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__ = '24.07.16'
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}, L{a_f2Tuple},
123 L{Datum}, 2-tuple (C{a, f})) or the equatorial radius (C{meter}).
124 @kwarg f: The ellipsoid's flattening (C{scalar}), required if B{C{a_earth}} is
125 is C{meter}, ignored otherwise.
126 @kwarg lon0: The central meridian (C{degrees180}).
127 @kwarg k0: Central scale factor (C{scalar}).
128 @kwarg raiser: If C{True}, throw a L{KTMError} for C{forward} singularities (C{bool}).
129 @kwarg TMorder_name: Optional C{B{name}=NN} (C{str}) and optional keyword argument
130 C{B{TMorder}=6} for the order of this L{KTransverseMercator}, see
131 property C{TMorder}.
133 @raise KTMError: Invalid B{C{a_earth}}, B{C{f}} or B{C{TMorder}}.
134 '''
135 if TMorder_name:
136 M = self._mTM
137 m, name = _xkwds_pop2(TMorder_name, TMorder=M)
138 if m != M:
139 self.TMorder = m
140 if name:
141 self.name = name
143 if f is not None:
144 self.ellipsoid = a_earth, f
145 elif a_earth in (_EWGS84, _WGS84, None):
146 pass
147 elif isinstance(a_earth, Datum):
148 self.datum = a_earth
149 else:
150 self.ellipsoid = a_earth
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 _cma(a, b0, b1, Cn):
430 '''(INTERNAL) Compute complex M{a * b0 - b1 + Cn} with complex
431 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} is no longer available.
437 '''
438 r = fsum1f_(a.real * b0.real, -a.imag * b0.imag, -b1.real, Cn)
439 j = fsum1f_(a.real * b0.imag, a.imag * b0.real, -b1.imag)
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 _c = _cma
465 while n > 0:
466 Cn = C[n]
467 y1 = _c(a, y0, y1, Cn)
468 z1 = _c(a, z0, z1, Cn * (n * 2))
469 n -= 1
470 Cn = C[n]
471 y0 = _c(a, y1, y0, Cn)
472 z0 = _c(a, z1, z0, Cn * (n * 2))
473 n -= 1
474 # assert n == 0
475 x = _c(s, y0, -x, _0_0)
476 c = _c(c, z0, z1, _1_0)
478 # Gauss-Schreiber to Gauss-Krueger TM
479 # C{cmath.polar} handles INF, NAN, etc.
480 k, g = polar(c)
481 g = degrees(g)
482 else: # E.isSpherical
483 g, k = _0_0, _1_0
485 return x.real, x.imag, g, k
488def _sinhcosh2(x):
489 '''(INTERNAL) Like C{sincos2}.
490 '''
491 return sinh(x), cosh(x)
494def _Xs(_Coeffs, m, E, RA=False): # in .rhumb.ekx
495 '''(INTERNAL) Compute the C{A}, C{B} or C{RA} terms of order
496 B{C{m}} for I{Krüger} series and I{rhumb.ekx._sincosSeries},
497 return a tuple with C{B{m} + 1} terms C{X}, C{X[0]==0}.
498 '''
499 Cs = _Coeffs[m]
500 assert len(Cs) == (((m + 1) * (m + 4)) if RA else
501 ((m + 3) * m)) // 2
502 n = n_ = E.n
503 if n: # isEllipsoidal
504 X = [0] # X[0] never used, it's just an integration
505 # constant, it cancels when evaluating a definite
506 # integral. Don't bother computing it, it is unused
507 # in C{_Cyxgk4} above and C{rhumb.ekx._sincosSeries}.
