Coverage for pygeodesy/ktm.py: 95%
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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>} (2008-2024)
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, atan2, _loneg, sincos2, sincos2d_
65from cmath import polar
66from math import asinh, cos, cosh, degrees, fabs, sin, sinh, sqrt, tanh
68__all__ = _ALL_LAZY.ktm
69__version__ = '24.11.24'
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 if lon0:
152 self.lon0 = lon0
153 self.k0 = k0
154 if raiser:
155 self.raiser = True
157 @Property_RO
158 def _Alp(self):
159 return _Xs(_AlpCoeffs, self.TMorder, self.ellipsoid)
161 @Property_RO
162 def _b1(self):
163 n = self.ellipsoid.n
164 if n: # isEllipsoidal
165 m = self.TMorder // 2
166 B1 = _B1Coeffs[m]
167 m += 1
168 b1 = _polynomial(n**2, B1, 0, m) / (B1[m] * (n + _1_0))
169 else: # isSpherical
170 b1 = _1_0 # B1[m - 1] / B1[m1] == 1, always
171 return b1
173 @Property_RO
174 def _Bet(self):
175 C = _Xs(_BetCoeffs, self.TMorder, self.ellipsoid)
176 return tuple(map(neg, C)) if self.f else C # negated if isEllipsoidal
178 @property
179 def datum(self):
180 '''Get this rhumb's datum (L{Datum}).
181 '''
182 return self._datum
184 @datum.setter # PYCHOK setter!
185 def datum(self, datum):
186 '''Set this rhumb's datum (L{Datum}).
187 '''
188 _xinstanceof(Datum, datum=datum)
189 if self._datum != datum:
190 _update_all(self)
191 self._datum = datum
193 @Property
194 def ellipsoid(self):
195 '''Get the ellipsoid (L{Ellipsoid}).
196 '''
197 return self.datum.ellipsoid
199 @ellipsoid.setter # PYCHOK setter!
200 def ellipsoid(self, a_earth_f):
201 '''Set this rhumb's ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum},
202 L{a_f2Tuple} or 2-tuple C{(a, f)}).
203 '''
204 self.datum = _spherical_datum(a_earth_f, Error=KTMError)
206 @Property_RO
207 def equatoradius(self):
208 '''Get the C{ellipsoid}'s equatorial radius, semi-axis (C{meter}).
209 '''
210 return self.ellipsoid.a
212 a = equatoradius
214 @Property_RO
215 def flattening(self):
216 '''Get the C{ellipsoid}'s flattening (C{scalar}).
217 '''
218 return self.ellipsoid.f
220 f = flattening
222 def forward(self, lat, lon, lon0=None, **name):
223 '''Forward projection, from geographic to transverse Mercator.
225 @arg lat: Latitude of point (C{degrees90}).
226 @arg lon: Longitude of point (C{degrees180}).
227 @arg lon0: Central meridian of the projection (C{degrees180}).
228 @kwarg name: Optional C{B{name}=NN} (C{str}).
230 @return: L{Forward4Tuple}C{(easting, northing, gamma, scale)} with
231 C{easting} and C{northing} in C{meter}, unfalsed, the
232 meridian convergence C{gamma} at point in C{degrees180}
233 and the C{scale} of projection at point C{scalar}. Any
234 value may be C{NAN}, C{NINF} or C{INF} for singularities.
236 @raise KTMError: For singularities, iff property C{raiser} is C{True}.
237 '''
238 lat, _lat = _unsigned2(_fix90(lat - self._lat0))
239 lon, _ = _diff182((self.lon0 if lon0 is None else lon0), lon)
240 lon, _lon = _unsigned2(lon)
241 backside = lon > 90
242 if backside: # PYCHOK no cover
243 lon = _loneg(lon)
244 if lat == 0:
245 _lat = True
247 sphi, cphi, slam, clam = sincos2d_(lat, lon)
248 E = self.ellipsoid
249 if cphi and lat != 90:
250 t = sphi / cphi
251 tp = E.es_taupf(t)
252 h = hypot(tp, clam)
253 if h:
254 xip = atan2(tp, clam)
255 etap = asinh(slam / h) # atanh(sin(lam) / cosh(psi))
256 g = _atan2d(slam * tp, clam * hypot1(tp)) # Krueger p 22 (44)
257 k = sqrt(cphi**2 * E.e2 + E.e21) * hypot1(t) / h
258 elif self.raiser:
259 raise KTMError(lat=lat, lon=lon, lon0=lon0, txt=_singular_)
260 else: # PYCHOK no cover
261 xip, etap = _0_0, _copysignINF(slam)
262 g, k = copysign0(_90_0, slam), INF
263 else: # PYCHOK no cover
264 xip, etap = PI_2, _0_0
265 g, k = lon, E.es_c
266 y, x, d, t = _Cyxgk4(E, xip, etap, self._Alp)
267 g -= d
268 k *= t * self._k0_b1
270 if backside: # PYCHOK no cover
271 y, g = (PI - y), _loneg(g)
272 y *= self._k0_a1
273 x *= self._k0_a1
274 if _lat:
275 y, g = neg_(y, g)
276 if _lon:
277 x, g = neg_(x, g)
278 return Forward4Tuple(x, y, _norm180(g), k, name=self._name__(name))
280 @property_doc_(''' the central scale factor (C{float}).''')
