Coverage for pygeodesy/rhumbx.py: 98%
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2# -*- coding: utf-8 -*-
4u'''A pure Python version of I{Karney}'s 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 2.0}.
9Class L{RhumbLine} has been enhanced with methods C{intersection2} and C{nearestOn4} to iteratively
10find the intersection of two rhumb lines, respectively the nearest point on a rumb line along a
11geodesic or perpendicular 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:Charles@Karney.com>} (2014-2022) 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 _ # PYCHOK semicolon
25from pygeodesy.basics import copysign0, neg, unsigned0, _zip
26from pygeodesy.constants import NAN, PI_2, _0_0s, _0_0, _0_5, \
27 _1_0, _2_0, _4_0, _720_0, _over
28from pygeodesy.errors import itemsorted, RhumbError, _Xorder
29from pygeodesy.fmath import hypot, hypot1
30# from pygeodesy.fsums import fsum1f_ # _MODS
31from pygeodesy.interns import NN, _COMMASPACE_
32from pygeodesy.karney import _atan2d, Caps, _diff182, _EWGS84, GDict, \
33 _GTuple, _norm180
34from pygeodesy.ktm import KTransverseMercator, _Xs, \
35 _AlpCoeffs, _BetCoeffs # PYCHOK used!
36from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS
37from pygeodesy.props import deprecated_method, Property, Property_RO, property_RO
38from pygeodesy.rhumbBase import RhumbBase, RhumbLineBase, pairs, _update_all_rls
39# from pygeodesy.streprs import pairs # from .rhumbBase
40from pygeodesy.units import Int
41from pygeodesy.utily import sincos2_
43from math import asinh, atan, cos, cosh, fabs, radians, sin, sinh, sqrt, tan
45__all__ = _ALL_LAZY.rhumbx
46__version__ = '23.08.09'
49class Rhumb(RhumbBase):
50 '''Class to solve the I{direct} and I{inverse rhumb} problems, based on
51 I{elliptic functions} or I{Krüger} series expansion.
53 @see: The U{Detailed Description<https://GeographicLib.SourceForge.io/C++/doc/
54 classGeographicLib_1_1Rhumb.html>} of I{Karney}'s C++ C{Rhumb Class}.
55 '''
56 _mRA = 6 # see .RAorder
58 def __init__(self, a_earth=_EWGS84, f=None, exact=True, name=NN, **RA_TMorder):
59 '''New C{rhumbx.Rhumb}.
61 @kwarg a_earth: This rhumb's earth model (L{Ellipsoid}, L{Ellipsoid2},
62 L{a_f2Tuple}, L{Datum}, 2-tuple C{(a, f)}) or the
63 (equatorial) radius (C{scalar}).
64 @kwarg f: The ellipsoid's flattening (C{scalar}), iff B{C{a_earth}} is
65 a C{scalar}, ignored otherwise.
66 @kwarg exact: If C{True}, use an addition theorem for elliptic integrals
67 to compute I{Divided differences}, otherwise use the I{Krüger}
68 series expansion (C{bool} or C{None}), see also properties
69 C{exact} and C{TMorder}.
70 @kwarg name: Optional name (C{str}).
71 @kwarg RA_TMorder: Optional keyword arguments B{C{RAorder}} and B{C{TMorder}}
72 to set the respective C{order}, see properties C{RAorder}
73 and C{TMorder} and method C{orders}.
75 @raise RhumbError: Invalid B{C{a_earth}}, B{C{f}} or B{C{RA_TMorder}}.
76 '''
77 RhumbBase.__init__(self, a_earth, f, exact, name)
78 if RA_TMorder:
79 self.orders(**RA_TMorder)
81 @Property_RO
82 def _A2(self): # Conformal2RectifyingCoeffs
83 m = self.TMorder
84 return _Xs(_AlpCoeffs, m, self.ellipsoid), m
86 @Property_RO
87 def _B2(self): # Rectifying2ConformalCoeffs
88 m = self.TMorder
89 return _Xs(_BetCoeffs, m, self.ellipsoid), m
91 def _DConformal2Rectifying(self, x, y): # radians
92 return _1_0 + (_sincosSeries(True, x, y, *self._A2) if self.f else _0_0)
94 def Direct(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE):
95 '''Solve the I{direct rhumb} problem, optionally with the area.
97 @arg lat1: Latitude of the first point (C{degrees90}).
98 @arg lon1: Longitude of the first point (C{degrees180}).
99 @arg azi12: Azimuth of the rhumb line (compass C{degrees}).
100 @arg s12: Distance along the rhumb line from the given to
101 the destination point (C{meter}), can be negative.
103 @return: L{GDict} with 2 up to 8 items C{lat2, lon2, a12, S12,
104 lat1, lon1, azi12, s12} with the destination point's
105 latitude C{lat2} and longitude C{lon2} in C{degrees},
106 the rhumb angle C{a12} in C{degrees} and area C{S12}
107 under the rhumb line in C{meter} I{squared}.
109 @note: If B{C{s12}} is large enough that the rhumb line crosses
110 a pole, the longitude of the second point is indeterminate
111 and C{NAN} is returned for C{lon2} and area C{S12}.
113 @note: If the given point is a pole, the cosine of its latitude is
114 taken to be C{sqrt(L{EPS})}. This position is extremely
115 close to the actual pole and allows the calculation to be
116 carried out in finite terms.
117 '''
118 rl = RhumbLine(self, lat1, lon1, azi12, caps=Caps.LINE_OFF,
119 name=self.name)
120 return rl.Position(s12, outmask | self._debug) # lat2, lon2, S12
122 @deprecated_method
123 def Direct7(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE_AREA):
124 '''DEPRECATED, use method L{Rhumb.Direct8}.
126 @return: A I{DEPRECATED} L{Rhumb7Tuple}.
127 '''
128 return self.Direct8(lat1, lon1, azi12, s12, outmask=outmask)._to7Tuple()
130 def _DIsometrict(self, phix, phiy, tphix, tphiy, _Dtan_phix_phiy):
131 E = self.ellipsoid
132 return _Dtan_phix_phiy * _Dasinh(tphix, tphiy) - \
133 _Dsin(phix, phiy) * _DeatanhE(sin(phix), sin(phiy), E)
135 def _DIsometric2Rectifyingd(self, psix, psiy): # degrees
136 if self.exact:
137 E = self.ellipsoid
138 phix, phiy, tphix, tphiy = _Eaux4(E.auxIsometric, psix, psiy)
139 t = _Dtant(phix - phiy, tphix, tphiy)
140 r = _over(self._DRectifyingt( tphix, tphiy, t),
141 self._DIsometrict(phix, phiy, tphix, tphiy, t))
142 else:
143 x, y = radians(psix), radians(psiy)
144 r = self._DConformal2Rectifying(_gd(x), _gd(y)) * _Dgd(x, y)
145 return r
147 def _DRectifyingt(self, tphix, tphiy, _Dtan_phix_phiy):
148 E = self.ellipsoid
149 tbetx = E.f1 * tphix
150 tbety = E.f1 * tphiy
151 return (E.f1 * _Dtan_phix_phiy * E.b * PI_2
152 * _DfEt( tbetx, tbety, self._eF)
153 * _Datan(tbetx, tbety)) / E.L
155 def _DRectifying2Conformal(self, x, y): # radians
156 return _1_0 - (_sincosSeries(True, x, y, *self._B2) if self.f else _0_0)
158 def _DRectifying2Isometricd(self, mux, muy): # degrees
159 E = self.ellipsoid
160 phix, phiy, tphix, tphiy = _Eaux4(E.auxRectifying, mux, muy)
161 if self.exact:
162 t = _Dtant(phix - phiy, tphix, tphiy)
163 r = _over(self._DIsometrict(phix, phiy, tphix, tphiy, t),
164 self._DRectifyingt( tphix, tphiy, t))
165 else:
166 r = self._DRectifying2Conformal(radians(mux), radians(muy)) * \
167 _Dgdinv(E.es_taupf(tphix), E.es_taupf(tphiy))
168 return r
170 @Property_RO
171 def _eF(self):
172 '''(INTERNAL) Get the ellipsoid's elliptic function.
173 '''
174 # .k2 = 0.006739496742276434
175 return self._E._elliptic_e12 # _MODS.elliptic.Elliptic(-self._E._e12)
177 def Inverse(self, lat1, lon1, lat2, lon2, outmask=Caps.AZIMUTH_DISTANCE):
178 '''Solve the I{inverse rhumb} problem.
180 @arg lat1: Latitude of the first point (C{degrees90}).
181 @arg lon1: Longitude of the first point (C{degrees180}).
182 @arg lat2: Latitude of the second point (C{degrees90}).
183 @arg lon2: Longitude of the second point (C{degrees180}).
185 @return: L{GDict} with 5 to 8 items C{azi12, s12, a12, S12,
186 lat1, lon1, lat2, lon2}, the rhumb line's azimuth C{azi12}
187 in compass C{degrees} between C{-180} and C{+180}, the
188 distance C{s12} and rhumb angle C{a12} between both points
189 in C{meter} respectively C{degrees} and the area C{S12}
190 under the rhumb line in C{meter} I{squared}.