508 _X, _p = X.append, _polynomial
509 i = (m + 2) if RA else 0
510 for r in _reverange(m): # [m-1 ... 0]
511 j = i + r + 1
512 _X(_p(n, Cs, i, j) * n_ / Cs[j])
513 i = j + 1
514 n_ *= n
515 X = tuple(X)
516 else: # isSpherical
517 X = _0_0s(m + 1)
518 return X
521# _Alp- and _BetCoeffs in .rhumb.ekx, .rhumb.bases
522_AlpCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
523 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4
524 164, 225, -480, 360, 720, # Alp[1]/n^1, polynomial(n), order 3
525 557, -864, 390, 1440, # Alp[2]/n^2, polynomial(n), order 2
526 -1236, 427, 1680, # PYCHOK Alp[3]/n^3, polynomial(n), order 1
527 49561, 161280), # Alp[4]/n^4, polynomial(n), order 0, count = 14
528 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5
529 -635, 328, 450, -960, 720, 1440, # Alp[1]/n^1, polynomial(n), order 4
530 4496, 3899, -6048, 2730, 10080, # PYCHOK Alp[2]/n^2, polynomial(n), order 3
531 15061, -19776, 6832, 26880, # PYCHOK Alp[3]/n^3, polynomial(n), order 2
532 -171840, 49561, 161280, # Alp[4]/n^4, polynomial(n), order 1
533 34729, 80640), # PYCHOK Alp[5]/n^5, polynomial(n), order 0, count = 20
534 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6
535 31564, -66675, 34440, 47250, -100800, 75600, 151200, # Alp[1]/n^1, polynomial(n), order 5
536 -1983433, 863232, 748608, -1161216, 524160, 1935360, # PYCHOK Alp[2]/n^2, polynomial(n), order 4
537 670412, 406647, -533952, 184464, 725760, # Alp[3]/n^3, polynomial(n), order 3
538 6601661, -7732800, 2230245, 7257600, # Alp[4]/n^4, polynomial(n), order 2
539 -13675556, 3438171, 7983360, # PYCHOK Alp[5]/n^5, polynomial(n), order 1
540 212378941, 319334400), # Alp[6]/n^6, polynomial(n), order 0, count = 27
541 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7
542 1804025, 2020096, -4267200, 2204160, 3024000, -6451200, 4838400, 9676800, # Alp[1]/n^1, polynomial(n), order 6
543 4626384, -9917165, 4316160, 3743040, -5806080, 2620800, 9676800, # Alp[2]/n^2, polynomial(n), order 5
544 -67102379, 26816480, 16265880, -21358080, 7378560, 29030400, # PYCHOK Alp[3]/n^3, polynomial(n), order 4
545 155912000, 72618271, -85060800, 24532695, 79833600, # Alp[4]/n^4, polynomial(n), order 3
546 102508609, -109404448, 27505368, 63866880, # Alp[5]/n^5, polynomial(n), order 2
547 -12282192400, 2760926233, 4151347200, # PYCHOK Alp[6]/n^6, polynomial(n), order 1
548 1522256789, 1383782400), # Alp[7]/n^7, polynomial(n), order 0, count = 35
549 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8
550 -75900428, 37884525, 42422016, -89611200, 46287360, 63504000, -135475200, 101606400, 203212800, # Alp[1]/n^1, polynomial(n), order 7
551 148003883, 83274912, -178508970, 77690880, 67374720, -104509440, 47174400, 174182400, # PYCHOK Alp[2]/n^2, polynomial(n), order 6
552 318729724, -738126169, 294981280, 178924680, -234938880, 81164160, 319334400, # PYCHOK Alp[3]/n^3, polynomial(n), order 5
553 -40176129013, 14967552000, 6971354016, -8165836800, 2355138720, 7664025600, # Alp[4]/n^4, polynomial(n), order 4
554 10421654396, 3997835751, -4266773472, 1072709352, 2490808320, # PYCHOK Alp[5]/n^5, polynomial(n), order 3
555 175214326799, -171950693600, 38652967262, 58118860800, # PYCHOK Alp[6]/n^6, polynomial(n), order 2
556 -67039739596, 13700311101, 12454041600, # PYCHOK Alp[7]/n^7, polynomial(n), order 1
557 1424729850961, 743921418240) # PYCHOK Alp[8]/n^8, polynomial(n), order 0, count = 44
558}
559_B1Coeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
560 2: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 2
561 1, 16, 64, 64), # b1 * (n + 1), polynomial(n2), order 2
562 3: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 3
563 1, 4, 64, 256, 256), # b1 * (n + 1), polynomial(n2), order 3
564 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 4
565 25, 64, 256, 4096, 16384, 16384) # PYCHOK b1 * (n + 1), polynomial(n2), order 4
566}
567_BetCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
568 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4
569 -4, 555, -960, 720, 1440, # Bet[1]/n^1, polynomial(n), order 3
570 -437, 96, 30, 1440, # Bet[2]/n^2, polynomial(n), order 2
