Coverage for pygeodesy/ecef.py: 95%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''I{Geocentric} Earth-Centered, Earth-Fixed (ECEF) coordinates. 

5 

6Geocentric conversions transcoded from I{Charles Karney}'s C++ class U{Geocentric 

7<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Geocentric.html>} 

8into pure Python class L{EcefKarney}, class L{EcefSudano} based on I{John Sudano}'s 

9U{paper<https://www.ResearchGate.net/publication/ 

103709199_An_exact_conversion_from_an_Earth-centered_coordinate_system_to_latitude_longitude_and_altitude>}, 

11class L{EcefVeness} transcoded from I{Chris Veness}' JavaScript classes U{LatLonEllipsoidal, 

12Cartesian<https://www.Movable-Type.co.UK/scripts/geodesy/docs/latlon-ellipsoidal.js.html>}, class L{EcefYou} 

13implementing I{Rey-Jer You}'s U{transformations<https://www.ResearchGate.net/publication/240359424>} and 

14classes L{EcefFarrell22} and L{EcefFarrell22} from I{Jay A. Farrell}'s U{Table 2.1 and 2.2 

15<https://Books.Google.com/books?id=fW4foWASY6wC>}, page 29-30. 

16 

17Following is a copy of I{Karney}'s U{Detailed Description 

18<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Geocentric.html>}. 

19 

20Convert between geodetic coordinates C{lat}-, C{lon}gitude and height C{h} (measured vertically 

21from the surface of the ellipsoid) to geocentric C{x}, C{y} and C{z} coordinates, also known as 

22I{Earth-Centered, Earth-Fixed} (U{ECEF<https://WikiPedia.org/wiki/ECEF>}). 

23 

24The origin of geocentric coordinates is at the center of the earth. The C{z} axis goes thru 

25the North pole, C{lat} = 90°. The C{x} axis goes thru C{lat} = 0°, C{lon} = 0°. 

26 

27The I{local (cartesian) origin} is at (C{lat0}, C{lon0}, C{height0}). The I{local} C{x} axis points 

28East, the I{local} C{y} axis points North and the I{local} C{z} axis is normal to the ellipsoid. The 

29plane C{z = -height0} is tangent to the ellipsoid, hence the alternate name I{local tangent plane}. 

30 

31Forward conversion from geodetic to geocentric (ECEF) coordinates is straightforward. 

32 

33For the reverse transformation we use Hugues Vermeille's U{I{Direct transformation from geocentric 

34coordinates to geodetic coordinates}<https://DOI.org/10.1007/s00190-002-0273-6>}, J. Geodesy 

35(2002) 76, page 451-454. 

36 

37Several changes have been made to ensure that the method returns accurate results for all finite 

38inputs (even if h is infinite). The changes are described in Appendix B of C. F. F. Karney 

39U{I{Geodesics on an ellipsoid of revolution}<https://ArXiv.org/abs/1102.1215v1>}, Feb. 2011, 85, 

40105-117 (U{preprint<https://ArXiv.org/abs/1102.1215v1>}). Vermeille similarly updated his method 

41in U{I{An analytical method to transform geocentric into geodetic coordinates} 

42<https://DOI.org/10.1007/s00190-010-0419-x>}, J. Geodesy (2011) 85, page 105-117. See U{Geocentric 

43coordinates<https://GeographicLib.SourceForge.io/C++/doc/geocentric.html>} for more information. 

44 

45The errors in these routines are close to round-off. Specifically, for points within 5,000 Km of 

46the surface of the ellipsoid (either inside or outside the ellipsoid), the error is bounded by 7 

47nm (7 nanometers) for the WGS84 ellipsoid. See U{Geocentric coordinates 

48<https://GeographicLib.SourceForge.io/C++/doc/geocentric.html>} for further information on the errors. 

49 

50@note: The C{reverse} methods of all C{Ecef...} classes return by default C{INT0} as the (geodetic) 

51longitude for I{polar} ECEF location C{x == y == 0}. Use keyword argument C{lon00} or property 

52C{lon00} to configure that value. 

53 

54@see: Module L{ltp} and class L{LocalCartesian}, a transcription of I{Charles Karney}'s C++ class 

55U{LocalCartesian<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1LocalCartesian.html>}, 

56for conversion between geodetic and I{local cartesian} coordinates in a I{local tangent plane} as 

57opposed to I{geocentric} (ECEF) ones. 

58''' 

59 

60from pygeodesy.basics import copysign0, isscalar, issubclassof, neg, map1, \ 

61 _xinstanceof, _xsubclassof 

62from pygeodesy.constants import EPS, EPS0, EPS02, EPS1, EPS2, EPS_2, INT0, PI, PI_2, \ 

63 _0_0, _0_0001, _0_01, _0_5, _1_0, _1_0_1T, _N_1_0, \ 

64 _2_0, _N_2_0, _3_0, _4_0, _6_0, _60_0, _90_0, _N_90_0, \ 

65 _100_0, _copysign_1_0, isnon0 # PYCHOK used! 

66from pygeodesy.datums import a_f2Tuple, _ellipsoidal_datum, _WGS84, _EWGS84 

67# from pygeodesy.ellipsoids import a_f2Tuple, _EWGS84 # from .datums 

68from pygeodesy.errors import _IndexError, LenError, _ValueError, _TypesError, \ 

69 _xattr, _xdatum, _xkwds, _xkwds_get 

70from pygeodesy.fmath import cbrt, fdot, hypot, hypot1, hypot2_ 

71from pygeodesy.fsums import Fsum, fsumf_, Fmt, unstr 

72from pygeodesy.interns import NN, _a_, _C_, _datum_, _ellipsoid_, _f_, _height_, \ 

73 _lat_, _lon_, _M_, _name_, _singular_, _SPACE_, \ 

74 _x_, _xyz_, _y_, _z_ 

75from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS 

76from pygeodesy.named import _NamedBase, _NamedTuple, notOverloaded, _Pass, _xnamed 

77from pygeodesy.namedTuples import LatLon2Tuple, LatLon3Tuple, \ 

78 PhiLam2Tuple, Vector3Tuple, Vector4Tuple 

79from pygeodesy.props import deprecated_method, Property_RO, property_RO, property_doc_ 

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

81from pygeodesy.units import Degrees, Height, Int, Lam, Lat, Lon, Meter, Phi, \ 

82 Scalar, Scalar_ 

83from pygeodesy.utily import atan1, atan1d, atan2d, degrees90, degrees180, sincos2, sincos2_, \ 

84 sincos2d, sincos2d_ 

85 

86from math import atan2, cos, degrees, fabs, radians, sqrt 

87 

88__all__ = _ALL_LAZY.ecef 

89__version__ = '23.09.29' 

90 

91_Ecef_ = 'Ecef' 

92_prolate_ = 'prolate' 

93_TRIPS = 17 # 8..9 sufficient, EcefSudano.reverse 

94_xyz_y_z = _xyz_, _y_, _z_ # _xargs_names(_xyzn4)[:3] 

95 

96 

97class EcefError(_ValueError): 

98 '''An ECEF or C{Ecef*} related issue. 

99 ''' 

100 pass 

101 

102 

103def _llhn4(latlonh, lon, height, suffix=NN, Error=EcefError, name=NN): # in .ltp.LocalCartesian.forward and -.reset 

104 '''(INTERNAL) Get C{lat, lon, h, name} as C{4-tuple}. 

105 ''' 

106 try: 

107 lat, lon = latlonh.lat, latlonh.lon 

108 h = _xattr(latlonh, height=_xattr(latlonh, h=height)) 

109 n = _xattr(latlonh, name=NN) 

110 except AttributeError: 

111 lat, h, n = latlonh, height, NN 

112 

113 try: 

114 llhn = Lat(lat), Lon(lon), Height(h), (name or n) 

115 except (TypeError, ValueError) as x: 

116 t = _lat_, _lon_, _height_ 

117 if suffix: 

118 t = (_ + suffix for _ in t) 

119 d = dict(zip(t, (lat, lon, h))) 

120 raise Error(cause=x, **d) 

121 return llhn 

122 

123 

124# kwd lon00 unused but will throw a TypeError if misspelled, etc. 

125def _xyzn4(xyz, y, z, Types, Error=EcefError, name=NN, # PYCHOK unused 

126 _xyz_y_z_names=_xyz_y_z, lon00=0): # in .ltp 

127 '''(INTERNAL) Get an C{(x, y, z, name)} 4-tuple. 

