Coverage for pygeodesy/ktm.py: 98%
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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_, _reverange
46from pygeodesy.constants import INF, _K0_UTM, PI, PI_2, _0_0s, _0_0, \
47 _1_0, _90_0, _copysignINF
48# from pygeodesy.datums import _spherical_datum # _MODS
49# from pygeodesy.ellipsoids import _EWGS84 # from .karney
50from pygeodesy.errors import _ValueError, _xkwds_get, _Xorder
51from pygeodesy.fmath import hypot, hypot1
52from pygeodesy.fsums import fsum1f_
53from pygeodesy.interns import NN, _COMMASPACE_, _singular_
54from pygeodesy.karney import _atan2d, _diff182, _fix90, _polynomial, \
55 _norm180, _unsigned2, _EWGS84, _NamedBase
56from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS, _pairs
57# from pygeodesy.named import _NamedBase # from .karney
58from pygeodesy.namedTuples import Forward4Tuple, Reverse4Tuple
59from pygeodesy.props import property_doc_, Property, Property_RO, \
60 _update_all
61# from pygeodesy.streprs import pairs as _pairs # from .lazily
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__ = '23.09.28'
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 _E = _EWGS84
111 _k0 = _K0_UTM # central scale factor
112 _lon0 = _0_0 # central meridian
113 _mTM = 6
114 _raiser = False # throw Error
116 def __init__(self, a_earth=_EWGS84, f=None, lon0=0, k0=_K0_UTM, name=NN,
117 raiser=False, **TMorder):
118 '''New L{KTransverseMercator}.
120 @kwarg a_earth: This rhumb's earth (L{Ellipsoid}, L{Ellipsoid2},
121 L{a_f2Tuple}, L{Datum}, 2-tuple (C{a, f})) or the
122 equatorial radius (C{scalar}, C{meter}).
123 @kwarg f: The ellipsoid's flattening (C{scalar}), iff B{C{a_earth}} is
124 a C{scalar}, ignored otherwise.
125 @kwarg lon0: The central meridian (C{degrees180}).
126 @kwarg k0: Central scale factor (C{scalar}).
127 @kwarg name: Optional name (C{str}).
128 @kwarg raiser: If C{True}, throw a L{KTMError} for C{forward}
129 singularities (C{bool}).
130 @kwarg TMorder: Keyword argument B{C{TMorder}}, see property C{TMorder}.
132 @raise KTMError: Invalid B{C{a_earth}}, B{C{f}} or B{C{TMorder}}.
133 '''
134 if f is not None:
135 self.ellipsoid = a_earth, f
136 elif a_earth not in (_EWGS84, None):
137 self.ellipsoid = a_earth
138 self.lon0 = lon0
139 self.k0 = k0
140 if name: # PYCHOK no cover
141 self.name = name
142 if raiser:
143 self.raiser = True
144 if TMorder:
145 self.TMorder = _xkwds_get(TMorder, TMorder=self._mTM)
147 @Property_RO
148 def _Alp(self):
149 return _Xs(_AlpCoeffs, self.TMorder, self.ellipsoid)
151 @Property_RO
152 def _b1(self):
153 n = self.ellipsoid.n
154 if n: # isEllipsoidal
155 m = self.TMorder // 2
156 B1 = _B1Coeffs[m]
157 m += 1
158 b1 = _polynomial(n**2, B1, 0, m) / (B1[m] * (n + _1_0))
159 else: # isSpherical
160 b1 = _1_0 # B1[m - 1] / B1[m1] == 1, always
161 return b1
163 @Property_RO
164 def _Bet(self):
165 C = _Xs(_BetCoeffs, self.TMorder, self.ellipsoid)
166 return tuple(map(neg, C)) if self.f else C # negated if isEllipsoidal
168 @Property
169 def ellipsoid(self):
170 '''Get the ellipsoid (L{Ellipsoid}).
171 '''
172 return self._E
174 @ellipsoid.setter # PYCHOK setter!
