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 hypot, hypot1
53from pygeodesy.fsums import fsum1f_
54from pygeodesy.interns import _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__ = '24.05.24'
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,
119 raiser=False, **TMorder_name):
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}}
126 is a C{scalar}, ignored otherwise.
127 @kwarg lon0: The central meridian (C{degrees180}).
128 @kwarg k0: Central scale factor (C{scalar}).
129 @kwarg raiser: If C{True}, throw a L{KTMError} for C{forward}
130 singularities (C{bool}).
131 @kwarg TMorder_name: Optional C{B{name}=NN} (C{str}) and optional
132 keyword argument C{B{TMorder}=6} for the order of
133 this L{KTransverseMercator}, see property C{TMorder}.
135 @raise KTMError: Invalid B{C{a_earth}}, B{C{f}} or B{C{TMorder}}.
136 '''
137 if TMorder_name:
138 M = self._mTM
139 m, name = _xkwds_pop2(TMorder_name, TMorder=M)
140 if m != M:
141 self.TMorder = m
142 if name:
143 self.name = name
145 if f is not None:
146 self.ellipsoid = a_earth, f
147 elif a_earth in (_EWGS84, _WGS84, None):
148 pass
149 elif isinstance(a_earth, Datum):
150 self.datum = a_earth
151 else:
152 self.ellipsoid = a_earth
154 self.lon0 = lon0
155 self.k0 = k0
156 if raiser:
157 self.raiser = True
159 @Property_RO
160 def _Alp(self):
161 return _Xs(_AlpCoeffs, self.TMorder, self.ellipsoid)
163 @Property_RO
164 def _b1(self):
165 n = self.ellipsoid.n
166 if n: # isEllipsoidal
167 m = self.TMorder // 2
168 B1 = _B1Coeffs[m]
169 m += 1
170 b1 = _polynomial(n**2, B1, 0, m) / (B1[m] * (n + _1_0))
171 else: # isSpherical
172 b1 = _1_0 # B1[m - 1] / B1[m1] == 1, always
173 return b1
175 @Property_RO
176 def _Bet(self):
177 C = _Xs(_BetCoeffs, self.TMorder, self.ellipsoid)
178 return tuple(map(neg, C)) if self.f else C # negated if isEllipsoidal
180 @property
181 def datum(self):
182 '''Get this rhumb's datum (L{Datum}).
183 '''
184 return self._datum
186 @datum.setter # PYCHOK setter!
187 def datum(self, datum):
188 '''Set this rhumb's datum (L{Datum}).
189 '''
190 _xinstanceof(Datum, datum=datum)
191 if self._datum != datum:
192 _update_all(self)
193 self._datum = datum
195 @Property
196 def ellipsoid(self):
197 '''Get the ellipsoid (L{Ellipsoid}).
198 '''
199 return self.datum.ellipsoid
201 @ellipsoid.setter # PYCHOK setter!
202 def ellipsoid(self, a_earth_f):
203 '''Set this rhumb's ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum},
204 L{a_f2Tuple} or 2-tuple C{(a, f)}).
205 '''
206 self.datum = _spherical_datum(a_earth_f, Error=KTMError)
208 @Property_RO
209 def equatoradius(self):
210 '''Get the C{ellipsoid}'s equatorial radius, semi-axis (C{meter}).
211 '''
212 return self.ellipsoid.a
214 a = equatoradius
216 @Property_RO
217 def flattening(self):
218 '''Get the C{ellipsoid}'s flattening (C{scalar}).
219 '''
220 return self.ellipsoid.f
222 f = flattening
224 def forward(self, lat, lon, lon0=None, **name):
225 '''Forward projection, from geographic to transverse Mercator.
227 @arg lat: Latitude of point (C{degrees90}).
228 @arg lon: Longitude of point (C{degrees180}).
229 @arg lon0: Central meridian of the projection (C{degrees180}).
230 @kwarg name: Optional C{B{name}=NN} (C{str}).
232 @return: L{Forward4Tuple}C{(easting, northing, gamma, scale)}
233 with C{easting} and C{northing} in C{meter}, unfalsed, the
234 meridian convergence C{gamma} at point in C{degrees180}
235 and the C{scale} of projection at point C{scalar}. Any
236 value may be C{NAN}, C{NINF} or C{INF} for singularities.
