Coverage for pygeodesy/rhumbaux.py: 96%
162 statements
« prev ^ index » next coverage.py v7.2.2, created at 2023-08-20 14:00 -0400
« prev ^ index » next coverage.py v7.2.2, created at 2023-08-20 14:00 -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:Karney@Alum.MIT.edu>} (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.ellipsoids import _EWGS84 # from .karney
35# from pygeodesy.errors import itemsorted, _xkwds_get # from .basics, ...
36from pygeodesy.karney import Caps, _diff182, GDict, _norm180, _EWGS84
37# from pygeodesy.fmath import hypot # from .auxilats.auxAngle
38from pygeodesy.interns import NN, _COMMASPACE_
39from pygeodesy.lazily import _ALL_LAZY, _ALL_DOCS, _ALL_MODS as _MODS
40# from pygeodesy.props import Property, Property_RO # from .rhumbBase
41from pygeodesy.rhumbBase import RhumbBase, RhumbLineBase, itemsorted, \
42 pairs, Property, Property_RO
43# from pygeodesy.streprs import pairs # from .rhumbBase
44# from pygeodesy.utily import atan2d # from .auxilats.auxAngle
46from math import ceil as _ceil, fabs, radians
48__all__ = _ALL_LAZY.rhumbaux
49__version__ = '23.08.20'
51# DIGITS = (sizeof(real) * 8) bits
52# = (ctypes.sizeof(ctypes.c_double(1.0)) * 8) bits
53# For |n| <= 0.99, actual max for doubles is 2163. This scales
54# as DIGITS and for long doubles (GEOGRAPHICLIB_PRECISION = 3,
55# DIGITS = 64), this becomes 2163 * 64 / 53 = 2612. Round this
56# up to 2^12 = 4096 and scale this by DIGITS//64 if DIGITS > 64.
57#
58# 64 = DIGITS for long double, 6 = 12 - _log2(64)
59_Lbits = 1 << (int(_ceil(_log2(max(MANT_DIG, 64)))) + 6)
62class RhumbAux(RhumbBase):
63 '''Class to solve the I{direct} and I{inverse rhumb} problems, based
64 on I{Auxiliary Latitudes} for accuracy near the poles.
66 @note: Package U{numpy<https://PyPI.org/project/numpy>} must be
67 installed, version 1.16 or later.
68 '''
70 def __init__(self, a_earth=_EWGS84, f=None, exact=True, name=NN, **TMorder): # PYCHOK signature
71 '''New C{rhumbaux.RhumbAux}.
73 @kwarg a_earth: This rhumb's earth model (L{Ellipsoid}, L{Ellipsoid2},
74 L{a_f2Tuple}, L{Datum}, 2-tuple C{(a, f)}) or the
75 (equatorial) radius (C{scalar}).
76 @kwarg f: The ellipsoid's flattening (C{scalar}), iff B{C{a_earth}} is
77 a C{scalar}, ignored otherwise.
78 @kwarg exact: If C{True}, use the exact expressions for the I{Auxiliary
79 Latitudes}, otherwise use the I{Fourier} series expansion
80 (C{bool}), see also property C{exact}.
81 @kwarg name: Optional name (C{str}).
82 @kwarg TMorder: Optional keyword argument B{C{TMorder}}, see property
83 C{TMorder}.
85 @raise ImportError: Package C{numpy} not found or not installed, only
86 required for area C{S12} when C{B{exact} is True}.
88 @raise RhumbError: Invalid B{C{a_earth}}, B{C{f}} or B{C{RA_TMorder}}.
89 '''
90 RhumbBase.__init__(self, a_earth, f, exact, name)
91 if TMorder:
92 self.Tmorder = _xkwds_get(TMorder, TMorder=RhumbBase._mTM)
94 def areaux(self, **exact):
95 '''Get this ellipsoid's B{C{exact}} surface area (C{meter} I{squared}).
97 @kwarg exact: Optional C{exact} (C{bool}), overriding this rhumb's
98 C{exact} setting, if C{True}, use the exact expression
99 for the authalic radius otherwise the I{Taylor} series.
