# Coverage for pygeodesy/ktm.py : 98%

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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 | # -*- coding: utf-8 -*- u'''A pure Python version of I{Karney}'s C++ class U{TransverseMercator <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1TransverseMercator.html>} based on I{Krüger} series. See also I{Karney}'s utility U{TransverseMercatorProj <https://GeographicLib.SourceForge.io/C++/doc/TransverseMercatorProj.1.html>}. Following and further below is a copy of I{Karney}'s U{TransverseMercator.hpp <https://GeographicLib.SourceForge.io/C++/doc/TransverseMercator_8hpp_source.html>} file C{Header}. This implementation follows closely JHS 154, ETRS89 - I{järjestelmään liittyvät karttaprojektiot, tasokoordinaatistot ja karttalehtijako} (Map projections, plane coordinates, and map sheet index for ETRS89), published by JUHTA, Finnish Geodetic Institute, and the National Land Survey of Finland (2006). The relevant section is available as the U{2008 PDF file <http://Docs.JHS-suositukset.FI/jhs-suositukset/JHS154/JHS154_liite1.pdf>}. This is a straight transcription of the formulas in this paper with the following exceptions: - Use of 6th order series instead of 4th order series. This reduces the error to about 5 nm for the UTM range of coordinates (instead of 200 nm), with a speed penalty of only 1%, - Use Newton's method instead of plain iteration to solve for latitude in terms of isometric latitude in the Reverse method, - Use of Horner's representation for evaluating polynomials and Clenshaw's method for summing trigonometric series, - Several modifications of the formulas to improve the numerical accuracy, - Evaluating the convergence and scale using the expression for the projection or its inverse. Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2008-2022) and licensed under the MIT/X11 License. For more information, see the U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation. ''' # make sure int/int division yields float quotient from __future__ import division as _; del _ # PYCHOK semicolon from pygeodesy.basics import copysign0, isint, isodd, neg, neg_ from pygeodesy.constants import INF, _K0_UTM, NINF, PI, PI_2, _0_0s, \ _0_0, _1_0, _90_0, _180_0 # from pygeodesy.datums import _spherical_datum # in KTransverseMercator.ellipsoid.setter from pygeodesy.errors import _or, _ValueError, _xkwds_get from pygeodesy.fmath import fsum1_, hypot, hypot1 # from pygeodesy.fsums import fsum1_ # from .fmath from pygeodesy.interns import NN, _COMMASPACE_, _not_, _singular_ from pygeodesy.karney import _atan2d, _diff182, _EWGS84, _fix90, \ _NamedBase, _norm180, _polynomial, _unsigned2 from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS, _pairs # from pygeodesy.named import _NamedBase # from .karney from pygeodesy.namedTuples import Forward4Tuple, Reverse4Tuple from pygeodesy.props import property_doc_, Property, Property_RO, \ _update_all # from pygeodesy.streprs import pairs as _pairs # from .lazily from pygeodesy.units import Degrees, Scalar_, _1mm as _TOL_10 # PYCHOK used! from pygeodesy.utily import atand, sincos2, sincos2d_ from cmath import phase from math import atan2, asinh, cos, cosh, degrees, sin, sinh, sqrt, tanh __all__ = _ALL_LAZY.ktm __version__ = '22.09.15' class KTMError(_ValueError): '''Error raised for L{KTransverseMercator} and L{KTransverseMercator.forward} issues. ''' pass class KTransverseMercator(_NamedBase): '''Transverse Mercator projection based on Krüger's method