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>}.
8Class L{RhumbLine} has been enhanced with methods C{intersection2} and C{nearestOn4} to find
9the intersection of two rhumb lines, respectively the nearest point on a rumb line.
11For more details, see the C++ U{GeographicLib<https://GeographicLib.SourceForge.io/C++/doc/index.html>}
12documentation, especially the U{Class List<https://GeographicLib.SourceForge.io/C++/doc/annotated.html>},
13the background information on U{Rhumb lines<https://GeographicLib.SourceForge.io/C++/doc/rhumb.html>},
14the utily U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>} and U{Online
15rhumb line calculations<https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve>}.
17Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2014-2022)
18and licensed under the MIT/X11 License. For more information, see the
19U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation.
20'''
21# make sure int/int division yields float quotient
22from __future__ import division as _; del _ # PYCHOK semicolon
24from pygeodesy.basics import copysign0, neg, _xinstanceof, _zip
25from pygeodesy.constants import INT0, _EPSqrt as _TOL, NAN, PI_2, isnan, _0_0s, \
26 _0_0, _0_5, _1_0, _2_0, _4_0, _90_0, _180_0, _720_0
27# from pygeodesy.datums import _spherical_datum # in Rhumb.ellipsoid.setter
28from pygeodesy.errors import IntersectionError, itemsorted, _ValueError, \
29 _xdatum, _xkwds
30# from pygeodesy.etm import ExactTransverseMercator # in ._RhumbLine.xTM
31from pygeodesy.fmath import euclid, favg, fsum1_, hypot, hypot1
32# from pygeodesy.fsums import fsum1_ # from .fmath
33from pygeodesy.interns import NN, _azi12_, _coincident_, _COMMASPACE_, \
34 _intersection_, _lat1_, _lat2_, _lon1_, _lon2_, \
35 _no_, _s12_, _S12_, _UNDER
36from pygeodesy.karney import _a12_, _atan2d, Caps, _CapsBase as _RhumbBase, \
37 _diff182, Direct9Tuple, _EWGS84, _fix90, GDict, \
38 _GTuple, Inverse10Tuple, _norm180
39from pygeodesy.ktm import KTransverseMercator, _Xorder, _Xs, \
40 _AlpCoeffs, _BetCoeffs # PYCHOK used!
41from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS
42from pygeodesy.namedTuples import Distance2Tuple, LatLon2Tuple, NearestOn4Tuple
43from pygeodesy.props import deprecated_method, Property, Property_RO, property_RO, \
44 _update_all
45from pygeodesy.streprs import Fmt, pairs, unstr
46from pygeodesy.units import Bearing as _Azi, Degrees as _Deg, Int, Lat, Lon, \
47 Meter as _M, Meter2 as _M2
48from pygeodesy.utily import sincos2_, sincos2d
49from pygeodesy.vector3d import _intersect3d3, Vector3d # in .intersection2 below
51from math import asinh, atan, cos, cosh, fabs, radians, sin, sinh, sqrt, tan
53__all__ = _ALL_LAZY.rhumbx
54__version__ = '23.04.20'
56_rls = [] # instances of C{RbumbLine} to be updated
57_TRIPS = 65 # .intersection2, 18+
60class _Lat(Lat):
61 '''(INTERNAL) Latitude B{C{lat}}.
62 '''
63 def __init__(self, *lat, **Error_name):
64 kwds = _xkwds(Error_name, clip=0, Error=RhumbError)
65 Lat.__new__(_Lat, *lat, **kwds)
68class _Lon(Lon):
69 '''(INTERNAL) Longitude B{C{lon}}.
70 '''
71 def __init__(self, *lon, **Error_name):
72 kwds = _xkwds(Error_name, clip=0, Error=RhumbError)
73 Lon.__new__(_Lon, *lon, **kwds)
76def _update_all_rls(r):
77 '''(INTERNAL) Zap cached/memoized C{Property[_RO]}s
78 of any L{RhumbLine} instances tied to the given
79 L{Rhumb} instance B{C{r}}.
80 '''
81 _xinstanceof(r, Rhumb)
82 _update_all(r)
83 for rl in _rls: # PYCHOK use weakref?
84 if rl._rhumb is r:
85 _update_all(rl)
88class Rhumb(_RhumbBase):
89 '''Class to solve of the I{direct} and I{inverse rhumb} problems, accurately.
91 @see: The U{Detailed Description<https://GeographicLib.SourceForge.io/C++/doc/
92 classGeographicLib_1_1Rhumb.html>} of I{Karney}'s C++ C{Rhumb Class}.
93 '''
94 _E = _EWGS84
95 _exact = True
96 _mRA = 6
97 _mTM = 6
99 def __init__(self, a_earth=_EWGS84, f=None, exact=True, name=NN, **RA_TMorder):
100 '''New L{Rhumb}.
102 @kwarg a_earth: This rhumb's earth (L{Ellipsoid}, L{Ellipsoid2},
103 L{a_f2Tuple}, L{Datum}, 2-tuple C{(a, f)}) or the
104 (equatorial) radius (C{scalar}).
105 @kwarg f: The ellipsoid's flattening (C{scalar}), iff B{C{a_earth}} is
106 a C{scalar}, ignored otherwise.
107 @kwarg exact: If C{True}, use an addition theorem for elliptic integrals
108 to compute I{Divided differences}, otherwise use the Krüger
109 series expansion (C{bool}), see also property C{exact}.
110 @kwarg name: Optional name (C{str}).
111 @kwarg RA_TMorder: Optional keyword arguments B{C{RAorder}} and B{C{TMorder}}
112 to set the respective C{order}, see properties C{RAorder}
113 and C{TMorder} and method C{orders}.
115 @raise RhumbError: Invalid B{C{a_earth}}, B{C{f}} or B{C{RA_TMorder}}.
116 '''
117 if f is not None:
118 self.ellipsoid = a_earth, f
119 elif a_earth not in (_EWGS84, None):
120 self.ellipsoid = a_earth
121 if not exact:
122 self._exact = False
123 if name:
124 self.name = name
125 if RA_TMorder:
126 self.orders(**RA_TMorder)
128 @Property_RO
129 def _A2(self): # Conformal2RectifyingCoeffs
130 m = self.TMorder
131 return _Xs(_AlpCoeffs, m, self.ellipsoid), m
133 @Property_RO
134 def _B2(self): # Rectifying2ConformalCoeffs
135 m = self.TMorder
136 return _Xs(_BetCoeffs, m, self.ellipsoid), m
138 def _DConformal2Rectifying(self, x, y): # radians
139 return _1_0 + (_sincosSeries(True, x, y, *self._A2) if self.f else _0_0)
141 def Direct(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE):
142 '''Solve the I{direct rhumb} problem, optionally with the area.
144 @arg lat1: Latitude of the first point (C{degrees90}).
145 @arg lon1: Longitude of the first point (C{degrees180}).
146 @arg azi12: Azimuth of the rhumb line (compass C{degrees}).
147 @arg s12: Distance along the rhumb line from the given to
148 the destination point (C{meter}), can be negative.
