1

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

3

4 u'''Albers Equal-Area projections.

5

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

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

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

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

10

11 See 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

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

14'''

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

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

17

18 from pygeodesy.basics import neg, neg_

19 from 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

22 from pygeodesy.datums import _ellipsoidal_datum, _WGS84

23 from pygeodesy.errors import _ValueError, _xkwds

24 from pygeodesy.fmath import hypot, hypot1, sqrt3

25 from pygeodesy.fsums import Fsum, fsum1f_

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

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

28 _negative_, _scale_, _SPACE_, _x_, _y_

29 from pygeodesy.karney import _diff182, _norm180, _signBit

30 from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY

31 from pygeodesy.named import _NamedBase, _NamedTuple, _Pass

32 from pygeodesy.props import deprecated_Property_RO, Property_RO, _update_all

33 from pygeodesy.streprs import Fmt, strs, unstr

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

35 from pygeodesy.utily import atan1, atan1d, degrees360, sincos2, sincos2d, \

36 sincos2d_

37

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

39

40__all__ = _ALL_LAZY.albers

41__version__ = '23.12.01'

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

50 def _ct2(s, c):

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

52 '''

53 c = max(EPS0, c)

54 return c, (s / c)

55

56

57 def _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

65 def _Ks(**name_k):

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

67 '''

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

69

70

71 def _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

78 def _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

87 class 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

95 class _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 * atan1d(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 C{ellipsoid}'s equatorial radius, semi-axis (C{meter}).

224 '''

225 return self.ellipsoid.a

226

227 a = equatoradius

228

229 @Property_RO

230 def flattening(self):

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

232 '''

233 return self.ellipsoid.f

234

235 f = flattening

236

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

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

239

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

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

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

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

244

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

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

247

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

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

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

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

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

253 '''

254 a = self.ellipsoid.a

255 s = self._sign

256

257 k0 = self._k0

258 n0 = self._n0

259 nrho0 = self._nrho0

260 txi0 = self._txi0

261

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

263

264 _, sxi, txi = self._cstxif3(ta)

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

266 (txi - txi0) * self._qZ

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

268

269 lon, _ = _diff182(lon0, lon)

270 x = radians(lon)

271 th = self._k02n0 * x

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

273 if n0:

274 x = sth / n0

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

276 y *= nrho0 / n0

277 else:

278 x *= self._k02

279 y = _0_0

280 t = nrho0 - n0 * drho

281 x = t * x / k0

282 y = s * (y + drho * cth) / k0

283

284 g = degrees360(s * th)

285 if t:

286 k0 = self._azik(t, ta)

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

288 name=name or self.name)

289

290 @Property_RO

291 def ispolar(self):

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

293 '''

294 return self._polar

295

296 isPolar = ispolar # synonym

297

298 def _k0s(self, k0):

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

300 '''

301 self._k0 = k0 = _Ks(k0=k0)

302 self._k02 = k02 = k0**2

303 self._k0n0 = k0 * self._n0

304 self._k02n0 = k02 * self._n0

305

306 @Property_RO

307 def lat0(self):

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

309

310 This is the latitude of minimum azimuthal scale and

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

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

313 2-parallel L{AlbersEqualArea2} and L{AlbersEqualArea4}

314 projections.

315 '''

316 return self._lat0

317

318 @Property_RO

319 def lat1(self):

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

321 '''

322 return self._lat1

323

324 @Property_RO

325 def lat2(self):

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

327

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

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

330 projections.

331 '''

332 return self._lat2

333

334 @deprecated_Property_RO

335 def majoradius(self): # PYCHOK no cover

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

337 return self.equatoradius

338

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

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

341

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

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

344

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

346

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

348 '''

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

350 if self._k0 != k0:

351 _update_all(self)

352 self._k0s(k0)

353

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

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

356

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

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

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

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

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

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

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

364

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

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

367

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

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

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

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

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

373 space the nearest pole is returned.

