Coverage for pygeodesy/rhumbaux.py: 95%
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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+} renamed as L{RhumbAux} respectively L{RhumbLineAux}.
8Class L{RhumbLineAux} 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 from an other point.
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, needed only for I{exact} area calculations C{S12} in L{RhumbAux}
23 and L{RhumbLineAux}.
24'''
25# make sure int/int division yields float quotient
26from __future__ import division as _; del _ # PYCHOK semicolon
28from pygeodesy.auxilats.auxAngle import AuxMu, AuxPhi, hypot
29from pygeodesy.auxilats.auxDLat import AuxDLat, _DClenshaw
30# from pygeodesy.auxilats.auxDST import AuxDST # _MODS
31from pygeodesy.auxilats.auxily import _Dlam, _Dp0Dpsi, _Ufloats
32from pygeodesy.basics import copysign0, _reverange, _zip, _xkwds_get
33from pygeodesy.constants import EPS_2, MANT_DIG, PI4, isinf, \
34 _0_0, _4_0, _720_0, _log2, _over
35# from pygeodesy.ellipsoids import _EWGS84 # from .karney
36# from pygeodesy.errors import itemsorted, _xkwds_get # from .basics, ...
37from pygeodesy.karney import Caps, _polynomial, _EWGS84
38# from pygeodesy.fmath import hypot # from .auxilats.auxAngle
39from pygeodesy.interns import NN, _COMMASPACE_
40from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS
41# from pygeodesy.props import Property_RO # from .rhumbBase
42from pygeodesy.rhumbBase import RhumbBase, RhumbLineBase, \
43 itemsorted, pairs, Property_RO
44# from pygeodesy.streprs import pairs # from .rhumbBase
46from math import ceil as _ceil, fabs, radians
48__all__ = _ALL_LAZY.rhumbaux
49__version__ = '23.09.20'
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 _DMu_DPsi(self, Phi1, Phi2, Chi1, Chi2):
126 xD = self._auxD
127 r = xD.DRectifying(Phi1, Phi2) if self.exact else \
128 xD.CRectifying(Chi1, Chi2)
129 if r:
130 r = _over(r, xD.DIsometric(Phi1, Phi2) if self.exact else
131 _Dlam(Chi1.tan, Chi2.tan)) # not Lambertian!
132 return r
134 def _Inverse4(self, lon12, r, outmask):
135 '''(INTERNAL) See method C{RhumbBase.Inverse}.
136 '''
137 psi1, Chi1, Phi1 = self._psiChiPhi3(r.lat1)
138 psi2, Chi2, Phi2 = self._psiChiPhi3(r.lat2)
139 psi12 = psi2 - psi1 # radians
140 lam12 = radians(lon12)
141 if (outmask & Caps.DISTANCE):
142 if isinf(psi1) or isinf(psi2): # PYCHOK no cover
143 d = Phi2.toMu(self).toRadians
144 d -= Phi1.toMu(self).toRadians
145 s = fabs(d)
146 else: # dmu/dpsi = dmu/dchi/dpsi/dchi
147 s = hypot(lam12, psi12)
148 if s:
149 s *= self._DMu_DPsi(Phi1, Phi2, Chi1, Chi2)
150 s *= self._rrm
151 a = _over(s, self._mpd)
152 r.set_(a12=copysign0(a, s), s12=s)
153 return lam12, psi12, Chi1, Chi2
155 def _latPhi2(self, mu):
156 Mu = AuxMu.fromDegrees(mu)
157 Phi = Mu.toPhi(self)
158 return Phi.toDegrees, Phi
160 @Property_RO
161 def _mpd(self): # meter per degree
162 return radians(self._rrm) # == self.ellipsoid._Lpd
164 def _psiChiPhi3(self, lat):
165 Phi = AuxPhi.fromDegrees(lat)
166 Chi = Phi.toChi(self)
167 psi = Chi.toLambertianRadians
168 return psi, Chi, Phi
170 @Property_RO
171 def _RA(self): # get the coefficients for area calculation
172 return tuple(_RAintegrate(self._auxD) if self.exact else
173 _RAseries(self._auxD))
175# _RhumbLine = RhumbLineAux # see further below
177 @Property_RO
178 def _rrm(self):
179 return self._auxD.RectifyingRadius(exact=self.exact)
181 def _S12d(self, Chix, Chiy, lon12): # degrees
182 '''(INTERNAL) Compute the area C{S12} from C{._meanSinXi(Chix, Chiy) * .c2 * lon12}.