175 def ellipsoid(self, a_earth_f):
176 '''Set this rhumb's ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum},
177 L{a_f2Tuple} or 2-tuple C{(a, f)}).
178 '''
179 E = _MODS.datums._spherical_datum(a_earth_f, Error=KTMError).ellipsoid
180 if self._E != E:
181 _update_all(self)
182 self._E = E
184 @Property_RO
185 def equatoradius(self):
186 '''Get the C{ellipsoid}'s equatorial radius, semi-axis (C{meter}).
187 '''
188 return self.ellipsoid.a
190 a = equatoradius
192 @Property_RO
193 def flattening(self):
194 '''Get the C{ellipsoid}'s flattening (C{scalar}).
195 '''
196 return self.ellipsoid.f
198 f = flattening
200 def forward(self, lat, lon, lon0=None, name=NN):
201 '''Forward projection, from geographic to transverse Mercator.
203 @arg lat: Latitude of point (C{degrees90}).
204 @arg lon: Longitude of point (C{degrees180}).
205 @arg lon0: Central meridian of the projection (C{degrees180}).
206 @kwarg name: Optional name (C{str}).
208 @return: L{Forward4Tuple}C{(easting, northing, gamma, scale)}
209 with C{easting} and C{northing} in C{meter}, unfalsed, the
210 meridian convergence C{gamma} at point in C{degrees180}
211 and the C{scale} of projection at point C{scalar}. Any
212 value may be C{NAN}, C{NINF} or C{INF} for singularities.
214 @raise KTMError: For singularities, iff property C{raiser} is
215 C{True}.
216 '''
217 lat, _lat = _unsigned2(_fix90(lat))
218 lon, _ = _diff182((self.lon0 if lon0 is None else lon0), lon)
219 lon, _lon = _unsigned2(lon)
220 backside = lon > 90
221 if backside: # PYCHOK no cover
222 lon = _loneg(lon)
223 if lat == 0:
224 _lat = True
226 sphi, cphi, slam, clam = sincos2d_(lat, lon)
227 E = self.ellipsoid
228 if cphi and lat != 90:
229 t = sphi / cphi
230 tp = E.es_taupf(t)
231 h = hypot(tp, clam)
232 if h:
233 xip = atan2(tp, clam)
234 etap = asinh(slam / h) # atanh(sin(lam) / cosh(psi))
235 g = _atan2d(slam * tp, clam * hypot1(tp)) # Krueger p 22 (44)
236 k = sqrt(cphi**2 * E.e2 + E.e21) * hypot1(t) / h
237 elif self.raiser:
238 raise KTMError(lat=lat, lon=lon, lon0=lon0, txt=_singular_)
239 else: # PYCHOK no cover
240 xip, etap = _0_0, _copysignINF(slam)
241 g, k = copysign0(_90_0, slam), INF
242 else: # PYCHOK no cover
243 xip, etap = PI_2, _0_0
244 g, k = lon, E.es_c
245 y, x, d, t = _Cyxgk4(E, xip, etap, self._Alp)
246 g -= d
247 k *= t * self._k0_b1
249 if backside: # PYCHOK no cover
250 y, g = (PI - y), _loneg(g)
251 y *= self._k0_a1
252 x *= self._k0_a1
253 if _lat:
254 y, g = neg_(y, g)
255 if _lon:
256 x, g = neg_(x, g)
258 return Forward4Tuple(x, y, _norm180(g), k, name=name or self.name)
260 @property_doc_(''' the central scale factor (C{float}).''')
261 def k0(self):
262 '''Get the central scale factor (C{float}), aka I{C{scale0}}.
263 '''
264 return self._k0 # aka scale0
266 @k0.setter # PYCHOK setter!
267 def k0(self, k0):
268 '''Set the central scale factor (C{float}), aka I{C{scale0}}.
270 @raise KTMError: Invalid B{C{k0}}.
271 '''
272 k0 = Scalar_(k0=k0, Error=KTMError, low=_TOL_10, high=_1_0)
273 if self._k0 != k0: # PYCHOK no cover
274 KTransverseMercator._k0_a1._update(self) # redo ._k0_a1
275 KTransverseMercator._k0_b1._update(self) # redo ._k0_b1
276 self._k0 = k0
278 @Property_RO
279 def _k0_a1(self):
280 '''(INTERNAL) Cache C{k0 * _b1 * equatoradius}.
