Coverage for pygeodesy/rhumbaux.py: 96%
163 statements
« prev ^ index » next coverage.py v7.2.2, created at 2023-08-06 15:27 -0400
« prev ^ index » next coverage.py v7.2.2, created at 2023-08-06 15:27 -0400
1# -*- coding: utf-8 -*-
3u'''A pure Python version of I{Karney}'s C++ classes U{Rhumb
4<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Rhumb.html>} and U{RhumbLine
5<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1RhumbLine.html>} from
6I{GeographicLib version 2.2+}.
8Class L{RhumbLine} has been enhanced with methods C{intersection2} and C{nearestOn4} to iteratively
9find the intersection of two rhumb lines, respectively the nearest point on a rumb line along a
10geodesic or perpendicular rhumb line.
12For more details, see the I{2.2} U{GeographicLib<https://GeographicLib.SourceForge.io/C++/doc/index.html>}
13documentation, especially the U{Class List<https://GeographicLib.SourceForge.io/C++/doc/annotated.html>},
14the background information on U{Rhumb lines<https://GeographicLib.SourceForge.io/C++/doc/rhumb.html>},
15utility U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>} and U{Online rhumb
16line calculations<https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve>}.
18Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2022-2023) and licensed under the MIT/X11
19License. For more information, see the U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation.
21@note: Class L{AuxDST} requires package U{numpy<https://PyPI.org/project/numpy>} to be installed,
22 version 1.16 or newer and needed for I{exact} area calculations.
23'''
24# make sure int/int division yields float quotient
25from __future__ import division as _; del _ # PYCHOK semicolon
27from pygeodesy.auxilats.auxAngle import AuxMu, AuxPhi, atan2d, hypot
28from pygeodesy.auxilats.auxDLat import AuxDLat, _DClenshaw
29# from pygeodesy.auxilats.auxDST import AuxDST # _MODS
30from pygeodesy.auxilats.auxily import _Dlam, _Dp0Dpsi, _Ufloats
31from pygeodesy.basics import _reverange, unsigned0, _zip, _xkwds_get
32from pygeodesy.constants import EPS_2, MANT_DIG, NAN, PI4, isinf, \
33 _0_0, _4_0, _720_0, _log2, _over
34# from pygeodesy.errors import itemsorted, _xkwds_get # from .basics, ...
35from pygeodesy.karney import Caps, _diff182, _EWGS84, GDict, _norm180
36# from pygeodesy.fmath import hypot # from .auxilats.auxAngle
37from pygeodesy.interns import NN, _COMMASPACE_
38from pygeodesy.lazily import _ALL_LAZY, _ALL_DOCS, _ALL_MODS as _MODS
39# from pygeodesy.props import Property, Property_RO # from .rhumbBase
40from pygeodesy.rhumbBase import RhumbBase, RhumbLineBase, itemsorted, \
41 pairs, Property, Property_RO
42# from pygeodesy.streprs import pairs # from .rhumbBase
43# from pygeodesy.utily import atan2d # from .auxilats.auxAngle
45from math import ceil as _ceil, fabs, radians
47__all__ = _ALL_LAZY.rhumbaux
48__version__ = '23.08.06'
50# DIGITS = (sizeof(real) * 8) bits # assert DIGITS > MANT_DIG
51# For |n| <= 0.99, actual max for doubles is 2163. This scales
52# as DIGITS and for long doubles (GEOGRAPHICLIB_PRECISION = 3,
53# DIGITS = 64), this becomes 2163 * 64 / 53 = 2612. Round this
54# up to 2^12 = 4096 and scale this by DIGITS//64 if DIGITS > 64.
55#
56# 64 = DIGITS for long double, 6 = 12 - _log2(64)
57_Lbits = 1 << (int(_ceil(_log2(max(MANT_DIG, 64)))) + 6)
58_rls = [] # instances of C{RbumbLineAux} to be updated
61class RhumbAux(RhumbBase):
62 '''Class to solve the I{direct} and I{inverse rhumb} problems, based
63 on I{Auxiliary Latitudes} for accuracy near the poles.
65 @note: Package U{numpy<https://PyPI.org/project/numpy>} must be
66 installed, version 1.16 or later.
67 '''
69 def __init__(self, a_earth=_EWGS84, f=None, exact=True, name=NN, **TMorder): # PYCHOK signature
70 '''New C{rhumbaux.RhumbAux}.
