1

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

3

4 u'''A pure Python version of I{Karney}'s C{Exact Transverse Mercator} (ETM) projection.

5

6 Classes L{Etm}, L{ETMError} and L{ExactTransverseMercator}, transcoded from I{Karney}'s

7 C++ class U{TransverseMercatorExact<https://GeographicLib.SourceForge.io/C++/doc/

8 classGeographicLib_1_1TransverseMercatorExact.html>}, abbreviated as C{TMExact} below.

9

10 Class L{ExactTransverseMercator} provides C{Exact Transverse Mercator} projections while

11 instances of class L{Etm} represent ETM C{(easting, northing)} locations. See also

12 I{Karney}'s utility U{TransverseMercatorProj<https://GeographicLib.SourceForge.io/C++/doc/

13 TransverseMercatorProj.1.html>} and use C{"python[3] -m pygeodesy.etm ..."} to compared

14 the results.

15

16 Following is a copy of I{Karney}'s U{TransverseMercatorExact.hpp

17<https://GeographicLib.SourceForge.io/C++/doc/TransverseMercatorExact_8hpp_source.html>}

18 file C{Header}.

19

20 Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2008-2023) and licensed

21 under the MIT/X11 License. For more information, see the U{GeographicLib<https://

22 GeographicLib.SourceForge.io>} documentation.

23

24 The method entails using the U{Thompson Transverse Mercator<https://WikiPedia.org/

25 wiki/Transverse_Mercator_projection>} as an intermediate projection. The projections

26 from the intermediate coordinates to C{phi, lam} and C{x, y} are given by elliptic

27 functions. The inverse of these projections are found by Newton's method with a

28 suitable starting guess.

29

30 The relevant section of L.P. Lee's paper U{Conformal Projections Based On Jacobian

31 Elliptic Functions<https://DOI.org/10.3138/X687-1574-4325-WM62>} in part V, pp

32 67-101. The C++ implementation and notation closely follow Lee, with the following

33 exceptions::

34

35 Lee here Description

36

37 x/a xi Northing (unit Earth)

38

39 y/a eta Easting (unit Earth)

40

41 s/a sigma xi + i * eta

42

43 y x Easting

44

45 x y Northing

46

47 k e Eccentricity

48

49 k^2 mu Elliptic function parameter

50

51 k'^2 mv Elliptic function complementary parameter

52

53 m k Scale

54

55 zeta zeta Complex longitude = Mercator = chi in paper

56

57 s sigma Complex GK = zeta in paper

58

59 Minor alterations have been made in some of Lee's expressions in an attempt to

60 control round-off. For example, C{atanh(sin(phi))} is replaced by C{asinh(tan(phi))}

61 which maintains accuracy near C{phi = pi/2}. Such changes are noted in the code.

62'''

63# make sure int/int division yields float quotient, see .basics

64 from __future__ import division as _; del _ # PYCHOK semicolon

65

66 from pygeodesy.basics import map1, neg, neg_, _xinstanceof

67 from pygeodesy.constants import EPS, EPS02, PI_2, PI_4, _K0_UTM, \

68 _1_EPS, isnear0, _0_0, _0_1, _0_5, \

69 _1_0, _2_0, _3_0, _4_0, _90_0, _180_0

70 from pygeodesy.datums import _ellipsoidal_datum, _WGS84

71 from pygeodesy.elliptic import _ALL_LAZY, Elliptic

72# from pygeodesy.errors import _incompatible # from .named

73 from pygeodesy.fmath import cbrt, hypot, hypot1, hypot2

74 from pygeodesy.fsums import Fsum, fsum1f_

75 from pygeodesy.interns import NN, _COMMASPACE_, _DASH_, _near_, _SPACE_, \

76 _spherical_, _usage

77 from pygeodesy.karney import _copyBit, _diff182, _fix90, _norm2, _norm180, \

78 _tand, _unsigned2

79# from pygeodesy.lazily import _ALL_LAZY # from .elliptic

80 from pygeodesy.named import callername, _incompatible, _NamedBase

81 from pygeodesy.namedTuples import Forward4Tuple, Reverse4Tuple

82 from pygeodesy.props import deprecated_method, deprecated_property_RO, \

83 Property_RO, property_RO, _update_all, \

84 property_doc_

85 from pygeodesy.streprs import Fmt, fstr, pairs, unstr

86 from pygeodesy.units import Degrees, Scalar_

87 from pygeodesy.utily import atand, atan2d, sincos2

88 from pygeodesy.utm import _cmlon, _LLEB, _parseUTM5, _toBand, _toXtm8, \

89 _to7zBlldfn, Utm, UTMError

90

91 from math import asinh, atan2, degrees, radians, sinh, sqrt

92

93__all__ = _ALL_LAZY.etm

94__version__ = '23.08.05'

95

96_OVERFLOW = _1_EPS**2 # about 2e+31

97_TAYTOL = pow(EPS, 0.6)

98_TAYTOL2 = _TAYTOL * _2_0

99_TOL_10 = EPS * _0_1

100_TRIPS = 21 # C++ 10

101

102

103 def _overflow(x):

104 '''(INTERNAL) Like C{copysign0(OVERFLOW, B{x})}.

105 '''

106 return _copyBit(_OVERFLOW, x)

107

108

109 class ETMError(UTMError):

110 '''Exact Transverse Mercator (ETM) parse, projection or other

111 L{Etm} issue or L{ExactTransverseMercator} conversion failure.

112 '''

113 pass

114

115

116 class Etm(Utm):

117 '''Exact Transverse Mercator (ETM) coordinate, a sub-class of L{Utm},

118 a Universal Transverse Mercator (UTM) coordinate using the

119 L{ExactTransverseMercator} projection for highest accuracy.

120

121 @note: Conversion of (geodetic) lat- and longitudes to/from L{Etm}

122 coordinates is 3-4 times slower than to/from L{Utm}.

123

124 @see: Karney's U{Detailed Description<https://GeographicLib.SourceForge.io/

125 html/classGeographicLib_1_1TransverseMercatorExact.html#details>}.

126 '''

127 _Error = ETMError # see utm.UTMError

128 _exactTM = None

129

130 __init__ = Utm.__init__

131 '''New L{Etm} Exact Transverse Mercator coordinate, raising L{ETMError}s.

132

133 @see: L{Utm.__init__} for more information.

