Coverage for pygeodesy/auxilats/auxLat.py: 92%
444 statements
« prev ^ index » next coverage.py v7.2.2, created at 2024-06-10 14:08 -0400
« prev ^ index » next coverage.py v7.2.2, created at 2024-06-10 14:08 -0400
2# -*- coding: utf-8 -*-
4u'''Class L{AuxLat} transcoded to Python from I{Karney}'s C++ class U{AuxLatitude
5<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AuxLatitude.html>}
6in I{GeographicLib version 2.2+}.
8Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2022-2023) and licensed
9under the MIT/X11 License. For more information, see the U{GeographicLib
10<https://GeographicLib.SourceForge.io>} documentation.
12@see: U{Auxiliary latitudes<https:#GeographicLib.SourceForge.io/C++/doc/auxlat.html>}
13 U{On auxiliary latitudes<https:#ArXiv.org/abs/2212.05818>}.
14'''
15# make sure int/int division yields float quotient, see .basics
16from __future__ import division as _; del _ # PYCHOK semicolon
18from pygeodesy.auxilats.auxAngle import AuxAngle, AuxBeta, AuxChi, _AuxClass, \
19 AuxMu, AuxPhi, AuxTheta, AuxXi
20from pygeodesy.auxilats.auxily import Aux, _sc, _sn, _Ufloats, atan1
21from pygeodesy.basics import _reverange, _xinstanceof, _passarg
22from pygeodesy.constants import INF, MAX_EXP, MIN_EXP, NAN, PI_2, PI_4, _EPSqrt, \
23 _0_0, _0_0s, _0_1, _0_25, _0_5, _1_0, _2_0, _3_0, \
24 _360_0, isfinite, isinf, isnan, _log2, _over
25from pygeodesy.datums import _ellipsoidal_datum, _WGS84, Ellipsoid, _name__
26# from pygeodesy.ellipsoids import Ellipsoid # from .datums
27from pygeodesy.elliptic import Elliptic as _Ef
28from pygeodesy.errors import AuxError, _xkwds_not, _xkwds_pop2, _Xorder
29# from pygeodesy.fmath import cbrt # from .karney
30from pygeodesy.fsums import Fsum, _Fsumf_, _sum
31# from pygeodesy.internals import _passarg # from .basics
32from pygeodesy.interns import NN, _DOT_, _not_scalar_, _UNDER_
33from pygeodesy.karney import _2cos2x, _polynomial, _ALL_DOCS, cbrt, _MODS
34# from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS # from .karney
35# from pygeodesy.named import _name__ # from .datums
36from pygeodesy.props import Property, Property_RO, _update_all
37from pygeodesy.units import _isDegrees, _isRadius, Degrees, Meter
38# from pygeodesy.utily import atan1 # from .auxily
40from math import asinh, atan2, copysign, cosh, fabs, sin, sinh, sqrt
41try:
42 from math import exp2 as _exp2
43except ImportError: # Python 3.11-
45 def _exp2(x):
46 return pow(_2_0, x)
48__all__ = ()
49__version__ = '24.05.24'
51_TRIPS = 1024 # XXX 2 or 3?
54class AuxLat(AuxAngle):
55 '''Base class for accurate conversion between I{Auxiliary} latitudes
56 on an ellipsoid.
58 Latitudes are represented by L{AuxAngle} instances in order to
59 maintain precision near the poles, I{Authalic} latitude C{Xi},
60 I{Conformal} C{Chi}, I{Geocentric} C{Theta}, I{Geographic} C{Phi},
61 I{Parametric} C{Beta} and I{Rectifying} C{Mu}.
63 @see: I{Karney}'s C++ class U{AuxLatitude
64 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AuxLatitude.html>}.
65 '''
66 _csc = dict() # global coeffs cache: [aL][k], upto max(k) * (4 + 6 + 8) floats
67 _E = _WGS84.ellipsoid
68# _Lmax = 0 # overwritten below
69 _mAL = 6 # 4, 6 or 8 aka Lmax
71 def __init__(self, a_earth=_WGS84, f=None, b=None, **ALorder_name):
72 '''New L{AuxLat} instance on an ellipsoid or datum.
74 @arg a_earth: Equatorial radius, semi-axis (C{meter}) or an
75 ellipsoid or datum (L{Datum}, L{Ellipsoid},
76 L{Ellipsoid2} or L{a_f2Tuple}).
77 @kwarg f: Flattening: M{(a - b) / a} (C{float}, near zero for
78 spherical), ignored if B{C{a_earth}} is not scalar.
79 @kwarg b: Optional polar radius, semi-axis (C{meter}, same
80 units as B{C{a_earth}}), ignored if B{C{a_earth}}
81 is not scalar.