72 @kwarg a_earth: This rhumb's earth model (L{Ellipsoid}, L{Ellipsoid2},
73 L{a_f2Tuple}, L{Datum}, 2-tuple C{(a, f)}) or the
74 (equatorial) radius (C{scalar}).
75 @kwarg f: The ellipsoid's flattening (C{scalar}), iff B{C{a_earth}} is
76 a C{scalar}, ignored otherwise.
77 @kwarg exact: If C{True}, use the exact expressions for the I{Auxiliary
78 Latitudes}, otherwise use the I{Fourier} series expansion
79 (C{bool}), see also property C{exact}.
80 @kwarg name: Optional name (C{str}).
81 @kwarg TMorder: Optional keyword argument B{C{TMorder}}, see property
82 C{TMorder}.
84 @raise ImportError: Package C{numpy} not found or not installed, only
85 required when C{B{exact} is True}.
87 @raise RhumbError: Invalid B{C{a_earth}}, B{C{f}} or B{C{RA_TMorder}}.
88 '''
89 RhumbBase.__init__(self, a_earth, f, exact, name)
90 if TMorder:
91 self.Tmorder = _xkwds_get(TMorder, TMorder=RhumbBase._mTM)
93 def areaux(self, **exact):
94 '''Get this ellipsoid's B{C{exact}} surface area (C{meter} I{squared}).
96 @kwarg exact: Optional C{exact} (C{bool}), overriding this rhumb's
97 C{exact} setting, if C{True}, use the exact expression
98 for the authalic radius otherwise the I{Taylor} series.
100 @return: The (signed?) surface area (C{meter} I{squared}).
102 @raise AuxError: If C{B{exact}=False} and C{abs(flattening)} exceeds
103 property C{f_max}.
105 @note: The area of a polygon encircling a pole can be found by adding
106 C{areaux / 2} to the sum of C{S12} for each side of the polygon.
108 @see: U{The area of rhumb polygons<https://ArXiv.org/pdf/2303.03219.pdf>}
109 and method L{auxilats.AuxLat.AuthalicRadius2}.
110 '''
111 x = _xkwds_get(exact, exact=self.exact)
112 a = (self._c2 * _720_0) if bool(x) is self.exact else \
113 (self._auxD.AuthalicRadius2(exact=x, f_max=self.f_max) * PI4)
114 return a
116 @Property_RO
117 def _auxD(self):
118 return AuxDLat(self.ellipsoid)
120 @Property_RO
121 def _c2(self): # radians makes _c2 a factor per degree
122 return radians(self._auxD.AuthalicRadius2(exact=self.exact, f_max=self.f_max))
124 def Direct(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE):
125 '''Solve the I{direct rhumb} problem, optionally with the area.
127 @arg lat1: Latitude of the first point (C{degrees90}).
128 @arg lon1: Longitude of the first point (C{degrees180}).
129 @arg azi12: Azimuth of the rhumb line (compass C{degrees}).
130 @arg s12: Distance along the rhumb line from the given to
131 the destination point (C{meter}), can be negative.
133 @return: L{GDict} with 2 up to 8 items C{lat2, lon2, a12, S12,
134 lat1, lon1, azi12, s12} with the destination point's
135 latitude C{lat2} and longitude C{lon2} in C{degrees},
136 the rhumb angle C{a12} in C{degrees} and area C{S12}
137 under the rhumb line in C{meter} I{squared}.
139 @note: If B{C{s12}} is large enough that the rhumb line crosses
140 a pole, the longitude of the second point is indeterminate
141 and C{NAN} is returned for C{lon2} and area C{S12}.
143 @note: If the given point is a pole, the cosine of its latitude is
144 taken to be C{sqrt(EPS)}. This position is extremely close
145 to the actual pole and allows the calculation to be carried
146 out in finite terms.
147 '''
148 rl = RhumbLineAux(self, lat1, lon1, azi12, caps=Caps.LINE_OFF,
149 name=self.name)
150 return rl.Position(s12, outmask) # lat2, lon2, S12
152 def _DMu_DPsi(self, Phi1, Phi2, Chi1, Chi2):
153 xD = self._auxD
154 return _over(xD.DRectifying(Phi1, Phi2),
155 xD.DIsometric( Phi1, Phi2)) if self.exact else \
156 _over(xD.DRectifyingZ(Chi1, Chi2),
157 _Dlam(Chi1.tan, Chi2.tan)) # not Lambertian!
