Coverage for pygeodesy/albers.py: 97%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''Albers Equal-Area projections. 

5 

6Classes L{AlbersEqualArea}, L{AlbersEqualArea2}, L{AlbersEqualArea4}, 

7L{AlbersEqualAreaCylindrical}, L{AlbersEqualAreaNorth}, L{AlbersEqualAreaSouth} 

8and L{AlbersError}, transcoded from I{Charles Karney}'s C++ class U{AlbersEqualArea 

9<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AlbersEqualArea.html>}. 

10 

11See also I{Albers Equal-Area Conic Projection} in U{John P. Snyder, "Map Projections 

12-- A Working Manual", 1987<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 98-106 

13and the Albers Conical Equal-Area examples on pp 291-294. 

14''' 

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

16from __future__ import division as _; del _ # PYCHOK semicolon 

17 

18from pygeodesy.basics import neg, neg_ 

19from pygeodesy.constants import EPS0, EPS02, _EPSqrt as _TOL, \ 

20 _0_0, _0_5, _1_0, _N_1_0, _2_0, \ 

21 _N_2_0, _4_0, _6_0, _90_0, _N_90_0 

22from pygeodesy.datums import _ellipsoidal_datum, _WGS84 

23from pygeodesy.errors import _ValueError, _xkwds 

24from pygeodesy.fmath import hypot, hypot1, sqrt3 

25from pygeodesy.fsums import Fsum, _Fsum1f_, fsum1f_ 

26from pygeodesy.interns import NN, _COMMASPACE_, _datum_, _gamma_, _k0_, \ 

27 _lat_, _lat1_, _lat2_, _lon_, _negative_, \ 

28 _scale_, _SPACE_, _x_, _y_ 

29from pygeodesy.karney import _diff182, _norm180, _signBit 

30from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY 

31from pygeodesy.named import _name2__, _NamedBase, _NamedTuple, _Pass 

32from pygeodesy.props import deprecated_Property_RO, Property_RO, _update_all 

33from pygeodesy.streprs import Fmt, strs, unstr 

34from pygeodesy.units import Bearing, Float_, Lat, Lat_, Lon, Meter, Scalar_ 

35from pygeodesy.utily import atan1, atan1d, atan2, degrees360, sincos2, \ 

36 sincos2d, sincos2d_ 

37 

38from math import atanh, degrees, fabs, radians, sqrt 

39 

40__all__ = _ALL_LAZY.albers 

41__version__ = '24.11.24' 

42 

43_k1_ = 'k1' 

44_NUMIT = 8 # XXX 4? 

45_NUMIT0 = 41 # XXX 21? 

46_TERMS = 31 # XXX 16? 

47_TOL0 = sqrt3(_TOL) 

48 

49 

50def _ct2(s, c): 

51 '''(INTERNAL) Avoid singularities at poles. 

52 ''' 

53 c = max(EPS0, c) 

54 return c, (s / c) 

55 

56 

57def _Ks(**name_k): 

58 '''(INTERNAL) Scale C{B{k} >= EPS0}. 

59 ''' 

60 return Scalar_(Error=AlbersError, low=EPS0, **name_k) # > 0 

61 

62 

63def _Lat(*lat, **Error_name_lat): 

64 '''(INTERNAL) Latitude C{-90 <= B{lat} <= 90}. 

65 ''' 

66 kwds = _xkwds(Error_name_lat, Error=AlbersError) 

67 return Lat_(*lat, **kwds) 

68 

69 

70def _qZx(albs): 

71 '''(INTERNAL) Set C{albs._qZ} and C{albs._qx}. 

72 ''' 

73 E = albs._datum.ellipsoid # _AlbersBase 

74 albs._qZ = qZ = _1_0 + E.e21 * _atanheE(_1_0, E) 

75 albs._qx = qZ / (_2_0 * E.e21) 

76 return qZ 

77 

78 

79class AlbersError(_ValueError): 

80 '''An L{AlbersEqualArea}, L{AlbersEqualArea2}, L{AlbersEqualArea4}, 

81 L{AlbersEqualAreaCylindrical}, L{AlbersEqualAreaNorth}, 

82 L{AlbersEqualAreaSouth} or L{Albers7Tuple} issue. 

83 ''' 

84 pass 

85 

86 

87class _AlbersBase(_NamedBase): 

88 '''(INTERNAL) Base class for C{AlbersEqualArea...} projections. 

