Coverage for pygeodesy/rhumb/bases.py: 94%
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2# -*- coding: utf-8 -*-
4u'''(INTERNAL) base classes C{RhumbBase} and C{RhumbLineBase}, pure Python version of I{Karney}'s
5C++ classes U{Rhumb<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Rhumb.html>}
6and U{RhumbLine<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1RhumbLine.html>}
7from I{GeographicLib versions 2.0} and I{2.2} and I{Karney}'s C++ example U{Rhumb intersect
8<https://SourceForge.net/p/geographiclib/discussion/1026620/thread/2ddc295e/>}.
10Class L{RhumbLineBase} has been enhanced with methods C{Intersecant2}, C{Intersection} and C{PlumbTo}
11to iteratively find the intersection of a rhumb line and a circle or an other rhumb line, respectively
12a perpendicular geodesic or other rhumb line.
14For more details, see the C++ U{GeographicLib<https://GeographicLib.SourceForge.io/C++/doc/index.html>}
15documentation, especially the U{Class List<https://GeographicLib.SourceForge.io/C++/doc/annotated.html>},
16the background information on U{Rhumb lines<https://GeographicLib.SourceForge.io/C++/doc/rhumb.html>},
17the utily U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>} and U{Online
18rhumb line calculations<https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve>}.
20Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2014-2023) and licensed under the MIT/X11
21License. For more information, see the U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation.
22'''
23# make sure int/int division yields float quotient
24from __future__ import division as _; del _ # PYCHOK semicolon
26from pygeodesy.basics import _copysign, itemsorted, unsigned0, _xinstanceof
27from pygeodesy.constants import EPS, EPS0, EPS1, INT0, NAN, _over, \
28 _EPSqrt as _TOL, _0_0, _0_01, _1_0, _90_0
29from pygeodesy.datums import Datum, _earth_datum, _spherical_datum, _WGS84
30from pygeodesy.errors import IntersectionError, RhumbError, _xdatum, \
31 _xkwds, _xkwds_pop2, _Xorder
32# from pygeodesy.etm import ExactTransverseMercator # _MODS
33from pygeodesy.fmath import euclid, favg, sqrt_a, Fsum
34# from pygeodesy.formy import opposing # _MODS
35# from pygeodesy.fsums import Fsum # from .fmath
36from pygeodesy.internals import _dunder_nameof, _under
37from pygeodesy.interns import NN, _coincident_, _COMMASPACE_, _Dash, \
38 _parallel_, _too_
39from pygeodesy.karney import _atan2d, Caps, _CapsBase, _diff182, _fix90, \
40 _norm180, GDict
41# from pygeodesy.ktm import KTransverseMercator, _AlpCoeffs # _MODS
42from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS
43from pygeodesy.namedTuples import Distance2Tuple, LatLon2Tuple
44from pygeodesy.props import deprecated_method, Property, Property_RO, \
45 property_RO, _update_all
46from pygeodesy.streprs import Fmt, pairs
47from pygeodesy.units import Float_, Lat, Lon, Meter, Radius_, Int # PYCHOK shared
48from pygeodesy.utily import acos1, _azireversed, _loneg, sincos2d, sincos2d_, \
49 _unrollon, _Wrap
50from pygeodesy.vector3d import _intersect3d3, Vector3d # in .Intersection below
52from math import cos, fabs
54__all__ = ()
55__version__ = '24.05.13'
57_anti_ = _Dash('anti')
58_rls = [] # instances of C{RbumbLine...} to be updated
59_TRIPS = 65 # .Intersection, .PlumbTo, 19+
62class _Lat(Lat):
63 '''(INTERNAL) Latitude B{C{lat}}.
64 '''
65 def __init__(self, *lat, **Error_name):
66 kwds = _xkwds(Error_name, clip=0, Error=RhumbError)
67 Lat.__new__(_Lat, *lat, **kwds)
70class _Lon(Lon):
71 '''(INTERNAL) Longitude B{C{lon}}.
72 '''
73 def __init__(self, *lon, **Error_name):
74 kwds = _xkwds(Error_name, clip=0, Error=RhumbError)
75 Lon.__new__(_Lon, *lon, **kwds)
78def _update_all_rls(r):
79 '''(INTERNAL) Zap cached/memoized C{Property[_RO]}s
80 of any C{RhumbLine} instances tied to the given
81 C{Rhumb} instance B{C{r}}.
82 '''
83 # _xinstanceof(_MODS.rhumb.aux_.RhumbAux, _MODS.rhumb.ekx.Rhumb, r=r)
84 _update_all(r)
85 for rl in _rls: # PYCHOK use weakref?
86 if rl._rhumb is r:
87 _update_all(rl)
90class RhumbBase(_CapsBase):
91 '''(INTERNAL) Base class for C{rhumb.aux_.RhumbAux} and C{rhumb.ekx.Rhumb}.
92 '''
93 _datum = _WGS84
94 _exact = True
95 _f_max = _0_01
96 _mTM = 6 # see .TMorder
98 def __init__(self, a_earth, f, exact, name):
99 '''New C{RhumbAux} or C{Rhumb}.
100 '''
101 _earth_datum(self, a_earth, f=f, name=name)
102 if not exact:
103 self.exact = False
104 if name:
105 self.name = name
107 @Property_RO
108 def a(self):
109 '''Get the C{ellipsoid}'s equatorial radius, semi-axis (C{meter}).
110 '''
111 return self.ellipsoid.a
113 equatoradius = a
115 def ArcDirect(self, lat1, lon1, azi12, a12, outmask=Caps.LATITUDE_LONGITUDE):
116 '''Solve the I{direct rhumb} problem, optionally with area.
118 @arg lat1: Latitude of the first point (C{degrees90}).
119 @arg lon1: Longitude of the first point (C{degrees180}).
120 @arg azi12: Azimuth of the rhumb line (compass C{degrees}).
121 @arg a12: Angle along the rhumb line from the given to the
122 destination point (C{degrees}), can be negative.
124 @return: L{GDict} with 2 up to 8 items C{lat2, lon2, a12, S12,
125 lat1, lon1, azi12, s12} with the destination point's
126 latitude C{lat2} and longitude C{lon2} in C{degrees},
127 the rhumb angle C{a12} in C{degrees} and area C{S12}
128 under the rhumb line in C{meter} I{squared}.
130 @raise ImportError: Package C{numpy} not found or not installed,
131 only required for area C{S12} when C{B{exact}
132 is True} and L{RhumbAux}.
134 @note: If B{C{a12}} is large enough that the rhumb line crosses
135 a pole, the longitude of the second point is indeterminate
136 and C{NAN} is returned for C{lon2} and area C{S12}.
138 @note: If the given point is a pole, the cosine of its latitude is
139 taken to be C{sqrt(L{EPS})}. This position is extremely
140 close to the actual pole and allows the calculation to be
141 carried out in finite terms.
142 '''
143 s12 = a12 * self._mpd
144 return self._DirectRhumb(lat1, lon1, azi12, a12, s12, outmask)
146 @Property_RO
147 def b(self):
148 '''Get the C{ellipsoid}'s polar radius, semi-axis (C{meter}).