159 def Inverse(self, lat1, lon1, lat2, lon2, outmask=Caps.AZIMUTH_DISTANCE):
160 '''Solve the I{inverse rhumb} problem.
162 @arg lat1: Latitude of the first point (C{degrees90}).
163 @arg lon1: Longitude of the first point (C{degrees180}).
164 @arg lat2: Latitude of the second point (C{degrees90}).
165 @arg lon2: Longitude of the second point (C{degrees180}).
167 @return: L{GDict} with 5 to 8 items C{azi12, s12, a12, S12,
168 lat1, lon1, lat2, lon2}, the rhumb line's azimuth C{azi12}
169 in compass C{degrees} between C{-180} and C{+180}, the
170 distance C{s12} and rhumb angle C{a12} between both points
171 in C{meter} respectively C{degrees} and the area C{S12}
172 under the rhumb line in C{meter} I{squared}.
174 @note: The shortest rhumb line is found. If the end points are
175 on opposite meridians, there are two shortest rhumb lines
176 and the East-going one is chosen.
178 @note: If either point is a pole, the cosine of its latitude is
179 taken to be C{epsilon}**-2 (where C{epsilon} is 2.0**-52).
180 This position is extremely close to the actual pole and
181 allows the calculation to be carried out in finite terms.
182 '''
183 r, Cs = GDict(name=self.name), Caps
184 if (outmask & Cs.AZIMUTH_DISTANCE_AREA):
185 psi1, Chi1, Phi1 = self._psiChiPhi3(lat1)
186 psi2, Chi2, Phi2 = self._psiChiPhi3(lat2)
188 psi12 = psi2 - psi1
189 lon12, _ = _diff182(lon1, lon2, K_2_0=True)
190 lam12 = radians(lon12)
191 if (outmask & Cs.AZIMUTH):
192 r.set_(azi12=atan2d(lam12, psi12))
193 if (outmask & Cs.DISTANCE):
194 if isinf(psi1) or isinf(psi2): # PYCHOK no cover
195 d = Phi2.toMu(self).toRadians
196 d -= Phi1.toMu(self).toRadians
197 s = fabs(d)
198 else: # dmu/dpsi = dmu/dchi/dpsi/dchi
199 s = self._DMu_DPsi(Phi1, Phi2, Chi1, Chi2)
200 s *= hypot(lam12, psi12)
201 r.set_(s12=self._rrm * s)
202 if (outmask & Cs.AREA):
203 S = self._c2SinXi(Chi1, Chi2)
204 r.set_(S12=unsigned0(S * lon12)) # like .gx
205 return r
207 def _c2SinXi(self, Chix, Chiy):
208 pP, xD = self._RA, self._auxD
210 tx, Phix = Chix.tan, Chix.toPhi(self)
211 ty, Phiy = Chiy.tan, Chiy.toPhi(self)
212 dD = _DClenshaw(False, Phix.toBeta(self).normalized,
213 Phiy.toBeta(self).normalized,
214 pP, min(len(pP), 8)) # Fsum
215 dD *= _over(xD.DParametric(Phix, Phiy),
216 xD.DIsometric( Phix, Phiy)) if self.exact else \
217 _over(xD.DParametricZ(Chix, Chiy), _Dlam(tx, ty)) # not Lambertian!
218 dD += _Dp0Dpsi(tx, ty)
219 dD *= self._c2
220 return float(dD)
222 def _psiChiPhi3(self, lat):
223 Phi = AuxPhi.fromDegrees(lat)
224 Chi = Phi.toChi(self)
225 psi = Chi.toLambertianRadians
226 return psi, Chi, Phi
228 @Property_RO
229 def _RA(self): # get the coefficients for area calculation
230 return tuple(_RAintegrate(self._auxD) if self.exact else
231 _RAseries(self._auxD))
233 @Property_RO
234 def _RhumbLine(self):
235 '''(INTERNAL) Get this module's C{RhumbLineAux} class.
236 '''
237 return RhumbLineAux
239 @Property_RO
240 def _rrm(self):
241 return self._auxD.RectifyingRadius(exact=self.exact)
243 @Property
244 def TMorder(self):
245 '''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
246 '''
247 return self._mTM
249 @TMorder.setter # PYCHOK setter!
250 def TMorder(self, order):
251 '''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
252 '''
253 self._TMorder(order)
255 def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature
256 '''Return this C{Rhumb} as string.
