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 _ # noqa: E702 ;
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 _Xs, _AlpCoeffs, _BetCoeffs # PYCHOK used!
35from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS
36from pygeodesy.props import deprecated_method, Property, Property_RO, \
37 property_RO
38from pygeodesy.rhumb.bases import RhumbBase, RhumbLineBase, \
39 Caps, _update_all_rls, _WGS84
40from pygeodesy.utily import atan1, sincos2_
42from math import asinh, atan, cos, cosh, radians, sin, sinh, sqrt, tan # as _tan
44__all__ = _ALL_LAZY.rhumb_ekx
45__version__ = '25.05.12'
48class Rhumb(RhumbBase):
49 '''Class to solve the I{direct} and I{inverse rhumb} problems, based on
50 I{elliptic functions} or I{Krüger series expansion}
52 @see: The U{Detailed Description<https://GeographicLib.SourceForge.io/C++/doc/
53 classGeographicLib_1_1Rhumb.html>} of I{Karney}'s C++ C{Rhumb Class}.
54 '''
55 _mRA = 6 # see .RAorder
57 def __init__(self, a_earth=_WGS84, f=None, exact=True, **RA_TMorder_name):
58 '''New C{Rhumb}.
60 @kwarg a_earth: This rhumb's earth model (L{Datum}, L{Ellipsoid},
61 L{Ellipsoid2}, L{a_f2Tuple}, 2-tuple C{(a, f)}) or
62 the (equatorial) radius (C{meter}, conventionally).
63 @kwarg f: The ellipsoid's flattening (C{scalar}), required if B{C{a_earth}}
64 is C{scalar}, ignored otherwise.
65 @kwarg exact: If C{True}, use an addition theorem for elliptic integrals
66 to compute I{Divided differences}, otherwise use the I{Krüger}
67 series expansion (C{bool} or C{None}), see also properties
68 C{exact} and C{TMorder}.
69 @kwarg RA_TMorder_name: Optional C{B{name}=NN} (C{str}) and optional keyword
70 arguments B{C{RAorder}=6} and B{C{TMorder}=6} to set the respective
71 C{order}, see properties C{RAorder} and C{TMorder}.
73 @raise RhumbError: Invalid B{C{a_earth}}, B{C{f}}, B{C{RAorder}} or B{C{TMorder}}.
74 '''
75 if RA_TMorder_name:
76 M = self._mRA
77 m, kwds = _xkwds_pop2(RA_TMorder_name, RAorder=M)
78 if m != M:
79 self.RAorder = m
80 else:
81 kwds = {}
82 RhumbBase.__init__(self, a_earth, f, exact, kwds)
84 @Property_RO
85 def _A2(self): # Conformal2RectifyingCoeffs
86 m = self.TMorder
87 return _Xs(_AlpCoeffs, m, self.ellipsoid), m
89 @Property_RO
90 def _B2(self): # Rectifying2ConformalCoeffs
91 m = self.TMorder
92 return _Xs(_BetCoeffs, m, self.ellipsoid), m
94 def _DConformal2Rectifying(self, x, y): # radians
95 return _1_0 + (_sincosSeries(True, x, y, *self._A2) if self.f else _0_0)
97 @deprecated_method
98 def Direct7(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE_AREA): # PYCHOK no cover
99 '''DEPRECATED, use method L{Rhumb.Direct8}.
101 @return: A I{DEPRECATED} L{Rhumb7Tuple}.
