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, isnon0 # PYCHOK used! 

66from pygeodesy.datums import a_f2Tuple, _ellipsoidal_datum, _WGS84 

67# from pygeodesy.ellipsoids import a_f2Tuple # 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_ 

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_ 

80from pygeodesy.streprs import Fmt, unstr 

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

82 Scalar, Scalar_ 

83from pygeodesy.utily import 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__ = '23.08.07' 

90 

91_Ecef_ = 'Ecef' 

92_EWGS84 = _WGS84.ellipsoid 

93_prolate_ = 'prolate' 

94_TRIPS = 17 # 8..9 sufficient, EcefSudano.reverse 

95_xyz_y_z = _xyz_, _y_, _z_ # _xargs_names(_xyzn4)[:3] 

96 

97 

98class EcefError(_ValueError): 

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

100 ''' 

101 pass 

102 

103 

104def _llhn4(latlonh, lon, height, suffix=NN, Error=EcefError, name=NN): # in .ltp.LocalCartesian.forward and -.reset 

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

106 ''' 

107 try: 

108 lat, lon = latlonh.lat, latlonh.lon 

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

110 n = _xattr(latlonh, name=NN) 

111 except AttributeError: 

112 lat, h, n = latlonh, height, NN 

113 

114 try: 

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

116 except (TypeError, ValueError) as x: 

117 t = _lat_, _lon_, _height_ 

118 if suffix: 

119 t = (_ + suffix for _ in t) 

120 d = dict(zip(t, (lat, lon, h))) 

121 raise Error(cause=x, **d) 

122 return llhn 

123 

124 

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

126def _xyzn4(xyz, y, z, Types, Error=EcefError, name=NN, # PYCHOK unused 

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

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

129 ''' 

130 try: 

131 try: 

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

133 if not isinstance(xyz, Types): 

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

135 except AttributeError: 

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

137 

138 except (TypeError, ValueError) as x: 

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

140 raise Error(cause=x, **d) 

141 return t 

142 

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

144 

145 

146class _EcefBase(_NamedBase): 

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

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

149 ''' 

150 _datum = _WGS84 

151 _E = _EWGS84 

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

153 

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

155 '''New C{Ecef*} converter. 

156 

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

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

159 equatorial radius (C{meter}). 

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

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

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

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

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

165 methods C{reverse}. 

166 

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

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

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

170 or B{C{f}} not less than 1.0. 

171 ''' 

172 try: 

173 E = a_ellipsoid 

174 if f is None: 

175 if E is _EWGS84 or E is _WGS84: 

176 raise AssertionError # "break" 

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

178 E = a_f2Tuple(E, f) 

179 else: 

180 raise ValueError # _invalid_ 

181 

182 d = _ellipsoidal_datum(E, name=name) 

183 E = d.ellipsoid 

184 if E.a < EPS or E.f > EPS1: 

185 raise ValueError # _invalid_ 

186 

187 self._datum = d 

188 self._E = E 

189 

190 except AssertionError: # "break" 

191 pass 

192 except (TypeError, ValueError) as x: 

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

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

195 

196 if name: 

197 self.name = name 

198 if lon00 is not INT0: 

199 self.lon00 = lon00 

200 

201 def __eq__(self, other): 

202 '''Compare this and an other Ecef. 

203 

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

205 

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

207 ''' 

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

209 other.ellipsoid == self.ellipsoid) 

210 

211 @Property_RO 

212 def equatoradius(self): 

213 '''Get the I{equatorial} radius, semi-axis (C{meter}). 

214 ''' 

215 return self.ellipsoid.a 

216 

217 a = equatorialRadius = equatoradius # Karney property 

218 

219 @Property_RO 

220 def datum(self): 

221 '''Get the datum (L{Datum}). 

222 ''' 

223 return self._datum 

224 

225 @Property_RO 

226 def ellipsoid(self): 

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

228 ''' 

229 return self._E 

230 

231 @Property_RO 

232 def flattening(self): # Karney property 

233 '''Get the I{flattening} (C{float}), M{(a - b) / a}, positive for 

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

235 ''' 

236 return self.ellipsoid.f 

237 

238 f = flattening 

239 

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

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

242 ''' 

243 if _philam: # lat, lon in radians 

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

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

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

247 else: 

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

249 

250 E = self.ellipsoid 

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

252 z = (h + n * E.e21) * sa 

253 x = (h + n) * ca 

254 

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

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

257 0, m, self.datum, 

258 name=name or self.name) 

259 

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

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

262 

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

264 latitude (C{degrees}). 

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

266 (C{degrees}). 

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

268 the surface of the ellipsoid. 

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

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

271 

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

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

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

275 and C{datum} if available. 

