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, _NamedTuple, _Pass, _xnamed 

77from pygeodesy.namedTuples import LatLon2Tuple, LatLon3Tuple, \ 

78 PhiLam2Tuple, Vector3Tuple, Vector4Tuple 

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

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

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

82 Phi, Scalar, Scalar_ 

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

84 sincos2, sincos2_, sincos2d, sincos2d_ 

85 

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

87 

88__all__ = _ALL_LAZY.ecef 

89__version__ = '24.10.28' 

90 

91_Ecef_ = 'Ecef' 

92_prolate_ = 'prolate' 

93_TRIPS = 33 # 8..9 sufficient, EcefSudano.reverse 

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

95 

96 

97class EcefError(_ValueError): 

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

99 ''' 

100 pass 

101 

102 

103class _EcefBase(_NamedBase): 

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

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

106 ''' 

107 _datum = _WGS84 

108 _E = _EWGS84 

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

110 

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

112 '''New C{Ecef*} converter. 

113 

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

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

116 equatorial radius (C{meter}). 

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

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

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

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

121 C{reverse} method. 

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

123 

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

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

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

127 or B{C{f}} not less than 1.0. 

128 ''' 

129 try: 

130 E = a_ellipsoid 

131 if f is None: 

132 pass 

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

134 E = a_f2Tuple(E, f) 

135 else: 

136 raise ValueError() # _invalid_ 

137 

138 if E not in (_EWGS84, _WGS84): 

139 d = _ellipsoidal_datum(E, **name) 

140 E = d.ellipsoid 

141 if E.a < EPS or E.f > EPS1: 

142 raise ValueError() # _invalid_ 

143 self._datum = d 

144 self._E = E 

145 

146 except (TypeError, ValueError) as x: 

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

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

149 

150 if name: 

151 self.name = name 

152 if lon00 is not INT0: 

153 self.lon00 = lon00 

154 

155 def __eq__(self, other): 

156 '''Compare this and an other Ecef. 

157 

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

159 

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

161 ''' 

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

163 other.ellipsoid == self.ellipsoid) 

164 

165 @Property_RO 

166 def datum(self): 

167 '''Get the datum (L{Datum}). 

168 ''' 

169 return self._datum 

170 

171 @Property_RO 

172 def ellipsoid(self): 

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

174 ''' 

175 return self._E 

176 

177 @Property_RO 

178 def equatoradius(self): 

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

180 ''' 

181 return self.ellipsoid.a 

182 

183 a = equatorialRadius = equatoradius # Karney property 

184 

185 @Property_RO 

186 def flattening(self): # Karney property 

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

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

189 ''' 

190 return self.ellipsoid.f 

191 

192 f = flattening 

193 

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

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

196 ''' 

197 if _philam: # lat, lon in radians 

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

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

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

201 else: 

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

203 

204 E = self.ellipsoid 

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

206 z = (h + n * E.e21) * sa 

207 x = (h + n) * ca 

208 

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

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

211 0, m, self.datum, 

212 name=self._name__(name)) 

213 

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

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

216 

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

218 latitude (C{degrees}). 

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

220 (C{degrees}). 

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

222 the surface of the ellipsoid. 

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

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

225 

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

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

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

229 and C{datum} if available. 

230 

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

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

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

234 

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

236 and avoid double angle conversions. 

237 ''' 

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

239 return self._forward(*llhn, M=M) 

240 

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

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

243 in I{radians}. 

244 

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

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

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

248 the surface of the ellipsoid. 

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

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

251 

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

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

254 C{degrees180(B{lam})}. 

255 

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

257 ''' 

258 try: # like function C{_llhn4} below 

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

260 except (TypeError, ValueError) as x: 

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

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

263 

264 @property_ROver 

265 def _Geocentrics(self): 

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

267 ''' 

268 return (Ecef9Tuple, # overwrite property_ROver 

269 _MODS.cartesianBase.CartesianBase) 

270 

271 @Property_RO 

272 def _isYou(self): 

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

274 ''' 

275 return isinstance(self, EcefYou) 

276 

277 @property 

278 def lon00(self): 

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

280 ''' 

281 return self._lon00 

282 

283 @lon00.setter # PYCHOK setter! 

284 def lon00(self, lon00): 

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

286 ''' 

287 self._lon00 = Degrees(lon00=lon00) 

288 

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

290 '''Creation a rotation matrix. 

