Coverage for pygeodesy/rhumb/ekx.py: 98%
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2# -*- coding: utf-8 -*-
4u'''A pure Python version of I{Karney}'s I{elliptic functions}, I{Krüger series expansion}, C++
5classes U{Rhumb<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Rhumb.html>} and
6and U{RhumbLine<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1RhumbLine.html>}
7from I{GeographicLib version 2.0}, kept for backward compatibility.
9Class L{RhumbLine} has been enhanced with methods C{Intersecant2}, C{Intersection} and C{PlumbTo} to
10iteratively find the intersection of a rhumb line and a circle or an other rhumb line, respectively
11a perpendicular geodesic or other rhumb line.
13For more details, see the C++ U{GeographicLib<https://GeographicLib.SourceForge.io/C++/doc/index.html>}
14documentation, especially the U{Class List<https://GeographicLib.SourceForge.io/C++/doc/annotated.html>},
15the background information on U{Rhumb lines<https://GeographicLib.SourceForge.io/C++/doc/rhumb.html>},
16the utily U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>} and U{Online
17rhumb line calculations<https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve>}.
19Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2014-2024) and licensed under the MIT/X11
20License. For more information, see the U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation.
21'''
22# make sure int/int division yields float quotient
23from __future__ import division as _; del _ # PYCHOK semicolon
25from pygeodesy.basics import copysign0, neg
26from pygeodesy.constants import PI_2, _0_0s, _0_0, _0_5, _1_0, \
27 _2_0, _4_0, _720_0, _over, _1_over
28# from pygeodesy.datums import _WGS84 # from .rhumb.bases
29# from pygeodesy.deprecated import RhumbOrder2Tuple # _MODS
30from pygeodesy.errors import RhumbError, _xkwds_pop2, _Xorder
31from pygeodesy.fmath import hypot, hypot1
32# from pygeodesy.fsums import fsum1f_ # _MODS
33# from pygeodesy.karney import Caps # from .rhumb.bases
34from pygeodesy.ktm import KTransverseMercator, _Xs, \
35 _AlpCoeffs, _BetCoeffs # PYCHOK used!
36from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS
37from pygeodesy.props import deprecated_method, Property, Property_RO, \
38 property_RO
39from pygeodesy.rhumb.bases import RhumbBase, RhumbLineBase, \
40 Caps, _update_all_rls, _WGS84
41from pygeodesy.utily import atan1, sincos2_
43from math import asinh, atan, cos, cosh, radians, sin, sinh, sqrt, tan # as _tan
45__all__ = _ALL_LAZY.rhumb_ekx
46__version__ = '24.11.26'
49class Rhumb(RhumbBase):
50 '''Class to solve the I{direct} and I{inverse rhumb} problems, based on
51 I{elliptic functions} or I{Krüger series expansion}
53 @see: The U{Detailed Description<https://GeographicLib.SourceForge.io/C++/doc/
54 classGeographicLib_1_1Rhumb.html>} of I{Karney}'s C++ C{Rhumb Class}.
55 '''
56 _mRA = 6 # see .RAorder
58 def __init__(self, a_earth=_WGS84, f=None, exact=True, **RA_TMorder_name):
59 '''New C{Rhumb}.
61 @kwarg a_earth: This rhumb's earth model (L{Datum}, L{Ellipsoid},
62 L{Ellipsoid2}, L{a_f2Tuple}, 2-tuple C{(a, f)}) or
63 the (equatorial) radius (C{meter}, conventionally).
64 @kwarg f: The ellipsoid's flattening (C{scalar}), required if B{C{a_earth}}
65 is C{scalar}, ignored otherwise.
66 @kwarg exact: If C{True}, use an addition theorem for elliptic integrals
67 to compute I{Divided differences}, otherwise use the I{Krüger}
68 series expansion (C{bool} or C{None}), see also properties
69 C{exact} and C{TMorder}.
70 @kwarg RA_TMorder_name: Optional C{B{name}=NN} (C{str}) and optional keyword
71 arguments B{C{RAorder}=6} and B{C{TMorder}=6} to set the respective
72 C{order}, see properties C{RAorder} and C{TMorder}.
74 @raise RhumbError: Invalid B{C{a_earth}}, B{C{f}}, B{C{RAorder}} or B{C{TMorder}}.