which evaluates the projection and its inverse in terms of a series. There's a singularity in the projection at I{phi = 0, lam - lam0 = +/- (1 - e) 90}, about +/- 82.6 degrees for WGS84, where I{e} is the eccentricity. Beyond this point, the series ceases to converge and the results from this method will be garbage. I{To be on the safe side, don't use this method if the angular distance from the central meridian exceeds (1 - 2e) x 90}, about 75 degrees for the WGS84 ellipsoid. Class L{ExactTransverseMercator} is an alternative implementation of the projection using I{exact} formulas which yield accurate (to 8 nm) results over the entire ellipsoid. The ellipsoid parameters and the central scale are set in the constructor. The central meridian (which is a trivial shift of the longitude) is specified as the C{lon0} keyword argument of the L{KTransverseMercator.forward} and L{KTransverseMercator.reverse} methods. The latitude of origin is taken to be the equator. There is no provision in this class for specifying a false easting or false northing or a different latitude of origin. However these are can be simply included by the calling function. The L{KTransverseMercator.forward} and L{KTransverseMercator.reverse} methods also return the meridian convergence C{gamma} and scale C{k}. The meridian convergence is the bearing of grid North, the C{y axis}, measured clockwise from true North. ''' _E = _EWGS84 _k0 = _K0_UTM # central scale factor _lon0 = _0_0 # central meridian _mTM = 6 _raiser = False # throw Error def __init__(self, a_earth=_EWGS84, f=None, lon0=0, k0=_K0_UTM, name=NN, raiser=False, **TMorder): '''New L{KTransverseMercator}. @kwarg a_earth: This rhumb's earth (L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple}, L{Datum}, 2-tuple (C{a, f})) or the equatorial radius (C{scalar}, C{meter}). @kwarg f: The ellipsoid's flattening (C{scalar}), iff B{C{a_earth}} is a C{scalar}, ignored otherwise. @kwarg lon0: The central meridian (C{degrees180}). @kwarg k0: Central scale factor (C{scalar}). @kwarg name: Optional name (C{str}). @kwarg raiser: If C{True}, throw a L{KTMError} for C{forward} singularities (C{bool}). @kwarg TMorder: Keyword argument B{C{TMorder}}, see property C{TMorder}. @raise KTMError: Invalid B{C{a_earth}}, B{C{f}} or B{C{TMorder}}. ''' if f is not None: self.ellipsoid = a_earth, f elif a_earth not in (_EWGS84, None): self.ellipsoid = a_earth self.lon0 = lon0 self.k0 = k0 if name: # PYCHOK no cover self.name = name if raiser: self.raiser = True if TMorder: self.TMorder = _xkwds_get(TMorder, TMorder=self._mTM) @Property_RO def _Alp(self): return _Xs(_AlpCoeffs, self.TMorder, self.ellipsoid) @Property_RO def _b1(self): n = self.ellipsoid.n if n: # isEllipsoidal m = self.TMorder // 2 B1 = _B1Coeffs[m] m += 1 b1 = _polynomial(n**2, B1, 0, m) / (B1[m] * (n + _1_0)) else: # isSpherical b1 = _1_0 # B1[m - 1] / B1[m1] == 1, always return b1 @Property_RO def _Bet(self): C = _Xs(_BetCoeffs, self.TMorder, self.ellipsoid) return tuple(map(neg, C)) if self.f else C # negated if isEllispoidal @Property def ellipsoid(self): '''Get the ellipsoid (L{Ellipsoid}). ''' return self._E @ellipsoid.setter # PYCHOK setter! def ellipsoid(self, a_earth_f): '''Set this rhumb's ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum}, L{a_f2Tuple} or 2-tuple C{(a, f)}). ''' E = _MODS.datums._spherical_datum(a_earth_f, Error=KTMError).ellipsoid if self._E != E: _update_all(self) self._E = E @Property_RO def equatoradius(self): '''Get the C{ellipsoid}'s