291 

292 @arg sa: C{sin(phi)} (C{float}). 

293 @arg ca: C{cos(phi)} (C{float}). 

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

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

296 

297 @return: An L{EcefMatrix}. 

298 ''' 

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

300 

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

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

303 ''' 

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

305 

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

307 '''I{Must be overloaded}.''' 

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

309 

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

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

312 

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

314 

315 @return: This C{Ecef*} (C{str}). 

316 ''' 

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

318 

319 

320class EcefFarrell21(_EcefBase): 

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

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

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

324 ''' 

325 

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

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

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

329 page 29. 

330 

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

332 coordinate (C{meter}). 

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

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

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

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

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

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

339 

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

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

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

343 if available. 

344 

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

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

347 zero division error. 

348 

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

350 ''' 

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

352 

353 E = self.ellipsoid 

354 a = E.a 

355 a2 = E.a2 

356 b2 = E.b2 

357 e2 = E.e2 

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

359 e4 = E.e4 

360 

361 try: # names as page 29 

362 z2 = z**2 

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

364 

365 p = hypot(x, y) 

366 p2 = p**2 

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

368 F = b2 * z2 * 54 

369 t = e4 * p2 * F / G**3 

370 t = cbrt(sqrt(t * (t + _2_0)) + t + _1_0) 

371 G *= fsumf_(t , _1_0, _1_0 / t) 

372 P = F / (G**2 * _3_0) 

373 Q = sqrt(_2_0 * e4 * P + _1_0) 

374 Q1 = Q + _1_0 

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

376 -P * ez / (Q * Q1), 

377 -P * p2 * _0_5)) 

378 r = p + e2 * r0 

379 v = b2 / (sqrt(r**2 + ez) * a) 

380 

381 h = hypot(r, z) * (_1_0 - v) 

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

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

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

385 

386 except (ValueError, ZeroDivisionError) as X: 

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

388 

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

390 1, None, self.datum, 

391 name=self._name__(name)) 

392 

393 

394class EcefFarrell22(_EcefBase): 

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

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

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

398 ''' 

399 

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

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

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

403 page 30. 

404 

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

406 coordinate (C{meter}). 

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

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

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

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

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

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

413 

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

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

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

417 if available. 

418 

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

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

421 zero division error. 

422 

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

424 ''' 

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

426 

427 E = self.ellipsoid 

428 a = E.a 

429 b = E.b 

430 

431 try: # see EcefVeness.reverse 

432 p = hypot(x, y) 

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

434 

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

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

437 p - c**3 * a * E.e2) 

438 

439 s, c = sincos2d(lat) 

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

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

442 else: # polar 

443 h = fabs(z) - b 

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

445 

446 except (ValueError, ZeroDivisionError) as e: 

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

448 

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

450 1, None, self.datum, 

451 name=self._name__(name)) 

452 

453 

454class EcefKarney(_EcefBase): 

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

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

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

458 

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

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

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

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

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

464 the rotation matrix. 

465 ''' 

466 

467 @Property_RO 

468 def hmax(self): 

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

470 ''' 

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

472 

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

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

475 

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

477 coordinate (C{meter}). 

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

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

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

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

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

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

484 

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

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

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

488 C{datum} if available. 

489 

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

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

492 

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

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

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

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

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

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

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

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

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

502 ''' 

503 def _norm3(y, x): 

504 h = hypot(y, x) # EPS0, EPS_2 

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

506 

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

508 

509 E = self.ellipsoid 

510 f = E.f 

511 

512 sb, cb, R = _norm3(y, x) 

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

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

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

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

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

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

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

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

521 sa, ca, _ = _norm3(z * _0_5, R) 

522 C = 1 

523 

524 elif E.e4: # E.isEllipsoidal 

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

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

527 p = (R / E.a)**2 

528 q = (z / E.a)**2 * E.e21 

529 if f < 0: 

530 p, q = q, p 

531 r = fsumf_(p, q, -E.e4) 

532 e = E.e4 * q 

533 if e or r > 0: 

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

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

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

537 u = r = r / _6_0 

538 r2 = r**2 

539 r3 = r2 * r 

540 t3 = r3 + s 

541 d *= t3 + r3 

542 if d < 0: 

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

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

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

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

547 else: 

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

549 # minimizes loss of precision due to cancellation. The 

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

551 # in definition of u. 

