Coverage for pygeodesy/auxilats/auxLat.py: 93%
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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 _passarg, _reverange, _xinstanceof
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
26# from pygeodesy.ellipsoids import Ellipsoid # from .datums
27from pygeodesy.elliptic import Elliptic as _Ef
28from pygeodesy.errors import AuxError, _xkwds, _xkwds_get, _Xorder
29# from pygeodesy.fmath import cbrt # from .karney
30from pygeodesy.fsums import Fsum, _Fsumf_, _sum
31from pygeodesy.karney import _2cos2x, _polynomial, _ALL_DOCS, cbrt, _MODS
32from pygeodesy.interns import NN, _DOT_, _UNDER_ # _earth_
33# from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS # from .karney
34from pygeodesy.props import Property, Property_RO, _update_all
35from pygeodesy.units import _isDegrees, _isRadius, Degrees, Meter
36# from pygeodesy.utily import atan1 # from .auxily
38from math import asinh, atan2, copysign, cosh, fabs, sin, sinh, sqrt
39try:
40 from math import exp2 as _exp2
41except ImportError: # Python 3.11-
43 def _exp2(x):
44 return pow(_2_0, x)
46__all__ = ()
47__version__ = '24.04.14'
49_TRIPS = 1024 # XXX 2 or 3?
52class AuxLat(AuxAngle):
53 '''Base class for accurate conversion between I{Auxiliary} latitudes
54 on an ellipsoid.
56 Latitudes are represented by L{AuxAngle} instances in order to
57 maintain precision near the poles, I{Authalic} latitude C{Xi},
58 I{Conformal} C{Chi}, I{Geocentric} C{Theta}, I{Geographic} C{Phi},
59 I{Parametric} C{Beta} and I{Rectifying} C{Mu}.
61 @see: I{Karney}'s C++ class U{AuxLatitude
62 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AuxLatitude.html>}.
63 '''
64 _csc = dict() # global coeffs cache: [aL][k], upto max(k) * (4 + 6 + 8) floats
65 _E = _WGS84.ellipsoid
66# _Lmax = 0 # overwritten below
67 _mAL = 6 # 4, 6 or 8 aka Lmax
69 def __init__(self, a_earth=_WGS84, f=None, b=None, name=NN, **ALorder):
70 '''New L{AuxLat} instance on an ellipsoid or datum.
72 @arg a_earth: Equatorial radius, semi-axis (C{meter}) or an
73 ellipsoid or datum (L{Datum}, L{Ellipsoid},
74 L{Ellipsoid2} or L{a_f2Tuple}).
75 @kwarg f: Flattening: M{(a - b) / a} (C{float}, near zero for
76 spherical), ignored if B{C{a_earth}} is not scalar.
77 @kwarg b: Optional polar radius, semi-axis (C{meter}, same
78 units as B{C{a_earth}}), ignored if B{C{a_earth}}
79 is not scalar.
80 @kwarg ALorder: Optional keyword arguments B{C{ALorder}} to
81 set L{AuxLat}'s C{order}, see property
82 C{ALorder}.
83 @kwarg name: Optional name (C{str}).
84 '''
85 if a_earth is not _WGS84:
86 n = name or AuxLat.__name__
87 try:
88 if b is f is None:
89 E = _ellipsoidal_datum(a_earth, name=n).ellipsoid # XXX raiser=_earth_
90 elif _isRadius(a_earth):
91 E = Ellipsoid(a_earth, f=f, b=b, name=_UNDER_(n))
92 else:
93 raise ValueError()
94 except Exception as x:
95 raise AuxError(a_earth=a_earth, f=f, b=b, cause=x)
96 self._E = E
98 if name:
99 self.name = name
100 if ALorder:
101 self.ALorder = _xkwds_get(ALorder, ALorder=AuxLat._mAL)
103 @Property_RO
104 def a(self):
105 '''Get the C{ellipsoid}'s equatorial radius (C{meter}, conventionally).
106 '''
107 return self.ellipsoid.a
109 equatoradius = a
111 @Property
112 def ALorder(self):
113 '''Get the I{AuxLat} order (C{int}, 4, 6 or 8).
114 '''
115 return self._mAL
117 @ALorder.setter # PYCHOK setter!
118 def ALorder(self, order):
119 '''Set the I{AuxLat} order (C{int}, 4, 6 or 8).
