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 

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

58''' 

59 

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

61 _xinstanceof, _xsubclassof # _args_kwds_names 

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, Fpowers, hypot, hypot1, hypot2_, sqrt0 

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 _name__, _name1__, _NamedBase, _NamedLocal, \ 

77 _NamedTuple, _Pass, _xnamed 

78from pygeodesy.namedTuples import LatLon2Tuple, LatLon3Tuple, \ 

79 PhiLam2Tuple, Vector3Tuple, Vector4Tuple 

80from pygeodesy.props import deprecated_method, Property_RO, property_RO, \ 

81 property_ROver, property_doc_ 

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

83from pygeodesy.units import _isRadius, Degrees, Height, Int, Lam, Lat, Lon, Meter, \ 

84 Phi, Scalar, Scalar_ 

85from pygeodesy.utily import atan1, atan1d, atan2, atan2d, degrees90, degrees180, \ 

86 sincos2, sincos2_, sincos2d, sincos2d_ 

87# from pygeodesy.vector3d import Vector3d # _MODS 

88 

89from math import cos, degrees, fabs, radians, sqrt 

90 

91__all__ = _ALL_LAZY.ecef 

92__version__ = '24.12.06' 

93 

94_Ecef_ = 'Ecef' 

95_prolate_ = 'prolate' 

96_TRIPS = 33 # 8..9 sufficient, EcefSudano.reverse 

97_xyz_y_z = _xyz_, _y_, _z_ # _args_kwds_names(_xyzn4)[:3] 

98 

99 

100class EcefError(_ValueError): 

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

102 ''' 

103 pass 

104 

105 

106class _EcefBase(_NamedBase): 

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

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

109 ''' 

110 _datum = _WGS84 

111 _E = _EWGS84 

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

113 

114 def __init__(self, a_ellipsoid=_EWGS84, f=None, lon00=INT0, **name): 

115 '''New C{Ecef*} converter. 

116 

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

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

119 equatorial radius (C{meter}). 

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

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

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

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

124 C{reverse} method. 

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

126 

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

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

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

130 or B{C{f}} not less than 1.0. 

131 ''' 

132 try: 

133 E = a_ellipsoid 

134 if f is None: 

135 pass 

136 elif _isRadius(E) and isscalar(f): 

137 E = a_f2Tuple(E, f) 

138 else: 

139 raise ValueError() # _invalid_ 

140 

141 if E not in (_EWGS84, _WGS84): 

142 d = _ellipsoidal_datum(E, **name) 

143 E = d.ellipsoid 

144 if E.a < EPS or E.f > EPS1: 

145 raise ValueError() # _invalid_ 

146 self._datum = d 

147 self._E = E 

148 

149 except (TypeError, ValueError) as x: 

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

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

152 

153 if name: 

154 self.name = name 

155 if lon00 is not INT0: 

156 self.lon00 = lon00 

157 

158 def __eq__(self, other): 

159 '''Compare this and an other Ecef. 

160 

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

162 

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

164 ''' 

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

166 other.ellipsoid == self.ellipsoid) 

167 

168 @Property_RO 

169 def datum(self): 

170 '''Get the datum (L{Datum}). 

171 ''' 

172 return self._datum 

173 

174 @Property_RO 

175 def ellipsoid(self): 

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

177 ''' 

178 return self._E 

179 

180 @Property_RO 

181 def equatoradius(self): 

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

183 ''' 

184 return self.ellipsoid.a 

185 

186 a = equatorialRadius = equatoradius # Karney property 

187 

188 @Property_RO 

189 def flattening(self): # Karney property 

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

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

192 ''' 

193 return self.ellipsoid.f 

194 

195 f = flattening 

196 

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

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

199 ''' 

200 if _philam: # lat, lon in radians 

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

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

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

204 else: 

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

206 

207 E = self.ellipsoid 

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

209 z = (h + n * E.e21) * sa 

210 x = (h + n) * ca 

211 

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

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

214 0, m, self.datum, 

215 name=self._name__(name)) 

216 

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

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

219 

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

221 latitude (C{degrees}). 

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

223 (C{degrees}). 

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

225 the surface of the ellipsoid. 

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

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

228 

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

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

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

232 and C{datum} if available. 

233 

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

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

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

237 

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

239 and avoid double angle conversions. 

240 ''' 

241 llhn = _llhn4(latlonh, lon, height, **name) 

242 return self._forward(*llhn, M=M) 

243 

244 def forward_(self, phi, lam, height=0, M=False, **name): 

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

246 in I{radians}. 

247 

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

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

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

251 the surface of the ellipsoid. 

