Coverage for pygeodesy/auxilats/auxLat.py: 87%
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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 _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, \
26 Ellipsoid, _name__, _EWGS84
27# from pygeodesy.ellipsoids import Ellipsoid, _EWGS84 # from .datums
28from pygeodesy.elliptic import Elliptic as _Ef
29from pygeodesy.errors import AuxError, _xkwds_not, _xkwds_pop2, _Xorder
30# from pygeodesy.fmath import cbrt # from .karney
31from pygeodesy.fsums import Fsum, _Fsumf_, _sum
32# from pygeodesy.internals import _passarg # from .basics
33from pygeodesy.interns import NN, _DOT_, _not_scalar_, _UNDER_
34from pygeodesy.karney import _2cos2x, _polynomial, _ALL_DOCS, cbrt, _MODS
35# from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS # from .karney
36# from pygeodesy.named import _name__ # from .datums
37from pygeodesy.props import Property, Property_RO, _update_all
38from pygeodesy.units import _isDegrees, _isRadius, Degrees, Meter
39# from pygeodesy.utily import atan1 # from .auxily
41from math import asinh, atan2, copysign, cosh, fabs, sin, sinh, sqrt
42try:
43 from math import exp2 as _exp2
44except ImportError: # Python 3.11-
46 def _exp2(x):
47 return pow(_2_0, x)
49__all__ = ()
50__version__ = '24.06.16'
52_TRIPS = 1024 # XXX 2 or 3?
55class AuxLat(AuxAngle):
56 '''Base class for accurate conversion between I{Auxiliary} latitudes
57 on an ellipsoid.
59 Latitudes are represented by L{AuxAngle} instances in order to
60 maintain precision near the poles, I{Authalic} latitude C{Xi},
61 I{Conformal} C{Chi}, I{Geocentric} C{Theta}, I{Geographic} C{Phi},
62 I{Parametric} C{Beta} and I{Rectifying} C{Mu}.
64 @see: I{Karney}'s C++ class U{AuxLatitude
65 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AuxLatitude.html>}.
66 '''
67 _csc = dict() # global coeffs cache: [aL][k], upto max(k) * (4 + 6 + 8) floats
68 _E = _EWGS84
69# _Lmax = 0 # overwritten below
70 _mAL = 6 # 4, 6 or 8 aka Lmax
72 def __init__(self, a_earth=_EWGS84, f=None, b=None, **ALorder_name):
73 '''New L{AuxLat} instance on an ellipsoid or datum.
75 @arg a_earth: Equatorial radius, semi-axis (C{meter}) or an ellipsoid or
76 datum (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}).
77 @kwarg f: Flattening: M{(a - b) / a} (C{float}, near zero for spherical),
78 required if B{C{a_earth}} is C{scalar} and C{B{b}=None}.
79 @kwarg b: Optional polar radius, semi-axis (C{meter}, required if B{C{a_earth}}
80 is C{scalar} and C{B{f}=None}.
81 @kwarg ALorder_name: Optional C{B{name}=NN} (C{str}) and optional order of
82 this L{AuxLat} C{B{ALorder}=6}, see property C{ALorder}.
83 '''
84 if ALorder_name:
85 M = self._mAL
86 m, name = _xkwds_pop2(ALorder_name, ALorder=M)
87 if m != M:
88 self.ALorder = m
89 else:
90 name = NN
91 try:
92 if a_earth not in (_EWGS84, _WGS84):
93 n = _name__(name, name__=AuxLat)
94 if b is f is None:
95 E = _ellipsoidal_datum(a_earth, name=n).ellipsoid # XXX raiser=_earth_
96 elif _isRadius(a_earth):
97 E = Ellipsoid(a_earth, f=f, b=b, name=_UNDER_(NN, n))
98 else:
99 raise ValueError(_not_scalar_)
100 self._E = E
101 elif not (b is f is None):
102 # turn _UnexpectedError into AuxError
103 name = _name__(name, **_xkwds_not(None, b=b, f=f))
105 if name:
106 self.name = name
107 except Exception as x:
108 raise AuxError(a_earth=a_earth, f=f, b=b, cause=x)
110 @Property_RO
111 def a(self):
112 '''Get the C{ellipsoid}'s equatorial radius (C{meter}, conventionally).
113 '''
114 return self.ellipsoid.a
116 equatoradius = a
118 @Property
119 def ALorder(self):
120 '''Get the I{AuxLat} order (C{int}, 4, 6 or 8).
121 '''
122 return self._mAL
124 @ALorder.setter # PYCHOK setter!
