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_ 

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

27 _lat_, _lat1_, _lat2_, _lon_, _name_, _not_, \ 

28 _negative_, _scale_, _SPACE_, _x_, _y_ 

29from pygeodesy.karney import _diff182, _norm180, _signBit 

30from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY 

31from pygeodesy.named import _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 atand, atan2d, degrees360, sincos2, \ 

36 sincos2d, sincos2d_ 

37 

38from math import atan, atan2, atanh, degrees, fabs, radians, sqrt 

39 

40__all__ = _ALL_LAZY.albers 

41__version__ = '23.08.18' 

42 

43_k1_ = 'k1' 

44_NUMIT = 8 # XXX 4? 

45_NUMIT0 = 41 # XXX 21? 

46_TERMS = 21 # 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 _Fsum1(a, b, c=_0_0): # floats=True 

58 '''(INTERNAL) C{Fsum} 1-primed. 

59 ''' 

60 S = Fsum() 

61 S._facc_(_1_0, a, b, c, _N_1_0, up=False) 

62 return S 

63 

64 

65def _Ks(**name_k): 

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

67 ''' 

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

69 

70 

71def _Lat(*lat, **Error_name_lat): 

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

73 ''' 

74 kwds = _xkwds(Error_name_lat, Error=AlbersError) 

75 return Lat_(*lat, **kwds) 

76 

77 

78def _qZx(albs): 

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

80 ''' 

81 E = albs._datum.ellipsoid # _AlbersBase 

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

83 albs._qx = qZ / (_2_0 * E.e21) 

84 return qZ 

85 

86 

87class AlbersError(_ValueError): 

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

89 L{AlbersEqualAreaCylindrical}, L{AlbersEqualAreaNorth}, 

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

91 ''' 

92 pass 

93 

94 

95class _AlbersBase(_NamedBase): 

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

97 

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

99 html/classGeographicLib_1_1AlbersEqualArea.html>}, method C{Init}. 

100 ''' 

101 _datum = _WGS84 

102 _k = NN # or _k0_ or _k1_ 

103 _k0 = _Ks(k0=_1_0) 

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

105 _k02 = _1_0 # (INTERNAL) k0**2 

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

107# _lat0 = None # lat origin 

108 _lat1 = None # let 1st parallel 

109 _lat2 = None # lat 2nd parallel 

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

111# _m02 = None # (INTERNAL) cached 

112# _n0 = None # (INTERNAL) cached 

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

114 _polar = False 

115 _qx = None # (INTERNAL) see _qZx 

116 _qZ = None # (INTERNAL) see _qZx 

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

118 _sign = +1 

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

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

121 

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

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

124 ''' 

125 qZ = self._qZ 

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

127 self._datum = _ellipsoidal_datum(datum, name=name) 

128 qZ = _qZx(self) 

129 elif qZ is None: 

130 qZ = _qZx(_AlbersBase) 

131 if name: 

132 self.name = name 

133 

134 E = self.ellipsoid 

135 c = min(ca1, ca2) 

136 if _signBit(c): 

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

138 polar = c < EPS0 # == 0 

139 

140 # determine hemisphere of tangent latitude 

141 if sa1 < 0: # and sa2 < 0: 

142 self._sign = -1 

143 # internally, tangent latitude positive 

144 sa1, sa2 = neg_(sa1, sa2) 

145 if sa1 > sa2: # make phi1 < phi2 

146 sa1, sa2 = sa2, sa1 

147 ca1, ca2 = ca2, ca1 

148 if sa1 < 0: # or sa2 < 0: 

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

150 ca1, ta1 = _ct2(sa1, ca1) 

151 ca2, ta2 = _ct2(sa2, ca2) 

152 

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

154 if par1 or polar: 

155 ta0, C = ta2, _1_0 

156 else: 

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

158 

159 self._lat0 = _Lat(lat0=self._sign * atand(ta0)) 

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

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

162 if polar: 

163 self._polar = True 

164# self._nrho0 = self._m0 = _0_0 

165 else: # m0 = nrho0 / E.a 

166 self._m0 = sqrt(m02) 

167 self._nrho0 = self._m0 * E.a 

168 t = self._txi0 = self._txif(ta0) 

169 h = hypot1(t) 

170 s = self._sxi0 = t / h 

171 if par1: 