281 '''
282 return self._k0_b1 * self.equatoradius
284 @Property_RO
285 def _k0_b1(self):
286 '''(INTERNAL) Cache C{k0 * _b1}.
287 '''
288 return self.k0 * self._b1
290 @property_doc_(''' the central meridian (C{degrees180}).''')
291 def lon0(self):
292 '''Get the central meridian (C{degrees180}).
293 '''
294 return self._lon0
296 @lon0.setter # PYCHOK setter!
297 def lon0(self, lon0):
298 '''Set the central meridian (C{degrees180}).
300 @raise KTMError: Invalid B{C{lon0}}.
301 '''
302 self._lon0 = _norm180(Degrees(lon0=lon0, Error=KTMError))
304 @property_doc_(''' raise a L{KTMError} for C{forward} singularities (C{bool}).''')
305 def raiser(self):
306 '''Get the error setting (C{bool}).
307 '''
308 return self._raiser
310 @raiser.setter # PYCHOK setter!
311 def raiser(self, raiser):
312 '''Set the error setting (C{bool}), to C{True} to throw a L{KTMError}
313 for C{forward} singularities.
314 '''
315 self._raiser = bool(raiser)
317 def reverse(self, x, y, lon0=None, name=NN):
318 '''Reverse projection, from transverse Mercator to geographic.
320 @arg x: Easting of point (C{meter}).
321 @arg y: Northing of point (C{meter}).
322 @arg lon0: Central meridian of the projection (C{degrees180}).
324 @return: L{Reverse4Tuple}C{(lat, lon, gamma, scale)} with
325 C{lat}- and C{lon}gitude in C{degrees}, I{unfalsed}.
326 '''
327 eta, _lon = _unsigned2(x / self._k0_a1)
328 xi, _lat = _unsigned2(y / self._k0_a1)
329 backside = xi > PI_2
330 if backside: # PYCHOK no cover
331 xi = PI - xi
333 E = self.ellipsoid
334 xip, etap, g, k = _Cyxgk4(E, xi, eta, self._Bet)
335 t = self._k0_b1
336 k = (t / k) if k else _copysignINF(t) # _over(t, k)
337 h, c = sinh(etap), cos(xip)
338 if c > 0:
339 r = hypot(h, c)
340 else: # PYCHOK no cover
341 r = fabs(h)
342 c = _0_0
343 if r:
344 lon = _atan2d(h, c) # Krueger p 17 (25)
345 s = sin(xip) # Newton for tau
346 t = E.es_tauf(s / r)
347 lat = atan1d(t)
348 g += _atan2d(s * tanh(etap), c) # Krueger p 19 (31)
349 k *= sqrt(E.e2 / (t**2 + _1_0) + E.e21) * hypot1(t) * r
350 else: # PYCHOK no cover
351 lat, lon = _90_0, _0_0
352 k *= E.es_c
354 if backside: # PYCHOK no cover
355 lon, g = _loneg(lon), _loneg(g)
356 if _lat:
357 lat, g = neg_(lat, g)
358 if _lon:
359 lon, g = neg_(lon, g)
361 lon += self.lon0 if lon0 is None else _norm180(lon0)
362 return Reverse4Tuple(lat, _norm180(lon), _norm180(g), k,
363 name=name or self.name)
365 @Property
366 def TMorder(self):
367 '''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
368 '''
369 return self._mTM
371 @TMorder.setter # PYCHOK setter!
372 def TMorder(self, order):
373 '''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
374 '''
375 m = _Xorder(_AlpCoeffs, KTMError, TMorder=order)
376 if self._mTM != m:
377 _update_all(self)
378 self._mTM = m
380 def toStr(self, **kwds):
381 '''Return a C{str} representation.
383 @arg kwds: Optional, overriding keyword arguments.
384 '''
385 d = dict(ellipsoid=self.ellipsoid, k0=self.k0, TMorder=self.TMorder)
386 if self.name: # PYCHOK no cover
387 d.update(name=self.name)
388 return _COMMASPACE_.join(_pairs(d, **kwds))
391def _cma(a, b0, b1, Cn):
392 '''(INTERNAL) Compute complex M{a * b0 - b1 + Cn} with complex
393 C{a}, C{b0} and C{b1} and scalar C{Cn}.
395 @see: CPython function U{_Py_c_prod<https://GitHub.com/python/
396 cpython/blob/main/Objects/complexobject.c>}.