102 '''
103 return self.Direct8(lat1, lon1, azi12, s12, outmask=outmask)._to7Tuple()
105 def _DIsometrict(self, phix, phiy, tphix, tphiy, _Dtan_phix_phiy):
106 E = self.ellipsoid
107 return _Dtan_phix_phiy * _Dasinh(tphix, tphiy) - \
108 _Dsin(phix, phiy) * _DeatanhE(sin(phix), sin(phiy), E)
110 def _DIsometric2Rectifyingd(self, psix, psiy): # degrees
111 if self.exact:
112 E = self.ellipsoid
113 phix, phiy, tphix, tphiy = _Eaux4(E.auxIsometric, psix, psiy)
114 t = _Dtant(phix - phiy, tphix, tphiy)
115 r = _over(self._DRectifyingt( tphix, tphiy, t),
116 self._DIsometrict(phix, phiy, tphix, tphiy, t))
117 else:
118 x, y = radians(psix), radians(psiy)
119 r = self._DConformal2Rectifying(_gd(x), _gd(y)) * _Dgd(x, y)
120 return r
122 def _DRectifyingt(self, tphix, tphiy, _Dtan_phix_phiy):
123 E = self.ellipsoid
124 tbetx = E.f1 * tphix
125 tbety = E.f1 * tphiy
126 return (E.f1 * _Dtan_phix_phiy * E.b * PI_2
127 * _DfEt( tbetx, tbety, self._eF)
128 * _Datan(tbetx, tbety)) / E.L
130 def _DRectifying2Conformal(self, x, y): # radians
131 return _1_0 - (_sincosSeries(True, x, y, *self._B2) if self.f else _0_0)
133 def _DRectifying2Isometricd(self, mux, muy): # degrees
134 E = self.ellipsoid
135 phix, phiy, tphix, tphiy = _Eaux4(E.auxRectifying, mux, muy)
136 if self.exact:
137 t = _Dtant(phix - phiy, tphix, tphiy)
138 r = _over(self._DIsometrict(phix, phiy, tphix, tphiy, t),
139 self._DRectifyingt( tphix, tphiy, t))
140 else:
141 r = self._DRectifying2Conformal(radians(mux), radians(muy)) * \
142 _Dgdinv(E.es_taupf(tphix), E.es_taupf(tphiy))
143 return r
145 @Property_RO
146 def _eF(self):
147 '''(INTERNAL) Get the ellipsoid's elliptic function.
148 '''
149 # .k2 = 0.006739496742276434
150 return self.ellipsoid._elliptic_e12 # _MODS.elliptic.Elliptic(-self.ellipsoid._e12)
152 def _Inverse4(self, lon12, r, outmask):
153 '''(INTERNAL) See method C{RhumbBase.Inverse}.
154 '''
155 E, Cs = self.ellipsoid, Caps
156 psi1 = E.auxIsometric(r.lat1)
157 psi2 = E.auxIsometric(r.lat2)
158 psi12 = psi2 - psi1 # degrees
159 if (outmask & Cs.DISTANCE):
160 a = s = hypot(lon12, psi12)
161 if a:
162 a *= self._DIsometric2Rectifyingd(psi2, psi1)
163 s = self._mpd * a # == E._Lpd
164 a = copysign0(a, s)
165 r.set_(a12=a, s12=s)
167 if ((outmask | self._debug) & Cs._DEBUG_INVERSE): # PYCHOK no cover
168 r.set_(a=E.a, f=E.f, f1=E.f1, L=E.L,
169 b=E.b, e=E.e, e2=E.e2, k2=self._eF.k2,
170 lon12=lon12, psi1=psi1, exact=self.exact,
171 psi12=psi12, psi2=psi2)
172 return lon12, psi12, psi1, psi2
174 @deprecated_method
175 def Inverse7(self, lat1, lon1, azi12, s12, outmask=Caps.AZIMUTH_DISTANCE_AREA): # PYCHOK no cover
176 '''DEPRECATED, use method L{Rhumb.Inverse8}.
178 @return: A I{DEPRECATED} L{Rhumb7Tuple}.
179 '''
180 return self.Inverse8(lat1, lon1, azi12, s12, outmask=outmask)._to7Tuple()
182 @Property_RO
183 def _mpd(self): # meter per degree
184 return self.ellipsoid._Lpd
186 @Property_RO
187 def _mpr(self): # meter per radian
188 return self.ellipsoid._Lpr # degrees(._Lpd)
190 @deprecated_method
191 def orders(self, RAorder=6, TMorder=6): # PYCHOK no cover
192 '''DEPRECATED, use properties C{RAorder} and/or C{TMorder}.
