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, NINF, PI, PI_2, _0_0s, \
47 _0_0, _1_0, _90_0, _180_0
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 fsum1_, hypot, hypot1
52# from pygeodesy.fsums import fsum1_ # from .fmath
53from pygeodesy.interns import NN, _COMMASPACE_, _singular_
54from pygeodesy.karney import _atan2d, _diff182, _fix90, _NamedBase, \
55 _norm180, _polynomial, _unsigned2, _EWGS84
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 atand, sincos2, sincos2d_
65from cmath import phase
66from math import atan2, asinh, cos, cosh, degrees, fabs, sin, sinh, sqrt, tanh
68__all__ = _ALL_LAZY.ktm
69__version__ = '23.08.20'
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 isEllispoidal
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{float}).
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 = _180_0 - 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(E.e21 + E.e2 * cphi**2) * 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, (NINF if slam < 0 else INF)
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, t, z = self._yxgk4(xip, etap, self._Alp)
246 g -= t
247 k *= z * self._k0_b1
249 if backside: # PYCHOK no cover
250 y, g = (PI - y), (_180_0 - 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 xip, etap, g, k = self._yxgk4(xi, eta, self._Bet)
334 t = self._k0_b1
335 k = (t / k) if k else (NINF if t < 0 else INF)
336 h, c = sinh(etap), cos(xip)
337 if c > 0:
338 r = hypot(h, c)
339 else: # PYCHOK no cover
340 r = fabs(h)
341 c = _0_0
342 E = self.ellipsoid
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 = atand(t)
348 g += _atan2d(s * tanh(etap), c) # Krueger p 19 (31)
349 k *= sqrt(E.e21 + E.e2 / (t**2 + _1_0)) * 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 = (_180_0 - lon), (_180_0 - 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))
390 def _yxgk4(self, xi_, eta_, C):
391 '''(INTERNAL) Complex Clenshaw summation with
392 C{B{C}=_Alp} or C{B{C}=-_Bet}, negated!
393 '''
394 def _sinhcosh2(x):
395 return sinh(x), cosh(x)
397 x = complex(xi_, eta_)
398 if self.f: # isEllipsoidal
399 s, c = sincos2( xi_ * 2)
400 sh, ch = _sinhcosh2(eta_ * 2)
401 n = -s
402 s = complex(s * ch, c * sh) # sin(zeta * 2)
403 c = complex(c * ch, n * sh) # cos(zeta * 2)
405 y0 = y1 = \
406 z0 = z1 = complex(0) # 0+j0
407 n = self.TMorder # == len(C) - 1
408 if isodd(n):
409 Cn = C[n]
410 y0 = complex(Cn) # +j0
411 z0 = complex(Cn * (n * 2))
412 n -= 1
413 a = c * 2 # cos(zeta * 2) * 2
414 _c = _C
415 while n > 0:
416 Cn = C[n]
417 y1 = _c(a, y0, y1, Cn)
418 z1 = _c(a, z0, z1, Cn * (n * 2))
419 n -= 1
420 Cn = C[n]
421 y0 = _c(a, y1, y0, Cn)
422 z0 = _c(a, z1, z0, Cn * (n * 2))
423 n -= 1
424 # assert n == 0
425 x = _c(s, y0, -x, _0_0)
426 c = _c(c, z0, z1, _1_0)
428 # Gauss-Schreiber to Gauss-Krueger TM
429 # C{cmath.phase} handles INF, NAN, etc.
430 g, k = degrees(phase(c)), abs(c)
431 else: # isSpherical
432 g, k = _0_0, _1_0
434 return x.real, x.imag, g, k
437def _C(a, b0, b1, Cn):
438 '''(INTERNAL) Accurately compute complex M{a * b0 - b1 + Cn}
439 with complex args C{a}, C{b0} and C{b1} and scalar C{Cn}.
441 @see: CPython function U{_Py_c_prod<https://GitHub.com/python/
442 cpython/blob/main/Objects/complexobject.c>}.
444 @note: Python function C{cmath.fsum} is no longer available,
445 but stil mentioned in Note 4 of the comments before
446 CPython function U{math_fsum<https://GitHub.com/python/
447 cpython/blob/main/Modules/mathmodule.c>}
448 '''
449 r = fsum1_(a.real * b0.real, -a.imag * b0.imag, -b1.real, Cn, floats=True)
450 j = fsum1_(a.real * b0.imag, a.imag * b0.real, -b1.imag, floats=True)
451 return complex(r, j)
454def _Xs(_Coeffs, m, E, RA=False): # in .rhumbx
455 '''(INTERNAL) Compute the C{A}, C{B} or C{RA} terms of order
456 B{C{m}} for I{Krüger} series and I{rhumbx._sincosSeries},
457 return a tuple with C{B{m} + 1} terms C{X}, C{X[0]==0}.