150 @return: L{GDict} with 2 up to 8 items C{lat2, lon2, a12, S12,
151 lat1, lon1, azi12, s12} with the destination point's
152 latitude C{lat2} and longitude C{lon2} in C{degrees},
153 the rhumb angle C{a12} in C{degrees} and area C{S12}
154 under the rhumb line in C{meter} I{squared}.
156 @note: If B{C{s12}} is large enough that the rhumb line crosses
157 a pole, the longitude of the second point is indeterminate
158 and C{NAN} is returned for C{lon2} and area C{S12}.
160 If the given point is a pole, the cosine of its latitude is
161 taken to be C{epsilon}**-2 (where C{epsilon} is 2.0**-52.
162 This position is extremely close to the actual pole and
163 allows the calculation to be carried out in finite terms.
164 '''
165 rl = _RhumbLine(self, lat1, lon1, azi12, caps=Caps.LINE_OFF,
166 name=self.name)
167 return rl.Position(s12, outmask | self._debug) # lat2, lon2, S12
169 @deprecated_method
170 def Direct7(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE_AREA):
171 '''DEPRECATED, use method L{Rhumb.Direct8}.
173 @return: A I{DEPRECATED} L{Rhumb7Tuple}.
174 '''
175 return self.Direct8(lat1, lon1, azi12, s12, outmask=outmask)._to7Tuple()
177 def Direct8(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE_AREA):
178 '''Like method L{Rhumb.Direct} but returning a L{Rhumb8Tuple} with area C{S12}.
179 '''
180 return self.Direct(lat1, lon1, azi12, s12, outmask=outmask).toRhumb8Tuple()
182 def DirectLine(self, lat1, lon1, azi12, name=NN, **caps): # caps=Caps.STANDARD
183 '''Define a L{RhumbLine} in terms of the I{direct} rhumb problem.
185 @arg lat1: Latitude of the first point (C{degrees90}).
186 @arg lon1: Longitude of the first point (C{degrees180}).
187 @arg azi12: Azimuth of the rhumb line (compass C{degrees}).
188 @kwarg caps: Optional C{caps}, see L{RhumbLine} C{B{caps}}.
190 @return: A L{RhumbLine} instance and invoke its method
191 L{RhumbLine.Position} to compute each point.
193 @note: Updates to this rhumb are reflected in the returned
194 rhumb line.
195 '''
196 return RhumbLine(self, lat1=lat1, lon1=lon1, azi12=azi12,
197 name=name or self.name, **caps)
199 def _DIsometrict(self, phix, phiy, tphix, tphiy, _Dtan_phix_phiy):
200 E = self.ellipsoid
201 return _Dtan_phix_phiy * _Dasinh(tphix, tphiy) - \
202 _Dsin(phix, phiy) * _DeatanhE(sin(phix), sin(phiy), E)
204 def _DIsometric2Rectifyingd(self, psix, psiy): # degrees
205 if self.exact:
206 E = self.ellipsoid
207 phix, phiy, tphix, tphiy = _Eaux4(E.auxIsometric, psix, psiy)
208 t = _Dtant(phix - phiy, tphix, tphiy)
209 r = self._DRectifyingt( tphix, tphiy, t) / \
210 self._DIsometrict(phix, phiy, tphix, tphiy, t)
211 else:
212 x, y = radians(psix), radians(psiy)
213 r = self._DConformal2Rectifying(_gd(x), _gd(y)) * _Dgd(x, y)
214 return r
216 def _DRectifyingt(self, tphix, tphiy, _Dtan_phix_phiy):
217 E = self.ellipsoid
218 tbetx = E.f1 * tphix
219 tbety = E.f1 * tphiy
220 return (E.f1 * _Dtan_phix_phiy * E.b * PI_2
221 * _DfEt( tbetx, tbety, self._eF)
222 * _Datan(tbetx, tbety)) / E.L
224 def _DRectifying2Conformal(self, x, y): # radians
225 return _1_0 - (_sincosSeries(True, x, y, *self._B2) if self.f else _0_0)
227 def _DRectifying2Isometricd(self, mux, muy): # degrees
228 E = self.ellipsoid
229 phix, phiy, tphix, tphiy = _Eaux4(E.auxRectifying, mux, muy)
230 if self.exact:
231 t = _Dtant(phix - phiy, tphix, tphiy)
232 r = self._DIsometrict(phix, phiy, tphix, tphiy, t) / \
233 self._DRectifyingt( tphix, tphiy, t)
234 else:
235 r = self._DRectifying2Conformal(radians(mux), radians(muy)) * \
236 _Dgdinv(E.es_taupf(tphix), E.es_taupf(tphiy))
237 return r
239 @Property_RO
240 def _eF(self):
241 '''(INTERNAL) Get the ellipsoid's elliptic function.
242 '''
243 # .k2 = 0.006739496742276434
244 return self._E._elliptic_e12 # _MODS.elliptic.Elliptic(-self._E._e12)
246 @Property
247 def ellipsoid(self):
248 '''Get this rhumb's ellipsoid (L{Ellipsoid}).
249 '''
250 return self._E
252 @ellipsoid.setter # PYCHOK setter!
253 def ellipsoid(self, a_earth_f):
254 '''Set this rhumb's ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum},
255 L{a_f2Tuple}, 2-tuple C{(a, f)}) or the (equatorial) radius (C{scalar}).
256 '''
257 E = _MODS.datums._spherical_datum(a_earth_f, Error=RhumbError).ellipsoid
258 if self._E != E:
259 _update_all_rls(self)
260 self._E = E
262 @property_RO
263 def equatoradius(self):
264 '''Get the C{ellipsoid}'s equatorial radius, semi-axis (C{meter}).
265 '''
266 return self.ellipsoid.a
268 a = equatoradius
270 @Property
271 def exact(self):
272 '''Get the I{exact} option (C{bool}).
273 '''
274 return self._exact
276 @exact.setter # PYCHOK setter!
277 def exact(self, exact):
278 '''Set the I{exact} option (C{bool}). If C{True}, use I{exact} rhumb
279 calculations, if C{False} results are less precise for more oblate
280 or more prolate ellipsoids with M{abs(flattening) > 0.01} (C{bool}).
282 @see: Option U{B{-s}<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>}
283 and U{ACCURACY<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html#ACCURACY>}.
284 '''
285 x = bool(exact)
286 if self._exact != x:
287 _update_all_rls(self)
288 self._exact = x
290 def flattening(self):
291 '''Get the C{ellipsoid}'s flattening (C{float}).
292 '''
293 return self.ellipsoid.f
295 f = flattening
297 def Inverse(self, lat1, lon1, lat2, lon2, outmask=Caps.AZIMUTH_DISTANCE):
298 '''Solve the I{inverse rhumb} problem.
300 @arg lat1: Latitude of the first point (C{degrees90}).
301 @arg lon1: Longitude of the first point (C{degrees180}).
302 @arg lat2: Latitude of the second point (C{degrees90}).
303 @arg lon2: Longitude of the second point (C{degrees180}).