82 @kwarg ALorder_name: Optional C{B{name}=NN} (C{str}) and optional
83 keyword argument C{B{ALorder}=6} for the order of
84 this L{AuxLat}, see property C{ALorder}.
85 '''
86 if ALorder_name:
87 M = self._mAL
88 m, name = _xkwds_pop2(ALorder_name, ALorder=M)
89 if m != M:
90 self.ALorder = m
91 else:
92 name = NN
93 try:
94 if a_earth is not _WGS84:
95 n = _name__(name, name__=AuxLat)
96 if b is f is None:
97 E = _ellipsoidal_datum(a_earth, name=n).ellipsoid # XXX raiser=_earth_
98 elif _isRadius(a_earth):
99 E = Ellipsoid(a_earth, f=f, b=b, name=_UNDER_(NN, n))
100 else:
101 raise ValueError(_not_scalar_)
102 self._E = E
103 elif not (b is f is None):
104 # _UnexpectedError into AuxError
105 name = _name__(name, **_xkwds_not(None, b=b, f=f))
107 if name:
108 self.name = name
109 except Exception as x:
110 raise AuxError(a_earth=a_earth, f=f, b=b, cause=x)
112 @Property_RO
113 def a(self):
114 '''Get the C{ellipsoid}'s equatorial radius (C{meter}, conventionally).
115 '''
116 return self.ellipsoid.a
118 equatoradius = a
120 @Property
121 def ALorder(self):
122 '''Get the I{AuxLat} order (C{int}, 4, 6 or 8).
123 '''
124 return self._mAL
126 @ALorder.setter # PYCHOK setter!
127 def ALorder(self, order):
128 '''Set the I{AuxLat} order (C{int}, 4, 6 or 8).
129 '''
130 m = _Xorder(_AR2Coeffs, AuxError, ALorder=order)
131 if self._mAL != m:
132 _update_all(self)
133 self._mAL = m
135 def _atanhee(self, tphi): # see Ellipsoid._es_atanh, .albers._atanhee
136 # atanh(e * sphi) = asinh(e' * sbeta)
137 f = self.f
138 s = _sn(self._fm1 * tphi) if f > 0 else _sn(tphi)
139 if f: # atanh(e * sphi) = asinh(e' * sbeta)
140 e = self._e
141 s = _over(atan1(e * s) if f < 0 else asinh(self._e1 * s), e)
142 return s
144 def Authalic(self, Phi, **diff_name):
145 '''Convert I{Geographic} to I{Aunthalic} latitude.
147 @arg Phi: Geographic latitude (L{AuxAngle}).
148 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}
149 optional C{B{name}=NN}.
151 @return: Parametric latitude, C{Beta} (L{AuxAngle}).
152 '''
153 _xinstanceof(AuxAngle, Phi=Phi)
154 # assert Phi._AUX == Aux.PHI
155 tphi = fabs(Phi.tan)
156 if isfinite(tphi) and tphi and self.f:
157 y, x = Phi._yx_normalized
158 q = self._q
159 qv = self._qf(tphi)
160 Dq2 = self._Dq(tphi)
161 Dq2 *= (q + qv) / (fabs(y) + _1_0) # _Dq(-tphi)
162 d, n = _diff_name2(Phi, **diff_name)
163 Xi = AuxXi(copysign(qv, Phi.y), x * sqrt(Dq2), name=n)
165 if d:
166 if isnan(tphi):
167 d = self._e2m1_sq2
168 else:
169 c = self.Parametric(Phi)._x_normalized
170 d = _over(c, Xi._x_normalized)**3
171 d *= _over(c, x) * _over(_2_0, q)
172 Xi._diff = d
173 else:
174 Xi = AuxXi(*Phi._yx) # diff default
175 # assert Xi._AUX == Aux.XI
176 return Xi
178 def AuthalicRadius2(self, exact=False, f_max=_0_1):
179 '''Get the I{Authalic} radius I{squared}.
181 @kwarg exact: If C{True}, use the exact expression, otherwise
182 the I{Taylor} series.
183 @kwarg f_max: C{Flattening} not to exceed (C{float}).
185 @return: Authalic radius I{squared} (C{meter} I{squared}, same
186 units as the ellipsoid axes).
188 @raise AuxError: If C{B{exact}=False} and C{abs(flattening)}
189 exceeds C{f_max}.
190 '''
191 f = self.f
192 if exact or not f:
193 c2 = self.ellipsoid.b2 * self._q # == ellipsoid.c2x * 2
194 elif fabs(f) < f_max:
195 # Using a * (a + b) / 2 as the multiplying factor leads to a rapidly
196 # converging series in n. Of course, using this series isn't really
197 # necessary, since the exact expression is simple to evaluate. However,
198 # we do it for consistency with RectifyingRadius and, presumably, the
199 # roundoff error is smaller compared to that for the exact expression.
