# Coverage for pygeodesy/etm.py : 94%

## 357 statements 336 run 21 missing 6 excluded

Hot-keys on this page

r m x p toggle line displays

j k next/prev highlighted chunk

0 (zero) top of page

1 (one) first highlighted chunk

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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 | # -*- coding: utf-8 -*- u'''Classes L{ETMError} and L{Etm}, a pure Python transcription of I{Charles Karney}'s C++ class U{TransverseMercatorExact <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1TransverseMercatorExact.html>}, abbreviated as C{TMExact} below. Python class L{ExactTransverseMercator} implements the C{Exact Transverse Mercator} (ETM) projection. Instances of class L{Etm} represent ETM C{easting, nothing} locations. Following is a copy of Karney's U{TransverseMercatorExact.hpp <https://GeographicLib.SourceForge.io/html/TransverseMercatorExact_8hpp_source.html>} file C{Header}. Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2008-2017) and licensed under the MIT/X11 License. For more information, see the U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation. The method entails using the C{Thompson Transverse Mercator} as an intermediate projection. The projections from the intermediate coordinates to C{phi, lam} and C{x, y} are given by elliptic functions. The inverse of these projections are found by Newton's method with a suitable starting guess. The relevant section of L.P. Lee's paper U{Conformal Projections Based On Jacobian Elliptic Functions<https://DOI.org/10.3138/X687-1574-4325-WM62>} is part V, pp 67--101. The C++ implementation and notation closely follow Lee, with the following exceptions:: Lee here Description x/a xi Northing (unit Earth) y/a eta Easting (unit Earth) s/a sigma xi + i * eta y x Easting x y Northing k e Eccentricity k^2 mu Elliptic function parameter k'^2 mv Elliptic function complementary parameter m k Scale zeta zeta Complex longitude = Mercator = chi in paper s sigma Complex GK = zeta in paper Minor alterations have been made in some of Lee's expressions in an attempt to control round-off. For example, C{atanh(sin(phi))} is replaced by C{asinh(tan(phi))} which maintains accuracy near C{phi = pi/2}. Such changes are noted in the code. ''' # make sure int/int division yields float quotient from __future__ import division division = 1 / 2 # double check int division, see .datum.py, .utily.py if not division: raise ImportError('%s 1/2 == %d' % ('division', division)) del division from pygeodesy.basics import EPS, PI_2, PI_4, property_doc_, \ property_RO, _xinstanceof from pygeodesy.datums import Datum, Datums from pygeodesy.elliptic import Elliptic, EllipticError, _TRIPS from pygeodesy.errors import _incompatible from pygeodesy.fmath import cbrt, Fsum, fsum_, hypot, hypot1, hypot2 from pygeodesy.interns import _COMMA_SPACE_, _convergence_, _easting_, \ _k0_, _lat_, _lon_, _lon0_, NN, _northing_, \ _no_convergence_, _scale_ # PYCHOK used! from pygeodesy.karney import _diff182, _fix90, _norm180 from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY from pygeodesy.named import _NamedBase, _NamedTuple, _xnamed from pygeodesy.streprs import pairs, unstr from pygeodesy.units import Lon, Scalar_ from pygeodesy.utily import sincos2 from pygeodesy.utm import _cmlon, _K0, _parseUTM5, Utm, UTMError, \ _toXtm8, _to7zBlldfn from pygeodesy.utmupsBase import _LLEB from math import asinh, atan, atan2, copysign, degrees, radians, \ sinh, sqrt, tan __all__ = _ALL_LAZY.etm __version__ = '20.08.24' _OVERFLOW = 1.0 / EPS**2 _TOL = EPS _TOL_10 = 0.1 * _TOL _TAYTOL = pow(_TOL, 0.6) _TAYTOL2 = 2.0 * _TAYTOL class EasNorExact4Tuple(_NamedTuple): '''4-Tuple C{(easting, northing, convergence, scale)} in C{meter}, C{meter}, C{degrees} and C{scalar}. ''' _Names_ = (_easting_, _northing_, _convergence_, _scale_) class ETMError(UTMError): '''Exact Transverse Mercator (ETM) parse, projection or other L{Etm} issue. ''' pass class Etm(Utm): '''Exact Transverse Mercator (ETM) coordinate, a sub-class of L{Utm}, a Universal Transverse Mercator (UTM) coordinate using the L{ExactTransverseMercator} projection for highest accuracy. @note: Conversion of L{Etm} coordinates to and from (geodetic) lat- and longitude is 3-4 times slower than L{Utm}. @see: Karney's U{Detailed Description<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1TransverseMercatorExact.html#details>}. ''' _Error = ETMError _exactTM = None def __init__(self, zone, hemisphere, easting, northing, band=NN, # PYCHOK expected datum=Datums.WGS84, falsed=True, convergence=None, scale=None, name=NN): '''New L{Etm} coordinate. @arg zone: Longitudinal UTM zone (C{int}, 1..60) or zone with/-out (latitudinal) Band letter (C{str}, '01C'..'60X'). @arg hemisphere: Northern or southern hemisphere (C{str}, C{'N[orth]'} or C{'S[outh]'}). @arg easting: Easting, see B{C{falsed}} (C{meter}). @arg northing: Northing, see B{C{falsed}} (C{meter}). @kwarg band: Optional, (latitudinal) band (C{str}, 'C'..'X'). @kwarg datum: Optional, this coordinate's datum (L{Datum}). @kwarg falsed: Both B{C{easting}} and B{C{northing}} are falsed (C{bool}). @kwarg convergence: Optional meridian convergence, bearing off grid North, clockwise from true North (C{degrees}) or C{None}. @kwarg scale: Optional grid scale factor (C{scalar}) or C{None}. @kwarg name: Optional name (C{str}). @raise ETMError: Invalid B{C{zone}}, B{C{hemishere}} or B{C{band}}. @example: >>> import pygeodesy >>> u = pygeodesy.Etm(31, 'N', 448251, 5411932) ''' Utm.__init__(self, zone, hemisphere, easting, northing, band=band, datum=datum, falsed=falsed, convergence=convergence, scale=scale, name=name) self.exactTM = self.datum.exactTM # ExactTransverseMercator(datum=self.datum) @property_doc_(''' the ETM projection (L{ExactTransverseMercator}).''') def exactTM(self): '''Get the ETM projection (L{ExactTransverseMercator}). ''' return self._exactTM @exactTM.setter # PYCHOK setter! def exactTM(self, exactTM): '''Set the ETM projection (L{ExactTransverseMercator}). ''' _xinstanceof(ExactTransverseMercator, exactTM=exactTM) E = self.datum.ellipsoid if exactTM._E != E or exactTM.majoradius != E.a \ or exactTM.flattening != E.f: raise ETMError(repr(exactTM), txt=_incompatible(repr(E))) self._exactTM = exactTM self._scale0 = exactTM.k0 def parse(self, strETM, name=NN): '''Parse a string to a similar L{Etm} instance. @arg strETM: The ETM coordinate (C{str}), see function L{parseETM5}. @kwarg name: Optional instance name (C{str}), overriding this name. @return: The instance (L{Etm}). @raise ETMError: Invalid B{C{strETM}}. @see: Function L{parseUPS5}, L{parseUTM5} and L{parseUTMUPS5}. ''' return parseETM5(strETM, datum=self.datum, Etm=self.classof, name=name or self.name) def parseETM(self, strETM): '''DEPRECATED, use method L{Etm.parse}. ''' return self.parse(strETM) def toLatLon(self, LatLon=None, unfalse=True, **unused): # PYCHOK expected '''Convert this ETM coordinate to an (ellipsoidal) geodetic point. @kwarg LatLon: Optional, ellipsoidal class to return the geodetic point (C{LatLon}) or C{None}. @kwarg unfalse: Unfalse B{C{easting}} and B{C{northing}} if falsed (C{bool}). @return: This ETM coordinate as (B{C{LatLon}}) or a L{LatLonDatum5Tuple}C{(lat, lon, datum, convergence, scale)} if