Coverage for pygeodesy/auxilats/auxLat.py: 93%
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« prev ^ index » next coverage.py v7.2.2, created at 2023-10-04 14:05 -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 isscalar, _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, _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 Degrees, Meter
36# from pygeodesy.utily import atan1, _passarg # from .auxily, _MODS
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__ = '23.09.26'
49_TRIPS = 1024 # XXX 2 or 3?
52class AuxLat(AuxAngle):
53 '''Accurate conversion between I{Auxiliary} latitudes on an ellipsoid.
55 Latitudes are represented by instances of L{AuxAngle} in order to
56 maintain precision close to the poles, I{Authalic} latitude C{Xi},
57 I{Conformal} C{Chi}, I{Geocentric} C{Theta}, I{Geographic} C{Phi},
58 I{Parametric} C{Beta} and I{Rectifying} C{Mu}.
60 @see: I{Karney}'s C++ class U{AuxLatitude
61 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AuxLatitude.html>}.
62 '''
63 _csc = dict() # global coeffs cache: [aL][k], upto max(k) * (4 + 6 + 8) floats
64 _E = _WGS84.ellipsoid
65# _Lmax = 0 # overwritten below
66 _mAL = 6 # 4, 6 or 8 aka Lmax
68 def __init__(self, a_earth=_WGS84, f=None, b=None, name=NN, **ALorder):
69 '''New L{AuxLat} instance on an ellipsoid or datum.
71 @arg a_earth: Equatorial radius, semi-axis (C{meter}) or an
72 ellipsoid or datum (L{Datum}, L{Ellipsoid},
73 L{Ellipsoid2} or L{a_f2Tuple}).
74 @kwarg f: Flattening: M{(a - b) / a} (C{float}, near zero for
75 spherical), ignored if B{C{a_earth}} is not scalar.
76 @kwarg b: Optional polar radius, semi-axis (C{meter}, same
77 units as B{C{a_earth}}), ignored if B{C{a_earth}}
78 is not scalar.
79 @kwarg ALorder: Optional keyword arguments B{C{ALorder}} to
80 set L{AuxLat}'s C{order}, see property
81 C{ALorder}.
82 @kwarg name: Optional name (C{str}).
83 '''
84 if a_earth is not _WGS84:
85 n = name or AuxLat.__name__
86 try:
87 if b is f is None:
88 E = _ellipsoidal_datum(a_earth, name=n).ellipsoid # XXX raiser=_earth_
89 elif isscalar(a_earth):
90 E = Ellipsoid(a_earth, f=f, b=b, name=_UNDER_(n))
91 else:
92 raise ValueError()
93 except Exception as x:
94 raise AuxError(a_earth=a_earth, f=f, b=b, cause=x)
95 self._E = E
97 if name:
98 self.name = name
99 if ALorder:
100 self.ALorder = _xkwds_get(ALorder, ALorder=AuxLat._mAL)
102 @Property_RO
103 def a(self):
104 '''Get the C{ellipsoid}'s equatorial radius (C{meter}, conventionally).
105 '''
106 return self.ellipsoid.a
108 equatoradius = a
110 @Property
111 def ALorder(self):
112 '''Get the I{AuxLat} order (C{int}, 4, 6 or 8).
113 '''
114 return self._mAL
116 @ALorder.setter # PYCHOK setter!
117 def ALorder(self, order):
118 '''Set the I{AuxLat} order (C{int}, 4, 6 or 8).
119 '''
120 m = _Xorder(_AR2Coeffs, AuxError, ALorder=order)
121 if self._mAL != m:
122 _update_all(self)
123 self._mAL = m
125 def _atanhee(self, tphi): # see Ellipsoid._es_atanh, .albers._atanhee
126 # atanh(e * sphi) = asinh(e' * sbeta)
127 f = self.f
128 s = _sn(self._fm1 * tphi) if f > 0 else _sn(tphi)
129 if f: # atanh(e * sphi) = asinh(e' * sbeta)
130 e = self._e
131 s = _over(atan1(e * s) if f < 0 else asinh(self._e1 * s), e)
132 return s
134 def Authalic(self, Phi, **diff_name):
135 '''Convert I{Geographic} to I{Aunthalic} latitude.
137 @arg Phi: Geographic latitude (L{AuxAngle}).
139 @return: Parametric latitude, C{Beta} (L{AuxAngle}).