125 def ALorder(self, order):
126 '''Set the I{AuxLat} order (C{int}, 4, 6 or 8).
127 '''
128 m = _Xorder(_AR2Coeffs, AuxError, ALorder=order)
129 if self._mAL != m:
130 _update_all(self)
131 self._mAL = m
133 def _atanhee(self, tphi): # see Ellipsoid._es_atanh, .albers._atanhee
134 # atanh(e * sphi) = asinh(e' * sbeta)
135 f = self.f
136 s = _sn(self._fm1 * tphi) if f > 0 else _sn(tphi)
137 if f: # atanh(e * sphi) = asinh(e' * sbeta)
138 e = self._e
139 s = _over(atan1(e * s) if f < 0 else asinh(self._e1 * s), e)
140 return s
142 def Authalic(self, Phi, **diff_name):
143 '''Convert I{Geographic} to I{Aunthalic} latitude.
145 @arg Phi: Geographic latitude (L{AuxAngle}).
146 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}
147 optional C{B{name}=NN}.
149 @return: Parametric latitude, C{Beta} (L{AuxAngle}).
150 '''
151 _xinstanceof(AuxAngle, Phi=Phi)
152 # assert Phi._AUX == Aux.PHI
153 tphi = fabs(Phi.tan)
154 if isfinite(tphi) and tphi and self.f:
155 y, x = Phi._yx_normalized
156 q = self._q
157 qv = self._qf(tphi)
158 Dq2 = self._Dq(tphi)
159 Dq2 *= (q + qv) / (fabs(y) + _1_0) # _Dq(-tphi)
160 d, n = _diff_name2(Phi, **diff_name)
161 Xi = AuxXi(copysign(qv, Phi.y), x * sqrt(Dq2), name=n)
163 if d:
164 if isnan(tphi):
165 d = self._e2m1_sq2
166 else:
167 c = self.Parametric(Phi)._x_normalized
168 d = _over(c, Xi._x_normalized)**3
169 d *= _over(c, x) * _over(_2_0, q)
170 Xi._diff = d
171 else:
172 Xi = AuxXi(*Phi._yx) # diff default
173 # assert Xi._AUX == Aux.XI
174 return Xi
176 def AuthalicRadius2(self, exact=False, f_max=_0_1):
177 '''Get the I{Authalic} radius I{squared}.
179 @kwarg exact: If C{True}, use the exact expression, otherwise
180 the I{Taylor} series.
181 @kwarg f_max: C{Flattening} not to exceed (C{float}).
183 @return: Authalic radius I{squared} (C{meter} I{squared}, same
184 units as the ellipsoid axes).
186 @raise AuxError: If C{B{exact}=False} and C{abs(flattening)}
187 exceeds C{f_max}.
188 '''
189 f = self.f
190 if exact or not f:
191 c2 = self.ellipsoid.b2 * self._q # == ellipsoid.c2x * 2
192 elif fabs(f) < f_max:
193 # Using a * (a + b) / 2 as the multiplying factor leads to a rapidly
194 # converging series in n. Of course, using this series isn't really
195 # necessary, since the exact expression is simple to evaluate. However,
196 # we do it for consistency with RectifyingRadius and, presumably, the
197 # roundoff error is smaller compared to that for the exact expression.
198 m = self.ALorder
199 c2 = _polynomial(self._n, _AR2Coeffs[m], 0, m)
200 c2 *= self.a * (self.a + self.b)
201 else:
202 raise AuxError(exact=exact, f=f, f_max=f_max)
203 return c2 * _0_5
205 @Property_RO
206 def b(self):
207 '''Get the C{ellipsoid}'s polar radius (C{meter}, conventionally).
208 '''
209 return self.ellipsoid.b # a * (_1_0 - f)
211 polaradius = b
213 def _coeffs(self, auxout, auxin):
214 # Get the polynomial coefficients as 4-, 6- or 8-tuple
215 aL = self.ALorder # aka Lmax
216 if auxout == auxin:
217 return _0_0s(aL) # uncached
219 k = Aux._1d(auxout, auxin)
220 try: # cached
221 return AuxLat._csc[aL][k]
222 except KeyError:
223 pass
225 Cx = _CXcoeffs(aL)
226 try:
227 Cx = Cx[auxout][auxin]
228 except KeyError as x:
229 raise AuxError(auxout=auxout, auxin=auxin, cause=x)
231 d = x = n = self._n
232 if Aux.use_n2(auxin) and Aux.use_n2(auxout):
233 x = self._n2
235 def _m(aL):
236 for m in _reverange(aL):
237 yield m // 2
238 else:
239 _m = _reverange # PYCHOK expected
241 i = 0
242 cs = []
243 _c = cs.append
244 _p = _polynomial
245 for m in _m(aL):
246 j = i + m + 1 # order m = j - i - 1
247 _c(_p(x, Cx, i, j) * d)
248 d *= n
249 i = j
250 # assert i == len(Cx) and len(cs) == aL
251 AuxLat._csc.setdefault(aL, {})[k] = cs = tuple(cs)
252 return cs
254 def Conformal(self, Phi, **diff_name):
255 '''Convert I{Geographic} to I{Conformal} latitude.