200 m = self.ALorder
201 c2 = _polynomial(self._n, _AR2Coeffs[m], 0, m)
202 c2 *= self.a * (self.a + self.b)
203 else:
204 raise AuxError(exact=exact, f=f, f_max=f_max)
205 return c2 * _0_5
207 @Property_RO
208 def b(self):
209 '''Get the C{ellipsoid}'s polar radius (C{meter}, conventionally).
210 '''
211 return self.ellipsoid.b # a * (_1_0 - f)
213 polaradius = b
215 def _coeffs(self, auxout, auxin):
216 # Get the polynomial coefficients as 4-, 6- or 8-tuple
217 aL = self.ALorder # aka Lmax
218 if auxout == auxin:
219 return _0_0s(aL) # uncached
221 k = Aux._1d(auxout, auxin)
222 try: # cached
223 return AuxLat._csc[aL][k]
224 except KeyError:
225 pass
227 Cx = _CXcoeffs(aL)
228 try:
229 Cx = Cx[auxout][auxin]
230 except KeyError as x:
231 raise AuxError(auxout=auxout, auxin=auxin, cause=x)
233 d = x = n = self._n
234 if Aux.use_n2(auxin) and Aux.use_n2(auxout):
235 x = self._n2
237 def _m(aL):
238 for m in _reverange(aL):
239 yield m // 2
240 else:
241 _m = _reverange # PYCHOK expected
243 i = 0
244 cs = []
245 _c = cs.append
246 _p = _polynomial
247 for m in _m(aL):
248 j = i + m + 1 # order m = j - i - 1
249 _c(_p(x, Cx, i, j) * d)
250 d *= n
251 i = j
252 # assert i == len(Cx) and len(cs) == aL
253 AuxLat._csc.setdefault(aL, {})[k] = cs = tuple(cs)
254 return cs
256 def Conformal(self, Phi, **diff_name):
257 '''Convert I{Geographic} to I{Conformal} latitude.
259 @arg Phi: Geographic latitude (L{AuxAngle}).
260 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}
261 and an optional C{B{name}=NN}.
263 @return: Conformal latitude, C{Chi} (L{AuxAngle}).
264 '''
265 _xinstanceof(AuxAngle, Phi=Phi)
266 # assert Phi._AUX == Aux.PHI
267 tphi = tchi = fabs(Phi.tan)
268 if isfinite(tphi) and tphi and self.f:
269 sig = sinh(self._atanhee(tphi) * self._e2)
270 scsig = _sc(sig)
271 scphi = _sc(tphi)
272 if self.f > 0:
273 # The general expression for tchi is
274 # tphi * scsig - sig * scphi
275 # This involves cancellation if f > 0, so change to
276 # (tphi - sig) * (tphi + sig) / (tphi * scsig + sig * scphi)
277 # To control overflow, write as (sigtphi = sig / tphi)
278 # (tphi - sig) * (1 + sigtphi) / (scsig + sigtphi * scphi)
279 sigtphi = sig / tphi
280 if sig < (tphi * _0_5):
281 t = tphi - sig
282 else:
283 def _asinh_2(x):
284 return asinh(x) * _0_5
285 # Still possibly dangerous cancellation in tphi - sig.
286 # Write tphi - sig = (1 - e) * Dg(1, e)
287 # Dg(x, y) = (g(x) - g(y)) / (x - y)
288 # g(x) = sinh(x * atanh(sphi * x))
289 # Note sinh(atanh(sphi)) = tphi
290 # Turn the crank on divided differences, substitute
291 # sphi = tphi / _sc(tphi)
292 # atanh(x) = asinh(x / sqrt(1 - x^2))
293 e = self._e
294 em1 = self._e2m1 / (_1_0 + e)
295 # assert em1 != 0
296 scb = self._scbeta(tphi)
297 scphib = scphi / scb # sec(phi) / sec(beta)
298 atphib = _asinh_2(tphi * e / scb) # atanh(e * sphi)
299 atphi = _asinh_2(tphi) # atanh(sphi)
300 t = _asinh_2(em1 * (tphi * scphib)) / em1
301 try:
302 Dg = _Fsumf_(atphi, atphib, t, e * t)
303 except ValueError: # Fsum(NAN) exception
304 Dg = _sum((atphi, atphib, t, e * t))
305 e *= atphib
306 t = atphi - e
307 if t: # sinh(0) == 0
308 Dg *= sinh(t) / t * cosh(atphi + e) * em1
309 t = float(Dg) # tphi - sig
310 tchi = _over(t * (_1_0 + sigtphi),
311 scsig + scphi * sigtphi) if t else _0_0
312 else:
313 tchi = tphi * scsig - sig * scphi
315 d, n = _diff_name2(Phi, **diff_name)
316 Chi = AuxChi(tchi, name=n).copyquadrant(Phi)
318 if d:
319 if isinf(tphi): # PYCHOK np cover
320 d = self._conformal_diff
321 else:
322 d = self.Parametric(Phi)._x_normalized
323 if d:
324 d = _over(d, Chi._x_normalized) * \
325 _over(d, Phi._x_normalized) * self._e2m1
326 Chi._diff = d
327 # assrt Chi._AUX == Aux.CHI
328 return Chi
330 @Property_RO
331 def _conformal_diff(self): # PYCHOK no cover
332 '''(INTERNAL) Constant I{Conformal} diff.
