Coverage for pygeodesy/rhumb/aux_.py: 96%
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2# -*- coding: utf-8 -*-
4u'''A pure Python version of I{Karney}'s I{Auxiliary Latitudes}, C++ classes U{Rhumb
5<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Rhumb.html>} and U{RhumbLine
6<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1RhumbLine.html>} from
7I{GeographicLib version 2.2+} renamed to L{RhumbAux} respectively L{RhumbLineAux}.
9Class L{RhumbLineAux} has been enhanced with methods C{Intersecant2}, C{Intersection} and C{PlumbTo}
10to iteratively 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 U{GeographicLib<https://GeographicLib.SourceForge.io/C++/doc/index.html>} I{2.2}
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>},
16utility U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>} and U{Online rhumb
17line calculations<https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve>}.
19Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2022-2024) and licensed under the MIT/X11
20License. For more information, see the U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation.
22@note: C{S12} area calculations in classes L{RhumbAux} and L{RhumbLineAux} depend on class L{AuxDST} which
23 requires U{numpy<https://PyPI.org/project/numpy>} to be installed, version 1.16 or newer.
25@note: Windows reserves file names U{AUX, COM[#], CON, LPT[#], NUL, PRN<https://learn.Microsoft.com/en-us/
26 windows/win32/fileio/naming-a-file#naming-conventions>} with and without extension.
27'''
28# make sure int/int division yields float quotient
29from __future__ import division as _; del _ # PYCHOK semicolon
31from pygeodesy.auxilats.auxAngle import AuxMu, AuxPhi, hypot
32from pygeodesy.auxilats.auxDLat import AuxDLat, _DClenshaw
33# from pygeodesy.auxilats.auxDST import AuxDST # _MODS
34from pygeodesy.auxilats.auxily import _Dlam, _Dp0Dpsi
35from pygeodesy.auxilats._CX_Rs import _Rdict, _Rkey, _Rtuple
36from pygeodesy.basics import copysign0, _reverange, _xkwds_get1
37from pygeodesy.constants import EPS_2, MANT_DIG, PI4, isinf, \
38 _0_0, _4_0, _720_0, _log2, _over
39# from pygeodesy.datums import _WGS84 # from .rhumb.bases
40# from pygeodesy.errors import _xkwds_get1 # from .basics
41from pygeodesy.karney import Caps, _polynomial
42# from pygeodesy.fmath import hypot # from .auxilats.auxAngle
43from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS
44# from pygeodesy.props import Property_RO # from .rhumb.bases
45from pygeodesy.rhumb.bases import RhumbBase, RhumbLineBase, \
46 Property_RO, _WGS84
48from math import ceil as _ceil, fabs, radians
50__all__ = _ALL_LAZY.rhumb_aux_
51__version__ = '25.01.15'
53# DIGITS = (sizeof(real) * 8) bits
54# = (ctypes.sizeof(ctypes.c_double(1.0)) * 8) bits
55# For |n| <= 0.99, actual max for doubles is 2163. This scales
56# as DIGITS and for long doubles (GEOGRAPHICLIB_PRECISION = 3,
57# DIGITS = 64), this becomes 2163 * 64 / 53 = 2612. Round this
58# up to 2^12 = 4096 and scale this by DIGITS//64 if DIGITS > 64.
59#
60# 64 = DIGITS for long double, 6 = 12 - _log2(64)
61_Lbits = 1 << (int(_ceil(_log2(max(MANT_DIG, 64)))) + 6)
64class RhumbAux(RhumbBase):
65 '''Class to solve the I{direct} and I{inverse rhumb} problems, based
66 on I{Auxiliary Latitudes} for accuracy near the poles.
68 @note: Package U{numpy<https://PyPI.org/project/numpy>} must be
69 installed, version 1.16 or later.
70 '''
72 def __init__(self, a_earth=_WGS84, f=None, exact=True, **TMorder_name): # PYCHOK signature
73 '''New C{RhumbAux}.