128 ''' 

129 try: 

130 try: 

131 t = xyz.x, xyz.y, xyz.z, _xattr(xyz, name=name) 

132 if not isinstance(xyz, Types): 

133 raise _TypesError(_xyz_y_z_names[0], xyz, *Types) 

134 except AttributeError: 

135 t = map1(float, xyz, y, z) + (name,) 

136 

137 except (TypeError, ValueError) as x: 

138 d = dict(zip(_xyz_y_z_names, (xyz, y, z))) 

139 raise Error(cause=x, **d) 

140 return t 

141 

142# assert _xyz_y_z == _xargs_names(_xyzn4)[:3] 

143 

144 

145class _EcefBase(_NamedBase): 

146 '''(INTERNAL) Base class for L{EcefFarrell21}, L{EcefFarrell22}, L{EcefKarney}, 

147 L{EcefSudano}, L{EcefVeness} and L{EcefYou}. 

148 ''' 

149 _datum = _WGS84 

150 _E = _EWGS84 

151 _lon00 = INT0 # arbitrary, "polar" lon for LocalCartesian, Ltp 

152 

153 def __init__(self, a_ellipsoid=_EWGS84, f=None, name=NN, lon00=INT0): 

154 '''New C{Ecef*} converter. 

155 

156 @arg a_ellipsoid: A (non-prolate) ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, 

157 L{Datum} or L{a_f2Tuple}) or C{scalar} ellipsoid's 

158 equatorial radius (C{meter}). 

159 @kwarg f: C{None} or the ellipsoid flattening (C{scalar}), required 

160 for C{scalar} B{C{a_ellipsoid}}, C{B{f}=0} represents a 

161 sphere, negative B{C{f}} a prolate ellipsoid. 

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

163 @kwarg lon00: An arbitrary, I{"polar"} longitude (C{degrees}), see the 

164 methods C{reverse}. 

165 

166 @raise EcefError: If B{C{a_ellipsoid}} not L{Ellipsoid}, L{Ellipsoid2}, 

167 L{Datum} or L{a_f2Tuple} or C{scalar} or B{C{f}} not 

168 C{scalar} or if C{scalar} B{C{a_ellipsoid}} not positive 

169 or B{C{f}} not less than 1.0. 

170 ''' 

171 try: 

172 E = a_ellipsoid 

173 if f is None: 

174 if E is _EWGS84 or E is _WGS84: 

175 raise AssertionError # "break" 

176 elif isscalar(E) and isscalar(f): 

177 E = a_f2Tuple(E, f) 

178 else: 

179 raise ValueError # _invalid_ 

180 

181 d = _ellipsoidal_datum(E, name=name) 

182 E = d.ellipsoid 

183 if E.a < EPS or E.f > EPS1: 

184 raise ValueError # _invalid_ 

185 

186 self._datum = d 

187 self._E = E 

188 

189 except AssertionError: # "break" 

190 pass 

191 except (TypeError, ValueError) as x: 

192 t = unstr(self.classname, a=a_ellipsoid, f=f) 

193 raise EcefError(_SPACE_(t, _ellipsoid_), cause=x) 

194 

195 if name: 

196 self.name = name 

197 if lon00 is not INT0: 

198 self.lon00 = lon00 

199 

200 def __eq__(self, other): 

201 '''Compare this and an other Ecef. 

202 

203 @arg other: The other ecef (C{Ecef*}). 

204 

205 @return: C{True} if equal, C{False} otherwise. 

206 ''' 

207 return other is self or (isinstance(other, self.__class__) and 

208 other.ellipsoid == self.ellipsoid) 

209 

210 @Property_RO 

211 def datum(self): 

212 '''Get the datum (L{Datum}). 

213 ''' 

214 return self._datum 

215 

216 @Property_RO 

217 def ellipsoid(self): 

218 '''Get the ellipsoid (L{Ellipsoid} or L{Ellipsoid2}). 

219 ''' 

220 return self._E 

221 

222 @Property_RO 

223 def equatoradius(self): 

224 '''Get the C{ellipsoid}'s equatorial radius, semi-axis (C{meter}). 

225 ''' 

226 return self.ellipsoid.a 

227 

228 a = equatorialRadius = equatoradius # Karney property 

229 

230 @Property_RO 

231 def flattening(self): # Karney property 

232 '''Get the C{ellipsoid}'s flattening (C{scalar}), positive for 

233 I{oblate}, negative for I{prolate} or C{0} for I{near-spherical}. 

234 ''' 

235 return self.ellipsoid.f 

236 

237 f = flattening 

238 

239 def _forward(self, lat, lon, h, name, M=False, _philam=False): # in .ltp.LocalCartesian.forward and -.reset 

240 '''(INTERNAL) Common for all C{Ecef*}. 

241 ''' 

242 if _philam: # lat, lon in radians 

243 sa, ca, sb, cb = sincos2_(lat, lon) 

244 lat = Lat(degrees90( lat), Error=EcefError) 

245 lon = Lon(degrees180(lon), Error=EcefError) 

246 else: 

247 sa, ca, sb, cb = sincos2d_(lat, lon) 

248 

249 E = self.ellipsoid 

250 n = E.roc1_(sa, ca) if self._isYou else E.roc1_(sa) 

251 z = (h + n * E.e21) * sa 

252 x = (h + n) * ca 

253 

254 m = self._Matrix(sa, ca, sb, cb) if M else None 

255 return Ecef9Tuple(x * cb, x * sb, z, lat, lon, h, 

256 0, m, self.datum, 

257 name=name or self.name) 

258 

259 def forward(self, latlonh, lon=None, height=0, M=False, name=NN): 

260 '''Convert from geodetic C{(lat, lon, height)} to geocentric C{(x, y, z)}. 

261 

262 @arg latlonh: Either a C{LatLon}, an L{Ecef9Tuple} or C{scalar} 

263 latitude (C{degrees}). 

264 @kwarg lon: Optional C{scalar} longitude for C{scalar} B{C{latlonh}} 

265 (C{degrees}). 

266 @kwarg height: Optional height (C{meter}), vertically above (or below) 

267 the surface of the ellipsoid. 

268 @kwarg M: Optionally, return the rotation L{EcefMatrix} (C{bool}). 

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

270 

271 @return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with 

272 geocentric C{(x, y, z)} coordinates for the given geodetic ones 

273 C{(lat, lon, height)}, case C{C} 0, optional C{M} (L{EcefMatrix}) 

274 and C{datum} if available. 

275 

276 @raise EcefError: If B{C{latlonh}} not C{LatLon}, L{Ecef9Tuple} or 

277 C{scalar} or B{C{lon}} not C{scalar} for C{scalar} 

278 B{C{latlonh}} or C{abs(lat)} exceeds 90°. 

279 

280 @note: Use method C{.forward_} to specify C{lat} and C{lon} in C{radians} 

281 and avoid double angle conversions. 

282 ''' 

283 llhn = _llhn4(latlonh, lon, height, name=name) 

284 return self._forward(*llhn, M=M) 

285 

286 def forward_(self, phi, lam, height=0, M=False, name=NN): 

287 '''Like method C{.forward} except with geodetic lat- and longitude given 

288 in I{radians}. 

289 

290 @arg phi: Latitude in I{radians} (C{scalar}). 

291 @arg lam: Longitude in I{radians} (C{scalar}). 

292 @kwarg height: Optional height (C{meter}), vertically above (or below) 

293 the surface of the ellipsoid. 

294 @kwarg M: Optionally, return the rotation L{EcefMatrix} (C{bool}). 

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

296 

297 @return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} 

298 with C{lat} set to C{degrees90(B{phi})} and C{lon} to 

299 C{degrees180(B{lam})}. 

300 

301 @raise EcefError: If B{C{phi}} or B{C{lam}} invalid or not C{scalar}. 

302 ''' 

303 try: # like function C{_llhn4} above 

304 plhn = Phi(phi), Lam(lam), Height(height), name 

305 except (TypeError, ValueError) as x: 

306 raise EcefError(phi=phi, lam=lam, height=height, cause=x) 

307 return self._forward(*plhn, M=M, _philam=True) 

308 

309 @property_RO 

310 def _Geocentrics(self): 

311 '''(INTERNAL) Get the valid geocentric classes. I{once}. 

312 ''' 

313 _EcefBase._Geocentrics = t = (Ecef9Tuple, # overwrite property_RO 

314 _MODS.cartesianBase.CartesianBase) 

315 return t 

316 

317 @Property_RO 

318 def _isYou(self): 

319 '''(INTERNAL) Is this an C{EcefYou}?. 

320 ''' 

321 return isinstance(self, EcefYou) 

322 

323 @property 

324 def lon00(self): 

325 '''Get the I{"polar"} longitude (C{degrees}), see method C{reverse}. 

326 ''' 

327 return self._lon00 

328 

329 @lon00.setter # PYCHOK setter! 

330 def lon00(self, lon00): 

331 '''Set the I{"polar"} longitude (C{degrees}), see method C{reverse}. 

332 ''' 

333 self._lon00 = Degrees(lon00=lon00) 

334 

335 def _Matrix(self, sa, ca, sb, cb): 

336 '''Creation a rotation matrix. 

337 

338 @arg sa: C{sin(phi)} (C{float}). 

339 @arg ca: C{cos(phi)} (C{float}). 

340 @arg sb: C{sin(lambda)} (C{float}). 

341 @arg cb: C{cos(lambda)} (C{float}). 

342 

343 @return: An L{EcefMatrix}. 