101 @return: The (signed?) surface area (C{meter} I{squared}).
103 @raise AuxError: If C{B{exact}=False} and C{abs(flattening)} exceeds
104 property C{f_max}.
106 @note: The area of a polygon encircling a pole can be found by adding
107 C{areaux / 2} to the sum of C{S12} for each side of the polygon.
109 @see: U{The area of rhumb polygons<https://ArXiv.org/pdf/2303.03219.pdf>}
110 and method L{auxilats.AuxLat.AuthalicRadius2}.
111 '''
112 x = _xkwds_get(exact, exact=self.exact)
113 a = (self._c2 * _720_0) if bool(x) is self.exact else \
114 (self._auxD.AuthalicRadius2(exact=x, f_max=self.f_max) * PI4)
115 return a
117 @Property_RO
118 def _auxD(self):
119 return AuxDLat(self.ellipsoid)
121 @Property_RO
122 def _c2(self): # radians makes _c2 a factor per degree
123 return radians(self._auxD.AuthalicRadius2(exact=self.exact, f_max=self.f_max))
125 def Direct(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE):
126 '''Solve the I{direct rhumb} problem, optionally with the area.
128 @arg lat1: Latitude of the first point (C{degrees90}).
129 @arg lon1: Longitude of the first point (C{degrees180}).
130 @arg azi12: Azimuth of the rhumb line (compass C{degrees}).
131 @arg s12: Distance along the rhumb line from the given to
132 the destination point (C{meter}), can be negative.
134 @return: L{GDict} with 2 up to 8 items C{lat2, lon2, a12, S12,
135 lat1, lon1, azi12, s12} with the destination point's
136 latitude C{lat2} and longitude C{lon2} in C{degrees},
137 the rhumb angle C{a12} in C{degrees} and area C{S12}
138 under the rhumb line in C{meter} I{squared}.
140 @raise ImportError: Package C{numpy} not found or not installed,
141 only required for area C{S12} when C{B{exact}
142 is True}.
144 @note: If B{C{s12}} is large enough that the rhumb line crosses
145 a pole, the longitude of the second point is indeterminate
146 and C{NAN} is returned for C{lon2} and area C{S12}.
148 @note: If the given point is a pole, the cosine of its latitude is
149 taken to be C{sqrt(L{EPS})}. This position is extremely
150 close to the actual pole and allows the calculation to be
151 carried out in finite terms.
152 '''
153 rl = RhumbLineAux(self, lat1, lon1, azi12, caps=Caps.LINE_OFF,
154 name=self.name)
155 return rl.Position(s12, outmask) # lat2, lon2, S12
157 def _DMu_DPsi(self, Phi1, Phi2, Chi1, Chi2):
158 xD = self._auxD
159 return _over(xD.DRectifying(Phi1, Phi2),
160 xD.DIsometric( Phi1, Phi2)) if self.exact else \
161 _over(xD.CRectifying(Chi1, Chi2),
162 _Dlam(Chi1.tan, Chi2.tan)) # not Lambertian!
164 def Inverse(self, lat1, lon1, lat2, lon2, outmask=Caps.AZIMUTH_DISTANCE):
165 '''Solve the I{inverse rhumb} problem.
167 @arg lat1: Latitude of the first point (C{degrees90}).
168 @arg lon1: Longitude of the first point (C{degrees180}).
169 @arg lat2: Latitude of the second point (C{degrees90}).
170 @arg lon2: Longitude of the second point (C{degrees180}).
172 @return: L{GDict} with 5 to 8 items C{azi12, s12, a12, S12,
173 lat1, lon1, lat2, lon2}, the rhumb line's azimuth C{azi12}
174 in compass C{degrees} between C{-180} and C{+180}, the
175 distance C{s12} and rhumb angle C{a12} between both points
176 in C{meter} respectively C{degrees} and the area C{S12}
177 under the rhumb line in C{meter} I{squared}.
179 @raise ImportError: Package C{numpy} not found or not installed,
180 only required for area C{S12} when C{B{exact}
181 is True}.
