1

2# -*- coding: utf-8 -*-

3

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

5

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

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

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

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

10 3709199_An_exact_conversion_from_an_Earth-centered_coordinate_system_to_latitude_longitude_and_altitude>},

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

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

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

14 classes 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

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

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

19

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

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

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

23

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

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

26

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

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

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

30

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

32

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

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

35(2002) 76, page 451-454.

36

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

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

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

40 105-117 (U{preprint<https://ArXiv.org/abs/1102.1215v1>}). Vermeille similarly updated his method

41 in 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

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

44

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

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

47 nm (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)

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

52 C{lon00} to configure that value.

53

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

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

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

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

58'''

59

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

61 _xinstanceof, _xsubclassof, typename # _args_kwds_names

62 from 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!

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

67 from pygeodesy.ecefLocals import _EcefLocal

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

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

70 _xattr, _xdatum, _xkwds, _xkwds_get

71 from pygeodesy.fmath import cbrt, fdot, hypot, hypot1, hypot2_, sqrt0

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

73# from pygeodesy.internals import typename # from .basics

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

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

76 _x_, _xyz_, _y_, _z_

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

78 from pygeodesy.named import _name__, _name1__, _NamedBase, _NamedTuple, _Pass, _xnamed

79 from pygeodesy.namedTuples import LatLon2Tuple, LatLon3Tuple, \

80 PhiLam2Tuple, Vector3Tuple, Vector4Tuple

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

82 property_ROver, property_doc_

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

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

85 Phi, Scalar, Scalar_

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

87 sincos2, sincos2_, sincos2d, sincos2d_

88# from pygeodesy.vector3d import Vector3d # _MODS

89

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

91

92__all__ = _ALL_LAZY.ecef

93__version__ = '25.04.28'

94

95_Ecef_ = 'Ecef'

96_prolate_ = 'prolate'

97_TRIPS = 33 # 8..9 sufficient, EcefSudano.reverse

98_xyz_y_z = _xyz_, _y_, _z_ # _args_kwds_names(_xyzn4)[:3]

99

100

101 class EcefError(_ValueError):

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

103 '''

104 pass

105

106

107 class _EcefBase(_NamedBase):

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

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

110 '''

111 _datum = _WGS84

112 _E = _EWGS84

113 _isYou = False

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

115

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

117 '''New C{Ecef*} converter.

118

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

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

121 equatorial radius (C{meter}).

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

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

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

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

126 C{reverse} method.

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

128

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

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

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

132 or B{C{f}} not less than 1.0.

133 '''

134 try:

135 E = a_ellipsoid

136 if f is None:

137 pass

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

139 E = a_f2Tuple(E, f)

140 else:

141 raise ValueError() # _invalid_

142

143 if not _isin(E, _EWGS84, _WGS84):

144 d = _ellipsoidal_datum(E, **name)

145 E = d.ellipsoid

146 if E.a < EPS or E.f > EPS1:

147 raise ValueError() # _invalid_

148 self._datum = d

149 self._E = E

150

151 except (TypeError, ValueError) as x:

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

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

154

155 if name:

156 self.name = name

157 if lon00 is not INT0:

158 self.lon00 = lon00

159

160 def __eq__(self, other):

161 '''Compare this and an other Ecef.

162

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

164

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

166 '''

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

168 other.ellipsoid == self.ellipsoid)

169

170 @Property_RO

171 def datum(self):

172 '''Get the datum (L{Datum}).

173 '''

174 return self._datum

175

176 @Property_RO

177 def ellipsoid(self):

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

179 '''

180 return self._E

181

182 @Property_RO

183 def equatoradius(self):

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

185 '''

186 return self.ellipsoid.a

187

188 a = equatorialRadius = equatoradius # Karney property

189

190 @Property_RO

191 def flattening(self): # Karney property

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

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

194 '''

195 return self.ellipsoid.f

196

197 f = flattening

198

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

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

201 '''

202 if _philam: # lat, lon in radians

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

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

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

206 else:

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

208

209 E = self.ellipsoid

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

211 z = (h + n * E.e21) * sa

212 x = (h + n) * ca

213

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

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

216 0, m, self.datum,

217 name=self._name__(name))

218

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

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

221

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

223 latitude (C{degrees}).

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

225 (C{degrees}).

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

227 the surface of the ellipsoid.