344 ''' 

345 return self._xnamed(EcefMatrix(sa, ca, sb, cb)) 

346 

347 def _polon(self, y, x, R, **name_lon00): 

348 '''(INTERNAL) Handle I{"polar"} longitude. 

349 ''' 

350 return atan2d(y, x) if R else _xkwds_get(name_lon00, lon00=self.lon00) 

351 

352 def reverse(self, xyz, y=None, z=None, M=False, **name_lon00): # PYCHOK no cover 

353 '''I{Must be overloaded}.''' 

354 notOverloaded(self, xyz, y=y, z=z, M=M, **name_lon00) 

355 

356 def toStr(self, prec=9, **unused): # PYCHOK signature 

357 '''Return this C{Ecef*} as a string. 

358 

359 @kwarg prec: Precision, number of decimal digits (0..9). 

360 

361 @return: This C{Ecef*} (C{str}). 

362 ''' 

363 return self.attrs(_a_, _f_, _datum_, _name_, prec=prec) # _ellipsoid_ 

364 

365 

366class EcefFarrell21(_EcefBase): 

367 '''Conversion between geodetic and geocentric, I{Earth-Centered, Earth-Fixed} (ECEF) 

368 coordinates based on I{Jay A. Farrell}'s U{Table 2.1<https://Books.Google.com/ 

369 books?id=fW4foWASY6wC>}, page 29. 

370 ''' 

371 

372 def reverse(self, xyz, y=None, z=None, M=None, **name_lon00): # PYCHOK unused M 

373 '''Convert from geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)} using 

374 I{Farrell}'s U{Table 2.1<https://Books.Google.com/books?id=fW4foWASY6wC>}, 

375 page 29. 

376 

377 @arg xyz: A geocentric (C{Cartesian}, L{Ecef9Tuple}) or C{scalar} ECEF C{x} 

378 coordinate (C{meter}). 

379 @kwarg y: ECEF C{y} coordinate for C{scalar} B{C{xyz}} and B{C{z}} (C{meter}). 

380 @kwarg z: ECEF C{z} coordinate for C{scalar} B{C{xyz}} and B{C{y}} (C{meter}). 

381 @kwarg M: I{Ignored}, rotation matrix C{M} not available. 

382 @kwarg name_lon00: Optional keyword arguments C{B{name}=NN} (C{str}) and 

383 I{"polar"} longitude C{B{lon00}=INT0} (C{degrees}), overriding 

384 the default and property C{lon00} setting and returned if 

385 C{B{x}=0} and C{B{y}=0}. 

386 

387 @return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with 

388 geodetic coordinates C{(lat, lon, height)} for the given geocentric 

389 ones C{(x, y, z)}, case C{C=1}, C{M=None} always and C{datum} 

390 if available. 

391 

392 @raise EcefError: Invalid B{C{xyz}} or C{scalar} C{x} or B{C{y}} and/or B{C{z}} 

393 not C{scalar} for C{scalar} B{C{xyz}} or C{sqrt} domain or 

394 zero division error. 

395 

396 @see: L{EcefFarrell22} and L{EcefVeness}. 

397 ''' 

398 x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, **name_lon00) 

399 

400 E = self.ellipsoid 

401 a = E.a 

402 a2 = E.a2 

403 b2 = E.b2 

404 e2 = E.e2 

405 e2_ = E.e2abs * E.a2_b2 # (E.e * E.a_b)**2 = 0.0820944... WGS84 

406 e4 = E.e4 

407 

408 try: # names as page 29 

409 z2 = z**2 

410 ez = z2 * (_1_0 - e2) # E.e2s2(z) 

411 

412 p = hypot(x, y) 

413 p2 = p**2 

414 G = p2 + ez - e2 * (a2 - b2) # p2 + ez - e4 * a2 

415 F = b2 * z2 * 54 

416 c = e4 * p2 * F / G**3 

417 s = cbrt(_1_0 + sqrt(c**2 + c + c) + c) 

418 G *= fsumf_(s, _1_0, _1_0 / s) 

419 P = F / (G**2 * _3_0) 

420 Q = sqrt(_2_0 * e4 * P + _1_0) 

421 Q1 = Q + _1_0 

422 r0 = P * p * e2 / Q1 - sqrt(fsumf_(a2 * (Q1 / Q) * _0_5, 

423 -P * ez / (Q * Q1), 

424 -P * p2 * _0_5)) 

425 r = p + e2 * r0 

426 v = b2 / (sqrt(r**2 + ez) * a) 

427 

428 h = hypot(r, z) * (_1_0 - v) 

429 lat = atan1d((e2_ * v + _1_0) * z, p) 

430 lon = self._polon(y, x, p, **name_lon00) 

431 # note, phi and lam are swapped on page 29 

432 

433 except (ValueError, ZeroDivisionError) as e: 

434 raise EcefError(x=x, y=y, z=z, cause=e) 

435 

436 return Ecef9Tuple(x, y, z, lat, lon, h, 

437 1, None, self.datum, 

438 name=name or self.name) 

439 

440 

441class EcefFarrell22(_EcefBase): 

442 '''Conversion between geodetic and geocentric, I{Earth-Centered, Earth-Fixed} (ECEF) 

443 coordinates based on I{Jay A. Farrell}'s U{Table 2.2<https://Books.Google.com/ 

444 books?id=fW4foWASY6wC>}, page 30. 

445 ''' 

446 

447 def reverse(self, xyz, y=None, z=None, M=None, **name_lon00): # PYCHOK unused M 

448 '''Convert from geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)} using 

449 I{Farrell}'s U{Table 2.2<https://Books.Google.com/books?id=fW4foWASY6wC>}, 

450 page 30. 

451 

452 @arg xyz: A geocentric (C{Cartesian}, L{Ecef9Tuple}) or C{scalar} ECEF C{x} 

453 coordinate (C{meter}). 

454 @kwarg y: ECEF C{y} coordinate for C{scalar} B{C{xyz}} and B{C{z}} (C{meter}). 

455 @kwarg z: ECEF C{z} coordinate for C{scalar} B{C{xyz}} and B{C{y}} (C{meter}). 

456 @kwarg M: I{Ignored}, rotation matrix C{M} not available. 

457 @kwarg name_lon00: Optional keyword arguments C{B{name}=NN} (C{str}) and 

458 I{"polar"} longitude C{B{lon00}=INT0} (C{degrees}), overriding 

459 the default and property C{lon00} setting and returned in case 

460 C{B{x}=0} and C{B{y}=0}. 

461 

462 @return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with 

463 geodetic coordinates C{(lat, lon, height)} for the given geocentric 

464 ones C{(x, y, z)}, case C{C=1}, C{M=None} always and C{datum} 

465 if available. 

466 

467 @raise EcefError: Invalid B{C{xyz}} or C{scalar} C{x} or B{C{y}} and/or B{C{z}} 

468 not C{scalar} for C{scalar} B{C{xyz}} or C{sqrt} domain or 

469 zero division error. 

470 

471 @see: L{EcefFarrell21} and L{EcefVeness}. 

472 ''' 

473 x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, **name_lon00) 

474 

475 E = self.ellipsoid 

476 a = E.a 

477 b = E.b 

478 

479 try: # see EcefVeness.reverse 

480 p = hypot(x, y) 

481 lon = self._polon(y, x, p, **name_lon00) 

482 

483 s, c = sincos2(atan2(z * a, p * b)) # == _norm3 

484 lat = atan1d(z + s**3 * b * E.e22, 

485 p - c**3 * a * E.e2) 

486 

487 s, c = sincos2d(lat) 

488 if c: # E.roc1_(s) = E.a / sqrt(1 - E.e2 * s**2) 

489 h = p / c - (E.roc1_(s) if s else a) 

490 else: # polar 

491 h = fabs(z) - b 

492 # note, phi and lam are swapped on page 30 

493 

494 except (ValueError, ZeroDivisionError) as e: 

495 raise EcefError(x=x, y=y, z=z, cause=e) 

496 

497 return Ecef9Tuple(x, y, z, lat, lon, h, 

498 1, None, self.datum, 

499 name=name or self.name) 

500 

501 

502class EcefKarney(_EcefBase): 

503 '''Conversion between geodetic and geocentric, I{Earth-Centered, Earth-Fixed} (ECEF) 

504 coordinates transcoded from I{Karney}'s C++ U{Geocentric<https://GeographicLib.SourceForge.io/ 

505 C++/doc/classGeographicLib_1_1Geocentric.html>} methods. 

506 

507 @note: On methods C{.forward} and C{.forwar_}, let C{v} be a unit vector located 

508 at C{(lat, lon, h)}. We can express C{v} as column vectors in one of two 

509 ways, C{v1} in East, North, Up (ENU) coordinates (where the components are 

510 relative to a local coordinate system at C{C(lat0, lon0, h0)}) or as C{v0} 

511 in geocentric C{x, y, z} coordinates. Then, M{v0 = M ⋅ v1} where C{M} is 

512 the rotation matrix. 