238 @raise KTMError: For singularities, iff property C{raiser} is
239 C{True}.
240 '''
241 lat, _lat = _unsigned2(_fix90(lat - self._lat0))
242 lon, _ = _diff182((self.lon0 if lon0 is None else lon0), lon)
243 lon, _lon = _unsigned2(lon)
244 backside = lon > 90
245 if backside: # PYCHOK no cover
246 lon = _loneg(lon)
247 if lat == 0:
248 _lat = True
250 sphi, cphi, slam, clam = sincos2d_(lat, lon)
251 E = self.ellipsoid
252 if cphi and lat != 90:
253 t = sphi / cphi
254 tp = E.es_taupf(t)
255 h = hypot(tp, clam)
256 if h:
257 xip = atan2(tp, clam)
258 etap = asinh(slam / h) # atanh(sin(lam) / cosh(psi))
259 g = _atan2d(slam * tp, clam * hypot1(tp)) # Krueger p 22 (44)
260 k = sqrt(cphi**2 * E.e2 + E.e21) * hypot1(t) / h
261 elif self.raiser:
262 raise KTMError(lat=lat, lon=lon, lon0=lon0, txt=_singular_)
263 else: # PYCHOK no cover
264 xip, etap = _0_0, _copysignINF(slam)
265 g, k = copysign0(_90_0, slam), INF
266 else: # PYCHOK no cover
267 xip, etap = PI_2, _0_0
268 g, k = lon, E.es_c
269 y, x, d, t = _Cyxgk4(E, xip, etap, self._Alp)
270 g -= d
271 k *= t * self._k0_b1
273 if backside: # PYCHOK no cover
274 y, g = (PI - y), _loneg(g)
275 y *= self._k0_a1
276 x *= self._k0_a1
277 if _lat:
278 y, g = neg_(y, g)
279 if _lon:
280 x, g = neg_(x, g)
281 return Forward4Tuple(x, y, _norm180(g), k, name=self._name__(name))
283 @property_doc_(''' the central scale factor (C{float}).''')
284 def k0(self):
285 '''Get the central scale factor (C{float}), aka I{C{scale0}}.
286 '''
287 return self._k0 # aka scale0
289 @k0.setter # PYCHOK setter!
290 def k0(self, k0):
291 '''Set the central scale factor (C{float}), aka I{C{scale0}}.
293 @raise KTMError: Invalid B{C{k0}}.
294 '''
295 k0 = Scalar_(k0=k0, Error=KTMError, low=_TOL_10, high=_1_0)
296 if self._k0 != k0: # PYCHOK no cover
297 KTransverseMercator._k0_a1._update(self) # redo ._k0_a1
298 KTransverseMercator._k0_b1._update(self) # redo ._k0_b1
299 self._k0 = k0
301 @Property_RO
302 def _k0_a1(self):
303 '''(INTERNAL) Cache C{k0 * _b1 * equatoradius}.
304 '''
305 return self._k0_b1 * self.equatoradius
307 @Property_RO
308 def _k0_b1(self):
309 '''(INTERNAL) Cache C{k0 * _b1}.
310 '''
311 return self.k0 * self._b1
313 @property_doc_(''' the central meridian (C{degrees180}).''')
314 def lon0(self):
315 '''Get the central meridian (C{degrees180}).
316 '''
317 return self._lon0
319 @lon0.setter # PYCHOK setter!
320 def lon0(self, lon0):
321 '''Set the central meridian (C{degrees180}).
323 @raise KTMError: Invalid B{C{lon0}}.
324 '''
325 self._lon0 = _norm180(Degrees(lon0=lon0, Error=KTMError))
327 @property_doc_(''' raise a L{KTMError} for C{forward} singularities (C{bool}).''')
328 def raiser(self):
329 '''Get the error setting (C{bool}).
330 '''
331 return self._raiser
333 @raiser.setter # PYCHOK setter!
334 def raiser(self, raiser):
335 '''Set the error setting (C{bool}), to C{True} to throw a L{KTMError}
336 for C{forward} singularities.