172 self._k0n0 = self._k02n0 = n0 

173 else: 

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

175 self._scxi0_ = h / (qZ * E.a2) 

176 

177 def _a_b_sxi3(self, *ca_sa_ta_scb_4s): 

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

179 ''' 

180 _1 = _1_0 

181 a = b = s = _0_0 

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

183 cxi, sxi, _ = self._cstxif3(ta) 

184 if sa > 0: 

185 sa += _1 

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

187 b += scb * ca**2 / sa 

188 else: 

189 sa = _1 - sa 

190 a += (_1 - sxi) / sa 

191 b += scb * sa 

192 s += sxi 

193 return a, b, s 

194 

195 def _azik(self, t, ta): 

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

197 ''' 

198 E = self.ellipsoid 

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

200 

201 def _cstxif3(self, ta): 

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

203 ''' 

204 t = self._txif(ta) 

205 c = _1_0 / hypot1(t) 

206 s = c * t 

207 return c, s, t 

208 

209 @Property_RO 

210 def datum(self): 

211 '''Get the datum (L{Datum}). 

212 ''' 

213 return self._datum 

214 

215 @Property_RO 

216 def ellipsoid(self): 

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

218 ''' 

219 return self.datum.ellipsoid 

220 

221 @Property_RO 

222 def equatoradius(self): 

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

224 ''' 

225 return self.ellipsoid.a 

226 

227 @Property_RO 

228 def flattening(self): 

229 '''Get the geodesic's flattening (C{float}). 

230 ''' 

231 return self.ellipsoid.f 

232 

233 def forward(self, lat, lon, lon0=0, name=NN): 

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

235 

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

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

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

239 @kwarg name: Optional name for the location (C{str}). 

240 

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

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

243 

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

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

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

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

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

249 ''' 

250 a = self.ellipsoid.a 

251 s = self._sign 

252 

253 k0 = self._k0 

254 n0 = self._n0 

255 nrho0 = self._nrho0 

256 txi0 = self._txi0 

257 

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

259 

260 _, sxi, txi = self._cstxif3(ta) 

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

262 (txi - txi0) * self._qZ 

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

264 

265 lon, _ = _diff182(lon0, lon) 

266 x = radians(lon) 

267 th = self._k02n0 * x 

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

269 if n0: 

270 x = sth / n0 

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

272 y *= nrho0 / n0 

273 else: 

274 x *= self._k02 

275 y = _0_0 

276 t = nrho0 - n0 * drho 

277 x = t * x / k0 

278 y = s * (y + drho * cth) / k0 

279 

280 g = degrees360(s * th) 

281 if t: 

282 k0 = self._azik(t, ta) 

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

284 name=name or self.name) 

285 

286 @Property_RO 

287 def ispolar(self): 

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

289 ''' 

290 return self._polar 

291 

292 isPolar = ispolar # synonym 

293 

294 def _k0s(self, k0): 

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

296 ''' 

297 self._k0 = k0 = _Ks(k0=k0) 

298 self._k02 = k02 = k0**2 

299 self._k0n0 = k0 * self._n0 

300 self._k02n0 = k02 * self._n0 

301 

302 @Property_RO 

303 def lat0(self): 

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

305 

306 This is the latitude of minimum azimuthal scale and 

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

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

309 2-parallel L{AlbersEqualArea2} and L{AlbersEqualArea4} 

310 projections. 

311 ''' 

312 return self._lat0 

313 

314 @Property_RO 

315 def lat1(self): 

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

317 ''' 

318 return self._lat1 

319 

320 @Property_RO 

321 def lat2(self): 

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

323 

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

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

326 projections. 

327 ''' 

328 return self._lat2 

329 

330 @deprecated_Property_RO 

331 def majoradius(self): # PYCHOK no cover 

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

333 return self.equatoradius 

334 

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

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

337 

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

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

340 

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

342 

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

344 ''' 

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

346 if self._k0 != k0: 

347 _update_all(self) 

348 self._k0s(k0) 

349 

350 def reverse(self, x, y, lon0=0, name=NN, LatLon=None, **LatLon_kwds): 

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

352 

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

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

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

356 @kwarg name: Optional name for the location (C{str}). 

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

358 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword 

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

360 

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

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

363 

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

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

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

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

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

369 space the nearest pole is returned. 