183 @note: The shortest rhumb line is found. If the end points are
184 on opposite meridians, there are two shortest rhumb lines
185 and the East-going one is chosen.
187 @note: If either point is a pole, the cosine of its latitude is
188 taken to be C{sqrt(L{EPS})}. This position is extremely
189 close to the actual pole and allows the calculation to be
190 carried out in finite terms.
191 '''
192 r, Cs = GDict(name=self.name), Caps
193 if (outmask & Cs.AZIMUTH_DISTANCE_AREA):
194 psi1, Chi1, Phi1 = self._psiChiPhi3(lat1)
195 psi2, Chi2, Phi2 = self._psiChiPhi3(lat2)
197 psi12 = psi2 - psi1
198 lon12, _ = _diff182(lon1, lon2, K_2_0=True)
199 lam12 = radians(lon12)
200 if (outmask & Cs.AZIMUTH):
201 r.set_(azi12=atan2d(lam12, psi12))
202 if (outmask & Cs.DISTANCE):
203 if isinf(psi1) or isinf(psi2): # PYCHOK no cover
204 d = Phi2.toMu(self).toRadians
205 d -= Phi1.toMu(self).toRadians
206 s = fabs(d)
207 else: # dmu/dpsi = dmu/dchi/dpsi/dchi
208 s = self._DMu_DPsi(Phi1, Phi2, Chi1, Chi2)
209 s *= hypot(lam12, psi12)
210 r.set_(s12=self._rrm * s)
211 if (outmask & Cs.AREA):
212 S = self._c2SinXi(Chi1, Chi2)
213 r.set_(S12=unsigned0(S * lon12)) # like .gx
214 return r
216 def _c2SinXi(self, Chix, Chiy):
217 pP, xD = self._RA, self._auxD
219 tx, Phix = Chix.tan, Chix.toPhi(self)
220 ty, Phiy = Chiy.tan, Chiy.toPhi(self)
221 dD = _DClenshaw(False, Phix.toBeta(self).normalized,
222 Phiy.toBeta(self).normalized,
223 pP, min(len(pP), 8)) # Fsum
224 dD *= _over(xD.DParametric(Phix, Phiy),
225 xD.DIsometric( Phix, Phiy)) if self.exact else \
226 _over(xD.CParametric(Chix, Chiy), _Dlam(tx, ty)) # not Lambertian!
227 dD += _Dp0Dpsi(tx, ty)
228 dD *= self._c2
229 return float(dD)
231 def _psiChiPhi3(self, lat):
232 Phi = AuxPhi.fromDegrees(lat)
233 Chi = Phi.toChi(self)
234 psi = Chi.toLambertianRadians
235 return psi, Chi, Phi
237 @Property_RO
238 def _RA(self): # get the coefficients for area calculation
239 return tuple(_RAintegrate(self._auxD) if self.exact else
240 _RAseries(self._auxD))
242 @Property_RO
243 def _RhumbLine(self):
244 '''(INTERNAL) Get this module's C{RhumbLineAux} class.
245 '''
246 return RhumbLineAux
248 @Property_RO
249 def _rrm(self):
250 return self._auxD.RectifyingRadius(exact=self.exact)
252 @Property
253 def TMorder(self):
254 '''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
255 '''
256 return self._mTM
258 @TMorder.setter # PYCHOK setter!
259 def TMorder(self, order):
260 '''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
261 '''
262 self._TMorder(order)
264 def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature
265 '''Return this C{Rhumb} as string.
267 @kwarg prec: The C{float} precision, number of decimal digits (0..9).
268 Trailing zero decimals are stripped for B{C{prec}} values
269 of 1 and above, but kept for negative B{C{prec}} values.
270 @kwarg sep: Separator to join (C{str}).
272 @return: Tuple items (C{str}).
273 '''
274 d = dict(ellipsoid=self.ellipsoid, exact=self.exact,
275 TMorder=self.TMorder)
276 return sep.join(pairs(itemsorted(d, asorted=False), prec=prec))
279class RhumbLineAux(RhumbLineBase):
280 '''Compute one or several points on a single rhumb line.