75 '''
76 if RA_TMorder_name:
77 M = self._mRA
78 m, kwds = _xkwds_pop2(RA_TMorder_name, RAorder=M)
79 if m != M:
80 self.RAorder = m
81 else:
82 kwds = {}
83 RhumbBase.__init__(self, a_earth, f, exact, kwds)
85 @Property_RO
86 def _A2(self): # Conformal2RectifyingCoeffs
87 m = self.TMorder
88 return _Xs(_AlpCoeffs, m, self.ellipsoid), m
90 @Property_RO
91 def _B2(self): # Rectifying2ConformalCoeffs
92 m = self.TMorder
93 return _Xs(_BetCoeffs, m, self.ellipsoid), m
95 def _DConformal2Rectifying(self, x, y): # radians
96 return _1_0 + (_sincosSeries(True, x, y, *self._A2) if self.f else _0_0)
98 @deprecated_method
99 def Direct7(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE_AREA): # PYCHOK no cover
100 '''DEPRECATED, use method L{Rhumb.Direct8}.
102 @return: A I{DEPRECATED} L{Rhumb7Tuple}.
103 '''
104 return self.Direct8(lat1, lon1, azi12, s12, outmask=outmask)._to7Tuple()
106 def _DIsometrict(self, phix, phiy, tphix, tphiy, _Dtan_phix_phiy):
107 E = self.ellipsoid
108 return _Dtan_phix_phiy * _Dasinh(tphix, tphiy) - \
109 _Dsin(phix, phiy) * _DeatanhE(sin(phix), sin(phiy), E)
111 def _DIsometric2Rectifyingd(self, psix, psiy): # degrees
112 if self.exact:
113 E = self.ellipsoid
114 phix, phiy, tphix, tphiy = _Eaux4(E.auxIsometric, psix, psiy)
115 t = _Dtant(phix - phiy, tphix, tphiy)
116 r = _over(self._DRectifyingt( tphix, tphiy, t),
117 self._DIsometrict(phix, phiy, tphix, tphiy, t))
118 else:
119 x, y = radians(psix), radians(psiy)
120 r = self._DConformal2Rectifying(_gd(x), _gd(y)) * _Dgd(x, y)
121 return r
123 def _DRectifyingt(self, tphix, tphiy, _Dtan_phix_phiy):
124 E = self.ellipsoid
125 tbetx = E.f1 * tphix
126 tbety = E.f1 * tphiy
127 return (E.f1 * _Dtan_phix_phiy * E.b * PI_2
128 * _DfEt( tbetx, tbety, self._eF)
129 * _Datan(tbetx, tbety)) / E.L
131 def _DRectifying2Conformal(self, x, y): # radians
132 return _1_0 - (_sincosSeries(True, x, y, *self._B2) if self.f else _0_0)
134 def _DRectifying2Isometricd(self, mux, muy): # degrees
135 E = self.ellipsoid
136 phix, phiy, tphix, tphiy = _Eaux4(E.auxRectifying, mux, muy)
137 if self.exact:
138 t = _Dtant(phix - phiy, tphix, tphiy)
139 r = _over(self._DIsometrict(phix, phiy, tphix, tphiy, t),
140 self._DRectifyingt( tphix, tphiy, t))
141 else:
142 r = self._DRectifying2Conformal(radians(mux), radians(muy)) * \
143 _Dgdinv(E.es_taupf(tphix), E.es_taupf(tphiy))
144 return r
146 @Property_RO
147 def _eF(self):
148 '''(INTERNAL) Get the ellipsoid's elliptic function.
149 '''
150 # .k2 = 0.006739496742276434
151 return self.ellipsoid._elliptic_e12 # _MODS.elliptic.Elliptic(-self.ellipsoid._e12)
153 def _Inverse4(self, lon12, r, outmask):
154 '''(INTERNAL) See method C{RhumbBase.Inverse}.