258 @kwarg prec: The C{float} precision, number of decimal digits (0..9).
259 Trailing zero decimals are stripped for B{C{prec}} values
260 of 1 and above, but kept for negative B{C{prec}} values.
261 @kwarg sep: Separator to join (C{str}).
263 @return: Tuple items (C{str}).
264 '''
265 d = dict(ellipsoid=self.ellipsoid, exact=self.exact,
266 TMorder=self.TMorder)
267 return sep.join(pairs(itemsorted(d, asorted=False), prec=prec))
270class RhumbLineAux(RhumbLineBase):
271 '''Compute one or several points on a single rhumb line.
273 Class C{RhumbLineAux} facilitates the determination of points
274 on a single rhumb line. The starting point (C{lat1}, C{lon1})
275 and the azimuth C{azi12} are specified once.
277 Method C{RhumbLineAux.Position} returns the location of an
278 other point and optionally the distance C{s12} along and the
279 area C{S12} under the rhumb line.
281 Method C{RhumbLineAux.intersection2} finds the intersection
282 between two rhumb lines.
284 Method C{RhumbLineAux.nearestOn4} computes the nearest point
285 on and the distance to a rhumb line in different ways.
286 '''
287 _Rhumb = RhumbAux # rhumbaux.RhumbAux
289 def __init__(self, rhumb, lat1=0, lon1=0, azi12=None, **caps_name): # PYCHOK signature
290 '''New C{rhumbaux.RhumbLineAux}.
292 @arg rhumb: The rhumb reference (C{rhumbaux.RhumbAux}).
293 @kwarg lat1: Latitude of the start point (C{degrees90}).
294 @kwarg lon1: Longitude of the start point (C{degrees180}).
295 @kwarg azi12: Azimuth of this rhumb line (compass C{degrees}).
296 @kwarg caps_name: Optional keyword arguments C{B{name}=NN} and
297 C{B{caps}=0}, a bit-or'ed combination of L{Caps}
298 values specifying the required capabilities. Include
299 C{Caps.LINE_OFF} if updates to the B{C{rhumb}} should
300 I{not} be reflected in this rhumb line.
301 '''
302 RhumbLineBase.__init__(self, rhumb, lat1, lon1, azi12, **caps_name)
304 @Property_RO
305 def _Chi1(self):
306 return self._Phi1.toChi(self.rhumb)
308 @Property_RO
309 def _mu1(self):
310 return self._Phi1.toMu(self.rhumb).toDegrees
312 @Property_RO
313 def _Phi1(self):
314 return AuxPhi.fromDegrees(self.lat1)
316 def Position(self, s12, outmask=Caps.LATITUDE_LONGITUDE):
317 '''Compute a point at a distance on this rhumb line.
319 @arg s12: The distance along this rhumb line between its origin
320 and the point (C{meters}), can be negative.
321 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying
322 the quantities to be returned.
324 @return: L{GDict} with 4 to 8 items C{azi12, a12, s12, S12, lat2,
325 lon2, lat1, lon1} with latitude C{lat2} and longitude
326 C{lon2} of the point in C{degrees}, the rhumb angle C{a12}
327 in C{degrees} from the start point of and the area C{S12}
328 under this rhumb line in C{meter} I{squared}.
330 @note: If B{C{s12}} is large enough that the rhumb line crosses a
331 pole, the longitude of the second point is indeterminate and
332 C{NAN} is returned for C{lon2} and area C{S12}.
334 If the first point is a pole, the cosine of its latitude is
335 taken to be C{epsilon}**-2 (where C{epsilon} is 2**-52).
336 This position is extremely close to the actual pole and
337 allows the calculation to be carried out in finite terms.