552 if d > 0: 

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

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

555 t = cbrt(t3) # t = r * t 

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

557 u = fsumf_(u, t, r2 / t) 

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

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

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

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

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

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

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

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

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

567 if f < 0: 

568 k1 -= E.e2 

569 else: 

570 k2 += E.e2 

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

572 h *= k1 - E.e21 

573 C = 2 

574 

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

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

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

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

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

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

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

582 q = E.e4 - p 

583 if f < 0: 

584 p, q = q, p 

585 e = E.a 

586 else: 

587 e = E.b2_a 

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

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

590 sa = neg(sa) 

591 h *= neg(e / E.e2abs) 

592 C = 3 

593 

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

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

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

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

598 h -= E.a 

599 C = 4 

600 

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

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

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

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

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

606 

607 

608class EcefSudano(_EcefBase): 

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

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

611 ''' 

612 _tol = EPS2 

613 

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

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

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

617 

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

619 coordinate (C{meter}). 

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

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

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

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

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

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

626 

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

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

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

630 

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

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

633 ''' 

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

635 

636 E = self.ellipsoid 

637 e = E.e2 * E.a 

638 R = hypot(x, y) # Rh 

639 d = e - R 

640 

641 lat = atan1d(z, R * E.e21) 

642 sa, ca = sincos2d(fabs(lat)) 

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

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

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

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

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

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

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

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

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

652 tol = self.tolerance 

653 _S2 = Fsum(sa).fsum2f_ 

654 for i in range(1, _TRIPS): 

655 ca2 = _1_0 - sa**2 

656 if ca2 < EPS_2: # PYCHOK no cover 

657 ca = _0_0 

658 break 

659 ca = sqrt(ca2) 

660 r = e / E.e2s2(sa) - R / ca2 

661 if fabs(r) < EPS_2: 

662 break 

663 lat = None 

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

665 if fabs(t) < tol: 

666 break 

667 else: 

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

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

670 

671 if lat is None: 

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

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

674 

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

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

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

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

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

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

681 

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

683 def tolerance(self): 

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

685 ''' 

686 return self._tol 

687 

688 @tolerance.setter # PYCHOK setter! 

689 def tolerance(self, tol): 

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

691 

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

693 ''' 

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

695 

696 

697class EcefVeness(_EcefBase): 

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

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

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

701 

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

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

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

705 ''' 

706 

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

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

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

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

711 

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

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

714 233668213>}, Survey Review, Vol 28, 218, Oct 1985. 

715 

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

717 coordinate (C{meter}). 

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

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

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

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

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

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

724 

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

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

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

728 

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

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

731 

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

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

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

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

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

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

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

739 publication/3709199>}. 

740 ''' 

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

742 

743 E = self.ellipsoid 

744 a = E.a 

745 

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

747 r = hypot(p, z) # polar radius 

748 if min(p, r) > EPS0: 

749 b = E.b * E.e22 

750 # parametric latitude (Bowring eqn 17, replaced) 

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

752 c = _1_0 / hypot1(t) 

753 s = c * t 

754 # geodetic latitude (Bowring eqn 18) 

755 lat = atan1d(z + s**3 * b, 

756 p - c**3 * a * E.e2) 

757 

758 # height above ellipsoid (Bowring eqn 7) 

759 sa, ca = sincos2d(lat) 

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

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

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

763 C = 1 

764 

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

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

767 C, lat, h = 2, _0_0, (p - a) 

768 

769 else: # polar lat, lon arbitrarily lon00 

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

771 

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

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

774 C, None, self.datum, # M=None 

775 name=self._name__(name)) 

776 

777 

778class EcefYou(_EcefBase): 

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

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

781 for I{non-prolate} ellipsoids. 

782 

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

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

785 11589/115114_9021_geod2ellip_final.pdf>} Studia Geophysica et Geodaetica, 2008, 52, 

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

787 ''' 

788 

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

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

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

792 

793 @staticmethod 

794 def _ee2(E): 

795 e2 = E.a2 - E.b2 

796 if e2 < 0 or E.f < 0: 

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

798 return sqrt0(e2), e2 

799 

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

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

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

803 

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

805 coordinate (C{meter}). 

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

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

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

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

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

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

812 

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

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

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

816 available. 