337 '''
338 self._raiser = bool(raiser)
340 def reset(self, lat0, lon0):
341 '''Set the central parallel and meridian.
343 @arg lat0: Latitude of the central parallel (C{degrees90}).
344 @arg lon0: Longitude of the central parallel (C{degrees180}).
346 @return: 2-Tuple C{(lat0, lon0)} of the previous central
347 parallel and meridian.
349 @raise KTMError: Invalid B{C{lat0}} or B{C{lon0}}.
350 '''
351 t = self._lat0, self.lon0
352 self._lat0 = _fix90(Degrees(lat0=lat0, Error=KTMError))
353 self. lon0 = lon0
354 return t
356 def reverse(self, x, y, lon0=None, **name):
357 '''Reverse projection, from transverse Mercator to geographic.
359 @arg x: Easting of point (C{meter}).
360 @arg y: Northing of point (C{meter}).
361 @arg lon0: Central meridian of the projection (C{degrees180}).
362 @kwarg name: Optional C{B{name}=NN} (C{str}).
364 @return: L{Reverse4Tuple}C{(lat, lon, gamma, scale)} with
365 C{lat}- and C{lon}gitude in C{degrees}, I{unfalsed}.
366 '''
367 eta, _lon = _unsigned2(x / self._k0_a1)
368 xi, _lat = _unsigned2(y / self._k0_a1)
369 backside = xi > PI_2
370 if backside: # PYCHOK no cover
371 xi = PI - xi
373 E = self.ellipsoid
374 xip, etap, g, k = _Cyxgk4(E, xi, eta, self._Bet)
375 t = self._k0_b1
376 k = (t / k) if k else _copysignINF(t) # _over(t, k)
377 h, c = sinh(etap), cos(xip)
378 if c > 0:
379 r = hypot(h, c)
380 else: # PYCHOK no cover
381 r = fabs(h)
382 c = _0_0
383 if r:
384 lon = _atan2d(h, c) # Krueger p 17 (25)
385 s = sin(xip) # Newton for tau
386 t = E.es_tauf(s / r)
387 lat = atan1d(t)
388 g += _atan2d(s * tanh(etap), c) # Krueger p 19 (31)
389 k *= sqrt(E.e2 / (t**2 + _1_0) + E.e21) * hypot1(t) * r
390 else: # PYCHOK no cover
391 lat = _90_0
392 lon = _0_0
393 k *= E.es_c
395 if backside: # PYCHOK no cover
396 lon, g = _loneg(lon), _loneg(g)
397 if _lat:
398 lat, g = neg_(lat, g)
399 if _lon:
400 lon, g = neg_(lon, g)
401 lat += self._lat0
402 lon += self._lon0 if lon0 is None else _norm180(lon0)
403 return Reverse4Tuple(lat, _norm180(lon), _norm180(g), k,
404 name=self._name__(name))
406 @Property
407 def TMorder(self):
408 '''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
409 '''
410 return self._mTM
412 @TMorder.setter # PYCHOK setter!
413 def TMorder(self, order):
414 '''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
415 '''
416 m = _Xorder(_AlpCoeffs, KTMError, TMorder=order)
417 if self._mTM != m:
418 _update_all(self)
419 self._mTM = m
421 def toStr(self, **kwds):
422 '''Return a C{str} representation.
424 @arg kwds: Optional, overriding keyword arguments.
425 '''
426 d = dict(ellipsoid=self.ellipsoid, k0=self.k0, TMorder=self.TMorder)
427 if self.name: # PYCHOK no cover
428 d.update(name=self.name)
429 return _COMMASPACE_.join(pairs(d, **kwds))
432def _cma(a, b0, b1, Cn):
433 '''(INTERNAL) Compute complex M{a * b0 - b1 + Cn} with complex
434 C{a}, C{b0} and C{b1} and scalar C{Cn}.
436 @see: CPython function U{_Py_c_prod<https://GitHub.com/python/
437 cpython/blob/main/Objects/complexobject.c>}.