370 ''' 

371 k0 = self._k0 

372 n0 = self._n0 

373 k0n0 = self._k0n0 

374 s = self._sign 

375 txi = self._txi0 

376 

377 x = Meter(x=x) 

378 nx = k0n0 * x 

379 y = Meter(y=y) 

380 y_ = s * y 

381 ny = k0n0 * y_ 

382 t = nrho0 = self._nrho0 

383 y1 = nrho0 - ny 

384 

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

386 if den: 

387 drho = _Fsum1(x * nx, y_ * nrho0 * _N_2_0, y_ * ny).fover(den / k0) 

388 # dsxia = scxi0 * dsxi 

389 t += drho * n0 

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

391 d_2 = (txi * _2_0 - d_) * d_ + _1_0 

392 txi = (txi - d_) / (sqrt(d_2) if d_2 > EPS02 else EPS0) 

393 

394 ta = self._tanf(txi) 

395 lat = atand(s * ta) 

396 

397 th = atan2(nx, y1) 

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

399 if lon0: 

400 lon += _norm180(lon0) 

401 lon = _norm180(lon) 

402 

403 n = name or self.name 

404 if LatLon is None: 

405 g = degrees360(s * th) 

406 if den: 

407 k0 = self._azik(t, ta) 

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

409 else: # PYCHOK no cover 

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

411 r = LatLon(lat, lon, **kwds) 

412 return r 

413 

414 @Property_RO 

415 def scale0(self): 

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

417 

418 This is the azimuthal scale on the latitude of origin 

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

420 ''' 

421 return self._k0 

422 

423 def _ta0(self, s1_qZ, ta0, E): 

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

425 ''' 

426 e2 = E.e2 

427 e21 = E.e21 

428 e22 = E.e22 # == e2 / e21 

429 tol = _tol(_TOL0, ta0) 

430 _Ta02 = Fsum(ta0).fsum2_ 

431 _fabs = fabs 

432 _fsum1 = fsum1f_ 

433 _sqrt = sqrt 

434 _1, _2 = _1_0, _2_0 

435 _4, _6 = _4_0, _6_0 

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

437 ta02 = ta0**2 

438 sca02 = ta02 + _1 

439 sca0 = _sqrt(sca02) 

440 sa0 = ta0 / sca0 

441 sa01 = sa0 + _1 

442 sa02 = sa0**2 

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

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

445 sa0m1 = sa0m / (_1 - e2 * sa0) 

446 sa021 = _1 - e2 * sa02 

447 

448 g = (_1 + ta02 * e21) * sa0 

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

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

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

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

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

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

455 u = _fsum1(s1_qZ * g, -D, g * BA) 

456 du = _fsum1(s1_qZ * dg, dD, dg * BA, g * d) 

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

458 if _fabs(d) < tol: 

459 return ta0 

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

461 

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

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

464 ''' 

465 E = self.ellipsoid 

466 f1, e2 = E.f1, E.e2 

467 _1 = _1_0 

468 

469 tb1 = f1 * ta1 

470 tb2 = f1 * ta2 

471 dtb12 = f1 * (tb1 + tb2) 

472 scb12 = _1 + tb1**2 

473 scb22 = _1 + tb2**2 

474 

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

476 sa12 = sa1 * sa2 

477 

478 esa1_2 = (_1 - e2 * sa1**2) \ 

479 * (_1 - e2 * sa2**2) 

480 esa12 = _1 + e2 * sa12 

481 

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

483 (ca2, sa2, ta2, scb22)) 

484 

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

486 C = _Fsum1(sxi * dtb12 / dsxi, scb22, scb12).fover(scb22 * scb12 * _2_0) 

487 

488 sa12 = fsum1f_(sa1, sa2, sa12) 

489 axi *= (sa12 * e2 + _1) / (sa12 + _1) 

490 bxi *= _Fsum1(sa1, sa2, esa12).fover(esa1_2) * e2 + _D2atanheE(sa1, sa2, E) * E.e21 

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

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

493 return ta0, C 

494 

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

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

497 ''' 

498 tol = _tol(_TOL, txi) 

499 e2 = self.ellipsoid.e2 

500 qx = self._qx 

501 

502 ta = txi 

503 _Ta2 = Fsum(ta).fsum2_ 

504 _fabs = fabs 

505 _sqrt3 = sqrt3 

506 _txif = self._txif 

507 _1 = _1_0 

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

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

510 # / (qZ * (1 - e^2 * sa^2)^2) 

511 ta2 = ta**2 

512 sca2 = _1 + ta2 

513 txia = _txif(ta) 

514 s3qx = _sqrt3(sca2 / (txia**2 + _1)) * qx # * _1_x21(txia) 

515 eta2 = (_1 - e2 * ta2 / sca2)**2 

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

517 if _fabs(d) < tol: 

518 return ta 

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

520 

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

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

523 

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

525 

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

527 (C{str}). 