305 @return: L{GDict} with 5 to 8 items C{azi12, s12, a12, S12,
306 lat1, lon1, lat2, lon2}, the rhumb line's azimuth C{azi12}
307 in compass C{degrees} between C{-180} and C{+180}, the
308 distance C{s12} and rhumb angle C{a12} between both points
309 in C{meter} respectively C{degrees} and the area C{S12}
310 under the rhumb line in C{meter} I{squared}.
312 @note: The shortest rhumb line is found. If the end points are
313 on opposite meridians, there are two shortest rhumb lines
314 and the East-going one is chosen.
316 If either point is a pole, the cosine of its latitude is
317 taken to be C{epsilon}**-2 (where C{epsilon} is 2.0**-52).
318 This position is extremely close to the actual pole and
319 allows the calculation to be carried out in finite terms.
320 '''
321 r, Cs = GDict(name=self.name), Caps
322 if (outmask & Cs.AZIMUTH_DISTANCE_AREA):
323 r.set_(lat1=lat1, lon1=lon1, lat2=lat2, lon2=lon2)
324 E = self.ellipsoid
325 psi1 = E.auxIsometric(lat1)
326 psi2 = E.auxIsometric(lat2)
327 psi12 = psi2 - psi1
328 lon12, _ = _diff182(lon1, lon2)
329 if (outmask & Cs.AZIMUTH):
330 r.set_(azi12=_atan2d(lon12, psi12))
331 if (outmask & Cs.DISTANCE):
332 a12 = hypot(lon12, psi12) * self._DIsometric2Rectifyingd(psi2, psi1)
333 s12 = a12 * E._L_90
334 r.set_(s12=s12, a12=copysign0(a12, s12))
335 if (outmask & Cs.AREA):
336 r.set_(S12=self._S12d(lon12, psi2, psi1))
337 if ((outmask | self._debug) & Cs._DEBUG_INVERSE): # PYCHOK no cover
338 r.set_(a=E.a, f=E.f, f1=E.f1, L=E.L,
339 b=E.b, e=E.e, e2=E.e2, k2=self._eF.k2,
340 lon12=lon12, psi1=psi1, exact=self.exact,
341 psi12=psi12, psi2=psi2)
342 return r
344# def Inverse3(self, lat1, lon1, lat2, lon2): # PYCHOK outmask
345# '''Return the distance in C{meter} and the forward and
346# reverse azimuths (initial and final bearing) in C{degrees}.
347#
348# @return: L{Distance3Tuple}C{(distance, initial, final)}.
349# '''
350# r = self.Inverse(lat1, lon1, lat2, lon2)
351# return Distance3Tuple(r.s12, r.azi12, r.azi12)
353 @deprecated_method
354 def Inverse7(self, lat1, lon1, azi12, s12, outmask=Caps.AZIMUTH_DISTANCE_AREA):
355 '''DEPRECATED, use method L{Rhumb.Inverse8}.
357 @return: A I{DEPRECATED} L{Rhumb7Tuple}.
358 '''
359 return self.Inverse8(lat1, lon1, azi12, s12, outmask=outmask)._to7Tuple()
361 def Inverse8(self, lat1, lon1, azi12, s12, outmask=Caps.AZIMUTH_DISTANCE_AREA):
362 '''Like method L{Rhumb.Inverse} but returning a L{Rhumb8Tuple} with area C{S12}.
363 '''
364 return self.Inverse(lat1, lon1, azi12, s12, outmask=outmask).toRhumb8Tuple()
366 def InverseLine(self, lat1, lon1, lat2, lon2, name=NN, **caps): # caps=Caps.STANDARD
367 '''Define a L{RhumbLine} in terms of the I{inverse} rhumb problem.
369 @arg lat1: Latitude of the first point (C{degrees90}).
370 @arg lon1: Longitude of the first point (C{degrees180}).
371 @arg lat2: Latitude of the second point (C{degrees90}).
372 @arg lon2: Longitude of the second point (C{degrees180}).
373 @kwarg caps: Optional C{caps}, see L{RhumbLine} C{B{caps}}.
375 @return: A L{RhumbLine} instance and invoke its method
376 L{RhumbLine.Position} to compute each point.
378 @note: Updates to this rhumb are reflected in the returned
379 rhumb line.
380 '''
381 r = self.Inverse(lat1, lon1, lat2, lon2, outmask=Caps.AZIMUTH)
382 return RhumbLine(self, lat1=lat1, lon1=lon1, azi12=r.azi12,
383 name=name or self.name, **caps)
385 Line = DirectLine # synonyms
387 def _MeanSinXi(self, x, y): # radians
388 s = _Dlog(cosh(x), cosh(y)) * _Dcosh(x, y)
389 if self.f:
390 s += _sincosSeries(False, _gd(x), _gd(y), *self._RA2) * _Dgd(x, y)
391 return s
393 def orders(self, RAorder=None, TMorder=None):
394 '''Get and set the I{RAorder} and/or I{TMorder}.
396 @kwarg RAorder: I{Rhumb Area} order (C{int}, 4, 5, 6, 7
397 or 8).
398 @kwarg TMorder: I{Transverse Mercator} order (C{int}, 4,
399 5, 6, 7 or 8).
401 @return: L{RhumbOrder2Tuple}C{(RAorder, TMorder)} with
402 the previous C{RAorder} and C{TMorder} setting.
403 '''
404 t = RhumbOrder2Tuple(self.RAorder, self.TMorder)
405 if RAorder not in (None, t.RAorder): # PYCHOK attr
406 self.RAorder = RAorder
407 if TMorder not in (None, t.TMorder): # PYCHOK attr
408 self.TMorder = TMorder
409 return t
411 @Property_RO
412 def _RA2(self):
413 # for WGS84: (0, -0.0005583633519275459, -3.743803759172812e-07, -4.633682270824446e-10,
414 # RAorder 6: -7.709197397676237e-13, -1.5323287106694307e-15, -3.462875359099873e-18)
415 m = self.RAorder
416 return _Xs(_RACoeffs, m, self.ellipsoid, RA=True), m
418 @Property
419 def RAorder(self):
420 '''Get the I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8).
421 '''
422 return self._mRA
424 @RAorder.setter # PYCHOK setter!
425 def RAorder(self, order):
426 '''Set the I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8).
427 '''
428 n = _Xorder(_RACoeffs, RhumbError, RAorder=order)
429 if self._mRA != n:
430 _update_all_rls(self)
431 self._mRA = n
433 def _S12d(self, lon12, psi2, psi1): # degrees
434 '''(INTERNAL) Compute the area C{S12}.
435 '''
436 r = (self.ellipsoid.areax if self.exact else
437 self.ellipsoid.area) * lon12 / _720_0
438 r *= self._MeanSinXi(radians(psi2), radians(psi1))
439 return r
441 @Property
442 def TMorder(self):
443 '''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
444 '''
445 return self._mTM
447 @TMorder.setter # PYCHOK setter!
448 def TMorder(self, order):
449 '''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
451 @note: Setting C{TMorder} turns property C{exact} off.
452 '''
453 n = _Xorder(_AlpCoeffs, RhumbError, TMorder=order)
454 if self._mTM != n:
455 _update_all_rls(self)
456 self._mTM = n
457 self.exact = False
459 def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature
460 '''Return this C{Rhumb} as string.