155 '''
156 E, Cs = self.ellipsoid, Caps
157 psi1 = E.auxIsometric(r.lat1)
158 psi2 = E.auxIsometric(r.lat2)
159 psi12 = psi2 - psi1 # degrees
160 if (outmask & Cs.DISTANCE):
161 a = s = hypot(lon12, psi12)
162 if a:
163 a *= self._DIsometric2Rectifyingd(psi2, psi1)
164 s = self._mpd * a # == E._Lpd
165 a = copysign0(a, s)
166 r.set_(a12=a, s12=s)
168 if ((outmask | self._debug) & Cs._DEBUG_INVERSE): # PYCHOK no cover
169 r.set_(a=E.a, f=E.f, f1=E.f1, L=E.L,
170 b=E.b, e=E.e, e2=E.e2, k2=self._eF.k2,
171 lon12=lon12, psi1=psi1, exact=self.exact,
172 psi12=psi12, psi2=psi2)
173 return lon12, psi12, psi1, psi2
175 @deprecated_method
176 def Inverse7(self, lat1, lon1, azi12, s12, outmask=Caps.AZIMUTH_DISTANCE_AREA): # PYCHOK no cover
177 '''DEPRECATED, use method L{Rhumb.Inverse8}.
179 @return: A I{DEPRECATED} L{Rhumb7Tuple}.
180 '''
181 return self.Inverse8(lat1, lon1, azi12, s12, outmask=outmask)._to7Tuple()
183 @Property_RO
184 def _mpd(self): # meter per degree
185 return self.ellipsoid._Lpd
187 @Property_RO
188 def _mpr(self): # meter per radian
189 return self.ellipsoid._Lpr # degrees(._Lpd)
191 @deprecated_method
192 def orders(self, RAorder=6, TMorder=6): # PYCHOK no cover
193 '''DEPRECATED, use properties C{RAorder} and/or C{TMorder}.
195 Get and set the I{RAorder} and/or I{TMorder}.
197 @kwarg RAorder: I{Rhumb Area} order (C{int}, 4, 5, 6, 7
198 or 8).
199 @kwarg TMorder: I{Transverse Mercator} order (C{int}, 4,
200 5, 6, 7 or 8).
202 @return: DEPRECATED L{RhumbOrder2Tuple}C{(RAorder, TMorder)}
203 with the previous C{RAorder} and C{TMorder} setting.
204 '''
205 t = _MODS.deprecated.classes.RhumbOrder2Tuple(self.RAorder, self.TMorder)
206 if RAorder != t.RAorder: # PYCHOK attr
207 self.RAorder = RAorder
208 if TMorder != t.TMorder: # PYCHOK attr
209 self.TMorder = TMorder
210 return t
212 @Property_RO
213 def _RA2(self):
214 # for WGS84: (0, -0.0005583633519275459, -3.743803759172812e-07, -4.633682270824446e-10,
215 # RAorder 6: -7.709197397676237e-13, -1.5323287106694307e-15, -3.462875359099873e-18)
216 m = self.RAorder
217 return _Xs(_RACoeffs, m, self.ellipsoid, RA=True), m
219 @Property
220 def RAorder(self):
221 '''Get the I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8).
222 '''
223 return self._mRA
225 @RAorder.setter # PYCHOK setter!
226 def RAorder(self, order):
227 '''Set the I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8).
228 '''
229 m = _Xorder(_RACoeffs, RhumbError, RAorder=order)
230 if self._mRA != m:
231 _update_all_rls(self)
232 self._mRA = m
234# _RhumbLine = RhumbLine # see further below
236 def _S12d(self, psi1, psi2, lon12): # degrees
237 '''(INTERNAL) Compute the area C{S12}.
238 '''
239 S = (self.ellipsoid.areax if self.exact else
240 self.ellipsoid.area) * lon12 / _720_0
241 if S:
242 x, y = radians(psi1), radians(psi2) # _meanSinXi(x, y)
243 s = _Dlog(cosh(x), cosh(y)) * _Dcosh(x, y)
244 if self.f:
245 s += _sincosSeries(False, _gd(x), _gd(y), *self._RA2) * _Dgd(x, y)
246 S *= s
247 return S
250class RhumbLine(RhumbLineBase):
251 '''Compute one or several points on a single rhumb line.
253 Class C{RhumbLine} facilitates the determination of points on
254 a single rhumb line. The starting point (C{lat1}, C{lon1})
255 and the azimuth C{azi12} are specified once.