398 @note: Python function C{cmath.fsum} is no longer available,
399 but stil mentioned in Note 4 of the comments before
400 CPython function U{math_fsum<https://GitHub.com/python/
401 cpython/blob/main/Modules/mathmodule.c>}
402 '''
403 r = fsum1f_(a.real * b0.real, -a.imag * b0.imag, -b1.real, Cn)
404 j = fsum1f_(a.real * b0.imag, a.imag * b0.real, -b1.imag)
405 return complex(r, j)
408def _Cyxgk4(E, xi_, eta_, C):
409 '''(INTERNAL) Complex Clenshaw summation with C{B{C}=._Alp}
410 or C{B{C}=-._Bet}.
411 '''
412 x = complex(xi_, eta_)
413 if E.f: # isEllipsoidal
414 s, c = sincos2( xi_ * 2)
415 sh, ch = _sinhcosh2(eta_ * 2)
416 n = -s
417 s = complex(s * ch, c * sh) # sin(zeta * 2)
418 c = complex(c * ch, n * sh) # cos(zeta * 2)
419 a = c * 2 # cos(zeta * 2) * 2
421 y0 = y1 = \
422 z0 = z1 = complex(0) # 0+0j
423 n = len(C) - 1 # == .TMorder
424 if isodd(n):
425 Cn = C[n]
426 y0 = complex(Cn) # +0j
427 z0 = complex(Cn * (n * 2))
428 n -= 1
429 _c = _cma
430 while n > 0:
431 Cn = C[n]
432 y1 = _c(a, y0, y1, Cn)
433 z1 = _c(a, z0, z1, Cn * (n * 2))
434 n -= 1
435 Cn = C[n]
436 y0 = _c(a, y1, y0, Cn)
437 z0 = _c(a, z1, z0, Cn * (n * 2))
438 n -= 1
439 # assert n == 0
440 x = _c(s, y0, -x, _0_0)
441 c = _c(c, z0, z1, _1_0)
443 # Gauss-Schreiber to Gauss-Krueger TM
444 # C{cmath.polar} handles INF, NAN, etc.
445 k, g = polar(c)
446 g = degrees(g)
447 else: # E.isSpherical
448 g, k = _0_0, _1_0
450 return x.real, x.imag, g, k
453def _sinhcosh2(x):
454 '''(INTERNAL) Like C{sincos2}.
455 '''
456 return sinh(x), cosh(x)
459def _Xs(_Coeffs, m, E, RA=False): # in .rhumbx
460 '''(INTERNAL) Compute the C{A}, C{B} or C{RA} terms of order
461 B{C{m}} for I{Krüger} series and I{rhumbx._sincosSeries},
462 return a tuple with C{B{m} + 1} terms C{X}, C{X[0]==0}.
463 '''
464 Cs = _Coeffs[m]
465 assert len(Cs) == (((m + 1) * (m + 4)) if RA else
466 ((m + 3) * m)) // 2
467 n = n_ = E.n
468 if n: # isEllipsoidal
469 X = [0] # X[0] never used, it's just an integration
470 # constant, it cancels when evaluating a definite
471 # integral. Don't bother computing it, it is unused
472 # in C{_Cyxgk4} above and C{rhumbx._sincosSeries}.