817 

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

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

820 ellipsoid is I{prolate}. 

821 ''' 

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

823 

824 E = self.ellipsoid 

825 e, e2 = self._ee2 

826 

827 q = hypot(x, y) # R 

828 u = Fpowers(2, x, y, z) - e2 

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

830 if u > EPS02: 

831 u = sqrt(u) 

832 p = hypot(u, e) 

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

834 sB, cB = sincos2(B) 

835 if cB and sB: 

836 p *= E.a 

837 d = (p / cB - e2 * cB) / sB 

838 if isnon0(d): 

839 B += fsumf_(u * E.b, -p, e2) / d 

840 sB, cB = sincos2(B) 

841 elif u < (-EPS2): 

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

843 else: 

844 sB, cB = _copysign_1_0(z), _0_0 

845 

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

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

848 

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

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

851 h = neg(h) # inside ellipsoid 

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

853 1, None, self.datum, # C=1, M=None 

854 name=self._name__(name)) 

855 

856 

857class EcefMatrix(_NamedTuple): 

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

859 

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

861 Geographic_coordinate_conversion#From_ECEF_to_ENU>} and 

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

863 ''' 

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

865 '_1_0_', '_1_1_', '_1_2_', 

866 '_2_0_', '_2_1_', '_2_2_') 

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

868 

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

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

871 ''' 

872 _NamedTuple._validate(self, underOK=True) 

873 

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

875 '''New L{EcefMatrix} matrix. 

876 

877 @arg sa: C{sin(phi)} (C{float}). 

878 @arg ca: C{cos(phi)} (C{float}). 

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

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

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

882 

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

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

885 ''' 

886 t = sa, ca, sb, cb 

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

888 t += _more # ... from .multiply 

889 

890 elif max(map(fabs, t)) > _1_0: 

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

892 

893 else: # build matrix from the following quaternion operations 

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

895 # or 

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

897 # where 

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

899 

900 # Local X axis (East) in geocentric coords 

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

902 # Local Y axis (North) in geocentric coords 

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

904 # Local Z axis (Up) in geocentric coords 

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

906 t = (-sb, -cb * sa, cb * ca, 

907 cb, -sb * sa, sb * ca, 

908 _0_0, ca, sa) 

909 

910 return _NamedTuple.__new__(cls, *t) 

911 

912 def column(self, column): 

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

914 ''' 

915 if 0 <= column < 3: 

916 return self[column::3] 

917 raise _IndexError(column=column) 

918 

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

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

921 

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

923 ''' 

924 return self.classof(*self) 

925 

926 __copy__ = __deepcopy__ = copy 

927 

928 @Property_RO 

929 def matrix3(self): 

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

931 ''' 

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

933 

934 @Property_RO 

935 def matrixTransposed3(self): 

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

937 ''' 

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

939 

940 def multiply(self, other): 

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

942 with an other matrix. 

943 

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

945 

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

947 

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

949 ''' 

950 _xinstanceof(EcefMatrix, other=other) 

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

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

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

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

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

956 

957 def rotate(self, xyz, *xyz0): 

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

959 

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

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

962 

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

964 

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

966 ''' 

967 if xyz0: 

968 if len(xyz0) != len(xyz): 

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

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

971 

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

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

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

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

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

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

978 

979 def row(self, row): 

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

981 ''' 

982 if 0 <= row < 3: 

983 r = row * 3 

984 return self[r:r+3] 

985 raise _IndexError(row=row) 

986 

987 def unrotate(self, xyz, *xyz0): 

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

989 

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

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

992 

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

994 

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

996 ''' 

997 if xyz0: 

998 if len(xyz0) != len(xyz): 

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

1000 _xyz = _1_0_1T + xyz 

1001 # x' = x0 + M[0] * x + M[1] * y + M[2] * z 

1002 # y' = y0 + M[3] * x + M[4] * y + M[5] * z 

1003 # z' = z0 + M[6] * x + M[7] * y + M[8] * z 

1004 xyz_ = (fdot(_xyz, xyz0[0], *self[0:3]), # .row(0) 

1005 fdot(_xyz, xyz0[1], *self[3:6]), # .row(1) 

1006 fdot(_xyz, xyz0[2], *self[6:9])) # .row(2) 

1007 else: 