149 '''
150 return self.ellipsoid.b
152 polaradius = b
154 @property
155 def datum(self):
156 '''Get this rhumb's datum (L{Datum}).
157 '''
158 return self._datum
160 @datum.setter # PYCHOK setter!
161 def datum(self, datum):
162 '''Set this rhumb's datum (L{Datum}).
164 @raise RhumbError: If C{abs(B{f}} exceeds non-zero C{f_max} and C{exact=False}.
165 '''
166 _xinstanceof(Datum, datum=datum)
167 if self._datum != datum:
168 self._exactest(self.exact, datum.ellipsoid, self.f_max)
169 _update_all_rls(self)
170 self._datum = datum
172 def _Direct(self, ll1, azi12, s12, **outmask):
173 '''(INTERNAL) Short-cut version, see .latlonBase.rhumb....
174 '''
175 return self.Direct(ll1.lat, ll1.lon, azi12, s12, **outmask)
177 def Direct(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE):
178 '''Solve the I{direct rhumb} problem, optionally with area.
180 @arg lat1: Latitude of the first point (C{degrees90}).
181 @arg lon1: Longitude of the first point (C{degrees180}).
182 @arg azi12: Azimuth of the rhumb line (compass C{degrees}).
183 @arg s12: Distance along the rhumb line from the given to
184 the destination point (C{meter}), can be negative.
186 @return: L{GDict} with 2 up to 8 items C{lat2, lon2, a12, S12,
187 lat1, lon1, azi12, s12} with the destination point's
188 latitude C{lat2} and longitude C{lon2} in C{degrees},
189 the rhumb angle C{a12} in C{degrees} and area C{S12}
190 under the rhumb line in C{meter} I{squared}.
192 @raise ImportError: Package C{numpy} not found or not installed,
193 only required for area C{S12} when C{B{exact}
194 is True} and L{RhumbAux}.
196 @note: If B{C{s12}} is large enough that the rhumb line crosses
197 a pole, the longitude of the second point is indeterminate
198 and C{NAN} is returned for C{lon2} and area C{S12}.
200 @note: If the given point is a pole, the cosine of its latitude is
201 taken to be C{sqrt(L{EPS})}. This position is extremely
202 close to the actual pole and allows the calculation to be
203 carried out in finite terms.
204 '''
205 a12 = _over(s12, self._mpd)
206 return self._DirectRhumb(lat1, lon1, azi12, a12, s12, outmask)
208 def Direct8(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE_AREA):
209 '''Like method L{Rhumb.Direct} but returning a L{Rhumb8Tuple} with area C{S12}.
210 '''
211 return self.Direct(lat1, lon1, azi12, s12, outmask=outmask).toRhumb8Tuple()
213 def _DirectLine(self, ll1, azi12, **caps_name):
214 '''(INTERNAL) Short-cut version, see .latlonBase.
215 '''
216 return self.DirectLine(ll1.lat, ll1.lon, azi12, **caps_name)
218 def DirectLine(self, lat1, lon1, azi12, **caps_name):
219 '''Define a C{RhumbLine} in terms of the I{direct} rhumb
220 problem to compute several points on a single rhumb line.
222 @arg lat1: Latitude of the first point (C{degrees90}).
223 @arg lon1: Longitude of the first point (C{degrees180}).
224 @arg azi12: Azimuth of the rhumb line (compass C{degrees}).
225 @kwarg caps_name: Optional keyword arguments C{B{name}=NN} and
226 C{B{caps}=Caps.STANDARD}, a bit-or'ed combination of
227 L{Caps} values specifying the required capabilities.
228 Include C{Caps.LINE_OFF} if updates to the B{C{rhumb}}
229 should I{not} be reflected in this rhumb line.
231 @return: A C{RhumbLine...} instance and invoke its method
232 C{.Position} to compute each point.
234 @note: Updates to this rhumb are reflected in the returned
235 rhumb line, unless C{B{caps} |= Caps.LINE_OFF}.
236 '''
237 return self._RhumbLine(self, lat1, lon1, azi12, **caps_name)
239 Line = DirectLine # synonyms
241 def _DirectRhumb(self, lat1, lon1, azi12, a12, s12, outmask):
242 '''(INTERNAL) See methods C{.ArcDirect} and C{.Direct}.
243 '''
244 rl = self._RhumbLine(self, lat1, lon1, azi12, caps=Caps.LINE_OFF,
245 name=self.name)
246 return rl._Position(a12, s12, outmask | self._debug) # lat2, lon2, S12
248 @Property
249 def ellipsoid(self):
250 '''Get this rhumb's ellipsoid (L{Ellipsoid}).
251 '''
252 return self.datum.ellipsoid
254 @ellipsoid.setter # PYCHOK setter!
255 def ellipsoid(self, a_earth_f):
256 '''Set this rhumb's ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or
257 L{a_f2Tuple}) or (equatorial) radius and flattening (2-tuple C{(a, f)}).
259 @raise RhumbError: If C{abs(B{f}} exceeds non-zero C{f_max} and C{exact=False}.
260 '''
261 self.datum = _spherical_datum(a_earth_f, Error=RhumbError)
263 @Property
264 def exact(self):
265 '''Get the I{exact} option (C{bool}).
266 '''
267 return self._exact
269 @exact.setter # PYCHOK setter!
270 def exact(self, exact):
271 '''Set the I{exact} option (C{bool}). If C{True}, use I{exact} rhumb
272 expressions, otherwise a series expansion (accurate for oblate or
273 prolate ellipsoids with C{abs(flattening)} below C{f_max}.
275 @raise RhumbError: If C{B{exact}=False} and C{abs(flattening})
276 exceeds non-zero C{f_max}.
278 @see: Option U{B{-s}<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>}
279 and U{ACCURACY<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html#ACCURACY>}.
280 '''
281 x = bool(exact)
282 if self._exact != x:
283 self._exactest(x, self.ellipsoid, self.f_max)
284 _update_all_rls(self)
285 self._exact = x
287 def _exactest(self, exact, ellipsoid, f_max):
288 # Helper for property setters C{ellipsoid}, C{exact} and C{f_max}
289 if fabs(ellipsoid.f) > f_max > 0 and not exact:
290 raise RhumbError(exact=exact, f=ellipsoid.f, f_max=f_max)
292 @Property_RO
293 def f(self):
294 '''Get the C{ellipsoid}'s flattening (C{float}).
295 '''
296 return self.ellipsoid.f
298 flattening = f
300 @property
301 def f_max(self):
302 '''Get the I{max.} flattening (C{float}).
303 '''
304 return self._f_max
306 @f_max.setter # PYCHOK setter!
307 def f_max(self, f_max): # PYCHOK no cover
308 '''Set the I{max.} flattening, not to exceed (C{float}).
310 @raise RhumbError: If C{exact=False} and C{abs(flattening})
311 exceeds non-zero C{f_max}.
312 '''
313 f = Float_(f_max=f_max, low=_0_0, high=EPS1)
314 if self._f_max != f:
315 self._exactest(self.exact, self.ellipsoid, f)
316 self._f_max = f
318 def _Inverse(self, ll1, ll2, wrap, **outmask):
319 '''(INTERNAL) Short-cut version, see .latlonBase.rhumb....