374 '''

375 k0 = self._k0

376 n0 = self._n0

377 k0n0 = self._k0n0

378 s = self._sign

379 txi = self._txi0

380

381 x = Meter(x=x)

382 nx = k0n0 * x

383 y = Meter(y=y)

384 y_ = s * y

385 ny = k0n0 * y_

386 t = nrho0 = self._nrho0

387 y1 = nrho0 - ny

388

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

390 if den:

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

392 # dsxia = scxi0 * dsxi

393 t += drho * n0

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

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

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

397

398 ta = self._tanf(txi)

399 lat = atan1d(s * ta)

400

401 th = atan2(nx, y1)

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

403 if lon0:

404 lon += _norm180(lon0)

405 lon = _norm180(lon)

406

407 n = name or self.name

408 if LatLon is None:

409 g = degrees360(s * th)

410 if den:

411 k0 = self._azik(t, ta)

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

413 else: # PYCHOK no cover

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

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

416 return r

417

418 @Property_RO

419 def scale0(self):

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

421

422 This is the azimuthal scale on the latitude of origin

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

424 '''

425 return self._k0

426

427 def _ta0(self, s1_qZ, ta0, E):

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

429 '''

430 e2 = E.e2

431 e21 = E.e21

432 e22 = E.e22 # == e2 / e21

433 tol = _tol(_TOL0, ta0)

434 _Ta02 = Fsum(ta0).fsum2_

435 _fabs = fabs

436 _fsum1 = fsum1f_

437 _sqrt = sqrt

438 _1, _2 = _1_0, _2_0

439 _4, _6 = _4_0, _6_0

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

441 ta02 = ta0**2

442 sca02 = ta02 + _1

443 sca0 = _sqrt(sca02)

444 sa0 = ta0 / sca0

445 sa01 = sa0 + _1

446 sa02 = sa0**2

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

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

449 sa0m1 = sa0m / (_1 - e2 * sa0)

450 sa021 = _1 - e2 * sa02

451

452 g = (_1 + ta02 * e21) * sa0

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

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

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

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

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

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

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

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

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

462 if _fabs(d) < tol:

463 return ta0

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

465

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

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

468 '''

469 E = self.ellipsoid

470 f1, e2 = E.f1, E.e2

471 _1 = _1_0

472

473 tb1 = f1 * ta1

474 tb2 = f1 * ta2

475 dtb12 = f1 * (tb1 + tb2)

476 scb12 = _1 + tb1**2

477 scb22 = _1 + tb2**2

478

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

480 sa12 = sa1 * sa2

481

482 esa1_2 = (_1 - e2 * sa1**2) \

483 * (_1 - e2 * sa2**2)

484 esa12 = _1 + e2 * sa12

485

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

487 (ca2, sa2, ta2, scb22))

488

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

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

491

492 sa12 = fsum1f_(sa1, sa2, sa12)

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

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

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

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

497 return ta0, C

498

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

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

501 '''

502 tol = _tol(_TOL, txi)

503 e2 = self.ellipsoid.e2

504 qx = self._qx

505

506 ta = txi

507 _Ta2 = Fsum(ta).fsum2_

508 _fabs = fabs

509 _sqrt3 = sqrt3

510 _txif = self._txif

511 _1 = _1_0

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

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

514 # / (qZ * (1 - e^2 * sa^2)^2)

515 ta2 = ta**2

516 sca2 = _1 + ta2

517 txia = _txif(ta)

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

519 eta2 = (_1 - e2 * ta2 / sca2)**2

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

521 if _fabs(d) < tol:

522 return ta

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

524

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

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

527

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

529

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

531 (C{str}).

532 '''

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

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

535

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

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

538

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

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

541

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

543 '''

544 k = self._k

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

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

547 (self.lat1,))

548 t = strs(t, prec=prec)

549 if k:

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

551 if self.datum != _WGS84:

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

553 if self.name:

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

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

556

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

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

559 '''

560 E = self.ellipsoid

561 _1 = _1_0

562

563 ca2 = _1_x21(ta)

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

565 sa1 = _1 + sa

566

567 es1 = sa * E.e2

568 es1m1 = sa1 * (_1 - es1)

569 es1p1 = sa1 / (_1 + es1)

570 es2m1 = _1 - sa * es1

571 es2m1a = es2m1 * E.e21 # e2m

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

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

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

575 return neg(t) if ta < 0 else t

576

577

578 class AlbersEqualArea(_AlbersBase):

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

580

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

582 '''

583 _k = _k0_

584

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

586 '''New L{AlbersEqualArea} projection.

587

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

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

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

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

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

593

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

595 '''

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

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

598 _AlbersBase.__init__(self, *args)

599

600

601 class AlbersEqualArea2(_AlbersBase):

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

603

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

605 '''

606 _k = _k1_

607

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

609 '''New L{AlbersEqualArea2} projection.

610

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

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

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

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

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

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

617

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

619 or no convergence.

620 '''

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

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

623 _AlbersBase.__init__(self, *args)

624

625

626 class AlbersEqualArea4(_AlbersBase):

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

628 and C{cos} of both standard parallels.

