983 if LatLon_and_kwds: 

984 c = C(r, datum=D) 

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

986 elif isinstance(r, C): 

987 r.height = h 

988 if i: 

989 r._iteration = i 

990 except Exception as x: 

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

992 **LatLon_and_kwds) 

993 return _xnamed(r, n) if n else r 

994 

995 

996def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

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

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

999 formula. 

1000 

1001 @arg lat1: Start latitude (C{degrees}). 

1002 @arg lon1: Start longitude (C{degrees}). 

1003 @arg lat2: End latitude (C{degrees}). 

1004 @arg lon2: End longitude (C{degrees}). 

1005 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1006 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1007 L{a_f2Tuple}) to use. 

1008 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1009 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1010 

1011 @return: Distance (C{meter}, same units as B{C{radius}}). 

1012 

1013 @raise TypeError: Invalid B{C{radius}}. 

1014 

1015 @see: U{Distance between two (spherical) points 

1016 <https://www.EdWilliams.org/avform.htm#Dist>}, functions 

1017 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1018 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1019 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, 

1020 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1021 

1022 @note: See note at function L{vincentys_}. 

1023 ''' 

1024 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2) 

1025 

1026 

1027def haversine_(phi2, phi1, lam21): 

1028 '''Compute the I{angular} distance between two (spherical) points 

1029 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} 

1030 formula. 

1031 

1032 @arg phi2: End latitude (C{radians}). 

1033 @arg phi1: Start latitude (C{radians}). 

1034 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1035 

1036 @return: Angular distance (C{radians}). 

1037 

1038 @see: Functions L{haversine}, L{cosineAndoyerLambert_}, 

1039 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1040 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_}, 

1041 L{thomas_} and L{vincentys_}. 

1042 

1043 @note: See note at function L{vincentys_}. 

1044 ''' 

1045 def _hsin(rad): 

1046 return sin(rad * _0_5)**2 

1047 

1048 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine 

1049 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2 

1050 

1051 

1052def heightOf(angle, distance, radius=R_M): 

1053 '''Determine the height above the (spherical) earth' surface after 

1054 traveling along a straight line at a given tilt. 

1055 

1056 @arg angle: Tilt angle above horizontal (C{degrees}). 

1057 @arg distance: Distance along the line (C{meter} or same units as 

1058 B{C{radius}}). 

1059 @kwarg radius: Optional mean earth radius (C{meter}). 

1060 

1061 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}). 

1062 

1063 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}. 

1064 

1065 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>} 

1066 (U{Shapiro et al. 2009, JTECH 

1067 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>} 

1068 and U{Potvin et al. 2012, JTECH 

1069 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}). 

1070 ''' 

1071 r = h = Radius(radius) 

1072 d = fabs(Distance(distance)) 

1073 if d > h: 

1074 d, h = h, d 

1075 

1076 if d > EPS0: # and h > EPS0 

1077 d = d / h # /= h chokes PyChecker 

1078 s = sin(Phid(angle=angle, clip=_180_0)) 

1079 s = fsumf_(_1_0, s * d * _2_0, d**2) 

1080 if s > 0: 

1081 return h * sqrt(s) - r 

1082 

1083 raise _ValueError(angle=angle, distance=distance, radius=radius) 

1084 

1085 

1086def heightOrthometric(h_ll, N): 

1087 '''Get the I{orthometric} height B{H}, the height above the geoid, earth surface. 

1088 

1089 @arg h_ll: The height above the ellipsoid (C{meter}) or an I{ellipsoidal} 

1090 location (C{LatLon} with a C{height} or C{h} attribute). 

1091 @arg N: The I{geoid} height (C{meter}), the height of the geoid above the 

1092 ellipsoid at the same B{C{h_ll}} location. 

1093 

1094 @return: I{Orthometric} height C{B{H} = B{h} - B{N}} (C{meter}, same units 

1095 as B{C{h}} and B{C{N}}). 

1096 

1097 @see: U{Ellipsoid, Geoid, and Othometric Heights<https://www.NGS.NOAA.gov/ 

1098 GEOID/PRESENTATIONS/2007_02_24_CCPS/Roman_A_PLSC2007notes.pdf>}, page 

1099 6 and module L{pygeodesy.geoids}. 

