# Coverage for pygeodesy/albers.py : 96%

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 | # -*- coding: utf-8 -*- u'''Classes L{AlbersEqualArea}, L{AlbersEqualArea2}, L{AlbersEqualArea4}, L{AlbersEqualAreaCylindrical}, L{AlbersEqualAreaNorth}, L{AlbersEqualAreaSouth} and L{AlbersError}, a transcription of I{Charles Karney}'s C++ class U{AlbersEqualArea <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1AlbersEqualArea.html>}. See also I{Albers Equal-Area Conic Projection} in U{John P. Snyder, "Map Projections -- A Working Manual", 1987<https://pubs.er.USGS.gov/djvu/PP/PP_1395.pdf>}, pp 98-106 and the Albers Conical Equal-Area examples on pp 291-294. @newfield example: Example, Examples ''' from pygeodesy.basics import property_RO, _xkwds from pygeodesy.datums import Datums, _ellipsoidal_datum from pygeodesy.errors import _ValueError from pygeodesy.fmath import Fsum, fsum_, hypot, hypot1, sqrt3 from pygeodesy.interns import EPS, NN, _datum_, _gamma_, _lat_, \ _lat0_, _lat1_, _lat2_, _lon_, _lon0_, \ _no_convergence_fmt_, _scale_, _x_, \ _y_, _0_0, _0_5, _1_0, _2_0, _3_0, _90_0 from pygeodesy.karney import _diff182, _norm180 from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY from pygeodesy.named import _NamedBase, _NamedTuple, _Pass from pygeodesy.units import Bearing, Float_, Lat, Lat_, Lon, Lon_, \ Meter, Scalar_ from pygeodesy.utily import atan2d, degrees360, sincos2, sincos2d from math import atan, atan2, atanh, degrees, radians, sqrt __all__ = _ALL_LAZY.albers __version__ = '20.10.02' _EPS__2 = EPS**2 _EPS__4 = EPS**4 _NUMIT = 8 # XXX 4? _NUMIT0 = 41 # XXX 21? _TERMS = 21 # XXX 9? _TOL = sqrt(EPS) _TOL0 = sqrt3(_TOL) class AlbersError(_ValueError): '''An L{AlbersEqualArea}, L{AlbersEqualArea2}, L{AlbersEqualArea4}, L{AlbersEqualAreaCylindrical}, L{AlbersEqualAreaNorth}, L{AlbersEqualAreaSouth} or L{Albers7Tuple} issue. ''' pass def _Dsn(x, y, sx, sy): '''(INTERNAL) Divided differences, defined as M{Df(x, y) = (f(x) - f(y)) / (x - y)} with M{sn(x) = x / sqrt(1 + x^2)}: M{Dsn(x, y) = (x + y) / ((sn(x) + sn(y)) * (1 + x^2) * (1 + y^2))}. @see: U{W. M. Kahan and R. J. Fateman, "Sympbolic omputation of divided differences"<https://people.EECS.Berkeley.EDU/~fateman/papers/divdiff.pdf>}, U{ACM SIGSAM Bulletin 33(2), 7-28 (1999)<https://DOI.org/10.1145/334714.334716>} and U{AlbersEqualArea.hpp <https://geographiclib.sourceforge.io/html/AlbersEqualArea_8hpp_source.html>}. ''' # sx = x / hypot1(x) d, t = _1_0, x * y if t > 0: s = sx + sy if s: d = (x + y) * ((sx * sy) / t)**2 / s else: t = x - y if t: d = (sx - sy) / t return d def _Ks(k, name, Error=AlbersError): '''(INTERNAL) Scale C{B{k} >= EPS}. ''' return Scalar_(k, name=name, Error=Error, low=EPS) # > 0 def _Lat(lat, name=_lat_, Error=AlbersError): '''(INTERNAL) Latitude C{-90 <= B{lat} <= 90}. ''' return Lat_(lat, name=name, Error=Error) def _Lon(lon, name=_lon_, Error=AlbersError): '''(INTERNAL) Longitude C{-180 <= B{lon} <= 180}. ''' return Lon_(lon, name=name, Error=Error) def _tol(tol, x): '''(INTERNAL) Converge tolerance. ''' return tol * max(1, abs(x)) class _AlbersBase(_NamedBase): '''(INTERNAL) Base class for C{AlbersEqualArea...