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 _isin, 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__ = '25.04.14' 

42 

43_k1_ = 'k1' 

44_NUMIT = 9 # 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 not _isin(datum, 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 ta0, C = (ta2, _1_0) if par1 or polar else \ 

147 self._ta0C2(ca1, sa1, ta1, ca2, sa2, ta2) 

148 

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

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

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

152 if polar: 

153 self._polar = True 

154# self._nrho0 = self._m0 = _0_0 

155 else: # m0 = nrho0 / E.a 

156 self._m0 = t = sqrt(m02) 

157 self._nrho0 = t * E.a 

158 t = self._txi0 = self._txif(ta0) 

159 h = hypot1(t) 

160 s = self._sxi0 = t / h 

161 if par1: 

162 self._k0n0 = self._k02n0 = n0 

163 else: 

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

165 self._scxi0_ = h / (qZ * E.a2) 

166 

167 def _a_b_sxi3(self, *ca_sa_ta_scb_4s): 

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

169 ''' 

170 _1 = _1_0 

171 a = b = s = _0_0 

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

173 cxi, sxi, _ = self._cstxif3(ta) 

174 if sa > 0: 

175 sa += _1 

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

177 b += scb * ca**2 / sa 

178 else: 

179 sa = _1 - sa 

180 a += (_1 - sxi) / sa 

181 b += scb * sa 

182 s += sxi 

183 return a, b, s 

184 

185 def _azik(self, t, ta): 

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

187 ''' 

188 E = self.ellipsoid 

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

190 

191 def _cstxif3(self, ta): 

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

193 ''' 

194 t = self._txif(ta) 

195 c = _1_0 / hypot1(t) 

196 s = c * t 

197 return c, s, t 

198 

199 @Property_RO 

200 def datum(self): 

201 '''Get the datum (L{Datum}). 

202 ''' 

203 return self._datum 

204 

205 @Property_RO 

206 def ellipsoid(self): 

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

208 ''' 

209 return self.datum.ellipsoid 

210 

211 @Property_RO 

212 def equatoradius(self): 

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

214 ''' 

215 return self.ellipsoid.a 

216 

217 a = equatoradius 

218 

219 @Property_RO 

220 def flattening(self): 

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

222 ''' 

223 return self.ellipsoid.f 

224 

225 f = flattening 

226 

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

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

229 

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

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

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

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

234 

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

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

237 

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

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

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

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

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

243 ''' 

244 a = self.ellipsoid.a 

245 s = self._sign 

246 

247 k0 = self._k0 

248 n0 = self._n0 

249 nrho0 = self._nrho0 

250 txi0 = self._txi0 

251 

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

253 

254 _, sxi, txi = self._cstxif3(ta) 

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

256 (txi - txi0) * self._qZ 

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

258 

259 lon, _ = _diff182(lon0, lon) 

260 x = radians(lon) 

261 th = self._k02n0 * x 

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

263 if n0: 

264 x = sth / n0 

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

266 y *= nrho0 / n0 

267 else: 

268 x *= self._k02 

269 y = _0_0 

270 t = nrho0 - n0 * drho 

271 x = t * x / k0 

272 y = s * (y + drho * cth) / k0 

273 

274 g = degrees360(s * th) 

275 if t: 

276 k0 = self._azik(t, ta) 

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

278 name=self._name__(name)) 

279 

280 @Property_RO 

281 def ispolar(self): 

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

283 ''' 

284 return self._polar 

285 

286 isPolar = ispolar # synonym 

287 

288 def _k0s(self, k0): 

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

290 ''' 

291 self._k0 = k0 = _Ks(k0=k0) 

292 self._k02 = k02 = k0**2 

293 self._k0n0 = k0 * self._n0 

294 self._k02n0 = k02 * self._n0 

295 

296 @Property_RO 

297 def lat0(self): 

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

299 

300 This is the latitude of minimum azimuthal scale and 

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

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

303 2-parallel L{AlbersEqualArea2} and L{AlbersEqualArea4} 

304 projections. 