256 '''
257 _Rhumb = Rhumb # rhumb.ekx.Rhumb
259 def __init__(self, rhumb, lat1=0, lon1=0, azi12=None, **caps_name): # PYCHOK signature
260 '''New C{RhumbLine}.
262 @arg rhumb: The rhumb reference (L{Rhumb}).
263 @kwarg lat1: Latitude of the start point (C{degrees90}).
264 @kwarg lon1: Longitude of the start point (C{degrees180}).
265 @kwarg azi12: Azimuth of this rhumb line (compass C{degrees}).
266 @kwarg caps_name: Optional keyword arguments C{B{name}=NN} and
267 C{B{caps}=0}, a bit-or'ed combination of L{Caps}
268 values specifying the required capabilities. Include
269 C{Caps.LINE_OFF} if updates to the B{C{rhumb}} should
270 I{not} be reflected in this rhumb line.
271 '''
272 RhumbLineBase.__init__(self, rhumb, lat1, lon1, azi12, **caps_name)
274 @Property_RO
275 def _dpm12(self): # PYCHOK no cover
276 '''(INTERNAL) Get this rhumb line's parallel I{circle radius} (C{degree per meter}).
277 '''
278 r = self._salp
279 if r:
280 r = _over(r, self.ellipsoid.circle4(self.lat1).radius)
281 return r
283 @Property_RO
284 def _mu1(self):
285 '''(INTERNAL) Get the I{rectifying auxiliary} latitude (C{degrees}).
286 '''
287 return self.ellipsoid.auxRectifying(self.lat1)
289 def _mu2lat(self, mu):
290 '''(INTERNAL) Get the inverse I{rectifying auxiliary} latitude (C{degrees}).
291 '''
292 return self.ellipsoid.auxRectifying(mu, inverse=True)
294 def _Position4(self, unused, mu2, s12, mu12):
295 '''(INTERNAL) See method C{RhumbLineBase._Position}.
296 '''
297 psi1 = psi2 = self._psi1
298 if mu12: # self._calp != 0
299 lat2 = self._mu2lat(mu2)
300 psi12 = self.rhumb._DRectifying2Isometricd(mu2, self._mu1) * mu12
301 lon2 = self._talp * psi12
302 psi2 += psi12
303 else: # meridional
304 lat2 = self.lat1
305 lon2 = self._dpm12 * s12
306 return lat2, lon2, psi1, psi2
308 @Property_RO
309 def _psi1(self):
310 '''(INTERNAL) Get the I{isometric auxiliary} latitude C{psi} (C{degrees}).
311 '''
312 return self.ellipsoid.auxIsometric(self.lat1)
314 @property_RO
315 def RAorder(self):
316 '''Get this rhumb line's I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8).
317 '''
318 return self.rhumb.RAorder
320Rhumb._RhumbLine = RhumbLine # PYCHOK see RhumbBase._RhumbLine
323# Use I{Divided Differences} to determine (mu2 - mu1) / (psi2 - psi1) accurately.
324# Definition: _Df(x,y,d) = (f(x) - f(y)) / (x - y), @see W. M. Kahan & R. J.
325# Fateman, "Symbolic computation of Divided Differences", SIGSAM Bull. 33(3),
326# 7-28 (1999). U{ACM<https://DL.ACM.org/doi/pdf/10.1145/334714.334716> and @see
327# U{UCB<https://www.CS.Berkeley.edu/~fateman/papers/divdiff.pdf>}, Dec 8, 1999.