257 @arg Phi: Geographic latitude (L{AuxAngle}).
258 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}
259 and an optional C{B{name}=NN}.
261 @return: Conformal latitude, C{Chi} (L{AuxAngle}).
262 '''
263 _xinstanceof(AuxAngle, Phi=Phi)
264 # assert Phi._AUX == Aux.PHI
265 tphi = tchi = fabs(Phi.tan)
266 if isfinite(tphi) and tphi and self.f:
267 sig = sinh(self._atanhee(tphi) * self._e2)
268 scsig = _sc(sig)
269 scphi = _sc(tphi)
270 if self.f > 0:
271 # The general expression for tchi is
272 # tphi * scsig - sig * scphi
273 # This involves cancellation if f > 0, so change to
274 # (tphi - sig) * (tphi + sig) / (tphi * scsig + sig * scphi)
275 # To control overflow, write as (sigtphi = sig / tphi)
276 # (tphi - sig) * (1 + sigtphi) / (scsig + sigtphi * scphi)
277 sigtphi = sig / tphi
278 if sig < (tphi * _0_5):
279 t = tphi - sig
280 else:
281 def _asinh_2(x):
282 return asinh(x) * _0_5
283 # Still possibly dangerous cancellation in tphi - sig.
284 # Write tphi - sig = (1 - e) * Dg(1, e)
285 # Dg(x, y) = (g(x) - g(y)) / (x - y)
286 # g(x) = sinh(x * atanh(sphi * x))
287 # Note sinh(atanh(sphi)) = tphi
288 # Turn the crank on divided differences, substitute
289 # sphi = tphi / _sc(tphi)
290 # atanh(x) = asinh(x / sqrt(1 - x^2))
291 e = self._e
292 em1 = self._e2m1 / (_1_0 + e)
293 # assert em1 != 0
294 scb = self._scbeta(tphi)
295 scphib = scphi / scb # sec(phi) / sec(beta)
296 atphib = _asinh_2(tphi * e / scb) # atanh(e * sphi)
297 atphi = _asinh_2(tphi) # atanh(sphi)
298 t = _asinh_2(em1 * (tphi * scphib)) / em1
299 try:
300 Dg = _Fsumf_(atphi, atphib, t, e * t)
301 except ValueError: # Fsum(NAN) exception
302 Dg = _sum((atphi, atphib, t, e * t))
303 e *= atphib
304 t = atphi - e
305 if t: # sinh(0) == 0
306 Dg *= sinh(t) / t * cosh(atphi + e) * em1
307 t = float(Dg) # tphi - sig
308 tchi = _over(t * (_1_0 + sigtphi),
309 scsig + scphi * sigtphi) if t else _0_0
310 else:
311 tchi = tphi * scsig - sig * scphi
313 d, n = _diff_name2(Phi, **diff_name)
314 Chi = AuxChi(tchi, name=n).copyquadrant(Phi)
316 if d:
317 if isinf(tphi): # PYCHOK np cover
318 d = self._conformal_diff
319 else:
320 d = self.Parametric(Phi)._x_normalized
321 if d:
322 d = _over(d, Chi._x_normalized) * \
323 _over(d, Phi._x_normalized) * self._e2m1
324 Chi._diff = d
325 # assrt Chi._AUX == Aux.CHI
326 return Chi
328 @Property_RO
329 def _conformal_diff(self): # PYCHOK no cover
330 '''(INTERNAL) Constant I{Conformal} diff.