equatorial radius, semi-axis (C{meter}). ''' return self.ellipsoid.a a = equatoradius @Property_RO def flattening(self): '''Get the C{ellipsoid}'s flattening (C{float}). ''' return self.ellipsoid.f f = flattening def forward(self, lat, lon, lon0=None, name=NN): '''Forward projection, from geographic to transverse Mercator. @arg lat: Latitude of point (C{degrees90}). @arg lon: Longitude of point (C{degrees180}). @arg lon0: Central meridian of the projection (C{degrees180}). @kwarg name: Optional name (C{str}). @return: L{Forward4Tuple}C{(easting, northing, gamma, scale)} with C{easting} and C{northing} in C{meter}, unfalsed, the meridian convergence C{gamma} at point in C{degrees180} and the C{scale} of projection at point C{scalar}. Any value may be C{NAN}, C{NINF} or C{INF} for singularities. @raise KTMError: For singularities, iff property C{raiser} is C{True}. ''' lat, _lat = _unsigned2(_fix90(lat)) lon, _ = _diff182((self.lon0 if lon0 is None else lon0), lon) lon, _lon = _unsigned2(lon) backside = lon > 90 if backside: # PYCHOK no cover lon = _180_0 - lon if lat == 0: _lat = True sphi, cphi, slam, clam = sincos2d_(lat, lon) E = self.ellipsoid if cphi and lat != 90: t = sphi / cphi tp = E.es_taupf(t) h = hypot(tp, clam) if h: xip = atan2(tp, clam) etap = asinh(slam / h) # atanh(sin(lam) / cosh(psi)) g = _atan2d(slam * tp, clam * hypot1(tp)) # Krueger p 22 (44) k = sqrt(E.e21 + E.e2 * cphi**2) * hypot1(t) / h elif self.raiser: raise KTMError(lat=lat, lon=lon, lon0=lon0, txt=_singular_) else: # PYCHOK no cover xip, etap = _0_0, (NINF if slam < 0 else INF) g, k = copysign0(_90_0, slam), INF else: # PYCHOK no cover xip, etap = PI_2, _0_0 g, k = lon, E.es_c y, x, t, z = self._yxgk4(xip, etap, self._Alp) g -= t k *= z * self._k0_b1 if backside: # PYCHOK no cover y, g = (PI - y), (_180_0 - g) y *= self._k0_a1 x *= self._k0_a1 if _lat: y, g = neg_(y, g) if _lon: x, g = neg_(x, g) return Forward4Tuple(x, y, _norm180(g), k, name=name or self.name) @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 KTMError: Invalid B{C{k0}}. ''' k0 = Scalar_(k0=k0, Error=KTMError, low=_TOL_10, high=_1_0) if self._k0 != k0: # PYCHOK no cover KTransverseMercator._k0_a1._update(self) # redo ._k0_a1 KTransverseMercator._k0_b1._update(self) # redo ._k0_b1 self._k0 = k0 @Property_RO def _k0_a1(self): '''(INTERNAL) Cache C{k0 * _b1 * equatoradius}. ''' return self._k0_b1 * self.equatoradius @Property_RO def _k0_b1(self): '''(INTERNAL) Cache C{k0 * _b1}. ''' return self.k0 * self._b1 @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 KTMError: Invalid B{C{lon0}}. ''' self._lon0 = _norm180(Degrees(lon0=lon0, Error=KTMError)) @property_doc_(''' raise a L{KTMError} for C{forward} singularities (C{bool}).''') def raiser(self): '''Get the error setting (C{bool}). ''' return self._raiser @raiser.setter # PYCHOK setter! def raiser(self, raiser): '''Set the error setting (C{bool}), to C{True} to throw a L{KTMError} for C{forward} singularities. ''' self._raiser = bool(raiser) def reverse(self, x, y, lon0=None, name=NN): '''Reverse projection, from transverse Mercator to geographic. @arg x: Easting of point (C{meter}). @arg y: Northing of point (C{meter}). @arg lon0: Central meridian of the projection (C{degrees180}). @return: L{Reverse4Tuple}C{(lat, lon, gamma, scale)} with C{lat}- and C{lon}gitude in C{degrees}, I{unfalsed}. ''' eta, _lon = _unsigned2(x / self._k0_a1) xi, _lat = _unsigned2(y / self._k0_a1) backside = xi > PI_2 if backside: # PYCHOK no cover xi = PI - xi xip, etap, g, k = self._yxgk4(xi, eta, self._Bet) t = self._k0_b1 k = (t / k) if k else (NINF if t < 0 else INF) h, c = sinh(etap), cos(xip) if c > 0: r = hypot(h, c) else: # PYCHOK no cover r = abs(h) c = _0_0 E = self.ellipsoid if r: lon = _atan2d(h, c) # Krueger p 17 (25) s = sin(xip) # Newton for tau t = E.es_tauf(s / r) lat = atand(t) g += _atan2d(s * tanh(etap), c) # Krueger p 19 (31) k *= sqrt(E.e21 + E.e2 / (t**2 + _1_0)) * hypot1(t) * r else: # PYCHOK no cover lat, lon = _90_0, _0_0 k *= E.es_c if backside: # PYCHOK no cover lon, g = (_180_0 - lon), (_180_0 - g) if _lat: lat, g = neg_(lat, g) if _lon: lon, g = neg_(lon, g) lon += self.lon0 if lon0 is None else _norm180(lon0) return Reverse4Tuple(lat, _norm180(lon), _norm180(g), k, name=name or self.name) @Property def TMorder(self): '''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8). ''' return self._mTM @TMorder.setter # PYCHOK setter! def TMorder(self, order): '''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8). ''' m = _Xorder(_AlpCoeffs, KTMError, TMorder=order) if self._mTM != m: _update_all(self) self._mTM = m def toStr(self, **kwds): '''Return a C{str} representation. @arg kwds: Optional, overriding keyword arguments. ''' d = dict(ellipsoid=self.ellipsoid, k0=self.k0, TMorder=self.TMorder) if self.name: # PYCHOK no cover d.update(name=self.name) return _COMMASPACE_.join(_pairs(d, **kwds)) def _yxgk4(self, xi_, eta_, C): '''(INTERNAL) Complex Clenshaw summation with C{B{C}=_Alp} or C{B{C}=-_Bet}, negated! ''' def _sinhcosh2(x): return sinh(x), cosh(x) x = complex(xi_, eta_) if self.f: # isEllipsoidal s, c = sincos2( xi_ * 2) sh, ch = _sinhcosh2(eta_ * 2) n = -s s = complex(s * ch, c * sh) # sin(zeta * 2) c = complex(c * ch, n * sh) # cos(zeta * 2) y0 = y1 = z0 = z1 = complex(0) # 0+j0 n = self.TMorder # == len(C) - 1 if isodd(n): Cn = C[n] y0 = complex(Cn) # +j0 z0 = complex(Cn * (n * 2)) n -= 1 a = c * 2 # cos(zeta * 2) * 2 while n > 0: Cn = C[n] y1 = _c(a, y0, y1, Cn) z1 = _c(a, z0, z1, Cn * (n * 2)) n -= 1 Cn = C[n] y0 = _c(a, y1, y0, Cn) z0 = _c(a, z1, z0, Cn * (n * 2)) n -= 1 # assert n == 0 x = _c(s, y0, -x, _0_0) c = _c(c, z0, z1, _1_0) # Gauss-Schreiber to Gauss-Krueger TM # C{cmath.phase} handles INF, NAN, etc. g, k = degrees(phase(c)), abs(c) else: # isSpherical g, k = _0_0, _1_0 return x.real, x.imag, g, k def _c(a, b0, b1, Cn): '''(INTERNAL) Accurately compute complex M{a * b0 - b1 + Cn} with complex args C{a}, C{b0} and C{b1} and scalar C{Cn}. @see: CPython function U{_Py_c_prod<https://GitHub.com/python/ cpython/blob/main/Objects/complexobject.c>}. @note: Python function C{cmath.fsum} is no longer available, but stil mentioned in Note 4 of the comments before CPython function U{math_fsum<https://GitHub.com/python/ cpython/blob/main/Modules/mathmodule.c>} ''' r = fsum1_(a.real * b0.real, -a.imag * b0.imag, -b1.real, Cn, floats=True) j = fsum1_(a.real * b0.imag, a.imag * b0.real, -b1.imag, floats=True) return complex(r, j) def _Xorder(_Coeffs, Error, **Xorder): # in .rhumbx '''(INTERNAL) Validate C{RAorder} or C{TMorder}. ''' X, m = Xorder.popitem() if m in _Coeffs and isint(m): return m t = sorted(map(str, _Coeffs.keys())) raise Error(X, m, txt=_not_(_or(*t))) def _Xs(_Coeffs, m, E, RA=False): # in .rhumbx '''(INTERNAL) Compute the C{A}, C{B} or C{RA} terms of order B{C{m}} for I{Krüger} series and I{rhumbx._sincosSeries}, return a tuple with C{B{m} + 1} terms C{X}, C{X[0]==0}. ''' Cs = _Coeffs[m] assert len(Cs) == (((m + 1) * (m + 4)) if RA else ((m + 3) * m)) // 2 n = n_ = E.n if n: # isEllipsoidal X = [0] # X[0] never used, it's just an integration # constant, it cancels when evaluating a definite # integral. Don't bother computing it, it is not # used in C{KTransverseMercator._yxgk4} above nor # in C{rhumbx._sincosSeries}. i = (m + 2) if RA else 0 for r in range(m - 1, -1, -1): # [m-1 ... 