} projections. @see: I{Karney}'s C++ class U{AlbersEqualArea<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1AlbersEqualArea.html>}, method C{Init}. ''' _datum = Datums.WGS84 _iteration = None _lat0 = None # lat origin _lat1 = None # let 1st parallel _lat2 = None # lat 2nd parallel _k0 = None # scale _k0n0 = None # (INTERNAL) k0 * no _k02 = None # (INTERNAL) k0**2 _k02n0 = None # (INTERNAL) k02 * n0 _m0 = _0_0 # if polar else sqrt(m02) _m02 = None # (INTERNAL) cached _n0 = None # (INTERNAL) cached _nrho0 = _0_0 # if polar else m0 * E.a _polar = False _qx = None # (INTERNAL) cached _qZ = None # (INTERNAL) cached _qZa2 = None # (INTERNAL) qZ * E.a**2 _scxi0 = None # (INTERNAL) sec(xi) _sign = +1 _sxi0 = None # (INTERNAL) sin(xi) _txi0 = None # (INTERNAL) tan(xi) def __init__(self, sa1, ca1, sa2, ca2, k, datum, name): '''(INTERNAL) New C{AlbersEqualArea...} instance. ''' if datum not in (None, self._datum): self._datum = _ellipsoidal_datum(datum, name=name) if name: self.name = name E = self.datum.ellipsoid b_a = E.b_a # fm = 1 - E.f e2 = E.e2 e12 = E.e12 # e2m = 1 - E.e2 self._qZ = qZ = _1_0 + e12 * self._atanhee(1) self._qZa2 = qZ * E.a2 self._qx = qZ / (_2_0 * e12) c = min(ca1, ca2) if c < 0: raise AlbersError(clat1=ca1, clat2=ca2) polar = c < _EPS__2 # == 0 # determine hemisphere of tangent latitude if sa1 < 0: # and sa2 < 0: self._sign = -1 # internally, tangent latitude positive sa1, sa2 = -sa1, -sa2 if sa1 > sa2: # make phi1 < phi2 sa1, sa2 = sa2, sa1 ca1, ca2 = ca2, ca1 if sa1 < 0: # or sa2 < 0: raise AlbersError(slat1=sa1, slat2=sa2) # avoid singularities at poles ca1, ca2 = max(_EPS__2, ca1), max(_EPS__2, ca2) ta1, ta2 = sa1 / ca1, sa2 / ca2 par1 = abs(ta1 - ta2) < _EPS__4 # ta1 == ta2 if par1 or polar: C, ta0 = _1_0, ta2 else: s1_qZ, C = self._s1_qZ_C2(ca1, sa1, ta1, ca2, sa2, ta2) ta0 = (ta2 + ta1) * _0_5 Ta0 = Fsum(ta0) tol = _tol(_TOL0, ta0) for self._iteration in range(1, _NUMIT0): ta02 = ta0**2 sca02 = ta02 + _1_0 sca0 = sqrt(sca02) sa0 = ta0 / sca0 sa01 = sa0 + _1_0 sa02 = sa0**2 # sa0m = 1 - sa0 = 1 / (sec(a0) * (tan(a0) + sec(a0))) sa0m = _1_0 / (sca0 * (ta0 + sca0)) # scb0^2 * sa0 g = (_1_0 + (b_a * ta0)**2) * sa0 dg = e12 * sca02 * (_1_0 + 2 * ta02) + e2 D = sa0m * (_1_0 - e2 * (_1_0 + sa01 * 2 * sa0)) / (e12 * sa01) # dD/dsa0 dD = -2 * (_1_0 - e2 * sa02 * (_3_0 + 2 * sa0)) / (e12 * sa01**2) sa02_ = _1_0 - e2 * sa02 sa0m_ = sa0m / (_1_0 - e2 * sa0) BA = sa0m_ * (self._atanhx1(e2 * sa0m_**2) * e12 - e2 * sa0m) \ - sa0m**2 * e2 * (2 + (_1_0 + e2) * sa0) / (e12 * sa02_) # == B + A dAB = 2 * e2 * (2 - e2 * (_1_0 + sa02)) / (e12 * sa02_**2 * sca02) u_du = fsum_(s1_qZ * g, -D, g * BA) \ / fsum_(s1_qZ * dg, -dD, dg * BA, g * dAB) # == u/du ta0, d = Ta0.fsum2_(-u_du * (sca0 * sca02)) if abs(d) < tol: break else: raise AlbersError(iteration=_NUMIT0, txt=_no_convergence_fmt_ % (tol,)) self._txi0 = txi0 = self._txif(ta0) self._scxi0 = hypot1(txi0) self._sxi0 = sxi0 = txi0 / self._scxi0 self._m02 = m02 = _1_0 / (_1_0 + (b_a * ta0)**2) self._n0 = n0 = ta0 / hypot1(ta0) if polar: self._polar = True self._nrho0 = self._m0 = _0_0 else: self._m0 = sqrt(m02) # == nrho0 / E.a self._nrho0 = E.a * self._m0 # == E.a * sqrt(m02) self._k0_(_1_0 if par1 else (k * sqrt(C / (m02 + n0 * qZ * sxi0)))) self._lat0 = _Lat(self._sign * degrees(atan(ta0)), name=_lat0_) def _s1_qZ_C2(self, ca1, sa1, ta1, ca2, sa2, ta2): '''(INTERNAL) Compute C{sm1 / (s / qZ)} and C{C} for .__init__. ''' E = self.datum.ellipsoid b_a = E.b_a e2 = E.e2 tb1 = b_a * ta1 tb2 = b_a * ta2 dtb12 = b_a * (tb1 + tb2) scb12 = _1_0 + tb1**2 scb22 = _1_0 + tb2**2 esa1_2 = (_1_0 - e2 * sa1**2) \ * (_1_0 - e2 * sa2**2) esa12 = _1_0 + e2 * sa1 * sa2 dsn = _Dsn(ta2, ta1, sa2, sa1) axi, bxi, sxi = self._a_b_sxi3((ca1, sa1, ta1, scb12), (ca2, sa2, ta2, scb22)) dsxi = (esa12 / esa1_2 + self._Datanhee(sa2, sa1)) * dsn / (_2_0 * self._qx) C = fsum_(sxi * dtb12 / dsxi, scb22, scb12) / (_2_0 * scb22 * scb12) sa12 = fsum_(sa1, sa2, sa1 * sa2) axi *= (_1_0 + e2 * sa12) / (_1_0 + sa12) bxi *= e2 * fsum_(sa1, sa2, esa12) / esa1_2 + E.e12 * self._D2atanhee(sa1, sa2) s1_qZ = dsn * (axi * self._qZ - bxi) / (_2_0 * dtb12) return s1_qZ, C def _a_b_sxi3(self, *ca_sa_ta_scb): '''(INTERNAL) Sum of C{sm1} terms and C{sin(xi)}s for ._s1_qZ_C2. ''' a = b = s = _0_0 for ca, sa, ta, scb in ca_sa_ta_scb: cxi, sxi, _ = self._cstxif3(ta) if sa > 0: sa += _1_0 a += (cxi / ca)**2 * sa / (_1_0 + sxi) b += scb * ca**2 / sa else: sa = _1_0 - sa a += (_1_0 - sxi) / sa b += scb * sa s += sxi return a, b, s @property_RO def datum(self): '''Get the datum (L{Datum}). ''' return self._datum @property_RO def equatoradius(self): '''Get the geodesic's equatorial (major) radius, semi-axis (C{meter}). ''' return self.datum.ellipsoid.a majoradius = equatoradius # for compatibility @property_RO def iteration(self): '''Get the iteration number (C{int}). ''' return self._iteration @property_RO def flattening(self): '''Get the geodesic's flattening (C{float}). ''' return self.datum.ellipsoid.f def forward(self, lat, lon, lon0=0, name=NN): '''Convert a geodetic location to east- and northing. @arg lat: Latitude of the location (C{degrees}). @arg lon: Longitude of the location (C{degrees}). @kwarg lon0: Optional central meridian longitude (C{degrees}). @kwarg name: Optional name for the location (C{str}). @return: An L{Albers7Tuple}C{(x, y, lat, lon, gamma, scale, datum)}. @note: The origin latitude is returned by C{property lat0}. No false easting or northing is added. The value of B{C{lat}} should be in the range C{[-90..90] degrees}. The returned values C{x} and C{y} will be large but finite for points projecting to infinity, i.e. one or both of the poles. ''' E = self.datum.ellipsoid s = self._sign k0 = self._k0 n0 = self._n0 nrho0 = self._nrho0 txi0 = self._txi0 sa, ca = sincos2d(_Lat(lat) * s) ca = max(_EPS__2, ca) ta = sa / ca _, sxi, txi = self._cstxif3(ta) dq = self._qZ * _Dsn(txi, txi0, sxi, self._sxi0) * (txi - txi0) drho = -E.a * dq / (sqrt(self._m02 - n0 * dq) + self._m0) lon = _Lon(lon) if lon0: lon, _ = _diff182(_Lon(lon0, name=_lon0_), lon) b = radians(lon) th = self._k02n0 * b sth, cth = sincos2(th) # XXX sin, cos if n0: x = sth / n0 y = (_1_0 - cth if cth < 0 else sth**2 / (_1_0 + cth)) / n0 else: x = self._k02 * b y = _0_0 t = nrho0 + n0 * drho x = t * x / k0 y = s * (nrho0 * y - drho * cth) / k0 g = degrees360(s * th) if t: k0 *= t * hypot1(E.b_a * ta) / E.a t = Albers7Tuple(x, y, lat, lon, g, k0, self.datum) return self._xnamed(t, name=name) @property_RO def ispolar(self): '''Is this projection polar (C{bool})? ''' return self._polar isPolar = ispolar # synonym @property_RO def lat0(self): '''Get the latitude of the projection origin (C{degrees}). This is the latitude of minimum azimuthal scale and equals the B{C{lat}} in the 1-parallel L{AlbersEqualArea} and lies between B{C{lat1}} and B{C{lat2}} for the 2-parallel L{AlbersEqualArea2} and L{AlbersEqualArea4} projections. ''' return self._lat0 @property_RO def lat1(self): '''Get the latitude of the first parallel (C{degrees}). ''' return self._lat1 @property_RO def lat2(self): '''Get the latitude of the second parallel (C{degrees}). @note: The second and first parallel latitudes are the same instance for 1-parallel C{AlbersEqualArea*} projections. ''' return self._lat2 def rescale0(self, lat, k=1): '''Set the azimuthal scale for this projection. @arg lat: Northern latitude (C{degrees}). @arg k: Azimuthal scale at latitude B{C{lat}} (C{scalar}). @raise AlbersError: Invalid B{C{lat}} or B{C{k}}. @note: This allows a I{latitude of conformality} to be specified. ''' self._k0_(_Ks(k, 'k') / self.forward(lat, _0_0).scale) def reverse(self, x, y, lon0=0, name=NN, LatLon=None, **LatLon_kwds): '''Convert an east- and northing location to geodetic lat- and longitude. @arg x: Easting of the location (C{meter}). @arg y: Northing of the location (C{meter}). @kwarg lon0: Optional central meridian longitude (C{degrees}). @kwarg name: Optional name for the location (C{str}). @kwarg LatLon: Class to use (C{LatLon}) or C{None}. @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments, ignored if C{B{LatLon}=None}. @return: The geodetic (C{LatLon}) or if B{C{LatLon}} is C{None} an L{Albers7Tuple}C{(x, y, lat, lon, gamma, scale, datum)}. @note: The origin latitude is returned by C{property lat0}. No false easting or northing is added. The returned value of C{lon} is in the range C{[-180..180] degrees} and C{lat} is in the range C{[-90..90] degrees}. If the given B{C{x}} or B{C{y}} point is outside the valid projected space the nearest pole is returned. ''' x = Meter(x=x) y = Meter(y=y) E = self.datum.ellipsoid k0 = self._k0 n0 = self._n0 k0n0 = self._k0n0 nrho0 = self._nrho0 txi0 = self._txi0 y_ = self._sign * y nx = k0n0 * x ny = k0n0 * y_ y1 = nrho0 - ny drho = den = nrho0 + hypot(nx, y1) # 0 implies origin with polar aspect if den: drho = fsum_(x * nx, -_2_0 * y_ * nrho0, y_ * ny) * k0 / den # dsxia = scxi0 * dsxi dsxia = -self._scxi0 * (_2_0 * nrho0 + n0 * drho) * drho / self._qZa2 t = _1_0 - dsxia * (_2_0 * txi0 + dsxia) txi = (txi0 + dsxia) / (sqrt(t) if t > _EPS__4 else _EPS__2) ta = self._tanf(txi) lat = degrees(atan(ta * self._sign)) b = atan2(nx, y1) lon = degrees(b / self._k02n0 if n0 else x / (y1 * k0)) if lon0: lon += _norm180(_Lon(lon0, name=_lon0_)) lon = _norm180(lon) if LatLon is None: g = degrees360(self._sign * b) if den: k0 *= (nrho0 + n0 * drho) * hypot1(E.b_a * ta) / E.a r = Albers7Tuple(x, y, lat, lon, g, k0, self.datum) else: kwds = _xkwds(LatLon_kwds, datum=self.datum) r = LatLon(lat, lon, **kwds) return self._xnamed(r, name=name) @property_RO def scale0(self): '''Get the central scale for the projection (C{float}). This is the azimuthal scale on the latitude of origin of the projection, see C{property lat0}. ''' return self._k0 def _atanhee(self, x): '''(INTERNAL) Function M{atanhee(x)}, defined as ... atanh( E.e * x) / E.e if f > 0 # oblate atan (sqrt(-E.e2) * x) / sqrt(-E.e2) if f < 0 # prolate x if f = 0. ''' E = self.datum.ellipsoid if E.isOblate: x = atanh(x * E.e) / E.e elif E.isProlate: x = atan2(abs(x) * E.e, -_1_0 if x < 0 else _1_0) / E.e return x def _atanhx1(self, x): '''(INTERNAL) Function M{atanh(sqrt(x)) / sqrt(x) - 1}. ''' s = abs(x) if s < _0_5: # for typical ... # x < E.e^2 = 2 * E.f use ... # x / 3 + x^2 / 5 + x^3 / 7 + ... y, k = x, 3 S = Fsum(y / k) for _ in range(_TERMS): y *= x # x**n k += 2 # 2*n + 1 s, d = S.fsum2_(y / k) if not d: break else: s = sqrt(s) s = (atanh(s) if x > 0 else atan(s)) / s - _1_0 return s def _cstxif3(self, ta): '''(INTERNAL) Get 3-tuple C{(cos, sin, tan)} of M{xi(ta)}. ''' t = self._txif(ta) c = _1_0 / hypot1(t) s = c * t return c, s, t def _Datanhee(self, x, y): '''(INTERNAL) Function M{Datanhee(x, y)}, defined as M{atanhee((x - y) / (1 - E.e^2 * x * y)) / (x - y)}. ''' E = self.datum.ellipsoid d = _1_0 - E.e2 * x * y t = x - y if t: s = self._atanhee(t / d) / t elif d: s = _1_0 / d else: raise AlbersError(x=x, y=y, d=d, txt=_AlbersBase._Datanhee.__name__) return s def _D2atanhee(self, x, y): '''(INTERNAL) Function M{D2atanhee(x, y)}, defined as M{(Datanhee(1, y) - Datanhee(1, x)) / (y - x)}. ''' e2 = self.datum.ellipsoid.e2 if (abs(x) + abs(y)) * e2 < _0_5: e = z = _1_0 k = 1 C = Fsum() S = Fsum() T = Fsum() # Taylor expansion for _ in range(_TERMS): T *= y; p = T.fsum_(z); z *= x # PYCHOK ; T *= y; t = T.fsum_(z); z *= x # PYCHOK ; e *= e2 k += 2 s, d = S.fsum2_(e * C.fsum_(p, t) / k) if not d: break elif (_1_0 - x): s = (self._Datanhee(_1_0, y) - self._Datanhee(x, y)) / (_1_0 - x) else: raise AlbersError(x=x, y=y, txt=_AlbersBase._D2atanhee.