281 def k0(self):
282 '''Get the central scale factor (C{float}), aka I{C{scale0}}.
283 '''
284 return self._k0 # aka scale0
286 @k0.setter # PYCHOK setter!
287 def k0(self, k0):
288 '''Set the central scale factor (C{float}), aka I{C{scale0}}.
290 @raise KTMError: Invalid B{C{k0}}.
291 '''
292 k0 = Scalar_(k0=k0, Error=KTMError, low=_TOL_10, high=_1_0)
293 if self._k0 != k0: # PYCHOK no cover
294 KTransverseMercator._k0_a1._update(self) # redo ._k0_a1
295 KTransverseMercator._k0_b1._update(self) # redo ._k0_b1
296 self._k0 = k0
298 @Property_RO
299 def _k0_a1(self):
300 '''(INTERNAL) Cache C{k0 * _b1 * equatoradius}.
301 '''
302 return self._k0_b1 * self.equatoradius
304 @Property_RO
305 def _k0_b1(self):
306 '''(INTERNAL) Cache C{k0 * _b1}.
307 '''
308 return self.k0 * self._b1
310 @property_doc_(''' the central meridian (C{degrees180}).''')
311 def lon0(self):
312 '''Get the central meridian (C{degrees180}).
313 '''
314 return self._lon0
316 @lon0.setter # PYCHOK setter!
317 def lon0(self, lon0):
318 '''Set the central meridian (C{degrees180}).
320 @raise KTMError: Invalid B{C{lon0}}.
321 '''
322 self._lon0 = _norm180(Degrees(lon0=lon0, Error=KTMError))
324 @property_doc_(''' raise a L{KTMError} for C{forward} singularities (C{bool}).''')
325 def raiser(self):
326 '''Get the error setting (C{bool}).
327 '''
328 return self._raiser
330 @raiser.setter # PYCHOK setter!
331 def raiser(self, raiser):
332 '''Set the error setting (C{bool}), to C{True} to throw a L{KTMError}
333 for C{forward} singularities.
334 '''
335 self._raiser = bool(raiser)
337 def reset(self, lat0, lon0):
338 '''Set the central parallel and meridian.
340 @arg lat0: Latitude of the central parallel (C{degrees90}).
341 @arg lon0: Longitude of the central parallel (C{degrees180}).
343 @return: 2-Tuple C{(lat0, lon0)} with the previous central
344 parallel and meridian.
346 @raise KTMError: Invalid B{C{lat0}} or B{C{lon0}}.
347 '''
348 t = self._lat0, self.lon0
349 self._lat0 = _fix90(Degrees(lat0=lat0, Error=KTMError))
350 self. lon0 = lon0
351 return t
353 def reverse(self, x, y, lon0=None, **name):
354 '''Reverse projection, from transverse Mercator to geographic.
356 @arg x: Easting of point (C{meter}).
357 @arg y: Northing of point (C{meter}).
358 @arg lon0: Central meridian of the projection (C{degrees180}).
359 @kwarg name: Optional C{B{name}=NN} (C{str}).