331 '''
332 e = self._e
333 if self.f > 0:
334 ss = sinh(asinh(self._e1) * e)
335 d = _over(_1_0, _sc(ss) + ss)
336 elif e:
337 ss = sinh(-atan1(e) * e)
338 d = _sc(ss) - ss
339 else:
340 d = _1_0
341 return d
343 def convert(self, auxout, Zeta_d, exact=False):
344 # Convert I{Auxiliary} or I{scalar} latitude
345 Z = d = Zeta_d
346 if isinstance(Z, AuxAngle):
347 A, auxin = _AuxClass(auxout), Z._AUX
348 if auxin == auxout:
349 pass
350 elif not (isfinite(Z.tan) and Z.tan): # XXX
351 Z = A(*Z._yx, aux=auxout, name=Z.name)
352 elif exact:
353 p = Aux.power(auxout, auxin)
354 if p is None:
355 P = self._fromAux(Z) # Phi
356 Z = self._toAux(auxout, P)
357 Z._iter = P.iteration
358 else:
359 y, x = Z._yx
360 if p:
361 y *= pow(self._fm1, p)
362 Z = A(y, x, aux=auxout, name=Z.name)
363 else:
364 cs = self._coeffs(auxout, auxin)
365 yx = Z._yx_normalized
366 Z = A(*yx, aux=auxout, name=Z.name)
367 # assert Z._yx == yx
368 r = _Clenshaw(True, Z, cs, self.ALorder)
369 Z += AuxAngle.fromRadians(r, aux=auxout)
370 # assert Z._AUX == auxout
371 return Z
373 elif _isDegrees(d):
374 Z = AuxPhi.fromDegrees(d)
375 d = round((d - Z.toDegrees) / _360_0) * _360_0
376 d += self.convert(auxout, Z, exact).toDegrees
377 return Degrees(d, name=Aux.Greek(auxout))
379 raise AuxError(auxout=auxout, Zeta_d=Zeta_d, exact=exact)
381 def _Dq(self, tphi):
382 # I{Divided Difference} of (q(1) - q(sphi)) / (1 - sphi).
383 sphi = _sn(tphi)
384 if tphi > 0:
385 scphi = _sc(tphi)
386 d = _1_0 / (scphi**2 * (_1_0 + sphi)) # XXX - sphi
387 if d:
388 # General expression for _Dq(1, sphi) is
389 # atanh(e * d / (1 - e2 * sphi)) / (e * d) +
390 # (1 + e2 * sphi) / ((1 - e2 * sphi * sphi) * e2m1)
391 # with atanh(e * d / (1 - e2 * sphi)) =
392 # atanh(e * d * scphi / (scphi - e2 * tphi))
393 e2m1, ed = self._e2m1, (self._e * d)
394 if e2m1 and ed:
395 e2 = self._e2
396 if e2 > 0: # assert self.f > 0
397 scb = self._scbeta(tphi)
398 q = scphib = scphi / scb
399 q *= (scphi + tphi * e2) / (e2m1 * scb)
400 q += asinh(self._e1 * d * scphib) / ed
401 else:
402 s2 = sphi * e2
403 q = (_1_0 + s2) / ((_1_0 - sphi * s2) * e2m1)
404 q += (atan2(ed, _1_0 - s2) / ed) if e2 < 0 else _1_0
405 else: # PYCHOK no cover
406 q = INF
407 else: # PYCHOK no cover
408 q = self._2_e2m12
409 else: # not reached, open-coded in .Authalic
410 q = _over(self._q - self._qf(tphi), _1_0 - sphi)
411 return q
413 @Property_RO
414 def _e(self): # unsigned, (1st) eccentricity
415 return self.ellipsoid.e # sqrt(fabs(self._e2))
417 @Property_RO
418 def _e1(self):
419 return sqrt(fabs(self._e12))
421 @Property_RO
422 def _e12(self):
423 return _over(self._e2, _1_0 - self._e2)
425 @Property_RO
426 def _e12p1(self):
427 return _1_0 / self._e2m1
429 @Property_RO
430 def _e2(self): # signed, (1st) eccentricity squared
431 return self.ellipsoid.e2
433 @Property_RO
434 def _e2m1(self): # 1 less 1st eccentricity squared
435 return self.ellipsoid.e21 # == ._fm1**2
437 @Property_RO
438 def _e2m1_sq2(self):
439 return self._e2m1 * sqrt(self._q * _0_5)
441 @Property_RO
442 def _2_e2m12(self):
443 return _2_0 / self._e2m1**2
445 @Property_RO
446 def _Ef_fRG_a2b2_PI_4(self):
447 E = self.ellipsoid
448 return _Ef.fRG(E.a2, E.b2) / PI_4
450 @Property_RO
451 def ellipsoid(self):
452 '''Get the ellipsoid (L{Ellipsoid}).
453 '''
454 return self._E
456 @Property_RO
457 def f(self):
458 '''Get the C{ellipsoid}'s flattening (C{scalar}).