89 

90 @see: I{Karney}'s C++ class U{AlbersEqualArea<https://GeographicLib.SourceForge.io/ 

91 C++/doc/classGeographicLib_1_1AlbersEqualArea.html>}, method C{Init}. 

92 ''' 

93 _datum = _WGS84 

94 _k = NN # or _k0_ or _k1_ 

95 _k0 = _Ks(k0=_1_0) 

96# _k0n0 = None # (INTERNAL) k0 * no 

97 _k02 = _1_0 # (INTERNAL) k0**2 

98# _k02n0 = None # (INTERNAL) k02 * n0 

99# _lat0 = None # lat origin 

100 _lat1 = None # let 1st parallel 

101 _lat2 = None # lat 2nd parallel 

102 _m0 = _0_0 # if polar else sqrt(m02) 

103# _m02 = None # (INTERNAL) cached 

104# _n0 = None # (INTERNAL) cached 

105 _nrho0 = _0_0 # if polar else m0 * E.a 

106 _polar = False 

107 _qx = None # (INTERNAL) see _qZx 

108 _qZ = None # (INTERNAL) see _qZx 

109# _scxi0_ = None # (INTERNAL) sec(xi) / (qZ * E.a2) 

110 _sign = +1 

111# _sxi0 = None # (INTERNAL) sin(xi) 

112# _txi0 = None # (INTERNAL) tan(xi) 

113 

114 def __init__(self, sa1, ca1, sa2, ca2, k, datum, **name): 

115 '''(INTERNAL) New C{AlbersEqualArea...} instance. 

116 ''' 

117 qZ = self._qZ 

118 if datum not in (None, self._datum): 

119 self._datum = _ellipsoidal_datum(datum, **name) 

120 qZ = _qZx(self) 

121 elif qZ is None: 

122 qZ = _qZx(_AlbersBase) 

123 if name: 

124 self.name = name 

125 

126 E = self.ellipsoid 

127 c = min(ca1, ca2) 

128 if _signBit(c): 

129 raise AlbersError(clat1=ca1, clat2=ca2, txt=_negative_) 

130 polar = c < EPS0 # == 0 

131 

132 # determine hemisphere of tangent latitude 

133 if sa1 < 0: # and sa2 < 0: 

134 self._sign = -1 

135 # internally, tangent latitude positive 

136 sa1, sa2 = neg_(sa1, sa2) 

137 if sa1 > sa2: # make phi1 < phi2 

138 sa1, sa2 = sa2, sa1 

139 ca1, ca2 = ca2, ca1 

140 if sa1 < 0: # or sa2 < 0: 

141 raise AlbersError(slat1=sa1, slat2=sa2, txt=_negative_) 

142 ca1, ta1 = _ct2(sa1, ca1) 

143 ca2, ta2 = _ct2(sa2, ca2) 

144 

145 par1 = fabs(ta1 - ta2) < EPS02 # ta1 == ta2 

146 if par1 or polar: 

147 ta0, C = ta2, _1_0 

148 else: 

149 ta0, C = self._ta0C2(ca1, sa1, ta1, ca2, sa2, ta2) 

150 

151 self._lat0 = _Lat(lat0=self._sign * atan1d(ta0)) 

152 self._m02 = m02 = _1_x21(E.f1 * ta0) 

153 self._n0 = n0 = ta0 / hypot1(ta0) 

154 if polar: 

155 self._polar = True 

156# self._nrho0 = self._m0 = _0_0 

157 else: # m0 = nrho0 / E.a 

158 self._m0 = sqrt(m02) 

159 self._nrho0 = self._m0 * E.a 

160 t = self._txi0 = self._txif(ta0) 

161 h = hypot1(t) 

162 s = self._sxi0 = t / h 

163 if par1: 

164 self._k0n0 = self._k02n0 = n0 

165 else: 

166 self._k0s(k * sqrt(C / (m02 + n0 * qZ * s))) 

167 self._scxi0_ = h / (qZ * E.a2) 

168 

169 def _a_b_sxi3(self, *ca_sa_ta_scb_4s): 

170 '''(INTERNAL) Sum of C{sm1} terms and C{sin(xi)}s for ._ta0C2. 

171 ''' 

172 _1 = _1_0 

173 a = b = s = _0_0 

174 for ca, sa, ta, scb in ca_sa_ta_scb_4s: 

175 cxi, sxi, _ = self._cstxif3(ta) 

176 if sa > 0: 

177 sa += _1 

178 a += (cxi / ca)**2 * sa / (sxi + _1) 

179 b += scb * ca**2 / sa 

180 else: 

181 sa = _1 - sa 

182 a += (_1 - sxi) / sa 

183 b += scb * sa 

184 s += sxi 

185 return a, b, s 

186 

187 def _azik(self, t, ta): 

188 '''(INTERNAL) Compute the azimuthal scale C{_Ks(k=k)}. 

189 ''' 

190 E = self.ellipsoid 

191 return _Ks(k=hypot1(E.b_a * ta) * self._k0 * t / E.a) 

192 

193 def _cstxif3(self, ta): 

194 '''(INTERNAL) Get 3-tuple C{(cos, sin, tan)} of M{xi(ta)}. 

195 ''' 

196 t = self._txif(ta) 

197 c = _1_0 / hypot1(t) 

198 s = c * t 

199 return c, s, t 

200 

201 @Property_RO 

202 def datum(self): 

203 '''Get the datum (L{Datum}). 

204 ''' 

205 return self._datum 

206 

207 @Property_RO 

208 def ellipsoid(self): 

209 '''Get the datum's ellipsoid (L{Ellipsoid}). 