183 '''
184 pP, xD = self._RA, self._auxD
186 tx, Phix = Chix.tan, Chix.toPhi(self)
187 ty, Phiy = Chiy.tan, Chiy.toPhi(self)
189 dD = xD.DParametric(Phix, Phiy) if self.exact else \
190 xD.CParametric(Chix, Chiy)
191 if dD:
192 dD = _over(dD, xD.DIsometric(Phix, Phiy) if self.exact else
193 _Dlam(tx, ty)) # not Lambertian!
194 dD *= _DClenshaw(False, Phix.toBeta(self).normalized,
195 Phiy.toBeta(self).normalized,
196 pP, min(len(pP), xD.ALorder)) # Fsum
197 dD += _Dp0Dpsi(tx, ty)
198 dD *= self._c2 * lon12
199 return float(dD)
201 def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature
202 '''Return this C{Rhumb} as string.
204 @kwarg prec: The C{float} precision, number of decimal digits (0..9).
205 Trailing zero decimals are stripped for B{C{prec}} values
206 of 1 and above, but kept for negative B{C{prec}} values.
207 @kwarg sep: Separator to join (C{str}).
209 @return: Tuple items (C{str}).
210 '''
211 d = dict(ellipsoid=self.ellipsoid, exact=self.exact,
212 TMorder=self.TMorder)
213 return sep.join(pairs(itemsorted(d, asorted=False), prec=prec))
216class RhumbLineAux(RhumbLineBase):
217 '''Compute one or several points on a single rhumb line.
219 Class C{RhumbLineAux} facilitates the determination of points
220 on a single rhumb line. The starting point (C{lat1}, C{lon1})
221 and the azimuth C{azi12} are specified once.
223 Method C{RhumbLineAux.Position} returns the location of an
224 other point and optionally the distance C{s12} along and the
225 area C{S12} under the rhumb line.
227 Method C{RhumbLineAux.intersection2} finds the intersection
228 between two rhumb lines.
230 Method C{RhumbLineAux.nearestOn4} computes the nearest point
231 on and the distance to a rhumb line in different ways.
232 '''
233 _Rhumb = RhumbAux # rhumbaux.RhumbAux
235 def __init__(self, rhumb, lat1=0, lon1=0, azi12=None, **caps_name): # PYCHOK signature
236 '''New C{rhumbaux.RhumbLineAux}.
238 @arg rhumb: The rhumb reference (C{rhumbaux.RhumbAux}).
239 @kwarg lat1: Latitude of the start point (C{degrees90}).
240 @kwarg lon1: Longitude of the start point (C{degrees180}).
241 @kwarg azi12: Azimuth of this rhumb line (compass C{degrees}).
242 @kwarg caps_name: Optional keyword arguments C{B{name}=NN} and
243 C{B{caps}=0}, a bit-or'ed combination of L{Caps}
244 values specifying the required capabilities. Include
245 C{Caps.LINE_OFF} if updates to the B{C{rhumb}} should
246 I{not} be reflected in this rhumb line.
247 '''
248 RhumbLineBase.__init__(self, rhumb, lat1, lon1, azi12, **caps_name)
250 @Property_RO
251 def _Chi1(self):
252 return self._Phi1.toChi(self.rhumb)
254 @Property_RO
255 def _mu1(self):
256 '''(INTERNAL) Get the I{rectifying auxiliary} latitude (C{degrees}).
257 '''
258 return self._Phi1.toMu(self.rhumb).toDegrees
260 def _mu2lat(self, mu):
261 '''(INTERNAL) Get the inverse I{rectifying auxiliary} latitude (C{degrees}).