439 @note: Python function C{cmath.fsum} is no longer available,
440 but stil mentioned in Note 4 of the comments before
441 CPython function U{math_fsum<https://GitHub.com/python/
442 cpython/blob/main/Modules/mathmodule.c>}
443 '''
444 r = fsum1f_(a.real * b0.real, -a.imag * b0.imag, -b1.real, Cn)
445 j = fsum1f_(a.real * b0.imag, a.imag * b0.real, -b1.imag)
446 return complex(r, j)
449def _Cyxgk4(E, xi_, eta_, C):
450 '''(INTERNAL) Complex Clenshaw summation with C{B{C}=._Alp}
451 or C{B{C}=-._Bet}.
452 '''
453 x = complex(xi_, eta_)
454 if E.f: # isEllipsoidal
455 s, c = sincos2( xi_ * 2)
456 sh, ch = _sinhcosh2(eta_ * 2)
457 n = -s
458 s = complex(s * ch, c * sh) # sin(zeta * 2)
459 c = complex(c * ch, n * sh) # cos(zeta * 2)
460 a = c * 2 # cos(zeta * 2) * 2
462 y0 = y1 = \
463 z0 = z1 = complex(0) # 0+0j
464 n = len(C) - 1 # == .TMorder
465 if isodd(n):
466 Cn = C[n]
467 y0 = complex(Cn) # +0j
468 z0 = complex(Cn * (n * 2))
469 n -= 1
470 _c = _cma
471 while n > 0:
472 Cn = C[n]
473 y1 = _c(a, y0, y1, Cn)
474 z1 = _c(a, z0, z1, Cn * (n * 2))
475 n -= 1
476 Cn = C[n]
477 y0 = _c(a, y1, y0, Cn)
478 z0 = _c(a, z1, z0, Cn * (n * 2))
479 n -= 1
480 # assert n == 0
481 x = _c(s, y0, -x, _0_0)
482 c = _c(c, z0, z1, _1_0)
484 # Gauss-Schreiber to Gauss-Krueger TM
485 # C{cmath.polar} handles INF, NAN, etc.
486 k, g = polar(c)
487 g = degrees(g)
488 else: # E.isSpherical
489 g, k = _0_0, _1_0
491 return x.real, x.imag, g, k
494def _sinhcosh2(x):
495 '''(INTERNAL) Like C{sincos2}.
496 '''
497 return sinh(x), cosh(x)
500def _Xs(_Coeffs, m, E, RA=False): # in .rhumb.ekx
501 '''(INTERNAL) Compute the C{A}, C{B} or C{RA} terms of order
502 B{C{m}} for I{Krüger} series and I{rhumb.ekx._sincosSeries},
503 return a tuple with C{B{m} + 1} terms C{X}, C{X[0]==0}.
504 '''
505 Cs = _Coeffs[m]
506 assert len(Cs) == (((m + 1) * (m + 4)) if RA else
507 ((m + 3) * m)) // 2
508 n = n_ = E.n
509 if n: # isEllipsoidal
510 X = [0] # X[0] never used, it's just an integration
511 # constant, it cancels when evaluating a definite
512 # integral. Don't bother computing it, it is unused
513 # in C{_Cyxgk4} above and C{rhumb.ekx._sincosSeries}.