210 ''' 

211 return self.datum.ellipsoid 

212 

213 @Property_RO 

214 def equatoradius(self): 

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

216 ''' 

217 return self.ellipsoid.a 

218 

219 a = equatoradius 

220 

221 @Property_RO 

222 def flattening(self): 

223 '''Get the C{ellipsoid}'s flattening (C{scalar}). 

224 ''' 

225 return self.ellipsoid.f 

226 

227 f = flattening 

228 

229 def forward(self, lat, lon, lon0=0, **name): 

230 '''Convert a geodetic location to east- and northing. 

231 

232 @arg lat: Latitude of the location (C{degrees}). 

233 @arg lon: Longitude of the location (C{degrees}). 

234 @kwarg lon0: Optional central meridian longitude (C{degrees}). 

235 @kwarg name: Optional C{B{name}=NN} for the location (C{str}). 

236 

237 @return: An L{Albers7Tuple}C{(x, y, lat, lon, gamma, scale, datum)}, 

238 with C{lon} offset by B{C{lon0}} and reduced C{[-180,180]}. 

239 

240 @note: The origin latitude is returned by C{property lat0}. No 

241 false easting or northing is added. The value of B{C{lat}} 

242 should be in the range C{[-90..90] degrees}. The returned 

243 values C{x} and C{y} will be large but finite for points 

244 projecting to infinity, i.e. one or both of the poles. 

245 ''' 

246 a = self.ellipsoid.a 

247 s = self._sign 

248 

249 k0 = self._k0 

250 n0 = self._n0 

251 nrho0 = self._nrho0 

252 txi0 = self._txi0 

253 

254 _, ta = _ct2(*sincos2d(s * _Lat(lat=lat))) 

255 

256 _, sxi, txi = self._cstxif3(ta) 

257 dq = _Dsn(txi, txi0, sxi, self._sxi0) * \ 

258 (txi - txi0) * self._qZ 

259 drho = a * dq / (sqrt(self._m02 - n0 * dq) + self._m0) 

260 

261 lon, _ = _diff182(lon0, lon) 

262 x = radians(lon) 

263 th = self._k02n0 * x 

264 sth, cth = sincos2(th) # XXX sin, cos 

265 if n0: 

266 x = sth / n0 

267 y = (_1_0 - cth) if cth < 0 else (sth**2 / (cth + _1_0)) 

268 y *= nrho0 / n0 

269 else: 

270 x *= self._k02 

271 y = _0_0 

272 t = nrho0 - n0 * drho 

273 x = t * x / k0 

274 y = s * (y + drho * cth) / k0 

275 

276 g = degrees360(s * th) 

277 if t: 

278 k0 = self._azik(t, ta) 

279 return Albers7Tuple(x, y, lat, lon, g, k0, self.datum, 

280 name=self._name__(name)) 

281 

282 @Property_RO 

283 def ispolar(self): 

284 '''Is this projection polar (C{bool})? 

285 ''' 

286 return self._polar 

287 

288 isPolar = ispolar # synonym 

289 

290 def _k0s(self, k0): 

291 '''(INTERNAL) Set C{._k0}, C{._k02}, etc. 

292 ''' 

293 self._k0 = k0 = _Ks(k0=k0) 

294 self._k02 = k02 = k0**2 

295 self._k0n0 = k0 * self._n0 

296 self._k02n0 = k02 * self._n0 

297 

298 @Property_RO 

299 def lat0(self): 

300 '''Get the latitude of the projection origin (C{degrees}). 

301 

302 This is the latitude of minimum azimuthal scale and 

303 equals the B{C{lat}} in the 1-parallel L{AlbersEqualArea} 

304 and lies between B{C{lat1}} and B{C{lat2}} for the 

305 2-parallel L{AlbersEqualArea2} and L{AlbersEqualArea4} 

306 projections. 

307 ''' 

308 return self._lat0 

309 

310 @Property_RO 

311 def lat1(self): 

312 '''Get the latitude of the first parallel (C{degrees}). 

313 ''' 

314 return self._lat1 

315 

316 @Property_RO 

317 def lat2(self): 

318 '''Get the latitude of the second parallel (C{degrees}). 

319 

320 @note: The second and first parallel latitudes are the 

321 same instance for 1-parallel C{AlbersEqualArea*} 

322 projections. 

323 ''' 

324 return self._lat2 

325 

326 @deprecated_Property_RO 

327 def majoradius(self): # PYCHOK no cover 

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

329 return self.equatoradius 

330 

331 def rescale0(self, lat, k=1): # PYCHOK no cover 

332 '''Set the azimuthal scale for this projection. 

333 

334 @arg lat: Northern latitude (C{degrees}). 

335 @arg k: Azimuthal scale at latitude B{C{lat}} (C{scalar}). 

336 

337 @raise AlbersError: Invalid B{C{lat}} or B{C{k}}. 

338 

339 @note: This allows a I{latitude of conformality} to be specified. 