333 '''
334 e = self._e
335 if self.f > 0:
336 ss = sinh(asinh(self._e1) * e)
337 d = _over(_1_0, _sc(ss) + ss)
338 elif e:
339 ss = sinh(-atan1(e) * e)
340 d = _sc(ss) - ss
341 else:
342 d = _1_0
343 return d
345 def convert(self, auxout, Zeta_d, exact=False):
346 # Convert I{Auxiliary} or I{scalar} latitude
347 Z = d = Zeta_d
348 if isinstance(Z, AuxAngle):
349 A, auxin = _AuxClass(auxout), Z._AUX
350 if auxin == auxout:
351 pass
352 elif not (isfinite(Z.tan) and Z.tan): # XXX
353 Z = A(*Z._yx, aux=auxout, name=Z.name)
354 elif exact:
355 p = Aux.power(auxout, auxin)
356 if p is None:
357 P = self._fromAux(Z) # Phi
358 Z = self._toAux(auxout, P)
359 Z._iter = P.iteration
360 else:
361 y, x = Z._yx
362 if p:
363 y *= pow(self._fm1, p)
364 Z = A(y, x, aux=auxout, name=Z.name)
365 else:
366 cs = self._coeffs(auxout, auxin)
367 yx = Z._yx_normalized
368 Z = A(*yx, aux=auxout, name=Z.name)
369 # assert Z._yx == yx
370 r = _Clenshaw(True, Z, cs, self.ALorder)
371 Z += AuxAngle.fromRadians(r, aux=auxout)
372 # assert Z._AUX == auxout
373 return Z
375 elif _isDegrees(d):
376 Z = AuxPhi.fromDegrees(d)
377 d = round((d - Z.toDegrees) / _360_0) * _360_0
378 d += self.convert(auxout, Z, exact).toDegrees
379 return Degrees(d, name=Aux.Greek(auxout))
381 raise AuxError(auxout=auxout, Zeta_d=Zeta_d, exact=exact)
383 def _Dq(self, tphi):
384 # I{Divided Difference} of (q(1) - q(sphi)) / (1 - sphi).
385 sphi = _sn(tphi)
386 if tphi > 0:
387 scphi = _sc(tphi)
388 d = _1_0 / (scphi**2 * (_1_0 + sphi)) # XXX - sphi
389 if d:
390 # General expression for _Dq(1, sphi) is
391 # atanh(e * d / (1 - e2 * sphi)) / (e * d) +
392 # (1 + e2 * sphi) / ((1 - e2 * sphi * sphi) * e2m1)
393 # with atanh(e * d / (1 - e2 * sphi)) =
394 # atanh(e * d * scphi / (scphi - e2 * tphi))
395 e2m1, ed = self._e2m1, (self._e * d)
396 if e2m1 and ed:
397 e2 = self._e2
398 if e2 > 0: # assert self.f > 0
399 scb = self._scbeta(tphi)
400 q = scphib = scphi / scb
401 q *= (scphi + tphi * e2) / (e2m1 * scb)
402 q += asinh(self._e1 * d * scphib) / ed
403 else:
404 s2 = sphi * e2
405 q = (_1_0 + s2) / ((_1_0 - sphi * s2) * e2m1)
406 q += (atan2(ed, _1_0 - s2) / ed) if e2 < 0 else _1_0
407 else: # PYCHOK no cover
408 q = INF
409 else: # PYCHOK no cover
410 q = self._2_e2m12
411 else: # not reached, open-coded in .Authalic
412 q = _over(self._q - self._qf(tphi), _1_0 - sphi)
413 return q
415 @Property_RO
416 def _e(self): # unsigned, (1st) eccentricity
417 return self.ellipsoid.e # sqrt(fabs(self._e2))
419 @Property_RO
420 def _e1(self):
421 return sqrt(fabs(self._e12))
423 @Property_RO
424 def _e12(self):
425 return _over(self._e2, _1_0 - self._e2)
427 @Property_RO
428 def _e12p1(self):
429 return _1_0 / self._e2m1
431 @Property_RO
432 def _e2(self): # signed, (1st) eccentricity squared
433 return self.ellipsoid.e2
435 @Property_RO
436 def _e2m1(self): # 1 less 1st eccentricity squared
437 return self.ellipsoid.e21 # == ._fm1**2
439 @Property_RO
440 def _e2m1_sq2(self):
441 return self._e2m1 * sqrt(self._q * _0_5)
443 @Property_RO
444 def _2_e2m12(self):
445 return _2_0 / self._e2m1**2
447 @Property_RO
448 def _Ef_fRG_a2b2_PI_4(self):
449 E = self.ellipsoid
450 return _Ef.fRG(E.a2, E.b2) / PI_4
452 @Property_RO
453 def ellipsoid(self):
454 '''Get the ellipsoid (L{Ellipsoid}).