194 Get and set the I{RAorder} and/or I{TMorder}.
196 @kwarg RAorder: I{Rhumb Area} order (C{int}, 4, 5, 6, 7
197 or 8).
198 @kwarg TMorder: I{Transverse Mercator} order (C{int}, 4,
199 5, 6, 7 or 8).
201 @return: DEPRECATED L{RhumbOrder2Tuple}C{(RAorder, TMorder)}
202 with the previous C{RAorder} and C{TMorder} setting.
203 '''
204 t = _MODS.deprecated.classes.RhumbOrder2Tuple(self.RAorder, self.TMorder)
205 if RAorder != t.RAorder: # PYCHOK attr
206 self.RAorder = RAorder
207 if TMorder != t.TMorder: # PYCHOK attr
208 self.TMorder = TMorder
209 return t
211 @Property_RO
212 def _RA2(self):
213 # for WGS84: (0, -0.0005583633519275459, -3.743803759172812e-07, -4.633682270824446e-10,
214 # RAorder 6: -7.709197397676237e-13, -1.5323287106694307e-15, -3.462875359099873e-18)
215 m = self.RAorder
216 return _Xs(_RACoeffs, m, self.ellipsoid, RA=True), m
218 @Property
219 def RAorder(self):
220 '''Get the I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8).
221 '''
222 return self._mRA
224 @RAorder.setter # PYCHOK setter!
225 def RAorder(self, order):
226 '''Set the I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8).
227 '''
228 m = _Xorder(_RACoeffs, RhumbError, RAorder=order)
229 if self._mRA != m:
230 _update_all_rls(self)
231 self._mRA = m
233# _RhumbLine = RhumbLine # see further below
235 def _S12d(self, psi1, psi2, lon12): # degrees
236 '''(INTERNAL) Compute the area C{S12}.
237 '''
238 S = (self.ellipsoid.areax if self.exact else
239 self.ellipsoid.area) * lon12 / _720_0
240 if S:
241 x, y = radians(psi1), radians(psi2) # _meanSinXi(x, y)
242 s = _Dlog(cosh(x), cosh(y)) * _Dcosh(x, y)
243 if self.f:
244 s += _sincosSeries(False, _gd(x), _gd(y), *self._RA2) * _Dgd(x, y)
245 S *= s
246 return S
249class RhumbLine(RhumbLineBase):
250 '''Compute one or several points on a single rhumb line.
252 Class C{RhumbLine} facilitates the determination of points on
253 a single rhumb line. The starting point (C{lat1}, C{lon1})
254 and the azimuth C{azi12} are specified once.
255 '''
256 _Rhumb = Rhumb # rhumb.ekx.Rhumb
258 def __init__(self, rhumb, lat1=0, lon1=0, azi12=None, **caps_name): # PYCHOK signature
259 '''New C{RhumbLine}.
261 @arg rhumb: The rhumb reference (L{Rhumb}).
262 @kwarg lat1: Latitude of the start point (C{degrees90}).
263 @kwarg lon1: Longitude of the start point (C{degrees180}).
264 @kwarg azi12: Azimuth of this rhumb line (compass C{degrees}).
265 @kwarg caps_name: Optional keyword arguments C{B{name}=NN} and C{B{caps}=0},
266 a bit-or'ed combination of L{Caps<pygeodesy.karney.Caps>} values
267 specifying the required capabilities. Include C{Caps.LINE_OFF}
268 if updates to the B{C{rhumb}} should I{not be reflected} in this
269 rhumb line.
270 '''
271 RhumbLineBase.__init__(self, rhumb, lat1, lon1, azi12, **caps_name)
273 @Property_RO
274 def _dpm12(self): # PYCHOK no cover
275 '''(INTERNAL) Get this rhumb line's parallel I{circle radius} (C{degree per meter}).