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

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

254 

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

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

257 C{degrees180(B{lam})}. 

258 

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

260 ''' 

261 try: # like function C{_llhn4} below 

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

263 except (TypeError, ValueError) as x: 

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

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

266 

267 @property_ROver 

268 def _Geocentrics(self): 

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

270 ''' 

271 return (Ecef9Tuple, # overwrite property_ROver 

272 _MODS.vector3d.Vector3d) # _MODS.cartesianBase.CartesianBase 

273 

274 @Property_RO 

275 def _isYou(self): 

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

277 ''' 

278 return isinstance(self, EcefYou) 

279 

280 @property 

281 def lon00(self): 

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

283 ''' 

284 return self._lon00 

285 

286 @lon00.setter # PYCHOK setter! 

287 def lon00(self, lon00): 

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

289 ''' 

290 self._lon00 = Degrees(lon00=lon00) 

291 

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

293 '''Creation a rotation matrix. 

294 

295 @arg sa: C{sin(phi)} (C{float}). 

296 @arg ca: C{cos(phi)} (C{float}). 

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

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

299 

300 @return: An L{EcefMatrix}. 

301 ''' 

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

303 

304 def _polon(self, y, x, R, **lon00_name): 

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

306 ''' 

307 return atan2d(y, x) if R else _xkwds_get(lon00_name, lon00=self.lon00) 

308 

309 def reverse(self, xyz, y=None, z=None, M=False, **lon00_name): # PYCHOK no cover 

310 '''I{Must be overloaded}.''' 

311 self._notOverloaded(xyz, y=y, z=z, M=M, **lon00_name) 

312 

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

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

315 

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

317 

318 @return: This C{Ecef*} (C{str}). 

319 ''' 

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

321 

322 

323class EcefFarrell21(_EcefBase): 

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

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

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

327 ''' 

328 

329 def reverse(self, xyz, y=None, z=None, M=None, **lon00_name): # PYCHOK unused M 

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

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

332 page 29, aka the I{Heikkinen application} of U{Ferrari's solution 

333 <https://WikiPedia.org/wiki/Geographic_coordinate_conversion>}. 

334 

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

336 coordinate (C{meter}). 

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

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

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

340 @kwarg lon00_name: Optional C{B{name}=NN} (C{str}) and optional keyword argument 

341 C{B{lon00}=INT0} (C{degrees}), an arbitrary I{"polar"} longitude 

342 returned if C{B{x}=0} and C{B{y}=0}, see property C{lon00}. 

343 

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

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

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

347 if available. 

348 

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

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

351 zero division error. 

352 

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

354 ''' 

355 x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, **lon00_name) 

356 

357 E = self.ellipsoid 

358 a = E.a 

359 a2 = E.a2 

360 b2 = E.b2 

361 e2 = E.e2 

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

363 e4 = E.e4 

364 

365 try: # names as page 29 

366 z2 = z**2 

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

368 

369 p = hypot(x, y) 

370 p2 = p**2 

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

372 F = b2 * z2 * 54 

373 c = e4 * p2 * F / G**3 

374 s = cbrt(sqrt(c * (c + _2_0)) + c + _1_0) 

375 G *= fsumf_(s , _1_0, _1_0 / s) # k 

376 P = F / (G**2 * _3_0) 

377 Q = sqrt(_2_0 * e4 * P + _1_0) 

378 Q1 = Q + _1_0 

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

380 -P * ez / (Q * Q1), 

381 -P * p2 * _0_5)) 

382 r = p + e2 * r0 

383 v = b2 / (sqrt(r**2 + ez) * a) # z0 / z 

384 

385 h = hypot(r, z) * (_1_0 - v) 

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

387 lon = self._polon(y, x, p, **lon00_name) 

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

389 

390 except (ValueError, ZeroDivisionError) as X: 

391 raise EcefError(x=x, y=y, z=z, cause=X) 

392 

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

394 1, None, self.datum, 

395 name=self._name__(name)) 

396 

397 

398class EcefFarrell22(_EcefBase): 

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

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

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

402 ''' 

403 

404 def reverse(self, xyz, y=None, z=None, M=None, **lon00_name): # PYCHOK unused M 

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

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

407 page 30. 

408 

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

410 coordinate (C{meter}). 

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

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

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

414 @kwarg lon00_name: Optional C{B{name}=NN} (C{str}) and optional keyword argument 

415 C{B{lon00}=INT0} (C{degrees}), an arbitrary I{"polar"} longitude 

416 returned if C{B{x}=0} and C{B{y}=0}, see property C{lon00}. 

417 

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

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

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

421 if available. 

422 

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

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

425 zero division error. 

426 

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

428 ''' 

429 x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, **lon00_name) 

430 

431 E = self.ellipsoid 

432 a = E.a 

433 b = E.b 

434 

435 try: # see EcefVeness.reverse 

436 p = hypot(x, y) 

437 lon = self._polon(y, x, p, **lon00_name) 

438 

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

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

441 p - c**3 * a * E.e2) 

442 

443 s, c = sincos2d(lat) 

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

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

446 else: # polar 

447 h = fabs(z) - b 

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

449 

450 except (ValueError, ZeroDivisionError) as e: 

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

452 

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

454 1, None, self.datum, 

455 name=self._name__(name)) 

456 

457 

458class EcefKarney(_EcefBase): 

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

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

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

462 

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

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

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

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

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

468 the rotation matrix. 