340 ''' 

341 k0 = _Ks(k=k) / self.forward(lat, _0_0).scale 

342 if self._k0 != k0: 

343 _update_all(self) 

344 self._k0s(k0) 

345 

346 def reverse(self, x, y, lon0=0, LatLon=None, **name_LatLon_kwds): 

347 '''Convert an east- and northing location to geodetic lat- and longitude. 

348 

349 @arg x: Easting of the location (C{meter}). 

350 @arg y: Northing of the location (C{meter}). 

351 @kwarg lon0: Optional central meridian longitude (C{degrees}). 

352 @kwarg LatLon: Class to use (C{LatLon}) or C{None}. 

353 @kwarg name_LatLon_kwds: Optional C{B{name}=NN} for the location 

354 and optional, additional B{C{LatLon}} keyword 

355 arguments, ignored if C{B{LatLon} is None}. 

356 

357 @return: The geodetic (C{LatLon}) or if C{B{LatLon} is None} an 

358 L{Albers7Tuple}C{(x, y, lat, lon, gamma, scale, datum)}. 

359 

360 @note: The origin latitude is returned by C{property lat0}. No 

361 false easting or northing is added. The returned value of 

362 C{lon} is in the range C{[-180..180] degrees} and C{lat} 

363 is in the range C{[-90..90] degrees}. If the given 

364 B{C{x}} or B{C{y}} point is outside the valid projected 

365 space the nearest pole is returned. 

366 ''' 

367 k0 = self._k0 

368 n0 = self._n0 

369 k0n0 = self._k0n0 

370 s = self._sign 

371 txi = self._txi0 

372 

373 x = Meter(x=x) 

374 nx = k0n0 * x 

375 y = Meter(y=y) 

376 y_ = s * y 

377 ny = k0n0 * y_ 

378 t = nrho0 = self._nrho0 

379 y1 = nrho0 - ny 

380 

381 den = hypot(nx, y1) + nrho0 # 0 implies origin with polar aspect 

382 if den: 

383 drho = _Fsum1f_(x * nx, y_ * nrho0 * _N_2_0, y_ * ny).fover(den / k0) 

384 # dsxia = scxi0 * dsxi 

385 t += drho * n0 # k0 below 

386 d_ = (nrho0 + t) * drho * self._scxi0_ # / (qZ * E.a2) 

387 t_ = txi - d_ 

388 d_ = (txi + t_) * d_ + _1_0 

389 txi = t_ / (sqrt(d_) if d_ > EPS02 else EPS0) 

390 

391 ta = self._tanf(txi) 

392 lat = atan1d(s * ta) 

393 

394 th = atan2(nx, y1) 

395 lon = degrees((th / self._k02n0) if n0 else (x / (y1 * k0))) 

396 if lon0: 

397 lon += _norm180(lon0) 

398 lon = _norm180(lon) 

399 

400 n, LatLon_kwds = _name2__(name_LatLon_kwds, _or_nameof=self) 

401 if LatLon is None: 

402 g = degrees360(s * th) 

403 if den: 

404 k0 = self._azik(t, ta) 

405 r = Albers7Tuple(x, y, lat, lon, g, k0, self.datum, name=n) 

406 else: # PYCHOK no cover 

407 kwds = _xkwds(LatLon_kwds, datum=self.datum, name=n) 

408 r = LatLon(lat, lon, **kwds) 

409 return r 

410 

411 @Property_RO 

412 def scale0(self): 

413 '''Get the central scale for the projection (C{float}). 

414 

415 This is the azimuthal scale on the latitude of origin 

416 of the projection, see C{property lat0}. 

417 ''' 

418 return self._k0 

419 

420 def _ta0(self, s1_qZ, ta0, E): 

421 '''(INTERNAL) Refine C{ta0} for C{._ta0C2}. 

422 ''' 

423 e2 = E.e2 

424 e21 = E.e21 

425 e22 = E.e22 # == e2 / e21 

426 tol = _tol(_TOL0, ta0) 

427 _Ta02 = Fsum(ta0).fsum2f_ 

428 _1, _2 = _1_0, _2_0 

429 _4, _6 = _4_0, _6_0 

430 for self._iteration in range(1, _NUMIT0): # 4 trips 

431 ta02 = ta0**2 

432 sca02 = ta02 + _1 

433 sca0 = sqrt(sca02) 

434 sa0 = ta0 / sca0 

435 sa01 = sa0 + _1 

436 sa02 = sa0**2 

437 # sa0m = 1 - sa0 = 1 / (sec(a0) * (tan(a0) + sec(a0))) 

438 sa0m = _1 / (sca0 * (ta0 + sca0)) # scb0^2 * sa0 

439 sa0m1 = sa0m / (_1 - e2 * sa0) 

440 sa021 = _1 - e2 * sa02 

441 

442 g = (_1 + ta02 * e21) * sa0 

443 dg = (_1 + ta02 * _2) * sca02 * e21 + e2 

444 D = (_1 - (_1 + sa0 * _2 * sa01) * e2) * sa0m / (e21 * sa01) # dD/dsa0 

445 dD = (_2 - (_6 + sa0 * _4) * sa02 * e2) / (e21 * sa01**2) 