276 '''
277 r = self._salp
278 if r:
279 r = _over(r, self.ellipsoid.circle4(self.lat1).radius)
280 return r
282 @Property_RO
283 def _mu1(self):
284 '''(INTERNAL) Get the I{rectifying auxiliary} latitude (C{degrees}).
285 '''
286 return self.ellipsoid.auxRectifying(self.lat1)
288 def _mu2lat(self, mu):
289 '''(INTERNAL) Get the inverse I{rectifying auxiliary} latitude (C{degrees}).
290 '''
291 return self.ellipsoid.auxRectifying(mu, inverse=True)
293 def _Position4(self, unused, mu2, s12, mu12):
294 '''(INTERNAL) See method C{RhumbLineBase._Position}.
295 '''
296 psi1 = psi2 = self._psi1
297 if mu12: # self._calp != 0
298 lat2 = self._mu2lat(mu2)
299 psi12 = self.rhumb._DRectifying2Isometricd(mu2, self._mu1) * mu12
300 lon2 = self._talp * psi12
301 psi2 += psi12
302 else: # meridional
303 lat2 = self.lat1
304 lon2 = self._dpm12 * s12
305 return lat2, lon2, psi1, psi2
307 @Property_RO
308 def _psi1(self):
309 '''(INTERNAL) Get the I{isometric auxiliary} latitude C{psi} (C{degrees}).
310 '''
311 return self.ellipsoid.auxIsometric(self.lat1)
313 @property_RO
314 def RAorder(self):
315 '''Get this rhumb line's I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8).
316 '''
317 return self.rhumb.RAorder
319Rhumb._RhumbLine = RhumbLine # PYCHOK see RhumbBase._RhumbLine
322# Use I{Divided Differences} to determine (mu2 - mu1) / (psi2 - psi1) accurately.
323# Definition: _Df(x,y,d) = (f(x) - f(y)) / (x - y), @see W. M. Kahan & R. J.
324# Fateman, "Symbolic computation of Divided Differences", SIGSAM Bull. 33(3),
325# 7-28 (1999). U{ACM<https://DL.ACM.org/doi/pdf/10.1145/334714.334716> and @see
326# U{UCB<https://www.CS.Berkeley.edu/~fateman/papers/divdiff.pdf>}, Dec 8, 1999.
328def _Dasinh(x, y):
329 hx = hypot1(x)
330 d = x - y
331 if d:
332 hx *= y
333 hy = x * hypot1(y)
334 t = (d * (x + y) / (hy + hx)) if (x * y) > 0 else (hy - hx)
335 r = asinh(t) / d
336 else:
337 r = _1_0 / hx
338 return r
341def _Datan(x, y):
342 xy = x * y
343 r = xy + _1_0
344 d = x - y
345 if d: # 2 * xy > -1 == 2 * xy + 1 > 0 == xy + r > 0 == xy > -r
346 r = (atan1(d, r) if xy > -r else (atan1(x) - atan1(y))) / d
347 else:
348 r = _1_over(r)
349 return r
352def _Dcosh(x, y):
353 return _Dsincos(x, y, sinh, sinh)
356def _DeatanhE(x, y, E): # see .albers._Datanhee
357 # Deatanhe(x, y) = eatanhe((x - y) / (1 - e^2 * x * y)) / (x - y)
358 e = _1_0 - E.e2 * x * y
359 if e: # assert not isnear0(e)
360 d = x - y
361 e = (E._es_atanh(d / e) / d) if d else (E.e2 / e)
362 return e
365def _DfEt(tx, ty, eF): # tangents
366 # eF = Elliptic(-E.e12) # -E.e2 / (1 - E.e2)
367 r, x, y, = _1_0, atan(tx), atan(ty)
368 d = x - y
369 if (x * y) > 0:
370 # See U{DLMF<https://DLMF.NIST.gov/19.11>}: 19.11.2 and 19.11.4
371 # letting theta -> x, phi -> -y, psi -> z