469 ''' 

470 

471 @Property_RO 

472 def hmax(self): 

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

474 ''' 

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

476 

477 def reverse(self, xyz, y=None, z=None, M=False, **lon00_name): 

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

479 

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

481 coordinate (C{meter}). 

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

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

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

485 @kwarg lon00_name: Optional C{B{name}=NN} (C{str}) and optional keyword argument 

486 C{B{lon00}=INT0} (C{degrees}), an arbitrary I{"polar"} longitude 

487 returned if C{B{x}=0} and C{B{y}=0}, see property C{lon00}. 

488 

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

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

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

492 C{datum} if available. 

493 

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

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

496 

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

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

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

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

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

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

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

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

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

506 ''' 

507 def _norm3(y, x): 

508 h = hypot(y, x) # EPS0, EPS_2 

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

510 

511 x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, **lon00_name) 

512 

513 E = self.ellipsoid 

514 f = E.f 

515 

516 sb, cb, R = _norm3(y, x) 

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

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

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

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

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

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

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

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

525 sa, ca, _ = _norm3(z * _0_5, R) 

526 C = 1 

527 

528 elif E.e4: # E.isEllipsoidal 

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

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

531 p = (R / E.a)**2 

532 q = (z / E.a)**2 * E.e21 

533 if f < 0: 

534 p, q = q, p 

535 r = fsumf_(p, q, -E.e4) 

536 e = E.e4 * q 

537 if e or r > 0: 

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

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

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

541 u = r = r / _6_0 

542 r2 = r**2 

543 r3 = r2 * r 

544 t3 = r3 + s 

545 d *= t3 + r3 

546 if d < 0: 

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

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

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

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

551 else: 

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

553 # minimizes loss of precision due to cancellation. The 

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

555 # in definition of u. 

556 if d > 0: 

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

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

559 t = cbrt(t3) # t = r * t 

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

561 u = fsumf_(u, t, r2 / t) 

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

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

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

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

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

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

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

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

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

571 if f < 0: 

572 k1 -= E.e2 

573 else: 

574 k2 += E.e2 

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

576 h *= k1 - E.e21 

577 C = 2 

578 

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

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

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

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

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

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

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

586 q = E.e4 - p 

587 if f < 0: 

588 p, q = q, p 

589 e = E.a 

590 else: 

591 e = E.b2_a 

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

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

594 sa = neg(sa) 

595 h *= neg(e / E.e2abs) 

596 C = 3 

597 

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

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

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

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

602 h -= E.a 

603 C = 4 

604 

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

606 lon = self._polon(sb, cb, R, **lon00_name) 

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

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

609 C, m, self.datum, name=self._name__(name)) 

610 

611 

612class EcefSudano(_EcefBase): 

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

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

615 ''' 

616 _tol = EPS2 

617 

618 def reverse(self, xyz, y=None, z=None, M=None, **lon00_name): # PYCHOK unused M 

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

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

621 

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

623 coordinate (C{meter}). 

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

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

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

627 @kwarg lon00_name: Optional C{B{name}=NN} (C{str}) and optional keyword argument 

628 C{B{lon00}=INT0} (C{degrees}), an arbitrary I{"polar"} longitude 

629 returned if C{B{x}=0} and C{B{y}=0}, see property C{lon00}. 

630 

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

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

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

634 

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

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

637 ''' 

638 x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, **lon00_name) 

639 

640 E = self.ellipsoid 

641 e = E.e2 * E.a 

642 R = hypot(x, y) # Rh 

643 d = e - R 

644 

645 lat = atan1d(z, R * E.e21) 

646 sa, ca = sincos2d(fabs(lat)) 

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

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

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

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

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

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

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

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

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

656 tol = self.tolerance 

657 _S2 = Fsum(sa).fsum2f_ 

658 for i in range(1, _TRIPS): 

659 ca2 = _1_0 - sa**2 

660 if ca2 < EPS_2: # PYCHOK no cover 

661 ca = _0_0 

662 break 

663 ca = sqrt(ca2) 

664 r = e / E.e2s2(sa) - R / ca2 

665 if fabs(r) < EPS_2: 

666 break 

667 lat = None 

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

669 if fabs(t) < tol: 

670 break 

671 else: 

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

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

674 

675 if lat is None: 

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

677 lon = self._polon(y, x, R, **lon00_name) 

678 

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

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

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

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

683 i, None, self.datum, # C=i, M=None 

684 iteration=i, name=self._name__(name)) 

685 

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

687 def tolerance(self): 

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

689 ''' 

690 return self._tol 

691 

692 @tolerance.setter # PYCHOK setter! 