513 ''' 

514 

515 @Property_RO 

516 def hmax(self): 

517 '''Get the distance or height limit (C{meter}, conventionally). 

518 ''' 

519 return self.equatoradius / EPS_2 # self.equatoradius * _2_EPS, 12M lighyears 

520 

521 def reverse(self, xyz, y=None, z=None, M=False, **name_lon00): 

522 '''Convert from geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)}. 

523 

524 @arg xyz: A geocentric (C{Cartesian}, L{Ecef9Tuple}) or C{scalar} ECEF C{x} 

525 coordinate (C{meter}). 

526 @kwarg y: ECEF C{y} coordinate for C{scalar} B{C{xyz}} and B{C{z}} (C{meter}). 

527 @kwarg z: ECEF C{z} coordinate for C{scalar} B{C{xyz}} and B{C{y}} (C{meter}). 

528 @kwarg M: Optionally, return the rotation L{EcefMatrix} (C{bool}). 

529 @kwarg name_lon00: Optional keyword arguments C{B{name}=NN} (C{str}) and 

530 I{"polar"} longitude C{B{lon00}=INT0} (C{degrees}), overriding 

531 the default and property C{lon00} setting and returned in case 

532 C{B{x}=0} and C{B{y}=0}. 

533 

534 @return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with 

535 geodetic coordinates C{(lat, lon, height)} for the given geocentric 

536 ones C{(x, y, z)}, case C{C}, optional C{M} (L{EcefMatrix}) and 

537 C{datum} if available. 

538 

539 @raise EcefError: Invalid B{C{xyz}} or C{scalar} C{x} or B{C{y}} and/or B{C{z}} 

540 not C{scalar} for C{scalar} B{C{xyz}}. 

541 

542 @note: In general, there are multiple solutions and the result which minimizes 

543 C{height} is returned, i.e., the C{(lat, lon)} corresponding to the 

544 closest point on the ellipsoid. If there are still multiple solutions 

545 with different latitudes (applies only if C{z} = 0), then the solution 

546 with C{lat} > 0 is returned. If there are still multiple solutions with 

547 different longitudes (applies only if C{x} = C{y} = 0), then C{lon00} is 

548 returned. The returned C{lon} is in the range [−180°, 180°] and C{height} 

549 is not below M{−E.a * (1 − E.e2) / sqrt(1 − E.e2 * sin(lat)**2)}. Like 

550 C{forward} above, M{v1 = Transpose(M) ⋅ v0}. 

551 ''' 

552 def _norm3(y, x): 

553 h = hypot(y, x) # EPS0, EPS_2 

554 return (y / h, x / h, h) if h > 0 else (_0_0, _1_0, h) 

555 

556 x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, **name_lon00) 

557 

558 E = self.ellipsoid 

559 f = E.f 

560 

561 sb, cb, R = _norm3(y, x) 

562 h = hypot(R, z) # distance to earth center 

563 if h > self.hmax: # PYCHOK no cover 

564 # We are really far away (> 12M light years). Treat the earth 

565 # as a point and h above as an acceptable approximation to the 

566 # height. This avoids overflow, e.g., in the computation of d 

567 # below. It's possible that h has overflowed to INF, that's OK. 

568 # Treat finite x, y, but R overflows to +INF by scaling by 2. 

569 sb, cb, R = _norm3(y * _0_5, x * _0_5) 

570 sa, ca, _ = _norm3(z * _0_5, R) 

571 C = 1 

572 

573 elif E.e4: # E.isEllipsoidal 

574 # Treat prolate spheroids by swapping R and Z here and by 

575 # switching the arguments to phi = atan2(...) at the end. 

576 p = (R / E.a)**2 

577 q = (z / E.a)**2 * E.e21 

578 if f < 0: 

579 p, q = q, p 

580 r = fsumf_(p, q, -E.e4) 

581 e = E.e4 * q 

582 if e or r > 0: 

583 # Avoid possible division by zero when r = 0 by multiplying 

584 # equations for s and t by r^3 and r, respectively. 

585 s = d = e * p / _4_0 # s = r^3 * s 

586 u = r = r / _6_0 

587 r2 = r**2 

588 r3 = r2 * r 

589 t3 = r3 + s 

590 d *= t3 + r3 

591 if d < 0: 

592 # t is complex, but the way u is defined, the result is real. 

593 # There are three possible cube roots. We choose the root 

594 # which avoids cancellation. Note, d < 0 implies r < 0. 

595 u += cos(atan2(sqrt(-d), -t3) / _3_0) * r * _2_0 

596 else: 

597 # Pick the sign on the sqrt to maximize abs(t3). This 

598 # minimizes loss of precision due to cancellation. The 

599 # result is unchanged because of the way the t is used 

600 # in definition of u. 

601 if d > 0: 

602 t3 += copysign0(sqrt(d), t3) # t3 = (r * t)^3 

603 # N.B. cbrt always returns the real root, cbrt(-8) = -2. 

604 t = cbrt(t3) # t = r * t 

605 if t: # t can be zero; but then r2 / t -> 0. 

606 u = fsumf_(u, t, r2 / t) 

607 v = sqrt(e + u**2) # guaranteed positive 

608 # Avoid loss of accuracy when u < 0. Underflow doesn't occur in 

609 # E.e4 * q / (v - u) because u ~ e^4 when q is small and u < 0. 

610 u = (e / (v - u)) if u < 0 else (u + v) # u+v, guaranteed positive 

611 # Need to guard against w going negative due to roundoff in u - q. 

612 w = E.e2abs * (u - q) / (_2_0 * v) 

613 # Rearrange expression for k to avoid loss of accuracy due to 

614 # subtraction. Division by 0 not possible because u > 0, w >= 0. 

615 k1 = k2 = (u / (sqrt(u + w**2) + w)) if w > 0 else sqrt(u) 

616 if f < 0: 

617 k1 -= E.e2 

618 else: 

619 k2 += E.e2 

620 sa, ca, h = _norm3(z / k1, R / k2) 

621 h *= k1 - E.e21 

622 C = 2 

623 

624 else: # e = E.e4 * q == 0 and r <= 0 

625 # This leads to k = 0 (oblate, equatorial plane) and k + E.e^2 = 0 

626 # (prolate, rotation axis) and the generation of 0/0 in the general 

627 # formulas for phi and h, using the general formula and division 

628 # by 0 in formula for h. Handle this case by taking the limits: 

629 # f > 0: z -> 0, k -> E.e2 * sqrt(q) / sqrt(E.e4 - p) 

630 # f < 0: r -> 0, k + E.e2 -> -E.e2 * sqrt(q) / sqrt(E.e4 - p) 

631 q = E.e4 - p 

632 if f < 0: 

633 p, q = q, p 

634 e = E.a 

635 else: 

636 e = E.b2_a 

637 sa, ca, h = _norm3(sqrt(q * E._1_e21), sqrt(p)) 

638 if z < 0: # for tiny negative z, not for prolate 

639 sa = neg(sa) 

640 h *= neg(e / E.e2abs) 

641 C = 3 

642 

643 else: # E.e4 == 0, spherical case 

644 # Dealing with underflow in the general case with E.e2 = 0 is 

645 # difficult. Origin maps to North pole, same as with ellipsoid. 

646 sa, ca, _ = _norm3((z if h else _1_0), R) 

647 h -= E.a 

648 C = 4 

649 

650 # lon00 <https://GitHub.com/mrJean1/PyGeodesy/issues/77> 

651 lon = self._polon(sb, cb, R, **name_lon00) 

652 m = self._Matrix(sa, ca, sb, cb) if M else None 

653 return Ecef9Tuple(x, y, z, atan1d(sa, ca), lon, h, 

654 C, m, self.datum, 

655 name=name or self.name) 

656 

657 

658class EcefSudano(_EcefBase): 

659 '''Conversion between geodetic and geocentric, I{Earth-Centered, Earth-Fixed} (ECEF) coordinates 

660 based on I{John J. Sudano}'s U{paper<https://www.ResearchGate.net/publication/3709199>}. 

661 ''' 

662 _tol = EPS2 

663 

664 def reverse(self, xyz, y=None, z=None, M=None, **name_lon00): # PYCHOK unused M 

665 '''Convert from geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)} using 

666 I{Sudano}'s U{iterative method<https://www.ResearchGate.net/publication/3709199>}. 

667 

668 @arg xyz: A geocentric (C{Cartesian}, L{Ecef9Tuple}) or C{scalar} ECEF C{x} 

669 coordinate (C{meter}). 

670 @kwarg y: ECEF C{y} coordinate for C{scalar} B{C{xyz}} and B{C{z}} (C{meter}). 

671 @kwarg z: ECEF C{z} coordinate for C{scalar} B{C{xyz}} and B{C{y}} (C{meter}). 

672 @kwarg M: I{Ignored}, rotation matrix C{M} not available. 