320 '''
321 if wrap:
322 ll2 = _unrollon(ll1, _Wrap.point(ll2))
323 return self.Inverse(ll1.lat, ll1.lon, ll2.lat, ll2.lon, **outmask)
325 def Inverse(self, lat1, lon1, lat2, lon2, outmask=Caps.AZIMUTH_DISTANCE):
326 '''Solve the I{inverse rhumb} problem.
328 @arg lat1: Latitude of the first point (C{degrees90}).
329 @arg lon1: Longitude of the first point (C{degrees180}).
330 @arg lat2: Latitude of the second point (C{degrees90}).
331 @arg lon2: Longitude of the second point (C{degrees180}).
333 @return: L{GDict} with 4 to 9 items C{lat1, lon1, lat2, lon2,
334 azi12, azi21, s12, a12, S12}, the rhumb line's azimuth
335 C{azi12} and I{reverse} azimuth C{azi21}, both in
336 compass C{degrees} between C{-180} and C{+180}, the
337 rhumb distance C{s12} and rhumb angle C{a12} between
338 both points in C{meter} respectively C{degrees} and
339 the area C{S12} under the rhumb line in C{meter}
340 I{squared}.
342 @raise ImportError: Package C{numpy} not found or not installed,
343 only required for L{RhumbAux} area C{S12}
344 when C{B{exact} is True}.
346 @note: The shortest rhumb line is found. If the end points are
347 on opposite meridians, there are two shortest rhumb lines
348 and the East-going one is chosen.
350 @note: If either point is a pole, the cosine of its latitude is
351 taken to be C{sqrt(L{EPS})}. This position is extremely
352 close to the actual pole and allows the calculation to be
353 carried out in finite terms.
354 '''
355 r = GDict(lat1=lat1, lon1=lon1, lat2=lat2, lon2=lon2, name=self.name)
356 Cs = Caps
357 if (outmask & Cs.AZIMUTH_DISTANCE_AREA):
358 lon12, _ = _diff182(lon1, lon2, K_2_0=True)
359 y, x, s1, s2 = self._Inverse4(lon12, r, outmask)
360 if (outmask & Cs.AZIMUTH):
361 z = _atan2d(y, x)
362 r.set_(azi12=z, azi21=_azireversed(z))
363 if (outmask & Cs.AREA):
364 S12 = self._S12d(s1, s2, lon12)
365 r.set_(S12=unsigned0(S12)) # like .gx
366 return r
368 def _Inverse4(self, lon12, r, outmask): # PYCHOK no cover
369 '''(INTERNAL) I{Must be overloaded}.'''
370 self._notOverloaded(lon12, r, Caps.toStr(outmask)) # underOK=True
372 def Inverse8(self, lat1, lon1, azi12, s12, outmask=Caps.AZIMUTH_DISTANCE_AREA):
373 '''Like method L{Rhumb.Inverse} but returning a L{Rhumb8Tuple} with area C{S12}.
374 '''
375 return self.Inverse(lat1, lon1, azi12, s12, outmask=outmask).toRhumb8Tuple()
377 def _InverseLine(self, ll1, ll2, wrap, **caps_name):
378 '''(INTERNAL) Short-cut version, see .latlonBase.
379 '''
380 if wrap:
381 ll2 = _unrollon(ll1, _Wrap.point(ll2))
382 return self.InverseLine(ll1.lat, ll1.lon, ll2.lat, ll2.lon, **caps_name)
384 def InverseLine(self, lat1, lon1, lat2, lon2, **caps_name):
385 '''Define a C{RhumbLine} in terms of the I{inverse} rhumb problem.
387 @arg lat1: Latitude of the first point (C{degrees90}).
388 @arg lon1: Longitude of the first point (C{degrees180}).
389 @arg lat2: Latitude of the second point (C{degrees90}).
390 @arg lon2: Longitude of the second point (C{degrees180}).
391 @kwarg caps_name: Optional keyword arguments C{B{name}=NN} and
392 C{B{caps}=Caps.STANDARD}, a bit-or'ed combination of
393 L{Caps} values specifying the required capabilities.
394 Include C{Caps.LINE_OFF} if updates to the B{C{rhumb}}
395 should I{not} be reflected in this rhumb line.
397 @return: A C{RhumbLine...} instance and invoke its method
398 C{ArcPosition} or C{Position} to compute points.
400 @note: Updates to this rhumb are reflected in the returned
401 rhumb line, unless C{B{caps} |= Caps.LINE_OFF}.
402 '''
403 r = self.Inverse(lat1, lon1, lat2, lon2, outmask=Caps.AZIMUTH)
404 return self._RhumbLine(self, lat1, lon1, r.azi12, **caps_name)
406 @Property_RO
407 def _mpd(self): # PYCHOK no cover
408 '''(INTERNAL) I{Must be overloaded}.'''
409 _MODS.named.notOverloaded(self)
411 @property_RO
412 def RAorder(self):
413 '''Get the I{Rhumb Area} order, C{None} always.
414 '''
415 return None
417 @property_RO
418 def _RhumbLine(self): # PYCHOK no cover
419 '''(INTERNAL) I{Must be overloaded}.'''
420 self._notOverloaded(underOK=True)
422 def _S12d(self, s1, s2, lon): # PYCHOK no cover
423 '''(INTERNAL) I{Must be overloaded}.'''
424 self._notOverloaded(s1, s2, lon) # underOK=True
426 @Property
427 def TMorder(self):
428 '''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
429 '''
430 return self._mTM
432 @TMorder.setter # PYCHOK setter!
433 def TMorder(self, order):
434 '''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
436 @note: Setting C{TMorder} turns property C{exact} off, but only
437 for L{Rhumb} instances.
438 '''
439 m = _Xorder(_MODS.ktm._AlpCoeffs, RhumbError, TMorder=order)
440 if self._mTM != m:
441 _update_all_rls(self)
442 self._mTM = m
443 if self.exact and isinstance(self, _MODS.rhumb.ekx.Rhumb):
444 self.exact = False
446 def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature
447 '''Return this C{Rhumb} as string.
449 @kwarg prec: The C{float} precision, number of decimal digits (0..9).
450 Trailing zero decimals are stripped for B{C{prec}} values
451 of 1 and above, but kept for negative B{C{prec}} values.
452 @kwarg sep: Separator to join (C{str}).
454 @return: Tuple items (C{str}).
455 '''
456 d = dict(ellipsoid=self.ellipsoid, RAorder=self.RAorder,
457 exact=self.exact, TMorder=self.TMorder)
458 return sep.join(pairs(itemsorted(d, asorted=False), prec=prec))
461class RhumbLineBase(_CapsBase):
462 '''(INTERNAL) Base class for C{rhumb.aux_.RhumbLineAux} and C{rhumb.ekx.RhumbLine}.
463 '''
464 _azi12 = _0_0
465 _calp = _1_0
466# _caps = \
467# _debug = 0
468# _lat1 = \
469# _lon1 = \
470# _lon12 = _0_0
471 _Rhumb = RhumbBase # compatible C{Rhumb} class
472 _rhumb = None # C{Rhumb} instance
473 _salp = \
474 _talp = _0_0
476 def __init__(self, rhumb, lat1, lon1, azi12, caps=Caps.STANDARD, name=NN):
477 '''New C{RhumbLine} or C{RhumbLineAux}.