1008 # x' = M[0] * x + M[1] * y + M[2] * z 

1009 # y' = M[3] * x + M[4] * y + M[5] * z 

1010 # z' = M[6] * x + M[7] * y + M[8] * z 

1011 xyz_ = (fdot(xyz, *self[0:3]), # .row(0) 

1012 fdot(xyz, *self[3:6]), # .row(1) 

1013 fdot(xyz, *self[6:9])) # .row(2) 

1014 return xyz_ 

1015 

1016 

1017class Ecef9Tuple(_NamedTuple): 

1018 '''9-Tuple C{(x, y, z, lat, lon, height, C, M, datum)} with I{geocentric} 

1019 C{x}, C{y} and C{z} plus I{geodetic} C{lat}, C{lon} and C{height}, case 

1020 C{C} (see the C{Ecef*.reverse} methods) and optionally, the rotation 

1021 matrix C{M} (L{EcefMatrix}) and C{datum}, with C{lat} and C{lon} in 

1022 C{degrees} and C{x}, C{y}, C{z} and C{height} in C{meter}, conventionally. 

1023 ''' 

1024 _Names_ = (_x_, _y_, _z_, _lat_, _lon_, _height_, _C_, _M_, _datum_) 

1025 _Units_ = ( Meter, Meter, Meter, Lat, Lon, Height, Int, _Pass, _Pass) 

1026 

1027 @property_ROver 

1028 def _CartesianBase(self): 

1029 '''(INTERNAL) Get class C{CartesianBase}, I{once}. 

1030 ''' 

1031 return _MODS.cartesianBase.CartesianBase # overwrite property_ROver 

1032 

1033 @deprecated_method 

1034 def convertDatum(self, datum2): # for backward compatibility 

1035 '''DEPRECATED, use method L{toDatum}.''' 

1036 return self.toDatum(datum2) 

1037 

1038 @Property_RO 

1039 def lam(self): 

1040 '''Get the longitude in C{radians} (C{float}). 

1041 ''' 

1042 return self.philam.lam 

1043 

1044 @Property_RO 

1045 def lamVermeille(self): 

1046 '''Get the longitude in C{radians} M{[-PI*3/2..+PI*3/2]} after U{Vermeille 

1047 <https://Search.ProQuest.com/docview/639493848>} (2004), page 95. 

1048 

1049 @see: U{Karney<https://GeographicLib.SourceForge.io/C++/doc/geocentric.html>}, 

1050 U{Vermeille<https://Search.ProQuest.com/docview/847292978>} 2011, pp 112-113, 116 

1051 and U{Featherstone, et.al.<https://Search.ProQuest.com/docview/872827242>}, page 7. 

1052 ''' 

1053 x, y = self.x, self.y 

1054 if y > EPS0: 

1055 r = atan2(x, hypot(y, x) + y) * _N_2_0 + PI_2 

1056 elif y < -EPS0: 

1057 r = atan2(x, hypot(y, x) - y) * _2_0 - PI_2 

1058 else: # y == 0 

1059 r = PI if x < 0 else _0_0 

1060 return Lam(Vermeille=r) 

1061 

1062 @Property_RO 

1063 def latlon(self): 

1064 '''Get the lat-, longitude in C{degrees} (L{LatLon2Tuple}C{(lat, lon)}). 

1065 ''' 

1066 return LatLon2Tuple(self.lat, self.lon, name=self.name) 

1067 

1068 @Property_RO 

1069 def latlonheight(self): 

1070 '''Get the lat-, longitude in C{degrees} and height (L{LatLon3Tuple}C{(lat, lon, height)}). 

1071 ''' 

1072 return self.latlon.to3Tuple(self.height) 

1073 

1074 @Property_RO 

1075 def latlonheightdatum(self): 

1076 '''Get the lat-, longitude in C{degrees} with height and datum (L{LatLon4Tuple}C{(lat, lon, height, datum)}). 

1077 ''' 

1078 return self.latlonheight.to4Tuple(self.datum) 

1079 

1080 @Property_RO 

1081 def latlonVermeille(self): 

1082 '''Get the latitude and I{Vermeille} longitude in C{degrees [-225..+225]} (L{LatLon2Tuple}C{(lat, lon)}). 

1083 

1084 @see: Property C{lonVermeille}. 