__name__) return s def _k0_(self, k): '''(INTERNAL) Set C{._k0}, C{._k02}, etc. ''' self._k0 = k = _Ks(k, 'k0') self._k02 = k2 = k**2 self._k0n0 = k * self._n0 self._k02n0 = k2 * self._n0 def _tanf(self, txi): # called from .Ellipsoid.auxAuthalic '''(INTERNAL) Function M{tan-phi from tan-xi}. ''' tol = _tol(_TOL, txi) e2 = self.datum.ellipsoid.e2 qx = self._qx ta = txi Ta = Fsum(ta) for self._iteration in range(1, _NUMIT): # max 2, mean 1.99 # dtxi/dta = (scxi / sca)^3 * 2 * (1 - e^2) / (qZ * (1 - e^2 * sa^2)^2) ta2 = ta**2 sca2 = ta2 + _1_0 txia = self._txif(ta) s3qx = sqrt3(sca2 / (_1_0 + txia**2)) * qx ta, d = Ta.fsum2_((txi - txia) * s3qx * (_1_0 - e2 * ta2 / sca2)**2) if abs(d) < tol: return ta raise AlbersError(iteration=_NUMIT, txt=_no_convergence_fmt_ % (tol,)) def _txif(self, ta): # called from .Ellipsoid.auxAuthalic '''(INTERNAL) Function M{tan-xi from tan-phi}. ''' E = self.datum.ellipsoid ca2 = _1_0 / (_1_0 + ta**2) sa = sqrt(ca2) * abs(ta) # enforce odd parity es1 = sa * E.e2 es2m1 = _1_0 - sa * es1 sp1 = _1_0 + sa es1p1 = sp1 / (_1_0 + es1) es1m1 = sp1 * (_1_0 - es1) es2m1a = es2m1 * E.e12 # e2m s = sqrt((ca2 / (es1p1 * es2m1a) + self._atanhee(ca2 / es1m1)) * (es1m1 / es2m1a + self._atanhee(es1p1))) t = (sa / es2m1 + self._atanhee(sa)) / s if ta < 0: t = -t return t class AlbersEqualArea(_AlbersBase): '''An Albers equal-area (authalic) projection with a single standard parallel. @see: L{AlbersEqualArea4} and L{AlbersEqualArea}. ''' def __init__(self, lat, k0=1, datum=Datums.WGS84, name=NN): '''New L{AlbersEqualArea} projection. @arg lat: Standard parallel (C{degrees}). @kwarg k0: Azimuthal scale on the standard parallel (C{scalar}). @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). @kwarg name: Optional name for the projection (C{str}). @raise AlbertError: Invalid B{C{lat}}, B{C{k0}} or no convergence. ''' self._lat1 = self._lat2 = lat = _Lat(lat, name=_lat1_) args = tuple(sincos2d(lat)) * 2 + (_Ks(k0, 'k0'), datum, name) _AlbersBase.__init__(self, *args) class AlbersEqualArea2(_AlbersBase): '''An Albers equal-area (authalic) projection with two standard parallels. @see: L{AlbersEqualArea4} and L{AlbersEqualArea}. ''' def __init__(self, lat1, lat2, k1=1, datum=Datums.WGS84, name=NN): '''New L{AlbersEqualArea2} projection. @arg lat1: First standard parallel (C{degrees}). @arg lat2: Second standard parallel (C{degrees}). @kwarg k1: Azimuthal scale on the standard parallels (C{scalar}). @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). @kwarg name: Optional name for the projection (C{str}). @raise AlbertError: Invalid B{C{lat1}}m B{C{lat2}}, B{C{k1}} or no convergence. ''' self._lat1, self._lat2 = lats = _Lat(lat1, name=_lat1_), \ _Lat(lat2, name=_lat2_) args = tuple(sincos2d(*lats)) + (_Ks(k1, 'k1'), datum, name) _AlbersBase.__init__(self, *args) class AlbersEqualArea4(_AlbersBase): '''An Albers equal-area (authalic) projection specified by the C{sin} and C{cos} of both standard parallels. @see: L{AlbersEqualArea2} and L{AlbersEqualArea}. ''' def __init__(self, slat1, clat1, slat2, clat2, k1=1, datum=Datums.WGS84, name=NN): '''New L{AlbersEqualArea4} projection. @arg slat1: Sine of first standard parallel (C{scalar}). @arg clat1: Cosine of first standard parallel (non-negative C{scalar}). @arg slat2: Sine of second