1100 ''' 

1101 h = h_ll if _isHeight(h_ll) else _xattr(h_ll, height=_xattr(h_ll, h=0)) 

1102 return Height(H=Height(h=h) - Height(N=N)) 

1103 

1104 

1105def horizon(height, radius=R_M, refraction=False): 

1106 '''Determine the distance to the horizon from a given altitude above the 

1107 (spherical) earth. 

1108 

1109 @arg height: Altitude (C{meter} or same units as B{C{radius}}). 

1110 @kwarg radius: Optional mean earth radius (C{meter}). 

1111 @kwarg refraction: Consider atmospheric refraction (C{bool}). 

1112 

1113 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}). 

1114 

1115 @raise ValueError: Invalid B{C{height}} or B{C{radius}}. 

1116 

1117 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}. 

1118 ''' 

1119 h, r = Height(height), Radius(radius) 

1120 if min(h, r) < 0: 

1121 raise _ValueError(height=height, radius=radius) 

1122 

1123 d2 = ((r * 2.415750694528) if refraction else # 2.0 / 0.8279 

1124 fsumf_(r, r, h)) * h 

1125 return sqrt0(d2) 

1126 

1127 

1128class _idllmn6(object): # see also .geodesicw._wargs, .latlonBase._toCartesian3, .vector2d._numpy 

1129 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}. 

1130 ''' 

1131 @contextmanager # <https://www.Python.org/dev/peps/pep-0343/> Examples 

1132 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds): 

1133 try: 

1134 if wrap: 

1135 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap) 

1136 kwds = _xkwds(kwds, wrap=wrap) # for _xError 

1137 m = small if small is _100km else Meter_(small=small) 

1138 n = _dunder_nameof(intersections2 if s else intersection2) 

1139 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m: 

1140 d, m = None, _MODS.vector3d 

1141 _i = m._intersects2 if s else m._intersect3d3 

1142 elif _isRadius(datum) and datum < 0 and not s: 

1143 d = _spherical_datum(-datum, name=n) 

1144 m = _MODS.sphericalNvector 

1145 _i = m.intersection 

1146 else: 

1147 d = _spherical_datum(datum, name=n) 

1148 if d.isSpherical: 

1149 m = _MODS.sphericalTrigonometry 

1150 _i = m._intersects2 if s else m._intersect 

1151 elif d.isEllipsoidal: 

1152 try: 

1153 if d.ellipsoid.geodesic: 

1154 pass 

1155 m = _MODS.ellipsoidalKarney 

1156 except ImportError: 

1157 m = _MODS.ellipsoidalExact 

1158 _i = m._intersections2 if s else m._intersection3 # ellispoidalBaseDI 

1159 else: 

1160 raise _TypeError(datum=datum) 

1161 yield _i, d, lat2, lon2, m, n 

1162 

1163 except (TypeError, ValueError) as x: 

1164 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum, 

1165 lat2=lat2, lon2=lon2, small=small, **kwds) 

1166 

1167_idllmn6 = _idllmn6() # PYCHOK singleton 

1168 

1169 

1170def intersection2(lat1, lon1, bearing1, 

1171 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True 

1172 '''I{Conveniently} compute the intersection of two lines each defined 

1173 by a (geodetic) point and a bearing from North, using either ... 

1174 

1175 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km 

1176 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1177 

1178 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}} 

1179 or a C{scalar B{datum}} representing the earth radius, conventionally 

1180 in C{meter} or ... 

1181 

1182 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative} 

1183 C{scalar}, (negative) earth radius, conventionally in C{meter} or ... 

1184 

1185 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}} 

1186 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1187 is installed, otherwise ... 

1188 

1189 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal. 

1190 

1191 @arg lat1: Latitude of the first point (C{degrees}). 

1192 @arg lon1: Longitude of the first point (C{degrees}). 

1193 @arg bearing1: Bearing at the first point (compass C{degrees}). 