528 ''' 

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

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

531 

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

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

534 

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

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

537 

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

539 ''' 

540 k = self._k 

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

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

543 (self.lat1,)) 

544 t = strs(t, prec=prec) 

545 if k: 

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

547 if self.datum != _WGS84: 

548 t += (Fmt.EQUAL(_datum_, self.datum),) 

549 if self.name: 

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

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

552 

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

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

555 ''' 

556 E = self.ellipsoid 

557 _1 = _1_0 

558 

559 ca2 = _1_x21(ta) 

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

561 sa1 = _1 + sa 

562 

563 es1 = sa * E.e2 

564 es1m1 = sa1 * (_1 - es1) 

565 es1p1 = sa1 / (_1 + es1) 

566 es2m1 = _1 - sa * es1 

567 es2m1a = es2m1 * E.e21 # e2m 

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

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

570 t = _Fsum1(sa / es2m1, _atanheE(sa, E)).fover(s) 

571 return neg(t) if ta < 0 else t 

572 

573 

574class AlbersEqualArea(_AlbersBase): 

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

576 

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

578 ''' 

579 _k = _k0_ 

580 

581 def __init__(self, lat, k0=1, datum=_WGS84, name=NN): 

582 '''New L{AlbersEqualArea} projection. 

583 

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

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

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

587 L{Ellipsoid2} or L{a_f2Tuple}). 

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

589 

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

591 ''' 

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

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

594 _AlbersBase.__init__(self, *args) 

595 

596 

597class AlbersEqualArea2(_AlbersBase): 

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

599 

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

601 ''' 

602 _k = _k1_ 

603 

604 def __init__(self, lat1, lat2, k1=1, datum=_WGS84, name=NN): 

605 '''New L{AlbersEqualArea2} projection. 

606 

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

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

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

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

611 L{Ellipsoid2} or L{a_f2Tuple}). 

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

613 

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

615 or no convergence. 

616 ''' 

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

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

619 _AlbersBase.__init__(self, *args) 

620 

621 

622class AlbersEqualArea4(_AlbersBase): 

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

624 and C{cos} of both standard parallels. 

625 

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

627 ''' 

628 _k = _k1_ 

629 

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

631 '''New L{AlbersEqualArea4} projection. 

632 

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

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

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

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

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

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

639 L{Ellipsoid2} or L{a_f2Tuple}). 

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

641 

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

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

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

645 ''' 

646 def _Lat_s_c3(name, s, c): 

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

648 L = _Lat(atan2d(s, c), name=name) 

649 return L, (s / r), (c / r) 

650 

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

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

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

654 

655 

656class AlbersEqualAreaCylindrical(_AlbersBase): 

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

658 to the cylindrical-equal-area projection. 

659 ''' 

660 _lat1 = _lat2 = _Lat(lat1=_0_0) 

661 

662 def __init__(self, lat=_0_0, datum=_WGS84, name=NN): 

663 '''New L{AlbersEqualAreaCylindrical} projection. 

664 

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

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

667 L{Ellipsoid2} or L{a_f2Tuple}). 

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

669 ''' 

670 _xlat(lat, _0_0, AlbersEqualAreaCylindrical) 

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

672 

673 

674class AlbersEqualAreaNorth(_AlbersBase): 

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

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

677 ''' 

678 _lat1 = _lat2 = _Lat(lat1=_90_0) 

679 

680 def __init__(self, lat=_90_0, datum=_WGS84, name=NN): 

681 '''New L{AlbersEqualAreaNorth} projection. 