458 '''
459 Cs = _Coeffs[m]
460 assert len(Cs) == (((m + 1) * (m + 4)) if RA else
461 ((m + 3) * m)) // 2
462 n = n_ = E.n
463 if n: # isEllipsoidal
464 X = [0] # X[0] never used, it's just an integration
465 # constant, it cancels when evaluating a definite
466 # integral. Don't bother computing it, it is not
467 # used in C{KTransverseMercator._yxgk4} above nor
468 # in C{rhumbx._sincosSeries}.
469 i = (m + 2) if RA else 0
470 for r in _reverange(m): # [m-1 ... 0]
471 j = i + r + 1
472 X.append(_polynomial(n, Cs, i, j) * n_ / Cs[j])
473 i = j + 1
474 n_ *= n
475 X = tuple(X)
476 else: # isSpherical
477 X = _0_0s(m + 1)
478 return X
481# _Alp- and _BetCoeffs in .rhumbx
482_AlpCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
483 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4
484 164, 225, -480, 360, 720, # Alp[1]/n^1, polynomial(n), order 3
485 557, -864, 390, 1440, # Alp[2]/n^2, polynomial(n), order 2
486 -1236, 427, 1680, # PYCHOK Alp[3]/n^3, polynomial(n), order 1
487 49561, 161280), # Alp[4]/n^4, polynomial(n), order 0, count = 14
488 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5
489 -635, 328, 450, -960, 720, 1440, # Alp[1]/n^1, polynomial(n), order 4
490 4496, 3899, -6048, 2730, 10080, # PYCHOK Alp[2]/n^2, polynomial(n), order 3
491 15061, -19776, 6832, 26880, # PYCHOK Alp[3]/n^3, polynomial(n), order 2
492 -171840, 49561, 161280, # Alp[4]/n^4, polynomial(n), order 1
493 34729, 80640), # PYCHOK Alp[5]/n^5, polynomial(n), order 0, count = 20
494 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6
495 31564, -66675, 34440, 47250, -100800, 75600, 151200, # Alp[1]/n^1, polynomial(n), order 5
496 -1983433, 863232, 748608, -1161216, 524160, 1935360, # PYCHOK Alp[2]/n^2, polynomial(n), order 4
497 670412, 406647, -533952, 184464, 725760, # Alp[3]/n^3, polynomial(n), order 3
498 6601661, -7732800, 2230245, 7257600, # Alp[4]/n^4, polynomial(n), order 2
499 -13675556, 3438171, 7983360, # PYCHOK Alp[5]/n^5, polynomial(n), order 1
500 212378941, 319334400), # Alp[6]/n^6, polynomial(n), order 0, count = 27
501 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7
502 1804025, 2020096, -4267200, 2204160, 3024000, -6451200, 4838400, 9676800, # Alp[1]/n^1, polynomial(n), order 6
503 4626384, -9917165, 4316160, 3743040, -5806080, 2620800, 9676800, # Alp[2]/n^2, polynomial(n), order 5
504 -67102379, 26816480, 16265880, -21358080, 7378560, 29030400, # PYCHOK Alp[3]/n^3, polynomial(n), order 4
505 155912000, 72618271, -85060800, 24532695, 79833600, # Alp[4]/n^4, polynomial(n), order 3
506 102508609, -109404448, 27505368, 63866880, # Alp[5]/n^5, polynomial(n), order 2
507 -12282192400, 2760926233, 4151347200, # PYCHOK Alp[6]/n^6, polynomial(n), order 1
508 1522256789, 1383782400), # Alp[7]/n^7, polynomial(n), order 0, count = 35
509 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8
510 -75900428, 37884525, 42422016, -89611200, 46287360, 63504000, -135475200, 101606400, 203212800, # Alp[1]/n^1, polynomial(n), order 7
511 148003883, 83274912, -178508970, 77690880, 67374720, -104509440, 47174400, 174182400, # PYCHOK Alp[2]/n^2, polynomial(n), order 6
512 318729724, -738126169, 294981280, 178924680, -234938880, 81164160, 319334400, # PYCHOK Alp[3]/n^3, polynomial(n), order 5
513 -40176129013, 14967552000, 6971354016, -8165836800, 2355138720, 7664025600, # Alp[4]/n^4, polynomial(n), order 4
514 10421654396, 3997835751, -4266773472, 1072709352, 2490808320, # PYCHOK Alp[5]/n^5, polynomial(n), order 3
515 175214326799, -171950693600, 38652967262, 58118860800, # PYCHOK Alp[6]/n^6, polynomial(n), order 2
516 -67039739596, 13700311101, 12454041600, # PYCHOK Alp[7]/n^7, polynomial(n), order 1
517 1424729850961, 743921418240) # PYCHOK Alp[8]/n^8, polynomial(n), order 0, count = 44
518}
519_B1Coeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
520 2: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 2
521 1, 16, 64, 64), # b1 * (n + 1), polynomial(n2), order 2
522 3: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 3
523 1, 4, 64, 256, 256), # b1 * (n + 1), polynomial(n2), order 3
524 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 4
525 25, 64, 256, 4096, 16384, 16384) # PYCHOK b1 * (n + 1), polynomial(n2), order 4
526}
527_BetCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00