134

135 @example:

136

137 >>> import pygeodesy

138 >>> u = pygeodesy.Etm(31, 'N', 448251, 5411932)

139 '''

140

141 @property_doc_(''' the ETM projection (L{ExactTransverseMercator}).''')

142 def exactTM(self):

143 '''Get the ETM projection (L{ExactTransverseMercator}).

144 '''

145 if self._exactTM is None:

146 self.exactTM = self.datum.exactTM # ExactTransverseMercator(datum=self.datum)

147 return self._exactTM

148

149 @exactTM.setter # PYCHOK setter!

150 def exactTM(self, exactTM):

151 '''Set the ETM projection (L{ExactTransverseMercator}).

152

153 @raise ETMError: The B{C{exacTM}}'s datum incompatible

154 with this ETM coordinate's C{datum}.

155 '''

156 _xinstanceof(ExactTransverseMercator, exactTM=exactTM)

157

158 E = self.datum.ellipsoid

159 if E != exactTM.ellipsoid: # may be None

160 raise ETMError(repr(exactTM), txt=_incompatible(repr(E)))

161 self._exactTM = exactTM

162 self._scale0 = exactTM.k0

163

164 def parse(self, strETM, name=NN):

165 '''Parse a string to a similar L{Etm} instance.

166

167 @arg strETM: The ETM coordinate (C{str}),

168 see function L{parseETM5}.

169 @kwarg name: Optional instance name (C{str}),

170 overriding this name.

171

172 @return: The instance (L{Etm}).

173

174 @raise ETMError: Invalid B{C{strETM}}.

175

176 @see: Function L{pygeodesy.parseUPS5}, L{pygeodesy.parseUTM5}

177 and L{pygeodesy.parseUTMUPS5}.

178 '''

179 return parseETM5(strETM, datum=self.datum, Etm=self.classof,

180 name=name or self.name)

181

182 @deprecated_method

183 def parseETM(self, strETM): # PYCHOK no cover

184 '''DEPRECATED, use method L{Etm.parse}.

185 '''

186 return self.parse(strETM)

187

188 def toLatLon(self, LatLon=None, unfalse=True, **unused): # PYCHOK expected

189 '''Convert this ETM coordinate to an (ellipsoidal) geodetic point.

190

191 @kwarg LatLon: Optional, ellipsoidal class to return the geodetic

192 point (C{LatLon}) or C{None}.

193 @kwarg unfalse: Unfalse B{C{easting}} and B{C{northing}} if

194 C{falsed} (C{bool}).

195

196 @return: This ETM coordinate as (B{C{LatLon}}) or a

197 L{LatLonDatum5Tuple}C{(lat, lon, datum, gamma,

198 scale)} if B{C{LatLon}} is C{None}.

199

200 @raise ETMError: This ETM coordinate's C{exacTM} and this C{datum}

201 incompatible or no convergence transforming to

202 lat- and longitude.

203

204 @raise TypeError: Invalid or non-ellipsoidal B{C{LatLon}}.

205

206 @example:

207

208 >>> from pygeodesy import ellipsoidalVincenty as eV, Etm

209 >>> u = Etm(31, 'N', 448251.795, 5411932.678)

210 >>> ll = u.toLatLon(eV.LatLon) # 48°51′29.52″N, 002°17′40.20″E

211 '''

212 if not self._latlon or self._latlon._toLLEB_args != (unfalse, self.exactTM):

213 self._toLLEB(unfalse=unfalse)

214 return self._latlon5(LatLon)

215

216 def _toLLEB(self, unfalse=True, **unused): # PYCHOK signature

217 '''(INTERNAL) Compute (ellipsoidal) lat- and longitude.

218 '''

219 xTM, d = self.exactTM, self.datum

220 # double check that this and exactTM's ellipsoid match

221 if xTM._E != d.ellipsoid: # PYCHOK no cover

222 t = repr(d.ellipsoid)

223 raise ETMError(repr(xTM._E), txt=_incompatible(t))

224

225 e, n = self.eastingnorthing2(falsed=not unfalse)

226 lon0 = _cmlon(self.zone) if bool(unfalse) == self.falsed else None

227 lat, lon, g, k = xTM.reverse(e, n, lon0=lon0)

228

229 ll = _LLEB(lat, lon, datum=d, name=self.name) # utm._LLEB

230 ll._gamma = g

231 ll._scale = k

232 self._latlon5args(ll, _toBand, unfalse, xTM)

233

234 def toUtm(self): # PYCHOK signature

235 '''Copy this ETM to a UTM coordinate.

236

237 @return: The UTM coordinate (L{Utm}).

238 '''

239 return self._xcopy2(Utm)

240

241

242 class ExactTransverseMercator(_NamedBase):

243 '''Pure Python version of Karney's C++ class U{TransverseMercatorExact

244 <https://GeographicLib.SourceForge.io/C++/doc/TransverseMercatorExact_8cpp_source.html>},

245 a numerically exact transverse Mercator projection, further referred to as C{TMExact}.

246 '''

247 _datum = None # Datum

248 _E = None # Ellipsoid

249 _extendp = False # use extended domain

250# _iteration = None # ._sigmaInv2 and ._zetaInv2

251 _k0 = _K0_UTM # central scale factor

252 _lon0 = _0_0 # central meridian

253 _mu = _0_0 # ._E.e2, 1st eccentricity squared

254 _mv = _1_0 # _1_0 - ._mu

255 _raiser = False # throw Error

256 _sigmaC = None # _sigmaInv04 case

257 _zetaC = None # _zetaInv04 case

258

259 def __init__(self, datum=_WGS84, lon0=0, k0=_K0_UTM, extendp=False, name=NN, raiser=False):

260 '''New L{ExactTransverseMercator} projection.

261

262 @kwarg datum: The I{non-spherical} datum or ellipsoid (L{Datum},

263 L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}).

264 @kwarg lon0: Central meridian, default (C{degrees180}).

265 @kwarg k0: Central scale factor (C{float}).

266 @kwarg extendp: Use the I{extended} domain (C{bool}), I{standard} otherwise.

267 @kwarg name: Optional name for the projection (C{str}).

268 @kwarg raiser: If C{True}, throw an L{ETMError} for convergence failures (C{bool}).

269

270 @raise ETMError: Near-spherical B{C{datum}} or C{ellipsoid} or invalid B{C{lon0}}

271 or B{C{k0}}.