629

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

631 '''

632 _k = _k1_

633

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

635 '''New L{AlbersEqualArea4} projection.

636

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

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

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

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

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

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

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

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

645

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

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

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

649 '''

650 def _Lat_s_c3(name, s, c):

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

652 L = _Lat(atan1d(s, c), name=name)

653 return L, (s / r), (c / r)

654

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

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

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

658

659

660 class AlbersEqualAreaCylindrical(_AlbersBase):

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

662 to the cylindrical-equal-area projection.

663 '''

664 _lat1 = _lat2 = _Lat(lat1=_0_0)

665

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

667 '''New L{AlbersEqualAreaCylindrical} projection.

668

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

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

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

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

673 '''

674 _xlat(lat, _0_0, AlbersEqualAreaCylindrical)

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

676

677

678 class AlbersEqualAreaNorth(_AlbersBase):

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

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

681 '''

682 _lat1 = _lat2 = _Lat(lat1=_90_0)

683

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

685 '''New L{AlbersEqualAreaNorth} projection.

686

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

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

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

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

691 '''

692 _xlat(lat, _90_0, AlbersEqualAreaNorth)

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

694

695

696 class AlbersEqualAreaSouth(_AlbersBase):

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

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

699 '''

700 _lat1 = _lat2 = _Lat(lat1=_N_90_0)

701

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

703 '''New L{AlbersEqualAreaSouth} projection.

704

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

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

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

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

709 '''

710 _xlat(lat, _N_90_0, AlbersEqualAreaSouth)

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

712

713

714 class Albers7Tuple(_NamedTuple):

715 '''7-Tuple C{(x, y, lat, lon, gamma, scale, datum)}, in C{meter},

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

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

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

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

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

721 the reciprocal C{1 / scale}.

722 '''

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

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

725

726

727 def _atanh1(x):

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

729 '''

730 s = fabs(x)

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

732 # x < E.e^2 == 2 * E.f use ...

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

734 y, k = x, 3

735 _S2 = Fsum(y / k).fsum2_

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

737 y *= x # x**n

738 k += 2 # 2*n + 1

739 s, d = _S2(y / k)

740 if not d:

741 break

742 elif s:

743 s = sqrt(s)

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

745 return s

746

747

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

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

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

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

752 x if f = 0.

753 '''

754 e = E.e # == sqrt(E.e2abs)

755 if e and x:

756 if E.f > 0: # .isOblate

757 x = atanh(x * e) / e

758 elif E.f < 0: # .isProlate

759 x = atan1(x * e) / e

760 return x

761

762

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

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

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

766 '''

767 e = _1_0 - E.e2 * x * y

768 if e:

769 d = x - y

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

771 return e

772

773

774 def _D2atanheE(x, y, E):

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

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

777 '''

778 s, e2 = _0_0, E.e2

779 if e2:

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

781 e = z = _1_0

782 k = 1

783 T = Fsum() # Taylor expansion

784 _T = T.fsum_

785 _C = Fsum().fsum_

786 _S2 = Fsum().fsum2_

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

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

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

790 e *= e2

791 k += 2

792 s, d = _S2(_C(p, t) * e / k)

793 if not d:

794 break

795 else: # PYCHOK no cover

796 s = _1_0 - x

797 if s:

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

799 return s

800

801

802 def _Dsn(x, y, sx, sy):

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

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

805 (1 + x^2) * (1 + y^2))}.

806

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

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

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

810 and U{AlbersEqualArea.hpp

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

812 '''

813 # sx = x / hypot1(x)

814 d, t = _1_0, (x * y)

815 if t > 0:

816 s = sx + sy

817 if s:

818 t = sx * sy / t

819 d = t**2 * (x + y) / s

820 elif x != y:

821 d = (sx - sy) / (x - y)

822 return d

823

824

825 def _tol(tol, x):

826 '''(INTERNAL) Converge tolerance.

827 '''

828 return tol * max(_1_0, fabs(x))

829

830

831 def _1_x21(x):

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

833 '''

834 return _1_0 / (x**2 + _1_0)

835

836

837 def _xlat(lat, f, where):

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

839 '''

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

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

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

843

844

845__all__ += _ALL_DOCS(_AlbersBase)

846

847# **) MIT License

848#

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

850#

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

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

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

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

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

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

857#

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

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

860#

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

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

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

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

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

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

867# OTHER DEALINGS IN THE SOFTWARE.