282 Class C{RhumbLineAux} facilitates the determination of points
283 on a single rhumb line. The starting point (C{lat1}, C{lon1})
284 and the azimuth C{azi12} are specified once.
286 Method C{RhumbLineAux.Position} returns the location of an
287 other point and optionally the distance C{s12} along and the
288 area C{S12} under the rhumb line.
290 Method C{RhumbLineAux.intersection2} finds the intersection
291 between two rhumb lines.
293 Method C{RhumbLineAux.nearestOn4} computes the nearest point
294 on and the distance to a rhumb line in different ways.
295 '''
296 _Rhumb = RhumbAux # rhumbaux.RhumbAux
298 def __init__(self, rhumb, lat1=0, lon1=0, azi12=None, **caps_name): # PYCHOK signature
299 '''New C{rhumbaux.RhumbLineAux}.
301 @arg rhumb: The rhumb reference (C{rhumbaux.RhumbAux}).
302 @kwarg lat1: Latitude of the start point (C{degrees90}).
303 @kwarg lon1: Longitude of the start point (C{degrees180}).
304 @kwarg azi12: Azimuth of this rhumb line (compass C{degrees}).
305 @kwarg caps_name: Optional keyword arguments C{B{name}=NN} and
306 C{B{caps}=0}, a bit-or'ed combination of L{Caps}
307 values specifying the required capabilities. Include
308 C{Caps.LINE_OFF} if updates to the B{C{rhumb}} should
309 I{not} be reflected in this rhumb line.
310 '''
311 RhumbLineBase.__init__(self, rhumb, lat1, lon1, azi12, **caps_name)
313 @Property_RO
314 def _Chi1(self):
315 return self._Phi1.toChi(self.rhumb)
317 @Property_RO
318 def _mu1(self):
319 return self._Phi1.toMu(self.rhumb).toDegrees
321 @Property_RO
322 def _Phi1(self):
323 return AuxPhi.fromDegrees(self.lat1)
325 def Position(self, s12, outmask=Caps.LATITUDE_LONGITUDE):
326 '''Compute a point at a distance on this rhumb line.
328 @arg s12: The distance along this rhumb line between its origin
329 and the point (C{meters}), can be negative.
330 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying
331 the quantities to be returned.
333 @return: L{GDict} with 4 to 8 items C{azi12, a12, s12, S12, lat2,
334 lon2, lat1, lon1} with latitude C{lat2} and longitude
335 C{lon2} of the point in C{degrees}, the rhumb angle C{a12}
336 in C{degrees} from the start point of and the area C{S12}
337 under this rhumb line in C{meter} I{squared}.
339 @raise ImportError: Package C{numpy} not found or not installed,
340 only required for area C{S12} when C{B{exact}
341 is True}.
343 @note: If B{C{s12}} is large enough that the rhumb line crosses a
344 pole, the longitude of the second point is indeterminate and
345 C{NAN} is returned for C{lon2} and area C{S12}.
347 If the first point is a pole, the cosine of its latitude is
348 taken to be C{sqrt(L{EPS})}. This position is extremely
349 close to the actual pole and allows the calculation to be
350 carried out in finite terms.