473 _X, _p = X.append, _polynomial
474 i = (m + 2) if RA else 0
475 for r in _reverange(m): # [m-1 ... 0]
476 j = i + r + 1
477 _X(_p(n, Cs, i, j) * n_ / Cs[j])
478 i = j + 1
479 n_ *= n
480 X = tuple(X)
481 else: # isSpherical
482 X = _0_0s(m + 1)
483 return X
486# _Alp- and _BetCoeffs in .rhumbx, .rhumbBase
487_AlpCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
488 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4
489 164, 225, -480, 360, 720, # Alp[1]/n^1, polynomial(n), order 3
490 557, -864, 390, 1440, # Alp[2]/n^2, polynomial(n), order 2
491 -1236, 427, 1680, # PYCHOK Alp[3]/n^3, polynomial(n), order 1
492 49561, 161280), # Alp[4]/n^4, polynomial(n), order 0, count = 14
493 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5
494 -635, 328, 450, -960, 720, 1440, # Alp[1]/n^1, polynomial(n), order 4
495 4496, 3899, -6048, 2730, 10080, # PYCHOK Alp[2]/n^2, polynomial(n), order 3
496 15061, -19776, 6832, 26880, # PYCHOK Alp[3]/n^3, polynomial(n), order 2
497 -171840, 49561, 161280, # Alp[4]/n^4, polynomial(n), order 1
498 34729, 80640), # PYCHOK Alp[5]/n^5, polynomial(n), order 0, count = 20
499 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6
500 31564, -66675, 34440, 47250, -100800, 75600, 151200, # Alp[1]/n^1, polynomial(n), order 5
501 -1983433, 863232, 748608, -1161216, 524160, 1935360, # PYCHOK Alp[2]/n^2, polynomial(n), order 4
502 670412, 406647, -533952, 184464, 725760, # Alp[3]/n^3, polynomial(n), order 3
503 6601661, -7732800, 2230245, 7257600, # Alp[4]/n^4, polynomial(n), order 2
504 -13675556, 3438171, 7983360, # PYCHOK Alp[5]/n^5, polynomial(n), order 1
505 212378941, 319334400), # Alp[6]/n^6, polynomial(n), order 0, count = 27
506 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7
507 1804025, 2020096, -4267200, 2204160, 3024000, -6451200, 4838400, 9676800, # Alp[1]/n^1, polynomial(n), order 6
508 4626384, -9917165, 4316160, 3743040, -5806080, 2620800, 9676800, # Alp[2]/n^2, polynomial(n), order 5
509 -67102379, 26816480, 16265880, -21358080, 7378560, 29030400, # PYCHOK Alp[3]/n^3, polynomial(n), order 4
510 155912000, 72618271, -85060800, 24532695, 79833600, # Alp[4]/n^4, polynomial(n), order 3
511 102508609, -109404448, 27505368, 63866880, # Alp[5]/n^5, polynomial(n), order 2
512 -12282192400, 2760926233, 4151347200, # PYCHOK Alp[6]/n^6, polynomial(n), order 1
513 1522256789, 1383782400), # Alp[7]/n^7, polynomial(n), order 0, count = 35
514 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8
515 -75900428, 37884525, 42422016, -89611200, 46287360, 63504000, -135475200, 101606400, 203212800, # Alp[1]/n^1, polynomial(n), order 7
516 148003883, 83274912, -178508970, 77690880, 67374720, -104509440, 47174400, 174182400, # PYCHOK Alp[2]/n^2, polynomial(n), order 6
517 318729724, -738126169, 294981280, 178924680, -234938880, 81164160, 319334400, # PYCHOK Alp[3]/n^3, polynomial(n), order 5
518 -40176129013, 14967552000, 6971354016, -8165836800, 2355138720, 7664025600, # Alp[4]/n^4, polynomial(n), order 4
519 10421654396, 3997835751, -4266773472, 1072709352, 2490808320, # PYCHOK Alp[5]/n^5, polynomial(n), order 3
520 175214326799, -171950693600, 38652967262, 58118860800, # PYCHOK Alp[6]/n^6, polynomial(n), order 2
521 -67039739596, 13700311101, 12454041600, # PYCHOK Alp[7]/n^7, polynomial(n), order 1
522 1424729850961, 743921418240) # PYCHOK Alp[8]/n^8, polynomial(n), order 0, count = 44
523}
524_B1Coeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
525 2: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 2
526 1, 16, 64, 64), # b1 * (n + 1), polynomial(n2), order 2
527 3: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 3
528 1, 4, 64, 256, 256), # b1 * (n + 1), polynomial(n2), order 3
529 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 4