B{C{LatLon}} is C{None}. @raise EllipticError: No convergence. @raise TypeError: If B{C{LatLon}} is not ellipsoidal. @example: >>> from pygeodesy import ellipsoidalVincenty as eV, Etm >>> u = Etm(31, 'N', 448251.795, 5411932.678) >>> ll = u.toLatLon(eV.LatLon) # 48°51′29.52″N, 002°17′40.20″E ''' xTM, d = self.exactTM, self.datum # double check that this and exactTM's ellipsoids stil match if xTM._E != d.ellipsoid: t = repr(d.ellipsoid) raise ETMError(repr(xTM._E), txt=_incompatible(t)) if self._latlon and self._latlon_args == (xTM, unfalse): return self._latlon5(LatLon) f = not unfalse e, n = self.to2en(falsed=f) # f = unfalse == self.falsed # == unfalse and self.falsed or (not unfalse and not self.falsed) # == unfalse if self.falsed else not unfalse # == unfalse if self.falsed else f if self.falsed: f = unfalse lon0 = _cmlon(self.zone) if f else None lat, lon, g, k = xTM.reverse(e, n, lon0=lon0) ll = _LLEB(lat, lon, datum=d, name=self.name) ll._convergence = g ll._scale = k self._latlon_to(ll, xTM, unfalse) return self._latlon5(LatLon) def _latlon_to(self, ll, xTM, unfalse): '''(INTERNAL) See C{.toLatLon}, C{toEtm8}, C{_toXtm8}. ''' self._latlon, self._latlon_args = ll, (xTM, unfalse) def toUtm(self): # PYCHOK signature '''Copy this ETM to a UTM coordinate. @return: The UTM coordinate (L{Utm}). ''' return self._xnamed(self._xcopy2(Utm)) class ExactTransverseMercator(_NamedBase): '''A Python version of Karney's U{TransverseMercatorExact <https://GeographicLib.SourceForge.io/html/TransverseMercatorExact_8cpp_source.html>} C++ class, a numerically exact transverse mercator projection, here referred to as C{TMExact}. @see: C{U{TMExact(real a, real f, real k0, bool extendp)<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1TransverseMercatorExact.html#a72ffcc89eee6f30a6d1f4d061518a6f1>}}. ''' _a = 0 # major radius _datum = None # Datum _e = 0 # eccentricity _E = None # Ellipsoid _extendp = True _f = 0 # flattening _iteration = 0 # ._sigmaInv and ._zetaInv _k0 = 1 # central scale factor _k0_a = 0 _lon0 = 0 # central meridian # see ._reset() below: # _e_PI_2 = _e * PI_2 # _e_PI_4 = _e * PI_4 # _e_TAYTOL = _e * _TAYTOL # _1_e_90 = (1 - _e) * 90 # _1_e_PI_2 = (1 - _e) * PI_2 # _1_e2_PI_2 = (1 - _e * 2) * PI_2 # _mu = _e**2 # _mu_2_1 = (_e**2 + 2) * 0.5 # _Eu = Elliptic(_mu) # _Eu_cE_1_4 = _Eu.cE * 0.25 # _Eu_cK_cE = _Eu.cK / _Eu.cE # _Eu_cK_PI_2 = _Eu.cK / PI_2 # _mv = 1 - _mu # _3_mv = 3.0 / _mv # _3_mv_e = _3_mv / _e # _Ev = Elliptic(_mv) # _Ev_cKE_3_4 = _Ev.cKE * 0.75 # _Ev_cKE_5_4 = _Ev.cKE * 1.25 def __init__(self, datum=Datums.WGS84, lon0=0, k0=_K0, extendp=True, name=NN): '''New L{ExactTransverseMercator} projection. @kwarg datum: The datum and ellipsoid to use (C{Datum}). @kwarg lon0: The central meridian (C{degrees180}). @kwarg k0: The central scale factor (C{float}). @kwarg extendp: Use the extended domain (C{bool}). @kwarg name: Optional name for the projection (C{str}). @raise EllipticError: No convergence. @raise ETMError: Invalid B{C{k0}}. @raise TypeError: Invalid B{C{datum}}. @raise ValueError: Invalid B{C{lon0}} or B{C{k0}}. @note: The maximum error for all 255.5K U{TMcoords.dat <https://Zenodo.org/record/32470>} tests (with C{0 <= lat <= 84} and C{0 <= lon}) is C{5.2e-08 .forward} or 52 nano-meter easting and northing and C{3.8e-13 .reverse} or 0.38 pico-degrees lat- and longitude (with Python 3.7.3, 2.7.16, PyPy6 3.5.3 and PyPy6 2.7.13, all in 64-bit on macOS 10.13.6 High Sierra). ''' if not extendp: self._extendp = False if name: self.name = name self.datum = datum self.lon0 = lon0 self.k0 = k0 @property_doc_(''' the datum (L{Datum}).''') def