140 '''
141 _xinstanceof(AuxAngle, Phi=Phi)
142 # assert Phi._AUX == Aux.PHI
143 tphi = fabs(Phi.tan)
144 if isfinite(tphi) and tphi and self.f:
145 y, x = Phi._yx_normalized
146 q = self._q
147 qv = self._qf(tphi)
148 Dq2 = self._Dq(tphi)
149 Dq2 *= (q + qv) / (fabs(y) + _1_0) # _Dq(-tphi)
150 Xi = AuxXi(copysign(qv, Phi.y), x * sqrt(Dq2),
151 name=_xkwds_get(diff_name, name=Phi.name))
153 if _xkwds_get(diff_name, diff=False):
154 if isnan(tphi):
155 d = self._e2m1_sq2
156 else:
157 c = self.Parametric(Phi)._x_normalized
158 d = _over(c, Xi._x_normalized)**3
159 d *= _over(c, x) * _over(_2_0, q)
160 Xi._diff = d
161 else:
162 Xi = AuxXi(*Phi._yx) # diff default
163 # assert Xi._AUX == Aux.XI
164 return Xi
166 def AuthalicRadius2(self, exact=False, f_max=_0_1):
167 '''Get the I{Authalic} radius I{squared}.
169 @kwarg exact: If C{True}, use the exact expression, otherwise
170 the I{Taylor} series.
171 @kwarg f_max: C{Flattening} not to exceed (C{float}).
173 @return: Authalic radius I{squared} (C{meter} I{squared}, same
174 units as the ellipsoid axes).
176 @raise AuxError: If C{B{exact}=False} and C{abs(flattening)}
177 exceeds C{f_max}.
178 '''
179 f = self.f
180 if exact or not f:
181 c2 = self.ellipsoid.b2 * self._q # == ellipsoid.c2x * 2
182 elif fabs(f) < f_max:
183 # Using a * (a + b) / 2 as the multiplying factor leads to a rapidly
184 # converging series in n. Of course, using this series isn't really
185 # necessary, since the exact expression is simple to evaluate. However,
186 # we do it for consistency with RectifyingRadius and, presumably, the
187 # roundoff error is smaller compared to that for the exact expression.
188 m = self.ALorder
189 c2 = _polynomial(self._n, _AR2Coeffs[m], 0, m)
190 c2 *= self.a * (self.a + self.b)
191 else:
192 raise AuxError(exact=exact, f=f, f_max=f_max)
193 return c2 * _0_5
195 @Property_RO
196 def b(self):
197 '''Get the C{ellipsoid}'s polar radius (C{meter}, conventionally).
198 '''
199 return self.ellipsoid.b # a * (_1_0 - f)
201 polaradius = b
203 def _coeffs(self, auxout, auxin):
204 # Get the polynomial coefficients as 4-, 6- or 8-tuple
205 aL = self.ALorder # aka Lmax
206 if auxout == auxin:
207 return _0_0s(aL) # uncached
209 k = Aux._1d(auxout, auxin)
210 try: # cached
211 return AuxLat._csc[aL][k]
212 except KeyError:
213 pass
215 Cx = _CXcoeffs(aL)
216 try:
217 Cx = Cx[auxout][auxin]
218 except KeyError as x:
219 raise AuxError(auxout=auxout, auxin=auxin, cause=x)
221 d = x = n = self._n
222 if Aux.use_n2(auxin) and Aux.use_n2(auxout):
223 x = self._n2
225 def _m(aL):
226 for m in _reverange(aL):
227 yield m // 2
228 else:
229 _m = _reverange # PYCHOK expected
231 i = 0
232 cs = []
233 _c = cs.append
234 _p = _polynomial
235 for m in _m(aL):
236 j = i + m + 1 # order m = j - i - 1
237 _c(_p(x, Cx, i, j) * d)
238 d *= n
239 i = j
240 # assert i == len(Cx) and len(cs) == aL
241 AuxLat._csc.setdefault(aL, {})[k] = cs = tuple(cs)
242 return cs
244 def Conformal(self, Phi, **diff_name):
245 '''Convert I{Geographic} to I{Conformal} latitude.
247 @arg Phi: Geographic latitude (L{AuxAngle}).
249 @return: Conformal latitude, C{Chi} (L{AuxAngle}).