693 def tolerance(self, tol): 

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

695 

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

697 ''' 

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

699 

700 

701class EcefVeness(_EcefBase): 

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

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

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

705 

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

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

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

709 ''' 

710 

711 def reverse(self, xyz, y=None, z=None, M=None, **lon00_name): # PYCHOK unused M 

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

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

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

715 

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

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

718 233668213>}, Survey Review, Vol 28, 218, Oct 1985. 

719 

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

721 coordinate (C{meter}). 

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

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

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

725 @kwarg lon00_name: Optional C{B{name}=NN} (C{str}) and optional keyword argument 

726 C{B{lon00}=INT0} (C{degrees}), an arbitrary I{"polar"} longitude 

727 returned if C{B{x}=0} and C{B{y}=0}, see property C{lon00}. 

728 

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

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

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

732 

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

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

735 

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

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

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

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

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

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

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

743 publication/3709199>}. 

744 ''' 

745 x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, **lon00_name) 

746 

747 E = self.ellipsoid 

748 a = E.a 

749 

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

751 r = hypot(p, z) # polar radius 

752 if min(p, r) > EPS0: 

753 b = E.b * E.e22 

754 # parametric latitude (Bowring eqn 17, replaced) 

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

756 c = _1_0 / hypot1(t) 

757 s = c * t 

758 # geodetic latitude (Bowring eqn 18) 

759 lat = atan1d(z + s**3 * b, 

760 p - c**3 * a * E.e2) 

761 

762 # height above ellipsoid (Bowring eqn 7) 

763 sa, ca = sincos2d(lat) 

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

765# h = p * ca + z * sa - (a * a / r) 

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

767 C = 1 

768 

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

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

771 C, lat, h = 2, _0_0, (p - a) 

772 

773 else: # polar lat, lon arbitrarily lon00 

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

775 

776 lon = self._polon(y, x, p, **lon00_name) 

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

778 C, None, self.datum, # M=None 

779 name=self._name__(name)) 

780 

781 

782class EcefYou(_EcefBase): 

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

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

785 for I{non-prolate} ellipsoids. 

786 

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

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

789 11589/115114_9021_geod2ellip_final.pdf>} Studia Geophysica et Geodaetica, 2008, 52, 

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

791 ''' 

792 

793 def __init__(self, a_ellipsoid=_EWGS84, f=None, **lon00_name): # PYCHOK signature 

794 _EcefBase.__init__(self, a_ellipsoid, f=f, **lon00_name) # inherited documentation 

795 self._ee2 = EcefYou._ee2(self.ellipsoid) 

796 

797 @staticmethod 

798 def _ee2(E): 

799 e2 = E.a2 - E.b2 

800 if e2 < 0 or E.f < 0: 

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

802 return sqrt0(e2), e2 

803 

804 def reverse(self, xyz, y=None, z=None, M=None, **lon00_name): # PYCHOK unused M 

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

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

807 

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

809 coordinate (C{meter}). 

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

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

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

813 @kwarg lon00_name: Optional C{B{name}=NN} (C{str}) and optional keyword argument 

814 C{B{lon00}=INT0} (C{degrees}), an arbitrary I{"polar"} longitude 

815 returned if C{B{x}=0} and C{B{y}=0}, see property C{lon00}. 

816 

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

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

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

820 available. 

821 

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

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

824 ellipsoid is I{prolate}. 

825 ''' 

826 x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, **lon00_name) 

827 

828 E = self.ellipsoid 

829 e, e2 = self._ee2 

830 

831 q = hypot(x, y) # R 

832 u = Fpowers(2, x, y, z) - e2 

833 u = u.fadd_(hypot(u, e * z * _2_0)).fover(_2_0) 

834 if u > EPS02: 

835 u = sqrt(u) 

836 p = hypot(u, e) 

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

838 sB, cB = sincos2(B) 

839 if cB and sB: 

840 p *= E.a 

841 d = (p / cB - e2 * cB) / sB 

842 if isnon0(d): 

843 B += fsumf_(u * E.b, -p, e2) / d 

844 sB, cB = sincos2(B) 

845 elif u < (-EPS2): 

846 raise EcefError(x=x, y=y, z=z, u=u, txt=_singular_) 

847 else: 

848 sB, cB = _copysign_1_0(z), _0_0 

849 

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

851 lon = self._polon(y, x, q, **lon00_name) 

852 

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

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

855 h = neg(h) # inside ellipsoid 

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

857 1, None, self.datum, # C=1, M=None 

858 name=self._name__(name)) 

859 

860 

861class EcefMatrix(_NamedTuple): 

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

863 

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

865 Geographic_coordinate_conversion#From_ECEF_to_ENU>} and 

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

867 ''' 

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

869 '_1_0_', '_1_1_', '_1_2_', 

870 '_2_0_', '_2_1_', '_2_2_') 

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

872 

873 def _validate(self, **unused): # PYCHOK unused 

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

875 ''' 

876 _NamedTuple._validate(self, underOK=True) 

877 

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

879 '''New L{EcefMatrix} matrix. 