305 ''' 

306 return self._lat0 

307 

308 @Property_RO 

309 def lat1(self): 

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

311 ''' 

312 return self._lat1 

313 

314 @Property_RO 

315 def lat2(self): 

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

317 

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

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

320 projections. 

321 ''' 

322 return self._lat2 

323 

324 @deprecated_Property_RO 

325 def majoradius(self): # PYCHOK no cover 

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

327 return self.equatoradius 

328 

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

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

331 

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

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

334 

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

336 

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

338 ''' 

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

340 if self._k0 != k0: 

341 _update_all(self) 

342 self._k0s(k0) 

343 

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

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

346 

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

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

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

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

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

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

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

354 

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

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

357 

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

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

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

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

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

363 space the nearest pole is returned. 

364 ''' 

365 k0 = self._k0 

366 n0 = self._n0 

367 k0n0 = self._k0n0 

368 s = self._sign 

369 txi = self._txi0 

370 

371 x = Meter(x=x) 

372 nx = k0n0 * x 

373 y = Meter(y=y) 

374 y_ = s * y 

375 ny = k0n0 * y_ 

376 t = nrho0 = self._nrho0 

377 y1 = nrho0 - ny 

378 

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

380 if den: 

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

382 # dsxia = scxi0 * dsxi 

383 t += drho * n0 # k0 below 

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

385 t_ = txi - d_ 

386 d_ = (txi + t_) * d_ + _1_0 

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

388 

389 ta = self._tanf(txi) 

390 lat = atan1d(s * ta) 

391 

392 th = atan2(nx, y1) 

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

394 if lon0: 

395 lon += _norm180(lon0) 

396 lon = _norm180(lon) 

397 

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

399 if LatLon is None: 

400 g = degrees360(s * th) 

401 if den: 

402 k0 = self._azik(t, ta) 

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

404 else: # PYCHOK no cover 

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

406 r = LatLon(lat, lon, **kwds) 

407 return r 

408 

409 @Property_RO 

410 def scale0(self): 

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

412 

413 This is the azimuthal scale on the latitude of origin 

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

415 ''' 

416 return self._k0 

417 

418 def _ta0(self, s1_qZ, ta0, E): 

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

420 ''' 

421 e2 = E.e2 

422 e21 = E.e21 

423 e22 = E.e22 # == e2 / e21 

424 tol = _tol(_TOL0, ta0) 

425 _Ta02 = Fsum(ta0).fsum2f_ 

426 _1, _2 = _1_0, _2_0 

427 _4, _6 = _4_0, _6_0 

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

429 ta02 = ta0**2 

430 sca02 = ta02 + _1 

431 sca0 = sqrt(sca02) 

432 sa0 = ta0 / sca0 

433 sa01 = sa0 + _1 

434 sa02 = sa0**2 

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

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

437 sa0m1 = sa0m / (_1 - e2 * sa0) 

438 sa021 = _1 - e2 * sa02 

439 

440 g = (_1 + ta02 * e21) * sa0 

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

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

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

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

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

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

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

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

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

450 if fabs(d) < tol: 

451 return ta0 

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

453 

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

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

456 ''' 

457 E = self.ellipsoid 

458 f1, e2 = E.f1, E.e2 

459 _1 = _1_0 

460 

461 tb1 = f1 * ta1 

462 tb2 = f1 * ta2 

463 dtb12 = f1 * (tb1 + tb2) 

464 scb12 = _1 + tb1**2 

465 scb22 = _1 + tb2**2 

466 

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

468 sa12 = sa1 * sa2 

469 

470 esa1_2 = (_1 - e2 * sa1**2) \ 

471 * (_1 - e2 * sa2**2) 

472 esa12 = _1 + e2 * sa12 

473 

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

475 (ca2, sa2, ta2, scb22)) 

476 

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

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

479 

480 S = _Fsum1f_(sa1, sa2, sa12) 

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

482 bxi *= _Fsum1f_( sa1, sa2, esa12).fover(esa1_2) * e2 + \ 

483 _D2atanheE(sa1, sa2, E) * E.e21 

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

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

486 return ta0, C 

487 

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

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

490 ''' 