1194 @arg lat2: Latitude of the second point (C{degrees}). 

1195 @arg lon2: Longitude of the second point (C{degrees}). 

1196 @arg bearing2: Bearing at the second point (compass C{degrees}). 

1197 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1198 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1199 radius (C{meter}, same units as B{C{radius1}} and 

1200 B{C{radius2}}) or C{None}. 

1201 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1202 and B{C{lon2}} (C{bool}). 

1203 @kwarg small: Upper limit for small distances (C{meter}). 

1204 

1205 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and 

1206 longitude of the intersection point. 

1207 

1208 @raise IntersectionError: Ambiguous or infinite intersection 

1209 or colinear, parallel or otherwise 

1210 non-intersecting lines. 

1211 

1212 @raise TypeError: Invalid B{C{datum}}. 

1213 

1214 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}}, 

1215 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}. 

1216 

1217 @see: Method L{RhumbLine.intersection2}. 

1218 

1219 @note: The returned intersections may be near-antipodal. 

1220 ''' 

1221 b1 = Bearing(bearing1=bearing1) 

1222 b2 = Bearing(bearing2=bearing2) 

1223 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1224 small, wrap, False, bearing1=b1, bearing2=b2) as t: 

1225 _i, d, lat2, lon2, m, n = t 

1226 if d is None: 

1227 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1, 

1228 m.Vector3d(lon2, lat2, 0), b2, useZ=False) 

1229 t = LatLon2Tuple(t.y, t.x, name=n) 

1230 

1231 else: 

1232 t = _i(m.LatLon(lat1, lon1, datum=d), b1, 

1233 m.LatLon(lat2, lon2, datum=d), b2, height=0, wrap=False) 

1234 if isinstance(t, Intersection3Tuple): # ellipsoidal 

1235 t, _, _ = t 

1236 t = LatLon2Tuple(t.lat, t.lon, name=n) 

1237 return t 

1238 

1239 

1240def intersections2(lat1, lon1, radius1, 

1241 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True 

1242 '''I{Conveniently} compute the intersections of two circles each defined 

1243 by a (geodetic) center point and a radius, using either ... 

1244 

1245 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km 

1246 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ... 

1247 

1248 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}} 

1249 or a C{scalar B{datum}} representing the earth radius, conventionally 

1250 in C{meter} or ... 

1251 

1252 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}} 

1253 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1254 is installed, otherwise ... 

1255 

1256 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal. 

1257 

1258 @arg lat1: Latitude of the first circle center (C{degrees}). 

1259 @arg lon1: Longitude of the first circle center (C{degrees}). 

1260 @arg radius1: Radius of the first circle (C{meter}, conventionally). 

1261 @arg lat2: Latitude of the second circle center (C{degrees}). 

1262 @arg lon2: Longitude of the second circle center (C{degrees}). 

1263 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}). 

1264 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1265 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth 

1266 radius (C{meter}, same units as B{C{radius1}} and 

1267 B{C{radius2}}) or C{None}. 

1268 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} 

1269 and B{C{lon2}} (C{bool}). 

1270 @kwarg small: Upper limit for small distances (C{meter}). 

1271 

1272 @return: 2-Tuple of the intersection points, each a 

1273 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the 

1274 points are the same instance, aka the I{radical center}. 

1275 

1276 @raise IntersectionError: Concentric, antipodal, invalid or 

1277 non-intersecting circles or no 

1278 convergence. 

1279 

1280 @raise TypeError: Invalid B{C{datum}}. 

1281 

1282 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}}, 

1283 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}. 