1085 ''' 

1086 return LatLon2Tuple(self.lat, self.lonVermeille, name=self.name) 

1087 

1088 @Property_RO 

1089 def lonVermeille(self): 

1090 '''Get the longitude in C{degrees [-225..+225]} after U{Vermeille 

1091 <https://Search.ProQuest.com/docview/639493848>} (2004), p 95. 

1092 

1093 @see: Property C{lamVermeille}. 

1094 ''' 

1095 return Lon(Vermeille=degrees(self.lamVermeille)) 

1096 

1097 @Property_RO 

1098 def phi(self): 

1099 '''Get the latitude in C{radians} (C{float}). 

1100 ''' 

1101 return self.philam.phi 

1102 

1103 @Property_RO 

1104 def philam(self): 

1105 '''Get the lat-, longitude in C{radians} (L{PhiLam2Tuple}C{(phi, lam)}). 

1106 ''' 

1107 return PhiLam2Tuple(radians(self.lat), radians(self.lon), name=self.name) 

1108 

1109 @Property_RO 

1110 def philamheight(self): 

1111 '''Get the lat-, longitude in C{radians} and height (L{PhiLam3Tuple}C{(phi, lam, height)}). 

1112 ''' 

1113 return self.philam.to3Tuple(self.height) 

1114 

1115 @Property_RO 

1116 def philamheightdatum(self): 

1117 '''Get the lat-, longitude in C{radians} with height and datum (L{PhiLam4Tuple}C{(phi, lam, height, datum)}). 

1118 ''' 

1119 return self.philamheight.to4Tuple(self.datum) 

1120 

1121 @Property_RO 

1122 def philamVermeille(self): 

1123 '''Get the latitude and I{Vermeille} longitude in C{radians [-PI*3/2..+PI*3/2]} (L{PhiLam2Tuple}C{(phi, lam)}). 

1124 

1125 @see: Property C{lamVermeille}. 

1126 ''' 

1127 return PhiLam2Tuple(radians(self.lat), self.lamVermeille, name=self.name) 

1128 

1129 def toCartesian(self, Cartesian=None, **Cartesian_kwds): 

1130 '''Return the geocentric C{(x, y, z)} coordinates as an ellipsoidal or spherical 

1131 C{Cartesian}. 

1132 

1133 @kwarg Cartesian: Optional class to return C{(x, y, z)} (L{ellipsoidalKarney.Cartesian}, 

1134 L{ellipsoidalNvector.Cartesian}, L{ellipsoidalVincenty.Cartesian}, 

1135 L{sphericalNvector.Cartesian} or L{sphericalTrigonometry.Cartesian}) 

1136 or C{None}. 

1137 @kwarg Cartesian_kwds: Optional, additional B{C{Cartesian}} keyword arguments, ignored 

1138 if C{B{Cartesian} is None}. 

1139 

1140 @return: A C{B{Cartesian}(x, y, z, **B{Cartesian_kwds})} instance or 

1141 a L{Vector4Tuple}C{(x, y, z, h)} if C{B{Cartesian} is None}. 

1142 

1143 @raise TypeError: Invalid B{C{Cartesian}} or B{C{Cartesian_kwds}}. 

1144 ''' 

1145 if Cartesian in (None, Vector4Tuple): 

1146 r = self.xyzh 

1147 elif Cartesian is Vector3Tuple: 

1148 r = self.xyz 

1149 else: 

1150 _xsubclassof(self._CartesianBase, Cartesian=Cartesian) 

1151 r = Cartesian(self, **_name1__(Cartesian_kwds, _or_nameof=self)) 

1152 return r 

1153 

1154 def toDatum(self, datum2, **name): 

1155 '''Convert this C{Ecef9Tuple} to an other datum. 

1156 

1157 @arg datum2: Datum to convert I{to} (L{Datum}). 

1158 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1159 

1160 @return: The converted 9-Tuple (C{Ecef9Tuple}). 

1161 

1162 @raise TypeError: The B{C{datum2}} is not a L{Datum}. 

1163 ''' 

1164 n = _name__(name, _or_nameof=self) 

1165 if self.datum in (None, datum2): # PYCHOK _Names_ 

1166 r = self.copy(name=n) 

1167 else: 

1168 c = self._CartesianBase(self, datum=self.datum, name=n) # PYCHOK _Names_ 

1169 # c.toLatLon converts datum, x, y, z, lat, lon, etc. 