276 

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

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

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

280 

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

282 and avoid double angle conversions. 

283 ''' 

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

285 return self._forward(*llhn, M=M) 

286 

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

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

289 in I{radians}. 

290 

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

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

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

294 the surface of the ellipsoid. 

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

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

297 

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

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

300 C{degrees180(B{lam})}. 

301 

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

303 ''' 

304 try: # like function C{_llhn4} above 

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

306 except (TypeError, ValueError) as x: 

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

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

309 

310 @property_RO 

311 def _Geocentrics(self): 

312 '''(INTERNAL) Valid geocentric classes. 

313 ''' 

314 t = Ecef9Tuple, _MODS.cartesianBase.CartesianBase 

315 _EcefBase._Geocentrics = t # overwrite the property 

316 return t 

317 

318 @Property_RO 

319 def _isYou(self): 

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

321 ''' 

322 return isinstance(self, EcefYou) 

323 

324 @property 

325 def lon00(self): 

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

327 ''' 

328 return self._lon00 

329 

330 @lon00.setter # PYCHOK setter! 

331 def lon00(self, lon00): 

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

333 ''' 

334 self._lon00 = Degrees(lon00=lon00) 

335 

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

337 '''Creation a rotation matrix. 

338 

339 @arg sa: C{sin(phi)} (C{float}). 

340 @arg ca: C{cos(phi)} (C{float}). 

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

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

343 

344 @return: An L{EcefMatrix}. 

345 ''' 

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

347 

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

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

350 ''' 

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

352 

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

354 '''(INTERNAL) I{Must be overloaded}, see function C{notOverloaded}. 

355 ''' 

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

357 

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

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

360 

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

362 

363 @return: This C{Ecef*} (C{str}). 

364 ''' 

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

366 

367 

368class EcefFarrell21(_EcefBase): 

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

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

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

372 ''' 

373 

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

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

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

377 page 29. 

378 

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

380 coordinate (C{meter}). 

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

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

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

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

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

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

387 C{B{x}=0} and C{B{y}=0}. 

388 

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

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

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

392 if available. 

393 

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

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

396 zero division error. 

397 

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

399 ''' 

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

401 

402 E = self.ellipsoid 

403 a = E.a 

404 a2 = E.a2 

405 b2 = E.b2 

406 e2 = E.e2 

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

408 e4 = E.e4 

409 

410 try: # names as page 29 

411 z2 = z**2 

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

413 

414 p = hypot(x, y) 

415 p2 = p**2 

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

417 F = b2 * z2 * 54 

418 c = e4 * p2 * F / G**3 

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

420 P = F / (_3_0 * (fsumf_(_1_0, s, _1_0 / s) * G)**2) 

421 Q = sqrt(_1_0 + _2_0 * e4 * P) 

422 Q1 = Q + _1_0 

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

424 -P * ez / (Q * Q1), 

425 -P * p2 * _0_5)) 

426 r = p + e2 * r0 

427 v = b2 / (sqrt(r**2 + ez) * a) 

428 

429 h = hypot(r, z) * (_1_0 - v) 

430 lat = atan2d((e2_ * v + _1_0) * z, p) 

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

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

433 

434 except (ValueError, ZeroDivisionError) as e: 

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

436 

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

438 1, None, self.datum, 

439 name=name or self.name) 

440 

441 

442class EcefFarrell22(_EcefBase): 

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

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

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

446 ''' 

447 

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

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

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

451 page 30. 

452 

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

454 coordinate (C{meter}). 

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

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

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

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

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

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

461 C{B{x}=0} and C{B{y}=0}. 

462 

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

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

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

466 if available. 

467 

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

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

470 zero division error. 

471 

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

473 ''' 

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

475 

476 E = self.ellipsoid 

477 a = E.a 

478 b = E.b 

479 

480 try: # see EcefVeness.reverse 

481 p = hypot(x, y) 

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

483 

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

485 lat = atan2d(z + s**3 * b * E.e22, 

486 p - c**3 * a * E.e2) 

487 

488 s, c = sincos2d(lat) 

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

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

491 else: # polar 

492 h = fabs(z) - b 

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

494 

495 except (ValueError, ZeroDivisionError) as e: 

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

497 

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

499 1, None, self.datum, 

500 name=name or self.name) 

501 

502 

503class EcefKarney(_EcefBase): 

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

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

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

507 

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

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

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

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

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

513 the rotation matrix. 

514 ''' 

515 

516 @Property_RO 

517 def hmax(self): 

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

519 ''' 

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

521 

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

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

524 

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

526 coordinate (C{meter}). 