478 '''
479 _xinstanceof(self._Rhumb, rhumb=rhumb)
481 self._lat1 = _Lat(lat1=_fix90(lat1))
482 self._lon1 = _Lon(lon1= lon1)
483 self._lon12 = _norm180(self._lon1)
484 if azi12: # non-zero, non-None
485 self.azi12 = _norm180(azi12)
487 n = name or rhumb.name
488 if n:
489 self.name=n
491 self._caps = caps
492 self._debug |= (caps | rhumb._debug) & Caps._DEBUG_DIRECT_LINE
493 if (caps & Caps.LINE_OFF): # copy to avoid updates
494 self._rhumb = rhumb.copy(deep=False, name=_under(rhumb.name))
495 else:
496 self._rhumb = rhumb
497 _rls.append(self)
499 def __del__(self): # XXX use weakref?
500 if _rls: # may be empty or None
501 try: # PYCHOK no cover
502 _rls.remove(self)
503 except (TypeError, ValueError):
504 pass
505 self._rhumb = None
506 # _update_all(self) # throws TypeError during Python 2 cleanup
508 def ArcPosition(self, a12, outmask=Caps.LATITUDE_LONGITUDE):
509 '''Compute a point at a given angular distance on this rhumb line.
511 @arg a12: The angle along this rhumb line from its origin to the
512 point (C{degrees}), can be negative.
513 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying
514 the quantities to be returned.
516 @return: L{GDict} with 4 to 8 items C{azi12, a12, s12, S12, lat2,
517 lon2, lat1, lon1} with latitude C{lat2} and longitude
518 C{lon2} of the point in C{degrees}, the rhumb distance
519 C{s12} in C{meter} from the start point of and the area
520 C{S12} under this rhumb line in C{meter} I{squared}.
522 @raise ImportError: Package C{numpy} not found or not installed,
523 only required for L{RhumbLineAux} area C{S12}
524 when C{B{exact} is True}.
526 @note: If B{C{a12}} is large enough that the rhumb line crosses a
527 pole, the longitude of the second point is indeterminate and
528 C{NAN} is returned for C{lon2} and area C{S12}.
530 If the first point is a pole, the cosine of its latitude is
531 taken to be C{sqrt(L{EPS})}. This position is extremely
532 close to the actual pole and allows the calculation to be
533 carried out in finite terms.
534 '''
535 return self._Position(a12, self.degrees2m(a12), outmask)
537 @Property
538 def azi12(self):
539 '''Get this rhumb line's I{azimuth} (compass C{degrees}).
540 '''
541 return self._azi12
543 @azi12.setter # PYCHOK setter!
544 def azi12(self, azi12):
545 '''Set this rhumb line's I{azimuth} (compass C{degrees}).
546 '''
547 z = _norm180(azi12)
548 if self._azi12 != z:
549 if self._rhumb:
550 _update_all(self)
551 self._azi12 = z
552 self._salp, self._calp = t = sincos2d(z) # no NEG0
553 self._talp = _over(*t)
555 @property_RO
556 def azi12_sincos2(self): # PYCHOK no cover
557 '''Get the sine and cosine of this rhumb line's I{azimuth} (2-tuple C{(sin, cos)}).
558 '''
559 return self._scalp, self._calp
561 @property_RO
562 def datum(self):
563 '''Get this rhumb line's datum (L{Datum}).
564 '''
565 return self.rhumb.datum
567 def degrees2m(self, angle):
568 '''Convert an angular distance along this rhumb line to C{meter}.
570 @arg angle: Angular distance (C{degrees}).
572 @return: Distance (C{meter}).
573 '''
574 return float(angle) * self.rhumb._mpd
576 @deprecated_method
577 def distance2(self, lat, lon): # PYCHOK no cover
578 '''DEPRECATED on 23.09.23, use method L{RhumbLineAux.Inverse} or L{RhumbLine.Inverse}.
580 @return: A L{Distance2Tuple}C{(distance, initial)} with the C{distance}
581 in C{meter} and C{initial} bearing (azimuth) in C{degrees}.
582 '''
583 r = self.Inverse(lat, lon)
584 return Distance2Tuple(r.s12, r.azi12)
586 @property_RO
587 def ellipsoid(self):
588 '''Get this rhumb line's ellipsoid (L{Ellipsoid}).
589 '''
590 return self.rhumb.ellipsoid
592 @property_RO
593 def exact(self):
594 '''Get this rhumb line's I{exact} option (C{bool}).
595 '''
596 return self.rhumb.exact
598 def Intersecant2(self, lat0, lon0, radius, napier=True, **tol_eps):
599 '''Compute the intersection(s) of this rhumb line and a circle.
601 @arg lat0: Latitude of the circle center (C{degrees}).
602 @arg lon0: Longitude of the circle center (C{degrees}).
603 @arg radius: Radius of the circle (C{meter}, conventionally).
604 @kwarg napier: If C{True}, apply I{Napier}'s spherical triangle
605 instead of planar trigonometry (C{bool}).
606 @kwarg tol_eps: Optional keyword arguments, see method
607 method L{Intersection} for further details.
609 @return: 2-Tuple C{(P, Q)} with both intersections (representing
610 a rhumb chord), each a L{GDict} from method L{Intersection}
611 extended to 18 items by C{lat3, lon3, azi03, a03, s03}
612 with azimuth C{azi03} of, distance C{a03} in C{degrees}
613 and C{s03} in C{meter} along the rhumb line from the circle
614 C{lat0, lon0} to the chord center C{lat3, lon3}. If this
615 rhumb line is tangential to the circle, both points
616 are the same L{GDict} instance with distances C{s02} and
617 C{s03} near-equal to the B{C{radius}}.
619 @raise IntersectionError: The circle and this rhumb line
620 do not intersect.
622 @raise UnitError: Invalid B{C{radius}}.
623 '''
624 r = Radius_(radius)
625 p = q = self.PlumbTo(lat0, lon0, exact=None, **tol_eps)
626 a = q.s02
627 t = dict(lat3=q.lat2, lon3=q.lon2, azi03=q.azi02, a03=q.a02, s03=a)
628 if a < r:
629 t.update(iteration=q.iteration, lat0=q.lat1, lon0=q.lon1, # or lat0, lon0
630 name=_dunder_nameof(self.Intersecant2, self.name))
631 if fabs(a) < EPS0: # coincident centers
632 d, h = _0_0, r
633 else:
634 d = q.s12
635 if napier: # Napier rule (R1) cos(b) = cos(c) / cos(a)
636 # <https://WikiPedia.org/wiki/Spherical_trigonometry>
637 m = self.rhumb._mpr
638 h = (acos1(cos(r / m) / cos(a / m)) * m) if m else _0_0
639 else:
640 h = _copysign(sqrt_a(r, a), a)
641 p = q = self.Position(d + h).set_(**t)
642 if h:
643 q = self.Position(d - h).set_(**t)
644 elif a > r:
645 t = _too_(Fmt.distant(a))
646 raise IntersectionError(self, lat0, lon0, radius,
647 txt=t, **tol_eps)
648 else: # tangential
649 q.set_(**t) # == p.set(_**t)
650 return p, q
652 @deprecated_method
653 def intersection2(self, other, **tol_eps): # PYCHOK no cover
654 '''DEPRECATED on 23.10.10, use method L{Intersection}.'''