120 '''
121 m = _Xorder(_AR2Coeffs, AuxError, ALorder=order)
122 if self._mAL != m:
123 _update_all(self)
124 self._mAL = m
126 def _atanhee(self, tphi): # see Ellipsoid._es_atanh, .albers._atanhee
127 # atanh(e * sphi) = asinh(e' * sbeta)
128 f = self.f
129 s = _sn(self._fm1 * tphi) if f > 0 else _sn(tphi)
130 if f: # atanh(e * sphi) = asinh(e' * sbeta)
131 e = self._e
132 s = _over(atan1(e * s) if f < 0 else asinh(self._e1 * s), e)
133 return s
135 def Authalic(self, Phi, **diff_name):
136 '''Convert I{Geographic} to I{Aunthalic} latitude.
138 @arg Phi: Geographic latitude (L{AuxAngle}).
140 @return: Parametric latitude, C{Beta} (L{AuxAngle}).
141 '''
142 _xinstanceof(AuxAngle, Phi=Phi)
143 # assert Phi._AUX == Aux.PHI
144 tphi = fabs(Phi.tan)
145 if isfinite(tphi) and tphi and self.f:
146 y, x = Phi._yx_normalized
147 q = self._q
148 qv = self._qf(tphi)
149 Dq2 = self._Dq(tphi)
150 Dq2 *= (q + qv) / (fabs(y) + _1_0) # _Dq(-tphi)
151 Xi = AuxXi(copysign(qv, Phi.y), x * sqrt(Dq2),
152 name=_xkwds_get(diff_name, name=Phi.name))
154 if _xkwds_get(diff_name, diff=False):
155 if isnan(tphi):
156 d = self._e2m1_sq2
157 else:
158 c = self.Parametric(Phi)._x_normalized
159 d = _over(c, Xi._x_normalized)**3
160 d *= _over(c, x) * _over(_2_0, q)
161 Xi._diff = d
162 else:
163 Xi = AuxXi(*Phi._yx) # diff default
164 # assert Xi._AUX == Aux.XI
165 return Xi
167 def AuthalicRadius2(self, exact=False, f_max=_0_1):
168 '''Get the I{Authalic} radius I{squared}.
170 @kwarg exact: If C{True}, use the exact expression, otherwise
171 the I{Taylor} series.
172 @kwarg f_max: C{Flattening} not to exceed (C{float}).
174 @return: Authalic radius I{squared} (C{meter} I{squared}, same
175 units as the ellipsoid axes).
177 @raise AuxError: If C{B{exact}=False} and C{abs(flattening)}
178 exceeds C{f_max}.
179 '''
180 f = self.f
181 if exact or not f:
182 c2 = self.ellipsoid.b2 * self._q # == ellipsoid.c2x * 2
183 elif fabs(f) < f_max:
184 # Using a * (a + b) / 2 as the multiplying factor leads to a rapidly
185 # converging series in n. Of course, using this series isn't really
186 # necessary, since the exact expression is simple to evaluate. However,
187 # we do it for consistency with RectifyingRadius and, presumably, the
188 # roundoff error is smaller compared to that for the exact expression.
189 m = self.ALorder
190 c2 = _polynomial(self._n, _AR2Coeffs[m], 0, m)
191 c2 *= self.a * (self.a + self.b)
192 else:
193 raise AuxError(exact=exact, f=f, f_max=f_max)
194 return c2 * _0_5
196 @Property_RO
197 def b(self):
198 '''Get the C{ellipsoid}'s polar radius (C{meter}, conventionally).
199 '''
200 return self.ellipsoid.b # a * (_1_0 - f)
202 polaradius = b
204 def _coeffs(self, auxout, auxin):
205 # Get the polynomial coefficients as 4-, 6- or 8-tuple
206 aL = self.ALorder # aka Lmax
207 if auxout == auxin:
208 return _0_0s(aL) # uncached
210 k = Aux._1d(auxout, auxin)
211 try: # cached
212 return AuxLat._csc[aL][k]
213 except KeyError:
214 pass
216 Cx = _CXcoeffs(aL)
217 try:
218 Cx = Cx[auxout][auxin]
219 except KeyError as x:
220 raise AuxError(auxout=auxout, auxin=auxin, cause=x)
222 d = x = n = self._n
223 if Aux.use_n2(auxin) and Aux.use_n2(auxout):
224 x = self._n2
226 def _m(aL):
227 for m in _reverange(aL):
228 yield m // 2
229 else:
230 _m = _reverange # PYCHOK expected
232 i = 0
233 cs = []
234 _c = cs.append
235 _p = _polynomial
236 for m in _m(aL):
237 j = i + m + 1 # order m = j - i - 1
238 _c(_p(x, Cx, i, j) * d)
239 d *= n
240 i = j
241 # assert i == len(Cx) and len(cs) == aL
242 AuxLat._csc.setdefault(aL, {})[k] = cs = tuple(cs)
243 return cs
245 def Conformal(self, Phi, **diff_name):
246 '''Convert I{Geographic} to I{Conformal} latitude.