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

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

230

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

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

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

234 and C{datum} if available.

235

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

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

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

239

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

241 and avoid double angle conversions.

242 '''

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

244 return self._forward(*llhn, M=M)

245

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

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

248 in I{radians}.

249

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

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

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

253 the surface of the ellipsoid.

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

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

256

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

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

259 C{degrees180(B{lam})}.

260

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

262 '''

263 try: # like function C{_llhn4} below

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

265 except (TypeError, ValueError) as x:

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

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

268

269 @property_ROver

270 def _Geocentrics(self):

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

272 '''

273 return (Ecef9Tuple, # overwrite property_ROver

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

275

276 @property

277 def lon00(self):

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

279 '''

280 return self._lon00

281

282 @lon00.setter # PYCHOK setter!

283 def lon00(self, lon00):

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

285 '''

286 self._lon00 = Degrees(lon00=lon00)

287

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

289 '''Creation a rotation matrix.

290

291 @arg sa: C{sin(phi)} (C{float}).

292 @arg ca: C{cos(phi)} (C{float}).

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

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

295

296 @return: An L{EcefMatrix}.

297 '''

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

299

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

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

302 '''

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

304

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

306 '''I{Must be overloaded}.'''

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

308

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

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

311

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

313

314 @return: This C{Ecef*} (C{str}).

315 '''

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

317

318

319 class EcefFarrell21(_EcefBase):

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

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

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

323 '''

324

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

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

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

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

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

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 c = e4 * p2 * F / G**3

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

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

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) # z0 / z

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

394 class 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

454 class 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

608 class 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

697 class 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

778 class 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 _isYou = True

789

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

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

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

793

794 @staticmethod

795 def _ee2(E):

796 e2 = E.a2 - E.b2

797 if e2 < 0 or E.f < 0:

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

799 return sqrt0(e2), e2

800

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

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

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

804

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

806 coordinate (C{meter}).

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

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

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

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

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

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

813

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

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

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

817 available.

818

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

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

821 ellipsoid is I{prolate}.

822 '''

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

824 q = hypot(x, y) # R

825

826 E = self.ellipsoid

827 e, e2 = self._ee2

828

829 u = hypot2_(x, y, z) - e2

830 u += hypot(u, e * z * _2_0)

831 u *= _0_5

832 if u > EPS02:

833 u = sqrt(u)

834 p = hypot(u, e)

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

836 sB, cB = sincos2(B)

837 if cB and sB:

838 p *= E.a

839 d = (p / cB - e2 * cB) / sB

840 if isnon0(d):

841 B += fsumf_(u * E.b, -p, e2) / d

842 sB, cB = sincos2(B)

843 elif u < (-EPS2):

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

845 else:

846 sB, cB = _copysign_1_0(z), _0_0

847

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

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

850

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

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

853 h = neg(h) # inside ellipsoid

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

855 1, None, self.datum, # C=1, M=None

856 name=self._name__(name))

857

858

859 class EcefMatrix(_NamedTuple):

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

861

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

863 Geographic_coordinate_conversion#From_ECEF_to_ENU>} and

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

865 '''

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

867 '_1_0_', '_1_1_', '_1_2_',

868 '_2_0_', '_2_1_', '_2_2_')

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

870

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

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

873 '''

874 _NamedTuple._validate(self, underOK=True)

875

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

877 '''New L{EcefMatrix} matrix.

878

879 @arg sa: C{sin(phi)} (C{float}).

880 @arg ca: C{cos(phi)} (C{float}).

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

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

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

884

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

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

887 '''

888 t = sa, ca, sb, cb

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

890 t += _more # ... from .multiply

891

892 elif max(map(fabs, t)) > _1_0:

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

894

895 else: # build matrix from the following quaternion operations

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

897 # or

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

899 # where

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

901

902 # Local X axis (East) in geocentric coords

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

904 # Local Y axis (North) in geocentric coords

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

906 # Local Z axis (Up) in geocentric coords

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

908 t = (-sb, -cb * sa, cb * ca,

909 cb, -sb * sa, sb * ca,

910 _0_0, ca, sa)

911

912 return _NamedTuple.__new__(cls, *t)

913

914 def column(self, column):

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

916 '''

917 if 0 <= column < 3:

918 return self[column::3]

919 raise _IndexError(column=column)

920

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

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

923

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

925 '''

926 return self.classof(*self)

927

928 __copy__ = __deepcopy__ = copy

929

930 @Property_RO

931 def matrix3(self):

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

933 '''

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

935

936 @Property_RO

937 def matrixTransposed3(self):

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

939 '''

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

941

942 def multiply(self, other):

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

944 with an other matrix.