192 @note: The shortest rhumb line is found. If the end points are
193 on opposite meridians, there are two shortest rhumb lines
194 and the East-going one is chosen.
196 @note: If either point is a pole, the cosine of its latitude is
197 taken to be C{sqrt(L{EPS})}. This position is extremely
198 close to the actual pole and allows the calculation to be
199 carried out in finite terms.
200 '''
201 r, Cs = GDict(name=self.name), Caps
202 if (outmask & Cs.AZIMUTH_DISTANCE_AREA):
203 r.set_(lat1=lat1, lon1=lon1, lat2=lat2, lon2=lon2)
204 E = self.ellipsoid
205 psi1 = E.auxIsometric(lat1)
206 psi2 = E.auxIsometric(lat2)
207 psi12 = psi2 - psi1
208 lon12, _ = _diff182(lon1, lon2)
209 if (outmask & Cs.AZIMUTH):
210 r.set_(azi12=_atan2d(lon12, psi12))
211 if (outmask & Cs.DISTANCE):
212 a12 = hypot(lon12, psi12) * self._DIsometric2Rectifyingd(psi2, psi1)
213 s12 = a12 * E._L_90
214 r.set_(s12=s12, a12=copysign0(a12, s12))
215 if (outmask & Cs.AREA):
216 r.set_(S12=self._S12d(lon12, psi2, psi1))
217 if ((outmask | self._debug) & Cs._DEBUG_INVERSE): # PYCHOK no cover
218 r.set_(a=E.a, f=E.f, f1=E.f1, L=E.L,
219 b=E.b, e=E.e, e2=E.e2, k2=self._eF.k2,
220 lon12=lon12, psi1=psi1, exact=self.exact,
221 psi12=psi12, psi2=psi2)
222 return r
224# def Inverse3(self, lat1, lon1, lat2, lon2): # PYCHOK outmask
225# '''Return the distance in C{meter} and the forward and
226# reverse azimuths (initial and final bearing) in C{degrees}.
227#
228# @return: L{Distance3Tuple}C{(distance, initial, final)}.
229# '''
230# r = self.Inverse(lat1, lon1, lat2, lon2)
231# return Distance3Tuple(r.s12, r.azi12, r.azi12)
233 @deprecated_method
234 def Inverse7(self, lat1, lon1, azi12, s12, outmask=Caps.AZIMUTH_DISTANCE_AREA):
235 '''DEPRECATED, use method L{Rhumb.Inverse8}.
237 @return: A I{DEPRECATED} L{Rhumb7Tuple}.
238 '''
239 return self.Inverse8(lat1, lon1, azi12, s12, outmask=outmask)._to7Tuple()
241 def _meanSinXi(self, x, y): # radians
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 return s
247 @deprecated_method
248 def orders(self, RAorder=None, TMorder=None): # PYCHOK expected
249 '''DEPRECATED, use properties C{RAorder} and/or C{TMorder}.
251 Get and set the I{RAorder} and/or I{TMorder}.
253 @kwarg RAorder: I{Rhumb Area} order (C{int}, 4, 5, 6, 7
254 or 8).
255 @kwarg TMorder: I{Transverse Mercator} order (C{int}, 4,
256 5, 6, 7 or 8).
258 @return: L{RhumbOrder2Tuple}C{(RAorder, TMorder)} with
259 the previous C{RAorder} and C{TMorder} setting.
260 '''
261 t = RhumbOrder2Tuple(self.RAorder, self.TMorder)
262 if RAorder not in (None, t.RAorder): # PYCHOK attr
263 self.RAorder = RAorder
264 if TMorder not in (None, t.TMorder): # PYCHOK attr
265 self.TMorder = TMorder
266 return t
268 @Property_RO
269 def _RA2(self):
270 # for WGS84: (0, -0.0005583633519275459, -3.743803759172812e-07, -4.633682270824446e-10,
271 # RAorder 6: -7.709197397676237e-13, -1.5323287106694307e-15, -3.462875359099873e-18)
272 m = self.RAorder
273 return _Xs(_RACoeffs, m, self.ellipsoid, RA=True), m
275 @Property
276 def RAorder(self):
277 '''Get the I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8).
278 '''
279 return self._mRA
281 @RAorder.setter # PYCHOK setter!
282 def RAorder(self, order):
283 '''Set the I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8).
284 '''
285 n = _Xorder(_RACoeffs, RhumbError, RAorder=order)
286 if self._mRA != n:
287 _update_all_rls(self)
288 self._mRA = n
290 @Property_RO
291 def _RhumbLine(self):
292 '''(INTERNAL) Get this module's C{RhumbLine} class.