491 e2 = self.ellipsoid.e2 

492 ta = txi 

493 tol = _tol(_TOL, ta) 

494 _Ta2 = Fsum(ta).fsum2f_ 

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

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

497 # / (qZ * (1 - e^2 * sa^2)^2) 

498 ta2 = ta**2 

499 sca2 = _1_0 + ta2 

500 txia = self._txif(ta) 

501 s3 = sqrt3(sca2 / (txia**2 + _1_0)) # * _1_x21(txia) 

502 s3 *= (e2 * ta2 / sca2 - _1_0)**2 * self._qx 

503 ta, d = _Ta2((txi - txia) * s3) 

504 if fabs(d) < tol: 

505 return ta 

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

507 

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

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

510 

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

512 

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

514 (C{str}). 

515 ''' 

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

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

518 

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

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

521 

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

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

524 

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

526 ''' 

527 k = self._k 

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

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

530 (self.lat1,)) 

531 t = strs(t, prec=prec) 

532 if k: 

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

534 if self.datum != _WGS84: 

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

536 if self.name: 

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

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

539 

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

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

542 ''' 

543 E = self.ellipsoid 

544 ca2 = _1_x21(ta) 

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

546 sa1 = _1_0 + sa 

547 

548 es1 = sa * E.e2 

549 es1m1 = sa1 * (_1_0 - es1) 

550 es1p1 = sa1 / (_1_0 + es1) 

551 es2m1 = _1_0 - sa * es1 

552 es2m1a = es2m1 * E.e21 # e2m 

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

554 s *= es1m1 / es2m1a + _atanheE( es1p1, E) 

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

556 return neg(t) if ta < 0 else t 

557 

558 

559class AlbersEqualArea(_AlbersBase): 

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

561 

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

563 ''' 

564 _k = _k0_ 

565 

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

567 '''New L{AlbersEqualArea} projection. 

568 

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

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

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

572 L{Ellipsoid2} or L{a_f2Tuple}). 

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

574 

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

576 ''' 

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

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

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

580 

581 

582class AlbersEqualArea2(_AlbersBase): 

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

584 

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

586 ''' 

587 _k = _k1_ 

588 

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

590 '''New L{AlbersEqualArea2} projection. 

591 

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

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

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

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

596 L{Ellipsoid2} or L{a_f2Tuple}). 

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

598 

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

600 or no convergence. 

601 ''' 

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

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

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

605 

606 

607class AlbersEqualArea4(_AlbersBase): 

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

609 and C{cos} of both standard parallels. 

610 

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

612 ''' 

613 _k = _k1_ 

614 

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

616 '''New L{AlbersEqualArea4} projection. 

617 

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

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

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

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

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

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

624 L{Ellipsoid2} or L{a_f2Tuple}). 

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

626 

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

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

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

630 ''' 

631 def _Lat_s_c3(n, s, c): 

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

633 L = _Lat( atan1d(s, c), name=n) 

634 return L, (s / r), (c / r) 

635 

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

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

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

639 

640 

641class AlbersEqualAreaCylindrical(_AlbersBase): 

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

643 to the cylindrical-equal-area projection. 

644 ''' 

645 _lat1 = _lat2 = _Lat(lat1=_0_0) 

646 

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

648 '''New L{AlbersEqualAreaCylindrical} projection. 

649 

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

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

652 L{Ellipsoid2} or L{a_f2Tuple}). 

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

654 ''' 

655 _xlat(lat, _0_0, AlbersEqualAreaCylindrical) 

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

657 

658 

659class AlbersEqualAreaNorth(_AlbersBase): 

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

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

662 ''' 

663 _lat1 = _lat2 = _Lat(lat1=_90_0) 

664 

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

666 '''New L{AlbersEqualAreaNorth} projection. 

