Coverage for pygeodesy/ktm.py: 97%
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2# -*- coding: utf-8 -*-
4u'''A pure Python version of I{Karney}'s C++ class U{TransverseMercator
5<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1TransverseMercator.html>}
6based on I{Krüger} series. See also I{Karney}'s utility U{TransverseMercatorProj
7<https://GeographicLib.SourceForge.io/C++/doc/TransverseMercatorProj.1.html>}.
9Following and further below is a copy of I{Karney}'s U{TransverseMercator.hpp
10<https://GeographicLib.SourceForge.io/C++/doc/TransverseMercator_8hpp_source.html>}
11file C{Header}.
13This implementation follows closely JHS 154, ETRS89 - I{järjestelmään liittyvät
14karttaprojektiot, tasokoordinaatistot ja karttalehtijako} (Map projections, plane
15coordinates, and map sheet index for ETRS89), published by JUHTA, Finnish Geodetic
16Institute, and the National Land Survey of Finland (2006). The relevant section
17is available as the U{2008 PDF file
18<http://Docs.JHS-suositukset.FI/jhs-suositukset/JHS154/JHS154_liite1.pdf>}.
20This is a straight transcription of the formulas in this paper with the
21following exceptions:
23 - Use of 6th order series instead of 4th order series. This reduces the
24 error to about 5 nm for the UTM range of coordinates (instead of 200 nm),
25 with a speed penalty of only 1%,
27 - Use Newton's method instead of plain iteration to solve for latitude
28 in terms of isometric latitude in the Reverse method,
30 - Use of Horner's representation for evaluating polynomials and Clenshaw's
31 method for summing trigonometric series,
33 - Several modifications of the formulas to improve the numerical accuracy,
35 - Evaluating the convergence and scale using the expression for the
36 projection or its inverse.
38Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2008-2023)
39and licensed under the MIT/X11 License. For more information, see the
40U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation.
41'''
42# make sure int/int division yields float quotient
43from __future__ import division as _; del _ # PYCHOK semicolon
45from pygeodesy.basics import copysign0, isodd, neg, neg_, \
46 _reverange, _xinstanceof
47from pygeodesy.constants import INF, _K0_UTM, PI, PI_2, _0_0s, _0_0, \
48 _1_0, _90_0, _copysignINF
49from pygeodesy.datums import Datum, _spherical_datum, _WGS84, _EWGS84
50# from pygeodesy.ellipsoids import _EWGS84 # from .datums
51from pygeodesy.errors import _ValueError, _xkwds_get, _Xorder
52from pygeodesy.fmath import hypot, hypot1
53from pygeodesy.fsums import fsum1f_
54from pygeodesy.interns import NN, _COMMASPACE_, _singular_
55from pygeodesy.karney import _atan2d, _diff182, _fix90, _norm180, \
56 _polynomial, _unsigned2
57# from pygeodesy.lazily import _ALL_LAZY # from .named
58from pygeodesy.named import _NamedBase, pairs, _ALL_LAZY
59from pygeodesy.namedTuples import Forward4Tuple, Reverse4Tuple
60from pygeodesy.props import property_doc_, Property, Property_RO, \
61 _update_all
62# from pygeodesy.streprs import pairs # from .named
63from pygeodesy.units import Degrees, Scalar_, _1mm as _TOL_10 # PYCHOK used!
64from pygeodesy.utily import atan1d, _loneg, sincos2, sincos2d_
66from cmath import polar
67from math import atan2, asinh, cos, cosh, degrees, fabs, sin, sinh, sqrt, tanh
69__all__ = _ALL_LAZY.ktm
70__version__ = '24.05.11'
73class KTMError(_ValueError):
74 '''Error raised for L{KTransverseMercator} and L{KTransverseMercator.forward} issues.
75 '''
76 pass
79class KTransverseMercator(_NamedBase):
80 '''I{Karney}'s C++ class U{TransverseMercator<https://GeographicLib.SourceForge.io/
81 C++/doc/classGeographicLib_1_1TransverseMercator.html>} transcoded to pure
82 Python, following is a partial copy of I{Karney}'s documentation.
84 Transverse Mercator projection based on Krüger's method which evaluates the
85 projection and its inverse in terms of a series.
87 There's a singularity in the projection at I{phi = 0, lam - lam0 = +/- (1 - e)
88 90}, about +/- 82.6 degrees for WGS84, where I{e} is the eccentricity. Beyond
89 this point, the series ceases to converge and the results from this method
90 will be garbage. I{To be on the safe side, don't use this method if the
91 angular distance from the central meridian exceeds (1 - 2e) x 90}, about 75
92 degrees for the WGS84 ellipsoid.
94 Class L{ExactTransverseMercator} is an alternative implementation of the
95 projection using I{exact} formulas which yield accurate (to 8 nm) results
96 over the entire ellipsoid.
98 The ellipsoid parameters and the central scale are set in the constructor.
99 The central meridian (which is a trivial shift of the longitude) is specified
100 as the C{lon0} keyword argument of the L{KTransverseMercator.forward} and
101 L{KTransverseMercator.reverse} methods. The latitude of origin is taken to
102 be the equator. There is no provision in this class for specifying a false
103 easting or false northing or a different latitude of origin. However these
104 are can be simply included by the calling function.
106 The L{KTransverseMercator.forward} and L{KTransverseMercator.reverse} methods
107 also return the meridian convergence C{gamma} and scale C{k}. The meridian
108 convergence is the bearing of grid North, the C{y axis}, measured clockwise
109 from true North.
110 '''
111 _datum = _WGS84
112 _k0 = _K0_UTM # central scale factor
113 _lat0 = _0_0 # central parallel
114 _lon0 = _0_0 # central meridian
115 _mTM = 6
116 _raiser = False # throw Error
118 def __init__(self, a_earth=_EWGS84, f=None, lon0=0, k0=_K0_UTM, name=NN,
119 raiser=False, **TMorder):
120 '''New L{KTransverseMercator}.
122 @kwarg a_earth: This rhumb's earth (L{Ellipsoid}, L{Ellipsoid2},
123 L{a_f2Tuple}, L{Datum}, 2-tuple (C{a, f})) or the
124 equatorial radius (C{scalar}, C{meter}).
125 @kwarg f: The ellipsoid's flattening (C{scalar}), iff B{C{a_earth}} is
126 a C{scalar}, ignored otherwise.
127 @kwarg lon0: The central meridian (C{degrees180}).
128 @kwarg k0: Central scale factor (C{scalar}).
129 @kwarg name: Optional name (C{str}).
130 @kwarg raiser: If C{True}, throw a L{KTMError} for C{forward}
131 singularities (C{bool}).
132 @kwarg TMorder: Keyword argument B{C{TMorder}}, see property C{TMorder}.
134 @raise KTMError: Invalid B{C{a_earth}}, B{C{f}} or B{C{TMorder}}.
135 '''
136 if f is not None:
137 self.ellipsoid = a_earth, f
138 elif a_earth in (_EWGS84, _WGS84, None):
139 pass
140 elif isinstance(a_earth, Datum):
141 self.datum = a_earth
142 else:
143 self.ellipsoid = a_earth
144 self.lon0 = lon0
145 self.k0 = k0
146 if name: # PYCHOK no cover
147 self.name = name
148 if raiser:
149 self.raiser = True
150 if TMorder:
151 self.TMorder = _xkwds_get(TMorder, TMorder=self._mTM)
153 @Property_RO
154 def _Alp(self):
155 return _Xs(_AlpCoeffs, self.TMorder, self.ellipsoid)
157 @Property_RO
158 def _b1(self):
159 n = self.ellipsoid.n
160 if n: # isEllipsoidal
161 m = self.TMorder // 2
162 B1 = _B1Coeffs[m]
163 m += 1
164 b1 = _polynomial(n**2, B1, 0, m) / (B1[m] * (n + _1_0))
165 else: # isSpherical
166 b1 = _1_0 # B1[m - 1] / B1[m1] == 1, always
167 return b1
169 @Property_RO
170 def _Bet(self):
171 C = _Xs(_BetCoeffs, self.TMorder, self.ellipsoid)
172 return tuple(map(neg, C)) if self.f else C # negated if isEllipsoidal
174 @property
175 def datum(self):
176 '''Get this rhumb's datum (L{Datum}).
177 '''
178 return self._datum
180 @datum.setter # PYCHOK setter!
181 def datum(self, datum):
182 '''Set this rhumb's datum (L{Datum}).
183 '''
184 _xinstanceof(Datum, datum=datum)
185 if self._datum != datum:
186 _update_all(self)
187 self._datum = datum
189 @Property
190 def ellipsoid(self):
191 '''Get the ellipsoid (L{Ellipsoid}).
192 '''
193 return self.datum.ellipsoid
195 @ellipsoid.setter # PYCHOK setter!
196 def ellipsoid(self, a_earth_f):
197 '''Set this rhumb's ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum},
198 L{a_f2Tuple} or 2-tuple C{(a, f)}).
199 '''
200 self.datum = _spherical_datum(a_earth_f, Error=KTMError)
202 @Property_RO
203 def equatoradius(self):
204 '''Get the C{ellipsoid}'s equatorial radius, semi-axis (C{meter}).
205 '''
206 return self.ellipsoid.a
208 a = equatoradius
210 @Property_RO
211 def flattening(self):
212 '''Get the C{ellipsoid}'s flattening (C{scalar}).
213 '''
214 return self.ellipsoid.f
216 f = flattening
218 def forward(self, lat, lon, lon0=None, name=NN):
219 '''Forward projection, from geographic to transverse Mercator.
221 @arg lat: Latitude of point (C{degrees90}).
222 @arg lon: Longitude of point (C{degrees180}).
223 @arg lon0: Central meridian of the projection (C{degrees180}).
224 @kwarg name: Optional name (C{str}).
226 @return: L{Forward4Tuple}C{(easting, northing, gamma, scale)}
227 with C{easting} and C{northing} in C{meter}, unfalsed, the
228 meridian convergence C{gamma} at point in C{degrees180}
229 and the C{scale} of projection at point C{scalar}. Any
230 value may be C{NAN}, C{NINF} or C{INF} for singularities.
232 @raise KTMError: For singularities, iff property C{raiser} is
233 C{True}.
234 '''
235 lat, _lat = _unsigned2(_fix90(lat - self._lat0))
236 lon, _ = _diff182((self.lon0 if lon0 is None else lon0), lon)
237 lon, _lon = _unsigned2(lon)
238 backside = lon > 90
239 if backside: # PYCHOK no cover
240 lon = _loneg(lon)
241 if lat == 0:
242 _lat = True
244 sphi, cphi, slam, clam = sincos2d_(lat, lon)
245 E = self.ellipsoid
246 if cphi and lat != 90:
247 t = sphi / cphi
248 tp = E.es_taupf(t)
249 h = hypot(tp, clam)
250 if h:
251 xip = atan2(tp, clam)
252 etap = asinh(slam / h) # atanh(sin(lam) / cosh(psi))
253 g = _atan2d(slam * tp, clam * hypot1(tp)) # Krueger p 22 (44)
254 k = sqrt(cphi**2 * E.e2 + E.e21) * hypot1(t) / h
255 elif self.raiser:
256 raise KTMError(lat=lat, lon=lon, lon0=lon0, txt=_singular_)
257 else: # PYCHOK no cover
258 xip, etap = _0_0, _copysignINF(slam)
259 g, k = copysign0(_90_0, slam), INF
260 else: # PYCHOK no cover
261 xip, etap = PI_2, _0_0
262 g, k = lon, E.es_c
263 y, x, d, t = _Cyxgk4(E, xip, etap, self._Alp)
264 g -= d
265 k *= t * self._k0_b1
267 if backside: # PYCHOK no cover
268 y, g = (PI - y), _loneg(g)
269 y *= self._k0_a1
270 x *= self._k0_a1
271 if _lat:
272 y, g = neg_(y, g)
273 if _lon:
274 x, g = neg_(x, g)
275 return Forward4Tuple(x, y, _norm180(g), k, name=name or self.name)
277 @property_doc_(''' the central scale factor (C{float}).''')
278 def k0(self):
279 '''Get the central scale factor (C{float}), aka I{C{scale0}}.
280 '''
281 return self._k0 # aka scale0
283 @k0.setter # PYCHOK setter!
284 def k0(self, k0):
285 '''Set the central scale factor (C{float}), aka I{C{scale0}}.
287 @raise KTMError: Invalid B{C{k0}}.
288 '''
289 k0 = Scalar_(k0=k0, Error=KTMError, low=_TOL_10, high=_1_0)
290 if self._k0 != k0: # PYCHOK no cover
291 KTransverseMercator._k0_a1._update(self) # redo ._k0_a1
292 KTransverseMercator._k0_b1._update(self) # redo ._k0_b1
293 self._k0 = k0
295 @Property_RO
296 def _k0_a1(self):
297 '''(INTERNAL) Cache C{k0 * _b1 * equatoradius}.
298 '''
299 return self._k0_b1 * self.equatoradius
301 @Property_RO
302 def _k0_b1(self):
303 '''(INTERNAL) Cache C{k0 * _b1}.
304 '''
305 return self.k0 * self._b1
307 @property_doc_(''' the central meridian (C{degrees180}).''')
308 def lon0(self):
309 '''Get the central meridian (C{degrees180}).
310 '''
311 return self._lon0
313 @lon0.setter # PYCHOK setter!
314 def lon0(self, lon0):
315 '''Set the central meridian (C{degrees180}).
317 @raise KTMError: Invalid B{C{lon0}}.
318 '''
319 self._lon0 = _norm180(Degrees(lon0=lon0, Error=KTMError))
321 @property_doc_(''' raise a L{KTMError} for C{forward} singularities (C{bool}).''')
322 def raiser(self):
323 '''Get the error setting (C{bool}).
324 '''
325 return self._raiser
327 @raiser.setter # PYCHOK setter!
328 def raiser(self, raiser):
329 '''Set the error setting (C{bool}), to C{True} to throw a L{KTMError}
330 for C{forward} singularities.
331 '''
332 self._raiser = bool(raiser)
334 def reset(self, lat0, lon0):
335 '''Set the central parallel and meridian.
337 @arg lat0: Latitude of the central parallel (C{degrees90}).
338 @arg lon0: Longitude of the central parallel (C{degrees180}).
340 @return: 2-Tuple C{(lat0, lon0)} of the previous central
341 parallel and meridian.
343 @raise KTMError: Invalid B{C{lat0}} or B{C{lon0}}.
344 '''
345 t = self._lat0, self.lon0
346 self._lat0 = _fix90(Degrees(lat0=lat0, Error=KTMError))
347 self. lon0 = lon0
348 return t
350 def reverse(self, x, y, lon0=None, name=NN):
351 '''Reverse projection, from transverse Mercator to geographic.
353 @arg x: Easting of point (C{meter}).
354 @arg y: Northing of point (C{meter}).
355 @arg lon0: Central meridian of the projection (C{degrees180}).
357 @return: L{Reverse4Tuple}C{(lat, lon, gamma, scale)} with
358 C{lat}- and C{lon}gitude in C{degrees}, I{unfalsed}.
359 '''
360 eta, _lon = _unsigned2(x / self._k0_a1)
361 xi, _lat = _unsigned2(y / self._k0_a1)
362 backside = xi > PI_2
363 if backside: # PYCHOK no cover
364 xi = PI - xi
366 E = self.ellipsoid
367 xip, etap, g, k = _Cyxgk4(E, xi, eta, self._Bet)
368 t = self._k0_b1
369 k = (t / k) if k else _copysignINF(t) # _over(t, k)
370 h, c = sinh(etap), cos(xip)
371 if c > 0:
372 r = hypot(h, c)
373 else: # PYCHOK no cover
374 r = fabs(h)
375 c = _0_0
376 if r:
377 lon = _atan2d(h, c) # Krueger p 17 (25)
378 s = sin(xip) # Newton for tau
379 t = E.es_tauf(s / r)
380 lat = atan1d(t)
381 g += _atan2d(s * tanh(etap), c) # Krueger p 19 (31)
382 k *= sqrt(E.e2 / (t**2 + _1_0) + E.e21) * hypot1(t) * r
383 else: # PYCHOK no cover
384 lat = _90_0
385 lon = _0_0
386 k *= E.es_c
388 if backside: # PYCHOK no cover
389 lon, g = _loneg(lon), _loneg(g)
390 if _lat:
391 lat, g = neg_(lat, g)
392 if _lon:
393 lon, g = neg_(lon, g)
394 lat += self._lat0
395 lon += self._lon0 if lon0 is None else _norm180(lon0)
396 return Reverse4Tuple(lat, _norm180(lon), _norm180(g), k,
397 name=name or self.name)
399 @Property
400 def TMorder(self):
401 '''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
402 '''
403 return self._mTM
405 @TMorder.setter # PYCHOK setter!
406 def TMorder(self, order):
407 '''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
408 '''
409 m = _Xorder(_AlpCoeffs, KTMError, TMorder=order)
410 if self._mTM != m:
411 _update_all(self)
412 self._mTM = m
414 def toStr(self, **kwds):
415 '''Return a C{str} representation.
417 @arg kwds: Optional, overriding keyword arguments.
418 '''
419 d = dict(ellipsoid=self.ellipsoid, k0=self.k0, TMorder=self.TMorder)
420 if self.name: # PYCHOK no cover
421 d.update(name=self.name)
422 return _COMMASPACE_.join(pairs(d, **kwds))
425def _cma(a, b0, b1, Cn):
426 '''(INTERNAL) Compute complex M{a * b0 - b1 + Cn} with complex
427 C{a}, C{b0} and C{b1} and scalar C{Cn}.
429 @see: CPython function U{_Py_c_prod<https://GitHub.com/python/
430 cpython/blob/main/Objects/complexobject.c>}.
432 @note: Python function C{cmath.fsum} is no longer available,
433 but stil mentioned in Note 4 of the comments before
434 CPython function U{math_fsum<https://GitHub.com/python/
435 cpython/blob/main/Modules/mathmodule.c>}
436 '''
437 r = fsum1f_(a.real * b0.real, -a.imag * b0.imag, -b1.real, Cn)
438 j = fsum1f_(a.real * b0.imag, a.imag * b0.real, -b1.imag)
439 return complex(r, j)
442def _Cyxgk4(E, xi_, eta_, C):
443 '''(INTERNAL) Complex Clenshaw summation with C{B{C}=._Alp}
444 or C{B{C}=-._Bet}.
445 '''
446 x = complex(xi_, eta_)
447 if E.f: # isEllipsoidal
448 s, c = sincos2( xi_ * 2)
449 sh, ch = _sinhcosh2(eta_ * 2)
450 n = -s
451 s = complex(s * ch, c * sh) # sin(zeta * 2)
452 c = complex(c * ch, n * sh) # cos(zeta * 2)
453 a = c * 2 # cos(zeta * 2) * 2
455 y0 = y1 = \
456 z0 = z1 = complex(0) # 0+0j
457 n = len(C) - 1 # == .TMorder
458 if isodd(n):
459 Cn = C[n]
460 y0 = complex(Cn) # +0j
461 z0 = complex(Cn * (n * 2))
462 n -= 1
463 _c = _cma
464 while n > 0:
465 Cn = C[n]
466 y1 = _c(a, y0, y1, Cn)
467 z1 = _c(a, z0, z1, Cn * (n * 2))
468 n -= 1
469 Cn = C[n]
470 y0 = _c(a, y1, y0, Cn)
471 z0 = _c(a, z1, z0, Cn * (n * 2))
472 n -= 1
473 # assert n == 0
474 x = _c(s, y0, -x, _0_0)
475 c = _c(c, z0, z1, _1_0)
477 # Gauss-Schreiber to Gauss-Krueger TM
478 # C{cmath.polar} handles INF, NAN, etc.
479 k, g = polar(c)
480 g = degrees(g)
481 else: # E.isSpherical
482 g, k = _0_0, _1_0
484 return x.real, x.imag, g, k
487def _sinhcosh2(x):
488 '''(INTERNAL) Like C{sincos2}.
489 '''
490 return sinh(x), cosh(x)
493def _Xs(_Coeffs, m, E, RA=False): # in .rhumb.ekx
494 '''(INTERNAL) Compute the C{A}, C{B} or C{RA} terms of order
495 B{C{m}} for I{Krüger} series and I{rhumb.ekx._sincosSeries},
496 return a tuple with C{B{m} + 1} terms C{X}, C{X[0]==0}.
497 '''
498 Cs = _Coeffs[m]
499 assert len(Cs) == (((m + 1) * (m + 4)) if RA else
500 ((m + 3) * m)) // 2
501 n = n_ = E.n
502 if n: # isEllipsoidal
503 X = [0] # X[0] never used, it's just an integration
504 # constant, it cancels when evaluating a definite
505 # integral. Don't bother computing it, it is unused
506 # in C{_Cyxgk4} above and C{rhumb.ekx._sincosSeries}.
507 _X, _p = X.append, _polynomial
508 i = (m + 2) if RA else 0
509 for r in _reverange(m): # [m-1 ... 0]
510 j = i + r + 1
511 _X(_p(n, Cs, i, j) * n_ / Cs[j])
512 i = j + 1
513 n_ *= n
514 X = tuple(X)
515 else: # isSpherical
516 X = _0_0s(m + 1)
517 return X
520# _Alp- and _BetCoeffs in .rhumb.ekx, .rhumb.bases
521_AlpCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
522 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4
523 164, 225, -480, 360, 720, # Alp[1]/n^1, polynomial(n), order 3
524 557, -864, 390, 1440, # Alp[2]/n^2, polynomial(n), order 2
525 -1236, 427, 1680, # PYCHOK Alp[3]/n^3, polynomial(n), order 1
526 49561, 161280), # Alp[4]/n^4, polynomial(n), order 0, count = 14
527 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5
528 -635, 328, 450, -960, 720, 1440, # Alp[1]/n^1, polynomial(n), order 4
529 4496, 3899, -6048, 2730, 10080, # PYCHOK Alp[2]/n^2, polynomial(n), order 3
530 15061, -19776, 6832, 26880, # PYCHOK Alp[3]/n^3, polynomial(n), order 2
531 -171840, 49561, 161280, # Alp[4]/n^4, polynomial(n), order 1
532 34729, 80640), # PYCHOK Alp[5]/n^5, polynomial(n), order 0, count = 20
533 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6
534 31564, -66675, 34440, 47250, -100800, 75600, 151200, # Alp[1]/n^1, polynomial(n), order 5
535 -1983433, 863232, 748608, -1161216, 524160, 1935360, # PYCHOK Alp[2]/n^2, polynomial(n), order 4
536 670412, 406647, -533952, 184464, 725760, # Alp[3]/n^3, polynomial(n), order 3
537 6601661, -7732800, 2230245, 7257600, # Alp[4]/n^4, polynomial(n), order 2
538 -13675556, 3438171, 7983360, # PYCHOK Alp[5]/n^5, polynomial(n), order 1