248 @arg Phi: Geographic latitude (L{AuxAngle}).
250 @return: Conformal latitude, C{Chi} (L{AuxAngle}).
251 '''
252 _xinstanceof(AuxAngle, Phi=Phi)
253 # assert Phi._AUX == Aux.PHI
254 tphi = tchi = fabs(Phi.tan)
255 if isfinite(tphi) and tphi and self.f:
256 sig = sinh(self._atanhee(tphi) * self._e2)
257 scsig = _sc(sig)
258 scphi = _sc(tphi)
259 if self.f > 0:
260 # The general expression for tchi is
261 # tphi * scsig - sig * scphi
262 # This involves cancellation if f > 0, so change to
263 # (tphi - sig) * (tphi + sig) / (tphi * scsig + sig * scphi)
264 # To control overflow, write as (sigtphi = sig / tphi)
265 # (tphi - sig) * (1 + sigtphi) / (scsig + sigtphi * scphi)
266 sigtphi = sig / tphi
267 if sig < (tphi * _0_5):
268 t = tphi - sig
269 else:
270 def _asinh_2(x):
271 return asinh(x) * _0_5
272 # Still possibly dangerous cancellation in tphi - sig.
273 # Write tphi - sig = (1 - e) * Dg(1, e)
274 # Dg(x, y) = (g(x) - g(y)) / (x - y)
275 # g(x) = sinh(x * atanh(sphi * x))
276 # Note sinh(atanh(sphi)) = tphi
277 # Turn the crank on divided differences, substitute
278 # sphi = tphi / _sc(tphi)
279 # atanh(x) = asinh(x / sqrt(1 - x^2))
280 e = self._e
281 em1 = self._e2m1 / (_1_0 + e)
282 # assert em1 != 0
283 scb = self._scbeta(tphi)
284 scphib = scphi / scb # sec(phi) / sec(beta)
285 atphib = _asinh_2(tphi * e / scb) # atanh(e * sphi)
286 atphi = _asinh_2(tphi) # atanh(sphi)
287 t = _asinh_2(em1 * (tphi * scphib)) / em1
288 try:
289 Dg = _Fsumf_(atphi, atphib, t, e * t)
290 except ValueError: # Fsum(NAN) exception
291 Dg = _sum((atphi, atphib, t, e * t))
292 e *= atphib
293 t = atphi - e
294 if t: # sinh(0) == 0
295 Dg *= sinh(t) / t * cosh(atphi + e) * em1
296 t = float(Dg) # tphi - sig
297 tchi = _over(t * (_1_0 + sigtphi),
298 scsig + scphi * sigtphi) if t else _0_0
299 else:
300 tchi = tphi * scsig - sig * scphi
302 n = _xkwds_get(diff_name, name=Phi.name)
303 Chi = AuxChi(tchi, name=n).copyquadrant(Phi)
305 if _xkwds_get(diff_name, diff=False):
306 if isinf(tphi): # PYCHOK np cover
307 d = self._conformal_diff
308 else:
309 d = self.Parametric(Phi)._x_normalized
310 if d:
311 d = _over(d, Chi._x_normalized) * \
312 _over(d, Phi._x_normalized) * self._e2m1
313 Chi._diff = d
314 # assrt Chi._AUX == Aux.CHI
315 return Chi
317 @Property_RO
318 def _conformal_diff(self): # PYCHOK no cover
319 '''(INTERNAL) Constant I{Conformal} diff.