514 _X, _p = X.append, _polynomial
515 i = (m + 2) if RA else 0
516 for r in _reverange(m): # [m-1 ... 0]
517 j = i + r + 1
518 _X(_p(n, Cs, i, j) * n_ / Cs[j])
519 i = j + 1
520 n_ *= n
521 X = tuple(X)
522 else: # isSpherical
523 X = _0_0s(m + 1)
524 return X
527# _Alp- and _BetCoeffs in .rhumb.ekx, .rhumb.bases
528_AlpCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
529 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4
530 164, 225, -480, 360, 720, # Alp[1]/n^1, polynomial(n), order 3
531 557, -864, 390, 1440, # Alp[2]/n^2, polynomial(n), order 2
532 -1236, 427, 1680, # PYCHOK Alp[3]/n^3, polynomial(n), order 1
533 49561, 161280), # Alp[4]/n^4, polynomial(n), order 0, count = 14
534 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5
535 -635, 328, 450, -960, 720, 1440, # Alp[1]/n^1, polynomial(n), order 4
536 4496, 3899, -6048, 2730, 10080, # PYCHOK Alp[2]/n^2, polynomial(n), order 3
537 15061, -19776, 6832, 26880, # PYCHOK Alp[3]/n^3, polynomial(n), order 2
538 -171840, 49561, 161280, # Alp[4]/n^4, polynomial(n), order 1
539 34729, 80640), # PYCHOK Alp[5]/n^5, polynomial(n), order 0, count = 20
540 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6
541 31564, -66675, 34440, 47250, -100800, 75600, 151200, # Alp[1]/n^1, polynomial(n), order 5
542 -1983433, 863232, 748608, -1161216, 524160, 1935360, # PYCHOK Alp[2]/n^2, polynomial(n), order 4
543 670412, 406647, -533952, 184464, 725760, # Alp[3]/n^3, polynomial(n), order 3
544 6601661, -7732800, 2230245, 7257600, # Alp[4]/n^4, polynomial(n), order 2
545 -13675556, 3438171, 7983360, # PYCHOK Alp[5]/n^5, polynomial(n), order 1
546 212378941, 319334400), # Alp[6]/n^6, polynomial(n), order 0, count = 27
547 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7
548 1804025, 2020096, -4267200, 2204160, 3024000, -6451200, 4838400, 9676800, # Alp[1]/n^1, polynomial(n), order 6
549 4626384, -9917165, 4316160, 3743040, -5806080, 2620800, 9676800, # Alp[2]/n^2, polynomial(n), order 5
550 -67102379, 26816480, 16265880, -21358080, 7378560, 29030400, # PYCHOK Alp[3]/n^3, polynomial(n), order 4
551 155912000, 72618271, -85060800, 24532695, 79833600, # Alp[4]/n^4, polynomial(n), order 3
552 102508609, -109404448, 27505368, 63866880, # Alp[5]/n^5, polynomial(n), order 2
553 -12282192400, 2760926233, 4151347200, # PYCHOK Alp[6]/n^6, polynomial(n), order 1
554 1522256789, 1383782400), # Alp[7]/n^7, polynomial(n), order 0, count = 35
555 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8
556 -75900428, 37884525, 42422016, -89611200, 46287360, 63504000, -135475200, 101606400, 203212800, # Alp[1]/n^1, polynomial(n), order 7
557 148003883, 83274912, -178508970, 77690880, 67374720, -104509440, 47174400, 174182400, # PYCHOK Alp[2]/n^2, polynomial(n), order 6
558 318729724, -738126169, 294981280, 178924680, -234938880, 81164160, 319334400, # PYCHOK Alp[3]/n^3, polynomial(n), order 5
559 -40176129013, 14967552000, 6971354016, -8165836800, 2355138720, 7664025600, # Alp[4]/n^4, polynomial(n), order 4
560 10421654396, 3997835751, -4266773472, 1072709352, 2490808320, # PYCHOK Alp[5]/n^5, polynomial(n), order 3
561 175214326799, -171950693600, 38652967262, 58118860800, # PYCHOK Alp[6]/n^6, polynomial(n), order 2
562 -67039739596, 13700311101, 12454041600, # PYCHOK Alp[7]/n^7, polynomial(n), order 1
563 1424729850961, 743921418240) # PYCHOK Alp[8]/n^8, polynomial(n), order 0, count = 44
564}
565_B1Coeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
566 2: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 2
567 1, 16, 64, 64), # b1 * (n + 1), polynomial(n2), order 2
568 3: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 3
569 1, 4, 64, 256, 256), # b1 * (n + 1), polynomial(n2), order 3
570 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 4
571 25, 64, 256, 4096, 16384, 16384) # PYCHOK b1 * (n + 1), polynomial(n2), order 4
572}
573_BetCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