329def _Dasinh(x, y):
330 hx = hypot1(x)
331 d = x - y
332 if d:
333 hx *= y
334 hy = x * hypot1(y)
335 t = (d * (x + y) / (hy + hx)) if (x * y) > 0 else (hy - hx)
336 r = asinh(t) / d
337 else:
338 r = _1_0 / hx
339 return r
342def _Datan(x, y):
343 xy = x * y
344 r = xy + _1_0
345 d = x - y
346 if d: # 2 * xy > -1 == 2 * xy + 1 > 0 == xy + r > 0 == xy > -r
347 r = (atan1(d, r) if xy > -r else (atan1(x) - atan1(y))) / d
348 else:
349 r = _1_over(r)
350 return r
353def _Dcosh(x, y):
354 return _Dsincos(x, y, sinh, sinh)
357def _DeatanhE(x, y, E): # see .albers._Datanhee
358 # Deatanhe(x, y) = eatanhe((x - y) / (1 - e^2 * x * y)) / (x - y)
359 e = _1_0 - E.e2 * x * y
360 if e: # assert not isnear0(e)
361 d = x - y
362 e = (E._es_atanh(d / e) / d) if d else (E.e2 / e)
363 return e
366def _DfEt(tx, ty, eF): # tangents
367 # eF = Elliptic(-E.e12) # -E.e2 / (1 - E.e2)
368 r, x, y, = _1_0, atan(tx), atan(ty)
369 d = x - y
370 if (x * y) > 0:
371 # See U{DLMF<https://DLMF.NIST.gov/19.11>}: 19.11.2 and 19.11.4
372 # letting theta -> x, phi -> -y, psi -> z
373 # (E(x) - E(y)) / d = E(z)/d - k2 * sin(x) * sin(y) * sin(z)/d
374 # tan(z/2) = (sin(x)*Delta(y) - sin(y)*Delta(x)) / (cos(x) + cos(y))
375 # = d * Dsin(x,y) * (sin(x) + sin(y))/(cos(x) + cos(y)) /
376 # (sin(x)*Delta(y) + sin(y)*Delta(x))
377 # = t = d * Dt
378 # sin(z) = 2*t/(1+t^2); cos(z) = (1-t^2)/(1+t^2)
379 # Alt (this only works for |z| <= pi/2 -- however, this conditions
380 # holds if x*y > 0):
381 # sin(z) = d * Dsin(x,y) * (sin(x) + sin(y)) /
382 # (sin(x)*cos(y)*Delta(y) + sin(y)*cos(x)*Delta(x))
383 # cos(z) = sqrt((1-sin(z))*(1+sin(z)))
384 sx, cx, sy, cy = sincos2_(x, y)
385 D = (cx + cy) * (eF.fDelta(sy, cy) * sx +
386 eF.fDelta(sx, cx) * sy)
387 D = (sx + sy) * _Dsin(x, y) / D
388 t = D * d
389 t2 = _1_0 + t**2
390 D *= _2_0 / t2
391 s = D * d
392 if s:
393 c = (t + _1_0) * (_1_0 - t) / t2
394 r = eF.fE(s, c, eF.fDelta(s, c)) / s
395 r = D * (r - eF.k2 * sx * sy)
396 elif d:
397 r = (eF.fE(x) - eF.fE(y)) / d
398 return r
401def _Dgd(x, y):
402 return _Datan(sinh(x), sinh(y)) * _Dsinh(x, y)
405def _Dgdinv(x, y): # x, y are tangents
406 return _Dasinh(x, y) / _Datan(x, y)
409def _Dlog(x, y):
410 d = (x - y) * _0_5
411 # Changed atanh(t / (x + y)) to asinh(t / (2 * sqrt(x*y))) to
412 # avoid taking atanh(1) when x is large and y is 1. This also
413 # fixes bogus results being returned for the area when an endpoint
414 # is at a pole. N.B. this routine is invoked with positive x
415 # and y, so the sqrt is always taken of a positive quantity.
416 return (asinh(d / sqrt(x * y)) / d) if d else _1_over(x)
419def _Dsin(x, y):
420 return _Dsincos(x, y, sin, cos)
423def _Dsincos(x, y, sin_, cos_):
424 r = cos_((x + y) * _0_5)
425 d = (x - y) * _0_5
426 if d:
427 r *= sin_(d) / d
428 return r
431def _Dsinh(x, y):
432 return _Dsincos(x, y, sinh, cosh)
435def _Dtan(x, y): # PYCHOK no cover
436 return _Dtant(x - y, tan(x), tan(y))
439def _Dtant(dxy, tx, ty):
440 txy = tx * ty
441 r = txy + _1_0
442 if dxy: # 2 * txy > -1 == 2 * txy + 1 > 0 == txy + r > 0 == txy > -r
443 r = ((tan(dxy) * r) if txy > -r else (tx - ty)) / dxy
444 return r
447def _Eaux4(E_aux, mu_psi_x, mu_psi_y): # degrees
448 # get inverse auxiliary lats in radians and tangents
449 phix = radians(E_aux(mu_psi_x, inverse=True))
450 phiy = radians(E_aux(mu_psi_y, inverse=True))
451 return phix, phiy, tan(phix), tan(phiy)
454def _gd(x):
455 return atan(sinh(x))
458def _sincosSeries(sinp, x, y, C, n):
459 # N.B. C[] has n+1 elements of which
460 # C[0] is ignored and n >= 0
461 # Use Clenshaw summation to evaluate
462 # m = (g(x) + g(y)) / 2 -- mean value
463 # s = (g(x) - g(y)) / (x - y) -- average slope
464 # where
465 # g(x) = sum(C[j] * SC(2 * j * x), j = 1..n)
466 # SC = sinp ? sin : cos
467 # CS = sinp ? cos : sin
468 # ...