320 '''
321 e = self._e
322 if self.f > 0:
323 ss = sinh(asinh(self._e1) * e)
324 d = _over(_1_0, _sc(ss) + ss)
325 elif e:
326 ss = sinh(-atan1(e) * e)
327 d = _sc(ss) - ss
328 else:
329 d = _1_0
330 return d
332 def convert(self, auxout, Zeta_d, exact=False):
333 # Convert I{Auxiliary} or I{scalar} latitude
334 Z = d = Zeta_d
335 if isinstance(Z, AuxAngle):
336 A, auxin = _AuxClass(auxout), Z._AUX
337 if auxin == auxout:
338 pass
339 elif not (isfinite(Z.tan) and Z.tan): # XXX
340 Z = A(*Z._yx, aux=auxout, name=Z.name)
341 elif exact:
342 p = Aux.power(auxout, auxin)
343 if p is None:
344 P = self._fromAux(Z) # Phi
345 Z = self._toAux(auxout, P)
346 Z._iter = P.iteration
347 else:
348 y, x = Z._yx
349 if p:
350 y *= pow(self._fm1, p)
351 Z = A(y, x, aux=auxout, name=Z.name)
352 else:
353 cs = self._coeffs(auxout, auxin)
354 yx = Z._yx_normalized
355 Z = A(*yx, aux=auxout, name=Z.name)
356 # assert Z._yx == yx
357 r = _Clenshaw(True, Z, cs, self.ALorder)
358 Z += AuxAngle.fromRadians(r, aux=auxout)
359 # assert Z._AUX == auxout
360 return Z
362 elif _isDegrees(d):
363 Z = AuxPhi.fromDegrees(d)
364 d = round((d - Z.toDegrees) / _360_0) * _360_0
365 d += self.convert(auxout, Z, exact).toDegrees
366 return Degrees(d, name=Aux.Greek(auxout))
368 raise AuxError(auxout=auxout, Zeta_d=Zeta_d, exact=exact)
370 def _Dq(self, tphi):
371 # I{Divided Difference} of (q(1) - q(sphi)) / (1 - sphi).
372 sphi = _sn(tphi)
373 if tphi > 0:
374 scphi = _sc(tphi)
375 d = _1_0 / (scphi**2 * (_1_0 + sphi)) # XXX - sphi
376 if d:
377 # General expression for _Dq(1, sphi) is
378 # atanh(e * d / (1 - e2 * sphi)) / (e * d) +
379 # (1 + e2 * sphi) / ((1 - e2 * sphi * sphi) * e2m1)
380 # with atanh(e * d / (1 - e2 * sphi)) =
381 # atanh(e * d * scphi / (scphi - e2 * tphi))
382 e2m1, ed = self._e2m1, (self._e * d)
383 if e2m1 and ed:
384 e2 = self._e2
385 if e2 > 0: # assert self.f > 0
386 scb = self._scbeta(tphi)
387 q = scphib = scphi / scb
388 q *= (scphi + tphi * e2) / (e2m1 * scb)
389 q += asinh(self._e1 * d * scphib) / ed
390 else:
391 s2 = sphi * e2
392 q = (_1_0 + s2) / ((_1_0 - sphi * s2) * e2m1)
393 q += (atan2(ed, _1_0 - s2) / ed) if e2 < 0 else _1_0
394 else: # PYCHOK no cover
395 q = INF
396 else: # PYCHOK no cover
397 q = self._2_e2m12
398 else: # not reached, open-coded in .Authalic
399 q = _over(self._q - self._qf(tphi), _1_0 - sphi)
400 return q
402 @Property_RO
403 def _e(self): # unsigned, (1st) eccentricity
404 return self.ellipsoid.e # sqrt(fabs(self._e2))
406 @Property_RO
407 def _e1(self):
408 return sqrt(fabs(self._e12))
410 @Property_RO
411 def _e12(self):
412 return _over(self._e2, _1_0 - self._e2)
414 @Property_RO
415 def _e12p1(self):
416 return _1_0 / self._e2m1
418 @Property_RO
419 def _e2(self): # signed, (1st) eccentricity squared
420 return self.ellipsoid.e2
422 @Property_RO
423 def _e2m1(self): # 1 less 1st eccentricity squared
424 return self.ellipsoid.e21 # == ._fm1**2
426 @Property_RO
427 def _e2m1_sq2(self):
428 return self._e2m1 * sqrt(self._q * _0_5)
430 @Property_RO
431 def _2_e2m12(self):
432 return _2_0 / self._e2m1**2
434 @Property_RO
435 def _Ef_fRG_a2b2_PI_4(self):
436 E = self.ellipsoid
437 return _Ef.fRG(E.a2, E.b2) / PI_4
439 @Property_RO
440 def ellipsoid(self):
441 '''Get the ellipsoid (L{Ellipsoid}).