372 # (E(x) - E(y)) / d = E(z)/d - k2 * sin(x) * sin(y) * sin(z)/d
373 # tan(z/2) = (sin(x)*Delta(y) - sin(y)*Delta(x)) / (cos(x) + cos(y))
374 # = d * Dsin(x,y) * (sin(x) + sin(y))/(cos(x) + cos(y)) /
375 # (sin(x)*Delta(y) + sin(y)*Delta(x))
376 # = t = d * Dt
377 # sin(z) = 2*t/(1+t^2); cos(z) = (1-t^2)/(1+t^2)
378 # Alt (this only works for |z| <= pi/2 -- however, this conditions
379 # holds if x*y > 0):
380 # sin(z) = d * Dsin(x,y) * (sin(x) + sin(y)) /
381 # (sin(x)*cos(y)*Delta(y) + sin(y)*cos(x)*Delta(x))
382 # cos(z) = sqrt((1-sin(z))*(1+sin(z)))
383 sx, cx, sy, cy = sincos2_(x, y)
384 D = (cx + cy) * (eF.fDelta(sy, cy) * sx +
385 eF.fDelta(sx, cx) * sy)
386 D = (sx + sy) * _Dsin(x, y) / D
387 t = D * d
388 t2 = _1_0 + t**2
389 D *= _2_0 / t2
390 s = D * d
391 if s:
392 c = (t + _1_0) * (_1_0 - t) / t2
393 r = eF.fE(s, c, eF.fDelta(s, c)) / s
394 r = D * (r - eF.k2 * sx * sy)
395 elif d:
396 r = (eF.fE(x) - eF.fE(y)) / d
397 return r
400def _Dgd(x, y):
401 return _Datan(sinh(x), sinh(y)) * _Dsinh(x, y)
404def _Dgdinv(x, y): # x, y are tangents
405 return _Dasinh(x, y) / _Datan(x, y)
408def _Dlog(x, y):
409 d = (x - y) * _0_5
410 # Changed atanh(t / (x + y)) to asinh(t / (2 * sqrt(x*y))) to
411 # avoid taking atanh(1) when x is large and y is 1. This also
412 # fixes bogus results being returned for the area when an endpoint
413 # is at a pole. N.B. this routine is invoked with positive x
414 # and y, so the sqrt is always taken of a positive quantity.
415 return (asinh(d / sqrt(x * y)) / d) if d else _1_over(x)
418def _Dsin(x, y):
419 return _Dsincos(x, y, sin, cos)
422def _Dsincos(x, y, sin_, cos_):
423 r = cos_((x + y) * _0_5)
424 d = (x - y) * _0_5
425 if d:
426 r *= sin_(d) / d
427 return r
430def _Dsinh(x, y):
431 return _Dsincos(x, y, sinh, cosh)
434def _Dtan(x, y): # PYCHOK no cover
435 return _Dtant(x - y, tan(x), tan(y))
438def _Dtant(dxy, tx, ty):
439 txy = tx * ty
440 r = txy + _1_0
441 if dxy: # 2 * txy > -1 == 2 * txy + 1 > 0 == txy + r > 0 == txy > -r
442 r = ((tan(dxy) * r) if txy > -r else (tx - ty)) / dxy
443 return r
446def _Eaux4(E_aux, mu_psi_x, mu_psi_y): # degrees
447 # get inverse auxiliary lats in radians and tangents
448 phix = radians(E_aux(mu_psi_x, inverse=True))
449 phiy = radians(E_aux(mu_psi_y, inverse=True))
450 return phix, phiy, tan(phix), tan(phiy)
453def _gd(x):
454 return atan(sinh(x))
457def _sincosSeries(sinp, x, y, C, n):
458 # N.B. C[] has n+1 elements of which
459 # C[0] is ignored and n >= 0
460 # Use Clenshaw summation to evaluate
461 # m = (g(x) + g(y)) / 2 -- mean value
462 # s = (g(x) - g(y)) / (x - y) -- average slope
463 # where
464 # g(x) = sum(C[j] * SC(2 * j * x), j = 1..n)
465 # SC = sinp ? sin : cos
466 # CS = sinp ? cos : sin
467 # ...