250 '''
251 _xinstanceof(AuxAngle, Phi=Phi)
252 # assert Phi._AUX == Aux.PHI
253 tphi = tchi = fabs(Phi.tan)
254 if isfinite(tphi) and tphi and self.f:
255 sig = sinh(self._atanhee(tphi) * self._e2)
256 scsig = _sc(sig)
257 scphi = _sc(tphi)
258 if self.f > 0:
259 # The general expression for tchi is
260 # tphi * scsig - sig * scphi
261 # This involves cancellation if f > 0, so change to
262 # (tphi - sig) * (tphi + sig) / (tphi * scsig + sig * scphi)
263 # To control overflow, write as (sigtphi = sig / tphi)
264 # (tphi - sig) * (1 + sigtphi) / (scsig + sigtphi * scphi)
265 sigtphi = sig / tphi
266 if sig < (tphi * _0_5):
267 t = tphi - sig
268 else:
269 def _asinh_2(x):
270 return asinh(x) * _0_5
271 # Still possibly dangerous cancellation in tphi - sig.
272 # Write tphi - sig = (1 - e) * Dg(1, e)
273 # Dg(x, y) = (g(x) - g(y)) / (x - y)
274 # g(x) = sinh(x * atanh(sphi * x))
275 # Note sinh(atanh(sphi)) = tphi
276 # Turn the crank on divided differences, substitute
277 # sphi = tphi / _sc(tphi)
278 # atanh(x) = asinh(x / sqrt(1 - x^2))
279 e = self._e
280 em1 = self._e2m1 / (_1_0 + e)
281 # assert em1 != 0
282 scb = self._scbeta(tphi)
283 scphib = scphi / scb # sec(phi) / sec(beta)
284 atphib = _asinh_2(tphi * e / scb) # atanh(e * sphi)
285 atphi = _asinh_2(tphi) # atanh(sphi)
286 t = _asinh_2(em1 * (tphi * scphib)) / em1
287 try:
288 Dg = Fsum(atphi, atphib, t, e * t)
289 except ValueError: # Fsum(NAN) exception
290 Dg = _sum((atphi, atphib, t, e * t))
291 e *= atphib
292 t = atphi - e
293 if t: # sinh(0) == 0
294 Dg *= sinh(t) / t * cosh(atphi + e) * em1
295 t = float(Dg) # tphi - sig
296 tchi = _over(t * (_1_0 + sigtphi),
297 scsig + scphi * sigtphi) if t else _0_0
298 else:
299 tchi = tphi * scsig - sig * scphi
301 n = _xkwds_get(diff_name, name=Phi.name)
302 Chi = AuxChi(tchi, name=n).copyquadrant(Phi)
304 if _xkwds_get(diff_name, diff=False):
305 if isinf(tphi): # PYCHOK np cover
306 d = self._conformal_diff
307 else:
308 d = self.Parametric(Phi)._x_normalized
309 if d:
310 d = _over(d, Chi._x_normalized) * \
311 _over(d, Phi._x_normalized) * self._e2m1
312 Chi._diff = d
313 # assrt Chi._AUX == Aux.CHI
314 return Chi
316 @Property_RO
317 def _conformal_diff(self): # PYCHOK no cover
318 '''(INTERNAL) Constant I{Conformal} diff.