462 @kwarg prec: The C{float} precision, number of decimal digits (0..9).
463 Trailing zero decimals are stripped for B{C{prec}} values
464 of 1 and above, but kept for negative B{C{prec}} values.
465 @kwarg sep: Separator to join (C{str}).
467 @return: Tuple items (C{str}).
468 '''
469 d = dict(ellipsoid=self.ellipsoid, RAorder=self.RAorder,
470 exact=self.exact, TMorder=self.TMorder)
471 return sep.join(pairs(itemsorted(d, asorted=False), prec=prec))
474class RhumbError(_ValueError):
475 '''Raised for a L{Rhumb} or L{RhumbLine} issue.
476 '''
477 pass
480class _RhumbLine(_RhumbBase):
481 '''(INTERNAL) Class L{RhumbLine}
482 '''
483 _azi12 = _0_0
484# _lat1 = _0_0
485# _lon1 = _0_0
486 _salp = _0_0
487 _calp = _1_0
488 _rhumb = None # L{Rhumb} instance
490 def __init__(self, rhumb, lat1, lon1, azi12, caps=0, name=NN): # case=Caps.?
491 '''New C{RhumbLine}.
492 '''
493 _xinstanceof(Rhumb, rhumb=rhumb)
494 self._lat1 = _Lat(lat1=_fix90(lat1))
495 self._lon1 = _Lon(lon1= lon1)
496 self._debug |= (caps | rhumb._debug) & Caps._DEBUG_DIRECT_LINE
497 if azi12: # non-zero
498 self.azi12 = azi12
499 self._caps = caps
500 if not (caps & Caps.LINE_OFF):
501 _rls.append(self)
502 n = name or rhumb.name
503 if n:
504 self.name=n
505 self._rhumb = rhumb # last
507 def __del__(self): # XXX use weakref?
508 if _rls: # may be empty or None
509 try: # PYCHOK no cover
510 _rls.remove(self)
511 except (TypeError, ValueError):
512 pass
513 self._rhumb = None
514 # _update_all(self) # throws TypeError during Python 2 cleanup
516 @Property
517 def azi12(self):
518 '''Get this rhumb line's I{azimuth} (compass C{degrees}).
519 '''
520 return self._azi12
522 @azi12.setter # PYCHOK setter!
523 def azi12(self, azi12):
524 '''Set this rhumb line's I{azimuth} (compass C{degrees}).
525 '''
526 z = _norm180(azi12)
527 if self._azi12 != z:
528 if self._rhumb:
529 _update_all(self)
530 self._azi12 = z
531 self._salp, self._calp = sincos2d(z) # no NEG0
533 def distance2(self, lat, lon):
534 '''Return the distance from and (initial) bearing at the given
535 point to this rhumb line's start point.
537 @arg lat: Latitude of the point (C{degrees}).
538 @arg lon: Longitude of the points (C{degrees}).
540 @return: A L{Distance2Tuple}C{(distance, initial)} with the C{distance}
541 in C{meter} and C{initial} bearing in C{degrees}.
543 @see: Methods L{RhumbLine.intersection2} and L{RhumbLine.nearestOn4}.
544 '''
545 r = self.rhumb.Inverse(self.lat1, self.lon1, lat, lon)
546# outmask=Caps.AZIMUTH_DISTANCE)
547 return Distance2Tuple(r.s12, r.azi12)
549 @Property_RO
550 def ellipsoid(self):
551 '''Get this rhumb line's ellipsoid (L{Ellipsoid}).
552 '''
553 return self.rhumb.ellipsoid
555 @property_RO
556 def exact(self):
557 '''Get this rhumb line's I{exact} option (C{bool}).
558 '''
559 return self.rhumb.exact
561 def intersection2(self, other, tol=_TOL, **eps):
562 '''Iteratively find the intersection of this and an other rhumb line.
564 @arg other: The other rhumb line (L{RhumbLine}).
565 @kwarg tol: Tolerance for longitudinal convergence (C{degrees}).
566 @kwarg eps: Tolerance for L{intersection3d3} (C{EPS}).
568 @return: A L{LatLon2Tuple}{(lat, lon)} with the C{lat}- and
569 C{lon}gitude of the intersection point.
571 @raise IntersectionError: No convergence for this B{C{tol}} or
572 no intersection for an other reason.
574 @see: Methods L{RhumbLine.distance2} and L{RhumbLine.nearestOn4}
575 and function L{pygeodesy.intersection3d3}.
577 @note: Each iteration involves a round trip to this rhumb line's
578 L{ExactTransverseMercator} or L{KTransverseMercator}
579 projection and invoking function L{intersection3d3} in
580 that domain.
581 '''
582 _xinstanceof(other, _RhumbLine)
583 _xdatum(self.rhumb, other.rhumb, Error=RhumbError)
584 try:
585 if other is self:
586 raise ValueError(_coincident_)
587 # make globals and invariants locals
588 _diff = euclid # approximate length
589 _i3d3 = _intersect3d3 # NOT .vector3d.intersection3d3
590 _LL2T = LatLon2Tuple
591 _xTMr = self.xTM.reverse # ellipsoidal or spherical
592 _s_3d, s_az = self._xTM3d, self.azi12
593 _o_3d, o_az = other._xTM3d, other.azi12
594 # use halfway point as initial estimate
595 p = _LL2T(favg(self.lat1, other.lat1),
596 favg(self.lon1, other.lon1))
597 for i in range(1, _TRIPS):
598 v = _i3d3(_s_3d(p), s_az, # point + bearing
599 _o_3d(p), o_az, useZ=False, **eps)[0]
600 t = _xTMr(v.x, v.y, lon0=p.lon) # PYCHOK Reverse4Tuple
601 d = _diff(t.lon - p.lon, t.lat) # PYCHOK t.lat + p.lat - p.lat
602 p = _LL2T(t.lat + p.lat, t.lon) # PYCHOK t.lon + p.lon = lon0
603 if d < tol:
604 return _LL2T(p.lat, p.lon, iteration=i, # PYCHOK p...
605 name=self.intersection2.__name__)
606 except Exception as x:
607 raise IntersectionError(_no_(_intersection_), cause=x)
608 t = unstr(self.intersection2, tol=tol, **eps)
609 raise IntersectionError(Fmt.no_convergence(d), txt=t)
611 @property_RO
612 def lat1(self):
613 '''Get this rhumb line's latitude (C{degrees90}).
614 '''
615 return self._lat1
617 @property_RO
618 def lon1(self):
619 '''Get this rhumb line's longitude (C{degrees180}).
620 '''
621 return self._lon1
623 @Property_RO
624 def latlon1(self):
625 '''Get this rhumb line's lat- and longitude (L{LatLon2Tuple}C{(lat, lon)}).
626 '''
627 return LatLon2Tuple(self.lat1, self.lon1)
629 @Property_RO
630 def _mu1(self):
631 '''(INTERNAL) Get the I{rectifying auxiliary} latitude C{mu} (C{degrees}).
