Coverage for pygeodesy/auxilats/auxLat.py: 93%
432 statements
« prev ^ index » next coverage.py v7.2.2, created at 2024-05-15 16:36 -0400
« prev ^ index » next coverage.py v7.2.2, created at 2024-05-15 16:36 -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
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
32# from pygeodesy.internals import _passarg # from .basics
33from pygeodesy.interns import NN, _DOT_, _UNDER_ # _earth_
34# from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS # from .karney
35from pygeodesy.props import Property, Property_RO, _update_all
36from pygeodesy.units import _isDegrees, _isRadius, Degrees, Meter
37# from pygeodesy.utily import atan1 # from .auxily
39from math import asinh, atan2, copysign, cosh, fabs, sin, sinh, sqrt
40try:
41 from math import exp2 as _exp2
42except ImportError: # Python 3.11-
44 def _exp2(x):
45 return pow(_2_0, x)
47__all__ = ()
48__version__ = '24.04.14'
50_TRIPS = 1024 # XXX 2 or 3?
53class AuxLat(AuxAngle):
54 '''Base class for accurate conversion between I{Auxiliary} latitudes
55 on an ellipsoid.
57 Latitudes are represented by L{AuxAngle} instances in order to
58 maintain precision near the poles, I{Authalic} latitude C{Xi},
59 I{Conformal} C{Chi}, I{Geocentric} C{Theta}, I{Geographic} C{Phi},
60 I{Parametric} C{Beta} and I{Rectifying} C{Mu}.
62 @see: I{Karney}'s C++ class U{AuxLatitude
63 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AuxLatitude.html>}.
64 '''
65 _csc = dict() # global coeffs cache: [aL][k], upto max(k) * (4 + 6 + 8) floats
66 _E = _WGS84.ellipsoid
67# _Lmax = 0 # overwritten below
68 _mAL = 6 # 4, 6 or 8 aka Lmax
70 def __init__(self, a_earth=_WGS84, f=None, b=None, name=NN, **ALorder):
71 '''New L{AuxLat} instance on an ellipsoid or datum.
73 @arg a_earth: Equatorial radius, semi-axis (C{meter}) or an
74 ellipsoid or datum (L{Datum}, L{Ellipsoid},
75 L{Ellipsoid2} or L{a_f2Tuple}).
76 @kwarg f: Flattening: M{(a - b) / a} (C{float}, near zero for
77 spherical), ignored if B{C{a_earth}} is not scalar.
78 @kwarg b: Optional polar radius, semi-axis (C{meter}, same
79 units as B{C{a_earth}}), ignored if B{C{a_earth}}
80 is not scalar.
81 @kwarg ALorder: Optional keyword arguments B{C{ALorder}} to
82 set L{AuxLat}'s C{order}, see property
83 C{ALorder}.
84 @kwarg name: Optional name (C{str}).
85 '''
86 if a_earth is not _WGS84:
87 n = name or AuxLat.__name__
88 try:
89 if b is f is None:
90 E = _ellipsoidal_datum(a_earth, name=n).ellipsoid # XXX raiser=_earth_
91 elif _isRadius(a_earth):
92 E = Ellipsoid(a_earth, f=f, b=b, name=_UNDER_(n))
93 else:
94 raise ValueError()
95 except Exception as x:
96 raise AuxError(a_earth=a_earth, f=f, b=b, cause=x)
97 self._E = E
99 if name:
100 self.name = name
101 if ALorder:
102 self.ALorder = _xkwds_get(ALorder, ALorder=AuxLat._mAL)
104 @Property_RO
105 def a(self):
106 '''Get the C{ellipsoid}'s equatorial radius (C{meter}, conventionally).
107 '''
108 return self.ellipsoid.a
110 equatoradius = a
112 @Property
113 def ALorder(self):
114 '''Get the I{AuxLat} order (C{int}, 4, 6 or 8).
115 '''
116 return self._mAL
118 @ALorder.setter # PYCHOK setter!
119 def ALorder(self, order):
120 '''Set the I{AuxLat} order (C{int}, 4, 6 or 8).