272

273 @see: U{Constructor TransverseMercatorExact<https://GeographicLib.SourceForge.io/

274 html/classGeographicLib_1_1TransverseMercatorExact.html>} for more details,

275 especially on B{X{extendp}}.

276

277 @note: For all 255.5K U{TMcoords.dat<https://Zenodo.org/record/32470>} tests (with

278 C{0 <= lat <= 84} and C{0 <= lon}) the maximum error is C{5.2e-08 .forward}

279 (or 52 nano-meter) easting and northing and C{3.8e-13 .reverse} (or 0.38

280 pico-degrees) lat- and longitude (with Python 3.7.3+, 2.7.16+, PyPy6 3.5.3

281 and PyPy6 2.7.13, all in 64-bit on macOS 10.13.6 High Sierra C{x86_64} and

282 12.2 Monterey C{arm64} and C{"arm64_x86_64"}).

283 '''

284 if extendp:

285 self._extendp = True

286 if name:

287 self.name = name

288 if raiser:

289 self.raiser = True

290

291 self.datum = datum # invokes ._reset

292 self.k0 = k0

293 self.lon0 = lon0

294

295 @property_doc_(''' the datum (L{Datum}).''')

296 def datum(self):

297 '''Get the datum (L{Datum}) or C{None}.

298 '''

299 return self._datum

300

301 @datum.setter # PYCHOK setter!

302 def datum(self, datum):

303 '''Set the datum and ellipsoid (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}).

304

305 @raise ETMError: Near-spherical B{C{datum}} or C{ellipsoid}.

306 '''

307 d = _ellipsoidal_datum(datum, name=self.name) # raiser=_datum_)

308 self._reset(d)

309 self._datum = d

310

311 @Property_RO

312 def _e(self):

313 '''(INTERNAL) Get and cache C{_e}.

314 '''

315 return self._E.e

316

317 @Property_RO

318 def _1_e_90(self): # PYCHOK no cover

319 '''(INTERNAL) Get and cache C{(1 - _e) * 90}.

320 '''

321 return (_1_0 - self._e) * _90_0

322

323 @property_RO

324 def ellipsoid(self):

325 '''Get the ellipsoid (L{Ellipsoid}).

326 '''

327 return self._E

328

329 @Property_RO

330 def _e_PI_2(self):

331 '''(INTERNAL) Get and cache C{_e * PI / 2}.

332 '''

333 return self._e * PI_2

334

335 @Property_RO

336 def _e_PI_4_(self):

337 '''(INTERNAL) Get and cache C{- _e * PI / 4}.

338 '''

339 return -self._e * PI_4

340

341 @Property_RO

342 def _1_e_PI_2(self):

343 '''(INTERNAL) Get and cache C{(1 - _e) * PI / 2}.

344 '''

345 return (_1_0 - self._e) * PI_2

346

347 @Property_RO

348 def _1_2e_PI_2(self):

349 '''(INTERNAL) Get and cache C{(1 - 2 * _e) * PI / 2}.

350 '''

351 return (_1_0 - self._e * _2_0) * PI_2

352

353 @property_RO

354 def equatoradius(self):

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

356 '''

357 return self._E.a

358

359 a = equatoradius

360

361 @Property_RO

362 def _e_TAYTOL(self):

363 '''(INTERNAL) Get and cache C{e * TAYTOL}.

364 '''

365 return self._e * _TAYTOL

366

367 @Property_RO

368 def _Eu(self):

369 '''(INTERNAL) Get and cache C{Elliptic(_mu)}.

370 '''

371 return Elliptic(self._mu)

372

373 @Property_RO

374 def _Eu_cE(self):

375 '''(INTERNAL) Get and cache C{_Eu.cE}.

376 '''

377 return self._Eu.cE

378

379 def _Eu_2cE_(self, xi):

380 '''(INTERNAL) Return C{_Eu.cE * 2 - B{xi}}.

381 '''

382 return self._Eu_cE * _2_0 - xi

383

384 @Property_RO

385 def _Eu_cE_4(self):

386 '''(INTERNAL) Get and cache C{_Eu.cE / 4}.

387 '''

388 return self._Eu_cE / _4_0

389

390 @Property_RO

391 def _Eu_cK(self):

392 '''(INTERNAL) Get and cache C{_Eu.cK}.

393 '''

394 return self._Eu.cK

395

396 @Property_RO

397 def _Eu_cK_cE(self):

398 '''(INTERNAL) Get and cache C{_Eu.cK / _Eu.cE}.

399 '''

400 return self._Eu_cK / self._Eu_cE

401

402 @Property_RO

403 def _Eu_2cK_PI(self):

404 '''(INTERNAL) Get and cache C{_Eu.cK * 2 / PI}.

405 '''

406 return self._Eu_cK / PI_2

407

408 @Property_RO

409 def _Ev(self):

410 '''(INTERNAL) Get and cache C{Elliptic(_mv)}.

411 '''

412 return Elliptic(self._mv)

413

414 @Property_RO

415 def _Ev_cK(self):

416 '''(INTERNAL) Get and cache C{_Ev.cK}.

417 '''

418 return self._Ev.cK

419

420 @Property_RO

421 def _Ev_cKE(self):

422 '''(INTERNAL) Get and cache C{_Ev.cKE}.

423 '''

424 return self._Ev.cKE

425

426 @Property_RO

427 def _Ev_3cKE_4(self):

428 '''(INTERNAL) Get and cache C{_Ev.cKE * 3 / 4}.

429 '''

430 return self._Ev_cKE * 0.75

431

432 @Property_RO

433 def _Ev_5cKE_4(self):

434 '''(INTERNAL) Get and cache C{_Ev.cKE * 5 / 4}.

435 '''

436 return self._Ev_cKE * 1.25

437

438 @Property_RO

439 def extendp(self):

440 '''Get the domain (C{bool}), I{extended} or I{standard}.

441 '''

442 return self._extendp

443

444 @property_RO

445 def flattening(self):

446 '''Get this C{ellipsoid}'s flattening (C{float}).

447 '''

448 return self._E.f

449

450 f = flattening

451

452 def forward(self, lat, lon, lon0=None, name=NN): # MCCABE 13

453 '''Forward projection, from geographic to transverse Mercator.

454

455 @arg lat: Latitude of point (C{degrees}).

456 @arg lon: Longitude of point (C{degrees}).

457 @kwarg lon0: Central meridian (C{degrees180}), overriding

458 the default if not C{None}.

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

460

461 @return: L{Forward4Tuple}C{(easting, northing, gamma, scale)}.