632 '''
633 return self.ellipsoid.auxRectifying(self.lat1)
635 def nearestOn4(self, lat, lon, tol=_TOL, **eps):
636 '''Iteratively locate the point on this rhumb line nearest to
637 the given point.
639 @arg lat: Latitude of the point (C{degrees}).
640 @arg lon: Longitude of the point (C{degrees}).
641 @kwarg tol: Longitudinal convergence tolerance (C{degrees}).
642 @kwarg eps: Tolerance for L{intersection3d3} (C{EPS}).
644 @return: A L{NearestOn4Tuple}C{(lat, lon, distance, normal)} with
645 the C{lat}- and C{lon}gitude of the nearest point on and
646 the C{distance} in C{meter} to this rhumb line and with the
647 azimuth of the C{normal}, perpendicular to this rhumb line.
649 @raise IntersectionError: No convergence for this B{C{eps}} or
650 no intersection for an other reason.
652 @see: Methods L{RhumbLine.distance2} and L{RhumbLine.intersection2}
653 and function L{intersection3d3}.
654 '''
655 z = _norm180(self.azi12 + _90_0) # perpendicular
656 r = _RhumbLine(self.rhumb, lat, lon, z, caps=Caps.LINE_OFF)
657 p = self.intersection2(r, tol=tol, **eps)
658 t = r.distance2(p.lat, p.lon)
659 return NearestOn4Tuple(p.lat, p.lon, t.distance, z,
660 iteration=p.iteration)
662 @Property_RO
663 def _psi1(self):
664 '''(INTERNAL) Get the I{isometric auxiliary} latitude C{psi} (C{degrees}).
665 '''
666 return self.ellipsoid.auxIsometric(self.lat1)
668 @property_RO
669 def RAorder(self):
670 '''Get this rhumb line's I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8).
671 '''
672 return self.rhumb.RAorder
674 @Property_RO
675 def _r1rad(self): # PYCHOK no cover
676 '''(INTERNAL) Get this rhumb line's parallel I{circle radius} (C{meter}).
677 '''
678 return radians(self.ellipsoid.circle4(self.lat1).radius)
680 @Property_RO
681 def rhumb(self):
682 '''Get this rhumb line's rhumb (L{Rhumb}).
683 '''
684 return self._rhumb
686 def Position(self, s12, outmask=Caps.LATITUDE_LONGITUDE):
687 '''Compute a position at a distance on this rhumb line.
689 @arg s12: The distance along this rhumb between its point and
690 the other point (C{meters}), can be negative.
691 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying
692 the quantities to be returned.
694 @return: L{GDict} with 4 to 8 items C{azi12, a12, s12, S12, lat2,
695 lon2, lat1, lon1} with latitude C{lat2} and longitude
696 C{lon2} of the other point in C{degrees}, the rhumb angle
697 C{a12} between both points in C{degrees} and the area C{S12}
698 under the rhumb line in C{meter} I{squared}.
700 @note: If B{C{s12}} is large enough that the rhumb line crosses a
701 pole, the longitude of the second point is indeterminate and
702 C{NAN} is returned for C{lon2} and area C{S12}.
704 If the first point is a pole, the cosine of its latitude is
705 taken to be C{epsilon}**-2 (where C{epsilon} is 2**-52).
706 This position is extremely close to the actual pole and
707 allows the calculation to be carried out in finite terms.
708 '''
709 r, Cs = GDict(name=self.name), Caps
710 if (outmask & Cs.LATITUDE_LONGITUDE_AREA):
711 E, R = self.ellipsoid, self.rhumb
712 mu12 = s12 * self._calp / E._L_90
713 mu2 = mu12 + self._mu1
714 if fabs(mu2) > 90: # PYCHOK no cover
715 mu2 = _norm180(mu2) # reduce to [-180, 180)
716 if fabs(mu2) > 90: # point on anti-meridian
717 mu2 = _norm180(_180_0 - mu2)
718 lat2x = E.auxRectifying(mu2, inverse=True)
719 lon2x = NAN
720 if (outmask & Cs.AREA):
721 r.set_(S12=NAN)
722 else:
723 psi2 = self._psi1
724 if self._calp:
725 lat2x = E.auxRectifying(mu2, inverse=True)
726 psi12 = R._DRectifying2Isometricd(mu2,
727 self._mu1) * mu12
728 lon2x = psi12 * self._salp / self._calp
729 psi2 += psi12
730 else: # PYCHOK no cover
731 lat2x = self.lat1
732 lon2x = self._salp * s12 / self._r1rad
733 if (outmask & Cs.AREA):
734 r.set_(S12=R._S12d(lon2x, self._psi1, psi2))
735 r.set_(s12=s12, azi12=self.azi12, a12=s12 / E._L_90)
736 if (outmask & Cs.LATITUDE):
737 r.set_(lat2=lat2x, lat1=self.lat1)
738 if (outmask & Cs.LONGITUDE):
739 if (outmask & Cs.LONG_UNROLL) and not isnan(lat2x):
740 lon2x += self.lon1
741 else:
742 lon2x = _norm180(_norm180(self.lon1) + lon2x)
743 r.set_(lon2=lon2x, lon1=self.lon1)
744 if ((outmask | self._debug) & Cs._DEBUG_DIRECT_LINE): # PYCHOK no cover
745 r.set_(a=E.a, f=E.f, f1=E.f1, L=E.L, exact=R.exact,
746 b=E.b, e=E.e, e2=E.e2, k2=R._eF.k2,
747 calp=self._calp, mu1 =self._mu1, mu12=mu12,
748 salp=self._salp, psi1=self._psi1, mu2=mu2)
749 return r
751 def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature
752 '''Return this C{RhumbLine} as string.
754 @kwarg prec: The C{float} precision, number of decimal digits (0..9).
755 Trailing zero decimals are stripped for B{C{prec}} values
756 of 1 and above, but kept for negative B{C{prec}} values.
757 @kwarg sep: Separator to join (C{str}).
759 @return: C{RhumbLine} (C{str}).
760 '''
761 d = dict(rhumb=self.rhumb, lat1=self.lat1, lon1=self.lon1,
762 azi12=self.azi12, exact=self.exact,
763 TMorder=self.TMorder, xTM=self.xTM)
764 return sep.join(pairs(itemsorted(d, asorted=False), prec=prec))
766 @property_RO
767 def TMorder(self):
768 '''Get this rhumb line's I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
769 '''
770 return self.rhumb.TMorder
772 @Property_RO
773 def xTM(self):
774 '''Get this rhumb line's I{Transverse Mercator} projection (L{ExactTransverseMercator}
775 if I{exact} and I{ellipsoidal}, otherwise L{KTransverseMercator}).
776 '''
777 E = self.ellipsoid
778 # ExactTransverseMercator doesn't handle spherical earth models
779 return _MODS.etm.ExactTransverseMercator(E) if self.exact and E.isEllipsoidal else \
780 KTransverseMercator(E, TMorder=self.TMorder)
782 def _xTM3d(self, latlon0, z=INT0, V3d=Vector3d):
783 '''(INTERNAL) C{xTM.forward} this C{latlon1} to C{V3d} with B{C{latlon0}}
784 as current intersection estimate and central meridian.