1284 ''' 

1285 r1 = Radius_(radius1=radius1) 

1286 r2 = Radius_(radius2=radius2) 

1287 with _idllmn6(datum, lat1, lon1, lat2, lon2, 

1288 small, wrap, True, radius1=r1, radius2=r2) as t: 

1289 _i, d, lat2, lon2, m, n = t 

1290 if d is None: 

1291 r1 = m2degrees(r1, radius=R_M, lat=lat1) 

1292 r2 = m2degrees(r2, radius=R_M, lat=lat2) 

1293 

1294 def _V2T(x, y, _, **unused): # _ == z unused 

1295 return LatLon2Tuple(y, x, name=n) 

1296 

1297 t = _i(m.Vector3d(lon1, lat1, 0), r1, 

1298 m.Vector3d(lon2, lat2, 0), r2, sphere=False, 

1299 Vector=_V2T) 

1300 else: 

1301 def _LL2T(lat, lon, **unused): 

1302 return LatLon2Tuple(lat, lon, name=n) 

1303 

1304 t = _i(m.LatLon(lat1, lon1, datum=d), r1, 

1305 m.LatLon(lat2, lon2, datum=d), r2, 

1306 LatLon=_LL2T, height=0, wrap=False) 

1307 return t 

1308 

1309 

1310def isantipode(lat1, lon1, lat2, lon2, eps=EPS): 

1311 '''Check whether two points are I{antipodal}, on diametrically 

1312 opposite sides of the earth. 

1313 

1314 @arg lat1: Latitude of one point (C{degrees}). 

1315 @arg lon1: Longitude of one point (C{degrees}). 

1316 @arg lat2: Latitude of the other point (C{degrees}). 

1317 @arg lon2: Longitude of the other point (C{degrees}). 

1318 @kwarg eps: Tolerance for near-equality (C{degrees}). 

1319 

1320 @return: C{True} if points are antipodal within the 

1321 B{C{eps}} tolerance, C{False} otherwise. 

1322 

1323 @see: Functions L{isantipode_} and L{antipode}. 

1324 ''' 

1325 return (fabs(lat1 + lat2) <= eps and 

1326 fabs(lon1 + lon2) <= eps) or _isequalTo( 

1327 normal(lat1, lon1), antipode(lat2, lon2), eps) 

1328 

1329 

1330def isantipode_(phi1, lam1, phi2, lam2, eps=EPS): 

1331 '''Check whether two points are I{antipodal}, on diametrically 

1332 opposite sides of the earth. 

1333 

1334 @arg phi1: Latitude of one point (C{radians}). 

1335 @arg lam1: Longitude of one point (C{radians}). 

1336 @arg phi2: Latitude of the other point (C{radians}). 

1337 @arg lam2: Longitude of the other point (C{radians}). 

1338 @kwarg eps: Tolerance for near-equality (C{radians}). 

1339 

1340 @return: C{True} if points are antipodal within the 

1341 B{C{eps}} tolerance, C{False} otherwise. 

1342 

1343 @see: Functions L{isantipode} and L{antipode_}. 

1344 ''' 

1345 return (fabs(phi1 + phi2) <= eps and 

1346 fabs(lam1 + lam2) <= eps) or _isequalTo_( 

1347 normal_(phi1, lam1), antipode_(phi2, lam2), eps) 

1348 

1349 

1350def _isequalTo(p1, p2, eps=EPS): 

1351 '''Compare 2 point lat-/lons ignoring C{class}. 

1352 ''' 

1353 return (fabs(p1.lat - p2.lat) <= eps and 

1354 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon) 

1355 

1356 

1357def _isequalTo_(p1, p2, eps=EPS): 

1358 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}. 

1359 ''' 

1360 return (fabs(p1.phi - p2.phi) <= eps and 

1361 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam) 

1362 

1363 

1364def isnormal(lat, lon, eps=0): 

1365 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their 

1366 respective I{normal} range in C{degrees}. 

1367 

1368 @arg lat: Latitude (C{degrees}). 

1369 @arg lon: Longitude (C{degrees}). 

1370 @kwarg eps: Optional tolerance C{degrees}). 

1371 

1372 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and 

1373 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise. 

1374 

1375 @see: Functions L{isnormal_} and L{normal}. 

1376 ''' 

1377 return (_90_0 - fabs(lat)) >= eps and _loneg(fabs(lon)) >= eps 

1378 

1379 

1380def isnormal_(phi, lam, eps=0): 

1381 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their 

1382 respective I{normal} range in C{radians}. 

1383 

1384 @arg phi: Latitude (C{radians}). 

1385 @arg lam: Longitude (C{radians}). 