361 @return: L{Reverse4Tuple}C{(lat, lon, gamma, scale)} with
362 C{lat}- and C{lon}gitude in C{degrees}, I{unfalsed}.
363 '''
364 eta, _lon = _unsigned2(x / self._k0_a1)
365 xi, _lat = _unsigned2(y / self._k0_a1)
366 backside = xi > PI_2
367 if backside: # PYCHOK no cover
368 xi = PI - xi
370 E = self.ellipsoid
371 xip, etap, g, k = _Cyxgk4(E, xi, eta, self._Bet)
372 t = self._k0_b1
373 k = (t / k) if k else _copysignINF(t) # _over(t, k)
374 h, c = sinh(etap), cos(xip)
375 if c > 0:
376 r = hypot(h, c)
377 else: # PYCHOK no cover
378 r = fabs(h)
379 c = _0_0
380 if r:
381 lon = _atan2d(h, c) # Krueger p 17 (25)
382 s = sin(xip) # Newton for tau
383 t = E.es_tauf(s / r)
384 lat = atan1d(t)
385 g += _atan2d(s * tanh(etap), c) # Krueger p 19 (31)
386 k *= sqrt(E.e2 / (t**2 + _1_0) + E.e21) * hypot1(t) * r
387 else: # PYCHOK no cover
388 lat = _90_0
389 lon = _0_0
390 k *= E.es_c
392 if backside: # PYCHOK no cover
393 lon, g = _loneg(lon), _loneg(g)
394 if _lat:
395 lat, g = neg_(lat, g)
396 if _lon:
397 lon, g = neg_(lon, g)
398 lat += self._lat0
399 lon += self._lon0 if lon0 is None else _norm180(lon0)
400 return Reverse4Tuple(lat, _norm180(lon), _norm180(g), k,
401 name=self._name__(name))
403 @Property
404 def TMorder(self):
405 '''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
406 '''
407 return self._mTM
409 @TMorder.setter # PYCHOK setter!
410 def TMorder(self, order):
411 '''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
412 '''
413 m = _Xorder(_AlpCoeffs, KTMError, TMorder=order)
414 if self._mTM != m:
415 _update_all(self)
416 self._mTM = m
418 def toStr(self, **kwds):
419 '''Return a C{str} representation.
421 @arg kwds: Optional, overriding keyword arguments.
422 '''
423 d = dict(ellipsoid=self.ellipsoid, k0=self.k0, TMorder=self.TMorder)
424 if self.name: # PYCHOK no cover
425 d.update(name=self.name)
426 return _COMMASPACE_.join(pairs(d, **kwds))
429def _cfma(a, b0, b1, Cn):
430 '''(INTERNAL) Complex fused-multiply-add M{a * b0 - b1 + Cn} with
431 complex C{a}, C{b0} and C{b1} and scalar C{Cn}.
433 @see: CPython function U{_Py_c_prod<https://GitHub.com/python/
434 cpython/blob/main/Objects/complexobject.c>}.
436 @note: Python function C{cmath.fsum} no longer exists.
437 '''
438 r = fdot_(a.real, b0.real, -a.imag, b0.imag, -b1.real, _1_0, start=Cn)
439 j = fdot_(a.real, b0.imag, a.imag, b0.real, -b1.imag, _1_0)
440 return complex(r, j)
443def _Cyxgk4(E, xi_, eta_, C):
444 '''(INTERNAL) Complex Clenshaw summation with C{B{C}=._Alp}
445 or C{B{C}=-._Bet}.
446 '''
447 x = complex(xi_, eta_)
448 if E.f: # isEllipsoidal
449 s, c = sincos2( xi_ * 2)
450 sh, ch = _sinhcosh2(eta_ * 2)
451 n = -s
452 s = complex(s * ch, c * sh) # sin(zeta * 2)
453 c = complex(c * ch, n * sh) # cos(zeta * 2)
454 a = c * 2 # cos(zeta * 2) * 2
456 y0 = y1 = \
457 z0 = z1 = complex(0) # 0+0j
458 n = len(C) - 1 # == .TMorder
459 if isodd(n):
460 Cn = C[n]
461 y0 = complex(Cn) # +0j
462 z0 = complex(Cn * (n * 2))
463 n -= 1
464 while n > 0:
465 Cn = C[n]
466 y1 = _cfma(a, y0, y1, Cn)
467 z1 = _cfma(a, z0, z1, Cn * (n * 2))
468 n -= 1
469 Cn = C[n]
470 y0 = _cfma(a, y1, y0, Cn)
471 z0 = _cfma(a, z1, z0, Cn * (n * 2))
472 n -= 1
473 # assert n == 0
474 x = _cfma(s, y0, -x, _0_0)
475 c = _cfma(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 i = (m + 2) if RA else 0
508 _p = _polynomial
509 for r in _reverange(m): # [m-1 ... 0]
510 j = i + r + 1
511 X.append(_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) 2022-2025 -- 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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