667 

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

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

670 L{Ellipsoid2} or L{a_f2Tuple}). 

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

672 ''' 

673 _xlat(lat, _90_0, AlbersEqualAreaNorth) 

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

675 

676 

677class AlbersEqualAreaSouth(_AlbersBase): 

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

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

680 ''' 

681 _lat1 = _lat2 = _Lat(lat1=_N_90_0) 

682 

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

684 '''New L{AlbersEqualAreaSouth} projection. 

685 

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

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

688 L{Ellipsoid2} or L{a_f2Tuple}). 

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

690 ''' 

691 _xlat(lat, _N_90_0, AlbersEqualAreaSouth) 

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

693 

694 

695class Albers7Tuple(_NamedTuple): 

696 '''7-Tuple C{(x, y, lat, lon, gamma, scale, datum)}, in C{meter}, 

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

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

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

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

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

702 the reciprocal C{1 / scale}. 

703 ''' 

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

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

706 

707 

708def _atanh1(x): 

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

710 ''' 

711 s = fabs(x) 

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

713 # x < E.e^2 == 2 * E.f use ... 

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

715 y, k = x, 3 

716 _S2 = Fsum(y / k).fsum2f_ 

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

718 y *= x # x**n 

719 k += 2 # 2*n + 1 

720 s, d = _S2(y / k) 

721 if not d: 

722 break 

723 elif s: 

724 s = sqrt(s) 

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

726 return s 

727 

728 

729def _atanheE(x, E): # see Ellipsoid._es_atanh, .AuxLat._atanhee 

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

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

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

733 x if f = 0. 

734 ''' 

735 e = E.e # == sqrt(E.e2abs) 

736 if e and x: 

737 if E.f > 0: # .isOblate 

738 x = atanh(x * e) / e 

739 elif E.f < 0: # .isProlate 

740 x = atan1(x * e) / e 

741 return x 

742 

743 

744def _DatanheE(x, y, E): # see .rhumb.ekx._DeatanhE 

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

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

747 ''' 

748 e = _1_0 - E.e2 * x * y 

749 if e: 

750 d = x - y 

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

752 return e 

753 

754 

755def _D2atanheE(x, y, E): 

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

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

758 ''' 

759 s, e2 = _0_0, E.e2 

760 if e2: 

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

762 e = z = _1_0 

763 k = 1 

764 T = Fsum() # Taylor expansion 

765 _T = T.Fsumf_ 

766 _C = Fsum().Fsum_ 

767 _S2 = Fsum().fsum2_ 

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

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

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

771 e *= e2 

772 k += 2 

773 s, d = _S2(_C(P, Q) * e / k) 

774 if not d: 

775 break 

776 else: # PYCHOK no cover 

777 s = _1_0 - x 

778 if s: 

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

780 return s 

781 

782 

783def _Dsn(x, y, sx, sy): 

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

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

786 (1 + x^2) * (1 + y^2))}. 

787 

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

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

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

791 and U{AlbersEqualArea.hpp 

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

793 ''' 

794 # sx = x / hypot1(x) 

795 d, t = _1_0, (x * y) 

796 if t > 0: 

797 s = sx + sy 

798 if s: 

799 t = sx * sy / t 

800 d = t**2 * (x + y) / s 

801 elif x != y: 

802 d = (sx - sy) / (x - y) 

803 return d 

804 

805 

806def _tol(tol, x): 

807 '''(INTERNAL) Converge tolerance. 

808 ''' 

809 return tol * max(_1_0, fabs(x)) 

810 

811 

812def _1_x21(x): 

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

814 ''' 

815 return _1_0 / (x**2 + _1_0) 

816 

817 

818def _xlat(lat, f, where): 

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

820 ''' 

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

822 t = unstr(where, lat=lat) 

823 raise AlbersError(t, txt_not_=f) 

824 

825 

826__all__ += _ALL_DOCS(_AlbersBase) 

827 

828# **) MIT License 

829# 

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

831# 

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

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

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

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

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

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

838# 

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

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

841# 

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

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

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

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

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

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

848# OTHER DEALINGS IN THE SOFTWARE.