462

463 @see: C{void TMExact::Forward(real lon0, real lat, real lon,

464 real &x, real &y,

465 real &gamma, real &k)}.

466

467 @raise ETMError: No convergence, thrown iff property

468 C{B{raiser}=True}.

469 '''

470 lat = _fix90(lat)

471 lon, _ = _diff182((self.lon0 if lon0 is None else lon0), lon)

472 if self.extendp:

473 backside = _lat = _lon = False

474 else: # enforce the parity

475 lat, _lat = _unsigned2(lat)

476 lon, _lon = _unsigned2(lon)

477 backside = lon > 90

478 if backside: # PYCHOK no cover

479 lon = _180_0 - lon

480 if lat == 0:

481 _lat = True

482

483 # u, v = coordinates for the Thompson TM, Lee 54

484 if lat == 90:

485 u = self._Eu_cK

486 v = self._iteration = self._zetaC = 0

487 elif lat == 0 and lon == self._1_e_90: # PYCHOK no cover

488 u = self._iteration = self._zetaC = 0

489 v = self._Ev_cK

490 else: # tau = tan(phi), taup = sinh(psi)

491 tau, lam = _tand(lat), radians(lon)

492 u, v = self._zetaInv2(self._E.es_taupf(tau), lam)

493

494 sncndn6 = self._sncndn6(u, v)

495 y, x, _ = self._sigma3(v, *sncndn6)

496 g, k = self._zetaScaled(sncndn6, ll=False) \

497 if lat != 90 else (lon, self.k0)

498

499 if backside:

500 y, g = self._Eu_2cE_(y), (_180_0 - g)

501 y *= self._k0_a

502 x *= self._k0_a

503 if _lat:

504 y, g = neg_(y, g)

505 if _lon:

506 x, g = neg_(x, g)

507 return Forward4Tuple(x, y, g, k, iteration=self._iteration,

508 name=name or self.name)

509

510 def _Inv03(self, psi, dlam, _3_mv_e): # (xi, deta, _3_mv)

511 '''(INTERNAL) Partial C{_zetaInv04} or C{_sigmaInv04}, Case 2

512 '''

513 # atan2(dlam-psi, psi+dlam) + 45d gives arg(zeta - zeta0) in

514 # range [-135, 225). Subtracting 180 (multiplier is negative)

515 # makes range [-315, 45). Multiplying by 1/3 (for cube root)

516 # gives range [-105, 15). In particular the range [-90, 180]

517 # in zeta space maps to [-90, 0] in w space as required.

518 a = atan2(dlam - psi, psi + dlam) / _3_0 - PI_4

519 s, c = sincos2(a)

520 h = hypot(psi, dlam)

521 r = cbrt(h * _3_mv_e)

522 u = r * c

523 v = r * s + self._Ev_cK

524 # Error using this guess is about 0.068 * rad^(5/3)

525 return u, v, h

526

527 @property_RO

528 def iteration(self):

529 '''Get the most recent C{ExactTransverseMercator.forward}

530 or C{ExactTransverseMercator.reverse} iteration number

531 (C{int}) or C{None} if not available/applicable.

532 '''

533 return self._iteration

534

535 @property_doc_(''' the central scale factor (C{float}).''')

536 def k0(self):

537 '''Get the central scale factor (C{float}), aka I{C{scale0}}.

538 '''

539 return self._k0 # aka scale0

540

541 @k0.setter # PYCHOK setter!

542 def k0(self, k0):

543 '''Set the central scale factor (C{float}), aka I{C{scale0}}.

544

545 @raise ETMError: Invalid B{C{k0}}.

546 '''

547 k0 = Scalar_(k0=k0, Error=ETMError, low=_TOL_10, high=_1_0)

548 if self._k0 != k0:

549 ExactTransverseMercator._k0_a._update(self) # redo ._k0_a

550 self._k0 = k0

551

552 @Property_RO

553 def _k0_a(self):

554 '''(INTERNAL) Get and cache C{k0 * equatoradius}.

555 '''

556 return self.k0 * self.equatoradius

557

558 @property_doc_(''' the central meridian (C{degrees180}).''')

559 def lon0(self):

560 '''Get the central meridian (C{degrees180}).

561 '''

562 return self._lon0

563

564 @lon0.setter # PYCHOK setter!

565 def lon0(self, lon0):

566 '''Set the central meridian (C{degrees180}).

567

568 @raise ETMError: Invalid B{C{lon0}}.

569 '''

570 self._lon0 = _norm180(Degrees(lon0=lon0, Error=ETMError))

571

572 @deprecated_property_RO

573 def majoradius(self): # PYCHOK no cover

574 '''DEPRECATED, use property C{equatoradius}.'''

575 return self.equatoradius

576

577 @Property_RO

578 def _1_mu_2(self):

579 '''(INTERNAL) Get and cache C{_mu / 2 + 1}.

580 '''

581 return _1_0 + self._mu * _0_5

582

583 @Property_RO

584 def _3_mv(self):

585 '''(INTERNAL) Get and cache C{3 / _mv}.

586 '''

587 return _3_0 / self._mv

588

589 @Property_RO

590 def _3_mv_e(self):

591 '''(INTERNAL) Get and cache C{3 / (_mv * _e)}.

592 '''

593 return _3_0 / (self._mv * self._e)

594

595 def _Newton2(self, taup, lam, u, v, C, *psi): # or (xi, eta, u, v)

596 '''(INTERNAL) Invert C{_zetaInv2} or C{_sigmaInv2} using Newton's method.

597

598 @return: 2-Tuple C{(u, v)}.

599

600 @raise ETMError: No convergence.