880 

881 @arg sa: C{sin(phi)} (C{float}). 

882 @arg ca: C{cos(phi)} (C{float}). 

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

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

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

886 

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

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

889 ''' 

890 t = sa, ca, sb, cb 

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

892 t += _more # ... from .multiply 

893 

894 elif max(map(fabs, t)) > _1_0: 

895 raise EcefError(unstr(EcefMatrix, *t)) 

896 

897 else: # build matrix from the following quaternion operations 

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

899 # or 

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

901 # where 

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

903 

904 # Local X axis (East) in geocentric coords 

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

906 # Local Y axis (North) in geocentric coords 

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

908 # Local Z axis (Up) in geocentric coords 

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

910 t = (-sb, -cb * sa, cb * ca, 

911 cb, -sb * sa, sb * ca, 

912 _0_0, ca, sa) 

913 

914 return _NamedTuple.__new__(cls, *t) 

915 

916 def column(self, column): 

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

918 ''' 

919 if 0 <= column < 3: 

920 return self[column::3] 

921 raise _IndexError(column=column) 

922 

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

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

925 

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

927 ''' 

928 return self.classof(*self) 

929 

930 __copy__ = __deepcopy__ = copy 

931 

932 @Property_RO 

933 def matrix3(self): 

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

935 ''' 

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

937 

938 @Property_RO 

939 def matrixTransposed3(self): 

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

941 ''' 

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

943 

944 def multiply(self, other): 

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

946 with an other matrix. 

947 

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

949 

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

951 

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

953 ''' 

954 _xinstanceof(EcefMatrix, other=other) 

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

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

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

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

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

960 

961 def rotate(self, xyz, *xyz0): 

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

963 

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

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

966 

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

968 

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

970 ''' 

971 if xyz0: 

972 if len(xyz0) != len(xyz): 

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

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

975 

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

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

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

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

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

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

982 

983 def row(self, row): 

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

985 ''' 

986 if 0 <= row < 3: 

987 r = row * 3 

988 return self[r:r+3] 

989 raise _IndexError(row=row) 

990 

991 def unrotate(self, xyz, *xyz0): 

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

993 

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

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

996 

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

998 

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

1000 ''' 

1001 if xyz0: 

1002 if len(xyz0) != len(xyz): 

1003 raise LenError(self.unrotate, xyz0=len(xyz0), xyz=len(xyz)) 

1004 _xyz = _1_0_1T + xyz 

1005 # x' = x0 + M[0] * x + M[1] * y + M[2] * z 

1006 # y' = y0 + M[3] * x + M[4] * y + M[5] * z 

1007 # z' = z0 + M[6] * x + M[7] * y + M[8] * z 

1008 xyz_ = (fdot(_xyz, xyz0[0], *self[0:3]), # .row(0) 

1009 fdot(_xyz, xyz0[1], *self[3:6]), # .row(1) 

1010 fdot(_xyz, xyz0[2], *self[6:9])) # .row(2) 

1011 else: 

1012 # x' = M[0] * x + M[1] * y + M[2] * z 

1013 # y' = M[3] * x + M[4] * y + M[5] * z 

1014 # z' = M[6] * x + M[7] * y + M[8] * z 

1015 xyz_ = (fdot(xyz, *self[0:3]), # .row(0) 

1016 fdot(xyz, *self[3:6]), # .row(1) 

1017 fdot(xyz, *self[6:9])) # .row(2) 

1018 return xyz_ 

1019 

1020 

1021class Ecef9Tuple(_NamedTuple, _NamedLocal): 

1022 '''9-Tuple C{(x, y, z, lat, lon, height, C, M, datum)} with I{geocentric} C{x}, 

1023 C{y} and C{z} plus I{geodetic} C{lat}, C{lon} and C{height}, case C{C} (see 

1024 the C{Ecef*.reverse} methods) and optionally, rotation matrix C{M} (L{EcefMatrix}) 

1025 and C{datum}, with C{lat} and C{lon} in C{degrees} and C{x}, C{y}, C{z} and 

1026 C{height} in C{meter}, conventionally. 

1027 ''' 

1028 _Names_ = (_x_, _y_, _z_, _lat_, _lon_, _height_, _C_, _M_, _datum_) 

1029 _Units_ = ( Meter, Meter, Meter, Lat, Lon, Height, Int, _Pass, _Pass) 

1030 

1031 @property_ROver 

1032 def _CartesianBase(self): 

1033 '''(INTERNAL) Get class C{CartesianBase}, I{once}. 

1034 ''' 

1035 return _MODS.cartesianBase.CartesianBase # overwrite property_ROver 

1036 

1037 @deprecated_method 

1038 def convertDatum(self, datum2): # for backward compatibility 

1039 '''DEPRECATED, use method L{toDatum}.''' 