528 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4
529 -4, 555, -960, 720, 1440, # Bet[1]/n^1, polynomial(n), order 3
530 -437, 96, 30, 1440, # Bet[2]/n^2, polynomial(n), order 2
531 -148, 119, 3360, # Bet[3]/n^3, polynomial(n), order 1
532 4397, 161280), # PYCHOK Bet[4]/n^4, polynomial(n), order 0, count = 14
533 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5
534 -3645, -64, 8880, -15360, 11520, 23040, # Bet[1]/n^1, polynomial(n), order 4
535 4416, -3059, 672, 210, 10080, # PYCHOK Bet[2]/n^2, polynomial(n), order 3
536 -627, -592, 476, 13440, # Bet[3]/n^3, polynomial(n), order 2
537 -3520, 4397, 161280, # Bet[4]/n^4, polynomial(n), order 1
538 4583, 161280), # PYCHOK Bet[5]/n^5, polynomial(n), order 0, count = 20
539 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6
540 384796, -382725, -6720, 932400, -1612800, 1209600, 2419200, # Bet[1]/n^1, polynomial(n), order 5
541 -1118711, 1695744, -1174656, 258048, 80640, 3870720, # PYCHOK Bet[2]/n^2, polynomial(n), order 4
542 22276, -16929, -15984, 12852, 362880, # Bet[3]/n^3, polynomial(n), order 3
543 -830251, -158400, 197865, 7257600, # PYCHOK Bet[4]/n^4, polynomial(n), order 2
544 -435388, 453717, 15966720, # PYCHOK Bet[5]/n^5, polynomial(n), order 1
545 20648693, 638668800), # Bet[6]/n^6, polynomial(n), order 0, count = 27
546 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7
547 -5406467, 6156736, -6123600, -107520, 14918400, -25804800, 19353600, 38707200, # Bet[1]/n^1, polynomial(n), order 6
548 829456, -5593555, 8478720, -5873280, 1290240, 403200, 19353600, # PYCHOK Bet[2]/n^2, polynomial(n), order 5
549 9261899, 3564160, -2708640, -2557440, 2056320, 58060800, # PYCHOK Bet[3]/n^3, polynomial(n), order 4
550 14928352, -9132761, -1742400, 2176515, 79833600, # PYCHOK Bet[4]/n^4, polynomial(n), order 3
551 -8005831, -1741552, 1814868, 63866880, # Bet[5]/n^5, polynomial(n), order 2
552 -261810608, 268433009, 8302694400, # Bet[6]/n^6, polynomial(n), order 1
553 219941297, 5535129600), # PYCHOK Bet[7]/n^7, polynomial(n), order 0, count = 35
554 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8
555 31777436, -37845269, 43097152, -42865200, -752640, 104428800, -180633600, 135475200, 270950400, # Bet[1]/n^1, polynomial(n), order 7
556 24749483, 14930208, -100683990, 152616960, -105719040, 23224320, 7257600, 348364800, # Bet[2]/n^2, polynomial(n), order 6
557 -232468668, 101880889, 39205760, -29795040, -28131840, 22619520, 638668800, # PYCHOK Bet[3]/n^3, polynomial(n), order 5
558 324154477, 1433121792, -876745056, -167270400, 208945440, 7664025600, # Bet[4]/n^4, polynomial(n), order 4
559 457888660, -312227409, -67920528, 70779852, 2490808320, # Bet[5]/n^5, polynomial(n), order 3
560 -19841813847, -3665348512, 3758062126, 116237721600, # PYCHOK Bet[6]/n^6, polynomial(n), order 2
561 -1989295244, 1979471673, 49816166400, # PYCHOK Bet[7]/n^7, polynomial(n), order 1
562 191773887257, 3719607091200) # Bet[8]/n^8, polynomial(n), order 0, count = 44
563}
565assert set(_AlpCoeffs.keys()) == set(_BetCoeffs.keys())
567if __name__ == '__main__':
569 from pygeodesy.interns import _usage
570 from sys import argv, exit as _exit
572 _exit(_usage(*argv).replace('.ktm', '.etm -series'))
574# **) MIT License
575#
576# Copyright (C) 2022-2023 -- mrJean1 at Gmail -- All Rights Reserved.
577#
578# Permission is hereby granted, free of charge, to any person obtaining a
579# copy of this software and associated documentation files (the "Software"),
580# to deal in the Software without restriction, including without limitation
581# the rights to use, copy, modify, merge, publish, distribute, sublicense,
582# and/or sell copies of the Software, and to permit persons to whom the
583# Software is furnished to do so, subject to the following conditions:
584#
585# The above copyright notice and this permission notice shall be included
586# in all copies or substantial portions of the Software.
587#
588# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
589# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
590# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
591# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
592# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
593# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
594# OTHER DEALINGS IN THE SOFTWARE.