standard parallel (C{scalar}). @arg clat2: Cosine of second standard parallel (non-negative C{scalar}). @kwarg k1: Azimuthal scale on the standard parallels (C{scalar}). @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). @kwarg name: Optional name for the projection (C{str}). @raise AlbertError: Negative B{C{clat1}} or B{C{clat2}}, B{C{slat1}} and B{C{slat2}} have opposite signs (hemispheres), invalid B{C{k1}} or no convergence. ''' def _Lat_s_c3(s, c, name): L = _Lat(atan2d(s, c), name=name) r = _1_0 / Float_(hypot(s, c), Error=AlbersError, name=name) return L, s * r, c * r self._lat1, sa1, ca1 = _Lat_s_c3(slat1, clat1, _lat1_) self._lat2, sa2, ca2 = _Lat_s_c3(slat2, clat2, _lat2_) _AlbersBase.__init__(self, sa1, ca1, sa2, ca2, _Ks(k1, 'k1'), datum, name) class AlbersEqualAreaCylindrical(_AlbersBase): '''An L{AlbersEqualArea} projection at C{lat=0} and C{k0=1} degenerating to the cylindrical-equal-area projection. ''' _lat1 = _lat2 = _Lat(_0_0, name=_lat1_) def __init__(self, datum=Datums.WGS84, name=NN): '''New L{AlbersEqualAreaCylindrical} projection. @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). @kwarg name: Optional name for the projection (C{str}). ''' _AlbersBase.__init__(self, _0_0, _1_0, _0_0, _1_0, _1_0, datum, name) class AlbersEqualAreaNorth(_AlbersBase): '''An azimuthal L{AlbersEqualArea} projection at C{lat=90} and C{k0=1} degenerating to the L{azimuthal} L{LambertEqualArea} projection. ''' _lat1 = _lat2 = _Lat(_90_0, name=_lat1_) def __init__(self, datum=Datums.WGS84, name=NN): '''New L{AlbersEqualAreaNorth} projection. @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). @kwarg name: Optional name for the projection (C{str}). ''' _AlbersBase.__init__(self, _1_0, _0_0, _1_0, _0_0, _1_0, datum, name) class AlbersEqualAreaSouth(_AlbersBase): '''An azimuthal L{AlbersEqualArea} projection at C{lat=-90} and C{k0=1} degenerating to the L{azimuthal} L{LambertEqualArea} projection. ''' _lat1 = _lat2 = _Lat(-_90_0, name=_lat1_) def __init__(self, datum=Datums.WGS84, name=NN): '''New L{AlbersEqualAreaSouth} projection. @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). @kwarg name: Optional name for the projection (C{str}). ''' _AlbersBase.__init__(self, -_1_0, _0_0, -_1_0, _0_0, _1_0, datum, name) class Albers7Tuple(_NamedTuple): '''7-Tuple C{(x, y, lat, lon, gamma, scale, datum)}, in C{meter}, C{meter}, C{degrees90}, C{degrees180}, C{degrees360}, C{scalar} and C{Datum} where C{(x, y)} is the projected, C{(lat, lon)} the geodetic location, C{gamma} the meridian convergence at point, the bearing of the y-axis measured clockwise from true North and C{scale} is the azimuthal scale of the projection at point. The radial scale is the reciprocal C{1 / scale}. ''' _Names_ = (_x_, _y_, _lat_, _lon_, _gamma_, _scale_, _datum_) _Units_ = ( Meter, Meter, Lat, Lon, Bearing, _Ks, _Pass) __all__ += _ALL_DOCS(Albers7Tuple, _AlbersBase) # **) MIT License # # Copyright (C) 2016-2020 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE.