Coverage for pygeodesy/ktm.py: 96%
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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_pop2, _Xorder
52from pygeodesy.fmath import fdot_, hypot, hypot1
53from pygeodesy.interns import _COMMASPACE_, _singular_
54from pygeodesy.karney import _atan2d, _diff182, _fix90, _norm180, \
55 _polynomial, _unsigned2
56# from pygeodesy.lazily import _ALL_LAZY # from .named
57from pygeodesy.named import _NamedBase, pairs, _ALL_LAZY
58from pygeodesy.namedTuples import Forward4Tuple, Reverse4Tuple
59from pygeodesy.props import property_doc_, Property, Property_RO, \
60 _update_all
61# from pygeodesy.streprs import pairs # from .named
62from pygeodesy.units import Degrees, Scalar_, _1mm as _TOL_10 # PYCHOK used!
63from pygeodesy.utily import atan1d, _loneg, sincos2, sincos2d_
65from cmath import polar
66from math import atan2, asinh, cos, cosh, degrees, fabs, sin, sinh, sqrt, tanh
68__all__ = _ALL_LAZY.ktm
69__version__ = '24.11.11'
72class KTMError(_ValueError):
73 '''Error raised for L{KTransverseMercator} and L{KTransverseMercator.forward} issues.
74 '''
75 pass
78class KTransverseMercator(_NamedBase):
79 '''I{Karney}'s C++ class U{TransverseMercator<https://GeographicLib.SourceForge.io/
80 C++/doc/classGeographicLib_1_1TransverseMercator.html>} transcoded to pure
81 Python, following is a partial copy of I{Karney}'s documentation.
83 Transverse Mercator projection based on Krüger's method which evaluates the
84 projection and its inverse in terms of a series.
86 There's a singularity in the projection at I{phi = 0, lam - lam0 = +/- (1 - e)
87 90}, about +/- 82.6 degrees for WGS84, where I{e} is the eccentricity. Beyond
88 this point, the series ceases to converge and the results from this method
89 will be garbage. I{To be on the safe side, don't use this method if the
90 angular distance from the central meridian exceeds (1 - 2e) x 90}, about 75
91 degrees for the WGS84 ellipsoid.
93 Class L{ExactTransverseMercator} is an alternative implementation of the
94 projection using I{exact} formulas which yield accurate (to 8 nm) results
95 over the entire ellipsoid.
97 The ellipsoid parameters and the central scale are set in the constructor.
98 The central meridian (which is a trivial shift of the longitude) is specified
99 as the C{lon0} keyword argument of the L{KTransverseMercator.forward} and
100 L{KTransverseMercator.reverse} methods. The latitude of origin is taken to
101 be the equator. There is no provision in this class for specifying a false
102 easting or false northing or a different latitude of origin. However these
103 are can be simply included by the calling function.
105 The L{KTransverseMercator.forward} and L{KTransverseMercator.reverse} methods
106 also return the meridian convergence C{gamma} and scale C{k}. The meridian
107 convergence is the bearing of grid North, the C{y axis}, measured clockwise
108 from true North.
109 '''
110 _datum = _WGS84
111 _k0 = _K0_UTM # central scale factor
112 _lat0 = _0_0 # central parallel
113 _lon0 = _0_0 # central meridian
114 _mTM = 6
115 _raiser = False # throw Error
117 def __init__(self, a_earth=_EWGS84, f=None, lon0=0, k0=_K0_UTM,
118 raiser=False, **TMorder_name):
119 '''New L{KTransverseMercator}.
121 @kwarg a_earth: This rhumb's earth (L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple},
122 L{Datum}, 2-tuple (C{a, f})) or the equatorial radius (C{meter}).
123 @kwarg f: The ellipsoid's flattening (C{scalar}), required if B{C{a_earth}} is
124 is C{meter}, ignored otherwise.
125 @kwarg lon0: The central meridian (C{degrees180}).
126 @kwarg k0: Central scale factor (C{scalar}).
127 @kwarg raiser: If C{True}, throw a L{KTMError} for C{forward} singularities (C{bool}).
128 @kwarg TMorder_name: Optional C{B{name}=NN} (C{str}) and optional keyword argument
129 C{B{TMorder}=6} for the order of this L{KTransverseMercator}, see
130 property C{TMorder}.
132 @raise KTMError: Invalid B{C{a_earth}}, B{C{f}} or B{C{TMorder}}.
133 '''
134 if TMorder_name:
135 M = self._mTM
136 m, name = _xkwds_pop2(TMorder_name, TMorder=M)
137 if m != M:
138 self.TMorder = m
139 if name:
140 self.name = name
142 if f is not None:
143 self.ellipsoid = a_earth, f
144 elif a_earth in (_EWGS84, _WGS84, None):
145 pass
146 elif isinstance(a_earth, Datum):
147 self.datum = a_earth
148 else:
149 self.ellipsoid = a_earth
151 self.lon0 = lon0
152 self.k0 = k0
153 if raiser:
154 self.raiser = True
156 @Property_RO
157 def _Alp(self):
158 return _Xs(_AlpCoeffs, self.TMorder, self.ellipsoid)
160 @Property_RO
161 def _b1(self):
162 n = self.ellipsoid.n
163 if n: # isEllipsoidal
164 m = self.TMorder // 2
165 B1 = _B1Coeffs[m]
166 m += 1
167 b1 = _polynomial(n**2, B1, 0, m) / (B1[m] * (n + _1_0))
168 else: # isSpherical
169 b1 = _1_0 # B1[m - 1] / B1[m1] == 1, always
170 return b1
172 @Property_RO
173 def _Bet(self):
174 C = _Xs(_BetCoeffs, self.TMorder, self.ellipsoid)
175 return tuple(map(neg, C)) if self.f else C # negated if isEllipsoidal
177 @property
178 def datum(self):
179 '''Get this rhumb's datum (L{Datum}).
180 '''
181 return self._datum
183 @datum.setter # PYCHOK setter!
184 def datum(self, datum):
185 '''Set this rhumb's datum (L{Datum}).
186 '''
187 _xinstanceof(Datum, datum=datum)
188 if self._datum != datum:
189 _update_all(self)
190 self._datum = datum
192 @Property
193 def ellipsoid(self):
194 '''Get the ellipsoid (L{Ellipsoid}).
195 '''
196 return self.datum.ellipsoid
198 @ellipsoid.setter # PYCHOK setter!
199 def ellipsoid(self, a_earth_f):
200 '''Set this rhumb's ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum},
201 L{a_f2Tuple} or 2-tuple C{(a, f)}).
202 '''
203 self.datum = _spherical_datum(a_earth_f, Error=KTMError)
205 @Property_RO
206 def equatoradius(self):
207 '''Get the C{ellipsoid}'s equatorial radius, semi-axis (C{meter}).
208 '''
209 return self.ellipsoid.a
211 a = equatoradius
213 @Property_RO
214 def flattening(self):
215 '''Get the C{ellipsoid}'s flattening (C{scalar}).
216 '''
217 return self.ellipsoid.f
219 f = flattening
221 def forward(self, lat, lon, lon0=None, **name):
222 '''Forward projection, from geographic to transverse Mercator.
224 @arg lat: Latitude of point (C{degrees90}).
225 @arg lon: Longitude of point (C{degrees180}).
226 @arg lon0: Central meridian of the projection (C{degrees180}).
227 @kwarg name: Optional C{B{name}=NN} (C{str}).
229 @return: L{Forward4Tuple}C{(easting, northing, gamma, scale)} with
230 C{easting} and C{northing} in C{meter}, unfalsed, the
231 meridian convergence C{gamma} at point in C{degrees180}
232 and the C{scale} of projection at point C{scalar}. Any
233 value may be C{NAN}, C{NINF} or C{INF} for singularities.
235 @raise KTMError: For singularities, iff property C{raiser} is C{True}.
236 '''
237 lat, _lat = _unsigned2(_fix90(lat - self._lat0))
238 lon, _ = _diff182((self.lon0 if lon0 is None else lon0), lon)
239 lon, _lon = _unsigned2(lon)
240 backside = lon > 90
241 if backside: # PYCHOK no cover
242 lon = _loneg(lon)
243 if lat == 0:
244 _lat = True
246 sphi, cphi, slam, clam = sincos2d_(lat, lon)
247 E = self.ellipsoid
248 if cphi and lat != 90:
249 t = sphi / cphi
250 tp = E.es_taupf(t)
251 h = hypot(tp, clam)
252 if h:
253 xip = atan2(tp, clam)
254 etap = asinh(slam / h) # atanh(sin(lam) / cosh(psi))
255 g = _atan2d(slam * tp, clam * hypot1(tp)) # Krueger p 22 (44)
256 k = sqrt(cphi**2 * E.e2 + E.e21) * hypot1(t) / h
257 elif self.raiser:
258 raise KTMError(lat=lat, lon=lon, lon0=lon0, txt=_singular_)
259 else: # PYCHOK no cover
260 xip, etap = _0_0, _copysignINF(slam)
261 g, k = copysign0(_90_0, slam), INF
262 else: # PYCHOK no cover
263 xip, etap = PI_2, _0_0
264 g, k = lon, E.es_c
265 y, x, d, t = _Cyxgk4(E, xip, etap, self._Alp)
266 g -= d
267 k *= t * self._k0_b1
269 if backside: # PYCHOK no cover
270 y, g = (PI - y), _loneg(g)
271 y *= self._k0_a1
272 x *= self._k0_a1
273 if _lat:
274 y, g = neg_(y, g)
275 if _lon:
276 x, g = neg_(x, g)
277 return Forward4Tuple(x, y, _norm180(g), k, name=self._name__(name))
279 @property_doc_(''' the central scale factor (C{float}).''')
280 def k0(self):
281 '''Get the central scale factor (C{float}), aka I{C{scale0}}.
282 '''
283 return self._k0 # aka scale0
285 @k0.setter # PYCHOK setter!
286 def k0(self, k0):
287 '''Set the central scale factor (C{float}), aka I{C{scale0}}.
289 @raise KTMError: Invalid B{C{k0}}.
290 '''
291 k0 = Scalar_(k0=k0, Error=KTMError, low=_TOL_10, high=_1_0)
292 if self._k0 != k0: # PYCHOK no cover
293 KTransverseMercator._k0_a1._update(self) # redo ._k0_a1
294 KTransverseMercator._k0_b1._update(self) # redo ._k0_b1
295 self._k0 = k0
297 @Property_RO
298 def _k0_a1(self):
299 '''(INTERNAL) Cache C{k0 * _b1 * equatoradius}.
300 '''
301 return self._k0_b1 * self.equatoradius
303 @Property_RO
304 def _k0_b1(self):
305 '''(INTERNAL) Cache C{k0 * _b1}.
306 '''
307 return self.k0 * self._b1
309 @property_doc_(''' the central meridian (C{degrees180}).''')
310 def lon0(self):
311 '''Get the central meridian (C{degrees180}).
312 '''
313 return self._lon0
315 @lon0.setter # PYCHOK setter!
316 def lon0(self, lon0):
317 '''Set the central meridian (C{degrees180}).
319 @raise KTMError: Invalid B{C{lon0}}.
320 '''
321 self._lon0 = _norm180(Degrees(lon0=lon0, Error=KTMError))
323 @property_doc_(''' raise a L{KTMError} for C{forward} singularities (C{bool}).''')
324 def raiser(self):
325 '''Get the error setting (C{bool}).
326 '''
327 return self._raiser
329 @raiser.setter # PYCHOK setter!
330 def raiser(self, raiser):
331 '''Set the error setting (C{bool}), to C{True} to throw a L{KTMError}
332 for C{forward} singularities.
333 '''
334 self._raiser = bool(raiser)
336 def reset(self, lat0, lon0):
337 '''Set the central parallel and meridian.
339 @arg lat0: Latitude of the central parallel (C{degrees90}).
340 @arg lon0: Longitude of the central parallel (C{degrees180}).
342 @return: 2-Tuple C{(lat0, lon0)} with the previous central
343 parallel and meridian.
345 @raise KTMError: Invalid B{C{lat0}} or B{C{lon0}}.
346 '''
347 t = self._lat0, self.lon0
348 self._lat0 = _fix90(Degrees(lat0=lat0, Error=KTMError))
349 self. lon0 = lon0
350 return t
352 def reverse(self, x, y, lon0=None, **name):
353 '''Reverse projection, from transverse Mercator to geographic.
355 @arg x: Easting of point (C{meter}).
356 @arg y: Northing of point (C{meter}).
357 @arg lon0: Central meridian of the projection (C{degrees180}).
358 @kwarg name: Optional C{B{name}=NN} (C{str}).
360 @return: L{Reverse4Tuple}C{(lat, lon, gamma, scale)} with
361 C{lat}- and C{lon}gitude in C{degrees}, I{unfalsed}.
362 '''
363 eta, _lon = _unsigned2(x / self._k0_a1)
364 xi, _lat = _unsigned2(y / self._k0_a1)
365 backside = xi > PI_2
366 if backside: # PYCHOK no cover
367 xi = PI - xi
369 E = self.ellipsoid
370 xip, etap, g, k = _Cyxgk4(E, xi, eta, self._Bet)
371 t = self._k0_b1
372 k = (t / k) if k else _copysignINF(t) # _over(t, k)
373 h, c = sinh(etap), cos(xip)
374 if c > 0:
375 r = hypot(h, c)
376 else: # PYCHOK no cover
377 r = fabs(h)
378 c = _0_0
379 if r:
380 lon = _atan2d(h, c) # Krueger p 17 (25)
381 s = sin(xip) # Newton for tau
382 t = E.es_tauf(s / r)
383 lat = atan1d(t)
384 g += _atan2d(s * tanh(etap), c) # Krueger p 19 (31)
385 k *= sqrt(E.e2 / (t**2 + _1_0) + E.e21) * hypot1(t) * r
386 else: # PYCHOK no cover
387 lat = _90_0
388 lon = _0_0
389 k *= E.es_c
391 if backside: # PYCHOK no cover
392 lon, g = _loneg(lon), _loneg(g)
393 if _lat:
394 lat, g = neg_(lat, g)
395 if _lon:
396 lon, g = neg_(lon, g)
397 lat += self._lat0
398 lon += self._lon0 if lon0 is None else _norm180(lon0)
399 return Reverse4Tuple(lat, _norm180(lon), _norm180(g), k,
400 name=self._name__(name))
402 @Property
403 def TMorder(self):
404 '''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
405 '''
406 return self._mTM
408 @TMorder.setter # PYCHOK setter!
409 def TMorder(self, order):
410 '''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
411 '''
412 m = _Xorder(_AlpCoeffs, KTMError, TMorder=order)
413 if self._mTM != m:
414 _update_all(self)
415 self._mTM = m
417 def toStr(self, **kwds):
418 '''Return a C{str} representation.
420 @arg kwds: Optional, overriding keyword arguments.
421 '''
422 d = dict(ellipsoid=self.ellipsoid, k0=self.k0, TMorder=self.TMorder)
423 if self.name: # PYCHOK no cover
424 d.update(name=self.name)
425 return _COMMASPACE_.join(pairs(d, **kwds))
428def _cma(a, b0, b1, Cn):
429 '''(INTERNAL) Compute complex M{a * b0 - b1 + Cn} with complex
430 C{a}, C{b0} and C{b1} and scalar C{Cn}.
432 @see: CPython function U{_Py_c_prod<https://GitHub.com/python/
433 cpython/blob/main/Objects/complexobject.c>}.
435 @note: Python function C{cmath.fsum} no longer exists.
436 '''
437 r = fdot_(a.real, b0.real, -a.imag, b0.imag, -b1.real, _1_0, start=Cn)
438 j = fdot_(a.real, b0.imag, a.imag, b0.real, -b1.imag, _1_0)
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 while n > 0:
464 Cn = C[n]
465 y1 = _cma(a, y0, y1, Cn)
466 z1 = _cma(a, z0, z1, Cn * (n * 2))
467 n -= 1
468 Cn = C[n]
469 y0 = _cma(a, y1, y0, Cn)
470 z0 = _cma(a, z1, z0, Cn * (n * 2))
471 n -= 1
472 # assert n == 0
473 x = _cma(s, y0, -x, _0_0)
474 c = _cma(c, z0, z1, _1_0)
476 # Gauss-Schreiber to Gauss-Krueger TM
477 # C{cmath.polar} handles INF, NAN, etc.
478 k, g = polar(c)
479 g = degrees(g)
480 else: # E.isSpherical
481 g, k = _0_0, _1_0
483 return x.real, x.imag, g, k
486def _sinhcosh2(x):
487 '''(INTERNAL) Like C{sincos2}.
488 '''
489 return sinh(x), cosh(x)
492def _Xs(_Coeffs, m, E, RA=False): # in .rhumb.ekx
493 '''(INTERNAL) Compute the C{A}, C{B} or C{RA} terms of order
494 B{C{m}} for I{Krüger} series and I{rhumb.ekx._sincosSeries},
495 return a tuple with C{B{m} + 1} terms C{X}, C{X[0]==0}.
496 '''
497 Cs = _Coeffs[m]
498 assert len(Cs) == (((m + 1) * (m + 4)) if RA else
499 ((m + 3) * m)) // 2
500 n = n_ = E.n
501 if n: # isEllipsoidal
502 X = [0] # X[0] never used, it's just an integration
503 # constant, it cancels when evaluating a definite
504 # integral. Don't bother computing it, it is unused
505 # in C{_Cyxgk4} above and C{rhumb.ekx._sincosSeries}.
506 i = (m + 2) if RA else 0
507 _p = _polynomial
508 for r in _reverange(m): # [m-1 ... 0]
509 j = i + r + 1
510 X.append(_p(n, Cs, i, j) * n_ / Cs[j])
511 i = j + 1
512 n_ *= n
513 X = tuple(X)
514 else: # isSpherical
515 X = _0_0s(m + 1)
516 return X
519# _Alp- and _BetCoeffs in .rhumb.ekx, .rhumb.bases
520_AlpCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
521 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4
522 164, 225, -480, 360, 720, # Alp[1]/n^1, polynomial(n), order 3
523 557, -864, 390, 1440, # Alp[2]/n^2, polynomial(n), order 2
524 -1236, 427, 1680, # PYCHOK Alp[3]/n^3, polynomial(n), order 1
525 49561, 161280), # Alp[4]/n^4, polynomial(n), order 0, count = 14
526 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5
527 -635, 328, 450, -960, 720, 1440, # Alp[1]/n^1, polynomial(n), order 4
528 4496, 3899, -6048, 2730, 10080, # PYCHOK Alp[2]/n^2, polynomial(n), order 3
529 15061, -19776, 6832, 26880, # PYCHOK Alp[3]/n^3, polynomial(n), order 2
530 -171840, 49561, 161280, # Alp[4]/n^4, polynomial(n), order 1
531 34729, 80640), # PYCHOK Alp[5]/n^5, polynomial(n), order 0, count = 20
532 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6
533 31564, -66675, 34440, 47250, -100800, 75600, 151200, # Alp[1]/n^1, polynomial(n), order 5
534 -1983433, 863232, 748608, -1161216, 524160, 1935360, # PYCHOK Alp[2]/n^2, polynomial(n), order 4
535 670412, 406647, -533952, 184464, 725760, # Alp[3]/n^3, polynomial(n), order 3
536 6601661, -7732800, 2230245, 7257600, # Alp[4]/n^4, polynomial(n), order 2
537 -13675556, 3438171, 7983360, # PYCHOK Alp[5]/n^5, polynomial(n), order 1
538 212378941, 319334400), # Alp[6]/n^6, polynomial(n), order 0, count = 27
539 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7
540 1804025, 2020096, -4267200, 2204160, 3024000, -6451200, 4838400, 9676800, # Alp[1]/n^1, polynomial(n), order 6
541 4626384, -9917165, 4316160, 3743040, -5806080, 2620800, 9676800, # Alp[2]/n^2, polynomial(n), order 5
542 -67102379, 26816480, 16265880, -21358080, 7378560, 29030400, # PYCHOK Alp[3]/n^3, polynomial(n), order 4
543 155912000, 72618271, -85060800, 24532695, 79833600, # Alp[4]/n^4, polynomial(n), order 3
544 102508609, -109404448, 27505368, 63866880, # Alp[5]/n^5, polynomial(n), order 2
545 -12282192400, 2760926233, 4151347200, # PYCHOK Alp[6]/n^6, polynomial(n), order 1
546 1522256789, 1383782400), # Alp[7]/n^7, polynomial(n), order 0, count = 35
547 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8
548 -75900428, 37884525, 42422016, -89611200, 46287360, 63504000, -135475200, 101606400, 203212800, # Alp[1]/n^1, polynomial(n), order 7
549 148003883, 83274912, -178508970, 77690880, 67374720, -104509440, 47174400, 174182400, # PYCHOK Alp[2]/n^2, polynomial(n), order 6
550 318729724, -738126169, 294981280, 178924680, -234938880, 81164160, 319334400, # PYCHOK Alp[3]/n^3, polynomial(n), order 5
551 -40176129013, 14967552000, 6971354016, -8165836800, 2355138720, 7664025600, # Alp[4]/n^4, polynomial(n), order 4
552 10421654396, 3997835751, -4266773472, 1072709352, 2490808320, # PYCHOK Alp[5]/n^5, polynomial(n), order 3
553 175214326799, -171950693600, 38652967262, 58118860800, # PYCHOK Alp[6]/n^6, polynomial(n), order 2
554 -67039739596, 13700311101, 12454041600, # PYCHOK Alp[7]/n^7, polynomial(n), order 1
555 1424729850961, 743921418240) # PYCHOK Alp[8]/n^8, polynomial(n), order 0, count = 44
556}
557_B1Coeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
558 2: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 2
559 1, 16, 64, 64), # b1 * (n + 1), polynomial(n2), order 2
560 3: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 3
561 1, 4, 64, 256, 256), # b1 * (n + 1), polynomial(n2), order 3
562 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 4
563 25, 64, 256, 4096, 16384, 16384) # PYCHOK b1 * (n + 1), polynomial(n2), order 4
564}
565_BetCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
566 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4
567 -4, 555, -960, 720, 1440, # Bet[1]/n^1, polynomial(n), order 3
568 -437, 96, 30, 1440, # Bet[2]/n^2, polynomial(n), order 2
569 -148, 119, 3360, # Bet[3]/n^3, polynomial(n), order 1
570 4397, 161280), # PYCHOK Bet[4]/n^4, polynomial(n), order 0, count = 14
571 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5
572 -3645, -64, 8880, -15360, 11520, 23040, # Bet[1]/n^1, polynomial(n), order 4
573 4416, -3059, 672, 210, 10080, # PYCHOK Bet[2]/n^2, polynomial(n), order 3
574 -627, -592, 476, 13440, # Bet[3]/n^3, polynomial(n), order 2
575 -3520, 4397, 161280, # Bet[4]/n^4, polynomial(n), order 1
576 4583, 161280), # PYCHOK Bet[5]/n^5, polynomial(n), order 0, count = 20
577 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6
578 384796, -382725, -6720, 932400, -1612800, 1209600, 2419200, # Bet[1]/n^1, polynomial(n), order 5
579 -1118711, 1695744, -1174656, 258048, 80640, 3870720, # PYCHOK Bet[2]/n^2, polynomial(n), order 4
580 22276, -16929, -15984, 12852, 362880, # Bet[3]/n^3, polynomial(n), order 3
581 -830251, -158400, 197865, 7257600, # PYCHOK Bet[4]/n^4, polynomial(n), order 2
582 -435388, 453717, 15966720, # PYCHOK Bet[5]/n^5, polynomial(n), order 1
583 20648693, 638668800), # Bet[6]/n^6, polynomial(n), order 0, count = 27
584 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7
585 -5406467, 6156736, -6123600, -107520, 14918400, -25804800, 19353600, 38707200, # Bet[1]/n^1, polynomial(n), order 6
586 829456, -5593555, 8478720, -5873280, 1290240, 403200, 19353600, # PYCHOK Bet[2]/n^2, polynomial(n), order 5
587 9261899, 3564160, -2708640, -2557440, 2056320, 58060800, # PYCHOK Bet[3]/n^3, polynomial(n), order 4
588 14928352, -9132761, -1742400, 2176515, 79833600, # PYCHOK Bet[4]/n^4, polynomial(n), order 3
589 -8005831, -1741552, 1814868, 63866880, # Bet[5]/n^5, polynomial(n), order 2
590 -261810608, 268433009, 8302694400, # Bet[6]/n^6, polynomial(n), order 1
591 219941297, 5535129600), # PYCHOK Bet[7]/n^7, polynomial(n), order 0, count = 35
592 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8
593 31777436, -37845269, 43097152, -42865200, -752640, 104428800, -180633600, 135475200, 270950400, # Bet[1]/n^1, polynomial(n), order 7
594 24749483, 14930208, -100683990, 152616960, -105719040, 23224320, 7257600, 348364800, # Bet[2]/n^2, polynomial(n), order 6
595 -232468668, 101880889, 39205760, -29795040, -28131840, 22619520, 638668800, # PYCHOK Bet[3]/n^3, polynomial(n), order 5
596 324154477, 1433121792, -876745056, -167270400, 208945440, 7664025600, # Bet[4]/n^4, polynomial(n), order 4
597 457888660, -312227409, -67920528, 70779852, 2490808320, # Bet[5]/n^5, polynomial(n), order 3
598 -19841813847, -3665348512, 3758062126, 116237721600, # PYCHOK Bet[6]/n^6, polynomial(n), order 2
599 -1989295244, 1979471673, 49816166400, # PYCHOK Bet[7]/n^7, polynomial(n), order 1
600 191773887257, 3719607091200) # Bet[8]/n^8, polynomial(n), order 0, count = 44
601}
603assert set(_AlpCoeffs.keys()) == set(_BetCoeffs.keys())
605if __name__ == '__main__':
607 from pygeodesy.internals import _usage
608 from sys import argv, exit as _exit
610 _exit(_usage(*argv).replace('.ktm', '.etm -series'))
612# **) MIT License
613#
614# Copyright (C) 2022-2024 -- mrJean1 at Gmail -- All Rights Reserved.
615#
616# Permission is hereby granted, free of charge, to any person obtaining a
617# copy of this software and associated documentation files (the "Software"),
618# to deal in the Software without restriction, including without limitation
619# the rights to use, copy, modify, merge, publish, distribute, sublicense,
620# and/or sell copies of the Software, and to permit persons to whom the
621# Software is furnished to do so, subject to the following conditions:
622#
623# The above copyright notice and this permission notice shall be included
624# in all copies or substantial portions of the Software.
625#
626# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
627# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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629# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
630# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
631# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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