601 '''

602 sca1, tol2 = _1_0, _TOL_10

603 if psi: # _zetaInv2

604 sca1 = sca1 / hypot1(taup) # /= chokes PyChecker

605 tol2 = tol2 / max(psi[0], _1_0)**2

606

607 _zeta3 = self._zeta3

608 _zetaDwd2 = self._zetaDwd2

609 else: # _sigmaInv2

610 _zeta3 = self._sigma3

611 _zetaDwd2 = self._sigmaDwd2

612

613 d2, r = tol2, self.raiser

614 _U_2_ = Fsum(u).fsum2_

615 _V_2_ = Fsum(v).fsum2_

616 # min iterations 2, max 6 or 7, mean 3.9 or 4.0

617 for i in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC

618 sncndn6 = self._sncndn6(u, v)

619 du, dv = _zetaDwd2(*sncndn6)

620 T, L, _ = _zeta3(v, *sncndn6)

621 T = (taup - T) * sca1

622 L -= lam

623 u, dU = _U_2_(T * du, L * dv)

624 v, dV = _V_2_(T * dv, -L * du)

625 if d2 < tol2:

626 r = False

627 break

628 d2 = hypot2(dU, dV)

629

630 self._iteration = i

631 if r: # PYCHOK no cover

632 i = callername(up=2, underOK=True)

633 t = unstr(i, taup, lam, u, v, C=C)

634 raise ETMError(Fmt.no_convergence(d2, tol2), txt=t)

635 return u, v

636

637 @property_doc_(''' raise an L{ETMError} for convergence failures (C{bool}).''')

638 def raiser(self):

639 '''Get the error setting (C{bool}).

640 '''

641 return self._raiser

642

643 @raiser.setter # PYCHOK setter!

644 def raiser(self, raiser):

645 '''Set the error setting (C{bool}), if C{True} throw an L{ETMError}

646 for convergence failures.

647 '''

648 self._raiser = bool(raiser)

649

650 def _reset(self, datum):

651 '''(INTERNAL) Set the ellipsoid and elliptic moduli.

652

653 @arg datum: Ellipsoidal datum (C{Datum}).

654

655 @raise ETMError: Near-spherical B{C{datum}} or C{ellipsoid}.

656 '''

657 E = datum.ellipsoid

658 mu = E.e2 # .eccentricity1st2

659 mv = E.e21 # _1_0 - mu

660 if isnear0(E.e) or isnear0(mu, eps0=EPS02) \

661 or isnear0(mv, eps0=EPS02): # or sqrt(mu) != E.e

662 raise ETMError(ellipsoid=E, txt=_near_(_spherical_))

663

664 if self._datum or self._E:

665 _i = ExactTransverseMercator.iteration._uname

666 _update_all(self, _i, '_sigmaC', '_zetaC') # _under

667

668 self._E = E

669 self._mu = mu

670 self._mv = mv

671

672 def reverse(self, x, y, lon0=None, name=NN):

673 '''Reverse projection, from Transverse Mercator to geographic.

674

675 @arg x: Easting of point (C{meters}).

676 @arg y: Northing of point (C{meters}).

677 @kwarg lon0: Central meridian (C{degrees180}), overriding

678 the default if not C{None}.

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

680

681 @return: L{Reverse4Tuple}C{(lat, lon, gamma, scale)}.

682

683 @see: C{void TMExact::Reverse(real lon0, real x, real y,

684 real &lat, real &lon,

685 real &gamma, real &k)}

686

687 @raise ETMError: No convergence, thrown iff property

688 C{B{raiser}=True}.

689 '''

690 # undoes the steps in .forward.

691 xi = y / self._k0_a

692 eta = x / self._k0_a

693 if self.extendp:

694 backside = _lat = _lon = False

695 else: # enforce the parity

696 eta, _lon = _unsigned2(eta)

697 xi, _lat = _unsigned2(xi)

698 backside = xi > self._Eu_cE

699 if backside: # PYCHOK no cover

700 xi = self._Eu_2cE_(xi)

701

702 # u, v = coordinates for the Thompson TM, Lee 54

703 if xi or eta != self._Ev_cKE:

704 u, v = self._sigmaInv2(xi, eta)

705 else: # PYCHOK no cover

706 u = self._iteration = self._sigmaC = 0

707 v = self._Ev_cK

708

709 if v or u != self._Eu_cK:

710 g, k, lat, lon = self._zetaScaled(self._sncndn6(u, v))

711 else: # PYCHOK no cover

712 g, k, lat, lon = _0_0, self.k0, _90_0, _0_0

713

714 if backside: # PYCHOK no cover

715 lon, g = (_180_0 - lon), (_180_0 - g)

716 if _lat:

717 lat, g = neg_(lat, g)

718 if _lon:

719 lon, g = neg_(lon, g)

720 lon += self.lon0 if lon0 is None else _norm180(lon0)

721 return Reverse4Tuple(lat, _norm180(lon), g, k, # _norm180(lat)

722 iteration=self._iteration,

723 name=name or self.name)

724

725 def _scaled2(self, tau, d2, snu, cnu, dnu, snv, cnv, dnv):

726 '''(INTERNAL) C{scaled}.

727

728 @note: Argument B{C{d2}} is C{_mu * cnu**2 + _mv * cnv**2}

729 from C{._zeta3}.

730

731 @return: 2-Tuple C{(convergence, scale)}.

732

733 @see: C{void TMExact::Scale(real tau, real /*lam*/,

734 real snu, real cnu, real dnu,

735 real snv, real cnv, real dnv,

736 real &gamma, real &k)}.

737 '''

738 mu, mv = self._mu, self._mv

739 cnudnv = cnu * dnv

740 # Lee 55.12 -- negated for our sign convention. g gives

741 # the bearing (clockwise from true north) of grid north

742 g = atan2d(mv * cnv * snv * snu, cnudnv * dnu)

743 # Lee 55.13 with nu given by Lee 9.1 -- in sqrt change

744 # the numerator from (1 - snu^2 * dnv^2) to (_mv * snv^2

745 # + cnu^2 * dnv^2) to maintain accuracy near phi = 90

746 # and change the denomintor from (dnu^2 + dnv^2 - 1) to

747 # (_mu * cnu^2 + _mv * cnv^2) to maintain accuracy near

748 # phi = 0, lam = 90 * (1 - e). Similarly rewrite sqrt in

749 # 9.1 as _mv + _mu * c^2 instead of 1 - _mu * sin(phi)^2

750 if d2 > 0:

751 # originally: sec2 = 1 + tau**2 # sec(phi)^2

752 # d2 = (mu * cnu**2 + mv * cnv**2)

753 # q2 = (mv * snv**2 + cnudnv**2) / d2

754 # k = sqrt(mv + mu / sec2) * sqrt(sec2) * sqrt(q2)

755 # = sqrt(mv * sec2 + mu) * sqrt(q2)

756 # = sqrt(mv + mv * tau**2 + mu) * sqrt(q2)

757 k2 = fsum1f_(mu, mv, mv * tau**2)

758 q2 = (mv * snv**2 + cnudnv**2) / d2

759 k = (sqrt(k2) * sqrt(q2) * self.k0) if \

760 (k2 > 0 and q2 > 0) else _0_0

761 else:

762 k = _OVERFLOW

763 return g, k

764

765 def _sigma3(self, v, snu, cnu, dnu, snv, cnv, dnv):

766 '''(INTERNAL) C{sigma}.