571 -148, 119, 3360, # Bet[3]/n^3, polynomial(n), order 1
572 4397, 161280), # PYCHOK Bet[4]/n^4, polynomial(n), order 0, count = 14
573 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5
574 -3645, -64, 8880, -15360, 11520, 23040, # Bet[1]/n^1, polynomial(n), order 4
575 4416, -3059, 672, 210, 10080, # PYCHOK Bet[2]/n^2, polynomial(n), order 3
576 -627, -592, 476, 13440, # Bet[3]/n^3, polynomial(n), order 2
577 -3520, 4397, 161280, # Bet[4]/n^4, polynomial(n), order 1
578 4583, 161280), # PYCHOK Bet[5]/n^5, polynomial(n), order 0, count = 20
579 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6
580 384796, -382725, -6720, 932400, -1612800, 1209600, 2419200, # Bet[1]/n^1, polynomial(n), order 5
581 -1118711, 1695744, -1174656, 258048, 80640, 3870720, # PYCHOK Bet[2]/n^2, polynomial(n), order 4
582 22276, -16929, -15984, 12852, 362880, # Bet[3]/n^3, polynomial(n), order 3
583 -830251, -158400, 197865, 7257600, # PYCHOK Bet[4]/n^4, polynomial(n), order 2
584 -435388, 453717, 15966720, # PYCHOK Bet[5]/n^5, polynomial(n), order 1
585 20648693, 638668800), # Bet[6]/n^6, polynomial(n), order 0, count = 27
586 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7
587 -5406467, 6156736, -6123600, -107520, 14918400, -25804800, 19353600, 38707200, # Bet[1]/n^1, polynomial(n), order 6
588 829456, -5593555, 8478720, -5873280, 1290240, 403200, 19353600, # PYCHOK Bet[2]/n^2, polynomial(n), order 5
589 9261899, 3564160, -2708640, -2557440, 2056320, 58060800, # PYCHOK Bet[3]/n^3, polynomial(n), order 4
590 14928352, -9132761, -1742400, 2176515, 79833600, # PYCHOK Bet[4]/n^4, polynomial(n), order 3
591 -8005831, -1741552, 1814868, 63866880, # Bet[5]/n^5, polynomial(n), order 2
592 -261810608, 268433009, 8302694400, # Bet[6]/n^6, polynomial(n), order 1
593 219941297, 5535129600), # PYCHOK Bet[7]/n^7, polynomial(n), order 0, count = 35
594 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8
595 31777436, -37845269, 43097152, -42865200, -752640, 104428800, -180633600, 135475200, 270950400, # Bet[1]/n^1, polynomial(n), order 7
596 24749483, 14930208, -100683990, 152616960, -105719040, 23224320, 7257600, 348364800, # Bet[2]/n^2, polynomial(n), order 6
597 -232468668, 101880889, 39205760, -29795040, -28131840, 22619520, 638668800, # PYCHOK Bet[3]/n^3, polynomial(n), order 5
598 324154477, 1433121792, -876745056, -167270400, 208945440, 7664025600, # Bet[4]/n^4, polynomial(n), order 4
599 457888660, -312227409, -67920528, 70779852, 2490808320, # Bet[5]/n^5, polynomial(n), order 3
600 -19841813847, -3665348512, 3758062126, 116237721600, # PYCHOK Bet[6]/n^6, polynomial(n), order 2
601 -1989295244, 1979471673, 49816166400, # PYCHOK Bet[7]/n^7, polynomial(n), order 1
602 191773887257, 3719607091200) # Bet[8]/n^8, polynomial(n), order 0, count = 44
603}
605assert set(_AlpCoeffs.keys()) == set(_BetCoeffs.keys())
607if __name__ == '__main__':
609 from pygeodesy.internals import _usage
610 from sys import argv, exit as _exit
612 _exit(_usage(*argv).replace('.ktm', '.etm -series'))
614# **) MIT License
615#
616# Copyright (C) 2022-2024 -- mrJean1 at Gmail -- All Rights Reserved.
617#
618# Permission is hereby granted, free of charge, to any person obtaining a
619# copy of this software and associated documentation files (the "Software"),
620# to deal in the Software without restriction, including without limitation
621# the rights to use, copy, modify, merge, publish, distribute, sublicense,
622# and/or sell copies of the Software, and to permit persons to whom the
623# Software is furnished to do so, subject to the following conditions:
624#
625# The above copyright notice and this permission notice shall be included
626# in all copies or substantial portions of the Software.
627#
628# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
629# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
630# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
631# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
632# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
633# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
634# OTHER DEALINGS IN THE SOFTWARE.