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

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

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

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

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

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

533 C{B{x}=0} and C{B{y}=0}. 

534 

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

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

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

538 C{datum} if available. 

539 

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

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

542 

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

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

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

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

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

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

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

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

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

552 ''' 

553 def _norm3(y, x): 

554 h = hypot(y, x) # EPS0, EPS_2 

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

556 

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

558 

559 E = self.ellipsoid 

560 f = E.f 

561 

562 sb, cb, R = _norm3(y, x) 

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

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

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

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

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

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

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

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

571 sa, ca, _ = _norm3(z * _0_5, R) 

572 C = 1 

573 

574 elif E.e4: # E.isEllipsoidal 

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

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

577 p = (R / E.a)**2 

578 q = (z / E.a)**2 * E.e21 

579 if f < 0: 

580 p, q = q, p 

581 r = fsumf_(p, q, -E.e4) 

582 e = E.e4 * q 

583 if e or r > 0: 

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

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

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

587 u = r = r / _6_0 

588 r2 = r**2 

589 r3 = r2 * r 

590 t3 = r3 + s 

591 d *= t3 + r3 

592 if d < 0: 

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

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

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

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

597 else: 

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

599 # minimizes loss of precision due to cancellation. The 

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

601 # in definition of u. 

602 if d > 0: 

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

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

605 t = cbrt(t3) # t = r * t 

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

607 u = fsumf_(u, t, r2 / t) 

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

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

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

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

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

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

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

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

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

617 if f < 0: 

618 k1 -= E.e2 

619 else: 

620 k2 += E.e2 

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

622 h *= k1 - E.e21 

623 C = 2 

624 

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

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

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

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

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

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

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

632 q = E.e4 - p 

633 if f < 0: 

634 p, q = q, p 

635 e = E.a 

636 else: 

637 e = E.b2_a 

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

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

640 sa = neg(sa) 

641 h *= neg(e / E.e2abs) 

642 C = 3 

643 

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

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

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

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

648 h -= E.a 

649 C = 4 

650 

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

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

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

654 return Ecef9Tuple(x, y, z, atan2d(sa, ca), lon, h, 

655 C, m, self.datum, 

656 name=name or self.name) 

657 

658 

659class EcefSudano(_EcefBase): 

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

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

662 ''' 

663 _tol = EPS2 

664 

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

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

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

668 

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

670 coordinate (C{meter}). 

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

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

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

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

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

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

677 C{B{x}=0} and C{B{y}=0}. 

678 

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

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

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

682 

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

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

685 ''' 

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

687 

688 E = self.ellipsoid 

689 e = E.e2 * E.a 

690 R = hypot(x, y) # Rh 

691 d = e - R 

692 

693 lat = atan2d(z, R * E.e21) 

694 sa, ca = sincos2d(fabs(lat)) 

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

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

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

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

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

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

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

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

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

704 tol = self.tolerance 

705 _S2 = Fsum(sa).fsum2_ 

706 for i in range(1, _TRIPS): 

707 ca2 = _1_0 - sa**2 

708 if ca2 < EPS_2: # PYCHOK no cover 

709 ca = _0_0 

710 break 

711 ca = sqrt(ca2) 

712 r = e / E.e2s2(sa) - R / ca2 

713 if fabs(r) < EPS_2: 

714 break 

715 lat = None 

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

717 if fabs(r) < tol: 

718 break 

719 else: 

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

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

722 

723 if lat is None: 

724 lat = copysign0(atan2d(fabs(sa), ca), z) 

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

726 

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

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

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

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

731 i, None, self.datum, # M=None 

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

733 return r 

734 

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

736 def tolerance(self): 

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

738 ''' 

739 return self._tol 

740 

741 @tolerance.setter # PYCHOK setter! 

742 def tolerance(self, tol): 

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

744 

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

746 ''' 

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

748 

749 

750class EcefVeness(_EcefBase): 

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

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

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

754 

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

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

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

758 ''' 

759 

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

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

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

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

764 

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

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

767 233668213>}, Survey Review, Vol 28, 218, Oct 1985. 

768 

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

770 coordinate (C{meter}). 

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

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

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

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

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

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

777 C{B{x}=0} and C{B{y}=0}. 

778 

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

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

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

782 

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

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

785 

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

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

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

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

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

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

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

793 publication/3709199>}. 