121 '''
122 m = _Xorder(_AR2Coeffs, AuxError, ALorder=order)
123 if self._mAL != m:
124 _update_all(self)
125 self._mAL = m
127 def _atanhee(self, tphi): # see Ellipsoid._es_atanh, .albers._atanhee
128 # atanh(e * sphi) = asinh(e' * sbeta)
129 f = self.f
130 s = _sn(self._fm1 * tphi) if f > 0 else _sn(tphi)
131 if f: # atanh(e * sphi) = asinh(e' * sbeta)
132 e = self._e
133 s = _over(atan1(e * s) if f < 0 else asinh(self._e1 * s), e)
134 return s
136 def Authalic(self, Phi, **diff_name):
137 '''Convert I{Geographic} to I{Aunthalic} latitude.
139 @arg Phi: Geographic latitude (L{AuxAngle}).
141 @return: Parametric latitude, C{Beta} (L{AuxAngle}).
142 '''
143 _xinstanceof(AuxAngle, Phi=Phi)
144 # assert Phi._AUX == Aux.PHI
145 tphi = fabs(Phi.tan)
146 if isfinite(tphi) and tphi and self.f:
147 y, x = Phi._yx_normalized
148 q = self._q
149 qv = self._qf(tphi)
150 Dq2 = self._Dq(tphi)
151 Dq2 *= (q + qv) / (fabs(y) + _1_0) # _Dq(-tphi)
152 Xi = AuxXi(copysign(qv, Phi.y), x * sqrt(Dq2),
153 name=_xkwds_get(diff_name, name=Phi.name))
155 if _xkwds_get(diff_name, diff=False):
156 if isnan(tphi):
157 d = self._e2m1_sq2
158 else:
159 c = self.Parametric(Phi)._x_normalized
160 d = _over(c, Xi._x_normalized)**3
161 d *= _over(c, x) * _over(_2_0, q)
162 Xi._diff = d
163 else:
164 Xi = AuxXi(*Phi._yx) # diff default
165 # assert Xi._AUX == Aux.XI
166 return Xi
168 def AuthalicRadius2(self, exact=False, f_max=_0_1):
169 '''Get the I{Authalic} radius I{squared}.
171 @kwarg exact: If C{True}, use the exact expression, otherwise
172 the I{Taylor} series.
173 @kwarg f_max: C{Flattening} not to exceed (C{float}).
175 @return: Authalic radius I{squared} (C{meter} I{squared}, same
176 units as the ellipsoid axes).
178 @raise AuxError: If C{B{exact}=False} and C{abs(flattening)}
179 exceeds C{f_max}.
180 '''
181 f = self.f
182 if exact or not f:
183 c2 = self.ellipsoid.b2 * self._q # == ellipsoid.c2x * 2
184 elif fabs(f) < f_max:
185 # Using a * (a + b) / 2 as the multiplying factor leads to a rapidly
186 # converging series in n. Of course, using this series isn't really
187 # necessary, since the exact expression is simple to evaluate. However,
188 # we do it for consistency with RectifyingRadius and, presumably, the
189 # roundoff error is smaller compared to that for the exact expression.
190 m = self.ALorder
191 c2 = _polynomial(self._n, _AR2Coeffs[m], 0, m)
192 c2 *= self.a * (self.a + self.b)
193 else:
194 raise AuxError(exact=exact, f=f, f_max=f_max)
195 return c2 * _0_5
197 @Property_RO
198 def b(self):
199 '''Get the C{ellipsoid}'s polar radius (C{meter}, conventionally).
200 '''
201 return self.ellipsoid.b # a * (_1_0 - f)
203 polaradius = b
205 def _coeffs(self, auxout, auxin):
206 # Get the polynomial coefficients as 4-, 6- or 8-tuple
207 aL = self.ALorder # aka Lmax
208 if auxout == auxin:
209 return _0_0s(aL) # uncached
211 k = Aux._1d(auxout, auxin)
212 try: # cached
213 return AuxLat._csc[aL][k]
214 except KeyError:
215 pass
217 Cx = _CXcoeffs(aL)
218 try:
219 Cx = Cx[auxout][auxin]
220 except KeyError as x:
221 raise AuxError(auxout=auxout, auxin=auxin, cause=x)
223 d = x = n = self._n
224 if Aux.use_n2(auxin) and Aux.use_n2(auxout):
225 x = self._n2
227 def _m(aL):
228 for m in _reverange(aL):
229 yield m // 2
230 else:
231 _m = _reverange # PYCHOK expected
233 i = 0
234 cs = []
235 _c = cs.append
236 _p = _polynomial
237 for m in _m(aL):
238 j = i + m + 1 # order m = j - i - 1
239 _c(_p(x, Cx, i, j) * d)
240 d *= n
241 i = j
242 # assert i == len(Cx) and len(cs) == aL
243 AuxLat._csc.setdefault(aL, {})[k] = cs = tuple(cs)
244 return cs
246 def Conformal(self, Phi, **diff_name):
247 '''Convert I{Geographic} to I{Conformal} latitude.