767

768 @return: 3-Tuple C{(xi, eta, d2)}.

769

770 @see: C{void TMExact::sigma(real /*u*/, real snu, real cnu, real dnu,

771 real v, real snv, real cnv, real dnv,

772 real &xi, real &eta)}.

773

774 @raise ETMError: No convergence.

775 '''

776 mu = self._mu * cnu

777 mv = self._mv * cnv

778 # Lee 55.4 writing

779 # dnu^2 + dnv^2 - 1 = _mu * cnu^2 + _mv * cnv^2

780 d2 = cnu * mu + cnv * mv

781 mu *= snu * dnu

782 mv *= snv * dnv

783 if d2 > 0: # /= chokes PyChecker

784 mu = mu / d2

785 mv = mv / d2

786 else:

787 mu, mv = map1(_overflow, mu, mv)

788 xi = self._Eu.fE(snu, cnu, dnu) - mu

789 v -= self._Ev.fE(snv, cnv, dnv) - mv

790 return xi, v, d2

791

792 def _sigmaDwd2(self, snu, cnu, dnu, snv, cnv, dnv):

793 '''(INTERNAL) C{sigmaDwd}.

794

795 @return: 2-Tuple C{(du, dv)}.

796

797 @see: C{void TMExact::dwdsigma(real /*u*/, real snu, real cnu, real dnu,

798 real /*v*/, real snv, real cnv, real dnv,

799 real &du, real &dv)}.

800 '''

801 snuv = snu * snv

802 # Reciprocal of 55.9: dw / ds = dn(w)^2/_mv,

803 # expanding complex dn(w) using A+S 16.21.4

804 d = self._mv * (cnv**2 + self._mu * snuv**2)**2

805 r = cnv * dnu * dnv

806 i = cnu * snuv * self._mu

807 du = (r**2 - i**2) / d

808 dv = neg(_2_0 * i * r / d)

809 return du, dv

810

811 def _sigmaInv2(self, xi, eta):

812 '''(INTERNAL) Invert C{sigma} using Newton's method.

813

814 @return: 2-Tuple C{(u, v)}.

815

816 @see: C{void TMExact::sigmainv(real xi, real eta,

817 real &u, real &v)}.

818

819 @raise ETMError: No convergence.

820 '''

821 u, v, t, self._sigmaC = self._sigmaInv04(xi, eta)

822 if not t:

823 u, v = self._Newton2(xi, eta, u, v, self._sigmaC)

824 return u, v

825

826 def _sigmaInv04(self, xi, eta):

827 '''(INTERNAL) Starting point for C{sigmaInv}.

828

829 @return: 4-Tuple C{(u, v, trip, Case)}.

830

831 @see: C{bool TMExact::sigmainv0(real xi, real eta,

832 real &u, real &v)}.

833 '''

834 t = False

835 d = eta - self._Ev_cKE

836 if eta > self._Ev_5cKE_4 or (xi < d and xi < -self._Eu_cE_4):

837 # sigma as a simple pole at

838 # w = w0 = Eu.K() + i * Ev.K()

839 # and sigma is approximated by

840 # sigma = (Eu.E() + i * Ev.KE()) + 1 / (w - w0)

841 u, v = _norm2(xi - self._Eu_cE, -d)

842 u += self._Eu_cK

843 v += self._Ev_cK

844 C = 1

845

846 elif (eta > self._Ev_3cKE_4 and xi < self._Eu_cE_4) or d > 0:

847 # At w = w0 = i * Ev.K(), we have

848 # sigma = sigma0 = i * Ev.KE()

849 # sigma' = sigma'' = 0

850 # including the next term in the Taylor series gives:

851 # sigma = sigma0 - _mv / 3 * (w - w0)^3

852 # When inverting this, we map arg(w - w0) = [-pi/2, -pi/6]

853 # to arg(sigma - sigma0) = [-pi/2, pi/2] mapping arg =

854 # [-pi/2, -pi/6] to [-pi/2, pi/2]

855 u, v, h = self._Inv03(xi, d, self._3_mv)

856 t = h < _TAYTOL2

857 C = 2

858

859 else: # use w = sigma * Eu.K/Eu.E (correct in limit _e -> 0)

860 u = v = self._Eu_cK_cE

861 u *= xi

862 v *= eta

863 C = 3

864

865 return u, v, t, C

866

867 def _sncndn6(self, u, v):

868 '''(INTERNAL) Get 6-tuple C{(snu, cnu, dnu, snv, cnv, dnv)}.

869 '''

870 # snu, cnu, dnu = self._Eu.sncndn(u)

871 # snv, cnv, dnv = self._Ev.sncndn(v)

872 return self._Eu.sncndn(u) + self._Ev.sncndn(v)

873

874 def toStr(self, joined=_COMMASPACE_, **kwds): # PYCHOK signature

875 '''Return a C{str} representation.

876

877 @kwarg joined: Separator to join the attribute strings

878 (C{str} or C{None} or C{NN} for non-joined).

879 @kwarg kwds: Optional, overriding keyword arguments.

880 '''

881 d = dict(datum=self.datum.name, lon0=self.lon0,

882 k0=self.k0, extendp=self.extendp)

883 if self.name:

884 d.update(name=self.name)

885 t = pairs(d, **kwds)

886 return joined.join(t) if joined else t

887

888 def _zeta3(self, unused, snu, cnu, dnu, snv, cnv, dnv): # _sigma3 signature

889 '''(INTERNAL) C{zeta}.

890

891 @return: 3-Tuple C{(taup, lambda, d2)}.