794 ''' 

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

796 

797 E = self.ellipsoid 

798 

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

800 r = hypot(p, z) # polar radius 

801 if min(p, r) > EPS0: 

802 b = E.b * E.e22 

803 # parametric latitude (Bowring eqn 17, replaced) 

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

805 c = _1_0 / hypot1(t) 

806 s = c * t 

807 

808 # geodetic latitude (Bowring eqn 18) 

809 lat = atan2d(z + b * s**3, 

810 p - E.e2 * E.a * c**3) 

811 

812 # height above ellipsoid (Bowring eqn 7) 

813 sa, ca = sincos2d(lat) 

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

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

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

817 C = 1 

818 

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

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

821 C, lat, h = 2, _0_0, (p - E.a) 

822 

823 else: # polar lat, lon arbitrarily lon00 

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

825 

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

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

828 C, None, self.datum, # M=None 

829 name=name or self.name) 

830 

831 

832class EcefYou(_EcefBase): 

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

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

835 for I{non-prolate} ellipsoids. 

836 

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

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

839 11589/115114_9021_geod2ellip_final.pdf>} Studia Geophysica et Geodaetica, 2008, 52, 

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

841 ''' 

842 

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

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

845 _ = EcefYou._e2(self.ellipsoid) 

846 

847 @staticmethod 

848 def _e2(E): 

849 e2 = E.a2 - E.b2 

850 if E.f < 0 or e2 < 0: 

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

852 return e2 

853 

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

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

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

857 

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

859 coordinate (C{meter}). 

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

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

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

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

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

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

866 C{B{x}=0} and C{B{y}=0}. 

867 

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

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

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

871 available. 

872 

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

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

875 ellipsoid is I{prolate}. 

876 ''' 

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

878 

879 E = self.ellipsoid 

880 e2 = EcefYou._e2(E) 

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

882 

883 q = hypot( x, y) # R 

884 r2 = hypot2_(x, y, z) 

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

886 if u > EPS02: 

887 u = sqrt(u) 

888 p = hypot(u, e) 

889 B = atan2(p * z, u * q) # beta0 = atan(p / u * z / q) 

890 sB, cB = sincos2(B) 

891 if cB and sB: 

892 p *= E.a 

893 d = (p / cB - e2 * cB) / sB 

894 if isnon0(d): 

895 B += fsumf_(u * E.b, -p, e2) / d 

896 sB, cB = sincos2(B) 

897 elif u < 0: 

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

899 else: 

900 sB, cB = (_N_1_0 if z < 0 else _1_0), _0_0 

901 

902 lat = atan2d(E.a * sB, E.b * cB) # atan(E.a_b * tan(B)) 

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

904 

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

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

907 h = neg(h) # inside ellipsoid 

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

909 1, None, self.datum, # C=1, M=None 

910 name=name or self.name) 

911 

912 

913class EcefMatrix(_NamedTuple): 

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

915 

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

917 Geographic_coordinate_conversion#From_ECEF_to_ENU>} and 

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

919 ''' 

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

921 '_1_0_', '_1_1_', '_1_2_', 

922 '_2_0_', '_2_1_', '_2_2_') 

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

924 

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

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

927 ''' 

928 _NamedTuple._validate(self, _OK=True) 

929 

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

931 '''New L{EcefMatrix} matrix. 

932 

933 @arg sa: C{sin(phi)} (C{float}). 

934 @arg ca: C{cos(phi)} (C{float}). 

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

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

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

938 

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

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

941 ''' 

942 t = sa, ca, sb, cb 

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

944 t += _more # ... from .multiply 

945 

946 elif max(map(fabs, t)) > _1_0: 

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

948 

949 else: # build matrix from the following quaternion operations 

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

951 # or 

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

953 # where 

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

955 

956 # Local X axis (East) in geocentric coords 

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

958 # Local Y axis (North) in geocentric coords 

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

960 # Local Z axis (Up) in geocentric coords 

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

962 t = (-sb, -cb * sa, cb * ca, 

963 cb, -sb * sa, sb * ca, 

964 _0_0, ca, sa) 

965 

966 return _NamedTuple.__new__(cls, *t) 

967 

968 def column(self, column): 

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

970 ''' 

971 if 0 <= column < 3: 

972 return self[column::3] 

973 raise _IndexError(column=column) 

974 

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

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

977 

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

979 ''' 

980 return self.classof(*self) 

981 

982 __copy__ = __deepcopy__ = copy 

983 

984 @Property_RO 

985 def matrix3(self): 

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

987 ''' 

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

989 

990 @Property_RO 

991 def matrixTransposed3(self): 

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

993 ''' 

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

995 

996 def multiply(self, other): 

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

998 with an other matrix. 

999 

1000 @arg other: The other matrix (L{EcefMatrix}). 

1001 

1002 @return: The matrix product (L{EcefMatrix}). 

1003 

1004 @raise TypeError: If B{C{other}} is not L{EcefMatrix}. 