945

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

947

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

949

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

951 '''

952 _xinstanceof(EcefMatrix, other=other)

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

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

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

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

957 return _xnamed(EcefMatrix(*X), typename(EcefMatrix.multiply))

958

959 def rotate(self, xyz, *xyz0):

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

961

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

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

964

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

966

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

968 '''

969 if xyz0:

970 if len(xyz0) != len(xyz):

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

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

973

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

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

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

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

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

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

980

981 def row(self, row):

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

983 '''

984 if 0 <= row < 3:

985 r = row * 3

986 return self[r:r+3]

987 raise _IndexError(row=row)

988

989 def unrotate(self, xyz, *xyz0):

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

991

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

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

994

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

996

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

998 '''

999 if xyz0:

1000 if len(xyz0) != len(xyz):

1001 raise LenError(self.unrotate, xyz0=len(xyz0), xyz=len(xyz))

1002 _xyz = _1_0_1T + xyz

1003 # x' = x0 + M[0] * x + M[1] * y + M[2] * z

1004 # y' = y0 + M[3] * x + M[4] * y + M[5] * z

1005 # z' = z0 + M[6] * x + M[7] * y + M[8] * z

1006 xyz_ = (fdot(_xyz, xyz0[0], *self[0:3]), # .row(0)

1007 fdot(_xyz, xyz0[1], *self[3:6]), # .row(1)

1008 fdot(_xyz, xyz0[2], *self[6:9])) # .row(2)

1009 else:

1010 # x' = M[0] * x + M[1] * y + M[2] * z

1011 # y' = M[3] * x + M[4] * y + M[5] * z

1012 # z' = M[6] * x + M[7] * y + M[8] * z

1013 xyz_ = (fdot(xyz, *self[0:3]), # .row(0)

1014 fdot(xyz, *self[3:6]), # .row(1)

1015 fdot(xyz, *self[6:9])) # .row(2)

1016 return xyz_

1017

1018

1019 class Ecef9Tuple(_NamedTuple, _EcefLocal):

1020 '''9-Tuple C{(x, y, z, lat, lon, height, C, M, datum)} with I{geocentric} C{x},

1021 C{y} and C{z} plus I{geodetic} C{lat}, C{lon} and C{height}, case C{C} (see

1022 the C{Ecef*.reverse} methods) and optionally, rotation matrix C{M} (L{EcefMatrix})

1023 and C{datum}, with C{lat} and C{lon} in C{degrees} and C{x}, C{y}, C{z} and

1024 C{height} in C{meter}, conventionally.

1025 '''

1026 _Names_ = (_x_, _y_, _z_, _lat_, _lon_, _height_, _C_, _M_, _datum_)

1027 _Units_ = ( Meter, Meter, Meter, Lat, Lon, Height, Int, _Pass, _Pass)

1028

1029 @property_ROver

1030 def _CartesianBase(self):

1031 '''(INTERNAL) Get class C{CartesianBase}, I{once}.

1032 '''

1033 return _MODS.cartesianBase.CartesianBase # overwrite property_ROver

1034

1035 @deprecated_method

1036 def convertDatum(self, datum2): # for backward compatibility

1037 '''DEPRECATED, use method L{toDatum}.'''

1038 return self.toDatum(datum2)

1039

1040 @property_RO

1041 def _ecef9(self): # in ._EcefLocal._Ltp_ecef2local

1042 return self

1043

1044 @Property_RO

1045 def lam(self):

1046 '''Get the longitude in C{radians} (C{float}).

1047 '''

1048 return self.philam.lam

1049

1050 @Property_RO

1051 def lamVermeille(self):

1052 '''Get the longitude in C{radians} M{[-PI*3/2..+PI*3/2]} after U{Vermeille

1053 <https://Search.ProQuest.com/docview/639493848>} (2004), page 95.

1054

1055 @see: U{Karney<https://GeographicLib.SourceForge.io/C++/doc/geocentric.html>},

1056 U{Vermeille<https://Search.ProQuest.com/docview/847292978>} 2011, pp 112-113, 116

1057 and U{Featherstone, et.al.<https://Search.ProQuest.com/docview/872827242>}, page 7.

