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 atand, atan2d, degrees360, sincos2, \

36 sincos2d, sincos2d_

37

38 from 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

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 * 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

574 class 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

597 class 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

622 class 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

656 class 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

674 class 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

692 class 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

710 class 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

723 def _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

744 def _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

759 def _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

770 def _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

798 def _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

821 def _tol(tol, x):

822 '''(INTERNAL) Converge tolerance.

823 '''

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

825

826

827 def _1_x21(x):

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

829 '''

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

831

832

833 def _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.