1005 ''' 

1006 _xinstanceof(EcefMatrix, other=other) 

1007 # like LocalCartesian.MatrixMultiply, C{self.matrixTransposed3 X other.matrix3} 

1008 # <https://GeographicLib.SourceForge.io/C++/doc/LocalCartesian_8cpp_source.html> 

1009 # X = (fdot(self.column(r), *other.column(c)) for r in (0,1,2) for c in (0,1,2)) 

1010 X = (fdot(self[r::3], *other[c::3]) for r in range(3) for c in range(3)) 

1011 return _xnamed(EcefMatrix(*X), EcefMatrix.multiply.__name__) 

1012 

1013 def rotate(self, xyz, *xyz0): 

1014 '''Forward rotation M{M0' ⋅ ([x, y, z] - [x0, y0, z0])'}. 

1015 

1016 @arg xyz: Local C{(x, y, z)} coordinates (C{3-tuple}). 

1017 @arg xyz0: Optional, local C{(x0, y0, z0)} origin (C{3-tuple}). 

1018 

1019 @return: Rotated C{(x, y, z)} location (C{3-tuple}). 

1020 

1021 @raise LenError: Unequal C{len(B{xyz})} and C{len(B{xyz0})}. 

1022 ''' 

1023 if xyz0: 

1024 if len(xyz0) != len(xyz): 

1025 raise LenError(self.rotate, xyz0=len(xyz0), xyz=len(xyz)) 

1026 xyz = tuple(c - c0 for c, c0 in zip(xyz, xyz0)) 

1027 

1028 # x' = M[0] * x + M[3] * y + M[6] * z 

1029 # y' = M[1] * x + M[4] * y + M[7] * z 

1030 # z' = M[2] * x + M[5] * y + M[8] * z 

1031 return (fdot(xyz, *self[0::3]), # .column(0) 

1032 fdot(xyz, *self[1::3]), # .column(1) 

1033 fdot(xyz, *self[2::3])) # .column(2) 

1034 

1035 def row(self, row): 

1036 '''Get this matrix' B{C{row}} 0, 1 or 2 as C{3-tuple}. 

1037 ''' 

1038 if 0 <= row < 3: 

1039 r = row * 3 

1040 return self[r:r+3] 

1041 raise _IndexError(row=row) 

1042 

1043 def unrotate(self, xyz, *xyz0): 

1044 '''Inverse rotation M{[x0, y0, z0] + M0 ⋅ [x,y,z]'}. 

1045 

1046 @arg xyz: Local C{(x, y, z)} coordinates (C{3-tuple}). 

1047 @arg xyz0: Optional, local C{(x0, y0, z0)} origin (C{3-tuple}). 

1048 

1049 @return: Unrotated C{(x, y, z)} location (C{3-tuple}). 

1050 

1051 @raise LenError: Unequal C{len(B{xyz})} and C{len(B{xyz0})}. 

1052 ''' 

1053 if xyz0: 

1054 if len(xyz0) != len(xyz): 

1055 raise LenError(self.unrotate, xyz0=len(xyz0), xyz=len(xyz)) 

1056 _xyz = _1_0_1T + xyz 

1057 # x' = x0 + M[0] * x + M[1] * y + M[2] * z 

1058 # y' = y0 + M[3] * x + M[4] * y + M[5] * z 

1059 # z' = z0 + M[6] * x + M[7] * y + M[8] * z 

1060 xyz_ = (fdot(_xyz, xyz0[0], *self[0:3]), # .row(0) 

1061 fdot(_xyz, xyz0[1], *self[3:6]), # .row(1) 

1062 fdot(_xyz, xyz0[2], *self[6:9])) # .row(2) 

1063 else: 

1064 # x' = M[0] * x + M[1] * y + M[2] * z 

1065 # y' = M[3] * x + M[4] * y + M[5] * z 

1066 # z' = M[6] * x + M[7] * y + M[8] * z 

1067 xyz_ = (fdot(xyz, *self[0:3]), # .row(0) 

1068 fdot(xyz, *self[3:6]), # .row(1) 

1069 fdot(xyz, *self[6:9])) # .row(2) 

1070 return xyz_ 

1071 

1072 

1073class Ecef9Tuple(_NamedTuple): 

1074 '''9-Tuple C{(x, y, z, lat, lon, height, C, M, datum)} with I{geocentric} 

1075 C{x}, C{y} and C{z} plus I{geodetic} C{lat}, C{lon} and C{height}, case 

1076 C{C} (see the C{Ecef*.reverse} methods) and optionally, the rotation 

1077 matrix C{M} (L{EcefMatrix}) and C{datum}, with C{lat} and C{lon} in 

1078 C{degrees} and C{x}, C{y}, C{z} and C{height} in C{meter}, conventionally. 

1079 ''' 

1080 _Names_ = (_x_, _y_, _z_, _lat_, _lon_, _height_, _C_, _M_, _datum_) 

1081 _Units_ = ( Meter, Meter, Meter, Lat, Lon, Height, Int, _Pass, _Pass) 

1082 

1083 @property_RO 

1084 def _CartesianBase(self): 

1085 '''(INTERNAL) Get/cache class C{CartesianBase}. 