673 @kwarg name_lon00: Optional keyword arguments C{B{name}=NN} (C{str}) and 

674 I{"polar"} longitude C{B{lon00}=INT0} (C{degrees}), overriding 

675 the default and property C{lon00} setting and returned in case 

676 C{B{x}=0} and C{B{y}=0}. 

677 

678 @return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with geodetic 

679 coordinates C{(lat, lon, height)} for the given geocentric ones C{(x, y, z)}, 

680 iteration C{C}, C{M=None} always and C{datum} if available. 

681 

682 @raise EcefError: Invalid B{C{xyz}} or C{scalar} C{x} or B{C{y}} and/or B{C{z}} 

683 not C{scalar} for C{scalar} B{C{xyz}} or no convergence. 

684 ''' 

685 x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, **name_lon00) 

686 

687 E = self.ellipsoid 

688 e = E.e2 * E.a 

689 R = hypot(x, y) # Rh 

690 d = e - R 

691 

692 lat = atan1d(z, R * E.e21) 

693 sa, ca = sincos2d(fabs(lat)) 

694 # Sudano's Eq (A-6) and (A-7) refactored/reduced, 

695 # replacing Rn from Eq (A-4) with n = E.a / ca: 

696 # N = ca**2 * ((z + E.e2 * n * sa) * ca - R * sa) 

697 # = ca**2 * (z * ca + E.e2 * E.a * sa - R * sa) 

698 # = ca**2 * (z * ca + (E.e2 * E.a - R) * sa) 

699 # D = ca**3 * (E.e2 * n / E.e2s2(sa)) - R 

700 # = ca**2 * (E.e2 * E.a / E.e2s2(sa) - R / ca**2) 

701 # N / D = (z * ca + (E.e2 * E.a - R) * sa) / 

702 # (E.e2 * E.a / E.e2s2(sa) - R / ca**2) 

703 tol = self.tolerance 

704 _S2 = Fsum(sa).fsum2_ 

705 for i in range(1, _TRIPS): 

706 ca2 = _1_0 - sa**2 

707 if ca2 < EPS_2: # PYCHOK no cover 

708 ca = _0_0 

709 break 

710 ca = sqrt(ca2) 

711 r = e / E.e2s2(sa) - R / ca2 

712 if fabs(r) < EPS_2: 

713 break 

714 lat = None 

715 sa, r = _S2(-z * ca / r, -d * sa / r) 

716 if fabs(r) < tol: 

717 break 

718 else: 

719 t = unstr(self.reverse, x=x, y=y, z=z) 

720 raise EcefError(Fmt.no_convergence(r, tol), txt=t) 

721 

722 if lat is None: 

723 lat = copysign0(atan1d(fabs(sa), ca), z) 

724 lon = self._polon(y, x, R, **name_lon00) 

725 

726 h = fsumf_(R * ca, fabs(z * sa), -E.a * E.e2s(sa)) # use Veness' 

727 # because Sudano's Eq (7) doesn't produce the correct height 

728 # h = (fabs(z) + R - E.a * cos(a + E.e21) * sa / ca) / (ca + sa) 

729 r = Ecef9Tuple(x, y, z, lat, lon, h, 

730 i, None, self.datum, # M=None 

731 iteration=i, name=name or self.name) 

732 return r 

733 

734 @property_doc_(''' the convergence tolerance (C{float}).''') 

735 def tolerance(self): 

736 '''Get the convergence tolerance (C{scalar}). 

737 ''' 

738 return self._tol 

739 

740 @tolerance.setter # PYCHOK setter! 

741 def tolerance(self, tol): 

742 '''Set the convergence tolerance (C{scalar}). 

743 

744 @raise EcefError: Non-scalar or invalid B{C{tol}}. 

745 ''' 

746 self._tol = Scalar_(tolerance=tol, low=EPS, Error=EcefError) 

747 

748 

749class EcefVeness(_EcefBase): 

750 '''Conversion between geodetic and geocentric, I{Earth-Centered, Earth-Fixed} (ECEF) coordinates 

751 transcoded from I{Chris Veness}' JavaScript classes U{LatLonEllipsoidal, Cartesian<https:// 

752 www.Movable-Type.co.UK/scripts/geodesy/docs/latlon-ellipsoidal.js.html>}. 

753 

754 @see: U{I{A Guide to Coordinate Systems in Great Britain}<https://www.OrdnanceSurvey.co.UK/ 

755 documents/resources/guide-coordinate-systems-great-britain.pdf>}, section I{B) Converting 

756 between 3D Cartesian and ellipsoidal latitude, longitude and height coordinates}. 

757 ''' 

758 

759 def reverse(self, xyz, y=None, z=None, M=None, **name_lon00): # PYCHOK unused M 

760 '''Conversion from geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)} 

761 transcoded from I{Chris Veness}' U{JavaScript<https://www.Movable-Type.co.UK/ 

762 scripts/geodesy/docs/latlon-ellipsoidal.js.html>}. 

763 

764 Uses B. R. Bowring’s formulation for μm precision in concise form U{I{The accuracy 

765 of geodetic latitude and height equations}<https://www.ResearchGate.net/publication/ 

766 233668213>}, Survey Review, Vol 28, 218, Oct 1985. 

767 

768 @arg xyz: A geocentric (C{Cartesian}, L{Ecef9Tuple}) or C{scalar} ECEF C{x} 

769 coordinate (C{meter}). 

770 @kwarg y: ECEF C{y} coordinate for C{scalar} B{C{xyz}} and B{C{z}} (C{meter}). 

771 @kwarg z: ECEF C{z} coordinate for C{scalar} B{C{xyz}} and B{C{y}} (C{meter}). 

772 @kwarg M: I{Ignored}, rotation matrix C{M} not available. 

773 @kwarg name_lon00: Optional keyword arguments C{B{name}=NN} (C{str}) and 

774 I{"polar"} longitude C{B{lon00}=INT0} (C{degrees}), overriding 

775 the default and property C{lon00} setting and returned in case 

776 C{B{x}=0} and C{B{y}=0}. 

777 

778 @return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with 

779 geodetic coordinates C{(lat, lon, height)} for the given geocentric 

780 ones C{(x, y, z)}, case C{C}, C{M=None} always and C{datum} if available. 

781 

782 @raise EcefError: Invalid B{C{xyz}} or C{scalar} C{x} or B{C{y}} and/or B{C{z}} 

783 not C{scalar} for C{scalar} B{C{xyz}}. 

784 

785 @see: Toms, Ralph M. U{I{An Efficient Algorithm for Geocentric to Geodetic 

786 Coordinate Conversion}<https://www.OSTI.gov/scitech/biblio/110235>}, 

787 Sept 1995 and U{I{An Improved Algorithm for Geocentric to Geodetic 

788 Coordinate Conversion}<https://www.OSTI.gov/scitech/servlets/purl/231228>}, 

789 Apr 1996, both from Lawrence Livermore National Laboratory (LLNL) and 

790 Sudano, John J, U{I{An exact conversion from an Earth-centered coordinate 

791 system to latitude longitude and altitude}<https://www.ResearchGate.net/ 

792 publication/3709199>}. 

793 ''' 

794 x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, **name_lon00) 

795 

796 E = self.ellipsoid 

797 

798 p = hypot(x, y) # distance from minor axis 

799 r = hypot(p, z) # polar radius 

800 if min(p, r) > EPS0: 

801 b = E.b * E.e22 

802 # parametric latitude (Bowring eqn 17, replaced) 

803 t = (E.b * z) / (E.a * p) * (_1_0 + b / r) 

804 c = _1_0 / hypot1(t) 

805 s = c * t 

806 

807 # geodetic latitude (Bowring eqn 18) 

808 lat = atan1d(z + b * s**3, 

809 p - E.e2 * E.a * c**3) 

810 

811 # height above ellipsoid (Bowring eqn 7) 

812 sa, ca = sincos2d(lat) 

813# r = E.a / E.e2s(sa) # length of normal terminated by minor axis 

814# h = p * ca + z * sa - (E.a * E.a / r) 

815 h = fsumf_(p * ca, z * sa, -E.a * E.e2s(sa)) 

816 C = 1 

817 

818 # see <https://GIS.StackExchange.com/questions/28446> 

819 elif p > EPS: # lat arbitrarily zero, equatorial lon 

820 C, lat, h = 2, _0_0, (p - E.a) 

821 

822 else: # polar lat, lon arbitrarily lon00 

823 C, lat, h = 3, (_N_90_0 if z < 0 else _90_0), (fabs(z) - E.b) 

824 

825 lon = self._polon(y, x, p, **name_lon00) 

826 return Ecef9Tuple(x, y, z, lat, lon, h, 

827 C, None, self.datum, # M=None 

828 name=name or self.name) 

829 

830 

831class EcefYou(_EcefBase): 

832 '''Conversion between geodetic and geocentric, I{Earth-Centered, Earth-Fixed} (ECEF) coordinates 

833 using I{Rey-Jer You}'s U{transformation<https://www.ResearchGate.net/publication/240359424>} 

834 for I{non-prolate} ellipsoids. 