293 '''
294 return RhumbLine
296 def _S12d(self, lon12, psi2, psi1): # degrees
297 '''(INTERNAL) Compute the area C{S12}.
298 '''
299 r = (self.ellipsoid.areax if self.exact else
300 self.ellipsoid.area) * lon12 / _720_0
301 r *= self._meanSinXi(radians(psi2), radians(psi1))
302 return r
304 @Property
305 def TMorder(self):
306 '''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
307 '''
308 return self._mTM
310 @TMorder.setter # PYCHOK setter!
311 def TMorder(self, order):
312 '''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
314 @note: Setting C{TMorder} turns property C{exact} off.
315 '''
316 self.exact = self._TMorder(order)
318 def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature
319 '''Return this C{Rhumb} as string.
321 @kwarg prec: The C{float} precision, number of decimal digits (0..9).
322 Trailing zero decimals are stripped for B{C{prec}} values
323 of 1 and above, but kept for negative B{C{prec}} values.
324 @kwarg sep: Separator to join (C{str}).
326 @return: Tuple items (C{str}).
327 '''
328 d = dict(ellipsoid=self.ellipsoid, RAorder=self.RAorder,
329 exact=self.exact, TMorder=self.TMorder)
330 return sep.join(pairs(itemsorted(d, asorted=False), prec=prec))
333class RhumbLine(RhumbLineBase):
334 '''Compute one or several points on a single rhumb line.
336 Class C{RhumbLine} facilitates the determination of points on
337 a single rhumb line. The starting point (C{lat1}, C{lon1})
338 and the azimuth C{azi12} are specified once.
340 Method C{RhumbLine.Position} returns the location of an other
341 point at distance C{s12} along and the area C{S12} under the
342 rhumb line.
344 Method C{RhumbLine.intersection2} finds the intersection between
345 two rhumb lines.
347 Method C{RhumbLine.nearestOn4} computes the nearest point on and
348 the distance to a rhumb line in different ways.
349 '''
350 _Rhumb = Rhumb # rhumbx.Rhumb
352 def __init__(self, rhumb, lat1=0, lon1=0, azi12=None, **caps_name): # PYCHOK signature
353 '''New C{rhumbx.RhumbLine}.
355 @arg rhumb: The rhumb reference (C{rhumbx.Rhumb}).
356 @kwarg lat1: Latitude of the start point (C{degrees90}).
357 @kwarg lon1: Longitude of the start point (C{degrees180}).
358 @kwarg azi12: Azimuth of this rhumb line (compass C{degrees}).
359 @kwarg caps_name: Optional keyword arguments C{B{name}=NN} and
360 C{B{caps}=0}, a bit-or'ed combination of L{Caps}
361 values specifying the required capabilities. Include
362 C{Caps.LINE_OFF} if updates to the B{C{rhumb}} should
363 I{not} be reflected in this rhumb line.
364 '''
365 RhumbLineBase.__init__(self, rhumb, lat1, lon1, azi12, **caps_name)
367 @Property_RO
368 def _mu1(self):
369 '''(INTERNAL) Get the I{rectifying auxiliary} latitude C{mu} (C{degrees}).
370 '''
371 return self.ellipsoid.auxRectifying(self.lat1)
373 def Position(self, s12, outmask=Caps.LATITUDE_LONGITUDE):
374 '''Compute a point at a distance on this rhumb line.
376 @arg s12: The distance along this rhumb between its point and
377 the other point (C{meters}), can be negative.
378 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying
379 the quantities to be returned.
381 @return: L{GDict} with 4 to 8 items C{azi12, a12, s12, S12, lat2,
382 lon2, lat1, lon1} with latitude C{lat2} and longitude
383 C{lon2} of the point in C{degrees}, the rhumb angle C{a12}
384 in C{degrees} from the start point of and the area C{S12}
385 under this rhumb line in C{meter} I{squared}.
387 @note: If B{C{s12}} is large enough that the rhumb line crosses a
388 pole, the longitude of the second point is indeterminate and
389 C{NAN} is returned for C{lon2} and area C{S12}.
391 If the first point is a pole, the cosine of its latitude is
392 taken to be C{sqrt(L{EPS})}. This position is extremely
393 close to the actual pole and allows the calculation to be
394 carried out in finite terms.
395 '''
396 r, Cs = GDict(name=self.name), Caps
397 if (outmask & Cs.LATITUDE_LONGITUDE_AREA):
398 E, R = self.ellipsoid, self.rhumb
399 a12 = s12 / E._L_90
400 mu12 = self._calp * a12
401 mu2, x90 = self._mu22(mu12, self._mu1)
402 if x90: # PYCHOK no cover
403 lat2 = E.auxRectifying(mu2, inverse=True)
404 lon2 = NAN
405 if (outmask & Cs.AREA):
406 r.set_(S12=NAN)
407 else:
408 psi2 = self._psi1
409 if self._calp:
410 lat2 = E.auxRectifying(mu2, inverse=True)
411 psi12 = R._DRectifying2Isometricd(mu2,
412 self._mu1) * mu12
413 lon2 = psi12 * self._salp / self._calp
414 psi2 += psi12
415 else: # PYCHOK no cover
416 lat2 = self.lat1
417 lon2 = self._salp * s12 / self._r1rad
418 if (outmask & Cs.AREA):
419 S12 = R._S12d(lon2, self._psi1, psi2)
420 r.set_(S12=unsigned0(S12)) # like .gx
421 if (outmask & Cs.LONGITUDE):
422 if (outmask & Cs.LONG_UNROLL):
423 lon2 += self.lon1
424 else:
425 lon2 = _norm180(self._lon12 + lon2)
426 r.set_(azi12=self.azi12, s12=s12, a12=a12)
427 if (outmask & Cs.LATITUDE):
428 r.set_(lat2=lat2, lat1=self.lat1)
429 if (outmask & Cs.LONGITUDE):
430 r.set_(lon2=lon2, lon1=self.lon1)
431 if ((outmask | self._debug) & Cs._DEBUG_DIRECT_LINE): # PYCHOK no cover
432 r.set_(a=E.a, f=E.f, f1=E.f1, L=E.L, exact=R.exact,
433 b=E.b, e=E.e, e2=E.e2, k2=R._eF.k2,
434 calp=self._calp, mu1 =self._mu1, mu12=mu12,
435 salp=self._salp, psi1=self._psi1, mu2=mu2)
436 return r
438 @Property_RO
439 def _psi1(self):
440 '''(INTERNAL) Get the I{isometric auxiliary} latitude C{psi} (C{degrees}).
441 '''
442 return self.ellipsoid.auxIsometric(self.lat1)
444 @property_RO
445 def RAorder(self):
446 '''Get this rhumb line's I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8).
447 '''
448 return self.rhumb.RAorder
450 @Property_RO
451 def _r1rad(self): # PYCHOK no cover
452 '''(INTERNAL) Get this rhumb line's parallel I{circle radius} (C{meter}).
453 '''
454 return radians(self.ellipsoid.circle4(self.lat1).radius)
457class RhumbOrder2Tuple(_GTuple):
458 '''2-Tuple C{(RAorder, TMorder)} with a I{Rhumb Area} and
459 I{Transverse Mercator} order, both C{int}, DEPRECATED.
460 '''
461 _Names_ = (Rhumb.RAorder.name, Rhumb.TMorder.name)
462 _Units_ = ( Int, Int)
465# Use I{Divided Differences} to determine (mu2 - mu1) / (psi2 - psi1) accurately.
466# Definition: _Df(x,y,d) = (f(x) - f(y)) / (x - y), @see W. M. Kahan & R. J.
467# Fateman, "Symbolic computation of Divided Differences", SIGSAM Bull. 33(3),
468# 7-28 (1999). U{ACM<https://DL.ACM.org/doi/pdf/10.1145/334714.334716> and @see
469# U{UCB<https://www.CS.Berkeley.edu/~fateman/papers/divdiff.pdf>}, Dec 8, 1999.