1170 # and returns another Ecef9Tuple iff LatLon is None 

1171 r = c.toLatLon(datum=datum2, LatLon=None) 

1172 return r 

1173 

1174 def toLatLon(self, LatLon=None, **LatLon_kwds): 

1175 '''Return the geodetic C{(lat, lon, height[, datum])} coordinates. 

1176 

1177 @kwarg LatLon: Optional class to return C{(lat, lon, height[, datum])} 

1178 or C{None}. 

1179 @kwarg LatLon_kwds: Optional B{C{height}}, B{C{datum}} and other 

1180 B{C{LatLon}} keyword arguments. 

1181 

1182 @return: An instance of C{B{LatLon}(lat, lon, **B{LatLon_kwds})} 

1183 or if C{B{LatLon} is None}, a L{LatLon3Tuple}C{(lat, lon, 

1184 height)} respectively L{LatLon4Tuple}C{(lat, lon, height, 

1185 datum)} depending on whether C{datum} is un-/specified. 

1186 

1187 @raise TypeError: Invalid B{C{LatLon}} or B{C{LatLon_kwds}}. 

1188 ''' 

1189 lat, lon, D = self.lat, self.lon, self.datum # PYCHOK Ecef9Tuple 

1190 kwds = _name1__(LatLon_kwds, _or_nameof=self) 

1191 kwds = _xkwds(kwds, height=self.height, datum=D) # PYCHOK Ecef9Tuple 

1192 d = kwds.get(_datum_, LatLon) 

1193 if LatLon is None: 

1194 r = LatLon3Tuple(lat, lon, kwds[_height_], name=kwds[_name_]) 

1195 if d is not None: 

1196 # assert d is not LatLon 

1197 r = r.to4Tuple(d) # checks type(d) 

1198 else: 

1199 if d is None: 

1200 _ = kwds.pop(_datum_) # remove None datum 

1201 r = LatLon(lat, lon, **kwds) 

1202 _xdatum(_xattr(r, datum=D), D) 

1203 return r 

1204 

1205 def toLocal(self, ltp, Xyz=None, **Xyz_kwds): 

1206 '''Convert this geocentric to I{local} C{x}, C{y} and C{z}. 

1207 

1208 @kwarg ltp: The I{local tangent plane} (LTP) to use (L{Ltp}). 

1209 @kwarg Xyz: Optional class to return C{x}, C{y} and C{z} 

1210 (L{XyzLocal}, L{Enu}, L{Ned}) or C{None}. 

1211 @kwarg Xyz_kwds: Optional, additional B{C{Xyz}} keyword 

1212 arguments, ignored if C{B{Xyz} is None}. 

1213 

1214 @return: An B{C{Xyz}} instance or if C{B{Xyz} is None}, 

1215 a L{Local9Tuple}C{(x, y, z, lat, lon, height, 

1216 ltp, ecef, M)} with C{M=None}, always. 

1217 

1218 @raise TypeError: Invalid B{C{ltp}}. 

1219 ''' 

1220 return _MODS.ltp._xLtp(ltp)._ecef2local(self, Xyz, Xyz_kwds) 

1221 

1222 def toVector(self, Vector=None, **Vector_kwds): 

1223 '''Return the geocentric C{(x, y, z)} coordinates as vector. 

1224 

1225 @kwarg Vector: Optional vector class to return C{(x, y, z)} or 

1226 C{None}. 

1227 @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword 

1228 arguments, ignored if C{B{Vector} is None}. 

1229 

1230 @return: A C{Vector}C{(x, y, z, **Vector_kwds)} instance or a 

1231 L{Vector3Tuple}C{(x, y, z)} if C{B{Vector} is None}. 

1232 

1233 @see: Propertes C{xyz} and C{xyzh} 

1234 ''' 

1235 return self.xyz if Vector is None else Vector( 

1236 *self.xyz, **_name1__(Vector_kwds, _or_nameof=self)) # PYCHOK Ecef9Tuple 

1237 

1238# def _T_x_M(self, T): 

1239# '''(INTERNAL) Update M{self.M = T.multiply(self.M)}. 

1240# ''' 

1241# return self.dup(M=T.multiply(self.M)) 

1242 

1243 @Property_RO 

1244 def xyz(self): 

1245 '''Get the geocentric C{(x, y, z)} coordinates (L{Vector3Tuple}C{(x, y, z)}). 