319 '''
320 e = self._e
321 if self.f > 0:
322 ss = sinh(asinh(self._e1) * e)
323 d = _over(_1_0, _sc(ss) + ss)
324 elif e:
325 ss = sinh(-atan1(e) * e)
326 d = _sc(ss) - ss
327 else:
328 d = _1_0
329 return d
331 def convert(self, auxout, Zeta_d, exact=False):
332 # Convert I{Auxiliary} or I{scalar} latitude
333 Z = d = Zeta_d
334 if isinstance(Z, AuxAngle):
335 A, auxin = _AuxClass(auxout), Z._AUX
336 if auxin == auxout:
337 pass
338 elif not (isfinite(Z.tan) and Z.tan): # XXX
339 Z = A(*Z._yx, aux=auxout, name=Z.name)
340 elif exact:
341 p = Aux.power(auxout, auxin)
342 if p is None:
343 P = self._fromAux(Z) # Phi
344 Z = self._toAux(auxout, P)
345 Z._iter = P.iteration
346 else:
347 y, x = Z._yx
348 if p:
349 y *= pow(self._fm1, p)
350 Z = A(y, x, aux=auxout, name=Z.name)
351 else:
352 cs = self._coeffs(auxout, auxin)
353 yx = Z._yx_normalized
354 Z = A(*yx, aux=auxout, name=Z.name)
355 # assert Z._yx == yx
356 r = _Clenshaw(True, Z, cs, self.ALorder)
357 Z += AuxAngle.fromRadians(r, aux=auxout)
358 # assert Z._AUX == auxout
359 return Z
361 elif isinstance(d, Degrees) or isscalar(d):
362 Z = AuxPhi.fromDegrees(d)
363 d = round((d - Z.toDegrees) / _360_0) * _360_0
364 d += self.convert(auxout, Z, exact).toDegrees
365 return Degrees(d, name=Aux.Greek(auxout))
367 raise AuxError(auxout=auxout, Zeta_d=Zeta_d, exact=exact)
369 def _Dq(self, tphi):
370 # I{Divided Difference} of (q(1) - q(sphi)) / (1 - sphi).
371 sphi = _sn(tphi)
372 if tphi > 0:
373 scphi = _sc(tphi)
374 d = _1_0 / (scphi**2 * (_1_0 + sphi)) # XXX - sphi
375 if d:
376 # General expression for _Dq(1, sphi) is
377 # atanh(e * d / (1 - e2 * sphi)) / (e * d) +
378 # (1 + e2 * sphi) / ((1 - e2 * sphi * sphi) * e2m1)
379 # with atanh(e * d / (1 - e2 * sphi)) =
380 # atanh(e * d * scphi / (scphi - e2 * tphi))
381 e2m1, ed = self._e2m1, (self._e * d)
382 if e2m1 and ed:
383 e2 = self._e2
384 if e2 > 0: # assert self.f > 0
385 scb = self._scbeta(tphi)
386 q = scphib = scphi / scb
387 q *= (scphi + tphi * e2) / (e2m1 * scb)
388 q += asinh(self._e1 * d * scphib) / ed
389 else:
390 s2 = sphi * e2
391 q = (_1_0 + s2) / ((_1_0 - sphi * s2) * e2m1)
392 q += (atan2(ed, _1_0 - s2) / ed) if e2 < 0 else _1_0
393 else: # PYCHOK no cover
394 q = INF
395 else: # PYCHOK no cover
396 q = self._2_e2m12
397 else: # not reached, open-coded in .Authalic
398 q = _over(self._q - self._qf(tphi), _1_0 - sphi)
399 return q
401 @Property_RO
402 def _e(self): # unsigned, (1st) eccentricity
403 return self.ellipsoid.e # sqrt(fabs(self._e2))
405 @Property_RO
406 def _e1(self):
407 return sqrt(fabs(self._e12))
409 @Property_RO
410 def _e12(self):
411 return _over(self._e2, _1_0 - self._e2)
413 @Property_RO
414 def _e12p1(self):
415 return _1_0 / self._e2m1
417 @Property_RO
418 def _e2(self): # signed, (1st) eccentricity squared
419 return self.ellipsoid.e2
421 @Property_RO
422 def _e2m1(self): # 1 less 1st eccentricity squared
423 return self.ellipsoid.e21 # == ._fm1**2
425 @Property_RO
426 def _e2m1_sq2(self):
427 return self._e2m1 * sqrt(self._q * _0_5)
429 @Property_RO
430 def _2_e2m12(self):
431 return _2_0 / self._e2m1**2
433 @Property_RO
434 def _Ef_fRG_a2b2_PI_4(self):
435 E = self.ellipsoid
436 return _Ef.fRG(E.a2, E.b2) / PI_4
438 @Property_RO
439 def ellipsoid(self):
440 '''Get the ellipsoid (L{Ellipsoid}).
441 '''
442 return self._E
444 @Property_RO
445 def f(self):
446 '''Get the C{ellipsoid}'s flattening (C{scalar}).
447 '''
448 return self.ellipsoid.f
450 flattening = f
452 @Property_RO
453 def _fm1(self): # 1 - flattening
454 return self.ellipsoid.f1
456 def _fromAux(self, Zeta, **name):
457 '''Convert I{Auxiliary} to I{Geographic} latitude.
459 @arg Zeta: Auxiliary latitude (L{AuxAngle}).