1058 '''

1059 x, y = self.x, self.y

1060 if y > EPS0:

1061 r = atan2(x, hypot(y, x) + y) * _N_2_0 + PI_2

1062 elif y < -EPS0:

1063 r = atan2(x, hypot(y, x) - y) * _2_0 - PI_2

1064 else: # y == 0

1065 r = PI if x < 0 else _0_0

1066 return Lam(Vermeille=r)

1067

1068 @Property_RO

1069 def latlon(self):

1070 '''Get the lat-, longitude in C{degrees} (L{LatLon2Tuple}C{(lat, lon)}).

1071 '''

1072 return LatLon2Tuple(self.lat, self.lon, name=self.name)

1073

1074 @Property_RO

1075 def latlonheight(self):

1076 '''Get the lat-, longitude in C{degrees} and height (L{LatLon3Tuple}C{(lat, lon, height)}).

1077 '''

1078 return self.latlon.to3Tuple(self.height)

1079

1080 @Property_RO

1081 def latlonheightdatum(self):

1082 '''Get the lat-, longitude in C{degrees} with height and datum (L{LatLon4Tuple}C{(lat, lon, height, datum)}).

1083 '''

1084 return self.latlonheight.to4Tuple(self.datum)

1085

1086 @Property_RO

1087 def latlonVermeille(self):

1088 '''Get the latitude and I{Vermeille} longitude in C{degrees [-225..+225]} (L{LatLon2Tuple}C{(lat, lon)}).

1089

1090 @see: Property C{lonVermeille}.

1091 '''

1092 return LatLon2Tuple(self.lat, self.lonVermeille, name=self.name)

1093

1094 @Property_RO

1095 def lonVermeille(self):

1096 '''Get the longitude in C{degrees [-225..+225]} after U{Vermeille

1097 <https://Search.ProQuest.com/docview/639493848>} 2004, p 95.

1098

1099 @see: Property C{lamVermeille}.

1100 '''

1101 return Lon(Vermeille=degrees(self.lamVermeille))

1102

1103 @Property_RO

1104 def phi(self):

1105 '''Get the latitude in C{radians} (C{float}).

1106 '''

1107 return self.philam.phi

1108

1109 @Property_RO

1110 def philam(self):

1111 '''Get the lat-, longitude in C{radians} (L{PhiLam2Tuple}C{(phi, lam)}).

1112 '''

1113 return PhiLam2Tuple(radians(self.lat), radians(self.lon), name=self.name)

1114

1115 @Property_RO

1116 def philamheight(self):

1117 '''Get the lat-, longitude in C{radians} and height (L{PhiLam3Tuple}C{(phi, lam, height)}).

1118 '''

1119 return self.philam.to3Tuple(self.height)

1120

1121 @Property_RO

1122 def philamheightdatum(self):

1123 '''Get the lat-, longitude in C{radians} with height and datum (L{PhiLam4Tuple}C{(phi, lam, height, datum)}).

1124 '''

1125 return self.philamheight.to4Tuple(self.datum)

1126

1127 @Property_RO

1128 def philamVermeille(self):

1129 '''Get the latitude and I{Vermeille} longitude in C{radians [-PI*3/2..+PI*3/2]} (L{PhiLam2Tuple}C{(phi, lam)}).

1130

1131 @see: Property C{lamVermeille}.

1132 '''

1133 return PhiLam2Tuple(radians(self.lat), self.lamVermeille, name=self.name)

1134

1135 phiVermeille = phi

1136

1137 def toCartesian(self, Cartesian=None, **Cartesian_kwds):

1138 '''Return the geocentric C{(x, y, z)} coordinates as an ellipsoidal or spherical

1139 C{Cartesian}.

1140

1141 @kwarg Cartesian: Optional class to return C{(x, y, z)} (L{ellipsoidalKarney.Cartesian},

1142 L{ellipsoidalNvector.Cartesian}, L{ellipsoidalVincenty.Cartesian},

1143 L{sphericalNvector.Cartesian} or L{sphericalTrigonometry.Cartesian})

1144 or C{None}.

1145 @kwarg Cartesian_kwds: Optionally, additional B{C{Cartesian}} keyword arguments, ignored

1146 if C{B{Cartesian} is None}.