1040 return self.toDatum(datum2) 

1041 

1042 @property_RO 

1043 def _ecef9(self): 

1044 return self 

1045 

1046 @Property_RO 

1047 def lam(self): 

1048 '''Get the longitude in C{radians} (C{float}). 

1049 ''' 

1050 return self.philam.lam 

1051 

1052 @Property_RO 

1053 def lamVermeille(self): 

1054 '''Get the longitude in C{radians} M{[-PI*3/2..+PI*3/2]} after U{Vermeille 

1055 <https://Search.ProQuest.com/docview/639493848>} (2004), page 95. 

1056 

1057 @see: U{Karney<https://GeographicLib.SourceForge.io/C++/doc/geocentric.html>}, 

1058 U{Vermeille<https://Search.ProQuest.com/docview/847292978>} 2011, pp 112-113, 116 

1059 and U{Featherstone, et.al.<https://Search.ProQuest.com/docview/872827242>}, page 7. 

1060 ''' 

1061 x, y = self.x, self.y 

1062 if y > EPS0: 

1063 r = atan2(x, hypot(y, x) + y) * _N_2_0 + PI_2 

1064 elif y < -EPS0: 

1065 r = atan2(x, hypot(y, x) - y) * _2_0 - PI_2 

1066 else: # y == 0 

1067 r = PI if x < 0 else _0_0 

1068 return Lam(Vermeille=r) 

1069 

1070 @Property_RO 

1071 def latlon(self): 

1072 '''Get the lat-, longitude in C{degrees} (L{LatLon2Tuple}C{(lat, lon)}). 

1073 ''' 

1074 return LatLon2Tuple(self.lat, self.lon, name=self.name) 

1075 

1076 @Property_RO 

1077 def latlonheight(self): 

1078 '''Get the lat-, longitude in C{degrees} and height (L{LatLon3Tuple}C{(lat, lon, height)}). 

1079 ''' 

1080 return self.latlon.to3Tuple(self.height) 

1081 

1082 @Property_RO 

1083 def latlonheightdatum(self): 

1084 '''Get the lat-, longitude in C{degrees} with height and datum (L{LatLon4Tuple}C{(lat, lon, height, datum)}). 

1085 ''' 

1086 return self.latlonheight.to4Tuple(self.datum) 

1087 

1088 @Property_RO 

1089 def latlonVermeille(self): 

1090 '''Get the latitude and I{Vermeille} longitude in C{degrees [-225..+225]} (L{LatLon2Tuple}C{(lat, lon)}). 

1091 

1092 @see: Property C{lonVermeille}. 

1093 ''' 

1094 return LatLon2Tuple(self.lat, self.lonVermeille, name=self.name) 

1095 

1096 @Property_RO 

1097 def lonVermeille(self): 

1098 '''Get the longitude in C{degrees [-225..+225]} after U{Vermeille 

1099 <https://Search.ProQuest.com/docview/639493848>} 2004, p 95. 

1100 

1101 @see: Property C{lamVermeille}. 

1102 ''' 

1103 return Lon(Vermeille=degrees(self.lamVermeille)) 

1104 

1105 def _ltp_toLocal(self, ltp, Xyz_kwds, **Xyz): # overloads C{_NamedLocal}'s 

1106 '''(INTERNAL) Invoke C{ltp._xLtp(ltp)._ecef2local}. 

1107 ''' 

1108 Xyz_ = self._ltp_toLocal2(Xyz_kwds, **Xyz) # in ._NamedLocal 

1109 ltp = self._ltp._xLtp(ltp, self._Ltp) # both in ._NamedLocal 

1110 return ltp._ecef2local(self, *Xyz_) 

1111 

1112 @Property_RO 

1113 def phi(self): 

1114 '''Get the latitude in C{radians} (C{float}). 

1115 ''' 

1116 return self.philam.phi 

1117 

1118 @Property_RO 

1119 def philam(self): 

1120 '''Get the lat-, longitude in C{radians} (L{PhiLam2Tuple}C{(phi, lam)}). 

1121 ''' 

1122 return PhiLam2Tuple(radians(self.lat), radians(self.lon), name=self.name) 

1123 

1124 @Property_RO 

1125 def philamheight(self): 

1126 '''Get the lat-, longitude in C{radians} and height (L{PhiLam3Tuple}C{(phi, lam, height)}). 

1127 ''' 

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

1129 

1130 @Property_RO 

1131 def philamheightdatum(self): 

1132 '''Get the lat-, longitude in C{radians} with height and datum (L{PhiLam4Tuple}C{(phi, lam, height, datum)}). 

1133 ''' 

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

1135 

1136 @Property_RO 

1137 def philamVermeille(self): 

1138 '''Get the latitude and I{Vermeille} longitude in C{radians [-PI*3/2..+PI*3/2]} (L{PhiLam2Tuple}C{(phi, lam)}). 

1139 

1140 @see: Property C{lamVermeille}. 