249 @arg Phi: Geographic latitude (L{AuxAngle}).
251 @return: Conformal latitude, C{Chi} (L{AuxAngle}).
252 '''
253 _xinstanceof(AuxAngle, Phi=Phi)
254 # assert Phi._AUX == Aux.PHI
255 tphi = tchi = fabs(Phi.tan)
256 if isfinite(tphi) and tphi and self.f:
257 sig = sinh(self._atanhee(tphi) * self._e2)
258 scsig = _sc(sig)
259 scphi = _sc(tphi)
260 if self.f > 0:
261 # The general expression for tchi is
262 # tphi * scsig - sig * scphi
263 # This involves cancellation if f > 0, so change to
264 # (tphi - sig) * (tphi + sig) / (tphi * scsig + sig * scphi)
265 # To control overflow, write as (sigtphi = sig / tphi)
266 # (tphi - sig) * (1 + sigtphi) / (scsig + sigtphi * scphi)
267 sigtphi = sig / tphi
268 if sig < (tphi * _0_5):
269 t = tphi - sig
270 else:
271 def _asinh_2(x):
272 return asinh(x) * _0_5
273 # Still possibly dangerous cancellation in tphi - sig.
274 # Write tphi - sig = (1 - e) * Dg(1, e)
275 # Dg(x, y) = (g(x) - g(y)) / (x - y)
276 # g(x) = sinh(x * atanh(sphi * x))
277 # Note sinh(atanh(sphi)) = tphi
278 # Turn the crank on divided differences, substitute
279 # sphi = tphi / _sc(tphi)
280 # atanh(x) = asinh(x / sqrt(1 - x^2))
281 e = self._e
282 em1 = self._e2m1 / (_1_0 + e)
283 # assert em1 != 0
284 scb = self._scbeta(tphi)
285 scphib = scphi / scb # sec(phi) / sec(beta)
286 atphib = _asinh_2(tphi * e / scb) # atanh(e * sphi)
287 atphi = _asinh_2(tphi) # atanh(sphi)
288 t = _asinh_2(em1 * (tphi * scphib)) / em1
289 try:
290 Dg = _Fsumf_(atphi, atphib, t, e * t)
291 except ValueError: # Fsum(NAN) exception
292 Dg = _sum((atphi, atphib, t, e * t))
293 e *= atphib
294 t = atphi - e
295 if t: # sinh(0) == 0
296 Dg *= sinh(t) / t * cosh(atphi + e) * em1
297 t = float(Dg) # tphi - sig
298 tchi = _over(t * (_1_0 + sigtphi),
299 scsig + scphi * sigtphi) if t else _0_0
300 else:
301 tchi = tphi * scsig - sig * scphi
303 n = _xkwds_get(diff_name, name=Phi.name)
304 Chi = AuxChi(tchi, name=n).copyquadrant(Phi)
306 if _xkwds_get(diff_name, diff=False):
307 if isinf(tphi): # PYCHOK np cover
308 d = self._conformal_diff
309 else:
310 d = self.Parametric(Phi)._x_normalized
311 if d:
312 d = _over(d, Chi._x_normalized) * \
313 _over(d, Phi._x_normalized) * self._e2m1
314 Chi._diff = d
315 # assrt Chi._AUX == Aux.CHI
316 return Chi
318 @Property_RO
319 def _conformal_diff(self): # PYCHOK no cover
320 '''(INTERNAL) Constant I{Conformal} diff.