530 25, 64, 256, 4096, 16384, 16384) # PYCHOK b1 * (n + 1), polynomial(n2), order 4
531}
532_BetCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
533 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4
534 -4, 555, -960, 720, 1440, # Bet[1]/n^1, polynomial(n), order 3
535 -437, 96, 30, 1440, # Bet[2]/n^2, polynomial(n), order 2
536 -148, 119, 3360, # Bet[3]/n^3, polynomial(n), order 1
537 4397, 161280), # PYCHOK Bet[4]/n^4, polynomial(n), order 0, count = 14
538 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5
539 -3645, -64, 8880, -15360, 11520, 23040, # Bet[1]/n^1, polynomial(n), order 4
540 4416, -3059, 672, 210, 10080, # PYCHOK Bet[2]/n^2, polynomial(n), order 3
541 -627, -592, 476, 13440, # Bet[3]/n^3, polynomial(n), order 2
542 -3520, 4397, 161280, # Bet[4]/n^4, polynomial(n), order 1
543 4583, 161280), # PYCHOK Bet[5]/n^5, polynomial(n), order 0, count = 20
544 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6
545 384796, -382725, -6720, 932400, -1612800, 1209600, 2419200, # Bet[1]/n^1, polynomial(n), order 5
546 -1118711, 1695744, -1174656, 258048, 80640, 3870720, # PYCHOK Bet[2]/n^2, polynomial(n), order 4
547 22276, -16929, -15984, 12852, 362880, # Bet[3]/n^3, polynomial(n), order 3
548 -830251, -158400, 197865, 7257600, # PYCHOK Bet[4]/n^4, polynomial(n), order 2
549 -435388, 453717, 15966720, # PYCHOK Bet[5]/n^5, polynomial(n), order 1
550 20648693, 638668800), # Bet[6]/n^6, polynomial(n), order 0, count = 27
551 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7
552 -5406467, 6156736, -6123600, -107520, 14918400, -25804800, 19353600, 38707200, # Bet[1]/n^1, polynomial(n), order 6
553 829456, -5593555, 8478720, -5873280, 1290240, 403200, 19353600, # PYCHOK Bet[2]/n^2, polynomial(n), order 5
554 9261899, 3564160, -2708640, -2557440, 2056320, 58060800, # PYCHOK Bet[3]/n^3, polynomial(n), order 4
555 14928352, -9132761, -1742400, 2176515, 79833600, # PYCHOK Bet[4]/n^4, polynomial(n), order 3
556 -8005831, -1741552, 1814868, 63866880, # Bet[5]/n^5, polynomial(n), order 2
557 -261810608, 268433009, 8302694400, # Bet[6]/n^6, polynomial(n), order 1
558 219941297, 5535129600), # PYCHOK Bet[7]/n^7, polynomial(n), order 0, count = 35
559 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8
560 31777436, -37845269, 43097152, -42865200, -752640, 104428800, -180633600, 135475200, 270950400, # Bet[1]/n^1, polynomial(n), order 7
561 24749483, 14930208, -100683990, 152616960, -105719040, 23224320, 7257600, 348364800, # Bet[2]/n^2, polynomial(n), order 6
562 -232468668, 101880889, 39205760, -29795040, -28131840, 22619520, 638668800, # PYCHOK Bet[3]/n^3, polynomial(n), order 5
563 324154477, 1433121792, -876745056, -167270400, 208945440, 7664025600, # Bet[4]/n^4, polynomial(n), order 4
564 457888660, -312227409, -67920528, 70779852, 2490808320, # Bet[5]/n^5, polynomial(n), order 3
565 -19841813847, -3665348512, 3758062126, 116237721600, # PYCHOK Bet[6]/n^6, polynomial(n), order 2
566 -1989295244, 1979471673, 49816166400, # PYCHOK Bet[7]/n^7, polynomial(n), order 1
567 191773887257, 3719607091200) # Bet[8]/n^8, polynomial(n), order 0, count = 44
568}
570assert set(_AlpCoeffs.keys()) == set(_BetCoeffs.keys())
572if __name__ == '__main__':
574 from pygeodesy.interns import _usage
575 from sys import argv, exit as _exit
577 _exit(_usage(*argv).replace('.ktm', '.etm -series'))
579# **) MIT License
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581# Copyright (C) 2022-2023 -- mrJean1 at Gmail -- All Rights Reserved.
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585# to deal in the Software without restriction, including without limitation
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588# Software is furnished to do so, subject to the following conditions:
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