539 212378941, 319334400), # Alp[6]/n^6, polynomial(n), order 0, count = 27
540 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7
541 1804025, 2020096, -4267200, 2204160, 3024000, -6451200, 4838400, 9676800, # Alp[1]/n^1, polynomial(n), order 6
542 4626384, -9917165, 4316160, 3743040, -5806080, 2620800, 9676800, # Alp[2]/n^2, polynomial(n), order 5
543 -67102379, 26816480, 16265880, -21358080, 7378560, 29030400, # PYCHOK Alp[3]/n^3, polynomial(n), order 4
544 155912000, 72618271, -85060800, 24532695, 79833600, # Alp[4]/n^4, polynomial(n), order 3
545 102508609, -109404448, 27505368, 63866880, # Alp[5]/n^5, polynomial(n), order 2
546 -12282192400, 2760926233, 4151347200, # PYCHOK Alp[6]/n^6, polynomial(n), order 1
547 1522256789, 1383782400), # Alp[7]/n^7, polynomial(n), order 0, count = 35
548 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8
549 -75900428, 37884525, 42422016, -89611200, 46287360, 63504000, -135475200, 101606400, 203212800, # Alp[1]/n^1, polynomial(n), order 7
550 148003883, 83274912, -178508970, 77690880, 67374720, -104509440, 47174400, 174182400, # PYCHOK Alp[2]/n^2, polynomial(n), order 6
551 318729724, -738126169, 294981280, 178924680, -234938880, 81164160, 319334400, # PYCHOK Alp[3]/n^3, polynomial(n), order 5
552 -40176129013, 14967552000, 6971354016, -8165836800, 2355138720, 7664025600, # Alp[4]/n^4, polynomial(n), order 4
553 10421654396, 3997835751, -4266773472, 1072709352, 2490808320, # PYCHOK Alp[5]/n^5, polynomial(n), order 3
554 175214326799, -171950693600, 38652967262, 58118860800, # PYCHOK Alp[6]/n^6, polynomial(n), order 2
555 -67039739596, 13700311101, 12454041600, # PYCHOK Alp[7]/n^7, polynomial(n), order 1
556 1424729850961, 743921418240) # PYCHOK Alp[8]/n^8, polynomial(n), order 0, count = 44
557}
558_B1Coeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
559 2: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 2
560 1, 16, 64, 64), # b1 * (n + 1), polynomial(n2), order 2
561 3: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 3
562 1, 4, 64, 256, 256), # b1 * (n + 1), polynomial(n2), order 3
563 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 4
564 25, 64, 256, 4096, 16384, 16384) # PYCHOK b1 * (n + 1), polynomial(n2), order 4
565}
566_BetCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
567 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4
568 -4, 555, -960, 720, 1440, # Bet[1]/n^1, polynomial(n), order 3
569 -437, 96, 30, 1440, # Bet[2]/n^2, polynomial(n), order 2
570 -148, 119, 3360, # Bet[3]/n^3, polynomial(n), order 1
571 4397, 161280), # PYCHOK Bet[4]/n^4, polynomial(n), order 0, count = 14
572 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5
573 -3645, -64, 8880, -15360, 11520, 23040, # Bet[1]/n^1, polynomial(n), order 4
574 4416, -3059, 672, 210, 10080, # PYCHOK Bet[2]/n^2, polynomial(n), order 3
575 -627, -592, 476, 13440, # Bet[3]/n^3, polynomial(n), order 2
576 -3520, 4397, 161280, # Bet[4]/n^4, polynomial(n), order 1
577 4583, 161280), # PYCHOK Bet[5]/n^5, polynomial(n), order 0, count = 20
578 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6
579 384796, -382725, -6720, 932400, -1612800, 1209600, 2419200, # Bet[1]/n^1, polynomial(n), order 5