321 '''
322 e = self._e
323 if self.f > 0:
324 ss = sinh(asinh(self._e1) * e)
325 d = _over(_1_0, _sc(ss) + ss)
326 elif e:
327 ss = sinh(-atan1(e) * e)
328 d = _sc(ss) - ss
329 else:
330 d = _1_0
331 return d
333 def convert(self, auxout, Zeta_d, exact=False):
334 # Convert I{Auxiliary} or I{scalar} latitude
335 Z = d = Zeta_d
336 if isinstance(Z, AuxAngle):
337 A, auxin = _AuxClass(auxout), Z._AUX
338 if auxin == auxout:
339 pass
340 elif not (isfinite(Z.tan) and Z.tan): # XXX
341 Z = A(*Z._yx, aux=auxout, name=Z.name)
342 elif exact:
343 p = Aux.power(auxout, auxin)
344 if p is None:
345 P = self._fromAux(Z) # Phi
346 Z = self._toAux(auxout, P)
347 Z._iter = P.iteration
348 else:
349 y, x = Z._yx
350 if p:
351 y *= pow(self._fm1, p)
352 Z = A(y, x, aux=auxout, name=Z.name)
353 else:
354 cs = self._coeffs(auxout, auxin)
355 yx = Z._yx_normalized
356 Z = A(*yx, aux=auxout, name=Z.name)
357 # assert Z._yx == yx
358 r = _Clenshaw(True, Z, cs, self.ALorder)
359 Z += AuxAngle.fromRadians(r, aux=auxout)
360 # assert Z._AUX == auxout
361 return Z
363 elif _isDegrees(d):
364 Z = AuxPhi.fromDegrees(d)
365 d = round((d - Z.toDegrees) / _360_0) * _360_0
366 d += self.convert(auxout, Z, exact).toDegrees
367 return Degrees(d, name=Aux.Greek(auxout))
369 raise AuxError(auxout=auxout, Zeta_d=Zeta_d, exact=exact)
371 def _Dq(self, tphi):
372 # I{Divided Difference} of (q(1) - q(sphi)) / (1 - sphi).
373 sphi = _sn(tphi)
374 if tphi > 0:
375 scphi = _sc(tphi)
376 d = _1_0 / (scphi**2 * (_1_0 + sphi)) # XXX - sphi
377 if d:
378 # General expression for _Dq(1, sphi) is
379 # atanh(e * d / (1 - e2 * sphi)) / (e * d) +
380 # (1 + e2 * sphi) / ((1 - e2 * sphi * sphi) * e2m1)
381 # with atanh(e * d / (1 - e2 * sphi)) =
382 # atanh(e * d * scphi / (scphi - e2 * tphi))
383 e2m1, ed = self._e2m1, (self._e * d)
384 if e2m1 and ed:
385 e2 = self._e2
386 if e2 > 0: # assert self.f > 0
387 scb = self._scbeta(tphi)
388 q = scphib = scphi / scb
389 q *= (scphi + tphi * e2) / (e2m1 * scb)
390 q += asinh(self._e1 * d * scphib) / ed
391 else:
392 s2 = sphi * e2
393 q = (_1_0 + s2) / ((_1_0 - sphi * s2) * e2m1)
394 q += (atan2(ed, _1_0 - s2) / ed) if e2 < 0 else _1_0
395 else: # PYCHOK no cover
396 q = INF
397 else: # PYCHOK no cover
398 q = self._2_e2m12
399 else: # not reached, open-coded in .Authalic
400 q = _over(self._q - self._qf(tphi), _1_0 - sphi)
401 return q
403 @Property_RO
404 def _e(self): # unsigned, (1st) eccentricity
405 return self.ellipsoid.e # sqrt(fabs(self._e2))
407 @Property_RO
408 def _e1(self):
409 return sqrt(fabs(self._e12))
411 @Property_RO
412 def _e12(self):
413 return _over(self._e2, _1_0 - self._e2)
415 @Property_RO
416 def _e12p1(self):
417 return _1_0 / self._e2m1
419 @Property_RO
420 def _e2(self): # signed, (1st) eccentricity squared
421 return self.ellipsoid.e2
423 @Property_RO
424 def _e2m1(self): # 1 less 1st eccentricity squared
425 return self.ellipsoid.e21 # == ._fm1**2
427 @Property_RO
428 def _e2m1_sq2(self):
429 return self._e2m1 * sqrt(self._q * _0_5)
431 @Property_RO
432 def _2_e2m12(self):
433 return _2_0 / self._e2m1**2
435 @Property_RO
436 def _Ef_fRG_a2b2_PI_4(self):
437 E = self.ellipsoid
438 return _Ef.fRG(E.a2, E.b2) / PI_4
440 @Property_RO
441 def ellipsoid(self):
442 '''Get the ellipsoid (L{Ellipsoid}).
443 '''
444 return self._E
446 @Property_RO
447 def f(self):
448 '''Get the C{ellipsoid}'s flattening (C{scalar}).