262 '''
263 lat, _ = self.rhumb._latPhi2(mu)
264 return lat
266 @Property_RO
267 def _Phi1(self):
268 return AuxPhi.fromDegrees(self.lat1)
270 def _Position4(self, a12, mu2, *unused): # PYCHOK s12, mu2
271 '''(INTERNAL) See method C{RhumbLineBase._Position}.
272 '''
273 R = self.rhumb
274 lat2, Phi2 = R._latPhi2(mu2)
275 Chi2 = Phi2.toChi(R)
276 Chi1 = self._Chi1
277 lon2 = self._salp * a12
278 if lon2:
279 m = R._DMu_DPsi(self._Phi1, Phi2, Chi1, Chi2)
280 lon2 = _over(lon2, m)
281 return lat2, lon2, Chi1, Chi2
283# @Property_RO
284# def _psi1(self):
285# return self._Chi1.toLambertianRadians
287RhumbAux._RhumbLine = RhumbLineAux # PYCHOK see RhumbBase._RhumbLine
290def _RAintegrate(auxD):
291 # Compute coefficients by Fourier transform of integrand
292 L = 2
293 fft = _MODS.auxilats.auxDST.AuxDST(L)
294 f = auxD._qIntegrand
295 c_ = fft.transform(f)
296 pP = []
297 _P = pP.append
298 # assert L < _Lbits
299 while L < _Lbits:
300 L = fft.reset(L) * 2
301 c_ = fft.refine(f, c_, _0_0) # sentine[L]
302 # assert len(c_) == L + 1
303 pP[:], k = [], -1
304 for j in range(1, L + 1):
305 # Compute Fourier coefficients of integral
306 p = (c_[j - 1] + c_[j]) / (_4_0 * j)
307 if fabs(p) > EPS_2:
308 k = -1 # run interrupted
309 else:
310 if k < 0:
311 k = 1 # mark as first small value
312 if (j - k) >= ((j + 7) // 8):
313 # run of at least (j - 1) // 8 small values
314 return pP[:j] # break while L loop
315 _P(-p)
316 return pP # no convergence, use pP as-is
319def _RAseries(auxD):
320 # Series expansions in n for Fourier coeffients of the integral
321 # @see: U{"Series expansions for computing rhumb areas"
322 # <https:#DOI.org/10.5281/zenodo.7685484>}.
323 d = n = auxD._n
324 i = 0
325 aL = auxD.ALorder
326 Cs = _RACoeffs[aL]
327 # assert len(Cs) == (aL * (aL + 1)) // 2
328 pP = []
329 _P = pP.append
330 _p = _polynomial
331 for m in _reverange(aL): # order
332 j = i + m + 1
333 _P(_p(n, Cs, i, j) * d)
334 d *= n
335 i = j
336 # assert i == len(pP)
337 return pP
340_f, _u = float, _Ufloats()
341_RACoeffs = { # Rhumb Area Coefficients in matrix Q
342 4: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4
343 596 / _f(2025), -398 / _f(945), 22 / _f(45), -1 / _f(3),
344 1543 / _f(4725), -118 / _f(315), 1 / _f(5),
345 152 / _f(945), -17 / _f(315),
346 5 / _f(252)),
347 5: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5
348 -102614 / _f(467775), 596 / _f(2025), -398 / _f(945), 22 / _f(45),
349 -1 / _f(3),
350 -24562 / _f(155925), 1543 / _f(4725), -118 / _f(315), 1 / _f(5),
351 -38068 / _f(155925), 152 / _f(945), -17 / _f(315),
352 -752 / _f(10395), 5 / _f(252),
353 -101 / _f(17325)),
354 6: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6
355 138734126 / _f(638512875), -102614 / _f(467775), 596 / _f(2025),
356 -398 / _f(945), 22 / _f(45), -1 / _f(3),
357 17749373 / _f(425675250), -24562 / _f(155925), 1543 / _f(4725),
358 -118 / _f(315), 1 / _f(5),
359 1882432 / _f(8513505), -38068 / _f(155925), 152 / _f(945),
360 -17 / _f(315),
361 268864 / _f(2027025), -752 / _f(10395), 5 / _f(252),
362 62464 / _f(2027025), -101 / _f(17325),
363 11537 / _f(4054050)),