461 @return: Geographic latitude, I{Phi} (L{AuxAngle}).
462 '''
463 _xinstanceof(AuxAngle, Zeta=Zeta)
464 aux = Zeta._AUX
465 n = _xkwds_get(name, name=Zeta.name)
466 f = self._fromAuxCase.get(aux, None)
467 if f is None:
468 Phi = AuxPhi(NAN, name=n)
469 elif callable(f):
470 t = self._fm1
471 t *= f(t)
472 Phi = _Newton(t, Zeta, self._toZeta(aux), name=n)
473 else: # assert isscalar(f)
474 y, x = Zeta._yx
475 Phi = AuxPhi(y / f, x, name=n)
476 # assert Phi._AUX == Aux.PHI
477 return Phi
479 @Property_RO
480 def _fromAuxCase(self):
481 '''(INTERNAL) switch(auxin): ...
482 '''
483 return {Aux.AUTHALIC: cbrt,
484 Aux.CONFORMAL: _MODS.utily._passarg,
485 Aux.GEOCENTRIC: self._e2m1,
486 Aux.GEOGRAPHIC: _1_0,
487 Aux.PARAMETRIC: self._fm1,
488 Aux.RECTIFYING: sqrt}
490 def Geocentric(self, Phi, **diff_name):
491 '''Convert I{Geographic} to I{Geocentric} latitude.
493 @arg Phi: Geographic latitude (L{AuxAngle}).
495 @return: Geocentric latitude, C{Phi} (L{AuxAngle}).
496 '''
497 _xinstanceof(AuxAngle, Phi=Phi)
498 # assert Phi._AUX == Aux.PHI
499 Theta = AuxTheta(Phi.y * self._e2m1, Phi.x,
500 name=_xkwds_get(diff_name, name=Phi.name))
501 if _xkwds_get(diff_name, diff=False):
502 Theta._diff = self._e2m1
503 return Theta
505 def Geodetic(self, Phi, **diff_name): # PYCHOK no cover
506 '''Convert I{Geographic} to I{Geodetic} latitude.
508 @arg Phi: Geographic latitude (L{AuxAngle}).
510 @return: Geodetic latitude, C{Phi} (L{AuxAngle}).
511 '''
512 _xinstanceof(AuxAngle, Phi=Phi)
513 # assert Phi._AUX == Aux.PHI
514 return AuxPhi(Phi, name=_xkwds_get(diff_name, name=Phi.name))
516 @Property_RO
517 def _n(self): # 3rd flattening
518 return self.ellipsoid.n
520 @Property_RO
521 def _n2(self):
522 return self._n**2
524 def Parametric(self, Phi, **diff_name):
525 '''Convert I{Geographic} to I{Parametric} latitude.
527 @arg Phi: Geographic latitude (L{AuxAngle}).
529 @return: Parametric latitude, C{Beta} (L{AuxAngle}).
530 '''
531 _xinstanceof(AuxAngle, Phi=Phi)
532 # assert Phi._AUX == Aux.PHI
533 Beta = AuxBeta(Phi.y * self._fm1, Phi.x,
534 name=_xkwds_get(diff_name, name=Phi.name))
535 if _xkwds_get(diff_name, diff=False):
536 Beta._diff = self._fm1
537 return Beta
539 Reduced = Parametric
541 @Property_RO
542 def _q(self): # constant _q
543 q, f = self._e12p1, self.f
544 if f:
545 e = self._e
546 q += _over(asinh(self._e1) if f > 0 else atan1(e), e)
547 else:
548 q += _1_0
549 return q
551 def _qf(self, tphi):
552 # function _q: atanh(e * sphi) / e + sphi / (1 - (e * sphi)^2)
553 scb = self._scbeta(tphi)
554 return self._atanhee(tphi) + (tphi / scb) * (_sc(tphi) / scb)
556 def _qIntegrand(self, beta):
557 # pbeta(beta) = integrate(q(beta), beta), with beta in radians
558 # q(beta) = (1-f) * (sin(xi) - sin(chi)) / cos(phi)
559 # = (1-f) * (cos(chi) - cos(xi)) / cos(phi) *
560 # (cos(xi) + cos(chi)) / (sin(xi) + sin(chi))
561 # Fit q(beta)/cos(beta) with Fourier transform
562 # q(beta)/cos(beta) = sum(c[k] * sin((2*k+1)*beta), k, 0, K-1)
563 # then the integral is
564 # pbeta = sum(d[k] * cos((2*k+2)*beta), k, 0, K-1)
565 # where
566 # d[k] = -1/(4*(k+1)) * (c[k] + c[k+1]) for k in 0..K-2
567 # d[K-1] = -1/(4*K) * c[K-1]
568 Beta = AuxBeta.fromRadians(beta)
569 if Beta.x: # and self._fm1:
570 Ax, _cv = Aux, self.convert
571 Phi = _cv(Ax.PHI, Beta, exact=True)
572 schi, cchi = _cv(Ax.CHI, Phi, exact=True)._yx_normalized
573 sxi, cxi = _cv(Ax.XI, Phi, exact=True)._yx_normalized
574 r = (sxi - schi) if fabs(schi) < fabs(cchi) else \
575 _over(_2cos2x(cchi, cxi), (sxi + schi) * _2_0)
576 r *= _over(self._fm1, Phi._x_normalized * Beta._x_normalized)
577 else: # beta == PI_2, PI3_2, ...