338 '''
339 r, Cs = GDict(name=self.name), Caps
340 if (outmask & Cs.LATITUDE_LONGITUDE_AREA):
341 E, R = self.ellipsoid, self.rhumb
342 r12 = _over(s12, radians(R._rrm))
343 mu2, x90 = self._mu22(self._calp * r12, self._mu1)
344 Mu2 = AuxMu.fromDegrees(mu2)
345 Phi2 = Mu2.toPhi(R)
346 lat2 = Phi2.toDegrees
347 if x90: # PYCHOK no cover
348 lon2 = NAN
349 if (outmask & Cs.AREA):
350 r.set_(S12=NAN)
351 else:
352 Chi2 = Phi2.toChi(R)
353 Chi1 = self._Chi1
354 lon2 = R._DMu_DPsi(self._Phi1, Phi2, Chi1, Chi2)
355 lon2 = _over(self._salp * r12, lon2)
356 if (outmask & Cs.AREA):
357 S = R._c2SinXi(Chi1, Chi2)
358 r.set_(S12=unsigned0(S * lon2)) # like .gx
359 if (outmask & Cs.LONGITUDE):
360 if (outmask & Cs.LONG_UNROLL):
361 lon2 += self.lon1
362 else:
363 lon2 = _norm180(self._lon12 + lon2)
364 r.set_(azi12=self.azi12, s12=s12, a12=s12 / E._L_90)
365 if (outmask & Cs.LATITUDE):
366 r.set_(lat2=lat2, lat1=self.lat1)
367 if (outmask & Cs.LONGITUDE):
368 r.set_(lon2=lon2, lon1=self.lon1)
369 return r
371# @Property_RO
372# def _psi1(self):
373# return self._Chi1.toLambertianRadians
376def _RAintegrate(auxD):
377 # Compute coefficients by Fourier transform of integrand
378 L = 2
379 fft = _MODS.auxilats.auxDST.AuxDST(L)
380 f = auxD._qIntegrand
381 c = fft.transform(f)
382 # assert L < _Lbits
383 while L < _Lbits:
384 fft.reset(L)
385 c = fft.refine(f, c)
386 L *= 2 # == len(c)
387 # assert len(c) == L
388 pP, k = [], -1
389 for j in range(1, L + 1):
390 # Compute Fourier coefficients of integral
391 p = -(c[j - 1] + (c[j] if j < L else _0_0)) / (_4_0 * j)
392 if fabs(p) > EPS_2:
393 k = -1 # run interrupted
394 else:
395 if k < 0:
396 k = 1 # mark as first small value
397 if (j - k) >= ((j + 7) // 8):
398 # run of at least (j - 1) // 8 small values
399 return pP[:j]
400 pP.append(p)
401 return pP # no convergence, use pP as-is
404def _RAseries(auxD):
405 # Series expansions in n for Fourier coeffients of the integral
406 # @see: U{"Series expansions for computing rhumb areas"
407 # <https:#DOI.org/10.5281/zenodo.7685484>}.
408 d = n = auxD._n
409 i = 0
410 pP = []
411 aL = auxD.ALorder
412 Cs = _RACoeffs[aL]
413 # assert len(Cs) == (aL * (aL + 1)) // 2
414 _p = _MODS.karney._polynomial
415 for m in _reverange(aL): # order
416 j = i + m + 1
417 pP.append(_p(n, Cs, i, j) * d)
418 d *= n
419 i = j
420 # assert i == len(pP)
421 return pP
424_f, _u = float, _Ufloats()
425_RACoeffs = { # Rhumb Area Coefficients in matrix Q
426 4: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4
427 596 / _f(2025), -398 / _f(945), 22 / _f(45), -1 / _f(3),
428 1543 / _f(4725), -118 / _f(315), 1 / _f(5),
429 152 / _f(945), -17 / _f(315),
430 5 / _f(252)),
431 5: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5
432 -102614 / _f(467775), 596 / _f(2025), -398 / _f(945), 22 / _f(45),
433 -1 / _f(3),
434 -24562 / _f(155925), 1543 / _f(4725), -118 / _f(315), 1 / _f(5),
435 -38068 / _f(155925), 152 / _f(945), -17 / _f(315),
436 -752 / _f(10395), 5 / _f(252),
437 -101 / _f(17325)),
438 6: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6
439 138734126 / _f(638512875), -102614 / _f(467775), 596 / _f(2025),
440 -398 / _f(945), 22 / _f(45), -1 / _f(3),
441 17749373 / _f(425675250), -24562 / _f(155925), 1543 / _f(4725),
442 -118 / _f(315), 1 / _f(5),
443 1882432 / _f(8513505), -38068 / _f(155925), 152 / _f(945),
444 -17 / _f(315),
445 268864 / _f(2027025), -752 / _f(10395), 5 / _f(252),
446 62464 / _f(2027025), -101 / _f(17325),
447 11537 / _f(4054050)),