455 '''
456 return self._E
458 @Property_RO
459 def f(self):
460 '''Get the C{ellipsoid}'s flattening (C{scalar}).
461 '''
462 return self.ellipsoid.f
464 flattening = f
466 @Property_RO
467 def _fm1(self): # 1 - flattening
468 return self.ellipsoid.f1
470 def _fromAux(self, Zeta, **name):
471 '''Convert I{Auxiliary} to I{Geographic} latitude.
473 @arg Zeta: Auxiliary latitude (L{AuxAngle}).
474 @kwarg name: Optional C{B{name}=NN} (C{str}).
476 @return: Geographic latitude, I{Phi} (L{AuxAngle}).
477 '''
478 _xinstanceof(AuxAngle, Zeta=Zeta)
479 aux = Zeta._AUX
480 n = _name__(name, _or_nameof=Zeta)
481 f = self._fromAuxCase.get(aux, None)
482 if f is None:
483 Phi = AuxPhi(NAN, name=n)
484 elif callable(f):
485 t = self._fm1
486 t *= f(t)
487 Phi = _Newton(t, Zeta, self._toZeta(aux), name=n)
488 else: # assert isscalar(f)
489 y, x = Zeta._yx
490 Phi = AuxPhi(y / f, x, name=n)
491 # assert Phi._AUX == Aux.PHI
492 return Phi
494 @Property_RO
495 def _fromAuxCase(self):
496 '''(INTERNAL) switch(auxin): ...
497 '''
498 return {Aux.AUTHALIC: cbrt,
499 Aux.CONFORMAL: _passarg,
500 Aux.GEOCENTRIC: self._e2m1,
501 Aux.GEOGRAPHIC: _1_0,
502 Aux.PARAMETRIC: self._fm1,
503 Aux.RECTIFYING: sqrt}
505 def Geocentric(self, Phi, **diff_name):
506 '''Convert I{Geographic} to I{Geocentric} latitude.
508 @arg Phi: Geographic latitude (L{AuxAngle}).
509 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}
510 and an optional C{B{name}=NN}.
512 @return: Geocentric latitude, C{Phi} (L{AuxAngle}).
513 '''
514 _xinstanceof(AuxAngle, Phi=Phi)
515 # assert Phi._AUX == Aux.PHI
516 d, n = _diff_name2(Phi, **diff_name)
517 Theta = AuxTheta(Phi.y * self._e2m1, Phi.x, name=n)
518 if d:
519 Theta._diff = self._e2m1
520 return Theta
522 def Geodetic(self, Phi, **name): # PYCHOK no cover
523 '''Convert I{Geographic} to I{Geodetic} latitude.
525 @arg Phi: Geographic latitude (L{AuxAngle}).
526 @kwarg name: Optional C{B{name}=NN} (C{str}).
528 @return: Geodetic latitude, C{Phi} (L{AuxAngle}).
529 '''
530 _xinstanceof(AuxAngle, Phi=Phi)
531 # assert Phi._AUX == Aux.PHI
532 _, n = _diff_name2(Phi, **name)
533 return AuxPhi(Phi, name=n)
535 @Property_RO
536 def _n(self): # 3rd flattening
537 return self.ellipsoid.n
539 @Property_RO
540 def _n2(self):
541 return self._n**2
543 def Parametric(self, Phi, **diff_name):
544 '''Convert I{Geographic} to I{Parametric} latitude.
546 @arg Phi: Geographic latitude (L{AuxAngle}).
547 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}
548 and an optional C{B{name}=NN}.
550 @return: Parametric latitude, C{Beta} (L{AuxAngle}).