578 r = _0_0 # XXX 0 avoids NAN summation exceptions
579 return r
581 def Rectifying(self, Phi, **diff_name):
582 '''Convert I{Geographic} to I{Rectifying} latitude.
584 @arg Phi: Geographic latitude (L{AuxAngle}).
586 @return: Rectifying latitude, C{Mu} (L{AuxAngle}).
587 '''
588 Beta = self.Parametric(Phi)
589 # assert Beta._AUX == Aux.BETA
590 sb, cb = map(fabs, Beta._yx_normalized)
591 a, ka, ka1 = _1_0, self._e2, self._e2m1
592 b, kb, kb1 = self._fm1, -self._e12, self._e12p1
593 if self.f < 0:
594 a, b = b, a
595 ka, kb = kb, ka
596 ka1, kb1 = kb1, ka1
597 sb, cb = cb, sb
598 # now a, b = larger/smaller semiaxis
599 # Beta measured from larger semiaxis
600 # kb, ka = modulus-squared for distance from Beta = 0, pi/2
601 # NB kb <= 0; 0 <= ka <= 1
602 # sa = b*E(Beta, sqrt(kb))
603 # sb = a*E(Beta',sqrt(ka))
604 # 1 - ka * (1 - sb2) = 1 - ka + ka*sb2
605 sb2 = sb**2
606 cb2 = cb**2
607 da2 = ka1 + ka * sb2
608 db2 = _1_0 - kb * sb2
609 # DLMF Eq. 19.25.9
610 my = b * sb * _Ef._RFRD(cb2, db2, _1_0, kb * sb2)
611 # DLMF Eq. 19.25.10 with complementary angles
612 mx = a * cb * (_Ef.fRF(sb2, da2, _1_0) * ka1 +
613 ka * cb2 * _Ef.fRD(sb2, _1_0, da2, _3_0) * ka1 +
614 ka * sb / sqrt(da2))
615 # my + mx = 2*_Ef.fRG(a*a, b*b) = a*E(e) = b*E(i*e')
616 # mr = a*E(e)*(2/pi) = b*E(i*e')*(2/pi)
617 if self.f < 0:
618 a, b = b, a
619 my, mx = mx, my
620 mr = (my + mx) / PI_2
621 if mr:
622 my = sin(my / mr)
623 mx = sin(mx / mr) # XXX zero?
624 else: # zero Mu
625 my, mx = _0_0, _1_0
626 n = _xkwds_get(diff_name, name=Phi.name)
627 Mu = AuxMu(my, mx, # normalized
628 name=n).copyquadrant(Phi)
630 if _xkwds_get(diff_name, diff=False):
631 d, x = _0_0, Beta._x_normalized
632 if x and mr:
633 if Mu.x and Phi.x and not isinf(Phi.tan):
634 d = b / mr * (x / Mu.x)**2 \
635 * (x / Phi._x_normalized)
636 else:
637 d = mr / a
638 Mu._diff = self._fm1 * d
639 return Mu
641 def RectifyingRadius(self, exact=False):
642 '''Get the I{Rectifying} radius.
644 @arg exact: If C{True}, use the exact expression,
645 otherwise the I{Taylor} series.