0] j = i + r + 1 X.append(_polynomial(n, Cs, i, j) * n_ / Cs[j]) i = j + 1 n_ *= n X = tuple(X) else: # isSpherical X = _0_0s(m + 1) return X # _Alp- and _BetCoeffs in .rhumbx _AlpCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4 164, 225, -480, 360, 720, # Alp[1]/n^1, polynomial(n), order 3 557, -864, 390, 1440, # Alp[2]/n^2, polynomial(n), order 2 -1236, 427, 1680, # PYCHOK Alp[3]/n^3, polynomial(n), order 1 49561, 161280), # Alp[4]/n^4, polynomial(n), order 0, count = 14 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5 -635, 328, 450, -960, 720, 1440, # Alp[1]/n^1, polynomial(n), order 4 4496, 3899, -6048, 2730, 10080, # PYCHOK Alp[2]/n^2, polynomial(n), order 3 15061, -19776, 6832, 26880, # PYCHOK Alp[3]/n^3, polynomial(n), order 2 -171840, 49561, 161280, # Alp[4]/n^4, polynomial(n), order 1 34729, 80640), # PYCHOK Alp[5]/n^5, polynomial(n), order 0, count = 20 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6 31564, -66675, 34440, 47250, -100800, 75600, 151200, # Alp[1]/n^1, polynomial(n), order 5 -1983433, 863232, 748608, -1161216, 524160, 1935360, # PYCHOK Alp[2]/n^2, polynomial(n), order 4 670412, 406647, -533952, 184464, 725760, # Alp[3]/n^3, polynomial(n), order 3 6601661, -7732800, 2230245, 7257600, # Alp[4]/n^4, polynomial(n), order 2 -13675556, 3438171, 7983360, # PYCHOK Alp[5]/n^5, polynomial(n), order 1 212378941, 319334400), # Alp[6]/n^6, polynomial(n), order 0, count = 27 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7 1804025, 2020096, -4267200, 2204160, 3024000, -6451200, 4838400, 9676800, # Alp[1]/n^1, polynomial(n), order 6 4626384, -9917165, 4316160, 3743040, -5806080, 2620800, 9676800, # Alp[2]/n^2, polynomial(n), order 5 -67102379, 26816480, 16265880, -21358080, 7378560, 29030400, # PYCHOK Alp[3]/n^3, polynomial(n), order 4 155912000, 72618271, -85060800, 24532695, 79833600, # Alp[4]/n^4, polynomial(n), order 3 102508609, -109404448, 27505368, 63866880, # Alp[5]/n^5, polynomial(n), order 2 -12282192400, 2760926233, 4151347200, # PYCHOK Alp[6]/n^6, polynomial(n), order 1 1522256789, 1383782400), # Alp[7]/n^7, polynomial(n), order 0, count = 35 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8 -75900428, 37884525, 42422016, -89611200, 46287360, 63504000, -135475200, 101606400, 203212800, # Alp[1]/n^1, polynomial(n), order 7 148003883, 83274912, -178508970, 77690880, 67374720, -104509440, 47174400, 174182400, # PYCHOK Alp[2]/n^2, polynomial(n), order 6 318729724, -738126169, 294981280, 178924680, -234938880, 81164160, 319334400, # PYCHOK Alp[3]/n^3, polynomial(n), order 5 -40176129013, 14967552000, 6971354016, -8165836800, 2355138720, 7664025600, # Alp[4]/n^4, polynomial(n), order 4 10421654396, 3997835751, -4266773472, 1072709352, 2490808320, # PYCHOK Alp[5]/n^5, polynomial(n), order 3 175214326799, -171950693600, 38652967262, 58118860800, # PYCHOK Alp[6]/n^6, polynomial(n), order 2 -67039739596, 13700311101, 12454041600, # PYCHOK Alp[7]/n^7, polynomial(n), order 1 1424729850961, 743921418240) # PYCHOK Alp[8]/n^8, polynomial(n), order 0, count = 44 } _B1Coeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00 2: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 2 1, 