551 '''
552 _xinstanceof(AuxAngle, Phi=Phi)
553 # assert Phi._AUX == Aux.PHI
554 d, n = _diff_name2(Phi, **diff_name)
555 Beta = AuxBeta(Phi.y * self._fm1, Phi.x, name=n)
556 if d:
557 Beta._diff = self._fm1
558 return Beta
560 Reduced = Parametric
562 @Property_RO
563 def _q(self): # constant _q
564 q, f = self._e12p1, self.f
565 if f:
566 e = self._e
567 q += _over(asinh(self._e1) if f > 0 else atan1(e), e)
568 else:
569 q += _1_0
570 return q
572 def _qf(self, tphi):
573 # function _q: atanh(e * sphi) / e + sphi / (1 - (e * sphi)^2)
574 scb = self._scbeta(tphi)
575 return self._atanhee(tphi) + (tphi / scb) * (_sc(tphi) / scb)
577 def _qIntegrand(self, beta):
578 # pbeta(beta) = integrate(q(beta), beta), with beta in radians
579 # q(beta) = (1-f) * (sin(xi) - sin(chi)) / cos(phi)
580 # = (1-f) * (cos(chi) - cos(xi)) / cos(phi) *
581 # (cos(xi) + cos(chi)) / (sin(xi) + sin(chi))
582 # Fit q(beta)/cos(beta) with Fourier transform
583 # q(beta)/cos(beta) = sum(c[k] * sin((2*k+1)*beta), k, 0, K-1)
584 # then the integral is
585 # pbeta = sum(d[k] * cos((2*k+2)*beta), k, 0, K-1)
586 # where
587 # d[k] = -1/(4*(k+1)) * (c[k] + c[k+1]) for k in 0..K-2
588 # d[K-1] = -1/(4*K) * c[K-1]
589 Beta = AuxBeta.fromRadians(beta)
590 if Beta.x: # and self._fm1:
591 Ax, _cv = Aux, self.convert
592 Phi = _cv(Ax.PHI, Beta, exact=True)
593 schi, cchi = _cv(Ax.CHI, Phi, exact=True)._yx_normalized
594 sxi, cxi = _cv(Ax.XI, Phi, exact=True)._yx_normalized
595 r = (sxi - schi) if fabs(schi) < fabs(cchi) else \
596 _over(_2cos2x(cchi, cxi), (sxi + schi) * _2_0)
597 r *= _over(self._fm1, Phi._x_normalized * Beta._x_normalized)
598 else: # beta == PI_2, PI3_2, ...
599 r = _0_0 # XXX 0 avoids NAN summation exceptions
600 return r
602 def Rectifying(self, Phi, **diff_name):
603 '''Convert I{Geographic} to I{Rectifying} latitude.
605 @arg Phi: Geographic latitude (L{AuxAngle}).
606 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}
607 and an optional C{B{name}=NN}.
609 @return: Rectifying latitude, C{Mu} (L{AuxAngle}).
610 '''
611 Beta = self.Parametric(Phi)
612 # assert Beta._AUX == Aux.BETA
613 sb, cb = map(fabs, Beta._yx_normalized)
614 a, ka, ka1 = _1_0, self._e2, self._e2m1
615 b, kb, kb1 = self._fm1, -self._e12, self._e12p1
616 if self.f < 0:
617 a, b = b, a
618 ka, kb = kb, ka
619 ka1, kb1 = kb1, ka1
620 sb, cb = cb, sb
621 # now a, b = larger/smaller semiaxis
622 # Beta measured from larger semiaxis
623 # kb, ka = modulus-squared for distance from Beta = 0, pi/2
624 # NB kb <= 0; 0 <= ka <= 1
625 # sa = b*E(Beta, sqrt(kb))
626 # sb = a*E(Beta',sqrt(ka))
627 # 1 - ka * (1 - sb2) = 1 - ka + ka*sb2
628 sb2 = sb**2
629 cb2 = cb**2
630 da2 = ka1 + ka * sb2
631 db2 = _1_0 - kb * sb2
632 # DLMF Eq. 19.25.9
633 my = b * sb * _Ef._RFRD(cb2, db2, _1_0, kb * sb2)
634 # DLMF Eq. 19.25.10 with complementary angles
635 mx = a * cb * (_Ef.fRF(sb2, da2, _1_0) * ka1 +
636 ka * cb2 * _Ef.fRD(sb2, _1_0, da2, _3_0) * ka1 +
637 ka * sb / sqrt(da2))
638 # my + mx = 2*_Ef.fRG(a*a, b*b) = a*E(e) = b*E(i*e')
639 # mr = a*E(e)*(2/pi) = b*E(i*e')*(2/pi)
640 if self.f < 0:
641 a, b = b, a
642 my, mx = mx, my
643 mr = (my + mx) / PI_2
644 if mr:
645 my = sin(my / mr)
646 mx = sin(mx / mr) # XXX zero?