442 '''
443 return self._E
445 @Property_RO
446 def f(self):
447 '''Get the C{ellipsoid}'s flattening (C{scalar}).
448 '''
449 return self.ellipsoid.f
451 flattening = f
453 @Property_RO
454 def _fm1(self): # 1 - flattening
455 return self.ellipsoid.f1
457 def _fromAux(self, Zeta, **name):
458 '''Convert I{Auxiliary} to I{Geographic} latitude.
460 @arg Zeta: Auxiliary latitude (L{AuxAngle}).
462 @return: Geographic latitude, I{Phi} (L{AuxAngle}).
463 '''
464 _xinstanceof(AuxAngle, Zeta=Zeta)
465 aux = Zeta._AUX
466 n = _xkwds_get(name, name=Zeta.name)
467 f = self._fromAuxCase.get(aux, None)
468 if f is None:
469 Phi = AuxPhi(NAN, name=n)
470 elif callable(f):
471 t = self._fm1
472 t *= f(t)
473 Phi = _Newton(t, Zeta, self._toZeta(aux), name=n)
474 else: # assert isscalar(f)
475 y, x = Zeta._yx
476 Phi = AuxPhi(y / f, x, name=n)
477 # assert Phi._AUX == Aux.PHI
478 return Phi
480 @Property_RO
481 def _fromAuxCase(self):
482 '''(INTERNAL) switch(auxin): ...
483 '''
484 return {Aux.AUTHALIC: cbrt,
485 Aux.CONFORMAL: _passarg,
486 Aux.GEOCENTRIC: self._e2m1,
487 Aux.GEOGRAPHIC: _1_0,
488 Aux.PARAMETRIC: self._fm1,
489 Aux.RECTIFYING: sqrt}
491 def Geocentric(self, Phi, **diff_name):
492 '''Convert I{Geographic} to I{Geocentric} latitude.
494 @arg Phi: Geographic latitude (L{AuxAngle}).
496 @return: Geocentric latitude, C{Phi} (L{AuxAngle}).
497 '''
498 _xinstanceof(AuxAngle, Phi=Phi)
499 # assert Phi._AUX == Aux.PHI
500 Theta = AuxTheta(Phi.y * self._e2m1, Phi.x,
501 name=_xkwds_get(diff_name, name=Phi.name))
502 if _xkwds_get(diff_name, diff=False):
503 Theta._diff = self._e2m1
504 return Theta
506 def Geodetic(self, Phi, **diff_name): # PYCHOK no cover
507 '''Convert I{Geographic} to I{Geodetic} latitude.
509 @arg Phi: Geographic latitude (L{AuxAngle}).
511 @return: Geodetic latitude, C{Phi} (L{AuxAngle}).
512 '''
513 _xinstanceof(AuxAngle, Phi=Phi)
514 # assert Phi._AUX == Aux.PHI
515 return AuxPhi(Phi, name=_xkwds_get(diff_name, name=Phi.name))
517 @Property_RO
518 def _n(self): # 3rd flattening
519 return self.ellipsoid.n
521 @Property_RO
522 def _n2(self):
523 return self._n**2
525 def Parametric(self, Phi, **diff_name):
526 '''Convert I{Geographic} to I{Parametric} latitude.
528 @arg Phi: Geographic latitude (L{AuxAngle}).
530 @return: Parametric latitude, C{Beta} (L{AuxAngle}).