835 

836 @see: Featherstone, W.E., Claessens, S.J. U{I{Closed-form transformation between geodetic and 

837 ellipsoidal coordinates}<https://Espace.Curtin.edu.AU/bitstream/handle/20.500.11937/ 

838 11589/115114_9021_geod2ellip_final.pdf>} Studia Geophysica et Geodaetica, 2008, 52, 

839 pages 1-18 and U{PyMap3D <https://PyPI.org/project/pymap3d>}. 

840 ''' 

841 

842 def __init__(self, a_ellipsoid=_EWGS84, f=None, **name_lon00): # PYCHOK signature 

843 _EcefBase.__init__(self, a_ellipsoid, f=f, **name_lon00) # inherited documentation 

844 _ = EcefYou._e2(self.ellipsoid) 

845 

846 @staticmethod 

847 def _e2(E): 

848 e2 = E.a2 - E.b2 

849 if E.f < 0 or e2 < 0: 

850 raise EcefError(ellipsoid=E, txt=_prolate_) 

851 return e2 

852 

853 def reverse(self, xyz, y=None, z=None, M=None, **name_lon00): # PYCHOK unused M 

854 '''Convert geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)} 

855 using I{Rey-Jer You}'s transformation. 

856 

857 @arg xyz: A geocentric (C{Cartesian}, L{Ecef9Tuple}) or C{scalar} ECEF C{x} 

858 coordinate (C{meter}). 

859 @kwarg y: ECEF C{y} coordinate for C{scalar} B{C{xyz}} and B{C{z}} (C{meter}). 

860 @kwarg z: ECEF C{z} coordinate for C{scalar} B{C{xyz}} and B{C{y}} (C{meter}). 

861 @kwarg M: I{Ignored}, rotation matrix C{M} not available. 

862 @kwarg name_lon00: Optional keyword arguments C{B{name}=NN} (C{str}) and 

863 I{"polar"} longitude C{B{lon00}=INT0} (C{degrees}), overriding 

864 the default and property C{lon00} setting and returned in case 

865 C{B{x}=0} and C{B{y}=0}. 

866 

867 @return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with 

868 geodetic coordinates C{(lat, lon, height)} for the given geocentric 

869 ones C{(x, y, z)}, case C{C=1}, C{M=None} always and C{datum} if 

870 available. 

871 

872 @raise EcefError: Invalid B{C{xyz}} or C{scalar} C{x} or B{C{y}} and/or 

873 B{C{z}} not C{scalar} for C{scalar} B{C{xyz}} or the 

874 ellipsoid is I{prolate}. 

875 ''' 

876 x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, **name_lon00) 

877 

878 E = self.ellipsoid 

879 e2 = EcefYou._e2(E) 

880 e = sqrt(e2) if e2 > 0 else _0_0 # XXX sqrt0(e2)? 

881 

882 q = hypot( x, y) # R 

883 r2 = hypot2_(x, y, z) 

884 u = fsumf_(r2, -e2, hypot(r2 - e2, e * z * _2_0)) * _0_5 

885 if u > EPS02: 

886 u = sqrt(u) 

887 p = hypot(u, e) 

888 B = atan1(p * z, u * q) # beta0 = atan(p / u * z / q) 

889 sB, cB = sincos2(B) 

890 if cB and sB: 

891 p *= E.a 

892 d = (p / cB - e2 * cB) / sB 

893 if isnon0(d): 

894 B += fsumf_(u * E.b, -p, e2) / d 

895 sB, cB = sincos2(B) 

896 elif u < 0: 

897 raise EcefError(x=x, y=y, z=z, txt=_singular_) 

898 else: 

899 sB, cB = _copysign_1_0(z), _0_0 

900 

901 lat = atan1d(E.a * sB, E.b * cB) # atan(E.a_b * tan(B)) 

902 lon = self._polon(y, x, q, **name_lon00) 

903 

904 h = hypot(z - E.b * sB, q - E.a * cB) 

905 if hypot2_(x, y, z * E.a_b) < E.a2: 

906 h = neg(h) # inside ellipsoid 

907 return Ecef9Tuple(x, y, z, lat, lon, h, 

908 1, None, self.datum, # C=1, M=None 

909 name=name or self.name) 

910 

911 

912class EcefMatrix(_NamedTuple): 

913 '''A rotation matrix known as I{East-North-Up (ENU) to ECEF}. 

914 

915 @see: U{From ENU to ECEF<https://WikiPedia.org/wiki/ 

916 Geographic_coordinate_conversion#From_ECEF_to_ENU>} and 

917 U{Issue #74<https://Github.com/mrJean1/PyGeodesy/issues/74>}. 

918 ''' 

919 _Names_ = ('_0_0_', '_0_1_', '_0_2_', # row-order 

920 '_1_0_', '_1_1_', '_1_2_', 

921 '_2_0_', '_2_1_', '_2_2_') 

922 _Units_ = (Scalar,) * len(_Names_) 

923 

924 def _validate(self, **_OK): # PYCHOK unused 

925 '''(INTERNAL) Allow C{_Names_} with leading underscore. 

926 ''' 

927 _NamedTuple._validate(self, _OK=True) 

928 

929 def __new__(cls, sa, ca, sb, cb, *_more): 

930 '''New L{EcefMatrix} matrix. 

931 

932 @arg sa: C{sin(phi)} (C{float}). 

933 @arg ca: C{cos(phi)} (C{float}). 

934 @arg sb: C{sin(lambda)} (C{float}). 

935 @arg cb: C{cos(lambda)} (C{float}). 

936 @arg _more: (INTERNAL) from C{.multiply}. 

937 

938 @raise EcefError: If B{C{sa}}, B{C{ca}}, B{C{sb}} or 

939 B{C{cb}} outside M{[-1.0, +1.0]}. 

940 ''' 

941 t = sa, ca, sb, cb 

942 if _more: # all 9 matrix elements ... 

943 t += _more # ... from .multiply 

944 

945 elif max(map(fabs, t)) > _1_0: 

946 raise EcefError(unstr(EcefMatrix.__name__, *t)) 

947 

948 else: # build matrix from the following quaternion operations 

949 # qrot(lam, [0,0,1]) * qrot(phi, [0,-1,0]) * [1,1,1,1]/2 

950 # or 

951 # qrot(pi/2 + lam, [0,0,1]) * qrot(-pi/2 + phi, [-1,0,0]) 

952 # where 

953 # qrot(t,v) = [cos(t/2), sin(t/2)*v[1], sin(t/2)*v[2], sin(t/2)*v[3]] 

954 

955 # Local X axis (East) in geocentric coords 

956 # M[0] = -slam; M[3] = clam; M[6] = 0; 

957 # Local Y axis (North) in geocentric coords 

958 # M[1] = -clam * sphi; M[4] = -slam * sphi; M[7] = cphi; 

959 # Local Z axis (Up) in geocentric coords 

960 # M[2] = clam * cphi; M[5] = slam * cphi; M[8] = sphi; 

961 t = (-sb, -cb * sa, cb * ca, 

962 cb, -sb * sa, sb * ca, 

963 _0_0, ca, sa) 

964 

965 return _NamedTuple.__new__(cls, *t) 

966 

967 def column(self, column): 

968 '''Get this matrix' B{C{column}} 0, 1 or 2 as C{3-tuple}. 

969 ''' 

970 if 0 <= column < 3: 

971 return self[column::3] 

972 raise _IndexError(column=column) 

973 

974 def copy(self, **unused): # PYCHOK signature 

975 '''Make a shallow or deep copy of this instance. 

976 

977 @return: The copy (C{This class} or subclass thereof). 

978 ''' 

979 return self.classof(*self) 

980 

981 __copy__ = __deepcopy__ = copy 

982 

983 @Property_RO 

984 def matrix3(self): 

985 '''Get this matrix' rows (C{3-tuple} of 3 C{3-tuple}s). 

986 ''' 

987 return tuple(map(self.row, range(3))) 

988 

989 @Property_RO 

990 def matrixTransposed3(self): 

991 '''Get this matrix' I{Transposed} rows (C{3-tuple} of 3 C{3-tuple}s). 

992 ''' 

993 return tuple(map(self.column, range(3))) 

994 

995 def multiply(self, other): 

996 '''Matrix multiply M{M0' ⋅ M} this matrix I{Transposed} 

997 with an other matrix. 

998 

999 @arg other: The other matrix (L{EcefMatrix}). 

1000 

1001 @return: The matrix product (L{EcefMatrix}). 

1002 

1003 @raise TypeError: If B{C{other}} is not L{EcefMatrix}. 

1004 ''' 

1005 _xinstanceof(EcefMatrix, other=other) 

1006 # like LocalCartesian.MatrixMultiply, C{self.matrixTransposed3 X other.matrix3} 

1007 # <https://GeographicLib.SourceForge.io/C++/doc/LocalCartesian_8cpp_source.html> 

1008 # X = (fdot(self.column(r), *other.column(c)) for r in (0,1,2) for c in (0,1,2)) 

1009 X = (fdot(self[r::3], *other[c::3]) for r in range(3) for c in range(3)) 

1010 return _xnamed(EcefMatrix(*X), EcefMatrix.multiply.__name__) 

1011 

1012 def rotate(self, xyz, *xyz0): 

1013 '''Forward rotation M{M0' ⋅ ([x, y, z] - [x0, y0, z0])'}. 