468 d, _neg = (x - y), neg
469 sp, cp, sd, cd = sincos2_(x + y, d)
470 sd = (sd / d) if d else _1_0
471 s = _neg(sp * sd) # negative
472 # 2x2 matrices in row-major order
473 a1 = s * d**2
474 a2 = s * _4_0
475 a0 = a3 = _2_0 * cp * cd # m
476 b2 = b1 = _0_0s(4)
477 if n > 0:
478 b1 = C[n], _0_0, _0_0, C[n]
480 _fsum = _MODS.fsums.fsum1f_
481 for j in range(n - 1, 0, -1): # C[0] unused
482 b1, b2, Cj = b2, b1, C[j]
483 # b1 = a * b2 - b1 + C[j] * I
484 m0, m1, m2, m3 = b2
485 n0, n1, n2, n3 = map(_neg, b1)
486 b1 = (_fsum(a0 * m0, a1 * m2, n0, Cj),
487 _fsum(a0 * m1, a1 * m3, n1),
488 _fsum(a2 * m0, a3 * m2, n2),
489 _fsum(a2 * m1, a3 * m3, n3, Cj))
490 # Here are the full expressions for m and s
491 # f01, f02, f11, f12 = (0, 0, cd * sp, 2 * sd * cp) if sinp else \
492 # (1, 0, cd * cp, -2 * sd * sp)
493 # m = -b2[1] * f02 + (C[0] - b2[0]) * f01 + b1[0] * f11 + b1[1] * f12
494 # s = -b2[2] * f01 + (C[0] - b2[3]) * f02 + b1[2] * f11 + b1[3] * f12
495 cd *= b1[2]
496 sd *= b1[3] * _2_0
497 s = _fsum(cd * sp, sd * cp) if sinp else \
498 _fsum(cd * cp, _neg(sd * sp), _neg(b2[2]))
499 return s
502_RACoeffs = { # Generated by Maxima on 2015-05-15 08:24:04-04:00
503 4: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4
504 691, 7860, -20160, 18900, 0, 56700, # R[0]/n^0, polynomial(n), order 4
505 1772, -5340, 6930, -4725, 14175, # R[1]/n^1, polynomial(n), order 3
506 -1747, 1590, -630, 4725, # PYCHOK R[2]/n^2, polynomial(n), order 2
507 104, -31, 315, # R[3]/n^3, polynomial(n), order 1
508 -41, 420), # PYCHOK R[4]/n^4, polynomial(n), order 0, count = 20
509 5: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5
510 -79036, 22803, 259380, -665280, 623700, 0, 1871100, # PYCHOK R[0]/n^0, polynomial(n), order 5
511 41662, 58476, -176220, 228690, -155925, 467775, # PYCHOK R[1]/n^1, polynomial(n), order 4
512 18118, -57651, 52470, -20790, 155925, # PYCHOK R[2]/n^2, polynomial(n), order 3
513 -23011, 17160, -5115, 51975, # PYCHOK R[3]/n^3, polynomial(n), order 2
514 5480, -1353, 13860, # PYCHOK R[4]/n^4, polynomial(n), order 1
515 -668, 5775), # PYCHOK R[5]/n^5, polynomial(n), order 0, count = 27
516 6: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6
517 128346268, -107884140, 31126095, 354053700, -908107200, 851350500, 0, 2554051500, # R[0]/n^0, polynomial(n), order 6
518 -114456994, 56868630, 79819740, -240540300, 312161850, -212837625, 638512875, # PYCHOK R[1]/n^1, polynomial(n), order 5
519 51304574, 24731070, -78693615, 71621550, -28378350, 212837625, # R[2]/n^2, polynomial(n), order 4
520 1554472, -6282003, 4684680, -1396395, 14189175, # R[3]/n^3, polynomial(n), order 3
521 -4913956, 3205800, -791505, 8108100, # PYCHOK R[4]/n^4, polynomial(n), order 2
522 1092376, -234468, 2027025, # R[5]/n^5, polynomial(n), order 1
523 -313076, 2027025), # PYCHOK R[6]/n^6, polynomial(n), order 0, count = 35
524 7: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7
525 -317195588, 385038804, -323652420, 93378285, 1062161100, -2724321600, 2554051500, 0, 7662154500, # PYCHOK R[0]/n^0, polynomial(n), order 7