1086 ''' 

1087 Ecef9Tuple._CartesianBase = C = _MODS.cartesianBase.CartesianBase # overwrite property 

1088 return C 

1089 

1090 @deprecated_method 

1091 def convertDatum(self, datum2): # for backward compatibility 

1092 '''DEPRECATED, use method L{toDatum}.''' 

1093 return self.toDatum(datum2) 

1094 

1095 @Property_RO 

1096 def lam(self): 

1097 '''Get the longitude in C{radians} (C{float}). 

1098 ''' 

1099 return self.philam.lam 

1100 

1101 @Property_RO 

1102 def lamVermeille(self): 

1103 '''Get the longitude in C{radians [-PI*3/2..+PI*3/2]} after U{Vermeille 

1104 <https://Search.ProQuest.com/docview/639493848>} (2004), page 95. 

1105 

1106 @see: U{Karney<https://GeographicLib.SourceForge.io/C++/doc/geocentric.html>}, 

1107 U{Vermeille<https://Search.ProQuest.com/docview/847292978>} 2011, pp 112-113, 116 

1108 and U{Featherstone, et.al.<https://Search.ProQuest.com/docview/872827242>}, page 7. 

1109 ''' 

1110 x, y = self.x, self.y 

1111 if y > EPS0: 

1112 r = atan2(x, hypot(y, x) + y) * _N_2_0 + PI_2 

1113 elif y < -EPS0: 

1114 r = atan2(x, hypot(y, x) - y) * _2_0 - PI_2 

1115 else: # y == 0 

1116 r = PI if x < 0 else _0_0 

1117 return Lam(Vermeille=r) 

1118 

1119 @Property_RO 

1120 def latlon(self): 

1121 '''Get the lat-, longitude in C{degrees} (L{LatLon2Tuple}C{(lat, lon)}). 

1122 ''' 

1123 return LatLon2Tuple(self.lat, self.lon, name=self.name) 

1124 

1125 @Property_RO 

1126 def latlonheight(self): 

1127 '''Get the lat-, longitude in C{degrees} and height (L{LatLon3Tuple}C{(lat, lon, height)}). 

1128 ''' 

1129 return self.latlon.to3Tuple(self.height) 

1130 

1131 @Property_RO 

1132 def latlonheightdatum(self): 

1133 '''Get the lat-, longitude in C{degrees} with height and datum (L{LatLon4Tuple}C{(lat, lon, height, datum)}). 

1134 ''' 

1135 return self.latlonheight.to4Tuple(self.datum) 

1136 

1137 @Property_RO 

1138 def latlonVermeille(self): 

1139 '''Get the latitude and I{Vermeille} longitude in C{degrees [-225..+225]} (L{LatLon2Tuple}C{(lat, lon)}). 

1140 

1141 @see: Property C{lonVermeille}. 

1142 ''' 

1143 return LatLon2Tuple(self.lat, self.lonVermeille, name=self.name) 

1144 

1145 @Property_RO 

1146 def lonVermeille(self): 

1147 '''Get the longitude in C{degrees [-225..+225]} after U{Vermeille 

1148 <https://Search.ProQuest.com/docview/639493848>} (2004), p 95. 

1149 

1150 @see: Property C{lamVermeille}. 

1151 ''' 

1152 return Lon(Vermeille=degrees(self.lamVermeille)) 

1153 

1154 @Property_RO 

1155 def phi(self): 

1156 '''Get the latitude in C{radians} (C{float}). 

1157 ''' 

1158 return self.philam.phi 

1159 

1160 @Property_RO 

1161 def philam(self): 

1162 '''Get the lat-, longitude in C{radians} (L{PhiLam2Tuple}C{(phi, lam)}). 

1163 ''' 

1164 return PhiLam2Tuple(radians(self.lat), radians(self.lon), name=self.name) 

1165 

1166 @Property_RO 

1167 def philamheight(self): 

1168 '''Get the lat-, longitude in C{radians} and height (L{PhiLam3Tuple}C{(phi, lam, height)}). 

1169 ''' 

1170 return self.philam.to3Tuple(self.height) 

1171 

1172 @Property_RO 

1173 def philamheightdatum(self): 

1174 '''Get the lat-, longitude in C{radians} with height and datum (L{PhiLam4Tuple}C{(phi, lam, height, datum)}). 