655 p = self.Intersection(other, **tol_eps)
656 r = LatLon2Tuple(p.lat2, p.lon2, name=self.intersection2.__name__)
657 r._iteration = p.iteration
658 return r
660 def Intersection(self, other, tol=_TOL, **eps):
661 '''I{Iteratively} find the intersection of this and an other rhumb line.
663 @arg other: The other rhumb line (C{RhumbLine}).
664 @kwarg tol: Tolerance for longitudinal convergence and parallel
665 error (C{degrees}).
666 @kwarg eps: Tolerance for L{pygeodesy.intersection3d3} (C{EPS}).
668 @return: The intersection point, a L{Position}-like L{GDict} with
669 13 items C{lat1, lon1, azi12, a12, s12, lat2, lon2, lat0,
670 lon0, azi02, a02, s02, at} with the rhumb angle C{a02}
671 and rhumb distance C{s02} between the start point C{lat0,
672 lon0} of the B{C{other}} rhumb line and the intersection
673 C{lat2, lon2}, the azimuth C{azi02} of the B{C{other}}
674 rhumb line and the angle C{at} between both rhumb lines.
675 See method L{Position} for further details.
677 @raise IntersectionError: No convergence for this B{C{tol}} or
678 no intersection for an other reason.
680 @see: Methods C{distance2} and C{PlumbTo} and function
681 L{pygeodesy.intersection3d3}.
683 @note: Each iteration involves a round trip to this rhumb line's
684 L{ExactTransverseMercator} or L{KTransverseMercator}
685 projection and function L{pygeodesy.intersection3d3} in
686 that domain.
687 '''
688 _xinstanceof(RhumbLineBase, other=other)
689 _xdatum(self.rhumb, other.rhumb, Error=RhumbError)
690 try:
691 if self.others(other) is self:
692 raise ValueError(_coincident_)
693 # make invariants and globals locals
694 _s_3d, s_az = self._xTM3d, self.azi12
695 _o_3d, o_az = other._xTM3d, other.azi12
696 p = _MODS.formy.opposing(s_az, o_az, margin=tol)
697 if p is not None: # == p in (True, False)
698 raise ValueError(_anti_(_parallel_) if p else _parallel_)
699 _diff = euclid # approximate length
700 _i3d3 = _intersect3d3 # NOT .vector3d.intersection3d3
701 _LL2T = LatLon2Tuple
702 _xTMr = self.xTM.reverse # ellipsoidal or spherical
703 # use halfway point as initial estimate
704 p = _LL2T(favg(self.lat1, other.lat1),
705 favg(self.lon1, other.lon1))
706 for i in range(1, _TRIPS):
707 v = _i3d3(_s_3d(p), s_az, # point + bearing
708 _o_3d(p), o_az, useZ=False, **eps)[0]
709 t = _xTMr(v.x, v.y, lon0=p.lon) # PYCHOK Reverse4Tuple
710 d = _diff(t.lon - p.lon, t.lat) # PYCHOK t.lat + p.lat - p.lat
711 p = _LL2T(t.lat + p.lat, t.lon) # PYCHOK t.lon + p.lon = lon0
712 if d < tol: # 19 trips
713 break
714 else:
715 raise ValueError(Fmt.no_convergence(d, tol))
717 P = GDict(lat1=self.lat1, lat2=p.lat, lat0=other.lat1,
718 lon1=self.lon1, lon2=p.lon, lon0=other.lon1,
719 name=_dunder_nameof(self.Intersection, self.name))
720 r = self.Inverse( p.lat, p.lon, outmask=Caps.DISTANCE)
721 t = other.Inverse(p.lat, p.lon, outmask=Caps.DISTANCE)
722 P.set_(azi12= self.azi12, a12=r.a12, s12=r.s12,
723 azi02=other.azi12, a02=t.a12, s02=t.s12,
724 at=other.azi12 - self.azi12, iteration=i)
725 except Exception as x:
726 raise IntersectionError(self, other, tol=tol,
727 eps=eps, cause=x)
728 return P
730 def Inverse(self, lat2, lon2, wrap=False, **outmask):
731 '''Return the rhumb angle, distance, azimuth, I{reverse} azimuth, etc. of
732 a rhumb line between the given point and this rhumb line's start point.
734 @arg lat2: Latitude of the point (C{degrees}).
735 @arg lon2: Longitude of the points (C{degrees}).
736 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
737 and B{C{lon2}} (C{bool}).
739 @return: L{GDict} with 8 items C{a12, s12, azi12, azi21, lat1, lon1,
740 lat2, lon2}, the rhumb angle C{a12} and rhumb distance C{s12}
741 between both points in C{degrees} respectively C{meter}, the
742 rhumb line's azimuth C{azi12} and I{reverse} azimuth C{azi21}
743 both in compass C{degrees} between C{-180} and C{+180}.
744 '''
745 if wrap:
746 _, lat2, lon2 = _Wrap.latlon3(self.lon1, _fix90(lat2), lon2, wrap)
747 r = self.rhumb.Inverse(self.lat1, self.lon1, lat2, lon2, **outmask)
748 return r
750 @Property_RO
751 def isLoxodrome(self):
752 '''Is this rhumb line a meridional (C{None}), a parallel
753 (C{False}) or a C{True} loxodrome?
755 @see: I{Osborne's} U{2.5 Rumb lines and loxodromes
756 <https://Zenodo.org/record/35392>}, page 37.
757 '''
758 return bool(self._salp) if self._calp else None
760 @Property_RO
761 def lat1(self):
762 '''Get this rhumb line's latitude (C{degrees90}).
763 '''
764 return self._lat1
766 @Property_RO
767 def lon1(self):
768 '''Get this rhumb line's longitude (C{degrees180}).
769 '''
770 return self._lon1
772 @Property_RO
773 def latlon1(self):
774 '''Get this rhumb line's lat- and longitude (L{LatLon2Tuple}C{(lat, lon)}).
775 '''
776 return LatLon2Tuple(self.lat1, self.lon1)
778 def m2degrees(self, distance):
779 '''Convert a distance along this rhumb line to an angular distance.
781 @arg distance: Distance (C{meter}).
783 @return: Angular distance (C{degrees}).
784 '''
785 return _over(float(distance), self.rhumb._mpd)
787 @property_RO
788 def _mu1(self): # PYCHOK no cover
789 '''(INTERNAL) I{Must be overloaded}.'''
790 self._notOverloaded(underOK=True)
792 def _mu2lat(self, mu2): # PYCHOK no cover
793 '''(INTERNAL) I{Must be overloaded}.'''
794 self._notOverloaded(mu2) # underOK=True
796 @deprecated_method
797 def nearestOn4(self, lat0, lon0, **exact_eps_est_tol): # PYCHOK no cover
798 '''DEPRECATED on 23.10.10, use method L{PlumbTo}.'''