526 258618446, -343370982, 170605890, 239459220, -721620900, 936485550, -638512875, 1915538625, # PYCHOK R[1]/n^1, polynomial(n), order 6
527 -248174686, 153913722, 74193210, -236080845, 214864650, -85135050, 638512875, # PYCHOK R[2]/n^2, polynomial(n), order 5
528 114450437, 23317080, -94230045, 70270200, -20945925, 212837625, # PYCHOK R[3]/n^3, polynomial(n), order 4
529 15445736, -103193076, 67321800, -16621605, 170270100, # PYCHOK R[4]/n^4, polynomial(n), order 3
530 -27766753, 16385640, -3517020, 30405375, # PYCHOK R[4]/n^4, polynomial(n), order 3
531 4892722, -939228, 6081075, # PYCHOK R[4]/n^4, polynomial(n), order 3
532 -3189007, 14189175), # PYCHOK R[7]/n^7, polynomial(n), order 0, count = 44
533 8: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8
534 71374704821, -161769749880, 196369790040, -165062734200, 47622925350, 541702161000, -1389404016000, 1302566265000, 0, 3907698795000, # R[0]/n^0, polynomial(n), order 8
535 -13691187484, 65947703730, -87559600410, 43504501950, 61062101100, -184013329500, 238803815250, -162820783125, 488462349375, # PYCHOK R[1]/n^1, polynomial(n), order 7
536 30802104839, -63284544930, 39247999110, 18919268550, -60200615475, 54790485750, -21709437750, 162820783125, # R[2]/n^2, polynomial(n), order 6
537 -8934064508, 5836972287, 1189171080, -4805732295, 3583780200, -1068242175, 10854718875, # PYCHOK R[3]/n^3, polynomial(n), order 5
538 50072287748, 3938662680, -26314234380, 17167059000, -4238509275, 43418875500, # R[4]/n^4, polynomial(n), order 4
539 359094172, -9912730821, 5849673480, -1255576140, 10854718875, # R[5]/n^5, polynomial(n), order 3
540 -16053944387, 8733508770, -1676521980, 10854718875, # PYCHOK R[6]/n^6, polynomial(n), order 2
541 930092876, -162639357, 723647925, # R[7]/n^7, polynomial(n), order 1
542 -673429061, 1929727800) # PYCHOK R[8]/n^8, polynomial(n), order 0, count = 54
543}
545__all__ += _ALL_DOCS(Caps)
547# **) MIT License
548#
549# Copyright (C) 2022-2025 -- mrJean1 at Gmail -- All Rights Reserved.
550#
551# Permission is hereby granted, free of charge, to any person obtaining a
552# copy of this software and associated documentation files (the "Software"),
553# to deal in the Software without restriction, including without limitation
554# the rights to use, copy, modify, merge, publish, distribute, sublicense,
555# and/or sell copies of the Software, and to permit persons to whom the
556# Software is furnished to do so, subject to the following conditions:
557#
558# The above copyright notice and this permission notice shall be included
559# in all copies or substantial portions of the Software.
560#
561# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
562# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
563# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
564# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
565# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
566# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
567# OTHER DEALINGS IN THE SOFTWARE.