459 '''
460 return self.ellipsoid.f
462 flattening = f
464 @Property_RO
465 def _fm1(self): # 1 - flattening
466 return self.ellipsoid.f1
468 def _fromAux(self, Zeta, **name):
469 '''Convert I{Auxiliary} to I{Geographic} latitude.
471 @arg Zeta: Auxiliary latitude (L{AuxAngle}).
472 @kwarg name: Optional C{B{name}=NN} (C{str}).
474 @return: Geographic latitude, I{Phi} (L{AuxAngle}).
475 '''
476 _xinstanceof(AuxAngle, Zeta=Zeta)
477 aux = Zeta._AUX
478 n = _name__(name, _or_nameof=Zeta)
479 f = self._fromAuxCase.get(aux, None)
480 if f is None:
481 Phi = AuxPhi(NAN, name=n)
482 elif callable(f):
483 t = self._fm1
484 t *= f(t)
485 Phi = _Newton(t, Zeta, self._toZeta(aux), name=n)
486 else: # assert isscalar(f)
487 y, x = Zeta._yx
488 Phi = AuxPhi(y / f, x, name=n)
489 # assert Phi._AUX == Aux.PHI
490 return Phi
492 @Property_RO
493 def _fromAuxCase(self):
494 '''(INTERNAL) switch(auxin): ...
495 '''
496 return {Aux.AUTHALIC: cbrt,
497 Aux.CONFORMAL: _passarg,
498 Aux.GEOCENTRIC: self._e2m1,
499 Aux.GEOGRAPHIC: _1_0,
500 Aux.PARAMETRIC: self._fm1,
501 Aux.RECTIFYING: sqrt}
503 def Geocentric(self, Phi, **diff_name):
504 '''Convert I{Geographic} to I{Geocentric} latitude.
506 @arg Phi: Geographic latitude (L{AuxAngle}).
507 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}
508 and an optional C{B{name}=NN}.
510 @return: Geocentric latitude, C{Phi} (L{AuxAngle}).
511 '''
512 _xinstanceof(AuxAngle, Phi=Phi)
513 # assert Phi._AUX == Aux.PHI
514 d, n = _diff_name2(Phi, **diff_name)
515 Theta = AuxTheta(Phi.y * self._e2m1, Phi.x, name=n)
516 if d:
517 Theta._diff = self._e2m1
518 return Theta
520 def Geodetic(self, Phi, **name): # PYCHOK no cover
521 '''Convert I{Geographic} to I{Geodetic} latitude.
523 @arg Phi: Geographic latitude (L{AuxAngle}).
524 @kwarg name: Optional C{B{name}=NN} (C{str}).
526 @return: Geodetic latitude, C{Phi} (L{AuxAngle}).
527 '''
528 _xinstanceof(AuxAngle, Phi=Phi)
529 # assert Phi._AUX == Aux.PHI
530 _, n = _diff_name2(Phi, **name)
531 return AuxPhi(Phi, name=n)
533 @Property_RO
534 def _n(self): # 3rd flattening
535 return self.ellipsoid.n
537 @Property_RO
538 def _n2(self):
539 return self._n**2
541 def Parametric(self, Phi, **diff_name):
542 '''Convert I{Geographic} to I{Parametric} latitude.
544 @arg Phi: Geographic latitude (L{AuxAngle}).
545 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}
546 and an optional C{B{name}=NN}.
548 @return: Parametric latitude, C{Beta} (L{AuxAngle}).