647 @return: Rectifying radius (L{Meter}, same units
648 as the ellipsoid axes).
649 '''
650 r = self._Ef_fRG_a2b2_PI_4 if exact else self._RectifyingR
651 return Meter(r, name=self.RectifyingRadius.__name__)
653 @Property_RO
654 def _RectifyingR(self):
655 m = self.ALorder
656 d = _polynomial(self._n2, _RRCoeffs[m], 0, m // 2)
657 return d * (self.a + self.b) * _0_5
659 def _scbeta(self, tphi):
660 return _sc(self._fm1 * tphi)
662 def _toAux(self, auxout, Phi, **diff_name):
663 '''Convert I{Geographic} to I{Auxiliary} latitude.
665 @arg auxout: I{Auxiliary} kind (C{Aux.KIND}).
666 @arg Phi: Geographic latitude (L{AuxLat}).
668 @return: Auxiliary latitude, I{Eta} (L{AuxLat}).
669 '''
670 _xinstanceof(AuxAngle, Phi=Phi)
671 # assert Phi._AUX == Aux.PHI
672 n = _xkwds_get(diff_name, name=Phi.name)
673 m = _toAuxCase.get(auxout, None)
674 if m: # callable
675 A = m(self, Phi, **_xkwds(diff_name, name=n))
676 elif auxout == Aux.GEODETIC: # == GEOGRAPHIC
677 A = AuxPhi(Phi, name=n)
678 else: # auxout?
679 A = AuxPhi(NAN, name=n)
680 # assert A._AUX == auxout
681 return A
683 def _toZeta(self, zetaux):
684 '''Return a (lean) function to create C{AuxPhi(tphi)} and
685 convert that into C{AuxAngle} of (fixed) kind C{zetaux}
686 for use only inside the C{_Newton} loop.
687 '''
688 class _AuxPhy(AuxPhi):
689 # lean C{AuxPhi} instance.
690 # _diff = _1_0
691 # _x = _1_0
693 def __init__(self, tphi): # PYCHOK signature
694 self._y = tphi
696 m = _toAuxCase.get(zetaux, None)
697 if m: # callable
699 def _toZeta(tphi):
700 return m(self, _AuxPhy(tphi), diff=True)
702 elif zetaux == Aux.GEODETIC: # GEOGRAPHIC
703 _toZeta = _AuxPhy
705 else: # zetaux?
707 def _toZeta(unused): # PYCHOK expected
708 return _AuxPhy(NAN)
710 return _toZeta
713# switch(auxout): ...
714_toAuxCase = {Aux.AUTHALIC: AuxLat.Authalic,
715 Aux.CONFORMAL: AuxLat.Conformal,
716 Aux.GEOCENTRIC: AuxLat.Geocentric,
717 Aux.PARAMETRIC: AuxLat.Parametric,
718 Aux.RECTIFYING: AuxLat.Rectifying}
721def _Clenshaw(sinp, Zeta, cs, K):
722 sz, cz = Zeta._yx # isnormal
723 # Evaluate sum(c[k] * sin((2*k+2) * Zeta)) if sinp else
724 # sum(c[k] * cos((2*k+2) * Zeta))
725 x = _2cos2x(cz, sz) # 2 * cos(2*Zeta)
726 if isfinite(x):
727 U0, U1 = Fsum(), Fsum()
728 # assert len(cs) == K
729 for r in _reverange(K):
730 U1 -= U0 * x + cs[r]
731 U1, U0 = U0, -U1
732 # u0*f0(Zeta) - u1*fm1(Zeta)
733 # f0 = sin(2*Zeta) if sinp else cos(2*Zeta)
734 # fm1 = 0 if sinp else 1
735 if sinp:
736 U0 *= sz * cz * _2_0
737 else:
738 U0 *= x * _0_5
739 U0 -= U1
740 x = float(U0)
741 return x
744def _CXcoeffs(aL): # PYCHOK in .auxilats.__main__
745 '''(INTERNAL) Get the C{CX_4}, C{_6} or C{_8} coefficients.