531 '''
532 _xinstanceof(AuxAngle, Phi=Phi)
533 # assert Phi._AUX == Aux.PHI
534 Beta = AuxBeta(Phi.y * self._fm1, Phi.x,
535 name=_xkwds_get(diff_name, name=Phi.name))
536 if _xkwds_get(diff_name, diff=False):
537 Beta._diff = self._fm1
538 return Beta
540 Reduced = Parametric
542 @Property_RO
543 def _q(self): # constant _q
544 q, f = self._e12p1, self.f
545 if f:
546 e = self._e
547 q += _over(asinh(self._e1) if f > 0 else atan1(e), e)
548 else:
549 q += _1_0
550 return q
552 def _qf(self, tphi):
553 # function _q: atanh(e * sphi) / e + sphi / (1 - (e * sphi)^2)
554 scb = self._scbeta(tphi)
555 return self._atanhee(tphi) + (tphi / scb) * (_sc(tphi) / scb)
557 def _qIntegrand(self, beta):
558 # pbeta(beta) = integrate(q(beta), beta), with beta in radians
559 # q(beta) = (1-f) * (sin(xi) - sin(chi)) / cos(phi)
560 # = (1-f) * (cos(chi) - cos(xi)) / cos(phi) *
561 # (cos(xi) + cos(chi)) / (sin(xi) + sin(chi))
562 # Fit q(beta)/cos(beta) with Fourier transform
563 # q(beta)/cos(beta) = sum(c[k] * sin((2*k+1)*beta), k, 0, K-1)
564 # then the integral is
565 # pbeta = sum(d[k] * cos((2*k+2)*beta), k, 0, K-1)
566 # where
567 # d[k] = -1/(4*(k+1)) * (c[k] + c[k+1]) for k in 0..K-2
568 # d[K-1] = -1/(4*K) * c[K-1]
569 Beta = AuxBeta.fromRadians(beta)
570 if Beta.x: # and self._fm1:
571 Ax, _cv = Aux, self.convert
572 Phi = _cv(Ax.PHI, Beta, exact=True)
573 schi, cchi = _cv(Ax.CHI, Phi, exact=True)._yx_normalized
574 sxi, cxi = _cv(Ax.XI, Phi, exact=True)._yx_normalized
575 r = (sxi - schi) if fabs(schi) < fabs(cchi) else \
576 _over(_2cos2x(cchi, cxi), (sxi + schi) * _2_0)
577 r *= _over(self._fm1, Phi._x_normalized * Beta._x_normalized)
578 else: # beta == PI_2, PI3_2, ...
579 r = _0_0 # XXX 0 avoids NAN summation exceptions
580 return r
582 def Rectifying(self, Phi, **diff_name):
583 '''Convert I{Geographic} to I{Rectifying} latitude.
585 @arg Phi: Geographic latitude (L{AuxAngle}).
587 @return: Rectifying latitude, C{Mu} (L{AuxAngle}).
588 '''
589 Beta = self.Parametric(Phi)
590 # assert Beta._AUX == Aux.BETA
591 sb, cb = map(fabs, Beta._yx_normalized)
592 a, ka, ka1 = _1_0, self._e2, self._e2m1
593 b, kb, kb1 = self._fm1, -self._e12, self._e12p1
594 if self.f < 0:
595 a, b = b, a
596 ka, kb = kb, ka
597 ka1, kb1 = kb1, ka1
598 sb, cb = cb, sb
599 # now a, b = larger/smaller semiaxis
600 # Beta measured from larger semiaxis
601 # kb, ka = modulus-squared for distance from Beta = 0, pi/2
602 # NB kb <= 0; 0 <= ka <= 1
603 # sa = b*E(Beta, sqrt(kb))
604 # sb = a*E(Beta',sqrt(ka))
605 # 1 - ka * (1 - sb2) = 1 - ka + ka*sb2
606 sb2 = sb**2
607 cb2 = cb**2
608 da2 = ka1 + ka * sb2
609 db2 = _1_0 - kb * sb2
610 # DLMF Eq. 19.25.9
611 my = b * sb * _Ef._RFRD(cb2, db2, _1_0, kb * sb2)
612 # DLMF Eq. 19.25.10 with complementary angles
613 mx = a * cb * (_Ef.fRF(sb2, da2, _1_0) * ka1 +
614 ka * cb2 * _Ef.fRD(sb2, _1_0, da2, _3_0) * ka1 +
615 ka * sb / sqrt(da2))
616 # my + mx = 2*_Ef.fRG(a*a, b*b) = a*E(e) = b*E(i*e')
617 # mr = a*E(e)*(2/pi) = b*E(i*e')*(2/pi)
618 if self.f < 0:
619 a, b = b, a
620 my, mx = mx, my
621 mr = (my + mx) / PI_2
622 if mr:
623 my = sin(my / mr)
624 mx = sin(mx / mr) # XXX zero?
625 else: # zero Mu
626 my, mx = _0_0, _1_0
627 n = _xkwds_get(diff_name, name=Phi.name)
628 Mu = AuxMu(my, mx, # normalized
629 name=n).copyquadrant(Phi)
631 if _xkwds_get(diff_name, diff=False):
632 d, x = _0_0, Beta._x_normalized
633 if x and mr:
634 if Mu.x and Phi.x and not isinf(Phi.tan):
635 d = b / mr * (x / Mu.x)**2 \
636 * (x / Phi._x_normalized)
637 else:
638 d = mr / a
639 Mu._diff = self._fm1 * d
640 return Mu
642 def RectifyingRadius(self, exact=False):
643 '''Get the I{Rectifying} radius.