448 7: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7
449 -565017322 / _f(1915538625), 138734126 / _f(638512875),
450 -102614 / _f(467775), 596 / _f(2025), -398 / _f(945), 22 / _f(45),
451 -1 / _f(3),
452 -1969276 / _f(58046625), 17749373 / _f(425675250), -24562 / _f(155925),
453 1543 / _f(4725), -118 / _f(315), 1 / _f(5),
454 -58573784 / _f(638512875), 1882432 / _f(8513505), -38068 / _f(155925),
455 152 / _f(945), -17 / _f(315),
456 -6975184 / _f(42567525), 268864 / _f(2027025), -752 / _f(10395),
457 5 / _f(252),
458 -112832 / _f(1447875), 62464 / _f(2027025), -101 / _f(17325),
459 -4096 / _f(289575), 11537 / _f(4054050),
460 -311 / _f(525525)),
461 8: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8
462 188270561816 / _f(488462349375), -565017322 / _f(1915538625),
463 138734126 / _f(638512875), -102614 / _f(467775), 596 / _f(2025),
464 -398 / _f(945), 22 / _f(45), -1 / _f(3),
465 2332829602 / _f(23260111875), -1969276 / _f(58046625),
466 17749373 / _f(425675250), -24562 / _f(155925), 1543 / _f(4725),
467 -118 / _f(315), 1 / _f(5),
468 -41570288 / _f(930404475), -58573784 / _f(638512875),
469 1882432 / _f(8513505), -38068 / _f(155925), 152 / _f(945),
470 -17 / _f(315),
471 1538774036 / _f(10854718875), -6975184 / _f(42567525),
472 268864 / _f(2027025), -752 / _f(10395), 5 / _f(252),
473 436821248 / _f(3618239625), -112832 / _f(1447875),
474 62464 / _f(2027025), -101 / _f(17325),
475 3059776 / _f(80405325), -4096 / _f(289575), 11537 / _f(4054050),
476 4193792 / _f(723647925), -311 / _f(525525),
477 1097653 / _f(1929727800))
478}
479del _f, _u, _Ufloats
482__all__ += _ALL_DOCS(Caps, RhumbAux, RhumbLineAux)
484if __name__ == '__main__':
486 from pygeodesy.lazily import printf
487 from pygeodesy import RhumbAux # PYCHOK RhumbAux is __main__.RhumbAux
489 def _re(fmt, r3, x3):
490 e3 = []
491 for r, x in _zip(r3, x3): # strict=True
492 e = fabs(r - x) / fabs(x)
493 e3.append('%.g' % (e,))
494 printf((fmt % r3) + ' rel errors: ' + ', '.join(e3))
496 # <https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolveå -p 9> version 2.2
497 rhumb = RhumbAux(exact=True) # WGS84 default
498 printf('# %r\n', rhumb)
499 r = rhumb.Direct8(40.6, -73.8, 51, 5.5e6) # from JFK about NE
500 _re('# JFK NE lat2=%.8f, lon2=%.8f, S12=%.1f', (r.lat2, r.lon2, r.S12), (71.688899882813, 0.2555198244234, 44095641862956.11))
501 r = rhumb.Inverse8(40.6, -73.8, 51.6, -0.5) # JFK to LHR
502 _re('# JFK-LHR azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (77.7683897102557, 5771083.38332803, 37395209100030.39))
503 r = rhumb.Inverse8(40.6, -73.8, 35.8, 140.3) # JFK to Tokyo Narita
504 _re('# JFK-NRT azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (-92.38888798169965, 12782581.067684170, -63760642939072.50))
506# % python3 -m pygeodesy.rhumbaux
508# RhumbAux(TMorder=6, ellipsoid=Ellipsoid(name='WGS84', a=6378137, b=6356752.31424518, f_=298.25722356, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e21=0.99330562, e22=0.0067395, e32=0.00335843, A=6367449.14582341, L=10001965.72931272, R1=6371008.77141506, R2=6371007.18091847, R3=6371000.79000916, Rbiaxial=6367453.63451633, Rtriaxial=6372797.5559594), exact=True)
510# JFK NE lat2=71.68889988, lon2=0.25551982, S12=44095641862956.1 rel errors: 4e-11, 2e-08, 5e-16
511# JFK-LHR azi12=77.76838971, s12=5771083.383 S12=37395209100030.3 rel errors: 3e-12, 5e-15, 6e-16
512# JFK-NRT azi12=-92.38888798, s12=12782581.068 S12=-63760642939072.5 rel errors: 2e-16, 3e-16, 6e-16