746 '''
747 try: # from pygeodesy.auxilats._CX_x import _coeffs_x as _coeffs
748 _CX_x = _DOT_(_MODS.auxilats.__name__, _UNDER_('_CX', aL))
749 _coeffs = _MODS.getattr(_CX_x, _UNDER_('_coeffs', aL))
750 except (AttributeError, ImportError, KeyError, TypeError) as x:
751 raise AuxError(ALorder=aL, cause=x)
752 # assert _coeffs.ALorder == aL
753 # assert _coeffs.n == Aux.len(aL)
754 return _coeffs
757def _Newton(tphi, Zeta, _toZeta, **name):
758 # Newton's method fro AuxLat._fromAux
759 try:
760 _lg2 = _log2
761 _abs = fabs
762 tz = _abs(Zeta.tan)
763 tphi = tz / tphi # **)
764 ltz = _lg2(tz) # **)
765 ltphi = _lg2(tphi) # **)
766 ltmin = min(ltphi, MIN_EXP)
767 ltmax = max(ltphi, MAX_EXP)
768# auxin = Zeta._AUX
769 s, n, __2 = 0, 3, _0_5 # n = i + 2
770 _TOL, _xp2 = _EPSqrt, _exp2
771 for i in range(1, _TRIPS): # up to 1 Ki!
772 # _toAux(auxin, AuxPhi(tphi), diff=True)
773 Z = _toZeta(tphi)
774 # assert Z._AUX == auxin
775 t, s_ = Z.tan, s
776 if t > tz:
777 ltmax, s = ltphi, +1
778 elif t < tz:
779 ltmin, s = ltphi, -1
780 else:
781 break
782 # derivative dtan(Z)/dtan(Phi)
783 # to dlog(tan(Z))/dlog(tan(Phi))
784 d = (ltz - _lg2(t)) * t # **)
785 if d:
786 d = d / (Z.diff * tphi) # **)
787 ltphi += d
788 tphi = _xp2(ltphi)
789 if _abs(d) < _TOL:
790 i += 1
791 # _toAux(auxin, AuxPhi(tphi), diff=True)
792 Z = _toZeta(tphi)
793 tphi -= _over(Z.tan - tz, Z.diff)
794 break
795 if (i > n and (s * s_) < 0) or not ltmin < ltphi < ltmax:
796 s, n = 0, (i + 2)
797 ltphi = (ltmin + ltmax) * __2
798 tphi = _xp2(ltphi)
799 else:
800 i = _TRIPS
801 Phi = AuxPhi(tphi, **name).copyquadrant(Zeta)
802 Phi._iter = i
803 except (ValueError, ZeroDivisionError): # **) zero t, tphi, tz or Z.diff
804 Phi = AuxPhi(Zeta, **name) # diff as-as
805 Phi._iter = 0
806 # assert Phi._AUX == Aux.PHI
807 return Phi
810_f, _u = float, _Ufloats()
811_1__f3 = -1 / _f(3) # XXX +1 / _f(3)
812_AR2Coeffs = {4: _u(4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3),
813 6: _u(4 / _f(1287), 4 / _f(693), 4 / _f(315), 4 / _f(105),
814 4 / _f(15), _1__f3),
815 8: _u(4 / _f(3315), 4 / _f(2145), 4 / _f(1287), 4 / _f(693),
816 4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3)}
817_RRCoeffs = {4: _u(1 / _f(64), _0_25),
818 6: _u(1 / _f(256), 1 / _f(64), _0_25),
819 8: _u(25 / _f(16384), 1 / _f(256), 1 / _f(64), _0_25)} # PYCHOK used!
820del _f, _u, _Ufloats, _1__f3
821# assert set(_AR2Coeffs.keys()) == set(_RRCoeffs.keys())
823# AuxLat._Lmax = max(_AR2Coeffs.keys()) # == max(ALorder)
825__all__ += _ALL_DOCS(AuxLat)
827# **) MIT License
828#
829# Copyright (C) 2023-2023 -- mrJean1 at Gmail -- All Rights Reserved.
830#
831# Permission is hereby granted, free of charge, to any person obtaining a
832# copy of this software and associated documentation files (the "Software"),
833# to deal in the Software without restriction, including without limitation
834# the rights to use, copy, modify, merge, publish, distribute, sublicense,
835# and/or sell copies of the Software, and to permit persons to whom the
836# Software is furnished to do so, subject to the following conditions:
837#
838# The above copyright notice and this permission notice shall be included
839# in all copies or substantial portions of the Software.
840#
841# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
842# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
843# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
844# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
845# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
846# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
847# OTHER DEALINGS IN THE SOFTWARE.