549 '''
550 _xinstanceof(AuxAngle, Phi=Phi)
551 # assert Phi._AUX == Aux.PHI
552 d, n = _diff_name2(Phi, **diff_name)
553 Beta = AuxBeta(Phi.y * self._fm1, Phi.x, name=n)
554 if d:
555 Beta._diff = self._fm1
556 return Beta
558 Reduced = Parametric
560 @Property_RO
561 def _q(self): # constant _q
562 q, f = self._e12p1, self.f
563 if f:
564 e = self._e
565 q += _over(asinh(self._e1) if f > 0 else atan1(e), e)
566 else:
567 q += _1_0
568 return q
570 def _qf(self, tphi):
571 # function _q: atanh(e * sphi) / e + sphi / (1 - (e * sphi)^2)
572 scb = self._scbeta(tphi)
573 return self._atanhee(tphi) + (tphi / scb) * (_sc(tphi) / scb)
575 def _qIntegrand(self, beta):
576 # pbeta(beta) = integrate(q(beta), beta), with beta in radians
577 # q(beta) = (1-f) * (sin(xi) - sin(chi)) / cos(phi)
578 # = (1-f) * (cos(chi) - cos(xi)) / cos(phi) *
579 # (cos(xi) + cos(chi)) / (sin(xi) + sin(chi))
580 # Fit q(beta)/cos(beta) with Fourier transform
581 # q(beta)/cos(beta) = sum(c[k] * sin((2*k+1)*beta), k, 0, K-1)
582 # then the integral is
583 # pbeta = sum(d[k] * cos((2*k+2)*beta), k, 0, K-1)
584 # where
585 # d[k] = -1/(4*(k+1)) * (c[k] + c[k+1]) for k in 0..K-2
586 # d[K-1] = -1/(4*K) * c[K-1]
587 Beta = AuxBeta.fromRadians(beta)
588 if Beta.x: # and self._fm1:
589 Ax, _cv = Aux, self.convert
590 Phi = _cv(Ax.PHI, Beta, exact=True)
591 schi, cchi = _cv(Ax.CHI, Phi, exact=True)._yx_normalized
592 sxi, cxi = _cv(Ax.XI, Phi, exact=True)._yx_normalized
593 r = (sxi - schi) if fabs(schi) < fabs(cchi) else \
594 _over(_2cos2x(cchi, cxi), (sxi + schi) * _2_0)
595 r *= _over(self._fm1, Phi._x_normalized * Beta._x_normalized)
596 else: # beta == PI_2, PI3_2, ...
597 r = _0_0 # XXX 0 avoids NAN summation exceptions
598 return r
600 def Rectifying(self, Phi, **diff_name):
601 '''Convert I{Geographic} to I{Rectifying} latitude.
603 @arg Phi: Geographic latitude (L{AuxAngle}).
604 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}
605 and an optional C{B{name}=NN}.
607 @return: Rectifying latitude, C{Mu} (L{AuxAngle}).
608 '''
609 Beta = self.Parametric(Phi)
610 # assert Beta._AUX == Aux.BETA
611 sb, cb = map(fabs, Beta._yx_normalized)
612 a, ka, ka1 = _1_0, self._e2, self._e2m1
613 b, kb, kb1 = self._fm1, -self._e12, self._e12p1
614 if self.f < 0:
615 a, b = b, a
616 ka, kb = kb, ka
617 ka1, kb1 = kb1, ka1
618 sb, cb = cb, sb
619 # now a, b = larger/smaller semiaxis
620 # Beta measured from larger semiaxis
621 # kb, ka = modulus-squared for distance from Beta = 0, pi/2
622 # NB kb <= 0; 0 <= ka <= 1
623 # sa = b*E(Beta, sqrt(kb))
624 # sb = a*E(Beta',sqrt(ka))
625 # 1 - ka * (1 - sb2) = 1 - ka + ka*sb2
626 sb2 = sb**2
627 cb2 = cb**2
628 da2 = ka1 + ka * sb2
629 db2 = _1_0 - kb * sb2
630 # DLMF Eq. 19.25.9
631 my = b * sb * _Ef._RFRD(cb2, db2, _1_0, kb * sb2)
632 # DLMF Eq. 19.25.10 with complementary angles
633 mx = a * cb * (_Ef.fRF(sb2, da2, _1_0) * ka1 +
634 ka * cb2 * _Ef.fRD(sb2, _1_0, da2, _3_0) * ka1 +
635 ka * sb / sqrt(da2))
636 # my + mx = 2*_Ef.fRG(a*a, b*b) = a*E(e) = b*E(i*e')
637 # mr = a*E(e)*(2/pi) = b*E(i*e')*(2/pi)
638 if self.f < 0:
639 a, b = b, a
640 my, mx = mx, my
641 mr = (my + mx) / PI_2
642 if mr:
643 my = sin(my / mr)
644 mx = sin(mx / mr) # XXX zero?
645 else: # zero Mu
646 my, mx = _0_0, _1_0
647 d, n = _diff_name2(Phi, **diff_name)
648 Mu = AuxMu(my, mx, # normalized
649 name=n).copyquadrant(Phi)
650 if d:
651 d, x = _0_0, Beta._x_normalized
652 if x and mr:
653 if Mu.x and Phi.x and not isinf(Phi.tan):
654 d = b / mr * (x / Mu.x)**2 \
655 * (x / Phi._x_normalized)
656 else:
657 d = mr / a
658 Mu._diff = self._fm1 * d
659 return Mu
661 def RectifyingRadius(self, exact=False):
662 '''Get the I{Rectifying} radius.