1386 @kwarg eps: Optional tolerance C{radians}). 

1387 

1388 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and 

1389 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise. 

1390 

1391 @see: Functions L{isnormal} and L{normal_}. 

1392 ''' 

1393 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps 

1394 

1395 

1396def latlon2n_xyz(lat, lon, **name): 

1397 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1398 earth's surface) X, Y and Z components. 

1399 

1400 @arg lat: Latitude (C{degrees}). 

1401 @arg lon: Longitude (C{degrees}). 

1402 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1403 

1404 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1405 

1406 @see: Function L{philam2n_xyz}. 

1407 

1408 @note: These are C{n-vector} x, y and z components, 

1409 I{NOT} geocentric ECEF x, y and z coordinates! 

1410 ''' 

1411 return _2n_xyz(name, *sincos2d_(lat, lon)) 

1412 

1413 

1414def _normal2(a, b, n_2, n, n2): 

1415 '''(INTERNAL) Helper for C{normal} and C{normal_}. 

1416 ''' 

1417 if fabs(b) > n: 

1418 b = remainder(b, n2) 

1419 if fabs(a) > n_2: 

1420 r = remainder(a, n) 

1421 if r != a: 

1422 a = -r 

1423 b -= n if b > 0 else -n 

1424 return float0_(a, b) 

1425 

1426 

1427def normal(lat, lon, **name): 

1428 '''Normalize a lat- I{and} longitude pair in C{degrees}. 

1429 

1430 @arg lat: Latitude (C{degrees}). 

1431 @arg lon: Longitude (C{degrees}). 

1432 @kwarg name: Optional, overriding C{B{name}="normal"} (C{str}). 

1433 

1434 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90} 

1435 and C{abs(lon) <= 180}. 

1436 

1437 @see: Functions L{normal_} and L{isnormal}. 

1438 ''' 

1439 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0), 

1440 name=_name__(name, name__=normal)) 

1441 

1442 

1443def normal_(phi, lam, **name): 

1444 '''Normalize a lat- I{and} longitude pair in C{radians}. 

1445 

1446 @arg phi: Latitude (C{radians}). 

1447 @arg lam: Longitude (C{radians}). 

1448 @kwarg name: Optional, overriding C{B{name}="normal_"} (C{str}). 

1449 

1450 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2} 

1451 and C{abs(lam) <= PI}. 

1452 

1453 @see: Functions L{normal} and L{isnormal_}. 

1454 ''' 

1455 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2), 

1456 name=_name__(name, name__=normal_)) 

1457 

1458 

1459def _2n_xyz(name, sa, ca, sb, cb): # name always **name 

1460 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}. 

1461 ''' 

1462 # Kenneth Gade eqn 3, but using right-handed 

1463 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N 

1464 return Vector3Tuple(ca * cb, ca * sb, sa, **name) 

1465 

1466 

1467def n_xyz2latlon(x, y, z, **name): 

1468 '''Convert C{n-vector} components to lat- and longitude in C{degrees}. 

1469 

1470 @arg x: X component (C{scalar}). 

1471 @arg y: Y component (C{scalar}). 

1472 @arg z: Z component (C{scalar}). 

1473 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1474 

1475 @return: A L{LatLon2Tuple}C{(lat, lon)}. 

1476 

1477 @see: Function L{n_xyz2philam}. 

1478 ''' 

1479 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), **name) 

1480 

1481 

1482def n_xyz2philam(x, y, z, **name): 

1483 '''Convert C{n-vector} components to lat- and longitude in C{radians}. 

1484 

1485 @arg x: X component (C{scalar}). 

1486 @arg y: Y component (C{scalar}). 

1487 @arg z: Z component (C{scalar}). 

1488 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1489 

1490 @return: A L{PhiLam2Tuple}C{(phi, lam)}. 

1491 

1492 @see: Function L{n_xyz2latlon}. 

1493 ''' 

1494 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), **name) 

1495 

1496 

1497def _opposes(d, m, n, n2): 

1498 '''(INTERNAL) Helper for C{opposing} and C{opposing_}. 