1175 ''' 

1176 return self.philamheight.to4Tuple(self.datum) 

1177 

1178 @Property_RO 

1179 def philamVermeille(self): 

1180 '''Get the latitude and I{Vermeille} longitude in C{radians [-PI*3/2..+PI*3/2]} (L{PhiLam2Tuple}C{(phi, lam)}). 

1181 

1182 @see: Property C{lamVermeille}. 

1183 ''' 

1184 return PhiLam2Tuple(radians(self.lat), self.lamVermeille, name=self.name) 

1185 

1186 def toCartesian(self, Cartesian=None, **Cartesian_kwds): 

1187 '''Return the geocentric C{(x, y, z)} coordinates as an ellipsoidal or spherical 

1188 C{Cartesian}. 

1189 

1190 @kwarg Cartesian: Optional class to return C{(x, y, z)} (L{ellipsoidalKarney.Cartesian}, 

1191 L{ellipsoidalNvector.Cartesian}, L{ellipsoidalVincenty.Cartesian}, 

1192 L{sphericalNvector.Cartesian} or L{sphericalTrigonometry.Cartesian}) 

1193 or C{None}. 

1194 @kwarg Cartesian_kwds: Optional, additional B{C{Cartesian}} keyword arguments, ignored 

1195 if C{B{Cartesian} is None}. 

1196 

1197 @return: A C{B{Cartesian}(x, y, z, **B{Cartesian_kwds})} instance or 

1198 a L{Vector4Tuple}C{(x, y, z, h)} if C{B{Cartesian} is None}. 

1199 

1200 @raise TypeError: Invalid B{C{Cartesian}} or B{C{Cartesian_kwds}}. 

1201 ''' 

1202 if Cartesian in (None, Vector4Tuple): 

1203 r = self.xyzh 

1204 elif Cartesian is Vector3Tuple: 

1205 r = self.xyz 

1206 else: 

1207 _xsubclassof(self._CartesianBase, Cartesian=Cartesian) 

1208 r = Cartesian(self, **_xkwds(Cartesian_kwds, name=self.name)) 

1209 return r 

1210 

1211 def toDatum(self, datum2): 

1212 '''Convert this C{Ecef9Tuple} to an other datum. 

1213 

1214 @arg datum2: Datum to convert I{to} (L{Datum}). 

1215 

1216 @return: The converted 9-Tuple (C{Ecef9Tuple}). 

1217 

1218 @raise TypeError: The B{C{datum2}} is not a L{Datum}. 

1219 ''' 

1220 if self.datum in (None, datum2): # PYCHOK _Names_ 

1221 r = self.copy() 

1222 else: 

1223 c = self._CartesianBase(self, datum=self.datum, name=self.name) # PYCHOK _Names_ 

1224 # c.toLatLon converts datum, x, y, z, lat, lon, etc. 

1225 # and returns another Ecef9Tuple iff LatLon is None 

1226 r = c.toLatLon(datum=datum2, LatLon=None) 

1227 return r 

1228 

1229 def toLatLon(self, LatLon=None, **LatLon_kwds): 

1230 '''Return the geodetic C{(lat, lon, height[, datum])} coordinates. 

1231 

1232 @kwarg LatLon: Optional class to return C{(lat, lon, height[, datum])} 

1233 or C{None}. 

1234 @kwarg LatLon_kwds: Optional B{C{height}}, B{C{datum}} and other 

1235 B{C{LatLon}} keyword arguments. 

1236 

1237 @return: An instance of C{B{LatLon}(lat, lon, **B{LatLon_kwds})} 

1238 or if B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat, lon, 

1239 height)} respectively L{LatLon4Tuple}C{(lat, lon, height, 

1240 datum)} depending on whether C{datum} is un-/specified. 

1241 

1242 @raise TypeError: Invalid B{C{LatLon}} or B{C{LatLon_kwds}}. 

1243 ''' 

1244 lat, lon, D = self.lat, self.lon, self.datum # PYCHOK Ecef9Tuple 

1245 kwds = _xkwds(LatLon_kwds, height=self.height, datum=D, name=self.name) # PYCHOK Ecef9Tuple 

1246 d = kwds.get(_datum_, LatLon) 

1247 if LatLon is None: 

1248 r = LatLon3Tuple(lat, lon, kwds[_height_], name=kwds[_name_]) 

1249 if d is not None: 

1250 # assert d is not LatLon 

1251 r = r.to4Tuple(d) # checks type(d) 

1252 else: 

1253 if d is None: 

1254 _ = kwds.pop(_datum_) # remove None datum 

1255 r = LatLon(lat, lon, **kwds) 

1256 _xdatum(_xattr(r, datum=D), D) 

1257 return r 

1258 

1259 def toLocal(self, ltp, Xyz=None, **Xyz_kwds): 

1260 '''Convert this geocentric to I{local} C{x}, C{y} and C{z}. 