647 else: # zero Mu
648 my, mx = _0_0, _1_0
649 d, n = _diff_name2(Phi, **diff_name)
650 Mu = AuxMu(my, mx, # normalized
651 name=n).copyquadrant(Phi)
652 if d:
653 d, x = _0_0, Beta._x_normalized
654 if x and mr:
655 if Mu.x and Phi.x and not isinf(Phi.tan):
656 d = b / mr * (x / Mu.x)**2 \
657 * (x / Phi._x_normalized)
658 else:
659 d = mr / a
660 Mu._diff = self._fm1 * d
661 return Mu
663 def RectifyingRadius(self, exact=False):
664 '''Get the I{Rectifying} radius.
666 @arg exact: If C{True}, use the exact expression,
667 otherwise the I{Taylor} series.
669 @return: Rectifying radius (L{Meter}, same units
670 as the ellipsoid axes).
671 '''
672 r = self._Ef_fRG_a2b2_PI_4 if exact else self._RectifyingR
673 return Meter(r, name__=self.RectifyingRadius)
675 @Property_RO
676 def _RectifyingR(self):
677 m = self.ALorder
678 d = _polynomial(self._n2, _RRCoeffs[m], 0, m // 2)
679 return d * (self.a + self.b) * _0_5
681 def _scbeta(self, tphi):
682 return _sc(self._fm1 * tphi)
684 def _toAux(self, auxout, Phi, **diff_name):
685 '''Convert I{Geographic} to I{Auxiliary} latitude.
687 @arg auxout: I{Auxiliary} kind (C{Aux.KIND}).
688 @arg Phi: Geographic latitude (L{AuxLat}).
689 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}
690 and an optional C{B{name}=NN}.
692 @return: Auxiliary latitude, I{Eta} (L{AuxLat}).
693 '''
694 _xinstanceof(AuxAngle, Phi=Phi)
695 # assert Phi._AUX == Aux.PHI
696 d, n = _diff_name2(Phi, **diff_name)
697 m = _toAuxCase.get(auxout, None)
698 if m: # callable
699 A = m(self, Phi, diff=d, name=n)
700 elif auxout == Aux.GEODETIC: # == GEOGRAPHIC
701 A = AuxPhi(Phi, name=n)
702 else: # auxout?
703 A = AuxPhi(NAN, name=n)
704 # assert A._AUX == auxout
705 return A
707 def _toZeta(self, zetaux):
708 '''Return a (lean) function to create C{AuxPhi(tphi)} and
709 convert that into C{AuxAngle} of (fixed) kind C{zetaux}
710 for use only inside the C{_Newton} loop.
711 '''
712 class _AuxPhy(AuxPhi):
713 # lean C{AuxPhi} instance.
714 # _diff = _1_0
715 # _x = _1_0
717 def __init__(self, tphi): # PYCHOK signature
718 self._y = tphi
720 m = _toAuxCase.get(zetaux, None)
721 if m: # callable
723 def _toZeta(tphi):
724 return m(self, _AuxPhy(tphi), diff=True)
726 elif zetaux == Aux.GEODETIC: # GEOGRAPHIC
727 _toZeta = _AuxPhy
729 else: # zetaux?
731 def _toZeta(unused): # PYCHOK expected
732 return _AuxPhy(NAN)
734 return _toZeta
737# switch(auxout): ...
738_toAuxCase = {Aux.AUTHALIC: AuxLat.Authalic,
739 Aux.CONFORMAL: AuxLat.Conformal,
740 Aux.GEOCENTRIC: AuxLat.Geocentric,
741 Aux.PARAMETRIC: AuxLat.Parametric,
742 Aux.RECTIFYING: AuxLat.Rectifying}
745def _Clenshaw(sinp, Zeta, cs, K):
746 sz, cz = Zeta._yx # isnormal
747 # Evaluate sum(c[k] * sin((2*k+2) * Zeta)) if sinp else
748 # sum(c[k] * cos((2*k+2) * Zeta))
749 x = _2cos2x(cz, sz) # 2 * cos(2*Zeta)
750 if isfinite(x):
751 U0, U1 = Fsum(), Fsum()
752 # assert len(cs) == K
753 for r in _reverange(K):
754 U1 -= U0 * x + cs[r]
755 U1, U0 = U0, -U1
756 # u0*f0(Zeta) - u1*fm1(Zeta)
757 # f0 = sin(2*Zeta) if sinp else cos(2*Zeta)
758 # fm1 = 0 if sinp else 1
759 if sinp:
760 U0 *= sz * cz * _2_0
761 else:
762 U0 *= x * _0_5
763 U0 -= U1
764 x = float(U0)
765 return x
768def _CXcoeffs(aL): # PYCHOK in .auxilats.__main__
769 '''(INTERNAL) Get the C{CX_4}, C{_6} or C{_8} coefficients.