75 @kwarg a_earth: This rhumb's earth model (L{Datum}, L{Ellipsoid},
76 L{Ellipsoid2}, L{a_f2Tuple}, 2-tuple C{(a, f)}) or
77 the (equatorial) radius (C{meter}, conventionally).
78 @kwarg f: The ellipsoid's flattening (C{scalar}), required if B{C{a_earth}}
79 is C{scalar}, ignored otherwise.
80 @kwarg exact: If C{True}, use the exact expressions for the I{Auxiliary
81 Latitudes}, otherwise use the I{Fourier} series expansion
82 (C{bool}), see also property C{exact}.
83 @kwarg TMorder_name: Optional C{B{name}=NN} (C{str}) and optional
84 keyword argument C{B{TMorder}=6}, the order of the
85 L{KTransverseMercator}, see property C{TMorder}.
87 @raise ImportError: Package C{numpy} not found or not installed, only
88 required for area C{S12} when C{B{exact} is True}.
90 @raise RhumbError: Invalid B{C{a_earth}}, B{C{f}} or B{C{TMorder}}.
91 '''
92 RhumbBase.__init__(self, a_earth, f, exact, TMorder_name)
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_get1(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 s = fabs(Phi2.toMu(self).toRadians -
144 Phi1.toMu(self).toRadians)
145 else: # dmu/dpsi = dmu/dchi/dpsi/dchi
146 s = hypot(lam12, psi12)
147 if s:
148 s *= self._DMu_DPsi(Phi1, Phi2, Chi1, Chi2)
149 s *= self._rrm
150 a = _over(s, self._mpd)
151 r.set_(a12=copysign0(a, s), s12=s)
152 return lam12, psi12, Chi1, Chi2
154 def _latPhi2(self, mu):
155 Mu = AuxMu.fromDegrees(mu)
156 Phi = Mu.toPhi(self)
157 return Phi.toDegrees, Phi
159 @Property_RO
160 def _mpd(self): # meter per degree
161 return radians(self._rrm) # == self.ellipsoid._Lpd
163 def _psiChiPhi3(self, lat):
164 Phi = AuxPhi.fromDegrees(lat)
165 Chi = Phi.toChi(self)
166 psi = Chi.toLambertianRadians
167 return psi, Chi, Phi
169 @Property_RO
170 def _RA(self): # get the coefficients for area calculation
171 return tuple(_RAintegrate(self._auxD) if self.exact else
172 _RAseries(self._auxD))
174# _RhumbLine = RhumbLineAux # see further below
176 @Property_RO
177 def _rrm(self):
178 return self._auxD.RectifyingRadius(exact=self.exact)
180 _mpr = _rrm # meter per radian, see _mpd
182 def _S12d(self, Chix, Chiy, lon12): # degrees
183 '''(INTERNAL) Compute the area C{S12} from C{._meanSinXi(Chix, Chiy) * .c2 * lon12}.
184 '''
185 pP, xD = self._RA, self._auxD
187 tx, Phix = Chix.tan, Chix.toPhi(self)
188 ty, Phiy = Chiy.tan, Chiy.toPhi(self)
190 dD = xD.DParametric(Phix, Phiy) if self.exact else \
191 xD.CParametric(Chix, Chiy)
192 if dD:
193 dD = _over(dD, xD.DIsometric(Phix, Phiy) if self.exact else
194 _Dlam(tx, ty)) # not Lambertian!
195 dD *= _DClenshaw(False, Phix.toBeta(self).normalized,
196 Phiy.toBeta(self).normalized,
197 pP, min(len(pP), xD.ALorder)) # Fsum
198 dD += _Dp0Dpsi(tx, ty)
199 dD *= self._c2 * lon12
200 return float(dD)
203class RhumbLineAux(RhumbLineBase):
204 '''Compute one or several points on a single rhumb line.
206 Class C{RhumbLineAux} facilitates the determination of points
207 on a single rhumb line. The starting point (C{lat1}, C{lon1})
208 and the azimuth C{azi12} are specified once.
209 '''
210 _Rhumb = RhumbAux # rhumb.aux_.RhumbAux
212 def __init__(self, rhumb, lat1=0, lon1=0, azi12=None, **caps_name): # PYCHOK signature
213 '''New C{RhumbLineAux}.
215 @arg rhumb: The rhumb reference (L{RhumbAux}).
216 @kwarg lat1: Latitude of the start point (C{degrees90}).
217 @kwarg lon1: Longitude of the start point (C{degrees180}).
218 @kwarg azi12: Azimuth of this rhumb line (compass C{degrees}).
219 @kwarg caps_name: Optional keyword arguments C{B{name}=NN} and
220 C{B{caps}=0}, a bit-or'ed combination of L{Caps}
221 values specifying the required capabilities. Include
222 C{Caps.LINE_OFF} if updates to the B{C{rhumb}} should
223 I{not} be reflected in this rhumb line.
224 '''
225 RhumbLineBase.__init__(self, rhumb, lat1, lon1, azi12, **caps_name)
227 @Property_RO
228 def _Chi1(self):
229 return self._Phi1.toChi(self.rhumb)
231 @Property_RO
232 def _mu1(self):
233 '''(INTERNAL) Get the I{rectifying auxiliary} latitude (C{degrees}).
234 '''
235 return self._Phi1.toMu(self.rhumb).toDegrees
237 def _mu2lat(self, mu):
238 '''(INTERNAL) Get the inverse I{rectifying auxiliary} latitude (C{degrees}).
239 '''
240 lat, _ = self.rhumb._latPhi2(mu)
241 return lat
243 @Property_RO
244 def _Phi1(self):
245 return AuxPhi.fromDegrees(self.lat1)
247 def _Position4(self, a12, mu2, *unused): # PYCHOK s12, mu2
248 '''(INTERNAL) See method C{RhumbLineBase._Position}.
249 '''
250 R = self.rhumb
251 lat2, Phi2 = R._latPhi2(mu2)
252 Chi2 = Phi2.toChi(R)
253 Chi1 = self._Chi1
254 lon2 = self._salp * a12
255 if lon2:
256 m = R._DMu_DPsi(self._Phi1, Phi2, Chi1, Chi2)
257 lon2 = _over(lon2, m)
258 return lat2, lon2, Chi1, Chi2
260# @Property_RO
261# def _psi1(self):
262# return self._Chi1.toLambertianRadians
264RhumbAux._RhumbLine = RhumbLineAux # PYCHOK see RhumbBase._RhumbLine
267def _RAintegrate(auxD):
268 # Compute coefficients by Fourier transform of integrand
269 L = 2
270 fft = _MODS.auxilats.auxDST.AuxDST(L)
271 f = auxD._qIntegrand
272 c_ = fft.transform(f)
273 pP = []
274 _P = pP.append
275 # assert L < _Lbits
276 while L < _Lbits:
277 L = fft.reset(L) * 2
278 c_ = fft.refine(f, c_, _0_0) # sentine[L]
279 # assert len(c_) == L + 1
280 pP[:], k = [], -1
281 for j in range(1, L + 1):
282 # Compute Fourier coefficients of integral
283 p = (c_[j - 1] + c_[j]) / (_4_0 * j)
284 if fabs(p) > EPS_2:
285 k = -1 # run interrupted
286 else:
287 if k < 0:
288 k = 1 # mark as first small value
289 if (j - k) >= ((j + 7) // 8):
290 # run of at least (j - 1) // 8 small values
291 return pP[:j] # break while L loop
292 _P(-p)
293 return pP # no convergence, use pP as-is
296def _RAseries(auxD):
297 # Series expansions in n for Fourier coeffients of the integral
298 # @see: U{"Series expansions for computing rhumb areas"
299 # <https:#DOI.org/10.5281/zenodo.7685484>}.