351 '''
352 r, Cs = GDict(name=self.name), Caps
353 if (outmask & Cs.LATITUDE_LONGITUDE_AREA):
354 E, R = self.ellipsoid, self.rhumb
355 r12 = _over(s12, radians(R._rrm))
356 mu2, x90 = self._mu22(self._calp * r12, self._mu1)
357 Mu2 = AuxMu.fromDegrees(mu2)
358 Phi2 = Mu2.toPhi(R)
359 lat2 = Phi2.toDegrees
360 if x90: # PYCHOK no cover
361 lon2 = NAN
362 if (outmask & Cs.AREA):
363 r.set_(S12=NAN)
364 else:
365 Chi2 = Phi2.toChi(R)
366 Chi1 = self._Chi1
367 lon2 = R._DMu_DPsi(self._Phi1, Phi2, Chi1, Chi2)
368 lon2 = _over(self._salp * r12, lon2)
369 if (outmask & Cs.AREA):
370 S = R._c2SinXi(Chi1, Chi2)
371 r.set_(S12=unsigned0(S * lon2)) # like .gx
372 if (outmask & Cs.LONGITUDE):
373 if (outmask & Cs.LONG_UNROLL):
374 lon2 += self.lon1
375 else:
376 lon2 = _norm180(self._lon12 + lon2)
377 r.set_(azi12=self.azi12, s12=s12, a12=s12 / E._L_90)
378 if (outmask & Cs.LATITUDE):
379 r.set_(lat2=lat2, lat1=self.lat1)
380 if (outmask & Cs.LONGITUDE):
381 r.set_(lon2=lon2, lon1=self.lon1)
382 return r
384# @Property_RO
385# def _psi1(self):
386# return self._Chi1.toLambertianRadians
389def _RAintegrate(auxD):
390 # Compute coefficients by Fourier transform of integrand
391 L = 2
392 fft = _MODS.auxilats.auxDST.AuxDST(L)
393 f = auxD._qIntegrand
394 c = fft.transform(f)
395 # assert L < _Lbits
396 while L < _Lbits:
397 fft.reset(L)
398 c = fft.refine(f, c)
399 L *= 2 # == len(c)
400 # assert len(c) == L
401 pP, k = [], -1
402 for j in range(1, L + 1):
403 # Compute Fourier coefficients of integral
404 p = -(c[j - 1] + (c[j] if j < L else _0_0)) / (_4_0 * j)
405 if fabs(p) > EPS_2:
406 k = -1 # run interrupted
407 else:
408 if k < 0:
409 k = 1 # mark as first small value
410 if (j - k) >= ((j + 7) // 8):
411 # run of at least (j - 1) // 8 small values
412 return pP[:j]
413 pP.append(p)
414 return pP # no convergence, use pP as-is
417def _RAseries(auxD):
418 # Series expansions in n for Fourier coeffients of the integral
419 # @see: U{"Series expansions for computing rhumb areas"
420 # <https:#DOI.org/10.5281/zenodo.7685484>}.
421 d = n = auxD._n
422 i = 0
423 pP = []
424 aL = auxD.ALorder
425 Cs = _RACoeffs[aL]
426 # assert len(Cs) == (aL * (aL + 1)) // 2
427 _p = _MODS.karney._polynomial
428 for m in _reverange(aL): # order
429 j = i + m + 1
430 pP.append(_p(n, Cs, i, j) * d)
431 d *= n
432 i = j
433 # assert i == len(pP)
434 return pP
437_f, _u = float, _Ufloats()
438_RACoeffs = { # Rhumb Area Coefficients in matrix Q
439 4: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4
440 596 / _f(2025), -398 / _f(945), 22 / _f(45), -1 / _f(3),
441 1543 / _f(4725), -118 / _f(315), 1 / _f(5),
442 152 / _f(945), -17 / _f(315),
443 5 / _f(252)),
444 5: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5
445 -102614 / _f(467775), 596 / _f(2025), -398 / _f(945), 22 / _f(45),
446 -1 / _f(3),
447 -24562 / _f(155925), 1543 / _f(4725), -118 / _f(315), 1 / _f(5),
448 -38068 / _f(155925), 152 / _f(945), -17 / _f(315),
449 -752 / _f(10395), 5 / _f(252),
450 -101 / _f(17325)),
451 6: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6
452 138734126 / _f(638512875), -102614 / _f(467775), 596 / _f(2025),
453 -398 / _f(945), 22 / _f(45), -1 / _f(3),
454 17749373 / _f(425675250), -24562 / _f(155925), 1543 / _f(4725),
455 -118 / _f(315), 1 / _f(5),
456 1882432 / _f(8513505), -38068 / _f(155925), 152 / _f(945),