664 @arg exact: If C{True}, use the exact expression,
665 otherwise the I{Taylor} series.
667 @return: Rectifying radius (L{Meter}, same units
668 as the ellipsoid axes).
669 '''
670 r = self._Ef_fRG_a2b2_PI_4 if exact else self._RectifyingR
671 return Meter(r, name__=self.RectifyingRadius)
673 @Property_RO
674 def _RectifyingR(self):
675 m = self.ALorder
676 d = _polynomial(self._n2, _RRCoeffs[m], 0, m // 2)
677 return d * (self.a + self.b) * _0_5
679 def _scbeta(self, tphi):
680 return _sc(self._fm1 * tphi)
682 def _toAux(self, auxout, Phi, **diff_name):
683 '''Convert I{Geographic} to I{Auxiliary} latitude.
685 @arg auxout: I{Auxiliary} kind (C{Aux.KIND}).
686 @arg Phi: Geographic latitude (L{AuxLat}).
687 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}
688 and an optional C{B{name}=NN}.
690 @return: Auxiliary latitude, I{Eta} (L{AuxLat}).
691 '''
692 _xinstanceof(AuxAngle, Phi=Phi)
693 # assert Phi._AUX == Aux.PHI
694 d, n = _diff_name2(Phi, **diff_name)
695 m = _toAuxCase.get(auxout, None)
696 if m: # callable
697 A = m(self, Phi, diff=d, name=n)
698 elif auxout == Aux.GEODETIC: # == GEOGRAPHIC
699 A = AuxPhi(Phi, name=n)
700 else: # auxout?
701 A = AuxPhi(NAN, name=n)
702 # assert A._AUX == auxout
703 return A
705 def _toZeta(self, zetaux):
706 '''Return a (lean) function to create C{AuxPhi(tphi)} and
707 convert that into C{AuxAngle} of (fixed) kind C{zetaux}
708 for use only inside the C{_Newton} loop.
709 '''
710 class _AuxPhy(AuxPhi):
711 # lean C{AuxPhi} instance.
712 # _diff = _1_0
713 # _x = _1_0
715 def __init__(self, tphi): # PYCHOK signature
716 self._y = tphi
718 m = _toAuxCase.get(zetaux, None)
719 if m: # callable
721 def _toZeta(tphi):
722 return m(self, _AuxPhy(tphi), diff=True)
724 elif zetaux == Aux.GEODETIC: # GEOGRAPHIC
725 _toZeta = _AuxPhy
727 else: # zetaux?
729 def _toZeta(unused): # PYCHOK expected
730 return _AuxPhy(NAN)
732 return _toZeta
735# switch(auxout): ...
736_toAuxCase = {Aux.AUTHALIC: AuxLat.Authalic,
737 Aux.CONFORMAL: AuxLat.Conformal,
738 Aux.GEOCENTRIC: AuxLat.Geocentric,
739 Aux.PARAMETRIC: AuxLat.Parametric,
740 Aux.RECTIFYING: AuxLat.Rectifying}
743def _Clenshaw(sinp, Zeta, cs, K):
744 sz, cz = Zeta._yx # isnormal
745 # Evaluate sum(c[k] * sin((2*k+2) * Zeta)) if sinp else
746 # sum(c[k] * cos((2*k+2) * Zeta))
747 x = _2cos2x(cz, sz) # 2 * cos(2*Zeta)
748 if isfinite(x):
749 U0, U1 = Fsum(), Fsum()
750 # assert len(cs) == K
751 for r in _reverange(K):
752 U1 -= U0 * x + cs[r]
753 U1, U0 = U0, -U1
754 # u0*f0(Zeta) - u1*fm1(Zeta)
755 # f0 = sin(2*Zeta) if sinp else cos(2*Zeta)
756 # fm1 = 0 if sinp else 1
757 if sinp:
758 U0 *= sz * cz * _2_0
759 else:
760 U0 *= x * _0_5
761 U0 -= U1
762 x = float(U0)
763 return x
766def _CXcoeffs(aL): # PYCHOK in .auxilats.__main__
767 '''(INTERNAL) Get the C{CX_4}, C{_6} or C{_8} coefficients.
768 '''
769 try: # from pygeodesy.auxilats._CX_x import _coeffs_x as _coeffs
770 _CX_x = _DOT_(_MODS.auxilats.__name__, _UNDER_('_CX', aL))
771 _coeffs = _MODS.getattr(_CX_x, _UNDER_('_coeffs', aL))
772 except (AttributeError, ImportError, KeyError, TypeError) as x:
773 raise AuxError(ALorder=aL, cause=x)
774 # assert _coeffs.ALorder == aL
775 # assert _coeffs.n == Aux.len(aL)
776 return _coeffs
779def _diff_name2(Phi, diff=False, **name):
780 '''(INTERNAL) Get C{{Bdiff}=False} and C{B{name}=NN}.