446 BA = (_atanh1(e2 * sa0m1**2) * e21 - e2 * sa0m) * sa0m1 \ 

447 - (_2 + (_1 + e2) * sa0) * sa0m**2 * e22 / sa021 # B + A 

448 d = (_4 - (_1 + sa02) * e2 * _2) * e22 / (sa021**2 * sca02) # dAB 

449 u = fsum1f_(s1_qZ * g, -D, g * BA) 

450 du = fsum1f_(s1_qZ * dg, dD, dg * BA, g * d) 

451 ta0, d = _Ta02(-u / du * (sca0 * sca02)) 

452 if fabs(d) < tol: 

453 return ta0 

454 raise AlbersError(Fmt.no_convergence(d, tol), txt=repr(self)) 

455 

456 def _ta0C2(self, ca1, sa1, ta1, ca2, sa2, ta2): 

457 '''(INTERNAL) Compute C{ta0} and C{C} for C{.__init__}. 

458 ''' 

459 E = self.ellipsoid 

460 f1, e2 = E.f1, E.e2 

461 _1 = _1_0 

462 

463 tb1 = f1 * ta1 

464 tb2 = f1 * ta2 

465 dtb12 = f1 * (tb1 + tb2) 

466 scb12 = _1 + tb1**2 

467 scb22 = _1 + tb2**2 

468 

469 dsn_2 = _Dsn(ta2, ta1, sa2, sa1) * _0_5 

470 sa12 = sa1 * sa2 

471 

472 esa1_2 = (_1 - e2 * sa1**2) \ 

473 * (_1 - e2 * sa2**2) 

474 esa12 = _1 + e2 * sa12 

475 

476 axi, bxi, sxi = self._a_b_sxi3((ca1, sa1, ta1, scb12), 

477 (ca2, sa2, ta2, scb22)) 

478 

479 dsxi = ((esa12 / esa1_2) + _DatanheE(sa2, sa1, E)) * dsn_2 / self._qx 

480 C = _Fsum1f_(sxi * dtb12 / dsxi, scb22, scb12).fover(scb22 * scb12 * _2_0) 

481 

482 S = _Fsum1f_(sa1, sa2, sa12) 

483 axi *= (S * e2 + _1).fover(S + _1, raiser=False) 

484 bxi *= _Fsum1f_(sa1, sa2, esa12).fover(esa1_2) * e2 + _D2atanheE(sa1, sa2, E) * E.e21 

485 s1_qZ = (axi * self._qZ - bxi) * dsn_2 / dtb12 

486 ta0 = self._ta0(s1_qZ, (ta1 + ta2) * _0_5, E) 

487 return ta0, C 

488 

489 def _tanf(self, txi): # in .Ellipsoid.auxAuthalic 

490 '''(INTERNAL) Function M{tan-phi from tan-xi}. 

491 ''' 

492 tol = _tol(_TOL, txi) 

493 e2 = self.ellipsoid.e2 

494 qx = self._qx 

495 

496 ta = txi 

497 _Ta2 = Fsum(ta).fsum2f_ 

498 _txif = self._txif 

499 _1 = _1_0 

500 for self._iteration in range(1, _NUMIT): # max 2, mean 1.99 

501 # dtxi / dta = (scxi / sca)^3 * 2 * (1 - e^2) 

502 # / (qZ * (1 - e^2 * sa^2)^2) 

503 ta2 = ta**2 

504 sca2 = _1 + ta2 

505 txia = _txif(ta) 

506 s3qx = sqrt3(sca2 / (txia**2 + _1)) * qx # * _1_x21(txia) 

507 eta2 = (_1 - e2 * ta2 / sca2)**2 

508 ta, d = _Ta2((txi - txia) * s3qx * eta2) 

509 if fabs(d) < tol: 

510 return ta 

511 raise AlbersError(Fmt.no_convergence(d, tol), txt=repr(self)) 

512 

513 def toRepr(self, prec=6, **unused): # PYCHOK expected 

514 '''Return a string representation of this projection. 

515 

516 @kwarg prec: Number of (decimal) digits, unstripped (C{int}). 

517 

518 @return: This projection as C{"<classname>(lat1, lat2, ...)"} 

519 (C{str}). 

520 ''' 

521 t = self.toStr(prec=prec, sep=_COMMASPACE_) 

522 return Fmt.PAREN(self.classname, t) 

523 

524 def toStr(self, prec=6, sep=_SPACE_, **unused): # PYCHOK expected 

525 '''Return a string representation of this projection. 

526 

527 @kwarg prec: Number of (decimal) digits, unstripped (C{int}). 

528 @kwarg sep: Separator to join (C{str}). 

529 

530 @return: This projection as C{"lat1 lat2"} (C{str}). 