799 P = self.PlumbTo(lat0, lon0, **exact_eps_est_tol)
800 r = _MODS.deprecated.classes.NearestOn4Tuple(P.lat2, P.lon2, P.s12, P.azi02,
801 name=self.nearestOn4.__name__)
802 r._iteration = P.iteration
803 return r
805 @deprecated_method
806 def NearestOn(self, lat0, lon0, **exact_eps_est_tol): # PYCHOK no cover
807 '''DEPRECATED on 23.10.30, use method L{PlumbTo}.'''
808 return self.PlumbTo(lat0, lon0, **exact_eps_est_tol)
810 def PlumbTo(self, lat0, lon0, exact=None, eps=EPS, est=None, tol=_TOL):
811 '''Compute the I{perpendicular} intersection of this rhumb line with a geodesic
812 from the given point (transcoded from I{Karney}'s C++ U{rhumb-intercept
813 <https://SourceForge.net/p/geographiclib/discussion/1026620/thread/2ddc295e/>}).
815 @arg lat0: Latitude of the point on the geodesic (C{degrees}).
816 @arg lon0: Longitude of the point on the geodesic (C{degrees}).
817 @kwarg exact: If C{None}, use a rhumb line perpendicular to this rhumb line,
818 otherwise use an I{exact} C{Geodesic...} from the given point
819 perpendicular to this rhumb line (C{bool} or C{Geodesic...}),
820 see method L{Ellipsoid.geodesic_}.
821 @kwarg eps: Optional tolerance for L{pygeodesy.intersection3d3} (C{EPS}),
822 used only if C{B{exact} is None}.
823 @kwarg est: Optionally, an initial estimate for the distance C{s12} of the
824 intersection I{along} this rhumb line (C{meter}), used only if
825 C{B{exact} is not None}.
826 @kwarg tol: Longitudinal convergence tolerance (C{degrees}) or distance
827 tolerance (C(meter)) when C{B{exact} is None}, respectively
828 C{not None}.
830 @return: The intersection point on this rhumb line, a L{GDict} from method
831 L{Intersection} if B{C{exact}=None}. If C{B{exact} is not None},
832 a L{Position}-like L{GDict} of 13 items C{azi12, a12, s12, lat2,
833 lat1, lat0, lon2, lon1, lon0, azi0, a02, s02, at} with distance
834 C{a02} in C{degrees} and C{s02} in C{meter} between the given point
835 C{lat0, lon0} and the intersection C{lat2, lon2}, geodesic azimuth
836 C{azi0} at the given point and the (perpendicular) angle C{at}
837 between the geodesic and this rhumb line at the intersection. The
838 I{geodesic} azimuth at the intersection is C{(at + azi12)}. See
839 method L{Position} for further details.
841 @raise ImportError: I{Karney}'s U{geographiclib
842 <https://PyPI.org/project/geographiclib>}
843 package not found or not installed.
845 @raise IntersectionError: No convergence for this B{C{eps}} or no
846 intersection for some other reason.
848 @see: Methods C{distance2}, C{Intersecant2} and C{Intersection}
849 and function L{pygeodesy.intersection3d3}.
850 '''
851 Cs, tol = Caps, Float_(tol=tol, low=EPS, high=None)
853# def _over(p, q): # see @note at method C{.Position}
854# if p:
855# p = (p / (q or _copysign(tol, q))) if isfinite(q) else NAN
856# return p
858 if exact is None:
859 z = _norm180(self.azi12 + _90_0) # perpendicular azimuth
860 rl = RhumbLineBase(self.rhumb, lat0, lon0, z, caps=Cs.LINE_OFF)
861 P = self.Intersection(rl, tol=tol, eps=eps)
863 else: # C{rhumb-intercept}
864 E = self.ellipsoid
865 _gI = E.geodesic_(exact=exact).Inverse
866 gm = Cs.STANDARD | Cs._REDUCEDLENGTH_GEODESICSCALE # ^ Cs.DISTANCE_IN
867 if est is None: # get an estimate from the "perpendicular" geodesic
868 r = _gI(self.lat1, self.lon1, lat0, lon0, outmask=Cs.AZIMUTH_DISTANCE)
869 d, _ = _diff182(r.azi2, self.azi12, K_2_0=True)
870 _, s12 = sincos2d(d)
871 s12 *= r.s12 # signed
872 else:
873 s12 = Meter(est=est)
874 try:
875 _abs = fabs
876 _d2 = _diff182
877 _ErT = E.rocPrimeVertical # aka rocTransverse
878 _ovr = _over
879 _S12 = Fsum(s12).fsum2f_
880 _scd = sincos2d_
881 for i in range(1, _TRIPS): # 9+, suffix 1 == C++ 2, 2 == C++ 3
882 P = self.Position(s12) # outmask=Cs.LATITUDE_LONGITUDE
883 r = _gI(lat0, lon0, P.lat2, P.lon2, outmask=gm)
884 d, _ = _d2(self.azi12, r.azi2, K_2_0=True)
885 s, c, s2, c2 = _scd(d, r.lat2)
886 c2 *= _ErT(r.lat2)
887 s *= _ovr(s2 * self._salp, c2) - _ovr(s * r.M21, r.m12)
888 s12, t = _S12(c / s) # XXX _ovr?
889 if _abs(t) < tol: # or _abs(c) < EPS
890 break
891 P.set_(azi0=r.azi1, a02=r.a12, s02=r.s12, # azi2=r.azi2,
892 lat0=lat0, lon0=lon0, iteration=i, at=r.azi2 - self.azi12,
893 name=_dunder_nameof(self.PlumbTo, self.name))
894 except Exception as x: # Fsum(NAN) Value-, ZeroDivisionError
895 raise IntersectionError(lat0, lon0, tol=tol, exact=exact,
896 eps=eps, est=est, iteration=i, cause=x)
898 return P
900 def Position(self, s12, outmask=Caps.LATITUDE_LONGITUDE):
901 '''Compute a point at a given distance on this rhumb line.
903 @arg s12: The distance along this rhumb line from its origin to
904 the point (C{meters}), can be negative.
905 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying
906 the quantities to be returned.
908 @return: L{GDict} with 4 to 8 items C{azi12, a12, s12, S12, lat2,
909 lat1, lon2, lon1} with latitude C{lat2} and longitude
910 C{lon2} of the point in C{degrees}, the rhumb angle C{a12}
911 in C{degrees} from the start point of and the area C{S12}
912 under this rhumb line in C{meter} I{squared}.
914 @raise ImportError: Package C{numpy} not found or not installed,
915 only required for L{RhumbLineAux} area C{S12}
916 when C{B{exact} is True}.
918 @note: If B{C{s12}} is large enough that the rhumb line crosses a
919 pole, the longitude of the second point is indeterminate and
920 C{NAN} is returned for C{lon2} and area C{S12}.
922 If the first point is a pole, the cosine of its latitude is
923 taken to be C{sqrt(L{EPS})}. This position is extremely
924 close to the actual pole and allows the calculation to be
925 carried out in finite terms.
926 '''
927 return self._Position(self.m2degrees(s12), s12, outmask)
929 def _Position(self, a12, s12, outmask):
930 '''(INTERNAL) C{Arc-/Position} helper.