770 '''
771 try: # from pygeodesy.auxilats._CX_x import _coeffs_x as _coeffs
772 _CX_x = _DOT_(_MODS.auxilats.__name__, _UNDER_('_CX', aL))
773 _coeffs = _MODS.getattr(_CX_x, _UNDER_('_coeffs', aL))
774 except (AttributeError, ImportError, KeyError, TypeError) as x:
775 raise AuxError(ALorder=aL, cause=x)
776 # assert _coeffs.ALorder == aL
777 # assert _coeffs.n == Aux.len(aL)
778 return _coeffs
781def _diff_name2(Phi, diff=False, **name):
782 '''(INTERNAL) Get C{{Bdiff}=False} and C{B{name}=NN}.
783 '''
784 n = _name__(name, _or_nameof=Phi) # if name else Phi.name
785 return diff, n
788def _Newton(tphi, Zeta, _toZeta, **name):
789 # Newton's method from AuxLat._fromAux
790 try:
791 _lg2 = _log2
792 _abs = fabs
793 tz = _abs(Zeta.tan)
794 tphi = tz / tphi # **)
795 ltz = _lg2(tz) # **)
796 ltphi = _lg2(tphi) # **)
797 ltmin = min(ltphi, MIN_EXP)
798 ltmax = max(ltphi, MAX_EXP)
799# auxin = Zeta._AUX
800 s, n, __2 = 0, 3, _0_5 # n = i + 2
801 _TOL, _xp2 = _EPSqrt, _exp2
802 for i in range(1, _TRIPS): # up to 1 Ki!
803 # _toAux(auxin, AuxPhi(tphi), diff=True)
804 Z = _toZeta(tphi)
805 # assert Z._AUX == auxin
806 t, s_ = Z.tan, s
807 if t > tz:
808 ltmax, s = ltphi, +1
809 elif t < tz:
810 ltmin, s = ltphi, -1
811 else:
812 break
813 # derivative dtan(Z)/dtan(Phi)
814 # to dlog(tan(Z))/dlog(tan(Phi))
815 d = (ltz - _lg2(t)) * t # **)
816 if d:
817 d = d / (Z.diff * tphi) # **)
818 ltphi += d
819 tphi = _xp2(ltphi)
820 if _abs(d) < _TOL:
821 i += 1
822 # _toAux(auxin, AuxPhi(tphi), diff=True)
823 Z = _toZeta(tphi)
824 tphi -= _over(Z.tan - tz, Z.diff)
825 break
826 if (i > n and (s * s_) < 0) or not ltmin < ltphi < ltmax:
827 s, n = 0, (i + 2)
828 ltphi = (ltmin + ltmax) * __2
829 tphi = _xp2(ltphi)
830 else:
831 i = _TRIPS
832 Phi = AuxPhi(tphi, **name).copyquadrant(Zeta)
833 Phi._iter = i
834 except (ValueError, ZeroDivisionError): # **) zero t, tphi, tz or Z.diff
835 Phi = AuxPhi(Zeta, **name) # diff as-as
836 Phi._iter = 0
837 # assert Phi._AUX == Aux.PHI
838 return Phi
841_f, _u = float, _Ufloats()
842_1__f3 = -1 / _f(3) # XXX +1 / _f(3)
843_AR2Coeffs = {4: _u(4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3),
844 6: _u(4 / _f(1287), 4 / _f(693), 4 / _f(315), 4 / _f(105),
845 4 / _f(15), _1__f3),
846 8: _u(4 / _f(3315), 4 / _f(2145), 4 / _f(1287), 4 / _f(693),
847 4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3)}
848_RRCoeffs = {4: _u(1 / _f(64), _0_25),
849 6: _u(1 / _f(256), 1 / _f(64), _0_25),
850 8: _u(25 / _f(16384), 1 / _f(256), 1 / _f(64), _0_25)} # PYCHOK used!
851del _f, _u, _Ufloats, _1__f3
852# assert set(_AR2Coeffs.keys()) == set(_RRCoeffs.keys())
854# AuxLat._Lmax = max(_AR2Coeffs.keys()) # == max(ALorder)
856__all__ += _ALL_DOCS(AuxLat)
858# **) MIT License
859#
860# Copyright (C) 2023-2024 -- mrJean1 at Gmail -- All Rights Reserved.
861#
862# Permission is hereby granted, free of charge, to any person obtaining a
863# copy of this software and associated documentation files (the "Software"),
864# to deal in the Software without restriction, including without limitation
865# the rights to use, copy, modify, merge, publish, distribute, sublicense,
866# and/or sell copies of the Software, and to permit persons to whom the
867# Software is furnished to do so, subject to the following conditions:
868#
869# The above copyright notice and this permission notice shall be included
870# in all copies or substantial portions of the Software.
871#
872# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
873# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
874# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
875# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
876# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
877# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
878# OTHER DEALINGS IN THE SOFTWARE.