785 '''
786 t = self.xTM.forward(self.lat1 - latlon0.lat, self.lon1, lon0=latlon0.lon)
787 return V3d(t.easting, t.northing, z)
790class RhumbLine(_RhumbLine):
791 '''Compute one or several points on a single rhumb line.
793 Class L{RhumbLine} facilitates the determination of points on
794 a single rhumb line. The starting point (C{lat1}, C{lon1})
795 and the azimuth C{azi12} are specified once.
797 Method L{RhumbLine.Position} returns the location of an other
798 point and optionally the distance C{s12} along the corresponding
799 area C{S12} under the rhumb line.
801 Method L{RhumbLine.intersection2} finds the intersection between
802 two rhumb lines.
804 Method L{RhumbLine.nearestOn4} computes the nearest point on and
805 its distance to a rhumb line.
806 '''
807 def __init__(self, rhumb, lat1=0, lon1=0, azi12=None, caps=0, name=NN): # case=Caps.?
808 '''New L{RhumbLine}.
810 @arg rhumb: The rhumb reference (L{Rhumb}).
811 @kwarg lat1: Latitude of the start point (C{degrees90}).
812 @kwarg lon1: Longitude of the start point (C{degrees180}).
813 @kwarg azi12: Azimuth of this rhumb line (compass C{degrees}).
814 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying
815 the capabilities. Include C{Caps.LINE_OFF} if
816 updates to B{C{rhumb}} should I{not} be reflected
817 in this rhumb line.
818 @kwarg name: Optional name (C{str}).
819 '''
820 if (caps & Caps.LINE_OFF): # copy to avoid updates
821 rhumb = rhumb.copy(deep=False, name=_UNDER(rhumb.name))
822 _RhumbLine.__init__(self, rhumb, lat1, lon1, azi12, caps=caps, name=name)
825class RhumbOrder2Tuple(_GTuple):
826 '''2-Tuple C{(RAorder, TMorder)} with a I{Rhumb Area} and
827 I{Transverse Mercator} order, both C{int}.
828 '''
829 _Names_ = (Rhumb.RAorder.name, Rhumb.TMorder.name)
830 _Units_ = ( Int, Int)
833class Rhumb8Tuple(_GTuple):
834 '''8-Tuple C{(lat1, lon1, lat2, lon2, azi12, s12, S12, a12)} with lat- C{lat1},
835 C{lat2} and longitudes C{lon1}, C{lon2} of both points, the azimuth of the
836 rhumb line C{azi12}, the distance C{s12}, the area C{S12} under the rhumb
837 line and the angular distance C{a12} between both points.
838 '''
839 _Names_ = (_lat1_, _lon1_, _lat2_, _lon2_, _azi12_, _s12_, _S12_, _a12_)
840 _Units_ = (_Lat, _Lon, _Lat, _Lon, _Azi, _M, _M2, _Deg)
842 def toDirect9Tuple(self, dflt=NAN, **a12_azi1_azi2_m12_M12_M21):
843 '''Convert this L{Rhumb8Tuple} result to a 9-tuple, like I{Karney}'s
844 method C{geographiclib.geodesic.Geodesic._GenDirect}.
846 @kwarg dflt: Default value for missing items (C{any}).
847 @kwarg a12_azi1_azi2_m12_M12_M21: Optional keyword arguments
848 to specify or override L{Inverse10Tuple} items.
850 @return: L{Direct9Tuple}C{(a12, lat2, lon2, azi2, s12,
851 m12, M12, M21, S12)}
852 '''
853 d = dict(azi1=self.azi12, M12=_1_0, m12=self.s12, # PYCHOK attr
854 azi2=self.azi12, M21=_1_0) # PYCHOK attr
855 if a12_azi1_azi2_m12_M12_M21:
856 d.update(a12_azi1_azi2_m12_M12_M21)
857 return self._toTuple(Direct9Tuple, dflt, d)
859 def toInverse10Tuple(self, dflt=NAN, **a12_m12_M12_M21_salp1_calp1_salp2_calp2):
860 '''Convert this L{Rhumb8Tuple} to a 10-tuple, like I{Karney}'s
861 method C{geographiclib.geodesic.Geodesic._GenInverse}.
863 @kwarg dflt: Default value for missing items (C{any}).
864 @kwarg a12_m12_M12_M21_salp1_calp1_salp2_calp2: Optional keyword
865 arguments to specify or override L{Inverse10Tuple} items.
867 @return: L{Inverse10Tuple}C{(a12, s12, salp1, calp1, salp2, calp2,
868 m12, M12, M21, S12)}.
869 '''
870 s, c = sincos2d(self.azi12) # PYCHOK attr
871 d = dict(salp1=s, calp1=c, M12=_1_0, m12=self.s12, # PYCHOK attr
872 salp2=s, calp2=c, M21=_1_0)
873 if a12_m12_M12_M21_salp1_calp1_salp2_calp2:
874 d.update(a12_m12_M12_M21_salp1_calp1_salp2_calp2)
875 return self._toTuple(Inverse10Tuple, dflt, d)
877 def _toTuple(self, nTuple, dflt, updates={}):
878 '''(INTERNAL) Convert this C{Rhumb8Tuple} to an B{C{nTuple}}.
879 '''
880 _g = self.toGDict(**updates).get
881 t = tuple(_g(n, dflt) for n in nTuple._Names_)
882 return nTuple(t, name=self.name)
884 @deprecated_method
885 def _to7Tuple(self):
886 '''DEPRECATED, do not use!
887 '''
888 return _MODS.deprecated.Rhumb7Tuple(self[:-1])
891# Use I{Divided Differences} to determine (mu2 - mu1) / (psi2 - psi1) accurately.
892# Definition: _Df(x,y,d) = (f(x) - f(y)) / (x - y), @see W. M. Kahan & R. J.
893# Fateman, "Symbolic computation of Divided Differences", SIGSAM Bull. 33(3),
894# 7-28 (1999). U{ACM<https://DL.ACM.org/doi/pdf/10.1145/334714.334716>, @see
895# U{UCB<https://www.CS.Berkeley.edu/~fateman/papers/divdiff.pdf>}, Dec 8, 1999.