1141 ''' 

1142 return PhiLam2Tuple(radians(self.lat), self.lamVermeille, name=self.name) 

1143 

1144 def toCartesian(self, Cartesian=None, **Cartesian_kwds): 

1145 '''Return the geocentric C{(x, y, z)} coordinates as an ellipsoidal or spherical 

1146 C{Cartesian}. 

1147 

1148 @kwarg Cartesian: Optional class to return C{(x, y, z)} (L{ellipsoidalKarney.Cartesian}, 

1149 L{ellipsoidalNvector.Cartesian}, L{ellipsoidalVincenty.Cartesian}, 

1150 L{sphericalNvector.Cartesian} or L{sphericalTrigonometry.Cartesian}) 

1151 or C{None}. 

1152 @kwarg Cartesian_kwds: Optionally, additional B{C{Cartesian}} keyword arguments, ignored 

1153 if C{B{Cartesian} is None}. 

1154 

1155 @return: A B{C{Cartesian}} instance or a L{Vector4Tuple}C{(x, y, z, h)} if C{B{Cartesian} 

1156 is None}. 

1157 

1158 @raise TypeError: Invalid B{C{Cartesian}} or B{C{Cartesian_kwds}} item. 

1159 ''' 

1160 if Cartesian in (None, Vector4Tuple): 

1161 r = self.xyzh 

1162 elif Cartesian is Vector3Tuple: 

1163 r = self.xyz 

1164 else: 

1165 _xsubclassof(self._CartesianBase, Cartesian=Cartesian) 

1166 r = Cartesian(self, **_name1__(Cartesian_kwds, _or_nameof=self)) 

1167 return r 

1168 

1169 def toDatum(self, datum2, **name): 

1170 '''Convert this C{Ecef9Tuple} to an other datum. 

1171 

1172 @arg datum2: Datum to convert I{to} (L{Datum}). 

1173 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1174 

1175 @return: The converted 9-Tuple (C{Ecef9Tuple}). 

1176 

1177 @raise TypeError: The B{C{datum2}} is not a L{Datum}. 

1178 ''' 

1179 n = _name__(name, _or_nameof=self) 

1180 if self.datum in (None, datum2): # PYCHOK _Names_ 

1181 r = self.copy(name=n) 

1182 else: 

1183 c = self._CartesianBase(self, datum=self.datum, name=n) # PYCHOK _Names_ 

1184 # c.toLatLon converts datum, x, y, z, lat, lon, etc. 

1185 # and returns another Ecef9Tuple iff LatLon is None 

1186 r = c.toLatLon(datum=datum2, LatLon=None) 

1187 return r 

1188 

1189 def toLatLon(self, LatLon=None, **LatLon_kwds): 

1190 '''Return the geodetic C{(lat, lon, height[, datum])} coordinates. 

1191 

1192 @kwarg LatLon: Optional class to return C{(lat, lon, height[, datum])} or C{None}. 

1193 @kwarg LatLon_kwds: Optional B{C{height}}, B{C{datum}} and other B{C{LatLon}} 

1194 keyword arguments. 

1195 

1196 @return: A B{C{LatLon}} instance or if C{B{LatLon} is None}, a L{LatLon4Tuple}C{(lat, 

1197 lon, height, datum)} or L{LatLon3Tuple}C{(lat, lon, height)} if C{datum} is 

1198 specified or not. 

1199 

1200 @raise TypeError: Invalid B{C{LatLon}} or B{C{LatLon_kwds}} item. 

1201 ''' 

1202 lat, lon, D = self.lat, self.lon, self.datum # PYCHOK Ecef9Tuple 

1203 kwds = _name1__(LatLon_kwds, _or_nameof=self) 

1204 kwds = _xkwds(kwds, height=self.height, datum=D) # PYCHOK Ecef9Tuple 

1205 d = kwds.get(_datum_, LatLon) 

1206 if LatLon is None: 

1207 r = LatLon3Tuple(lat, lon, kwds[_height_], name=kwds[_name_]) 

1208 if d is not None: 

1209 # assert d is not LatLon 

1210 r = r.to4Tuple(d) # checks type(d) 

1211 else: 

1212 if d is None: 

1213 _ = kwds.pop(_datum_) # remove None datum 

1214 r = LatLon(lat, lon, **kwds) 

1215 _xdatum(_xattr(r, datum=D), D) 

1216 return r 

1217 

1218 def toVector(self, Vector=None, **Vector_kwds): 

1219 '''Return these geocentric C{(x, y, z)} coordinates as vector. 

1220 

1221 @kwarg Vector: Optional vector class to return C{(x, y, z)} or C{None}. 

1222 @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword arguments, 

1223 ignored if C{B{Vector} is None}. 

1224 

1225 @return: A B{C{Vector}} instance or a L{Vector3Tuple}C{(x, y, z)} if 

1226 C{B{Vector} is None}. 

1227 

1228 @raise TypeError: Invalid B{C{Vector}} or B{C{Vector_kwds}} item. 