531 ''' 

532 k = self._k 

533 t = (self.lat1, self.lat2, self._k0) if k is _k1_ else ( 

534 (self.lat1, self._k0) if k is _k0_ else 

535 (self.lat1,)) 

536 t = strs(t, prec=prec) 

537 if k: 

538 t = t[:-1] + (Fmt.EQUAL(k, t[-1]),) 

539 if self.datum != _WGS84: 

540 t += Fmt.EQUAL(datum=self.datum), 

541 if self.name: 

542 t += Fmt.EQUAL(name=repr(self.name)), 

543 return t if sep is None else sep.join(t) 

544 

545 def _txif(self, ta): # in .Ellipsoid.auxAuthalic 

546 '''(INTERNAL) Function M{tan-xi from tan-phi}. 

547 ''' 

548 E = self.ellipsoid 

549 _1 = _1_0 

550 

551 ca2 = _1_x21(ta) 

552 sa = sqrt(ca2) * fabs(ta) # enforce odd parity 

553 sa1 = _1 + sa 

554 

555 es1 = sa * E.e2 

556 es1m1 = sa1 * (_1 - es1) 

557 es1p1 = sa1 / (_1 + es1) 

558 es2m1 = _1 - sa * es1 

559 es2m1a = es2m1 * E.e21 # e2m 

560 s = sqrt((ca2 / (es1p1 * es2m1a) + _atanheE(ca2 / es1m1, E)) 

561 * (es1m1 / es2m1a + _atanheE(es1p1, E))) 

562 t = _Fsum1f_(sa / es2m1, _atanheE(sa, E)).fover(s) 

563 return neg(t) if ta < 0 else t 

564 

565 

566class AlbersEqualArea(_AlbersBase): 

567 '''An Albers equal-area (authalic) projection with a single standard parallel. 

568 

569 @see: L{AlbersEqualArea2} and L{AlbersEqualArea4}. 

570 ''' 

571 _k = _k0_ 

572 

573 def __init__(self, lat, k0=1, datum=_WGS84, **name): 

574 '''New L{AlbersEqualArea} projection. 

575 

576 @arg lat: Standard parallel (C{degrees}). 

577 @kwarg k0: Azimuthal scale on the standard parallel (C{scalar}). 

578 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, 

579 L{Ellipsoid2} or L{a_f2Tuple}). 

580 @kwarg name: Optional C{B{name}=NN} for the projection (C{str}). 

581 

582 @raise AlbersError: Invalid B{C{lat}}, B{C{k0}} or no convergence. 

583 ''' 

584 self._lat1 = self._lat2 = lat = _Lat(lat1=lat) 

585 args = tuple(sincos2d(lat)) * 2 + (_Ks(k0=k0), datum) 

586 _AlbersBase.__init__(self, *args, **name) 

587 

588 

589class AlbersEqualArea2(_AlbersBase): 

590 '''An Albers equal-area (authalic) projection with two standard parallels. 

591 

592 @see: L{AlbersEqualArea} and L{AlbersEqualArea4}. 

593 ''' 

594 _k = _k1_ 

595 

596 def __init__(self, lat1, lat2, k1=1, datum=_WGS84, **name): 

597 '''New L{AlbersEqualArea2} projection. 

598 

599 @arg lat1: First standard parallel (C{degrees}). 

600 @arg lat2: Second standard parallel (C{degrees}). 

601 @kwarg k1: Azimuthal scale on the standard parallels (C{scalar}). 

602 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, 

603 L{Ellipsoid2} or L{a_f2Tuple}). 

604 @kwarg name: Optional C{B{name}=NN} for the projection (C{str}). 

605 

606 @raise AlbersError: Invalid B{C{lat1}}m B{C{lat2}}, B{C{k1}} 

607 or no convergence. 

608 ''' 

609 self._lat1, self._lat2 = lats = _Lat(lat1=lat1), _Lat(lat2=lat2) 

610 args = tuple(sincos2d_(*lats)) + (_Ks(k1=k1), datum) 

611 _AlbersBase.__init__(self, *args, **name) 

612 

613 

614class AlbersEqualArea4(_AlbersBase): 

615 '''An Albers equal-area (authalic) projection specified by the C{sin} 

616 and C{cos} of both standard parallels. 

617 

618 @see: L{AlbersEqualArea} and L{AlbersEqualArea2}. 

619 ''' 

620 _k = _k1_ 

621 

622 def __init__(self, slat1, clat1, slat2, clat2, k1=1, datum=_WGS84, **name): 

623 '''New L{AlbersEqualArea4} projection. 

624 

625 @arg slat1: Sine of first standard parallel (C{scalar}). 

626 @arg clat1: Cosine of first standard parallel (non-negative C{scalar}). 

627 @arg slat2: Sine of second standard parallel (C{scalar}). 

628 @arg clat2: Cosine of second standard parallel (non-negative C{scalar}). 

629 @kwarg k1: Azimuthal scale on the standard parallels (C{scalar}). 

630 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, 

631 L{Ellipsoid2} or L{a_f2Tuple}). 

632 @kwarg name: Optional C{B{name}=NN} for the projection (C{str}). 