457 -17 / _f(315),
458 268864 / _f(2027025), -752 / _f(10395), 5 / _f(252),
459 62464 / _f(2027025), -101 / _f(17325),
460 11537 / _f(4054050)),
461 7: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7
462 -565017322 / _f(1915538625), 138734126 / _f(638512875),
463 -102614 / _f(467775), 596 / _f(2025), -398 / _f(945), 22 / _f(45),
464 -1 / _f(3),
465 -1969276 / _f(58046625), 17749373 / _f(425675250), -24562 / _f(155925),
466 1543 / _f(4725), -118 / _f(315), 1 / _f(5),
467 -58573784 / _f(638512875), 1882432 / _f(8513505), -38068 / _f(155925),
468 152 / _f(945), -17 / _f(315),
469 -6975184 / _f(42567525), 268864 / _f(2027025), -752 / _f(10395),
470 5 / _f(252),
471 -112832 / _f(1447875), 62464 / _f(2027025), -101 / _f(17325),
472 -4096 / _f(289575), 11537 / _f(4054050),
473 -311 / _f(525525)),
474 8: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8
475 188270561816 / _f(488462349375), -565017322 / _f(1915538625),
476 138734126 / _f(638512875), -102614 / _f(467775), 596 / _f(2025),
477 -398 / _f(945), 22 / _f(45), -1 / _f(3),
478 2332829602 / _f(23260111875), -1969276 / _f(58046625),
479 17749373 / _f(425675250), -24562 / _f(155925), 1543 / _f(4725),
480 -118 / _f(315), 1 / _f(5),
481 -41570288 / _f(930404475), -58573784 / _f(638512875),
482 1882432 / _f(8513505), -38068 / _f(155925), 152 / _f(945),
483 -17 / _f(315),
484 1538774036 / _f(10854718875), -6975184 / _f(42567525),
485 268864 / _f(2027025), -752 / _f(10395), 5 / _f(252),
486 436821248 / _f(3618239625), -112832 / _f(1447875),
487 62464 / _f(2027025), -101 / _f(17325),
488 3059776 / _f(80405325), -4096 / _f(289575), 11537 / _f(4054050),
489 4193792 / _f(723647925), -311 / _f(525525),
490 1097653 / _f(1929727800))
491}
492del _f, _u, _Ufloats
495__all__ += _ALL_DOCS(Caps, RhumbAux, RhumbLineAux)
497if __name__ == '__main__':
499 from pygeodesy.lazily import printf
500 from pygeodesy import RhumbAux # PYCHOK RhumbAux is __main__.RhumbAux
502 def _re(fmt, r3, x3):
503 e3 = []
504 for r, x in _zip(r3, x3): # strict=True
505 e = fabs(r - x) / fabs(x)
506 e3.append('%.g' % (e,))
507 printf((fmt % r3) + ' rel errors: ' + ', '.join(e3))
509 # <https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolveå -p 9> version 2.2
510 rhumb = RhumbAux(exact=True) # WGS84 default
511 printf('# %r\n', rhumb)
512 r = rhumb.Direct8(40.6, -73.8, 51, 5.5e6) # from JFK about NE
513 _re('# JFK NE lat2=%.8f, lon2=%.8f, S12=%.1f', (r.lat2, r.lon2, r.S12), (71.688899882813, 0.2555198244234, 44095641862956.11))
514 r = rhumb.Inverse8(40.6, -73.8, 51.6, -0.5) # JFK to LHR
515 _re('# JFK-LHR azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (77.7683897102557, 5771083.38332803, 37395209100030.39))
516 r = rhumb.Inverse8(40.6, -73.8, 35.8, 140.3) # JFK to Tokyo Narita
517 _re('# JFK-NRT azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (-92.38888798169965, 12782581.067684170, -63760642939072.50))
519# % python3 -m pygeodesy.rhumbaux
521# 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)
523# JFK NE lat2=71.68889988, lon2=0.25551982, S12=44095641862956.1 rel errors: 4e-11, 2e-08, 5e-16
524# JFK-LHR azi12=77.76838971, s12=5771083.383 S12=37395209100030.3 rel errors: 3e-12, 5e-15, 6e-16
525# JFK-NRT azi12=-92.38888798, s12=12782581.068 S12=-63760642939072.5 rel errors: 2e-16, 3e-16, 6e-16