449 '''
450 return self.ellipsoid.f
452 flattening = f
454 @Property_RO
455 def _fm1(self): # 1 - flattening
456 return self.ellipsoid.f1
458 def _fromAux(self, Zeta, **name):
459 '''Convert I{Auxiliary} to I{Geographic} latitude.
461 @arg Zeta: Auxiliary latitude (L{AuxAngle}).
463 @return: Geographic latitude, I{Phi} (L{AuxAngle}).
464 '''
465 _xinstanceof(AuxAngle, Zeta=Zeta)
466 aux = Zeta._AUX
467 n = _xkwds_get(name, name=Zeta.name)
468 f = self._fromAuxCase.get(aux, None)
469 if f is None:
470 Phi = AuxPhi(NAN, name=n)
471 elif callable(f):
472 t = self._fm1
473 t *= f(t)
474 Phi = _Newton(t, Zeta, self._toZeta(aux), name=n)
475 else: # assert isscalar(f)
476 y, x = Zeta._yx
477 Phi = AuxPhi(y / f, x, name=n)
478 # assert Phi._AUX == Aux.PHI
479 return Phi
481 @Property_RO
482 def _fromAuxCase(self):
483 '''(INTERNAL) switch(auxin): ...
484 '''
485 return {Aux.AUTHALIC: cbrt,
486 Aux.CONFORMAL: _passarg,
487 Aux.GEOCENTRIC: self._e2m1,
488 Aux.GEOGRAPHIC: _1_0,
489 Aux.PARAMETRIC: self._fm1,
490 Aux.RECTIFYING: sqrt}
492 def Geocentric(self, Phi, **diff_name):
493 '''Convert I{Geographic} to I{Geocentric} latitude.
495 @arg Phi: Geographic latitude (L{AuxAngle}).
497 @return: Geocentric latitude, C{Phi} (L{AuxAngle}).
498 '''
499 _xinstanceof(AuxAngle, Phi=Phi)
500 # assert Phi._AUX == Aux.PHI
501 Theta = AuxTheta(Phi.y * self._e2m1, Phi.x,
502 name=_xkwds_get(diff_name, name=Phi.name))
503 if _xkwds_get(diff_name, diff=False):
504 Theta._diff = self._e2m1
505 return Theta
507 def Geodetic(self, Phi, **diff_name): # PYCHOK no cover
508 '''Convert I{Geographic} to I{Geodetic} latitude.
510 @arg Phi: Geographic latitude (L{AuxAngle}).
512 @return: Geodetic latitude, C{Phi} (L{AuxAngle}).
513 '''
514 _xinstanceof(AuxAngle, Phi=Phi)
515 # assert Phi._AUX == Aux.PHI
516 return AuxPhi(Phi, name=_xkwds_get(diff_name, name=Phi.name))
518 @Property_RO
519 def _n(self): # 3rd flattening
520 return self.ellipsoid.n
522 @Property_RO
523 def _n2(self):
524 return self._n**2
526 def Parametric(self, Phi, **diff_name):
527 '''Convert I{Geographic} to I{Parametric} latitude.
529 @arg Phi: Geographic latitude (L{AuxAngle}).
531 @return: Parametric latitude, C{Beta} (L{AuxAngle}).