892

893 @see: C{void TMExact::zeta(real /*u*/, real snu, real cnu, real dnu,

894 real /*v*/, real snv, real cnv, real dnv,

895 real &taup, real &lam)}

896 '''

897 e, cnu2, mv = self._e, cnu**2, self._mv

898 # Overflow value like atan(overflow) = pi/2

899 t1 = t2 = _overflow(snu)

900 # Lee 54.17 but write

901 # atanh(snu * dnv) = asinh(snu * dnv / sqrt(cnu^2 + _mv * snu^2 * snv^2))

902 # atanh(_e * snu / dnv) = asinh(_e * snu / sqrt(_mu * cnu^2 + _mv * cnv^2))

903 d1 = cnu2 + mv * (snu * snv)**2

904 if d1 > EPS02: # _EPSmin

905 t1 = snu * dnv / sqrt(d1)

906 else:

907 d1 = 0

908 d2 = self._mu * cnu2 + mv * cnv**2

909 if d2 > EPS02: # _EPSmin

910 t2 = sinh(e * asinh(e * snu / sqrt(d2)))

911 else:

912 d2 = 0

913 # psi = asinh(t1) - asinh(t2)

914 # taup = sinh(psi)

915 taup = t1 * hypot1(t2) - t2 * hypot1(t1)

916 lam = (atan2(dnu * snv, cnu * cnv) -

917 atan2(cnu * snv * e, dnu * cnv) * e) if d1 and d2 else _0_0

918 return taup, lam, d2

919

920 def _zetaDwd2(self, snu, cnu, dnu, snv, cnv, dnv):

921 '''(INTERNAL) C{zetaDwd}.

922

923 @return: 2-Tuple C{(du, dv)}.

924

925 @see: C{void TMExact::dwdzeta(real /*u*/, real snu, real cnu, real dnu,

926 real /*v*/, real snv, real cnv, real dnv,

927 real &du, real &dv)}.

928 '''

929 cnu2 = cnu**2 * self._mu

930 cnv2 = cnv**2

931 dnuv = dnu * dnv

932 dnuv2 = dnuv**2

933 snuv = snu * snv

934 snuv2 = snuv**2 * self._mu

935 # Lee 54.21 but write (see A+S 16.21.4)

936 # (1 - dnu^2 * snv^2) = (cnv^2 + _mu * snu^2 * snv^2)

937 d = self._mv * (cnv2 + snuv2)**2 # max(d, EPS02)?

938 du = cnu * dnuv * (cnv2 - snuv2) / d

939 dv = cnv * snuv * (cnu2 + dnuv2) / d

940 return du, neg(dv)

941

942 def _zetaInv2(self, taup, lam):

943 '''(INTERNAL) Invert C{zeta} using Newton's method.

944

945 @return: 2-Tuple C{(u, v)}.

946

947 @see: C{void TMExact::zetainv(real taup, real lam,

948 real &u, real &v)}.

949

950 @raise ETMError: No convergence.

951 '''

952 psi = asinh(taup)

953 u, v, t, self._zetaC = self._zetaInv04(psi, lam)

954 if not t:

955 u, v = self._Newton2(taup, lam, u, v, self._zetaC, psi)

956 return u, v

957

958 def _zetaInv04(self, psi, lam):

959 '''(INTERNAL) Starting point for C{zetaInv}.

960

961 @return: 4-Tuple C{(u, v, trip, Case)}.

962

963 @see: C{bool TMExact::zetainv0(real psi, real lam, # radians

964 real &u, real &v)}.

965 '''

966 if lam > self._1_2e_PI_2:

967 d = lam - self._1_e_PI_2

968 if psi < d and psi < self._e_PI_4_: # PYCHOK no cover

969 # N.B. this branch is normally *not* taken because psi < 0

970 # is converted psi > 0 by .forward. There's a log singularity

971 # at w = w0 = Eu.K() + i * Ev.K(), corresponding to the south

972 # pole, where we have, approximately

973 # psi = _e + i * pi/2 - _e * atanh(cos(i * (w - w0)/(1 + _mu/2)))

974 # Inverting this gives:

975 e = self._e # eccentricity

976 s, c = sincos2((PI_2 - lam) / e)

977 h, r = sinh(_1_0 - psi / e), self._1_mu_2

978 u = self._Eu_cK - r * asinh(s / hypot(c, h))

979 v = self._Ev_cK - r * atan2(c, h)

980 return u, v, False, 1

981

982 elif psi < self._e_PI_2:

983 # At w = w0 = i * Ev.K(), we have

984 # zeta = zeta0 = i * (1 - _e) * pi/2

985 # zeta' = zeta'' = 0

986 # including the next term in the Taylor series gives:

987 # zeta = zeta0 - (_mv * _e) / 3 * (w - w0)^3

988 # When inverting this, we map arg(w - w0) = [-90, 0]

989 # to arg(zeta - zeta0) = [-90, 180]

990 u, v, h = self._Inv03(psi, d, self._3_mv_e)

991 return u, v, (h < self._e_TAYTOL), 2

992

993 # Use spherical TM, Lee 12.6 -- writing C{atanh(sin(lam) /

994 # cosh(psi)) = asinh(sin(lam) / hypot(cos(lam), sinh(psi)))}.

995 # This takes care of the log singularity at C{zeta = Eu.K()},

996 # corresponding to the north pole.

997 s, c = sincos2(lam)

998 h, r = sinh(psi), self._Eu_2cK_PI

999 # But scale to put 90, 0 on the right place

1000 u = r * atan2(h, c)

1001 v = r * asinh(s / hypot(h, c))

1002 return u, v, False, 3

1003

1004 def _zetaScaled(self, sncndn6, ll=True):

1005 '''(INTERNAL) Recompute (T, L) from (u, v) to improve accuracy of Scale.

1006

1007 @arg sncndn6: 6-Tuple C{(snu, cnu, dnu, snv, cnv, dnv)}.

1008

1009 @return: 2-Tuple C{(g, k)} if not C{B{ll}} else

1010 4-tuple C{(g, k, lat, lon)}.

1011 '''

1012 t, lam, d2 = self._zeta3(None, *sncndn6)

1013 tau = self._E.es_tauf(t)

1014 g_k = self._scaled2(tau, d2, *sncndn6)

1015 if ll:

1016 g_k += atand(tau), degrees(lam)

1017 return g_k # or (g, k, lat, lon)

1018

1019

1020 def parseETM5(strUTM, datum=_WGS84, Etm=Etm, falsed=True, name=NN):

1021 '''Parse a string representing a UTM coordinate, consisting

1022 of C{"zone[band] hemisphere easting northing"}.

1023

1024 @arg strUTM: A UTM coordinate (C{str}).

1025 @kwarg datum: Optional datum to use (L{Datum}, L{Ellipsoid},

1026 L{Ellipsoid2} or L{a_f2Tuple}).

1027 @kwarg Etm: Optional class to return the UTM coordinate

1028 (L{Etm}) or C{None}.