1014 

1015 @arg xyz: Local C{(x, y, z)} coordinates (C{3-tuple}). 

1016 @arg xyz0: Optional, local C{(x0, y0, z0)} origin (C{3-tuple}). 

1017 

1018 @return: Rotated C{(x, y, z)} location (C{3-tuple}). 

1019 

1020 @raise LenError: Unequal C{len(B{xyz})} and C{len(B{xyz0})}. 

1021 ''' 

1022 if xyz0: 

1023 if len(xyz0) != len(xyz): 

1024 raise LenError(self.rotate, xyz0=len(xyz0), xyz=len(xyz)) 

1025 xyz = tuple(c - c0 for c, c0 in zip(xyz, xyz0)) 

1026 

1027 # x' = M[0] * x + M[3] * y + M[6] * z 

1028 # y' = M[1] * x + M[4] * y + M[7] * z 

1029 # z' = M[2] * x + M[5] * y + M[8] * z 

1030 return (fdot(xyz, *self[0::3]), # .column(0) 

1031 fdot(xyz, *self[1::3]), # .column(1) 

1032 fdot(xyz, *self[2::3])) # .column(2) 

1033 

1034 def row(self, row): 

1035 '''Get this matrix' B{C{row}} 0, 1 or 2 as C{3-tuple}. 

1036 ''' 

1037 if 0 <= row < 3: 

1038 r = row * 3 

1039 return self[r:r+3] 

1040 raise _IndexError(row=row) 

1041 

1042 def unrotate(self, xyz, *xyz0): 

1043 '''Inverse rotation M{[x0, y0, z0] + M0 ⋅ [x,y,z]'}. 

1044 

1045 @arg xyz: Local C{(x, y, z)} coordinates (C{3-tuple}). 

1046 @arg xyz0: Optional, local C{(x0, y0, z0)} origin (C{3-tuple}). 

1047 

1048 @return: Unrotated C{(x, y, z)} location (C{3-tuple}). 

1049 

1050 @raise LenError: Unequal C{len(B{xyz})} and C{len(B{xyz0})}. 

1051 ''' 

1052 if xyz0: 

1053 if len(xyz0) != len(xyz): 

1054 raise LenError(self.unrotate, xyz0=len(xyz0), xyz=len(xyz)) 

1055 _xyz = _1_0_1T + xyz 

1056 # x' = x0 + M[0] * x + M[1] * y + M[2] * z 

1057 # y' = y0 + M[3] * x + M[4] * y + M[5] * z 

1058 # z' = z0 + M[6] * x + M[7] * y + M[8] * z 

1059 xyz_ = (fdot(_xyz, xyz0[0], *self[0:3]), # .row(0) 

1060 fdot(_xyz, xyz0[1], *self[3:6]), # .row(1) 

1061 fdot(_xyz, xyz0[2], *self[6:9])) # .row(2) 

1062 else: 

1063 # x' = M[0] * x + M[1] * y + M[2] * z 

1064 # y' = M[3] * x + M[4] * y + M[5] * z 

1065 # z' = M[6] * x + M[7] * y + M[8] * z 

1066 xyz_ = (fdot(xyz, *self[0:3]), # .row(0) 

1067 fdot(xyz, *self[3:6]), # .row(1) 

1068 fdot(xyz, *self[6:9])) # .row(2) 

1069 return xyz_ 

1070 

1071 

1072class Ecef9Tuple(_NamedTuple): 

1073 '''9-Tuple C{(x, y, z, lat, lon, height, C, M, datum)} with I{geocentric} 

1074 C{x}, C{y} and C{z} plus I{geodetic} C{lat}, C{lon} and C{height}, case 

1075 C{C} (see the C{Ecef*.reverse} methods) and optionally, the rotation 

1076 matrix C{M} (L{EcefMatrix}) and C{datum}, with C{lat} and C{lon} in 

1077 C{degrees} and C{x}, C{y}, C{z} and C{height} in C{meter}, conventionally. 

1078 ''' 

1079 _Names_ = (_x_, _y_, _z_, _lat_, _lon_, _height_, _C_, _M_, _datum_) 

1080 _Units_ = ( Meter, Meter, Meter, Lat, Lon, Height, Int, _Pass, _Pass) 

1081 

1082 @property_RO 

1083 def _CartesianBase(self): 

1084 '''(INTERNAL) Get class C{CartesianBase}, I{once}. 

1085 ''' 

1086 Ecef9Tuple._CartesianBase = C = _MODS.cartesianBase.CartesianBase # overwrite property_RO 

1087 return C 

1088 

1089 @deprecated_method 

1090 def convertDatum(self, datum2): # for backward compatibility 

1091 '''DEPRECATED, use method L{toDatum}.''' 

1092 return self.toDatum(datum2) 

1093 

1094 @Property_RO 

1095 def lam(self): 

1096 '''Get the longitude in C{radians} (C{float}). 

1097 ''' 

1098 return self.philam.lam 

1099 

1100 @Property_RO 

1101 def lamVermeille(self): 

1102 '''Get the longitude in C{radians} M{[-PI*3/2..+PI*3/2]} after U{Vermeille 

1103 <https://Search.ProQuest.com/docview/639493848>} (2004), page 95. 

1104 

1105 @see: U{Karney<https://GeographicLib.SourceForge.io/C++/doc/geocentric.html>}, 

1106 U{Vermeille<https://Search.ProQuest.com/docview/847292978>} 2011, pp 112-113, 116 

1107 and U{Featherstone, et.al.<https://Search.ProQuest.com/docview/872827242>}, page 7. 

1108 ''' 

1109 x, y = self.x, self.y 

1110 if y > EPS0: 

1111 r = atan2(x, hypot(y, x) + y) * _N_2_0 + PI_2 

1112 elif y < -EPS0: 

1113 r = atan2(x, hypot(y, x) - y) * _2_0 - PI_2 

1114 else: # y == 0 

1115 r = PI if x < 0 else _0_0 

1116 return Lam(Vermeille=r) 

1117 

1118 @Property_RO 

1119 def latlon(self): 

1120 '''Get the lat-, longitude in C{degrees} (L{LatLon2Tuple}C{(lat, lon)}). 

1121 ''' 

1122 return LatLon2Tuple(self.lat, self.lon, name=self.name) 

1123 

1124 @Property_RO 

1125 def latlonheight(self): 

1126 '''Get the lat-, longitude in C{degrees} and height (L{LatLon3Tuple}C{(lat, lon, height)}). 

1127 ''' 

1128 return self.latlon.to3Tuple(self.height) 

1129 

1130 @Property_RO 

1131 def latlonheightdatum(self): 

1132 '''Get the lat-, longitude in C{degrees} with height and datum (L{LatLon4Tuple}C{(lat, lon, height, datum)}). 

1133 ''' 

1134 return self.latlonheight.to4Tuple(self.datum) 

1135 

1136 @Property_RO 

1137 def latlonVermeille(self): 

1138 '''Get the latitude and I{Vermeille} longitude in C{degrees [-225..+225]} (L{LatLon2Tuple}C{(lat, lon)}). 

1139 

1140 @see: Property C{lonVermeille}. 

1141 ''' 

1142 return LatLon2Tuple(self.lat, self.lonVermeille, name=self.name) 

1143 

1144 @Property_RO 

1145 def lonVermeille(self): 

1146 '''Get the longitude in C{degrees [-225..+225]} after U{Vermeille 

1147 <https://Search.ProQuest.com/docview/639493848>} (2004), p 95. 

1148 

1149 @see: Property C{lamVermeille}. 

1150 ''' 

1151 return Lon(Vermeille=degrees(self.lamVermeille)) 

1152 

1153 @Property_RO 

1154 def phi(self): 

1155 '''Get the latitude in C{radians} (C{float}). 

1156 ''' 

1157 return self.philam.phi 

1158 

1159 @Property_RO 

1160 def philam(self): 

1161 '''Get the lat-, longitude in C{radians} (L{PhiLam2Tuple}C{(phi, lam)}). 

1162 ''' 

1163 return PhiLam2Tuple(radians(self.lat), radians(self.lon), name=self.name) 

1164 

1165 @Property_RO 

1166 def philamheight(self): 

1167 '''Get the lat-, longitude in C{radians} and height (L{PhiLam3Tuple}C{(phi, lam, height)}). 

1168 ''' 

1169 return self.philam.to3Tuple(self.height) 

1170 

1171 @Property_RO 

1172 def philamheightdatum(self): 

1173 '''Get the lat-, longitude in C{radians} with height and datum (L{PhiLam4Tuple}C{(phi, lam, height, datum)}). 