471def _Dasinh(x, y):
472 hx = hypot1(x)
473 d = x - y
474 if d:
475 hx *= y
476 hy = x * hypot1(y)
477 t = (d * (x + y) / (hy + hx)) if (x * y) > 0 else (hy - hx)
478 r = asinh(t) / d
479 else:
480 r = _1_0 / hx
481 return r
484def _Datan(x, y):
485 xy = x * y
486 r = xy + _1_0
487 d = x - y
488 if d: # 2 * xy > -1 == 2 * xy + 1 > 0 == xy + r > 0 == xy > -r
489 r = (atan(d / r) if xy > -r else (atan(x) - atan(y))) / d
490 else:
491 r = _1_0 / r
492 return r
495def _Dcosh(x, y):
496 return _Dsincos(x, y, sinh, sinh)
499def _DeatanhE(x, y, E): # see .albers._Datanhee
500 # Deatanhe(x, y) = eatanhe((x - y) / (1 - e^2 * x * y)) / (x - y)
501 e = _1_0 - E.e2 * x * y
502 if e: # assert not isnear0(e)
503 d = x - y
504 e = (E._es_atanh(d / e) / d) if d else (E.e2 / e)
505 return e
508def _DfEt(tx, ty, eF): # tangents
509 # eF = Elliptic(-E.e12) # -E.e2 / (1 - E.e2)
510 r, x, y, = _1_0, atan(tx), atan(ty)
511 d = x - y
512 if (x * y) > 0:
513 # See U{DLMF<https://DLMF.NIST.gov/19.11>}: 19.11.2 and 19.11.4
514 # letting theta -> x, phi -> -y, psi -> z
515 # (E(x) - E(y)) / d = E(z)/d - k2 * sin(x) * sin(y) * sin(z)/d
516 # tan(z/2) = (sin(x)*Delta(y) - sin(y)*Delta(x)) / (cos(x) + cos(y))
517 # = d * Dsin(x,y) * (sin(x) + sin(y))/(cos(x) + cos(y)) /
518 # (sin(x)*Delta(y) + sin(y)*Delta(x))
519 # = t = d * Dt
520 # sin(z) = 2*t/(1+t^2); cos(z) = (1-t^2)/(1+t^2)
521 # Alt (this only works for |z| <= pi/2 -- however, this conditions
522 # holds if x*y > 0):
523 # sin(z) = d * Dsin(x,y) * (sin(x) + sin(y)) /
524 # (sin(x)*cos(y)*Delta(y) + sin(y)*cos(x)*Delta(x))
525 # cos(z) = sqrt((1-sin(z))*(1+sin(z)))
526 sx, cx, sy, cy = sincos2_(x, y)
527 D = (cx + cy) * (eF.fDelta(sy, cy) * sx +
528 eF.fDelta(sx, cx) * sy)
529 D = (sx + sy) * _Dsin(x, y) / D
530 t = D * d
531 t2 = _1_0 + t**2
532 D *= _2_0 / t2
533 s = D * d
534 if s:
535 c = (t + _1_0) * (_1_0 - t) / t2
536 r = eF.fE(s, c, eF.fDelta(s, c)) / s
537 r = D * (r - eF.k2 * sx * sy)
538 elif d:
539 r = (eF.fE(x) - eF.fE(y)) / d
540 return r
543def _Dgd(x, y):
544 return _Datan(sinh(x), sinh(y)) * _Dsinh(x, y)
547def _Dgdinv(x, y): # x, y are tangents
548 return _Dasinh(x, y) / _Datan(x, y)
551def _Dlog(x, y):
552 d = (x - y) * _0_5
553 # Changed atanh(t / (x + y)) to asinh(t / (2 * sqrt(x*y))) to
554 # avoid taking atanh(1) when x is large and y is 1. This also
555 # fixes bogus results being returned for the area when an endpoint
556 # is at a pole. N.B. this routine is invoked with positive x
557 # and y, so the sqrt is always taken of a positive quantity.
558 return (asinh(d / sqrt(x * y)) / d) if d else (_1_0 / x)
561def _Dsin(x, y):
562 return _Dsincos(x, y, sin, cos)
565def _Dsincos(x, y, sin_, cos_):
566 r = cos_((x + y) * _0_5)
567 d = (x - y) * _0_5
568 if d:
569 r *= sin_(d) / d
570 return r
573def _Dsinh(x, y):
574 return _Dsincos(x, y, sinh, cosh)
577def _Dtan(x, y): # PYCHOK no cover
578 return _Dtant(x - y, tan(x), tan(y))
581def _Dtant(dxy, tx, ty):
582 txy = tx * ty
583 r = txy + _1_0
584 if dxy: # 2 * txy > -1 == 2 * txy + 1 > 0 == txy + r > 0 == txy > -r
585 r = ((tan(dxy) * r) if txy > -r else (tx - ty)) / dxy
586 return r
589def _Eaux4(E_aux, mu_psi_x, mu_psi_y): # degrees
590 # get inverse auxiliary lats in radians and tangents
591 phix = radians(E_aux(mu_psi_x, inverse=True))
592 phiy = radians(E_aux(mu_psi_y, inverse=True))
593 return phix, phiy, tan(phix), tan(phiy)
596def _gd(x):
597 return atan(sinh(x))
600def _sincosSeries(sinp, x, y, C, n):
601 # N.B. C[] has n+1 elements of which
602 # C[0] is ignored and n >= 0
603 # Use Clenshaw summation to evaluate
604 # m = (g(x) + g(y)) / 2 -- mean value
605 # s = (g(x) - g(y)) / (x - y) -- average slope
606 # where
607 # g(x) = sum(C[j] * SC(2 * j * x), j = 1..n)
608 # SC = sinp ? sin : cos
609 # CS = sinp ? cos : sin
610 # ...