datum(self): '''Get the datum (L{Datum}) or C{None}. ''' return self._datum @datum.setter # PYCHOK setter! def datum(self, datum): '''Set the datum and ellipsoid (L{Datum}). @raise EllipticError: No convergence. @raise TypeError: Invalid B{C{datum}}. ''' _xinstanceof(Datum, datum=datum) E = datum.ellipsoid self._reset(E.e, E.e2) self._a = E.a self._f = E.f # flattening = (a - b) / a self._datum = datum self._E = E @property_RO def equatoradius(self): '''Get the equatorial (major) radius, semi-axis (C{meter}). ''' return self._a majoradius = equatoradius # for backward compatibility @property_RO def extendp(self): '''Get using the extended domain (C{bool}). ''' return self._extendp @property_RO def flattening(self): '''Get the flattening (C{float}). ''' return self._f def forward(self, lat, lon, lon0=None): # MCCABE 13 '''Forward projection, from geographic to transverse Mercator. @arg lat: Latitude of point (C{degrees}). @arg lon: Longitude of point (C{degrees}). @kwarg lon0: Central meridian of the projection (C{degrees}). @return: L{EasNorExact4Tuple}C{(easting, northing, convergence, scale)} in C{meter}, C{meter}, C{degrees} and C{scalar}. @see: C{void TMExact::Forward(real lon0, real lat, real lon, real &x, real &y, real &gamma, real &k)}. @raise EllipticError: No convergence. ''' lat = _fix90(lat) lon, _ = _diff182((self._lon0 if lon0 is None else lon0), lon) # Explicitly enforce the parity backside = _lat = _lon = False if not self.extendp: if lat < 0: _lat, lat = True, -lat if lon < 0: _lon, lon = True, -lon if lon > 90: backside = True if lat == 0: _lat = True lon = 180 - lon # u,v = coordinates for the Thompson TM, Lee 54 if lat == 90: u = self._Eu.cK v = self._iteration = 0 elif lat == 0 and lon == self._1_e_90: u = self._iteration = 0 v = self._Ev.cK else: # tau = tan(phi), taup = sinh(psi) tau, lam = tan(radians(lat)), radians(lon) u, v = self._zetaInv(self._E.es_taupf(tau), lam) sncndn6 = self._sncndn6(u, v) xi, eta, _ = self._sigma3(v, *sncndn6) if backside: xi = 2 * self._Eu.cE - xi y = xi * self._k0_a x = eta * self._k0_a if lat == 90: g, k = lon, self._k0 else: g, k = self._zetaScaled(sncndn6, ll=False) if backside: g = 180 - g if _lat: y, g = -y, -g if _lon: x, g = -x, -g r = EasNorExact4Tuple(x, y, g, k) r._iteration = self._iteration return r @property_RO def iteration(self): '''Get the most recent C{ExactTransverseMercator.forward} or C{ExactTransverseMercator.reverse} iteration number (C{int} or C{0} if not available/applicable). ''' return self._iteration @property_doc_(''' the central scale factor (C{float}).''') def k0(self): '''Get the central scale factor (C{float}), aka I{C{scale0}}. ''' return self._k0 # aka scale0 @k0.setter # PYCHOK setter! def k0(self, k0): '''Set the central scale factor (C{float}), aka I{C{scale0}}. @raise ETMError: Invalid B{C{k0}}. ''' self._k0 = Scalar_(k0, name=_k0_, Error=ETMError, low=_TOL_10, high=1.0) # if not self._k0 > 0: # raise Scalar_.Error_(Scalar_, k0, name=_k0_, Error=ETMError) self._k0_a = self._k0 * self._a @property_doc_(''' the central meridian (C{degrees180}).''') def lon0(self): '''Get the central meridian (C{degrees180}). ''' return self._lon0 @lon0.setter # PYCHOK setter! def lon0(self, lon0): '''Set the central meridian (C{degrees180}). @raise ValueError: Invalid B{C{lon0}}. ''' self._lon0 = _norm180(Lon(lon0, name=_lon0_)) def _reset(self, e, e2): '''(INTERNAL) Get elliptic functions and pre-compute some frequently used values. @arg e: Eccentricity (C{float}). @arg e2: Eccentricity squared (C{float}). @raise EllipticError: No convergence. ''' # assert e2 == e**2 self._e = e self._e_PI_2 = e * PI_2 self._e_PI_4 = e * PI_4 self._e_TAYTOL = e * _TAYTOL self._1_e_90 = (1 - e) * 90 self._1_e_PI_2 = (1 - e) * PI_2 self._1_e2_PI_2 = (1 - e * 2) * PI_2 self._mu = e2 self._mu_2_1 = (e2 + 2) * 0.5 self._Eu = Elliptic(self._mu) self._Eu_cE_1_4 = self._Eu.cE * 0.25 self._Eu_cK_cE = self._Eu.cK / self._Eu.cE self._Eu_cK_PI_2 = self._Eu.cK / PI_2 self._mv = 1 - e2 self._3_mv = 3.0 / self._mv self._3_mv_e = self._3_mv / e self._Ev = Elliptic(self._mv) self._Ev_cKE_3_4 = self._Ev.cKE * 0.75 self._Ev_cKE_5_4 = self._Ev.cKE * 1.25 self._iteration = 0 def reverse(self, x, y, lon0=None): '''Reverse projection, from Transverse Mercator to geographic. @arg x: Easting of point (C{meters}). @arg y: Northing of point (C{meters}). @kwarg lon0: Central meridian of the projection (C{degrees}). @return: L{LatLonExact4Tuple}C{(lat, lon, convergence, scale)} in C{degrees}, C{degrees180}, C{degrees} and C{scalar}. @see: C{void TMExact::Reverse(real lon0, real x, real y, real &lat, real &lon, real &gamma, real &k)} @raise EllipticError: No convergence. ''' # undoes the steps in .forward. xi = y / self._k0_a eta = x / self._k0_a backside = _lat = _lon = False if not self.extendp: # enforce the parity if y < 0: _lat, xi = True, -xi if x < 0: _lon, eta = True, -eta if xi > self._Eu.cE: xi = 2 * self._Eu.cE - xi backside = True # u,v = coordinates for the Thompson TM, Lee 54 if xi != 0 or eta != self._Ev.cKE: u, v = self._sigmaInv(xi, eta) else: u = self._iteration = 0 v = self._Ev.cK if v != 0 or u != self._Eu.cK: g, k, lat, lon = self._zetaScaled(self._sncndn6(u, v)) else: g, k, lat, lon = 0, self._k0, 90, 0 if backside: lon, g = (180 - lon), (180 - g) if _lat: lat, g = -lat, -g if _lon: lon, g = -lon, -g lon += self._lon0 if lon0 is None else _norm180(lon0) r = LatLonExact4Tuple(_norm180(lat), _norm180(lon), g, k) r._iteration = self._iteration return r def _scaled(self, tau, d2, snu, cnu, dnu, snv, cnv, dnv): '''(INTERNAL) C{scaled}. @note: Argument B{C{d2}} is C{_mu * cnu**2 + _mv * cnv**2} from C{._sigma3} or C{._zeta3}. @return: 2-Tuple C{(convergence, scale)}. @see: C{void TMExact::Scale(real tau, real /*lam*/, real snu, real cnu, real dnu, real snv, real cnv, real dnv, real &gamma, real &k)}. ''' mu, mv = self._mu, self._mv cnudnv = cnu * dnv # Lee 55.12 -- negated for our sign convention. g gives # the bearing (clockwise from true north) of grid north g = atan2(mv * cnv * snv * snu, cnudnv * dnu) # Lee 55.13 with nu given by Lee 9.1 -- in sqrt change # the numerator from # # (1 - snu^2 * dnv^2) to (_mv * snv^2 + cnu^2 * dnv^2) # # to maintain accuracy near phi = 90 and change the # denomintor from # (dnu^2 + dnv^2 - 1) to (_mu * cnu^2 + _mv * cnv^2) # # to maintain accuracy near phi = 0, lam = 90 * (1 - e). # Similarly rewrite sqrt term in 9.1 as # # _mv + _mu * c^2 instead of 1 - _mu * sin(phi)^2 q2 = (mv * snv**2 + cnudnv**2) / d2 # originally: sec2 = 1 + tau**2 # sec(phi)^2 # k = sqrt(mv + mu / sec2) * sqrt(sec2) * sqrt(q2) # = sqrt(mv + mv * tau**2 + mu) * sqrt(q2) k = sqrt(fsum_(mu, mv, mv * tau**2)) * sqrt(q2) return degrees(g), k * self._k0 def _sigma3(self, v, snu, cnu, dnu, snv, cnv, dnv): # PYCHOK unused '''(INTERNAL) C{sigma}. @return: 3-Tuple C{(xi, eta, d2)}. @see: C{void TMExact::sigma(real /*u*/, real snu, real cnu, real dnu, real v, real snv, real cnv, real dnv, real &xi, real &eta)}. @raise EllipticError: No convergence. ''' # Lee 55.4 writing # dnu^2 + dnv^2 - 1 = _mu * cnu^2 + _mv * cnv^2 d2 = self._mu * cnu**2 + self._mv * cnv**2 xi = self._Eu.fE(snu, cnu, dnu) - self._mu * snu * cnu * dnu / d2 eta = v - self._Ev.fE(snv, cnv, dnv) + self._mv * snv * cnv * dnv / d2 return xi, eta, d2 def _sigmaDwd(self, snu, cnu, dnu, snv, cnv, dnv): '''(INTERNAL) C{sigmaDwd}. @return: 2-Tuple C{(du, dv)}. @see: C{void TMExact::dwdsigma(real /*u*/, real snu, real cnu, real dnu, real /*v*/, real snv, real cnv, real dnv, real &du, real &dv)}. ''' snuv = snu * snv # Reciprocal of 55.9: dw / ds = dn(w)^2/_mv, # expanding complex dn(w) using A+S 16.21.4 d = self._mv * (cnv**2 + self._mu * snuv**2)**2 r = cnv * dnu * dnv i = -cnu * snuv * self._mu du = (r**2 - i**2) / d dv = 2 * r * i / d return du, dv def _sigmaInv(self, xi, eta): '''(INTERNAL) Invert C{sigma} using Newton's method. @return: 2-Tuple C{(u, v)}. @see: C{void TMExact::sigmainv(real xi, real eta, real &u, real &v)}. @raise EllipticError: No convergence. ''' u, v, trip = self._sigmaInv0(xi, eta) if trip: self._iteration = 0 else: U, V = Fsum(u), Fsum(v) # min iterations = 2, max = 7, mean = 3.9 for self._iteration in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC sncndn6 = self._sncndn6(u, v) X, E, _ = self._sigma3(v, *sncndn6) dw, dv = self._sigmaDwd( *sncndn6) X = xi - X E -= eta u, du = U.fsum2_(X * dw, E * dv) v, dv = V.fsum2_(X * dv, -E * dw) if trip: break trip = hypot2(du, dv) < _TOL_10 else: t = unstr(self._sigmaInv.__name__, xi, eta) raise EllipticError(_no_convergence_, txt=t) return u, v def _sigmaInv0(self, xi, eta): '''(INTERNAL) Starting point for C{sigmaInv}. @return: 3-Tuple C{(u, v, trip)}. @see: C{bool TMExact::sigmainv0(real xi, real eta, real &u, real &v)}. ''' trip = False if eta > self._Ev_cKE_5_4 or xi < min(- self._Eu_cE_1_4, eta - self._Ev.cKE): # sigma as a simple pole at # w = w0 = Eu.K() + i * Ev.K() # and sigma is approximated by # sigma = (Eu.E() + i * Ev.KE()) + 1 / (w - w0) x = xi - self._Eu.cE y = eta - self._Ev.cKE d = hypot2(x, y) u = self._Eu.cK + x / d v = self._Ev.cK - y / d elif eta > self._Ev.cKE or (xi < self._Eu_cE_1_4 and eta > self._Ev_cKE_3_4): # At w = w0 = i * Ev.K(), we have # sigma = sigma0 = i * Ev.KE() # sigma' = sigma'' = 0 # # including the next term in the Taylor series gives: # sigma = sigma0 - _mv / 3 * (w - w0)^3 # # When inverting this, we map arg(w - w0) = [-pi/2, -pi/6] # to arg(sigma - sigma0) = [-pi/2, pi/2] # mapping arg = [-pi/2, -pi/6] to [-pi/2, pi/2] d = eta - self._Ev.cKE r = hypot(xi, d) # Error using this guess is about 0.068 * rad^(5/3) trip = r < _TAYTOL2 # Map the range [-90, 180] in sigma space to [-90, 0] in # w space. See discussion in zetainv0 on the cut for ang. r = cbrt(r * self._3_mv) a = atan2(d - xi, xi + d) / 3.0 - PI_4 s, c = sincos2(a) u = r * c v = r * s + self._Ev.cK else: # use w = sigma * Eu.K/Eu.E (correct in the limit _e -> 0) u = xi * self._Eu_cK_cE v = eta * self._Eu_cK_cE return u, v, trip def _sncndn6(self, u, v): '''(INTERNAL) Get 6-tuple C{(snu, cnu, dnu, snv, cnv, dnv)}. ''' # snu, cnu, dnu = self._Eu.sncndn(u) # snv, cnv, dnv = self._Ev.sncndn(v) return self._Eu.sncndn(u) + self._Ev.sncndn(v) def toStr(self, **kwds): '''Return a C{str} representation. @arg kwds: Optional, overriding keyword arguments. ''' d = dict(name=self.name) if self.name else {} d = dict(datum=self.datum.name, lon0=self.lon0, k0=self.k0, extendp=self.extendp, **d) return _COMMA_SPACE_.join(pairs(d, **kwds)) def _zeta3(self, snu, cnu, dnu, snv, cnv, dnv): '''(INTERNAL) C{zeta}. @return: 3-Tuple C{(taup, lambda, d2)}. @see: C{void TMExact::zeta(real /*u*/, real snu, real cnu, real dnu, real /*v*/, real snv, real cnv, real dnv, real &taup, real &lam)} ''' e = self._e # Lee 54.17 but write # atanh(snu * dnv) = asinh(snu * dnv / sqrt(cnu^2 + _mv * snu^2 * snv^2)) # atanh(_e * snu / dnv) = asinh(_e * snu / sqrt(_mu * cnu^2 + _mv * cnv^2)) d1 = cnu**2 + self._mv * (snu * snv)**2 d2 = self._mu * cnu**2 + self._mv * cnv**2 # Overflow value s.t. atan(overflow) = pi/2 t1 = t2 = copysign(_OVERFLOW, snu) if d1 > 0: t1 = snu * dnv / sqrt(d1) lam = 0 if d2 > 0: t2 = sinh(e * asinh(e * snu / sqrt(d2))) if d1 > 0: lam = atan2(dnu * snv , cnu * cnv) - \ atan2(cnu * snv * e, dnu * cnv) * e # psi = asinh(t1) - asinh(t2) # taup = sinh(psi) taup = t1 * hypot1(t2) - t2 * hypot1(t1) return taup, lam, d2 def _zetaDwd(self, snu, cnu, dnu, snv, cnv, dnv): '''(INTERNAL) C{zetaDwd}. @return: 2-Tuple C{(du, dv)}. @see: C{void TMExact::dwdzeta(real /*u*/, real snu, real cnu, real dnu, real /*v*/, real snv, real cnv, real dnv, real &du, real &dv)}. ''' cnu2 = cnu**2 * self._mu cnv2 = cnv**2 dnuv = dnu * dnv dnuv2 = dnuv**2 snuv = snu * snv snuv2 = snuv**2 * self._mu # Lee 54.21 but write # (1 - dnu^2 * snv^2) = (cnv^2 + _mu * snu^2 * snv^2) # (see A+S 16.21.4) d = self._mv * (cnv2 + snuv2)**2 du = cnu * dnuv * (cnv2 - snuv2) / d dv = -cnv * snuv * (cnu2 + dnuv2) / d return du, dv def _zetaInv(self, taup, lam): '''(INTERNAL) Invert C{zeta} using Newton's method. @return: 2-Tuple C{(u, v)}. @see: C{void TMExact::zetainv(real taup, real lam, real &u, real &v)}. @raise EllipticError: No convergence. ''' psi = asinh(taup) sca = 1.0 / hypot1(taup) u, v, trip = self._zetaInv0(psi, lam) if trip: self._iteration = 0 else: stol2 = _TOL_10 / max(psi**2, 1.0) U, V = Fsum(u), Fsum(v) # min iterations = 2, max = 6, mean = 4.0 for self._iteration in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC sncndn6 = self._sncndn6(u, v) T, L, _ = self._zeta3( *sncndn6) dw, dv = self._zetaDwd(*sncndn6) T = (taup - T) * sca L -= lam u, du = U.fsum2_(T * dw, L * dv) v, dv = V.fsum2_(T * dv, -L * dw) if trip: break trip = hypot2(du, dv) < stol2 else: t = unstr(self._zetaInv.__name__, taup, lam) raise EllipticError(_no_convergence_, txt=t) return u, v def _zetaInv0(self, psi, lam): '''(INTERNAL) Starting point for C{zetaInv}. @return: 3-Tuple C{(u, v, trip)}. @see: C{bool TMExact::zetainv0(real psi, real lam, # radians real &u, real &v)}. ''' trip = False if psi < -self._e_PI_4 and lam > self._1_e2_PI_2 \ and psi < (lam - self._1_e_PI_2): # N.B. this branch is normally not taken because psi < 0 # is converted psi > 0 by Forward. # # There's a log singularity at w = w0 = Eu.K() + i * Ev.K(), # corresponding to the south pole, where we have, approximately # # psi = _e + i * pi/2 - _e * atanh(cos(i * (w - w0)/(1 + _mu/2))) # # Inverting this gives: h = sinh(1 - psi / self._e) a = (PI_2 - lam) / self._e s, c = sincos2(a) u = self._Eu.cK - asinh(s / hypot(c, h)) * self._mu_2_1 v = self._Ev.cK - atan2(c, h) * self._mu_2_1 elif psi < self._e_PI_2 and lam > self._1_e2_PI_2: # At w = w0 = i * Ev.K(), we have # # zeta = zeta0 = i * (1 - _e) * pi/2 # zeta' = zeta'' = 0 # # including the next term in the Taylor series gives: # # zeta = zeta0 - (_mv * _e) / 3 * (w - w0)^3 # # When inverting this, we map arg(w - w0) = [-90, 0] to # arg(zeta - zeta0) = [-90, 180] d = lam - self._1_e_PI_2 r = hypot(psi, d) # Error using this guess is about 0.21 * (rad/e)^(5/3) trip = r < self._e_TAYTOL # atan2(dlam-psi, psi+dlam) + 45d gives arg(zeta - zeta0) # in range [-135, 225). Subtracting 180 (since multiplier # is negative) makes range [-315, 45). Multiplying by 1/3 # (for cube root) gives range [-105, 15). In particular # the range [-90, 180] in zeta space maps to [-90, 0] in # w space as required. r = cbrt(r * self._3_mv_e) a = atan2(d - psi, psi + d) / 3.0 - PI_4 s, c = sincos2(a) u = r * c v = r * s + self._Ev.cK else: # Use spherical TM, Lee 