931 '''
932 r = GDict(azi12=self.azi12, a12=a12, s12=s12, name=self.name)
933 Cs = Caps
934 if (outmask & Cs.LATITUDE_LONGITUDE_AREA):
935 if a12 or s12:
936 mu12 = self._calp * a12
937 mu2 = self._mu1 + mu12
938 if fabs(mu2) > 90: # past pole
939 mu2 = _norm180(mu2) # reduce to [-180, 180)
940 if fabs(mu2) > 90: # point on anti-meridian
941 mu2 = _norm180(_loneg(mu2))
942 lat2 = self._mu2lat(mu2)
943 lon2 = S12 = NAN
944 else:
945 lat2, lon2, S1, S2 = self._Position4(a12, mu2, s12, mu12)
946 if (outmask & Cs.AREA):
947 S12 = self.rhumb._S12d(S1, S2, lon2)
948 S12 = unsigned0(S12) # like .gx
949# else:
950# S12 = None # unused
951 if (outmask & Cs.LONGITUDE):
952 if (outmask & Cs.LONG_UNROLL):
953 lon2 += self.lon1
954 else:
955 lon2 = _norm180(self._lon12 + lon2)
956 else: # coincident
957 lat2, lon2 = self.latlon1
958 S12 = _0_0
960 if (outmask & Cs.AREA):
961 r.set_(S12=S12)
962 if (outmask & Cs.LATITUDE):
963 r.set_(lat2=lat2, lat1=self.lat1)
964 if (outmask & Cs.LONGITUDE):
965 r.set_(lon2=lon2, lon1=self.lon1)
966 return r
968 def _Position4(self, a12, mu2, s12, mu12): # PYCHOK no cover
969 '''(INTERNAL) I{Must be overloaded}.'''
970 self._notOverloaded(a12, s12, mu2, mu12) # underOK=True
972 @Property_RO
973 def rhumb(self):
974 '''Get this rhumb line's rhumb (L{RhumbAux} or L{Rhumb}).
975 '''
976 return self._rhumb
978 def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature
979 '''Return this C{RhumbLine} as string.
981 @kwarg prec: The C{float} precision, number of decimal digits (0..9).
982 Trailing zero decimals are stripped for B{C{prec}} values
983 of 1 and above, but kept for negative B{C{prec}} values.
984 @kwarg sep: Separator to join (C{str}).
986 @return: C{RhumbLine} (C{str}).
987 '''
988 d = dict(rhumb=self.rhumb, lat1=self.lat1, lon1=self.lon1,
989 azi12=self.azi12, exact=self.exact,
990 TMorder=self.TMorder, xTM=self.xTM)
991 return sep.join(pairs(itemsorted(d, asorted=False), prec=prec))
993 @property_RO
994 def TMorder(self):
995 '''Get this rhumb line's I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
996 '''
997 return self.rhumb.TMorder
999 @Property_RO
1000 def xTM(self):
1001 '''Get this rhumb line's I{Transverse Mercator} projection (L{ExactTransverseMercator}
1002 if I{exact} and I{ellipsoidal}, otherwise L{KTransverseMercator} for C{TMorder}).
1003 '''
1004 E = self.ellipsoid
1005 # ExactTransverseMercator doesn't handle spherical earth models
1006 return _MODS.etm.ExactTransverseMercator(E) if self.exact and E.isEllipsoidal else \
1007 _MODS.ktm.KTransverseMercator(E, TMorder=self.TMorder)
1009 def _xTM3d(self, latlon0, z=INT0, V3d=Vector3d):
1010 '''(INTERNAL) C{xTM.forward} this C{latlon1} to C{V3d} with B{C{latlon0}}
1011 as current intersection estimate and central meridian.
1012 '''
1013 t = self.xTM.forward(self.lat1 - latlon0.lat, self.lon1, lon0=latlon0.lon)
1014 return V3d(t.easting, t.northing, z)
1017class _PseudoRhumbLine(RhumbLineBase):
1018 '''(INTERNAL) Pseudo-rhumb line for a geodesic (line), see C{geodesicw._PlumbTo}.
1019 '''
1020 def __init__(self, gl, name=NN):
1021 R = RhumbBase(gl.geodesic.ellipsoid, None, True, name)
1022 RhumbLineBase.__init__(self, R, gl.lat1, gl.lon1, 0, caps=Caps.LINE_OFF)
1023 self._azi1 = self.azi12 = gl.azi1
1024 self._gl = gl
1025 self._gD = gl.geodesic.Direct
1027 def PlumbTo(self, lat0, lon0, **exact_eps_est_tol): # PYCHOK signature
1028 P = RhumbLineBase.PlumbTo(self, lat0, lon0, **exact_eps_est_tol)
1029 z, P = _xkwds_pop2(P, azi12=None)
1030 P.set_(azi1=self._gl.azi1, azi2=z)
1031 return P # geodesic L{Position}
1033 def Position(self, s12, **unused): # PYCHOK signature
1034 r = self._gD(self.lat1, self.lon1, self._azi1, s12)
1035 self._azi1 = r.azi1
1036 self.azi12 = z = r.azi2
1037 self._salp, _ = sincos2d(z)
1038 return r.set_(azi12=z)
1041__all__ += _ALL_DOCS(RhumbBase, RhumbLineBase)
1043if __name__ == '__main__':
1045 from pygeodesy import printf, Rhumb as Rh, RhumbAux as Ah
1046 from pygeodesy.basics import _zip
1047 from pygeodesy.ellipsoids import _EWGS84
1049 Al = Ah(_EWGS84).Line(30, 0, 45)
1050 Rl = Rh(_EWGS84).Line(30, 0, 45)
1052 for i in range(1, 10):
1053 s = .5e6 + 1e6 / i
1054 a = Al.Position(s).lon2
1055 r = Rl.Position(s).lon2
1056 e = (fabs(a - r) / a) if a else 0
1057 printf('# Position.lon2 %.14f vs %.14f, diff %g', r, a, e)
1059 for exact in (None, False, True):
1060 for est in (None, 1e6):
1061 a = Al.PlumbTo(60, 0, exact=exact, est=est)
1062 r = Rl.PlumbTo(60, 0, exact=exact, est=est)
1063 printf('# %s, iteration=%s, exact=%s, est=%s\n# %s, iteration=%s',
1064 a.toRepr(), a.iteration, exact, est,
1065 r.toRepr(), r.iteration, nl=1)
1067 NE_=(71.688899882813, 0.2555198244234, 44095641862956.11)
1068 LHR=(77.7683897102557, 5771083.38332803, 37395209100030.39)
1069 NRT=(-92.38888798169965, 12782581.067684170, -63760642939072.50)
1071 def _ref(fmt, r3, x3):
1072 e3 = []
1073 for r, x in _zip(r3, x3): # strict=True
1074 e = fabs(r - x) / fabs(x)
1075 e3.append('%.g' % (e,))
1076 printf((fmt % r3) + ', rel errors: ' + ', '.join(e3))
1078 for R in (Ah, Rh): # <https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve -p 9> version 2.2
1079 rh = R(exact=True) # WGS84 default
1080 printf('# %r', rh, nl=1)
1081 r = rh.Direct8(40.6, -73.8, 51, 5.5e6) # from JFK about NE
1082 _ref('# JFK NE lat2=%.12f, lon2=%.12f, S12=%.1f', (r.lat2, r.lon2, r.S12), NE_)
1083 r = rh.Inverse8(40.6, -73.8, 51.6, -0.5) # JFK to LHR
1084 _ref('# JFK-LHR azi12=%.12f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), LHR)