1174 ''' 

1175 return self.philamheight.to4Tuple(self.datum) 

1176 

1177 @Property_RO 

1178 def philamVermeille(self): 

1179 '''Get the latitude and I{Vermeille} longitude in C{radians [-PI*3/2..+PI*3/2]} (L{PhiLam2Tuple}C{(phi, lam)}). 

1180 

1181 @see: Property C{lamVermeille}. 

1182 ''' 

1183 return PhiLam2Tuple(radians(self.lat), self.lamVermeille, name=self.name) 

1184 

1185 def toCartesian(self, Cartesian=None, **Cartesian_kwds): 

1186 '''Return the geocentric C{(x, y, z)} coordinates as an ellipsoidal or spherical 

1187 C{Cartesian}. 

1188 

1189 @kwarg Cartesian: Optional class to return C{(x, y, z)} (L{ellipsoidalKarney.Cartesian}, 

1190 L{ellipsoidalNvector.Cartesian}, L{ellipsoidalVincenty.Cartesian}, 

1191 L{sphericalNvector.Cartesian} or L{sphericalTrigonometry.Cartesian}) 

1192 or C{None}. 

1193 @kwarg Cartesian_kwds: Optional, additional B{C{Cartesian}} keyword arguments, ignored 

1194 if C{B{Cartesian} is None}. 

1195 

1196 @return: A C{B{Cartesian}(x, y, z, **B{Cartesian_kwds})} instance or 

1197 a L{Vector4Tuple}C{(x, y, z, h)} if C{B{Cartesian} is None}. 

1198 

1199 @raise TypeError: Invalid B{C{Cartesian}} or B{C{Cartesian_kwds}}. 

1200 ''' 

1201 if Cartesian in (None, Vector4Tuple): 

1202 r = self.xyzh 

1203 elif Cartesian is Vector3Tuple: 

1204 r = self.xyz 

1205 else: 

1206 _xsubclassof(self._CartesianBase, Cartesian=Cartesian) 

1207 r = Cartesian(self, **_xkwds(Cartesian_kwds, name=self.name)) 

1208 return r 

1209 

1210 def toDatum(self, datum2): 

1211 '''Convert this C{Ecef9Tuple} to an other datum. 

1212 

1213 @arg datum2: Datum to convert I{to} (L{Datum}). 

1214 

1215 @return: The converted 9-Tuple (C{Ecef9Tuple}). 

1216 

1217 @raise TypeError: The B{C{datum2}} is not a L{Datum}. 

1218 ''' 

1219 if self.datum in (None, datum2): # PYCHOK _Names_ 

1220 r = self.copy() 

1221 else: 

1222 c = self._CartesianBase(self, datum=self.datum, name=self.name) # PYCHOK _Names_ 

1223 # c.toLatLon converts datum, x, y, z, lat, lon, etc. 

1224 # and returns another Ecef9Tuple iff LatLon is None 

1225 r = c.toLatLon(datum=datum2, LatLon=None) 

1226 return r 

1227 

1228 def toLatLon(self, LatLon=None, **LatLon_kwds): 

1229 '''Return the geodetic C{(lat, lon, height[, datum])} coordinates. 

1230 

1231 @kwarg LatLon: Optional class to return C{(lat, lon, height[, datum])} 

1232 or C{None}. 

1233 @kwarg LatLon_kwds: Optional B{C{height}}, B{C{datum}} and other 

1234 B{C{LatLon}} keyword arguments. 

1235 

1236 @return: An instance of C{B{LatLon}(lat, lon, **B{LatLon_kwds})} 

1237 or if B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat, lon, 

1238 height)} respectively L{LatLon4Tuple}C{(lat, lon, height, 

1239 datum)} depending on whether C{datum} is un-/specified. 

1240 

1241 @raise TypeError: Invalid B{C{LatLon}} or B{C{LatLon_kwds}}. 

1242 ''' 

1243 lat, lon, D = self.lat, self.lon, self.datum # PYCHOK Ecef9Tuple 

1244 kwds = _xkwds(LatLon_kwds, height=self.height, datum=D, name=self.name) # PYCHOK Ecef9Tuple 

1245 d = kwds.get(_datum_, LatLon) 

1246 if LatLon is None: 

1247 r = LatLon3Tuple(lat, lon, kwds[_height_], name=kwds[_name_]) 

1248 if d is not None: 

1249 # assert d is not LatLon 

1250 r = r.to4Tuple(d) # checks type(d) 

1251 else: 

1252 if d is None: 

1253 _ = kwds.pop(_datum_) # remove None datum 

1254 r = LatLon(lat, lon, **kwds) 

1255 _xdatum(_xattr(r, datum=D), D) 

1256 return r 

1257 

1258 def toLocal(self, ltp, Xyz=None, **Xyz_kwds): 

1259 '''Convert this geocentric to I{local} C{x}, C{y} and C{z}. 

1260 

1261 @kwarg ltp: The I{local tangent plane} (LTP) to use (L{Ltp}). 

1262 @kwarg Xyz: Optional class to return C{x}, C{y} and C{z} 

1263 (L{XyzLocal}, L{Enu}, L{Ned}) or C{None}. 

1264 @kwarg Xyz_kwds: Optional, additional B{C{Xyz}} keyword 

1265 arguments, ignored if C{B{Xyz} is None}. 

1266 

1267 @return: An B{C{Xyz}} instance or if C{B{Xyz} is None}, 

1268 a L{Local9Tuple}C{(x, y, z, lat, lon, height, 

1269 ltp, ecef, M)} with C{M=None}, always. 

1270 

1271 @raise TypeError: Invalid B{C{ltp}}. 

1272 ''' 

1273 return _MODS.ltp._xLtp(ltp)._ecef2local(self, Xyz, Xyz_kwds) 

1274 

1275 def toVector(self, Vector=None, **Vector_kwds): 

1276 '''Return the geocentric C{(x, y, z)} coordinates as vector. 

1277 

1278 @kwarg Vector: Optional vector class to return C{(x, y, z)} or 

1279 C{None}. 

1280 @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword 

1281 arguments, ignored if C{B{Vector} is None}. 

1282 

1283 @return: A C{Vector}C{(x, y, z, **Vector_kwds)} instance or a 

1284 L{Vector3Tuple}C{(x, y, z)} if B{C{Vector}} is C{None}. 

1285 

1286 @see: Propertes C{xyz} and C{xyzh} 

1287 ''' 

1288 return self.xyz if Vector is None else self._xnamed( 

1289 Vector(*self.xyz, **Vector_kwds)) # PYCHOK Ecef9Tuple 

1290 

1291# def _T_x_M(self, T): 

1292# '''(INTERNAL) Update M{self.M = T.multiply(self.M)}. 

1293# ''' 

1294# return self.dup(M=T.multiply(self.M)) 

1295 

1296 @Property_RO 

1297 def xyz(self): 

1298 '''Get the geocentric C{(x, y, z)} coordinates (L{Vector3Tuple}C{(x, y, z)}). 

1299 ''' 

1300 return Vector3Tuple(self.x, self.y, self.z, name=self.name) 

1301 

1302 @Property_RO 

1303 def xyzh(self): 

1304 '''Get the geocentric C{(x, y, z)} coordinates and C{height} (L{Vector4Tuple}C{(x, y, z, h)}) 

1305 ''' 

1306 return self.xyz.to4Tuple(self.height) 

1307 

1308 

1309def _4Ecef(this, Ecef): # in .datums.Datum.ecef, .ellipsoids.Ellipsoid.ecef 

1310 '''Return an ECEF converter for C{this} L{Datum} or L{Ellipsoid}. 

1311 ''' 

1312 if Ecef is None: 

1313 Ecef = EcefKarney 

1314 else: 

1315 _xinstanceof(*_Ecefs, Ecef=Ecef) 

1316 return Ecef(this, name=this.name) 

1317 

1318 

1319def _xEcef(Ecef): # PYCHOK .latlonBase.py 

1320 '''(INTERNAL) Validate B{C{Ecef}} I{class}. 

1321 ''' 

1322 if issubclassof(Ecef, _EcefBase): 

1323 return Ecef 

1324 raise _TypesError(_Ecef_, Ecef, *_Ecefs) 

1325 

1326 

1327_Ecefs = (EcefKarney, EcefSudano, EcefVeness, EcefYou, 

1328 EcefFarrell21, EcefFarrell22) 

1329 

1330__all__ += _ALL_DOCS(_EcefBase) 

1331 

1332# **) MIT License 

1333# 

1334# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved. 

1335# 

1336# Permission is hereby granted, free of charge, to any person obtaining a 

1337# copy of this software and associated documentation files (the "Software"), 

1338# to deal in the Software without restriction, including without limitation 

1339# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1340# and/or sell copies of the Software, and to permit persons to whom the 

1341# Software is furnished to do so, subject to the following conditions: 

1342# 

1343# The above copyright notice and this permission notice shall be included 

1344# in all copies or substantial portions of the Software. 

1345# 

1346# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1347# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1348# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1349# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1350# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1351# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1352# OTHER DEALINGS IN THE SOFTWARE.