1499 ''' 

1500 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1 

1501 return False if d < m or d > (n2 - m) else ( 

1502 True if (n - m) < d < (n + m) else None) 

1503 

1504 

1505def opposing(bearing1, bearing2, margin=_90_0): 

1506 '''Compare the direction of two bearings given in C{degrees}. 

1507 

1508 @arg bearing1: First bearing (compass C{degrees}). 

1509 @arg bearing2: Second bearing (compass C{degrees}). 

1510 @kwarg margin: Optional, interior angle bracket (C{degrees}). 

1511 

1512 @return: C{True} if both bearings point in opposite, C{False} if 

1513 in similar or C{None} if in I{perpendicular} directions. 

1514 

1515 @see: Function L{opposing_}. 

1516 ''' 

1517 m = Degrees_(margin=margin, low=EPS0, high=_90_0) 

1518 return _opposes(bearing2 - bearing1, m, _180_0, _360_0) 

1519 

1520 

1521def opposing_(radians1, radians2, margin=PI_2): 

1522 '''Compare the direction of two bearings given in C{radians}. 

1523 

1524 @arg radians1: First bearing (C{radians}). 

1525 @arg radians2: Second bearing (C{radians}). 

1526 @kwarg margin: Optional, interior angle bracket (C{radians}). 

1527 

1528 @return: C{True} if both bearings point in opposite, C{False} if 

1529 in similar or C{None} if in perpendicular directions. 

1530 

1531 @see: Function L{opposing}. 

1532 ''' 

1533 m = Radians_(margin=margin, low=EPS0, high=PI_2) 

1534 return _opposes(radians2 - radians1, m, PI, PI2) 

1535 

1536 

1537def philam2n_xyz(phi, lam, **name): 

1538 '''Convert lat-, longitude to C{n-vector} (I{normal} to the 

1539 earth's surface) X, Y and Z components. 

1540 

1541 @arg phi: Latitude (C{radians}). 

1542 @arg lam: Longitude (C{radians}). 

1543 @kwarg name: Optional name (C{str}). 

1544 

1545 @return: A L{Vector3Tuple}C{(x, y, z)}. 

1546 

1547 @see: Function L{latlon2n_xyz}. 

1548 

1549 @note: These are C{n-vector} x, y and z components, 

1550 I{NOT} geocentric ECEF x, y and z coordinates! 

1551 ''' 

1552 return _2n_xyz(name, *sincos2_(phi, lam)) 

1553 

1554 

1555def _Propy(func, nargs, kwds): 

1556 '''(INTERNAL) Helper for the C{frechet.[-]Frechet**} and 

1557 C{hausdorff.[-]Hausdorff*} classes. 

1558 ''' 

1559 try: 

1560 _xcallable(distance=func) 

1561 # assert _args_kwds_count2(func)[0] == nargs + int(ismethod(func)) 

1562 _ = func(*_0_0s(nargs), **kwds) 

1563 except Exception as x: 

1564 t = unstr(func, **kwds) 

1565 raise _TypeError(t, cause=x) 

1566 return func 

1567 

1568 

1569def _radical2(d, r1, r2, **name): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d 

1570 # (INTERNAL) See C{radical2} below 

1571 # assert d > EPS0 

1572 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5 

1573 n = _name__(name, name__=radical2) 

1574 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d, name=n) 

1575 

1576 

1577def radical2(distance, radius1, radius2, **name): 

1578 '''Compute the I{radical ratio} and I{radical line} of two 

1579 U{intersecting circles<https://MathWorld.Wolfram.com/ 

1580 Circle-CircleIntersection.html>}. 

1581 

1582 The I{radical line} is perpendicular to the axis thru the 

1583 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}. 

1584 

1585 @arg distance: Distance between the circle centers (C{scalar}). 

1586 @arg radius1: Radius of the first circle (C{scalar}). 

1587 @arg radius2: Radius of the second circle (C{scalar}). 

1588 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1589 

1590 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <= 

1591 ratio <= 1.0} and C{xline} is along the B{C{distance}}. 