633 

634 @raise AlbersError: Negative B{C{clat1}} or B{C{clat2}}, B{C{slat1}} 

635 and B{C{slat2}} have opposite signs (hemispheres), 

636 invalid B{C{k1}} or no convergence. 

637 ''' 

638 def _Lat_s_c3(n, s, c): 

639 r = Float_(hypot(s, c), name=n, Error=AlbersError) 

640 L = _Lat( atan1d(s, c), name=n) 

641 return L, (s / r), (c / r) 

642 

643 self._lat1, sa1, ca1 = _Lat_s_c3(_lat1_, slat1, clat1) 

644 self._lat2, sa2, ca2 = _Lat_s_c3(_lat2_, slat2, clat2) 

645 _AlbersBase.__init__(self, sa1, ca1, sa2, ca2, _Ks(k1=k1), datum, **name) 

646 

647 

648class AlbersEqualAreaCylindrical(_AlbersBase): 

649 '''An L{AlbersEqualArea} projection at C{lat=0} and C{k0=1} degenerating 

650 to the cylindrical-equal-area projection. 

651 ''' 

652 _lat1 = _lat2 = _Lat(lat1=_0_0) 

653 

654 def __init__(self, lat=_0_0, datum=_WGS84, **name): 

655 '''New L{AlbersEqualAreaCylindrical} projection. 

656 

657 @kwarg lat: Standard parallel (C{0 degrees} I{fixed}). 

658 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, 

659 L{Ellipsoid2} or L{a_f2Tuple}). 

660 @kwarg name: Optional C{B{name}=NN} for the projection (C{str}). 

661 ''' 

662 _xlat(lat, _0_0, AlbersEqualAreaCylindrical) 

663 _AlbersBase.__init__(self, _0_0, _1_0, _0_0, _1_0, 1, datum, **name) 

664 

665 

666class AlbersEqualAreaNorth(_AlbersBase): 

667 '''An azimuthal L{AlbersEqualArea} projection at C{lat=90} and C{k0=1} 

668 degenerating to the L{azimuthal} L{LambertEqualArea} projection. 

669 ''' 

670 _lat1 = _lat2 = _Lat(lat1=_90_0) 

671 

672 def __init__(self, lat=_90_0, datum=_WGS84, **name): 

673 '''New L{AlbersEqualAreaNorth} projection. 

674 

675 @kwarg lat: Standard parallel (C{90 degrees} I{fixed}). 

676 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, 

677 L{Ellipsoid2} or L{a_f2Tuple}). 

678 @kwarg name: Optional C{B{name}=NN} for the projection (C{str}). 

679 ''' 

680 _xlat(lat, _90_0, AlbersEqualAreaNorth) 

681 _AlbersBase.__init__(self, _1_0, _0_0, _1_0, _0_0, 1, datum, **name) 

682 

683 

684class AlbersEqualAreaSouth(_AlbersBase): 

685 '''An azimuthal L{AlbersEqualArea} projection at C{lat=-90} and C{k0=1} 

686 degenerating to the L{azimuthal} L{LambertEqualArea} projection. 

687 ''' 

688 _lat1 = _lat2 = _Lat(lat1=_N_90_0) 

689 

690 def __init__(self, lat=_N_90_0, datum=_WGS84, **name): 

691 '''New L{AlbersEqualAreaSouth} projection. 

692 

693 @kwarg lat: Standard parallel (C{-90 degrees} I{fixed}). 

694 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, 

695 L{Ellipsoid2} or L{a_f2Tuple}). 

696 @kwarg name: Optional C{B{name}=NN} for the projection (C{str}). 

697 ''' 

698 _xlat(lat, _N_90_0, AlbersEqualAreaSouth) 

699 _AlbersBase.__init__(self, _N_1_0, _0_0, _N_1_0, _0_0, 1, datum, **name) 

700 

701 

702class Albers7Tuple(_NamedTuple): 

703 '''7-Tuple C{(x, y, lat, lon, gamma, scale, datum)}, in C{meter}, 

704 C{meter}, C{degrees90}, C{degrees180}, C{degrees360}, C{scalar} and 

705 C{Datum} where C{(x, y)} is the projected, C{(lat, lon)} the geodetic 

706 location, C{gamma} the meridian convergence at point, the bearing of 

707 the y-axis measured clockwise from true North and C{scale} is the 

708 azimuthal scale of the projection at point. The radial scale is 

709 the reciprocal C{1 / scale}. 

710 ''' 

711 _Names_ = (_x_, _y_, _lat_, _lon_, _gamma_, _scale_, _datum_) 

712 _Units_ = ( Meter, Meter, Lat, Lon, Bearing, _Pass, _Pass) 

713 

714 

715def _atanh1(x): 

716 '''(INTERNAL) Function M{atanh(sqrt(x)) / sqrt(x) - 1}. 

717 ''' 

718 s = fabs(x) 

719 if 0 < s < _0_5: # for typical ... 

720 # x < E.e^2 == 2 * E.f use ... 

721 # x / 3 + x^2 / 5 + x^3 / 7 + ... 