580 -1118711, 1695744, -1174656, 258048, 80640, 3870720, # PYCHOK Bet[2]/n^2, polynomial(n), order 4
581 22276, -16929, -15984, 12852, 362880, # Bet[3]/n^3, polynomial(n), order 3
582 -830251, -158400, 197865, 7257600, # PYCHOK Bet[4]/n^4, polynomial(n), order 2
583 -435388, 453717, 15966720, # PYCHOK Bet[5]/n^5, polynomial(n), order 1
584 20648693, 638668800), # Bet[6]/n^6, polynomial(n), order 0, count = 27
585 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7
586 -5406467, 6156736, -6123600, -107520, 14918400, -25804800, 19353600, 38707200, # Bet[1]/n^1, polynomial(n), order 6
587 829456, -5593555, 8478720, -5873280, 1290240, 403200, 19353600, # PYCHOK Bet[2]/n^2, polynomial(n), order 5
588 9261899, 3564160, -2708640, -2557440, 2056320, 58060800, # PYCHOK Bet[3]/n^3, polynomial(n), order 4
589 14928352, -9132761, -1742400, 2176515, 79833600, # PYCHOK Bet[4]/n^4, polynomial(n), order 3
590 -8005831, -1741552, 1814868, 63866880, # Bet[5]/n^5, polynomial(n), order 2
591 -261810608, 268433009, 8302694400, # Bet[6]/n^6, polynomial(n), order 1
592 219941297, 5535129600), # PYCHOK Bet[7]/n^7, polynomial(n), order 0, count = 35
593 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8
594 31777436, -37845269, 43097152, -42865200, -752640, 104428800, -180633600, 135475200, 270950400, # Bet[1]/n^1, polynomial(n), order 7
595 24749483, 14930208, -100683990, 152616960, -105719040, 23224320, 7257600, 348364800, # Bet[2]/n^2, polynomial(n), order 6
596 -232468668, 101880889, 39205760, -29795040, -28131840, 22619520, 638668800, # PYCHOK Bet[3]/n^3, polynomial(n), order 5
597 324154477, 1433121792, -876745056, -167270400, 208945440, 7664025600, # Bet[4]/n^4, polynomial(n), order 4
598 457888660, -312227409, -67920528, 70779852, 2490808320, # Bet[5]/n^5, polynomial(n), order 3
599 -19841813847, -3665348512, 3758062126, 116237721600, # PYCHOK Bet[6]/n^6, polynomial(n), order 2
600 -1989295244, 1979471673, 49816166400, # PYCHOK Bet[7]/n^7, polynomial(n), order 1
601 191773887257, 3719607091200) # Bet[8]/n^8, polynomial(n), order 0, count = 44
602}
604assert set(_AlpCoeffs.keys()) == set(_BetCoeffs.keys())
606if __name__ == '__main__':
608 from pygeodesy.internals import _usage
609 from sys import argv, exit as _exit
611 _exit(_usage(*argv).replace('.ktm', '.etm -series'))
613# **) MIT License
614#
615# Copyright (C) 2022-2024 -- mrJean1 at Gmail -- All Rights Reserved.
616#
617# Permission is hereby granted, free of charge, to any person obtaining a
618# copy of this software and associated documentation files (the "Software"),
619# to deal in the Software without restriction, including without limitation
620# the rights to use, copy, modify, merge, publish, distribute, sublicense,
621# and/or sell copies of the Software, and to permit persons to whom the
622# Software is furnished to do so, subject to the following conditions:
623#
624# The above copyright notice and this permission notice shall be included
625# in all copies or substantial portions of the Software.
626#
627# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
628# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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632# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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