532 '''
533 _xinstanceof(AuxAngle, Phi=Phi)
534 # assert Phi._AUX == Aux.PHI
535 Beta = AuxBeta(Phi.y * self._fm1, Phi.x,
536 name=_xkwds_get(diff_name, name=Phi.name))
537 if _xkwds_get(diff_name, diff=False):
538 Beta._diff = self._fm1
539 return Beta
541 Reduced = Parametric
543 @Property_RO
544 def _q(self): # constant _q
545 q, f = self._e12p1, self.f
546 if f:
547 e = self._e
548 q += _over(asinh(self._e1) if f > 0 else atan1(e), e)
549 else:
550 q += _1_0
551 return q
553 def _qf(self, tphi):
554 # function _q: atanh(e * sphi) / e + sphi / (1 - (e * sphi)^2)
555 scb = self._scbeta(tphi)
556 return self._atanhee(tphi) + (tphi / scb) * (_sc(tphi) / scb)
558 def _qIntegrand(self, beta):
559 # pbeta(beta) = integrate(q(beta), beta), with beta in radians
560 # q(beta) = (1-f) * (sin(xi) - sin(chi)) / cos(phi)
561 # = (1-f) * (cos(chi) - cos(xi)) / cos(phi) *
562 # (cos(xi) + cos(chi)) / (sin(xi) + sin(chi))
563 # Fit q(beta)/cos(beta) with Fourier transform
564 # q(beta)/cos(beta) = sum(c[k] * sin((2*k+1)*beta), k, 0, K-1)
565 # then the integral is
566 # pbeta = sum(d[k] * cos((2*k+2)*beta), k, 0, K-1)
567 # where
568 # d[k] = -1/(4*(k+1)) * (c[k] + c[k+1]) for k in 0..K-2
569 # d[K-1] = -1/(4*K) * c[K-1]
570 Beta = AuxBeta.fromRadians(beta)
571 if Beta.x: # and self._fm1:
572 Ax, _cv = Aux, self.convert
573 Phi = _cv(Ax.PHI, Beta, exact=True)
574 schi, cchi = _cv(Ax.CHI, Phi, exact=True)._yx_normalized
575 sxi, cxi = _cv(Ax.XI, Phi, exact=True)._yx_normalized
576 r = (sxi - schi) if fabs(schi) < fabs(cchi) else \
577 _over(_2cos2x(cchi, cxi), (sxi + schi) * _2_0)
578 r *= _over(self._fm1, Phi._x_normalized * Beta._x_normalized)
579 else: # beta == PI_2, PI3_2, ...
580 r = _0_0 # XXX 0 avoids NAN summation exceptions
581 return r
583 def Rectifying(self, Phi, **diff_name):
584 '''Convert I{Geographic} to I{Rectifying} latitude.
586 @arg Phi: Geographic latitude (L{AuxAngle}).
588 @return: Rectifying latitude, C{Mu} (L{AuxAngle}).
589 '''
590 Beta = self.Parametric(Phi)
591 # assert Beta._AUX == Aux.BETA
592 sb, cb = map(fabs, Beta._yx_normalized)
593 a, ka, ka1 = _1_0, self._e2, self._e2m1
594 b, kb, kb1 = self._fm1, -self._e12, self._e12p1
595 if self.f < 0:
596 a, b = b, a
597 ka, kb = kb, ka
598 ka1, kb1 = kb1, ka1
599 sb, cb = cb, sb
600 # now a, b = larger/smaller semiaxis
601 # Beta measured from larger semiaxis
602 # kb, ka = modulus-squared for distance from Beta = 0, pi/2
603 # NB kb <= 0; 0 <= ka <= 1
604 # sa = b*E(Beta, sqrt(kb))
605 # sb = a*E(Beta',sqrt(ka))
606 # 1 - ka * (1 - sb2) = 1 - ka + ka*sb2
607 sb2 = sb**2
608 cb2 = cb**2
609 da2 = ka1 + ka * sb2
610 db2 = _1_0 - kb * sb2
611 # DLMF Eq. 19.25.9
612 my = b * sb * _Ef._RFRD(cb2, db2, _1_0, kb * sb2)
613 # DLMF Eq. 19.25.10 with complementary angles
614 mx = a * cb * (_Ef.fRF(sb2, da2, _1_0) * ka1 +
615 ka * cb2 * _Ef.fRD(sb2, _1_0, da2, _3_0) * ka1 +
616 ka * sb / sqrt(da2))
617 # my + mx = 2*_Ef.fRG(a*a, b*b) = a*E(e) = b*E(i*e')
618 # mr = a*E(e)*(2/pi) = b*E(i*e')*(2/pi)
619 if self.f < 0:
620 a, b = b, a
621 my, mx = mx, my
622 mr = (my + mx) / PI_2
623 if mr:
624 my = sin(my / mr)
625 mx = sin(mx / mr) # XXX zero?
626 else: # zero Mu
627 my, mx = _0_0, _1_0
628 n = _xkwds_get(diff_name, name=Phi.name)
629 Mu = AuxMu(my, mx, # normalized
630 name=n).copyquadrant(Phi)
632 if _xkwds_get(diff_name, diff=False):
633 d, x = _0_0, Beta._x_normalized
634 if x and mr:
635 if Mu.x and Phi.x and not isinf(Phi.tan):
636 d = b / mr * (x / Mu.x)**2 \
637 * (x / Phi._x_normalized)
638 else:
639 d = mr / a
640 Mu._diff = self._fm1 * d
641 return Mu
643 def RectifyingRadius(self, exact=False):
644 '''Get the I{Rectifying} radius.