300 d = n = auxD._n
301 i = 0
302 aL = auxD.ALorder
303 Cs = _RACoeffs[aL]
304 # assert len(Cs) == (aL * (aL + 1)) // 2
305 pP = []
306 _p = _polynomial
307 for m in _reverange(aL): # order
308 j = i + m + 1
309 pP.append(_p(n, Cs, i, j) * d)
310 d *= n
311 i = j
312 # assert i == len(pP)
313 return pP
316_RACoeffs = _Rdict(110, # Rhumb Area Coefficients in matrix Q
317 _Rtuple(_Rkey(4), 10, # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4
318 '596/2025, -398/945, 22/45, -1/3',
319 '1543/4725, -118/315, 1/5',
320 '152/945, -17/315',
321 '5/252'),
322 _Rtuple(_Rkey(5), 15, # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5
323 '-102614/467775, 596/2025, -398/945, 22/45, -1/3',
324 '-24562/155925, 1543/4725, -118/315, 1/5',
325 '-38068/155925, 152/945, -17/315',
326 '-752/10395, 5/252',
327 '-101/17325'),
328 _Rtuple(_Rkey(6), 21, # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6
329 '138734126/638512875, -102614/467775, 596/2025, -398/945, 22/45, -1/3',
330 '17749373/425675250, -24562/155925, 1543/4725, -118/315, 1/5',
331 '1882432/8513505, -38068/155925, 152/945, -17/315',
332 '268864/2027025, -752/10395, 5/252',
333 '62464/2027025, -101/17325',
334 '11537/4054050'),
335 _Rtuple(_Rkey(7), 28, # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7
336 '-565017322/1915538625, 138734126/638512875, -102614/467775, 596/2025, -398/945, 22/45, -1/3',
337 '-1969276/58046625, 17749373/425675250, -24562/155925, 1543/4725, -118/315, 1/5',
338 '-58573784/638512875, 1882432/8513505, -38068/155925, 152/945, -17/315',
339 '-6975184/42567525, 268864/2027025, -752/10395, 5/252',
340 '-112832/1447875, 62464/2027025, -101/17325',
341 '-4096/289575, 11537/4054050',
342 '-311/525525'),
343 _Rtuple(_Rkey(8), 36, # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8
344 '188270561816/488462349375, -565017322/1915538625, 138734126/638512875, -102614/467775, 596/2025, -398/945, 22/45, -1/3',
345 '2332829602/23260111875, -1969276/58046625, 17749373/425675250, -24562/155925, 1543/4725, -118/315, 1/5',
346 '-41570288/930404475, -58573784/638512875, 1882432/8513505, -38068/155925, 152/945, -17/315',
347 '1538774036/10854718875, -6975184/42567525, 268864/2027025, -752/10395, 5/252',
348 '436821248/3618239625, -112832/1447875, 62464/2027025, -101/17325',
349 '3059776/80405325, -4096/289575, 11537/4054050',
350 '4193792/723647925, -311/525525',
351 '1097653/1929727800')
352)
353del _Rdict, _Rkey, _Rtuple
355__all__ += _ALL_DOCS(Caps)
357# **) MIT License
358#
359# Copyright (C) 2023-2025 -- mrJean1 at Gmail -- All Rights Reserved.
360#
361# Permission is hereby granted, free of charge, to any person obtaining a
362# copy of this software and associated documentation files (the "Software"),
363# to deal in the Software without restriction, including without limitation
364# the rights to use, copy, modify, merge, publish, distribute, sublicense,
365# and/or sell copies of the Software, and to permit persons to whom the
366# Software is furnished to do so, subject to the following conditions:
367#
368# The above copyright notice and this permission notice shall be included
369# in all copies or substantial portions of the Software.
370#
371# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
372# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
373# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
374# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
375# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
376# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
377# OTHER DEALINGS IN THE SOFTWARE.