722 y, k = x, 3 

723 _S2 = Fsum(y / k).fsum2f_ 

724 for _ in range(_TERMS): # 9 terms 

725 y *= x # x**n 

726 k += 2 # 2*n + 1 

727 s, d = _S2(y / k) 

728 if not d: 

729 break 

730 elif s: 

731 s = sqrt(s) 

732 s = (atanh(s) if x > 0 else atan1(s)) / s - _1_0 

733 return s 

734 

735 

736def _atanheE(x, E): # see Ellipsoid._es_atanh, .AuxLat._atanhee 

737 '''(INTERNAL) Function M{atanhee(x)}, defined as ... 

738 atanh( E.e * x) / E.e if f > 0 # oblate 

739 atan (sqrt(-E.e2) * x) / sqrt(-E.e2) if f < 0 # prolate 

740 x if f = 0. 

741 ''' 

742 e = E.e # == sqrt(E.e2abs) 

743 if e and x: 

744 if E.f > 0: # .isOblate 

745 x = atanh(x * e) / e 

746 elif E.f < 0: # .isProlate 

747 x = atan1(x * e) / e 

748 return x 

749 

750 

751def _DatanheE(x, y, E): # see .rhumb.ekx._DeatanhE 

752 '''(INTERNAL) Function M{Datanhee(x, y)}, defined as 

753 M{atanhee((x - y) / (1 - E.e^2 * x * y)) / (x - y)}. 

754 ''' 

755 e = _1_0 - E.e2 * x * y 

756 if e: 

757 d = x - y 

758 e = (_atanheE(d / e, E) / d) if d else (_1_0 / e) 

759 return e 

760 

761 

762def _D2atanheE(x, y, E): 

763 '''(INTERNAL) Function M{D2atanhee(x, y)}, defined as 

764 M{(Datanhee(1, y) - Datanhee(1, x)) / (y - x)}. 

765 ''' 

766 s, e2 = _0_0, E.e2 

767 if e2: 

768 if ((fabs(x) + fabs(y)) * e2) < _0_5: 

769 e = z = _1_0 

770 k = 1 

771 T = Fsum() # Taylor expansion 

772 _T = T.Fsumf_ 

773 _C = Fsum().Fsum_ 

774 _S2 = Fsum().fsum2_ 

775 for _ in range(_TERMS): # 15 terms 

776 T *= y; P = _T(z); z *= x # PYCHOK ; 

777 T *= y; Q = _T(z); z *= x # PYCHOK ; 

778 e *= e2 

779 k += 2 

780 s, d = _S2(_C(P, Q) * e / k) 

781 if not d: 

782 break 

783 else: # PYCHOK no cover 

784 s = _1_0 - x 

785 if s: 

786 s = (_DatanheE(_1_0, y, E) - _DatanheE(x, y, E)) / s 

787 return s 

788 

789 

790def _Dsn(x, y, sx, sy): 

791 '''(INTERNAL) Divided differences, defined as M{Df(x, y) = (f(x) - f(y)) / (x - y)} 

792 with M{sn(x) = x / sqrt(1 + x^2)}: M{Dsn(x, y) = (x + y) / ((sn(x) + sn(y)) * 

793 (1 + x^2) * (1 + y^2))}. 

794 

795 @see: U{W. M. Kahan and R. J. Fateman, "Sympbolic Computation of Divided 

796 Differences"<https://People.EECS.Berkeley.EDU/~fateman/papers/divdiff.pdf>}, 

797 U{ACM SIGSAM Bulletin 33(2), 7-28 (1999)<https://DOI.org/10.1145/334714.334716>} 

798 and U{AlbersEqualArea.hpp 

799 <https://GeographicLib.SourceForge.io/C++/doc/AlbersEqualArea_8hpp_source.html>}. 

800 ''' 

801 # sx = x / hypot1(x) 

802 d, t = _1_0, (x * y) 

803 if t > 0: 

804 s = sx + sy 

805 if s: 

806 t = sx * sy / t 

807 d = t**2 * (x + y) / s 

808 elif x != y: 

809 d = (sx - sy) / (x - y) 

810 return d 

811 

812 

813def _tol(tol, x): 

814 '''(INTERNAL) Converge tolerance. 

815 ''' 

816 return tol * max(_1_0, fabs(x)) 

817 

818 

819def _1_x21(x): 

820 '''(INTERNAL) Return M{1 / (x**2 + 1)}. 

821 ''' 

822 return _1_0 / (x**2 + _1_0) 

823 

824 

825def _xlat(lat, f, where): 

826 '''(INTERNAL) check fixed C{lat}. 

827 ''' 

828 if lat is not f and _Lat(lat=lat) != f: 

829 t = unstr(where, lat=lat) 

830 raise AlbersError(t, txt_not_=f) 

831 

832 

833__all__ += _ALL_DOCS(_AlbersBase) 

834 

835# **) MIT License 

836# 

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

838# 

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

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

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

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

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

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

845# 

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

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

848# 

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

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

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

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

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

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

855# OTHER DEALINGS IN THE SOFTWARE.