1147

1148 @return: A B{C{Cartesian}} instance or a L{Vector4Tuple}C{(x, y, z, h)} if C{B{Cartesian}

1149 is None}.

1150

1151 @raise TypeError: Invalid B{C{Cartesian}} or B{C{Cartesian_kwds}} item.

1152 '''

1153 if _isin(Cartesian, None, Vector4Tuple):

1154 r = self.xyzh

1155 elif Cartesian is Vector3Tuple:

1156 r = self.xyz

1157 else:

1158 _xsubclassof(self._CartesianBase, Cartesian=Cartesian)

1159 r = Cartesian(self, **_name1__(Cartesian_kwds, _or_nameof=self))

1160 return r

1161

1162 def toDatum(self, datum2, **name):

1163 '''Convert this C{Ecef9Tuple} to an other datum.

1164

1165 @arg datum2: Datum to convert I{to} (L{Datum}).

1166 @kwarg name: Optional C{B{name}=NN} (C{str}).

1167

1168 @return: The converted 9-Tuple (C{Ecef9Tuple}).

1169

1170 @raise TypeError: The B{C{datum2}} is not a L{Datum}.

1171 '''

1172 n = _name__(name, _or_nameof=self)

1173 if _isin(self.datum, None, datum2): # PYCHOK _Names_

1174 r = self.copy(name=n)

1175 else:

1176 c = self._CartesianBase(self, datum=self.datum, name=n) # PYCHOK _Names_

1177 # c.toLatLon converts datum, x, y, z, lat, lon, etc.

1178 # and returns another Ecef9Tuple iff LatLon is None

1179 r = c.toLatLon(datum=datum2, LatLon=None)

1180 return r

1181

1182 def toLatLon(self, LatLon=None, **LatLon_kwds):

1183 '''Return the geodetic C{(lat, lon, height[, datum])} coordinates.

1184

1185 @kwarg LatLon: Optional class to return C{(lat, lon, height[, datum])} or C{None}.

1186 @kwarg LatLon_kwds: Optional B{C{height}}, B{C{datum}} and other B{C{LatLon}}

1187 keyword arguments.

1188

1189 @return: A B{C{LatLon}} instance or if C{B{LatLon} is None}, a L{LatLon4Tuple}C{(lat,

1190 lon, height, datum)} or L{LatLon3Tuple}C{(lat, lon, height)} if C{datum} is

1191 specified or not.

1192

1193 @raise TypeError: Invalid B{C{LatLon}} or B{C{LatLon_kwds}} item.

1194 '''

1195 lat, lon, D = self.lat, self.lon, self.datum # PYCHOK Ecef9Tuple

1196 kwds = _name1__(LatLon_kwds, _or_nameof=self)

1197 kwds = _xkwds(kwds, height=self.height, datum=D) # PYCHOK Ecef9Tuple

1198 d = kwds.get(_datum_, LatLon)

1199 if LatLon is None:

1200 r = LatLon3Tuple(lat, lon, kwds[_height_], name=kwds[_name_])

1201 if d is not None:

1202 # assert d is not LatLon

1203 r = r.to4Tuple(d) # checks type(d)

1204 else:

1205 if d is None:

1206 _ = kwds.pop(_datum_) # remove None datum

1207 r = LatLon(lat, lon, **kwds)

1208 _xdatum(_xattr(r, datum=D), D)

1209 return r

1210

1211 def toVector(self, Vector=None, **Vector_kwds):

1212 '''Return these geocentric C{(x, y, z)} coordinates as vector.

1213

1214 @kwarg Vector: Optional vector class to return C{(x, y, z)} or C{None}.

1215 @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword arguments,

1216 ignored if C{B{Vector} is None}.

1217

1218 @return: A B{C{Vector}} instance or a L{Vector3Tuple}C{(x, y, z)} if

1219 C{B{Vector} is None}.

1220

1221 @raise TypeError: Invalid B{C{Vector}} or B{C{Vector_kwds}} item.

1222

1223 @see: Propertes C{xyz} and C{xyzh}

1224 '''

1225 return self.xyz if Vector is None else Vector(

1226 *self.xyz, **_name1__(Vector_kwds, _or_nameof=self)) # PYCHOK Ecef9Tuple

1227

1228# def _T_x_M(self, T):

1229# '''(INTERNAL) Update M{self.M = T.multiply(self.M)}.