469 d, _neg = (x - y), neg
470 sp, cp, sd, cd = sincos2_(x + y, d)
471 sd = (sd / d) if d else _1_0
472 s = _neg(sp * sd) # negative
473 # 2x2 matrices in row-major order
474 a1 = s * d**2
475 a2 = s * _4_0
476 a0 = a3 = _2_0 * cp * cd # m
477 b2 = b1 = _0_0s(4)
478 if n > 0:
479 b1 = C[n], _0_0, _0_0, C[n]
481 _fsum = _MODS.fsums.fsum1f_
482 for j in range(n - 1, 0, -1): # C[0] unused
483 b1, b2, Cj = b2, b1, C[j]
484 # b1 = a * b2 - b1 + C[j] * I
485 m0, m1, m2, m3 = b2
486 n0, n1, n2, n3 = map(_neg, b1)
487 b1 = (_fsum(a0 * m0, a1 * m2, n0, Cj),
488 _fsum(a0 * m1, a1 * m3, n1),
489 _fsum(a2 * m0, a3 * m2, n2),
490 _fsum(a2 * m1, a3 * m3, n3, Cj))
491 # Here are the full expressions for m and s
492 # f01, f02, f11, f12 = (0, 0, cd * sp, 2 * sd * cp) if sinp else \
493 # (1, 0, cd * cp, -2 * sd * sp)
494 # m = -b2[1] * f02 + (C[0] - b2[0]) * f01 + b1[0] * f11 + b1[1] * f12
495 # s = -b2[2] * f01 + (C[0] - b2[3]) * f02 + b1[2] * f11 + b1[3] * f12
496 cd *= b1[2]
497 sd *= b1[3] * _2_0
498 s = _fsum(cd * sp, sd * cp) if sinp else \
499 _fsum(cd * cp, _neg(sd * sp), _neg(b2[2]))
500 return s
503_RACoeffs = { # Generated by Maxima on 2015-05-15 08:24:04-04:00
504 4: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4
505 691, 7860, -20160, 18900, 0, 56700, # R[0]/n^0, polynomial(n), order 4
506 1772, -5340, 6930, -4725, 14175, # R[1]/n^1, polynomial(n), order 3
507 -1747, 1590, -630, 4725, # PYCHOK R[2]/n^2, polynomial(n), order 2
508 104, -31, 315, # R[3]/n^3, polynomial(n), order 1
509 -41, 420), # PYCHOK R[4]/n^4, polynomial(n), order 0, count = 20
510 5: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5
511 -79036, 22803, 259380, -665280, 623700, 0, 1871100, # PYCHOK R[0]/n^0, polynomial(n), order 5
512 41662, 58476, -176220, 228690, -155925, 467775, # PYCHOK R[1]/n^1, polynomial(n), order 4
513 18118, -57651, 52470, -20790, 155925, # PYCHOK R[2]/n^2, polynomial(n), order 3
514 -23011, 17160, -5115, 51975, # PYCHOK R[3]/n^3, polynomial(n), order 2
515 5480, -1353, 13860, # PYCHOK R[4]/n^4, polynomial(n), order 1
516 -668, 5775), # PYCHOK R[5]/n^5, polynomial(n), order 0, count = 27
517 6: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6
518 128346268, -107884140, 31126095, 354053700, -908107200, 851350500, 0, 2554051500, # R[0]/n^0, polynomial(n), order 6
519 -114456994, 56868630, 79819740, -240540300, 312161850, -212837625, 638512875, # PYCHOK R[1]/n^1, polynomial(n), order 5
520 51304574, 24731070, -78693615, 71621550, -28378350, 212837625, # R[2]/n^2, polynomial(n), order 4
521 1554472, -6282003, 4684680, -1396395, 14189175, # R[3]/n^3, polynomial(n), order 3
522 -4913956, 3205800, -791505, 8108100, # PYCHOK R[4]/n^4, polynomial(n), order 2
523 1092376, -234468, 2027025, # R[5]/n^5, polynomial(n), order 1
524 -313076, 2027025), # PYCHOK R[6]/n^6, polynomial(n), order 0, count = 35
525 7: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7
526 -317195588, 385038804, -323652420, 93378285, 1062161100, -2724321600, 2554051500, 0, 7662154500, # PYCHOK R[0]/n^0, polynomial(n), order 7