682 

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

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

685 L{Ellipsoid2} or L{a_f2Tuple}). 

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

687 ''' 

688 _xlat(lat, _90_0, AlbersEqualAreaNorth) 

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

690 

691 

692class AlbersEqualAreaSouth(_AlbersBase): 

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

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

695 ''' 

696 _lat1 = _lat2 = _Lat(lat1=_N_90_0) 

697 

698 def __init__(self, lat=_N_90_0, datum=_WGS84, name=NN): 

699 '''New L{AlbersEqualAreaSouth} projection. 

700 

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

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

703 L{Ellipsoid2} or L{a_f2Tuple}). 

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

705 ''' 

706 _xlat(lat, _N_90_0, AlbersEqualAreaSouth) 

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

708 

709 

710class Albers7Tuple(_NamedTuple): 

711 '''7-Tuple C{(x, y, lat, lon, gamma, scale, datum)}, in C{meter}, 

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

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

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

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

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

717 the reciprocal C{1 / scale}. 

718 ''' 

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

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

721 

722 

723def _atanh1(x): 

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

725 ''' 

726 s = fabs(x) 

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

728 # x < E.e^2 == 2 * E.f use ... 

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

730 y, k = x, 3 

731 _S2 = Fsum(y / k).fsum2_ 

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

733 y *= x # x**n 

734 k += 2 # 2*n + 1 

735 s, d = _S2(y / k) 

736 if not d: 

737 break 

738 elif s: 

739 s = sqrt(s) 

740 s = (atanh(s) if x > 0 else atan(s)) / s - _1_0 

741 return s 

742 

743 

744def _atanheE(x, E): # see Ellipsoid._es_atanh, .AuxLat._atanhee 

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

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

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

748 x if f = 0. 

749 ''' 

750 e = E.e 

751 if E.f > 0: # .isOblate 

752 x = atanh( x * e) / e 

753 elif E.f < 0: # .isProlate 

754 x = (atan2(-x * e, _N_1_0) if x < 0 else 

755 atan2( x * e, _1_0)) / e 

756 return x 

757 

758 

759def _DatanheE(x, y, E): # see .rhumbx._DeatanhE 

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

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

762 ''' 

763 e = _1_0 - E.e2 * x * y 

764 if e: 

765 d = x - y 

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

767 return e 

768 

769 

770def _D2atanheE(x, y, E): 

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

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

773 ''' 

774 s, e2 = _0_0, E.e2 

775 if e2: 

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

777 e = z = _1_0 

778 k = 1 

779 T = Fsum() # Taylor expansion 

780 _T = T.fsum_ 

781 _C = Fsum().fsum_ 

782 _S2 = Fsum().fsum2_ 

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

784 T *= y; p = _T(z); z *= x # PYCHOK ; 

785 T *= y; t = _T(z); z *= x # PYCHOK ; 

786 e *= e2 

787 k += 2 

788 s, d = _S2(_C(p, t) * e / k) 

789 if not d: 

790 break 

791 else: # PYCHOK no cover 

792 s = _1_0 - x 

793 if s: 

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

795 return s 

796 

797 

798def _Dsn(x, y, sx, sy): 

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

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

801 (1 + x^2) * (1 + y^2))}. 

802 

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

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

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

806 and U{AlbersEqualArea.hpp 

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

808 ''' 

809 # sx = x / hypot1(x) 

810 d, t = _1_0, (x * y) 

811 if t > 0: 

812 s = sx + sy 

813 if s: 

814 t = sx * sy / t 

815 d = t**2 * (x + y) / s 

816 elif x != y: 

817 d = (sx - sy) / (x - y) 

818 return d 

819 

820 

821def _tol(tol, x): 

822 '''(INTERNAL) Converge tolerance. 

823 ''' 

824 return tol * max(_1_0, fabs(x)) 

825 

826 

827def _1_x21(x): 

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

829 ''' 

830 return _1_0 / (x**2 + _1_0) 

831 

832 

833def _xlat(lat, f, where): 

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

835 ''' 

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

837 t = unstr(where.__name__, lat=lat) 

838 raise AlbersError(t, txt=_not_(f)) 

839 

840 

841__all__ += _ALL_DOCS(_AlbersBase) 

842 

843# **) MIT License 

844# 

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

846# 

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

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

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

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

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

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

853# 

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

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

856# 

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

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

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

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

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

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

863# OTHER DEALINGS IN THE SOFTWARE.