1261 

1262 @kwarg ltp: The I{local tangent plane} (LTP) to use (L{Ltp}). 

1263 @kwarg Xyz: Optional class to return C{x}, C{y} and C{z} 

1264 (L{XyzLocal}, L{Enu}, L{Ned}) or C{None}. 

1265 @kwarg Xyz_kwds: Optional, additional B{C{Xyz}} keyword 

1266 arguments, ignored if C{B{Xyz} is None}. 

1267 

1268 @return: An B{C{Xyz}} instance or if C{B{Xyz} is None}, 

1269 a L{Local9Tuple}C{(x, y, z, lat, lon, height, 

1270 ltp, ecef, M)} with C{M=None}, always. 

1271 

1272 @raise TypeError: Invalid B{C{ltp}}. 

1273 ''' 

1274 return _MODS.ltp._xLtp(ltp)._ecef2local(self, Xyz, Xyz_kwds) 

1275 

1276 def toVector(self, Vector=None, **Vector_kwds): 

1277 '''Return the geocentric C{(x, y, z)} coordinates as vector. 

1278 

1279 @kwarg Vector: Optional vector class to return C{(x, y, z)} or 

1280 C{None}. 

1281 @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword 

1282 arguments, ignored if C{B{Vector} is None}. 

1283 

1284 @return: A C{Vector}C{(x, y, z, **Vector_kwds)} instance or a 

1285 L{Vector3Tuple}C{(x, y, z)} if B{C{Vector}} is C{None}. 

1286 

1287 @see: Propertes C{xyz} and C{xyzh} 

1288 ''' 

1289 return self.xyz if Vector is None else self._xnamed( 

1290 Vector(self.x, self.y, self.z, **Vector_kwds)) # PYCHOK Ecef9Tuple 

1291 

1292# def _T_x_M(self, T): 

1293# '''(INTERNAL) Update M{self.M = T.multiply(self.M)}. 

1294# ''' 

1295# return self.dup(M=T.multiply(self.M)) 

1296 

1297 @Property_RO 

1298 def xyz(self): 

1299 '''Get the geocentric C{(x, y, z)} coordinates (L{Vector3Tuple}C{(x, y, z)}). 

1300 ''' 

1301 return Vector3Tuple(self.x, self.y, self.z, name=self.name) 

1302 

1303 @Property_RO 

1304 def xyzh(self): 

1305 '''Get the geocentric C{(x, y, z)} coordinates and C{height} (L{Vector4Tuple}C{(x, y, z, h)}) 

1306 ''' 

1307 return self.xyz.to4Tuple(self.height) 

1308 

1309 

1310def _4Ecef(this, Ecef): # in .datums.Datum.ecef, .ellipsoids.Ellipsoid.ecef 

1311 '''Return an ECEF converter for C{this} L{Datum} or L{Ellipsoid}. 

1312 ''' 

1313 if Ecef is None: 

1314 Ecef = EcefKarney 

1315 else: 

1316 _xinstanceof(*_Ecefs, Ecef=Ecef) 

1317 return Ecef(this, name=this.name) 

1318 

1319 

1320def _xEcef(Ecef): # PYCHOK .latlonBase.py 

1321 '''(INTERNAL) Validate B{C{Ecef}} I{class}. 

1322 ''' 

1323 if issubclassof(Ecef, _EcefBase): 

1324 return Ecef 

1325 raise _TypesError(_Ecef_, Ecef, *_Ecefs) 

1326 

1327 

1328_Ecefs = (EcefKarney, EcefSudano, EcefVeness, EcefYou, 

1329 EcefFarrell21, EcefFarrell22) 

1330 

1331__all__ += _ALL_DOCS(_EcefBase) 

1332 

1333# **) MIT License 

1334# 

1335# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved. 

1336# 

1337# Permission is hereby granted, free of charge, to any person obtaining a 

1338# copy of this software and associated documentation files (the "Software"), 

1339# to deal in the Software without restriction, including without limitation 

1340# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1341# and/or sell copies of the Software, and to permit persons to whom the 

1342# Software is furnished to do so, subject to the following conditions: 

1343# 

1344# The above copyright notice and this permission notice shall be included 

1345# in all copies or substantial portions of the Software. 

1346# 

1347# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1348# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1349# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1350# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1351# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1352# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1353# OTHER DEALINGS IN THE SOFTWARE.