897def _Dasinh(x, y):
898 hx = hypot1(x)
899 d = x - y
900 if d:
901 hx *= y
902 hy = x * hypot1(y)
903 t = (d * (x + y) / (hy + hx)) if (x * y) > 0 else (hy - hx)
904 r = asinh(t) / d
905 else:
906 r = _1_0 / hx
907 return r
910def _Datan(x, y):
911 xy = x * y
912 r = xy + _1_0
913 d = x - y
914 if d: # 2 * xy > -1 == 2 * xy + 1 > 0 == xy + r > 0 == xy > -r
915 r = (atan(d / r) if xy > -r else (atan(x) - atan(y))) / d
916 else:
917 r = _1_0 / r
918 return r
921def _Dcosh(x, y):
922 return _Dsincos(x, y, sinh, sinh)
925def _DeatanhE(x, y, E): # see .albers._Datanhee
926 # Deatanhe(x, y) = eatanhe((x - y) / (1 - e^2 * x * y)) / (x - y)
927 e = _1_0 - E.e2 * x * y
928 # assert not isnear0(e)
929 d = x - y
930 return (E._es_atanh(d / e) / d) if d else (E.e2 / e)
933def _DfEt(tx, ty, eF): # tangents
934 # eF = Elliptic(-E.e12) # -E.e2 / (1 - E.e2)
935 r, x, y, = _1_0, atan(tx), atan(ty)
936 d = x - y
937 if (x * y) > 0:
938 # See U{DLMF<https://DLMF.NIST.gov/19.11>}: 19.11.2 and 19.11.4
939 # letting theta -> x, phi -> -y, psi -> z
940 # (E(x) - E(y)) / d = E(z)/d - k2 * sin(x) * sin(y) * sin(z)/d
941 # tan(z/2) = (sin(x)*Delta(y) - sin(y)*Delta(x)) / (cos(x) + cos(y))
942 # = d * Dsin(x,y) * (sin(x) + sin(y))/(cos(x) + cos(y)) /
943 # (sin(x)*Delta(y) + sin(y)*Delta(x))
944 # = t = d * Dt
945 # sin(z) = 2*t/(1+t^2); cos(z) = (1-t^2)/(1+t^2)
946 # Alt (this only works for |z| <= pi/2 -- however, this conditions
947 # holds if x*y > 0):
948 # sin(z) = d * Dsin(x,y) * (sin(x) + sin(y)) /
949 # (sin(x)*cos(y)*Delta(y) + sin(y)*cos(x)*Delta(x))
950 # cos(z) = sqrt((1-sin(z))*(1+sin(z)))
951 sx, cx, sy, cy = sincos2_(x, y)
952 D = (cx + cy) * (eF.fDelta(sy, cy) * sx +
953 eF.fDelta(sx, cx) * sy)
954 D = (sx + sy) * _Dsin(x, y) / D
955 t = D * d
956 t2 = t**2 + _1_0
957 D *= _2_0 / t2
958 s = D * d
959 if s:
960 c = (t + _1_0) * (_1_0 - t) / t2
961 r = eF.fE(s, c, eF.fDelta(s, c)) / s
962 r = D * (r - eF.k2 * sx * sy)
963 elif d:
964 r = (eF.fE(x) - eF.fE(y)) / d
965 return r
968def _Dgd(x, y):
969 return _Datan(sinh(x), sinh(y)) * _Dsinh(x, y)
972def _Dgdinv(x, y): # x, y are tangents
973 return _Dasinh(x, y) / _Datan(x, y)
976def _Dlog(x, y):
977 d = (x - y) * _0_5
978 # Changed atanh(t / (x + y)) to asinh(t / (2 * sqrt(x*y))) to
979 # avoid taking atanh(1) when x is large and y is 1. This also
980 # fixes bogus results being returned for the area when an endpoint
981 # is at a pole. N.B. this routine is invoked with positive x
982 # and y, so the sqrt is always taken of a positive quantity.
983 return (asinh(d / sqrt(x * y)) / d) if d else (_1_0 / x)
986def _Dsin(x, y):
987 return _Dsincos(x, y, sin, cos)
990def _Dsincos(x, y, sin_, cos_):
991 r = cos_((x + y) * _0_5)
992 d = (x - y) * _0_5
993 if d:
994 r *= sin_(d) / d
995 return r
998def _Dsinh(x, y):
999 return _Dsincos(x, y, sinh, cosh)
1002def _Dtan(x, y): # PYCHOK no cover
1003 return _Dtant(x - y, tan(x), tan(y))
1006def _Dtant(dxy, tx, ty):
1007 txy = tx * ty
1008 r = txy + _1_0
1009 if dxy: # 2 * txy > -1 == 2 * txy + 1 > 0 == txy + r > 0 == txy > -r
1010 r = ((tan(dxy) * r) if txy > -r else (tx - ty)) / dxy
1011 return r
1014def _Eaux4(E_aux, mu_psi_x, mu_psi_y): # degrees
1015 # get inverse auxiliary lats in radians and tangents
1016 phix = radians(E_aux(mu_psi_x, inverse=True))
1017 phiy = radians(E_aux(mu_psi_y, inverse=True))
1018 return phix, phiy, tan(phix), tan(phiy)
1021def _gd(x):
1022 return atan(sinh(x))
1025def _sincosSeries(sinp, x, y, C, n):
1026 # N.B. C[] has n+1 elements of which
1027 # C[0] is ignored and n >= 0
1028 # Use Clenshaw summation to evaluate
1029 # m = (g(x) + g(y)) / 2 -- mean value
1030 # s = (g(x) - g(y)) / (x - y) -- average slope
1031 # where
1032 # g(x) = sum(C[j] * SC(2 * j * x), j = 1..n)
1033 # SC = sinp ? sin : cos
1034 # CS = sinp ? cos : sin
1035 # ...
1036 d = x - y
1037 sp, cp, sd, cd = sincos2_(x + y, d)
1038 sd = (sd / d) if d else _1_0
1039 m = cp * cd * _2_0
1040 s = neg(sp * sd) # negative
1041 # 2x2 matrices in row-major order
1042 a0, a1 = m, (s * d**2)
1043 a2, a3 = (s * _4_0), m
1044 b2 = b1 = _0_0s(4)
1045 if n > 0:
1046 b1 = C[n], _0_0, _0_0, C[n]
1047 _fsum1_, _neg = fsum1_, neg
1048 for j in range(n - 1, 0, -1):
1049 b1, b2, Cj = b2, b1, C[j] # C[0] unused
1050 # b1 = a * b2 - b1 + C[j] * I
1051 m0, m1, m2, m3 = b2
1052 n0, n1, n2, n3 = map(_neg, b1)
1053 b1 = (_fsum1_(a0 * m0, a1 * m2, n0, Cj, floats=True),
1054 _fsum1_(a0 * m1, a1 * m3, n1, floats=True),
1055 _fsum1_(a2 * m0, a3 * m2, n2, floats=True),
1056 _fsum1_(a2 * m1, a3 * m3, n3, Cj, floats=True))
1057 # Here are the full expressions for m and s
1058 # f01, f02, f11, f12 = (0, 0, cd * sp, 2 * sd * cp) if sinp else \
1059 # (1, 0, cd * cp, -2 * sd * sp)
1060 # m = -b2[1] * f02 + (C[0] - b2[0]) * f01 + b1[0] * f11 + b1[1] * f12
1061 # s = -b2[2] * f01 + (C[0] - b2[3]) * f02 + b1[2] * f11 + b1[3] * f12
1062 cd *= b1[2]
1063 sd *= b1[3] * _2_0
1064 s = _fsum1_(cd * sp, sd * cp, floats=True) if sinp else \
1065 _fsum1_(cd * cp, _neg(sd * sp), _neg(b2[2]), floats=True)
1066 return s
1069_RACoeffs = { # Generated by Maxima on 2015-05-15 08:24:04-04:00
1070 4: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4
1071 691, 7860, -20160, 18900, 0, 56700, # R[0]/n^0, polynomial(n), order 4
1072 1772, -5340, 6930, -4725, 14175, # R[1]/n^1, polynomial(n), order 3
1073 -1747, 1590, -630, 4725, # PYCHOK R[2]/n^2, polynomial(n), order 2
1074 104, -31, 315, # R[3]/n^3, polynomial(n), order 1
1075 -41, 420), # PYCHOK R[4]/n^4, polynomial(n), order 0, count = 20
1076 5: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5
1077 -79036, 22803, 259380, -665280, 623700, 0, 1871100, # PYCHOK R[0]/n^0, polynomial(n), order 5
1078 41662, 58476, -176220, 228690, -155925, 467775, # PYCHOK R[1]/n^1, polynomial(n), order 4