645 @arg exact: If C{True}, use the exact expression,
646 otherwise the I{Taylor} series.
648 @return: Rectifying radius (L{Meter}, same units
649 as the ellipsoid axes).
650 '''
651 r = self._Ef_fRG_a2b2_PI_4 if exact else self._RectifyingR
652 return Meter(r, name=self.RectifyingRadius.__name__)
654 @Property_RO
655 def _RectifyingR(self):
656 m = self.ALorder
657 d = _polynomial(self._n2, _RRCoeffs[m], 0, m // 2)
658 return d * (self.a + self.b) * _0_5
660 def _scbeta(self, tphi):
661 return _sc(self._fm1 * tphi)
663 def _toAux(self, auxout, Phi, **diff_name):
664 '''Convert I{Geographic} to I{Auxiliary} latitude.
666 @arg auxout: I{Auxiliary} kind (C{Aux.KIND}).
667 @arg Phi: Geographic latitude (L{AuxLat}).
669 @return: Auxiliary latitude, I{Eta} (L{AuxLat}).
670 '''
671 _xinstanceof(AuxAngle, Phi=Phi)
672 # assert Phi._AUX == Aux.PHI
673 n = _xkwds_get(diff_name, name=Phi.name)
674 m = _toAuxCase.get(auxout, None)
675 if m: # callable
676 A = m(self, Phi, **_xkwds(diff_name, name=n))
677 elif auxout == Aux.GEODETIC: # == GEOGRAPHIC
678 A = AuxPhi(Phi, name=n)
679 else: # auxout?
680 A = AuxPhi(NAN, name=n)
681 # assert A._AUX == auxout
682 return A
684 def _toZeta(self, zetaux):
685 '''Return a (lean) function to create C{AuxPhi(tphi)} and
686 convert that into C{AuxAngle} of (fixed) kind C{zetaux}
687 for use only inside the C{_Newton} loop.
688 '''
689 class _AuxPhy(AuxPhi):
690 # lean C{AuxPhi} instance.
691 # _diff = _1_0
692 # _x = _1_0
694 def __init__(self, tphi): # PYCHOK signature
695 self._y = tphi
697 m = _toAuxCase.get(zetaux, None)
698 if m: # callable
700 def _toZeta(tphi):
701 return m(self, _AuxPhy(tphi), diff=True)
703 elif zetaux == Aux.GEODETIC: # GEOGRAPHIC
704 _toZeta = _AuxPhy
706 else: # zetaux?
708 def _toZeta(unused): # PYCHOK expected
709 return _AuxPhy(NAN)
711 return _toZeta
714# switch(auxout): ...
715_toAuxCase = {Aux.AUTHALIC: AuxLat.Authalic,
716 Aux.CONFORMAL: AuxLat.Conformal,
717 Aux.GEOCENTRIC: AuxLat.Geocentric,
718 Aux.PARAMETRIC: AuxLat.Parametric,
719 Aux.RECTIFYING: AuxLat.Rectifying}
722def _Clenshaw(sinp, Zeta, cs, K):
723 sz, cz = Zeta._yx # isnormal
724 # Evaluate sum(c[k] * sin((2*k+2) * Zeta)) if sinp else
725 # sum(c[k] * cos((2*k+2) * Zeta))
726 x = _2cos2x(cz, sz) # 2 * cos(2*Zeta)
727 if isfinite(x):
728 U0, U1 = Fsum(), Fsum()
729 # assert len(cs) == K
730 for r in _reverange(K):
731 U1 -= U0 * x + cs[r]
732 U1, U0 = U0, -U1
733 # u0*f0(Zeta) - u1*fm1(Zeta)
734 # f0 = sin(2*Zeta) if sinp else cos(2*Zeta)
735 # fm1 = 0 if sinp else 1
736 if sinp:
737 U0 *= sz * cz * _2_0
738 else:
739 U0 *= x * _0_5
740 U0 -= U1
741 x = float(U0)
742 return x
745def _CXcoeffs(aL): # PYCHOK in .auxilats.__main__
746 '''(INTERNAL) Get the C{CX_4}, C{_6} or C{_8} coefficients.