527 258618446, -343370982, 170605890, 239459220, -721620900, 936485550, -638512875, 1915538625, # PYCHOK R[1]/n^1, polynomial(n), order 6
528 -248174686, 153913722, 74193210, -236080845, 214864650, -85135050, 638512875, # PYCHOK R[2]/n^2, polynomial(n), order 5
529 114450437, 23317080, -94230045, 70270200, -20945925, 212837625, # PYCHOK R[3]/n^3, polynomial(n), order 4
530 15445736, -103193076, 67321800, -16621605, 170270100, # PYCHOK R[4]/n^4, polynomial(n), order 3
531 -27766753, 16385640, -3517020, 30405375, # PYCHOK R[4]/n^4, polynomial(n), order 3
532 4892722, -939228, 6081075, # PYCHOK R[4]/n^4, polynomial(n), order 3
533 -3189007, 14189175), # PYCHOK R[7]/n^7, polynomial(n), order 0, count = 44
534 8: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8
535 71374704821, -161769749880, 196369790040, -165062734200, 47622925350, 541702161000, -1389404016000, 1302566265000, 0, 3907698795000, # R[0]/n^0, polynomial(n), order 8
536 -13691187484, 65947703730, -87559600410, 43504501950, 61062101100, -184013329500, 238803815250, -162820783125, 488462349375, # PYCHOK R[1]/n^1, polynomial(n), order 7
537 30802104839, -63284544930, 39247999110, 18919268550, -60200615475, 54790485750, -21709437750, 162820783125, # R[2]/n^2, polynomial(n), order 6
538 -8934064508, 5836972287, 1189171080, -4805732295, 3583780200, -1068242175, 10854718875, # PYCHOK R[3]/n^3, polynomial(n), order 5
539 50072287748, 3938662680, -26314234380, 17167059000, -4238509275, 43418875500, # R[4]/n^4, polynomial(n), order 4
540 359094172, -9912730821, 5849673480, -1255576140, 10854718875, # R[5]/n^5, polynomial(n), order 3
541 -16053944387, 8733508770, -1676521980, 10854718875, # PYCHOK R[6]/n^6, polynomial(n), order 2
542 930092876, -162639357, 723647925, # R[7]/n^7, polynomial(n), order 1
543 -673429061, 1929727800) # PYCHOK R[8]/n^8, polynomial(n), order 0, count = 54
544}
546__all__ += _ALL_DOCS(Caps)
548# **) MIT License
549#
550# Copyright (C) 2022-2025 -- mrJean1 at Gmail -- All Rights Reserved.
551#
552# Permission is hereby granted, free of charge, to any person obtaining a
553# copy of this software and associated documentation files (the "Software"),
554# to deal in the Software without restriction, including without limitation
555# the rights to use, copy, modify, merge, publish, distribute, sublicense,
556# and/or sell copies of the Software, and to permit persons to whom the
557# Software is furnished to do so, subject to the following conditions:
558#
559# The above copyright notice and this permission notice shall be included
560# in all copies or substantial portions of the Software.
561#
562# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
563# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
564# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
565# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
566# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
567# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
568# OTHER DEALINGS IN THE SOFTWARE.