1229 

1230 @see: Propertes C{xyz} and C{xyzh} 

1231 ''' 

1232 return self.xyz if Vector is None else Vector( 

1233 *self.xyz, **_name1__(Vector_kwds, _or_nameof=self)) # PYCHOK Ecef9Tuple 

1234 

1235# def _T_x_M(self, T): 

1236# '''(INTERNAL) Update M{self.M = T.multiply(self.M)}. 

1237# ''' 

1238# return self.dup(M=T.multiply(self.M)) 

1239 

1240 @Property_RO 

1241 def xyz(self): 

1242 '''Get the geocentric C{(x, y, z)} coordinates (L{Vector3Tuple}C{(x, y, z)}). 

1243 ''' 

1244 return Vector3Tuple(self.x, self.y, self.z, name=self.name) 

1245 

1246 @Property_RO 

1247 def xyzh(self): 

1248 '''Get the geocentric C{(x, y, z)} coordinates and C{height} (L{Vector4Tuple}C{(x, y, z, h)}) 

1249 ''' 

1250 return self.xyz.to4Tuple(self.height) 

1251 

1252 

1253def _4Ecef(this, Ecef): # in .datums.Datum.ecef, .ellipsoids.Ellipsoid.ecef 

1254 '''Return an ECEF converter for C{this} L{Datum} or L{Ellipsoid}. 

1255 ''' 

1256 if Ecef is None: 

1257 Ecef = EcefKarney 

1258 else: 

1259 _xinstanceof(*_Ecefs, Ecef=Ecef) 

1260 return Ecef(this, name=this.name) 

1261 

1262 

1263def _llhn4(latlonh, lon, height, suffix=NN, Error=EcefError, **name): # in .ltp 

1264 '''(INTERNAL) Get a C{(lat, lon, h, name)} 4-tuple. 

1265 ''' 

1266 try: 

1267 lat, lon = latlonh.lat, latlonh.lon 

1268 h = _xattr(latlonh, height=_xattr(latlonh, h=height)) 

1269 n = _name__(name, _or_nameof=latlonh) # == latlonh._name__(name) 

1270 except AttributeError: 

1271 lat, h, n = latlonh, height, _name__(**name) 

1272 try: 

1273 return Lat(lat), Lon(lon), Height(h), n 

1274 except (TypeError, ValueError) as x: 

1275 t = _lat_, _lon_, _height_ 

1276 if suffix: 

1277 t = (_ + suffix for _ in t) 

1278 d = dict(zip(t, (lat, lon, h))) 

1279 raise Error(cause=x, **d) 

1280 

1281 

1282def _xEcef(Ecef): # PYCHOK .latlonBase 

1283 '''(INTERNAL) Validate B{C{Ecef}} I{class}. 

1284 ''' 

1285 if issubclassof(Ecef, _EcefBase): 

1286 return Ecef 

1287 raise _TypesError(_Ecef_, Ecef, *_Ecefs) 

1288 

1289 

1290# kwd lon00 unused but will throw a TypeError if misspelled, etc. 

1291def _xyzn4(xyz, y, z, Types, Error=EcefError, lon00=0, # PYCHOK unused 

1292 _xyz_y_z_names=_xyz_y_z, **name): # in .ltp 

1293 '''(INTERNAL) Get an C{(x, y, z, name)} 4-tuple. 

1294 ''' 

1295 try: 

1296 n = _name__(name, _or_nameof=xyz) # == xyz._name__(name) 

1297 try: 

1298 t = xyz.x, xyz.y, xyz.z, n 

1299 if not isinstance(xyz, Types): 

1300 raise _TypesError(_xyz_y_z_names[0], xyz, *Types) 

1301 except AttributeError: 

1302 t = map1(float, xyz, y, z) + (n,) 

1303 except (TypeError, ValueError) as x: 

1304 d = dict(zip(_xyz_y_z_names, (xyz, y, z))) 

1305 raise Error(cause=x, **d) 

1306 return t 

1307# assert _xyz_y_z == _args_kwds_names(_xyzn4)[:3] 

1308 

1309 

1310_Ecefs = (EcefKarney, EcefSudano, EcefVeness, EcefYou, 

1311 EcefFarrell21, EcefFarrell22) 

1312__all__ += _ALL_DOCS(_EcefBase) 

1313 

1314# **) MIT License 

1315# 

1316# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved. 

1317# 

1318# Permission is hereby granted, free of charge, to any person obtaining a 

1319# copy of this software and associated documentation files (the "Software"), 

1320# to deal in the Software without restriction, including without limitation 

1321# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1322# and/or sell copies of the Software, and to permit persons to whom the 

1323# Software is furnished to do so, subject to the following conditions: 

1324# 

1325# The above copyright notice and this permission notice shall be included 

1326# in all copies or substantial portions of the Software. 

1327# 

1328# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1329# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1330# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1331# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1332# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1333# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1334# OTHER DEALINGS IN THE SOFTWARE.