574 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4
575 -4, 555, -960, 720, 1440, # Bet[1]/n^1, polynomial(n), order 3
576 -437, 96, 30, 1440, # Bet[2]/n^2, polynomial(n), order 2
577 -148, 119, 3360, # Bet[3]/n^3, polynomial(n), order 1
578 4397, 161280), # PYCHOK Bet[4]/n^4, polynomial(n), order 0, count = 14
579 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5
580 -3645, -64, 8880, -15360, 11520, 23040, # Bet[1]/n^1, polynomial(n), order 4
581 4416, -3059, 672, 210, 10080, # PYCHOK Bet[2]/n^2, polynomial(n), order 3
582 -627, -592, 476, 13440, # Bet[3]/n^3, polynomial(n), order 2
583 -3520, 4397, 161280, # Bet[4]/n^4, polynomial(n), order 1
584 4583, 161280), # PYCHOK Bet[5]/n^5, polynomial(n), order 0, count = 20
585 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6
586 384796, -382725, -6720, 932400, -1612800, 1209600, 2419200, # Bet[1]/n^1, polynomial(n), order 5
587 -1118711, 1695744, -1174656, 258048, 80640, 3870720, # PYCHOK Bet[2]/n^2, polynomial(n), order 4
588 22276, -16929, -15984, 12852, 362880, # Bet[3]/n^3, polynomial(n), order 3
589 -830251, -158400, 197865, 7257600, # PYCHOK Bet[4]/n^4, polynomial(n), order 2
590 -435388, 453717, 15966720, # PYCHOK Bet[5]/n^5, polynomial(n), order 1
591 20648693, 638668800), # Bet[6]/n^6, polynomial(n), order 0, count = 27
592 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7
593 -5406467, 6156736, -6123600, -107520, 14918400, -25804800, 19353600, 38707200, # Bet[1]/n^1, polynomial(n), order 6
594 829456, -5593555, 8478720, -5873280, 1290240, 403200, 19353600, # PYCHOK Bet[2]/n^2, polynomial(n), order 5
595 9261899, 3564160, -2708640, -2557440, 2056320, 58060800, # PYCHOK Bet[3]/n^3, polynomial(n), order 4
596 14928352, -9132761, -1742400, 2176515, 79833600, # PYCHOK Bet[4]/n^4, polynomial(n), order 3
597 -8005831, -1741552, 1814868, 63866880, # Bet[5]/n^5, polynomial(n), order 2
598 -261810608, 268433009, 8302694400, # Bet[6]/n^6, polynomial(n), order 1
599 219941297, 5535129600), # PYCHOK Bet[7]/n^7, polynomial(n), order 0, count = 35
600 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8
601 31777436, -37845269, 43097152, -42865200, -752640, 104428800, -180633600, 135475200, 270950400, # Bet[1]/n^1, polynomial(n), order 7
602 24749483, 14930208, -100683990, 152616960, -105719040, 23224320, 7257600, 348364800, # Bet[2]/n^2, polynomial(n), order 6
603 -232468668, 101880889, 39205760, -29795040, -28131840, 22619520, 638668800, # PYCHOK Bet[3]/n^3, polynomial(n), order 5
604 324154477, 1433121792, -876745056, -167270400, 208945440, 7664025600, # Bet[4]/n^4, polynomial(n), order 4
605 457888660, -312227409, -67920528, 70779852, 2490808320, # Bet[5]/n^5, polynomial(n), order 3
606 -19841813847, -3665348512, 3758062126, 116237721600, # PYCHOK Bet[6]/n^6, polynomial(n), order 2
607 -1989295244, 1979471673, 49816166400, # PYCHOK Bet[7]/n^7, polynomial(n), order 1
608 191773887257, 3719607091200) # Bet[8]/n^8, polynomial(n), order 0, count = 44
609}
611assert set(_AlpCoeffs.keys()) == set(_BetCoeffs.keys())
613if __name__ == '__main__':
615 from pygeodesy.internals import _usage
616 from sys import argv, exit as _exit
618 _exit(_usage(*argv).replace('.ktm', '.etm -series'))
620# **) MIT License
621#
622# Copyright (C) 2022-2024 -- mrJean1 at Gmail -- All Rights Reserved.
623#
624# Permission is hereby granted, free of charge, to any person obtaining a
625# copy of this software and associated documentation files (the "Software"),
626# to deal in the Software without restriction, including without limitation
627# the rights to use, copy, modify, merge, publish, distribute, sublicense,
628# and/or sell copies of the Software, and to permit persons to whom the
629# Software is furnished to do so, subject to the following conditions:
630#
631# The above copyright notice and this permission notice shall be included
632# in all copies or substantial portions of the Software.
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