1592 

1593 @raise IntersectionError: The B{C{distance}} exceeds the sum 

1594 of B{C{radius1}} and B{C{radius2}}. 

1595 

1596 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or 

1597 B{C{radius2}}. 

1598 

1599 @see: U{Circle-Circle Intersection 

1600 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}. 

1601 ''' 

1602 d = Distance_(distance, low=_0_0) 

1603 r1 = Radius_(radius1=radius1) 

1604 r2 = Radius_(radius2=radius2) 

1605 if d > (r1 + r2): 

1606 raise IntersectionError(distance=d, radius1=r1, radius2=r2, 

1607 txt=_too_(_distant_)) 

1608 return _radical2(d, r1, r2, **name) if d > EPS0 else \ 

1609 Radical2Tuple(_0_5, _0_0, **name) 

1610 

1611 

1612class Radical2Tuple(_NamedTuple): 

1613 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and 

1614 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0} 

1615 ''' 

1616 _Names_ = (_ratio_, _xline_) 

1617 _Units_ = ( Scalar, Scalar) 

1618 

1619 

1620def _radistance(inst): 

1621 '''(INTERNAL) Helper for the L{frechet._FrechetMeterRadians} 

1622 and L{hausdorff._HausdorffMeterRedians} classes. 

1623 ''' 

1624 wrap_, kwds_ = _xkwds_pop2(inst._kwds, wrap=False) 

1625 func_ = inst._func_ 

1626 try: # calling lower-overhead C{func_} 

1627 func_(0, _0_25, _0_5, **kwds_) 

1628 wrap_ = _Wrap._philamop(wrap_) 

1629 except TypeError: 

1630 return inst.distance 

1631 

1632 def _philam(p): 

1633 try: 

1634 return p.phi, p.lam 

1635 except AttributeError: # no .phi or .lam 

1636 return radians(p.lat), radians(p.lon) 

1637 

1638 def _func_wrap(point1, point2): 

1639 phi1, lam1 = wrap_(*_philam(point1)) 

1640 phi2, lam2 = wrap_(*_philam(point2)) 

1641 return func_(phi2, phi1, lam2 - lam1, **kwds_) 

1642 

1643 inst._units = inst._units_ 

1644 return _func_wrap 

1645 

1646 

1647def _scale_deg(lat1, lat2): # degrees 

1648 # scale factor cos(mean of lats) for delta lon 

1649 m = fabs(lat1 + lat2) * _0_5 

1650 return cos(radians(m)) if m < 90 else _0_0 

1651 

1652 

1653def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights 

1654 # scale factor cos(mean of phis) for delta lam 

1655 m = fabs(phi1 + phi2) * _0_5 

1656 return cos(m) if m < PI_2 else _0_0 

1657 

1658 

1659def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw 

1660 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine. 

1661 ''' 

1662 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21) 

1663 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21 

1664 

1665 

1666def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

1667 '''Compute the distance between two (ellipsoidal) points using 

1668 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1669 formula. 

1670 

1671 @arg lat1: Start latitude (C{degrees}). 

1672 @arg lon1: Start longitude (C{degrees}). 

1673 @arg lat2: End latitude (C{degrees}). 

1674 @arg lon2: End longitude (C{degrees}). 

1675 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, 

1676 L{Ellipsoid2} or L{a_f2Tuple}) to use. 

1677 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1678 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1679 

1680 @return: Distance (C{meter}, same units as the B{C{datum}}'s 

1681 ellipsoid axes). 

1682 

1683 @raise TypeError: Invalid B{C{datum}}. 

1684 

1685 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

1686 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, 

1687 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}. 

1688 ''' 

1689 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2) 

1690 

1691 

1692def thomas_(phi2, phi1, lam21, datum=_WGS84): 

1693 '''Compute the I{angular} distance between two (ellipsoidal) points using 

1694 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

1695 formula. 

1696 

1697 @arg phi2: End latitude (C{radians}). 

1698 @arg phi1: Start latitude (C{radians}). 

1699 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1700 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, 

1701 L{Ellipsoid2} or L{a_f2Tuple}). 

1702 

1703 @return: Angular distance (C{radians}). 