781 '''
782 n = _name__(name, _or_nameof=Phi) # if name else Phi.name
783 return diff, n
786def _Newton(tphi, Zeta, _toZeta, **name):
787 # Newton's method from AuxLat._fromAux
788 try:
789 _lg2 = _log2
790 _abs = fabs
791 tz = _abs(Zeta.tan)
792 tphi = tz / tphi # **)
793 ltz = _lg2(tz) # **)
794 ltphi = _lg2(tphi) # **)
795 ltmin = min(ltphi, MIN_EXP)
796 ltmax = max(ltphi, MAX_EXP)
797# auxin = Zeta._AUX
798 s, n, __2 = 0, 3, _0_5 # n = i + 2
799 _TOL, _xp2 = _EPSqrt, _exp2
800 for i in range(1, _TRIPS): # up to 1 Ki!
801 # _toAux(auxin, AuxPhi(tphi), diff=True)
802 Z = _toZeta(tphi)
803 # assert Z._AUX == auxin
804 t, s_ = Z.tan, s
805 if t > tz:
806 ltmax, s = ltphi, +1
807 elif t < tz:
808 ltmin, s = ltphi, -1
809 else:
810 break
811 # derivative dtan(Z)/dtan(Phi)
812 # to dlog(tan(Z))/dlog(tan(Phi))
813 d = (ltz - _lg2(t)) * t # **)
814 if d:
815 d = d / (Z.diff * tphi) # **)
816 ltphi += d
817 tphi = _xp2(ltphi)
818 if _abs(d) < _TOL:
819 i += 1
820 # _toAux(auxin, AuxPhi(tphi), diff=True)
821 Z = _toZeta(tphi)
822 tphi -= _over(Z.tan - tz, Z.diff)
823 break
824 if (i > n and (s * s_) < 0) or not ltmin < ltphi < ltmax:
825 s, n = 0, (i + 2)
826 ltphi = (ltmin + ltmax) * __2
827 tphi = _xp2(ltphi)
828 else:
829 i = _TRIPS
830 Phi = AuxPhi(tphi, **name).copyquadrant(Zeta)
831 Phi._iter = i
832 except (ValueError, ZeroDivisionError): # **) zero t, tphi, tz or Z.diff
833 Phi = AuxPhi(Zeta, **name) # diff as-as
834 Phi._iter = 0
835 # assert Phi._AUX == Aux.PHI
836 return Phi
839_f, _u = float, _Ufloats()
840_1__f3 = -1 / _f(3) # XXX +1 / _f(3)
841_AR2Coeffs = {4: _u(4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3),
842 6: _u(4 / _f(1287), 4 / _f(693), 4 / _f(315), 4 / _f(105),
843 4 / _f(15), _1__f3),
844 8: _u(4 / _f(3315), 4 / _f(2145), 4 / _f(1287), 4 / _f(693),
845 4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3)}
846_RRCoeffs = {4: _u(1 / _f(64), _0_25),
847 6: _u(1 / _f(256), 1 / _f(64), _0_25),
848 8: _u(25 / _f(16384), 1 / _f(256), 1 / _f(64), _0_25)} # PYCHOK used!
849del _f, _u, _Ufloats, _1__f3
850# assert set(_AR2Coeffs.keys()) == set(_RRCoeffs.keys())
852# AuxLat._Lmax = max(_AR2Coeffs.keys()) # == max(ALorder)
854__all__ += _ALL_DOCS(AuxLat)
856# **) MIT License
857#
858# Copyright (C) 2023-2024 -- mrJean1 at Gmail -- All Rights Reserved.
859#
860# Permission is hereby granted, free of charge, to any person obtaining a
861# copy of this software and associated documentation files (the "Software"),
862# to deal in the Software without restriction, including without limitation
863# the rights to use, copy, modify, merge, publish, distribute, sublicense,
864# and/or sell copies of the Software, and to permit persons to whom the
865# Software is furnished to do so, subject to the following conditions:
866#
867# The above copyright notice and this permission notice shall be included
868# in all copies or substantial portions of the Software.
869#
870# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
871# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
872# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
873# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
874# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
875# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
876# OTHER DEALINGS IN THE SOFTWARE.