611 d, _neg = (x - y), neg
612 sp, cp, sd, cd = sincos2_(x + y, d)
613 sd = (sd / d) if d else _1_0
614 s = _neg(sp * sd) # negative
615 # 2x2 matrices in row-major order
616 a1 = s * d**2
617 a2 = s * _4_0
618 a0 = a3 = _2_0 * cp * cd # m
619 b2 = b1 = _0_0s(4)
620 if n > 0:
621 b1 = C[n], _0_0, _0_0, C[n]
623 _fsum = _MODS.fsums.fsum1f_
624 for j in range(n - 1, 0, -1): # C[0] unused
625 b1, b2, Cj = b2, b1, C[j]
626 # b1 = a * b2 - b1 + C[j] * I
627 m0, m1, m2, m3 = b2
628 n0, n1, n2, n3 = map(_neg, b1)
629 b1 = (_fsum(a0 * m0, a1 * m2, n0, Cj),
630 _fsum(a0 * m1, a1 * m3, n1),
631 _fsum(a2 * m0, a3 * m2, n2),
632 _fsum(a2 * m1, a3 * m3, n3, Cj))
633 # Here are the full expressions for m and s
634 # f01, f02, f11, f12 = (0, 0, cd * sp, 2 * sd * cp) if sinp else \
635 # (1, 0, cd * cp, -2 * sd * sp)
636 # m = -b2[1] * f02 + (C[0] - b2[0]) * f01 + b1[0] * f11 + b1[1] * f12
637 # s = -b2[2] * f01 + (C[0] - b2[3]) * f02 + b1[2] * f11 + b1[3] * f12
638 cd *= b1[2]
639 sd *= b1[3] * _2_0
640 s = _fsum(cd * sp, sd * cp) if sinp else \
641 _fsum(cd * cp, _neg(sd * sp), _neg(b2[2]))
642 return s
645_RACoeffs = { # Generated by Maxima on 2015-05-15 08:24:04-04:00
646 4: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4
647 691, 7860, -20160, 18900, 0, 56700, # R[0]/n^0, polynomial(n), order 4
648 1772, -5340, 6930, -4725, 14175, # R[1]/n^1, polynomial(n), order 3
649 -1747, 1590, -630, 4725, # PYCHOK R[2]/n^2, polynomial(n), order 2
650 104, -31, 315, # R[3]/n^3, polynomial(n), order 1
651 -41, 420), # PYCHOK R[4]/n^4, polynomial(n), order 0, count = 20
652 5: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5
653 -79036, 22803, 259380, -665280, 623700, 0, 1871100, # PYCHOK R[0]/n^0, polynomial(n), order 5
654 41662, 58476, -176220, 228690, -155925, 467775, # PYCHOK R[1]/n^1, polynomial(n), order 4
655 18118, -57651, 52470, -20790, 155925, # PYCHOK R[2]/n^2, polynomial(n), order 3
656 -23011, 17160, -5115, 51975, # PYCHOK R[3]/n^3, polynomial(n), order 2
657 5480, -1353, 13860, # PYCHOK R[4]/n^4, polynomial(n), order 1
658 -668, 5775), # PYCHOK R[5]/n^5, polynomial(n), order 0, count = 27
659 6: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6
660 128346268, -107884140, 31126095, 354053700, -908107200, 851350500, 0, 2554051500, # R[0]/n^0, polynomial(n), order 6
661 -114456994, 56868630, 79819740, -240540300, 312161850, -212837625, 638512875, # PYCHOK R[1]/n^1, polynomial(n), order 5
662 51304574, 24731070, -78693615, 71621550, -28378350, 212837625, # R[2]/n^2, polynomial(n), order 4
663 1554472, -6282003, 4684680, -1396395, 14189175, # R[3]/n^3, polynomial(n), order 3
664 -4913956, 3205800, -791505, 8108100, # PYCHOK R[4]/n^4, polynomial(n), order 2
665 1092376, -234468, 2027025, # R[5]/n^5, polynomial(n), order 1
666 -313076, 2027025), # PYCHOK R[6]/n^6, polynomial(n), order 0, count = 35
667 7: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7
668 -317195588, 385038804, -323652420, 93378285, 1062161100, -2724321600, 2554051500, 0, 7662154500, # PYCHOK R[0]/n^0, polynomial(n), order 7
669 258618446, -343370982, 170605890, 239459220, -721620900, 936485550, -638512875, 1915538625, # PYCHOK R[1]/n^1, polynomial(n), order 6
670 -248174686, 153913722, 74193210, -236080845, 214864650, -85135050, 638512875, # PYCHOK R[2]/n^2, polynomial(n), order 5
671 114450437, 23317080, -94230045, 70270200, -20945925, 212837625, # PYCHOK R[3]/n^3, polynomial(n), order 4
672 15445736, -103193076, 67321800, -16621605, 170270100, # PYCHOK R[4]/n^4, polynomial(n), order 3
673 -27766753, 16385640, -3517020, 30405375, # PYCHOK R[4]/n^4, polynomial(n), order 3
674 4892722, -939228, 6081075, # PYCHOK R[4]/n^4, polynomial(n), order 3
675 -3189007, 14189175), # PYCHOK R[7]/n^7, polynomial(n), order 0, count = 44