1246 ''' 

1247 return Vector3Tuple(self.x, self.y, self.z, name=self.name) 

1248 

1249 @Property_RO 

1250 def xyzh(self): 

1251 '''Get the geocentric C{(x, y, z)} coordinates and C{height} (L{Vector4Tuple}C{(x, y, z, h)}) 

1252 ''' 

1253 return self.xyz.to4Tuple(self.height) 

1254 

1255 

1256def _4Ecef(this, Ecef): # in .datums.Datum.ecef, .ellipsoids.Ellipsoid.ecef 

1257 '''Return an ECEF converter for C{this} L{Datum} or L{Ellipsoid}. 

1258 ''' 

1259 if Ecef is None: 

1260 Ecef = EcefKarney 

1261 else: 

1262 _xinstanceof(*_Ecefs, Ecef=Ecef) 

1263 return Ecef(this, name=this.name) 

1264 

1265 

1266def _llhn4(latlonh, lon, height, suffix=NN, Error=EcefError, **name): # in .ltp 

1267 '''(INTERNAL) Get a C{(lat, lon, h, name)} 4-tuple. 

1268 ''' 

1269 try: 

1270 lat, lon = latlonh.lat, latlonh.lon 

1271 h = _xattr(latlonh, height=_xattr(latlonh, h=height)) 

1272 n = _name__(name, _or_nameof=latlonh) # == latlonh._name__(name) 

1273 except AttributeError: 

1274 lat, h, n = latlonh, height, _name__(**name) 

1275 try: 

1276 return Lat(lat), Lon(lon), Height(h), n 

1277 except (TypeError, ValueError) as x: 

1278 t = _lat_, _lon_, _height_ 

1279 if suffix: 

1280 t = (_ + suffix for _ in t) 

1281 d = dict(zip(t, (lat, lon, h))) 

1282 raise Error(cause=x, **d) 

1283 

1284 

1285def _xEcef(Ecef): # PYCHOK .latlonBase 

1286 '''(INTERNAL) Validate B{C{Ecef}} I{class}. 

1287 ''' 

1288 if issubclassof(Ecef, _EcefBase): 

1289 return Ecef 

1290 raise _TypesError(_Ecef_, Ecef, *_Ecefs) 

1291 

1292 

1293# kwd lon00 unused but will throw a TypeError if misspelled, etc. 

1294def _xyzn4(xyz, y, z, Types, Error=EcefError, lon00=0, # PYCHOK unused 

1295 _xyz_y_z_names=_xyz_y_z, **name): # in .ltp 

1296 '''(INTERNAL) Get an C{(x, y, z, name)} 4-tuple. 

1297 ''' 

1298 try: 

1299 n = _name__(name, _or_nameof=xyz) # == xyz._name__(name) 

1300 try: 

1301 t = xyz.x, xyz.y, xyz.z, n 

1302 if not isinstance(xyz, Types): 

1303 raise _TypesError(_xyz_y_z_names[0], xyz, *Types) 

1304 except AttributeError: 

1305 t = map1(float, xyz, y, z) + (n,) 

1306 except (TypeError, ValueError) as x: 

1307 d = dict(zip(_xyz_y_z_names, (xyz, y, z))) 

1308 raise Error(cause=x, **d) 

1309 return t 

1310# assert _xyz_y_z == _args_kwds_names(_xyzn4)[:3] 

1311 

1312 

1313_Ecefs = (EcefKarney, EcefSudano, EcefVeness, EcefYou, 

1314 EcefFarrell21, EcefFarrell22) 

1315__all__ += _ALL_DOCS(_EcefBase) 

1316 

1317# **) MIT License 

1318# 

1319# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

1320# 

1321# Permission is hereby granted, free of charge, to any person obtaining a 

1322# copy of this software and associated documentation files (the "Software"), 

1323# to deal in the Software without restriction, including without limitation 

1324# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1325# and/or sell copies of the Software, and to permit persons to whom the 

1326# Software is furnished to do so, subject to the following conditions: 

1327# 

1328# The above copyright notice and this permission notice shall be included 

1329# in all copies or substantial portions of the Software. 

1330# 

1331# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1332# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1333# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1334# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1335# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1336# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1337# OTHER DEALINGS IN THE SOFTWARE.