12.6 -- writing C{atanh(sin(lam) / # cosh(psi)) = asinh(sin(lam) / hypot(cos(lam), sinh(psi)))}. # This takes care of the log singularity at C{zeta = Eu.K()}, # corresponding to the north pole. s, c = sincos2(lam) h, r = sinh(psi), self._Eu_cK_PI_2 # But scale to put 90, 0 on the right place u = r * atan2(h, c) v = r * asinh(s / hypot(c, h)) return u, v, trip def _zetaScaled(self, sncndn6, ll=True): '''(INTERNAL) Recompute (T, L) from (u, v) to improve accuracy of Scale. @arg sncndn6: 6-Tuple C{(snu, cnu, dnu, snv, cnv, dnv)}. @return: 2-Tuple C{(g, k)} if B{C{ll}} is C{False} else 4-tuple C{(g, k, lat, lon)}. ''' t, lam, d2 = self._zeta3( *sncndn6) tau = self._E.es_tauf(t) r = self._scaled(tau, d2, *sncndn6) if ll: r += degrees(atan(tau)), degrees(lam) return r class LatLonExact4Tuple(_NamedTuple): '''4-Tuple C{(lat, lon, convergence, scale)} in C{degrees180}, C{degrees}, C{degrees} and C{scalar}. ''' _Names_ = (_lat_, _lon_, _convergence_, _scale_) def parseETM5(strUTM, datum=Datums.WGS84, Etm=Etm, falsed=True, name=NN): '''Parse a string representing a UTM coordinate, consisting of C{"zone[band] hemisphere easting northing"}. @arg strUTM: A UTM coordinate (C{str}). @kwarg datum: Optional datum to use (L{Datum}). @kwarg Etm: Optional class to return the UTM coordinate (L{Etm}) or C{None}. @kwarg falsed: Both easting and northing are falsed (C{bool}). @kwarg name: Optional B{C{Etm}} name (C{str}). @return: The UTM coordinate (B{C{Etm}}) or if B{C{Etm}} is C{None}, a L{UtmUps5Tuple}C{(zone, hemipole, easting, northing, band)}. The C{hemipole} is the hemisphere C{'N'|'S'}. @raise ETMError: Invalid B{C{strUTM}}. @example: >>> u = parseETM5('31 N 448251 5411932') >>> u.toStr2() # [Z:31, H:N, E:448251, N:5411932] >>> u = parseETM5('31 N 448251.8 5411932.7') >>> u.toStr() # 31 N 448252 5411933 ''' r = _parseUTM5(strUTM, datum, Etm, falsed, Error=ETMError) return _xnamed(r, name, force=True) def toEtm8(latlon, lon=None, datum=None, Etm=Etm, falsed=True, name=NN, zone=None, **cmoff): '''Convert a lat-/longitude point to an ETM coordinate. @arg latlon: Latitude (C{degrees}) or an (ellipsoidal) geodetic C{LatLon} point. @kwarg lon: Optional longitude (C{degrees}) or C{None}. @kwarg datum: Optional datum for this ETM coordinate, overriding B{C{latlon}}'s datum (C{Datum}). @kwarg Etm: Optional class to return the ETM coordinate (L{Etm}) or C{None}. @kwarg falsed: False both easting and northing (C{bool}). @kwarg name: Optional B{C{Utm}} name (C{str}). @kwarg zone: Optional UTM zone to enforce (C{int} or C{str}). @kwarg cmoff: DEPRECATED, use B{C{falsed}}. Offset longitude from the zone's central meridian (C{bool}). @return: The ETM coordinate (B{C{Etm}}) or a L{UtmUps8Tuple}C{(zone, hemipole, easting, northing, band, datum, convergence, scale)} if B{C{Etm}} is C{None} or not B{C{falsed}}. The C{hemipole} is the C{'N'|'S'} hemisphere. @raise EllipticError: No convergence. @raise ETMError: Invalid B{C{zone}}. @raise TypeError: If B{C{latlon}} is not ellipsoidal. @raise RangeError: If B{C{lat}} outside the valid UTM bands or if B{C{lat}} or B{C{lon}} outside the valid range and L{rangerrors} set to C{True}. @raise ValueError: If B{C{lon}} value is missing or if B{C{latlon}} is invalid. ''' z, B, lat, lon, d, f, name = _to7zBlldfn(latlon, lon, datum, falsed, name, zone, ETMError, **cmoff) lon0 = _cmlon(z) if f else None x, y, g, k = d.exactTM.forward(lat, lon, lon0=lon0) return _toXtm8(Etm, z, lat, x, y, B, d, g, k, f, name, latlon, d.exactTM, Error=ETMError) __all__ += _ALL_DOCS(EasNorExact4Tuple, LatLonExact4Tuple) # **) 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.