646 @arg exact: If C{True}, use the exact expression,
647 otherwise the I{Taylor} series.
649 @return: Rectifying radius (L{Meter}, same units
650 as the ellipsoid axes).
651 '''
652 r = self._Ef_fRG_a2b2_PI_4 if exact else self._RectifyingR
653 return Meter(r, name=self.RectifyingRadius.__name__)
655 @Property_RO
656 def _RectifyingR(self):
657 m = self.ALorder
658 d = _polynomial(self._n2, _RRCoeffs[m], 0, m // 2)
659 return d * (self.a + self.b) * _0_5
661 def _scbeta(self, tphi):
662 return _sc(self._fm1 * tphi)
664 def _toAux(self, auxout, Phi, **diff_name):
665 '''Convert I{Geographic} to I{Auxiliary} latitude.
667 @arg auxout: I{Auxiliary} kind (C{Aux.KIND}).
668 @arg Phi: Geographic latitude (L{AuxLat}).
670 @return: Auxiliary latitude, I{Eta} (L{AuxLat}).
671 '''
672 _xinstanceof(AuxAngle, Phi=Phi)
673 # assert Phi._AUX == Aux.PHI
674 n = _xkwds_get(diff_name, name=Phi.name)
675 m = _toAuxCase.get(auxout, None)
676 if m: # callable
677 A = m(self, Phi, **_xkwds(diff_name, name=n))
678 elif auxout == Aux.GEODETIC: # == GEOGRAPHIC
679 A = AuxPhi(Phi, name=n)
680 else: # auxout?
681 A = AuxPhi(NAN, name=n)
682 # assert A._AUX == auxout
683 return A
685 def _toZeta(self, zetaux):
686 '''Return a (lean) function to create C{AuxPhi(tphi)} and
687 convert that into C{AuxAngle} of (fixed) kind C{zetaux}
688 for use only inside the C{_Newton} loop.
689 '''
690 class _AuxPhy(AuxPhi):
691 # lean C{AuxPhi} instance.
692 # _diff = _1_0
693 # _x = _1_0
695 def __init__(self, tphi): # PYCHOK signature
696 self._y = tphi
698 m = _toAuxCase.get(zetaux, None)
699 if m: # callable
701 def _toZeta(tphi):
702 return m(self, _AuxPhy(tphi), diff=True)
704 elif zetaux == Aux.GEODETIC: # GEOGRAPHIC
705 _toZeta = _AuxPhy
707 else: # zetaux?
709 def _toZeta(unused): # PYCHOK expected
710 return _AuxPhy(NAN)
712 return _toZeta
715# switch(auxout): ...
716_toAuxCase = {Aux.AUTHALIC: AuxLat.Authalic,
717 Aux.CONFORMAL: AuxLat.Conformal,
718 Aux.GEOCENTRIC: AuxLat.Geocentric,
719 Aux.PARAMETRIC: AuxLat.Parametric,
720 Aux.RECTIFYING: AuxLat.Rectifying}
723def _Clenshaw(sinp, Zeta, cs, K):
724 sz, cz = Zeta._yx # isnormal
725 # Evaluate sum(c[k] * sin((2*k+2) * Zeta)) if sinp else
726 # sum(c[k] * cos((2*k+2) * Zeta))
727 x = _2cos2x(cz, sz) # 2 * cos(2*Zeta)
728 if isfinite(x):
729 U0, U1 = Fsum(), Fsum()
730 # assert len(cs) == K
731 for r in _reverange(K):
732 U1 -= U0 * x + cs[r]
733 U1, U0 = U0, -U1
734 # u0*f0(Zeta) - u1*fm1(Zeta)
735 # f0 = sin(2*Zeta) if sinp else cos(2*Zeta)
736 # fm1 = 0 if sinp else 1
737 if sinp:
738 U0 *= sz * cz * _2_0
739 else:
740 U0 *= x * _0_5
741 U0 -= U1
742 x = float(U0)
743 return x
746def _CXcoeffs(aL): # PYCHOK in .auxilats.__main__
747 '''(INTERNAL) Get the C{CX_4}, C{_6} or C{_8} coefficients.