364 7: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7
365 -565017322 / _f(1915538625), 138734126 / _f(638512875),
366 -102614 / _f(467775), 596 / _f(2025), -398 / _f(945), 22 / _f(45),
367 -1 / _f(3),
368 -1969276 / _f(58046625), 17749373 / _f(425675250), -24562 / _f(155925),
369 1543 / _f(4725), -118 / _f(315), 1 / _f(5),
370 -58573784 / _f(638512875), 1882432 / _f(8513505), -38068 / _f(155925),
371 152 / _f(945), -17 / _f(315),
372 -6975184 / _f(42567525), 268864 / _f(2027025), -752 / _f(10395),
373 5 / _f(252),
374 -112832 / _f(1447875), 62464 / _f(2027025), -101 / _f(17325),
375 -4096 / _f(289575), 11537 / _f(4054050),
376 -311 / _f(525525)),
377 8: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8
378 188270561816 / _f(488462349375), -565017322 / _f(1915538625),
379 138734126 / _f(638512875), -102614 / _f(467775), 596 / _f(2025),
380 -398 / _f(945), 22 / _f(45), -1 / _f(3),
381 2332829602 / _f(23260111875), -1969276 / _f(58046625),
382 17749373 / _f(425675250), -24562 / _f(155925), 1543 / _f(4725),
383 -118 / _f(315), 1 / _f(5),
384 -41570288 / _f(930404475), -58573784 / _f(638512875),
385 1882432 / _f(8513505), -38068 / _f(155925), 152 / _f(945),
386 -17 / _f(315),
387 1538774036 / _f(10854718875), -6975184 / _f(42567525),
388 268864 / _f(2027025), -752 / _f(10395), 5 / _f(252),
389 436821248 / _f(3618239625), -112832 / _f(1447875),
390 62464 / _f(2027025), -101 / _f(17325),
391 3059776 / _f(80405325), -4096 / _f(289575), 11537 / _f(4054050),
392 4193792 / _f(723647925), -311 / _f(525525),
393 1097653 / _f(1929727800))
394}
395del _f, _u, _Ufloats
397__all__ += _ALL_DOCS(Caps, RhumbAux, RhumbLineAux)
399if __name__ == '__main__':
401 from pygeodesy import printf, RhumbAux # PYCHOK RhumbAux is __main__.RhumbAux
403 def _re(fmt, r3, x3):
404 e3 = []
405 for r, x in _zip(r3, x3): # strict=True
406 e = fabs(r - x) / fabs(x)
407 e3.append('%.g' % (e,))
408 printf((fmt % r3) + ' rel errors: ' + ', '.join(e3))
410 # <https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve -p 9> version 2.2
411 rhumb = RhumbAux(exact=True) # WGS84 default
412 printf('# %r\n', rhumb)
413 r = rhumb.Direct8(40.6, -73.8, 51, 5.5e6) # from JFK about NE
414 _re('# JFK NE lat2=%.8f, lon2=%.8f, S12=%.1f', (r.lat2, r.lon2, r.S12), (71.688899882813, 0.2555198244234, 44095641862956.11))
415 r = rhumb.Inverse8(40.6, -73.8, 51.6, -0.5) # JFK to LHR
416 _re('# JFK-LHR azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (77.7683897102557, 5771083.38332803, 37395209100030.39))
417 r = rhumb.Inverse8(40.6, -73.8, 35.8, 140.3) # JFK to Tokyo Narita
418 _re('# JFK-NRT azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (-92.38888798169965, 12782581.067684170, -63760642939072.50))
420# % python3 -m pygeodesy.rhumbaux
422# 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)
424# JFK NE lat2=71.68889988, lon2=0.25551982, S12=44095641862956.1 rel errors: 4e-11, 2e-08, 5e-16
425# JFK-LHR azi12=77.76838971, s12=5771083.383 S12=37395209100030.3 rel errors: 3e-12, 5e-15, 6e-16
426# JFK-NRT azi12=-92.38888798, s12=12782581.068 S12=-63760642939072.5 rel errors: 2e-16, 3e-16, 6e-16