16, 64, 64), # b1 * (n + 1), polynomial(n2), order 2 3: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 3 1, 4, 64, 256, 256), # b1 * (n + 1), polynomial(n2), order 3 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 4 25, 64, 256, 4096, 16384, 16384) # PYCHOK b1 * (n + 1), polynomial(n2), order 4 } _BetCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4 -4, 555, -960, 720, 1440, # Bet[1]/n^1, polynomial(n), order 3 -437, 96, 30, 1440, # Bet[2]/n^2, polynomial(n), order 2 -148, 119, 3360, # Bet[3]/n^3, polynomial(n), order 1 4397, 161280), # PYCHOK Bet[4]/n^4, polynomial(n), order 0, count = 14 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5 -3645, -64, 8880, -15360, 11520, 23040, # Bet[1]/n^1, polynomial(n), order 4 4416, -3059, 672, 210, 10080, # PYCHOK Bet[2]/n^2, polynomial(n), order 3 -627, -592, 476, 13440, # Bet[3]/n^3, polynomial(n), order 2 -3520, 4397, 161280, # Bet[4]/n^4, polynomial(n), order 1 4583, 161280), # PYCHOK Bet[5]/n^5, polynomial(n), order 0, count = 20 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6 384796, -382725, -6720, 932400, -1612800, 1209600, 2419200, # Bet[1]/n^1, polynomial(n), order 5 -1118711, 1695744, -1174656, 258048, 80640, 3870720, # PYCHOK Bet[2]/n^2, polynomial(n), order 4 22276, -16929, -15984, 12852, 362880, # Bet[3]/n^3, polynomial(n), order 3 -830251, -158400, 197865, 7257600, # PYCHOK Bet[4]/n^4, polynomial(n), order 2 -435388, 453717, 15966720, # PYCHOK Bet[5]/n^5, polynomial(n), order 1 20648693, 638668800), # Bet[6]/n^6, polynomial(n), order 0, count = 27 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7 -5406467, 6156736, -6123600, -107520, 14918400, -25804800, 19353600, 38707200, # Bet[1]/n^1, polynomial(n), order 6 829456, -5593555, 8478720, -5873280, 1290240, 403200, 19353600, # PYCHOK Bet[2]/n^2, polynomial(n), order 5 9261899, 3564160, -2708640, -2557440, 2056320, 58060800, # PYCHOK Bet[3]/n^3, polynomial(n), order 4 14928352, -9132761, -1742400, 2176515, 79833600, # PYCHOK Bet[4]/n^4, polynomial(n), order 3 -8005831, -1741552, 1814868, 63866880, # Bet[5]/n^5, polynomial(n), order 2 -261810608, 268433009, 8302694400, # Bet[6]/n^6, polynomial(n), order 1 219941297, 5535129600), # PYCHOK Bet[7]/n^7, polynomial(n), order 0, count = 35 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8 31777436, -37845269, 43097152, -42865200, -752640, 104428800, -180633600, 135475200, 270950400, # Bet[1]/n^1, polynomial(n), order 7 24749483, 14930208, -100683990, 152616960, -105719040, 23224320, 7257600, 348364800, # Bet[2]/n^2, polynomial(n), order 6 -232468668, 101880889, 39205760, -29795040, -28131840, 22619520, 638668800, # PYCHOK Bet[3]/n^3, polynomial(n), order 5 324154477, 1433121792, -876745056, -167270400, 208945440, 7664025600, # Bet[4]/n^4, polynomial(n), order 4 457888660, -312227409, -67920528, 70779852, 2490808320, # Bet[5]/n^5, polynomial(n), order 3 -19841813847, -3665348512, 3758062126, 116237721600, # PYCHOK Bet[6]/n^6, polynomial(n), order 2 -1989295244, 1979471673, 49816166400, # PYCHOK Bet[7]/n^7, polynomial(n), order 1 191773887257, 3719607091200) # Bet[8]/n^8, polynomial(n), order 0, count = 44 } assert set(_AlpCoeffs.keys()) == set(_BetCoeffs.keys()) if __name__ == '__main__': from pygeodesy.interns import _usage from sys import argv, exit as _exit _exit(_usage(*argv).replace('.ktm', '.etm -series')) # **) MIT License # # Copyright (C) 2022-2022 -- 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: # 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