748 '''
749 try: # from pygeodesy.auxilats._CX_x import _coeffs_x as _coeffs
750 _CX_x = _DOT_(_MODS.auxilats.__name__, _UNDER_('_CX', aL))
751 _coeffs = _MODS.getattr(_CX_x, _UNDER_('_coeffs', aL))
752 except (AttributeError, ImportError, KeyError, TypeError) as x:
753 raise AuxError(ALorder=aL, cause=x)
754 # assert _coeffs.ALorder == aL
755 # assert _coeffs.n == Aux.len(aL)
756 return _coeffs
759def _Newton(tphi, Zeta, _toZeta, **name):
760 # Newton's method fro AuxLat._fromAux
761 try:
762 _lg2 = _log2
763 _abs = fabs
764 tz = _abs(Zeta.tan)
765 tphi = tz / tphi # **)
766 ltz = _lg2(tz) # **)
767 ltphi = _lg2(tphi) # **)
768 ltmin = min(ltphi, MIN_EXP)
769 ltmax = max(ltphi, MAX_EXP)
770# auxin = Zeta._AUX
771 s, n, __2 = 0, 3, _0_5 # n = i + 2
772 _TOL, _xp2 = _EPSqrt, _exp2
773 for i in range(1, _TRIPS): # up to 1 Ki!
774 # _toAux(auxin, AuxPhi(tphi), diff=True)
775 Z = _toZeta(tphi)
776 # assert Z._AUX == auxin
777 t, s_ = Z.tan, s
778 if t > tz:
779 ltmax, s = ltphi, +1
780 elif t < tz:
781 ltmin, s = ltphi, -1
782 else:
783 break
784 # derivative dtan(Z)/dtan(Phi)
785 # to dlog(tan(Z))/dlog(tan(Phi))
786 d = (ltz - _lg2(t)) * t # **)
787 if d:
788 d = d / (Z.diff * tphi) # **)
789 ltphi += d
790 tphi = _xp2(ltphi)
791 if _abs(d) < _TOL:
792 i += 1
793 # _toAux(auxin, AuxPhi(tphi), diff=True)
794 Z = _toZeta(tphi)
795 tphi -= _over(Z.tan - tz, Z.diff)
796 break
797 if (i > n and (s * s_) < 0) or not ltmin < ltphi < ltmax:
798 s, n = 0, (i + 2)
799 ltphi = (ltmin + ltmax) * __2
800 tphi = _xp2(ltphi)
801 else:
802 i = _TRIPS
803 Phi = AuxPhi(tphi, **name).copyquadrant(Zeta)
804 Phi._iter = i
805 except (ValueError, ZeroDivisionError): # **) zero t, tphi, tz or Z.diff
806 Phi = AuxPhi(Zeta, **name) # diff as-as
807 Phi._iter = 0
808 # assert Phi._AUX == Aux.PHI
809 return Phi
812_f, _u = float, _Ufloats()
813_1__f3 = -1 / _f(3) # XXX +1 / _f(3)
814_AR2Coeffs = {4: _u(4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3),
815 6: _u(4 / _f(1287), 4 / _f(693), 4 / _f(315), 4 / _f(105),
816 4 / _f(15), _1__f3),
817 8: _u(4 / _f(3315), 4 / _f(2145), 4 / _f(1287), 4 / _f(693),
818 4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3)}
819_RRCoeffs = {4: _u(1 / _f(64), _0_25),
820 6: _u(1 / _f(256), 1 / _f(64), _0_25),
821 8: _u(25 / _f(16384), 1 / _f(256), 1 / _f(64), _0_25)} # PYCHOK used!
822del _f, _u, _Ufloats, _1__f3
823# assert set(_AR2Coeffs.keys()) == set(_RRCoeffs.keys())
825# AuxLat._Lmax = max(_AR2Coeffs.keys()) # == max(ALorder)
827__all__ += _ALL_DOCS(AuxLat)
829# **) MIT License
830#
831# Copyright (C) 2023-2024 -- mrJean1 at Gmail -- All Rights Reserved.
832#
833# Permission is hereby granted, free of charge, to any person obtaining a
834# copy of this software and associated documentation files (the "Software"),
835# to deal in the Software without restriction, including without limitation
836# the rights to use, copy, modify, merge, publish, distribute, sublicense,
837# and/or sell copies of the Software, and to permit persons to whom the
838# Software is furnished to do so, subject to the following conditions:
839#
840# The above copyright notice and this permission notice shall be included
841# in all copies or substantial portions of the Software.
842#
843# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
844# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
845# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
846# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
847# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
848# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
849# OTHER DEALINGS IN THE SOFTWARE.