1029 @kwarg falsed: Both easting and northing are C{falsed} (C{bool}).

1030 @kwarg name: Optional B{C{Etm}} name (C{str}).

1031

1032 @return: The UTM coordinate (B{C{Etm}}) or if B{C{Etm}} is

1033 C{None}, a L{UtmUps5Tuple}C{(zone, hemipole, easting,

1034 northing, band)}. The C{hemipole} is the hemisphere

1035 C{'N'|'S'}.

1036

1037 @raise ETMError: Invalid B{C{strUTM}}.

1038

1039 @raise TypeError: Invalid or near-spherical B{C{datum}}.

1040

1041 @example:

1042

1043 >>> u = parseETM5('31 N 448251 5411932')

1044 >>> u.toRepr() # [Z:31, H:N, E:448251, N:5411932]

1045 >>> u = parseETM5('31 N 448251.8 5411932.7')

1046 >>> u.toStr() # 31 N 448252 5411933

1047 '''

1048 r = _parseUTM5(strUTM, datum, Etm, falsed, Error=ETMError, name=name)

1049 return r

1050

1051

1052 def toEtm8(latlon, lon=None, datum=None, Etm=Etm, falsed=True,

1053 name=NN, strict=True,

1054 zone=None, **cmoff):

1055 '''Convert a lat-/longitude point to an ETM coordinate.

1056

1057 @arg latlon: Latitude (C{degrees}) or an (ellipsoidal)

1058 geodetic C{LatLon} point.

1059 @kwarg lon: Optional longitude (C{degrees}) or C{None}.

1060 @kwarg datum: Optional datum for this ETM coordinate,

1061 overriding B{C{latlon}}'s datum (L{Datum},

1062 L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}).

1063 @kwarg Etm: Optional class to return the ETM coordinate

1064 (L{Etm}) or C{None}.

1065 @kwarg falsed: False both easting and northing (C{bool}).

1066 @kwarg name: Optional B{C{Utm}} name (C{str}).

1067 @kwarg strict: Restrict B{C{lat}} to UTM ranges (C{bool}).

1068 @kwarg zone: Optional UTM zone to enforce (C{int} or C{str}).

1069 @kwarg cmoff: DEPRECATED, use B{C{falsed}}. Offset longitude

1070 from the zone's central meridian (C{bool}).

1071

1072 @return: The ETM coordinate (B{C{Etm}}) or a

1073 L{UtmUps8Tuple}C{(zone, hemipole, easting, northing,

1074 band, datum, gamma, scale)} if B{C{Etm}} is C{None}

1075 or not B{C{falsed}}. The C{hemipole} is theC{'N'|'S'}

1076 hemisphere.

1077

1078 @raise ETMError: No convergence transforming to ETM east-

1079 and northing.

1080

1081 @raise ETMError: Invalid B{C{zone}} or near-spherical or

1082 incompatible B{C{datum}} or C{ellipsoid}.

1083

1084 @raise RangeError: If B{C{lat}} outside the valid UTM bands or

1085 if B{C{lat}} or B{C{lon}} outside the valid

1086 range and L{pygeodesy.rangerrors} set to C{True}.

1087

1088 @raise TypeError: Invalid or near-spherical B{C{datum}} or

1089 B{C{latlon}} not ellipsoidal.

1090

1091 @raise ValueError: The B{C{lon}} value is missing or B{C{latlon}}

1092 is invalid.

1093 '''

1094 z, B, lat, lon, d, f, name = _to7zBlldfn(latlon, lon, datum,

1095 falsed, name, zone,

1096 strict, ETMError, **cmoff)

1097 lon0 = _cmlon(z) if f else None

1098 x, y, g, k = d.exactTM.forward(lat, lon, lon0=lon0)

1099

1100 return _toXtm8(Etm, z, lat, x, y, B, d, g, k, f,

1101 name, latlon, d.exactTM, Error=ETMError)

1102

1103

1104 if __name__ == '__main__': # MCCABE 13

1105

1106 from pygeodesy.lazily import _ALL_MODS as _MODS, printf

1107 from sys import argv, exit as _exit

1108

1109 # mimick some of I{Karney}'s utility C{TransverseMercatorProj}

1110 _f = _r = _s = _t = False

1111 _as = argv[1:]

1112 while _as and _as[0].startswith(_DASH_):

1113 _a = _as.pop(0)

1114 if len(_a) < 2:

1115 _exit('%s: option %r invalid' % (_usage(*argv), _a))

1116 elif '-forward'.startswith(_a):

1117 _f, _r = True, False

1118 elif '-reverse'.startswith(_a):

1119 _f, _r = False, True

1120 elif '-series'.startswith(_a):

1121 _s, _t = True, False

1122 elif _a == '-t':

1123 _s, _t = False, True

1124 elif '-help'.startswith(_a):

1125 _exit(_usage(argv[0], '[-s | -t]',

1126 '[-f[orward] <lat> <lon>',

1127 '| -r[everse] <easting> <northing>',

1128 '| <lat> <lon>]',

1129 '| -h[elp]'))

1130 else:

1131 _exit('%s: option %r not supported' % (_usage(*argv), _a))

1132 if len(_as) > 1:

1133 f2 = map1(float, *_as[:2])

1134 else:

1135 _exit('%s ...: incomplete' % (_usage(*argv),))

1136

1137 if _s:

1138 tm = _MODS.ktm.KTransverseMercator()

1139 else:

1140 tm = ExactTransverseMercator(extendp=_t)

1141

1142 if _f:

1143 t = tm.forward(*f2)

1144 elif _r:

1145 t = tm.reverse(*f2)

1146 else:

1147 t = tm.forward(*f2)

1148 printf(fstr(t, sep=_SPACE_))

1149 t = tm.reverse(t.easting, t.northing)

1150 printf(fstr(t, sep=_SPACE_))

1151

1152# **) MIT License

1153#

1154# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.

1155#

1156# Permission is hereby granted, free of charge, to any person obtaining a

1157# copy of this software and associated documentation files (the "Software"),

1158# to deal in the Software without restriction, including without limitation

1159# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1160# and/or sell copies of the Software, and to permit persons to whom the

1161# Software is furnished to do so, subject to the following conditions:

1162#

1163# The above copyright notice and this permission notice shall be included

1164# in all copies or substantial portions of the Software.

1165#

1166# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1167# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1168# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1169# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1170# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1171# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1172# OTHER DEALINGS IN THE SOFTWARE.