747 '''
748 try: # from pygeodesy.auxilats._CX_x import _coeffs_x as _coeffs
749 _CX_x = _DOT_(_MODS.auxilats.__name__, _UNDER_('_CX', aL))
750 _coeffs = _MODS.getattr(_CX_x, _UNDER_('_coeffs', aL))
751 except (AttributeError, ImportError, KeyError, TypeError) as x:
752 raise AuxError(ALorder=aL, cause=x)
753 # assert _coeffs.ALorder == aL
754 # assert _coeffs.n == Aux.len(aL)
755 return _coeffs
758def _Newton(tphi, Zeta, _toZeta, **name):
759 # Newton's method fro AuxLat._fromAux
760 try:
761 _lg2 = _log2
762 _abs = fabs
763 tz = _abs(Zeta.tan)
764 tphi = tz / tphi # **)
765 ltz = _lg2(tz) # **)
766 ltphi = _lg2(tphi) # **)
767 ltmin = min(ltphi, MIN_EXP)
768 ltmax = max(ltphi, MAX_EXP)
769# auxin = Zeta._AUX
770 s, n, __2 = 0, 3, _0_5 # n = i + 2
771 _TOL, _xp2 = _EPSqrt, _exp2
772 for i in range(1, _TRIPS): # up to 1 Ki!
773 # _toAux(auxin, AuxPhi(tphi), diff=True)
774 Z = _toZeta(tphi)
775 # assert Z._AUX == auxin
776 t, s_ = Z.tan, s
777 if t > tz:
778 ltmax, s = ltphi, +1
779 elif t < tz:
780 ltmin, s = ltphi, -1
781 else:
782 break
783 # derivative dtan(Z)/dtan(Phi)
784 # to dlog(tan(Z))/dlog(tan(Phi))
785 d = (ltz - _lg2(t)) * t # **)
786 if d:
787 d = d / (Z.diff * tphi) # **)
788 ltphi += d
789 tphi = _xp2(ltphi)
790 if _abs(d) < _TOL:
791 i += 1
792 # _toAux(auxin, AuxPhi(tphi), diff=True)
793 Z = _toZeta(tphi)
794 tphi -= _over(Z.tan - tz, Z.diff)
795 break
796 if (i > n and (s * s_) < 0) or not ltmin < ltphi < ltmax:
797 s, n = 0, (i + 2)
798 ltphi = (ltmin + ltmax) * __2
799 tphi = _xp2(ltphi)
800 else:
801 i = _TRIPS
802 Phi = AuxPhi(tphi, **name).copyquadrant(Zeta)
803 Phi._iter = i
804 except (ValueError, ZeroDivisionError): # **) zero t, tphi, tz or Z.diff
805 Phi = AuxPhi(Zeta, **name) # diff as-as
806 Phi._iter = 0
807 # assert Phi._AUX == Aux.PHI
808 return Phi
811_f, _u = float, _Ufloats()
812_1__f3 = -1 / _f(3) # XXX +1 / _f(3)
813_AR2Coeffs = {4: _u(4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3),
814 6: _u(4 / _f(1287), 4 / _f(693), 4 / _f(315), 4 / _f(105),
815 4 / _f(15), _1__f3),
816 8: _u(4 / _f(3315), 4 / _f(2145), 4 / _f(1287), 4 / _f(693),
817 4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3)}
818_RRCoeffs = {4: _u(1 / _f(64), _0_25),
819 6: _u(1 / _f(256), 1 / _f(64), _0_25),
820 8: _u(25 / _f(16384), 1 / _f(256), 1 / _f(64), _0_25)} # PYCHOK used!
821del _f, _u, _Ufloats, _1__f3
822# assert set(_AR2Coeffs.keys()) == set(_RRCoeffs.keys())
824# AuxLat._Lmax = max(_AR2Coeffs.keys()) # == max(ALorder)
826__all__ += _ALL_DOCS(AuxLat)
828# **) MIT License
829#
830# Copyright (C) 2023-2024 -- mrJean1 at Gmail -- All Rights Reserved.
831#
832# Permission is hereby granted, free of charge, to any person obtaining a
833# copy of this software and associated documentation files (the "Software"),
834# to deal in the Software without restriction, including without limitation
835# the rights to use, copy, modify, merge, publish, distribute, sublicense,
836# and/or sell copies of the Software, and to permit persons to whom the
837# Software is furnished to do so, subject to the following conditions:
838#
839# The above copyright notice and this permission notice shall be included
840# in all copies or substantial portions of the Software.
841#
842# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
843# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
844# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
845# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
846# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
847# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
848# OTHER DEALINGS IN THE SOFTWARE.