676 8: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8
677 71374704821, -161769749880, 196369790040, -165062734200, 47622925350, 541702161000, -1389404016000, 1302566265000, 0, 3907698795000, # R[0]/n^0, polynomial(n), order 8
678 -13691187484, 65947703730, -87559600410, 43504501950, 61062101100, -184013329500, 238803815250, -162820783125, 488462349375, # PYCHOK R[1]/n^1, polynomial(n), order 7
679 30802104839, -63284544930, 39247999110, 18919268550, -60200615475, 54790485750, -21709437750, 162820783125, # R[2]/n^2, polynomial(n), order 6
680 -8934064508, 5836972287, 1189171080, -4805732295, 3583780200, -1068242175, 10854718875, # PYCHOK R[3]/n^3, polynomial(n), order 5
681 50072287748, 3938662680, -26314234380, 17167059000, -4238509275, 43418875500, # R[4]/n^4, polynomial(n), order 4
682 359094172, -9912730821, 5849673480, -1255576140, 10854718875, # R[5]/n^5, polynomial(n), order 3
683 -16053944387, 8733508770, -1676521980, 10854718875, # PYCHOK R[6]/n^6, polynomial(n), order 2
684 930092876, -162639357, 723647925, # R[7]/n^7, polynomial(n), order 1
685 -673429061, 1929727800) # PYCHOK R[8]/n^8, polynomial(n), order 0, count = 54
686}
688__all__ += _ALL_DOCS(Caps, Rhumb, RhumbLine)
690if __name__ == '__main__':
692 from pygeodesy.lazily import printf
694 def _re(fmt, r3, x3):
695 e3 = []
696 for r, x in _zip(r3, x3): # strict=True
697 e = fabs(r - x) / fabs(x)
698 e3.append('%.g' % (e,))
699 printf((fmt % r3) + ' rel errors: ' + ', '.join(e3))
701 # <https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve> version 2.0
702 rhumb = Rhumb(exact=True) # WGS84 default
703 printf('# %r\n', rhumb)
704 r = rhumb.Direct8(40.6, -73.8, 51, 5.5e6) # from JFK about NE
705 _re('# JFK NE lat2=%.8f, lon2=%.8f, S12=%.1f', (r.lat2, r.lon2, r.S12), (71.68889988, 0.25551982, 44095641862956.148438))
706 r = rhumb.Inverse8(40.6, -73.8, 51.6, -0.5) # JFK to LHR
707 _re('# JFK-LHR azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (77.76838971, 5771083.383328, 37395209100030.367188))
708 r = rhumb.Inverse8(40.6, -73.8, 35.8, 140.3) # JFK to Tokyo Narita
709 _re('# JFK-NRT azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (-92.388887981699639, 12782581.0676841792, -63760642939072.492))
711# % python3 -m pygeodesy.rhumbx
713# Rhumb(RAorder=6, 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)
715# JFK NE lat2=71.68889988, lon2=0.25551982, S12=44095641862956.1 rel errors: 4e-11, 2e-08, 5e-16
716# JFK-LHR azi12=77.76838971, s12=5771083.383 S12=37395209100030.4 rel errors: 3e-12, 5e-15, 0
717# JFK-NRT azi12=-92.38888798, s12=12782581.068 S12=-63760642939072.5 rel errors: 2e-16, 3e-16, 0
719# **) MIT License
720#
721# Copyright (C) 2022-2023 -- mrJean1 at Gmail -- All Rights Reserved.
722#
723# Permission is hereby granted, free of charge, to any person obtaining a
724# copy of this software and associated documentation files (the "Software"),
725# to deal in the Software without restriction, including without limitation
726# the rights to use, copy, modify, merge, publish, distribute, sublicense,
727# and/or sell copies of the Software, and to permit persons to whom the
728# Software is furnished to do so, subject to the following conditions:
729#
730# The above copyright notice and this permission notice shall be included
731# in all copies or substantial portions of the Software.
732#
733# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
734# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
735# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
736# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
737# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
738# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
739# OTHER DEALINGS IN THE SOFTWARE.