1079 18118, -57651, 52470, -20790, 155925, # PYCHOK R[2]/n^2, polynomial(n), order 3
1080 -23011, 17160, -5115, 51975, # PYCHOK R[3]/n^3, polynomial(n), order 2
1081 5480, -1353, 13860, # PYCHOK R[4]/n^4, polynomial(n), order 1
1082 -668, 5775), # PYCHOK R[5]/n^5, polynomial(n), order 0, count = 27
1083 6: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6
1084 128346268, -107884140, 31126095, 354053700, -908107200, 851350500, 0, 2554051500, # R[0]/n^0, polynomial(n), order 6
1085 -114456994, 56868630, 79819740, -240540300, 312161850, -212837625, 638512875, # PYCHOK R[1]/n^1, polynomial(n), order 5
1086 51304574, 24731070, -78693615, 71621550, -28378350, 212837625, # R[2]/n^2, polynomial(n), order 4
1087 1554472, -6282003, 4684680, -1396395, 14189175, # R[3]/n^3, polynomial(n), order 3
1088 -4913956, 3205800, -791505, 8108100, # PYCHOK R[4]/n^4, polynomial(n), order 2
1089 1092376, -234468, 2027025, # R[5]/n^5, polynomial(n), order 1
1090 -313076, 2027025), # PYCHOK R[6]/n^6, polynomial(n), order 0, count = 35
1091 7: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7
1092 -317195588, 385038804, -323652420, 93378285, 1062161100, -2724321600, 2554051500, 0, 7662154500, # PYCHOK R[0]/n^0, polynomial(n), order 7
1093 258618446, -343370982, 170605890, 239459220, -721620900, 936485550, -638512875, 1915538625, # PYCHOK R[1]/n^1, polynomial(n), order 6
1094 -248174686, 153913722, 74193210, -236080845, 214864650, -85135050, 638512875, # PYCHOK R[2]/n^2, polynomial(n), order 5
1095 114450437, 23317080, -94230045, 70270200, -20945925, 212837625, # PYCHOK R[3]/n^3, polynomial(n), order 4
1096 15445736, -103193076, 67321800, -16621605, 170270100, # PYCHOK R[4]/n^4, polynomial(n), order 3
1097 -27766753, 16385640, -3517020, 30405375, # PYCHOK R[4]/n^4, polynomial(n), order 3
1098 4892722, -939228, 6081075, # PYCHOK R[4]/n^4, polynomial(n), order 3
1099 -3189007, 14189175), # PYCHOK R[7]/n^7, polynomial(n), order 0, count = 44
1100 8: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8
1101 71374704821, -161769749880, 196369790040, -165062734200, 47622925350, 541702161000, -1389404016000, 1302566265000, 0, 3907698795000, # R[0]/n^0, polynomial(n), order 8
1102 -13691187484, 65947703730, -87559600410, 43504501950, 61062101100, -184013329500, 238803815250, -162820783125, 488462349375, # PYCHOK R[1]/n^1, polynomial(n), order 7
1103 30802104839, -63284544930, 39247999110, 18919268550, -60200615475, 54790485750, -21709437750, 162820783125, # R[2]/n^2, polynomial(n), order 6
1104 -8934064508, 5836972287, 1189171080, -4805732295, 3583780200, -1068242175, 10854718875, # PYCHOK R[3]/n^3, polynomial(n), order 5
1105 50072287748, 3938662680, -26314234380, 17167059000, -4238509275, 43418875500, # R[4]/n^4, polynomial(n), order 4
1106 359094172, -9912730821, 5849673480, -1255576140, 10854718875, # R[5]/n^5, polynomial(n), order 3
1107 -16053944387, 8733508770, -1676521980, 10854718875, # PYCHOK R[6]/n^6, polynomial(n), order 2
1108 930092876, -162639357, 723647925, # R[7]/n^7, polynomial(n), order 1
1109 -673429061, 1929727800) # PYCHOK R[8]/n^8, polynomial(n), order 0, count = 54
1110}
1112__all__ += _ALL_DOCS(Caps, _RhumbLine)
1114if __name__ == '__main__':
1116 def _re(fmt, r3, x3):
1117 e3 = []
1118 for r, x in _zip(r3, x3): # strict=True
1119 e = fabs(r - x) / fabs(x)
1120 e3.append('%.g' % (e,))
1121 print((fmt % r3) + ' rel errors: ' + ', '.join(e3))
1123 # <https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve>
1124 rhumb = Rhumb(exact=True) # WGS84 default
1125 print('# %r\n' % rhumb)
1126 r = rhumb.Direct8(40.6, -73.8, 51, 5.5e6) # from JFK about NE
1127 _re('# JFK NE lat2=%.8f, lon2=%.8f, S12=%.1f', (r.lat2, r.lon2, r.S12), (71.68889988, 0.25551982, 44095641862956.148438))
1128 r = rhumb.Inverse8(40.6, -73.8, 51.6, -0.5) # JFK to LHR
1129 _re('# JFK-LHR azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (77.76838971, 5771083.383328, 37395209100030.367188))
1130 r = rhumb.Inverse8(40.6, -73.8, 35.8, 140.3) # JFK to Tokyo Narita
1131 _re('# JFK-NRT azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (-92.388887981699639, 12782581.0676841792, -63760642939072.492))
1133# % python3 -m pygeodesy.rhumbx
1135# 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, 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)
1137# JFK NE lat2=71.68889988, lon2=0.25551982, S12=44095641862956.1 rel errors: 4e-11, 2e-08, 2e-15
1138# JFK-LHR azi12=77.76838971, s12=5771083.383 S12=37395209100030.4 rel errors: 3e-12, 5e-15, 2e-16
1139# JFK-NRT azi12=-92.38888798, s12=12782581.068 S12=-63760642939072.5 rel errors: 2e-16, 3e-16, 0
1141# **) MIT License
1142#
1143# Copyright (C) 2022-2023 -- mrJean1 at Gmail -- All Rights Reserved.
1144#
1145# Permission is hereby granted, free of charge, to any person obtaining a
1146# copy of this software and associated documentation files (the "Software"),
1147# to deal in the Software without restriction, including without limitation
1148# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1149# and/or sell copies of the Software, and to permit persons to whom the
1150# Software is furnished to do so, subject to the following conditions:
1151#
1152# The above copyright notice and this permission notice shall be included
1153# in all copies or substantial portions of the Software.
1154#
1155# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1156# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1157# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1158# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1159# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1160# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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