1085 r = rh.Inverse8(40.6, -73.8, 35.8, 140.3) # JFK to Tokyo Narita
1086 _ref('# JFK-NRT azi12=%.12f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), NRT)
1088# % python3.10 -m pygeodesy3.rhumb.Bases
1090# Position.lon2 11.61455846901637 vs 11.61455846901637, diff 3.05885e-16
1091# Position.lon2 7.58982302826842 vs 7.58982302826842, diff 2.34045e-16
1092# Position.lon2 6.28526067416369 vs 6.28526067416369, diff 2.82623e-16
1093# Position.lon2 5.63938995325146 vs 5.63938995325146, diff 1.57495e-16
1094# Position.lon2 5.25385527435707 vs 5.25385527435707, diff 0
1095# Position.lon2 4.99764604290380 vs 4.99764604290380, diff 8.88597e-16
1096# Position.lon2 4.81503363740473 vs 4.81503363740473, diff 1.84459e-16
1097# Position.lon2 4.67828821748836 vs 4.67828821748835, diff 5.69553e-16
1098# Position.lon2 4.57205667906283 vs 4.57205667906283, diff 5.82787e-16
1100# Intersection(a02=17.798332, a12=19.521356, at=90.0, azi02=135.0, azi12=45.0, lat0=60.0, lat1=30.0, lat2=45.0, lon0=0.0, lon1=0.0, lon2=15.830286, name='Intersection', s02=1977981.142985, s12=2169465.957531), iteration=9, exact=None, est=None
1101# Intersection(a02=17.798332, a12=19.521356, at=90.0, azi02=135.0, azi12=45.0, lat0=60.0, lat1=30.0, lat2=45.0, lon0=0.0, lon1=0.0, lon2=15.830286, name='Intersection', s02=1977981.142985, s12=2169465.957531), iteration=9
1103# Intersection(a02=17.798332, a12=19.521356, at=90.0, azi02=135.0, azi12=45.0, lat0=60.0, lat1=30.0, lat2=45.0, lon0=0.0, lon1=0.0, lon2=15.830286, name='Intersection', s02=1977981.142985, s12=2169465.957531), iteration=9, exact=None, est=1000000.0
1104# Intersection(a02=17.798332, a12=19.521356, at=90.0, azi02=135.0, azi12=45.0, lat0=60.0, lat1=30.0, lat2=45.0, lon0=0.0, lon1=0.0, lon2=15.830286, name='Intersection', s02=1977981.142985, s12=2169465.957531), iteration=9
1106# PlumbTo(a02=17.967658, a12=27.74256, at=90.0, azi0=113.73626, azi12=45.0, lat0=60, lat1=30.0, lat2=49.634582, lon0=0, lon1=0.0, lon2=25.767876, name='PlumbTo', s02=1997960.116871, s12=3083112.636236), iteration=5, exact=False, est=None
1107# PlumbTo(a02=17.967658, a12=27.74256, at=90.0, azi0=113.73626, azi12=45.0, lat0=60, lat1=30.0, lat2=49.634582, lon0=0, lon1=0.0, lon2=25.767876, name='PlumbTo', s02=1997960.116871, s12=3083112.636236), iteration=5
1109# PlumbTo(a02=17.967658, a12=27.74256, at=90.0, azi0=113.73626, azi12=45.0, lat0=60, lat1=30.0, lat2=49.634582, lon0=0, lon1=0.0, lon2=25.767876, name='PlumbTo', s02=1997960.116871, s12=3083112.636236), iteration=7, exact=False, est=1000000.0
1110# PlumbTo(a02=17.967658, a12=27.74256, at=90.0, azi0=113.73626, azi12=45.0, lat0=60, lat1=30.0, lat2=49.634582, lon0=0, lon1=0.0, lon2=25.767876, name='PlumbTo', s02=1997960.116871, s12=3083112.636236), iteration=7
1112# PlumbTo(a02=17.967658, a12=27.74256, at=90.0, azi0=113.73626, azi12=45.0, lat0=60, lat1=30.0, lat2=49.634582, lon0=0, lon1=0.0, lon2=25.767876, name='PlumbTo', s02=1997960.116871, s12=3083112.636236), iteration=5, exact=True, est=None
1113# PlumbTo(a02=17.967658, a12=27.74256, at=90.0, azi0=113.73626, azi12=45.0, lat0=60, lat1=30.0, lat2=49.634582, lon0=0, lon1=0.0, lon2=25.767876, name='PlumbTo', s02=1997960.116871, s12=3083112.636236), iteration=5
1115# PlumbTo(a02=17.967658, a12=27.74256, at=90.0, azi0=113.73626, azi12=45.0, lat0=60, lat1=30.0, lat2=49.634582, lon0=0, lon1=0.0, lon2=25.767876, name='PlumbTo', s02=1997960.116871, s12=3083112.636236), iteration=7, exact=True, est=1000000.0
1116# PlumbTo(a02=17.967658, a12=27.74256, at=90.0, azi0=113.73626, azi12=45.0, lat0=60, lat1=30.0, lat2=49.634582, lon0=0, lon1=0.0, lon2=25.767876, name='PlumbTo', s02=1997960.116871, s12=3083112.636236), iteration=7
1118# RhumbAux(RAorder=None, 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)
1119# JFK NE lat2=71.688899882813, lon2=0.255519824423, S12=44095641862956.1, rel errors: 4e-16, 2e-13, 4e-16
1120# JFK-LHR azi12=77.768389710256, s12=5771083.383 S12=37395209100030.4, rel errors: 5e-16, 3e-16, 8e-16
1121# JFK-NRT azi12=-92.388887981700, s12=12782581.068 S12=-63760642939072.5, rel errors: 0, 1e-16, 7e-16
1123# 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)
1124# JFK NE lat2=71.688899882813, lon2=0.255519824423, S12=44095641862956.1, rel errors: 2e-16, 1e-13, 5e-16
1125# JFK-LHR azi12=77.768389710256, s12=5771083.383 S12=37395209100030.4, rel errors: 4e-16, 3e-16, 6e-16
1126# JFK-NRT azi12=-92.388887981700, s12=12782581.068 S12=-63760642939072.5, rel errors: 0, 1e-16, 1e-16
1128# **) MIT License
1129#
1130# Copyright (C) 2022-2024 -- mrJean1 at Gmail -- All Rights Reserved.
1131#
1132# Permission is hereby granted, free of charge, to any person obtaining a
1133# copy of this software and associated documentation files (the "Software"),
1134# to deal in the Software without restriction, including without limitation
1135# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1136# and/or sell copies of the Software, and to permit persons to whom the
1137# Software is furnished to do so, subject to the following conditions:
1138#
1139# The above copyright notice and this permission notice shall be included
1140# in all copies or substantial portions of the Software.
1141#
1142# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1143# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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1145# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1146# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1147# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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