1704 

1705 @raise TypeError: Invalid B{C{datum}}. 

1706 

1707 @see: Functions L{thomas}, L{cosineAndoyerLambert_}, 

1708 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1709 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_}, 

1710 L{haversine_} and L{vincentys_} and U{Geodesy-PHP 

1711 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/ 

1712 Distance/ThomasFormula.php>}. 

1713 ''' 

1714 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21) 

1715 if r and isnon0(c1) and isnon0(c2): 

1716 E = _ellipsoidal(datum, thomas_) 

1717 if E.f: 

1718 r1 = atan2(E.b_a * s1, c1) 

1719 r2 = atan2(E.b_a * s2, c2) 

1720 

1721 j = (r2 + r1) * _0_5 

1722 k = (r2 - r1) * _0_5 

1723 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5) 

1724 

1725 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2) 

1726 u = _1_0 - h 

1727 if isnon0(u) and isnon0(h): 

1728 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h) 

1729 sr, cr = sincos2(r) 

1730 if isnon0(sr): 

1731 u = 2 * (sj * ck)**2 / u 

1732 h = 2 * (sk * cj)**2 / h 

1733 x = u + h 

1734 y = u - h 

1735 

1736 s = r / sr 

1737 e = 4 * s**2 

1738 d = 2 * cr 

1739 a = e * d 

1740 b = 2 * r 

1741 c = s - (a - d) * _0_5 

1742 f = E.f * _0_25 

1743 

1744 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y) 

1745 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr 

1746 return r 

1747 

1748 

1749def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

1750 '''Compute the distance between two (spherical) points using 

1751 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1752 spherical formula. 

1753 

1754 @arg lat1: Start latitude (C{degrees}). 

1755 @arg lon1: Start longitude (C{degrees}). 

1756 @arg lat2: End latitude (C{degrees}). 

1757 @arg lon2: End longitude (C{degrees}). 

1758 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) 

1759 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or 

1760 L{a_f2Tuple}) to use. 

1761 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1762 B{C{lat2}} and B{C{lon2}} (C{bool}). 

1763 

1764 @return: Distance (C{meter}, same units as B{C{radius}}). 

1765 

1766 @raise UnitError: Invalid B{C{radius}}. 

1767 

1768 @see: Functions L{vincentys_}, L{cosineAndoyerLambert}, 

1769 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular}, 

1770 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, 

1771 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2}, 

1772 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

1773 

1774 @note: See note at function L{vincentys_}. 

1775 ''' 

1776 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2) 

1777 

1778 

1779def vincentys_(phi2, phi1, lam21): 

1780 '''Compute the I{angular} distance between two (spherical) points using 

1781 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

1782 spherical formula. 

1783 

1784 @arg phi2: End latitude (C{radians}). 

1785 @arg phi1: Start latitude (C{radians}). 

1786 @arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

1787 

1788 @return: Angular distance (C{radians}). 

1789 

1790 @see: Functions L{vincentys}, L{cosineAndoyerLambert_}, 

1791 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

1792 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_}, 

1793 L{haversine_} and L{thomas_}. 

1794 

1795 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_} 

1796 produce equivalent results, but L{vincentys_} is suitable 

1797 for antipodal points and slightly more expensive (M{3 cos, 

1798 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_} 

1799 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and 

1800 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}). 

1801 ''' 

1802 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21) 

1803 

1804 c = c2 * c21 

1805 x = s1 * s2 + c1 * c 

1806 y = c1 * s2 - s1 * c 

1807 return atan2(hypot(c2 * s21, y), x) 

1808 

1809# **) MIT License 

1810# 

1811# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

1812# 

1813# Permission is hereby granted, free of charge, to any person obtaining a 

1814# copy of this software and associated documentation files (the "Software"), 

1815# to deal in the Software without restriction, including without limitation 

1816# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1817# and/or sell copies of the Software, and to permit persons to whom the 

1818# Software is furnished to do so, subject to the following conditions: 

1819# 

1820# The above copyright notice and this permission notice shall be included 

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1822# 

1823# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

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