1230# '''

1231# return self.dup(M=T.multiply(self.M))

1232

1233 @Property_RO

1234 def xyz(self):

1235 '''Get the geocentric C{(x, y, z)} coordinates (L{Vector3Tuple}C{(x, y, z)}).

1236 '''

1237 return Vector3Tuple(self.x, self.y, self.z, name=self.name)

1238

1239 @Property_RO

1240 def xyzh(self):

1241 '''Get the geocentric C{(x, y, z)} coordinates and C{height} (L{Vector4Tuple}C{(x, y, z, h)})

1242 '''

1243 return self.xyz.to4Tuple(self.height)

1244

1245

1246 def _4Ecef(this, Ecef): # in .datums.Datum.ecef, .ellipsoids.Ellipsoid.ecef

1247 '''Return an ECEF converter for C{this} L{Datum} or L{Ellipsoid}.

1248 '''

1249 if Ecef is None:

1250 Ecef = EcefKarney

1251 else:

1252 _xinstanceof(*_Ecefs, Ecef=Ecef)

1253 return Ecef(this, name=this.name)

1254

1255

1256 def _llhn4(latlonh, lon, height, suffix=NN, Error=EcefError, **name): # in .ltp

1257 '''(INTERNAL) Get a C{(lat, lon, h, name)} 4-tuple.

1258 '''

1259 try:

1260 lat, lon = latlonh.lat, latlonh.lon

1261 h = _xattr(latlonh, height=_xattr(latlonh, h=height))

1262 n = _name__(name, _or_nameof=latlonh) # == latlonh._name__(name)

1263 except AttributeError:

1264 lat, h, n = latlonh, height, _name__(**name)

1265 try:

1266 return Lat(lat), Lon(lon), Height(h), n

1267 except (TypeError, ValueError) as x:

1268 t = _lat_, _lon_, _height_

1269 if suffix:

1270 t = (_ + suffix for _ in t)

1271 d = dict(zip(t, (lat, lon, h)))

1272 raise Error(cause=x, **d)

1273

1274

1275 def _xEcef(Ecef): # PYCHOK .latlonBase

1276 '''(INTERNAL) Validate B{C{Ecef}} I{class}.

1277 '''

1278 if issubclassof(Ecef, _EcefBase):

1279 return Ecef

1280 raise _TypesError(_Ecef_, Ecef, *_Ecefs)

1281

1282

1283# kwd lon00 unused but will throw a TypeError if misspelled, etc.

1284 def _xyzn4(xyz, y, z, Types, Error=EcefError, lon00=0, # PYCHOK unused

1285 _xyz_y_z_names=_xyz_y_z, **name): # in .ltp

1286 '''(INTERNAL) Get an C{(x, y, z, name)} 4-tuple.

1287 '''

1288 try:

1289 n = _name__(name, _or_nameof=xyz) # == xyz._name__(name)

1290 try:

1291 t = xyz.x, xyz.y, xyz.z, n

1292 if not isinstance(xyz, Types):

1293 raise _TypesError(_xyz_y_z_names[0], xyz, *Types)

1294 except AttributeError:

1295 t = map1(float, xyz, y, z) + (n,)

1296 except (TypeError, ValueError) as x:

1297 d = dict(zip(_xyz_y_z_names, (xyz, y, z)))

1298 raise Error(cause=x, **d)

1299 return t

1300# assert _xyz_y_z == _args_kwds_names(_xyzn4)[:3]

1301

1302

1303_Ecefs = (EcefKarney, EcefSudano, EcefVeness, EcefYou,

1304 EcefFarrell21, EcefFarrell22)

1305__all__ += _ALL_DOCS(_EcefBase)

1306

1307# **) MIT License

1308#

1309# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.

1310#

1311# Permission is hereby granted, free of charge, to any person obtaining a

1312# copy of this software and associated documentation files (the "Software"),

1313# to deal in the Software without restriction, including without limitation

1314# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1315# and/or sell copies of the Software, and to permit persons to whom the

1316# Software is furnished to do so, subject to the following conditions:

1317#

1318# The above copyright notice and this permission notice shall be included

1319# in all copies or substantial portions of the Software.

1320#

1321# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1322# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1323# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1324# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1325# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1326# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1327# OTHER DEALINGS IN THE SOFTWARE.