Coverage for pygeodesy/albers.py: 97%
413 statements
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2# -*- coding: utf-8 -*-
4u'''Albers Equal-Area projections.
6Classes L{AlbersEqualArea}, L{AlbersEqualArea2}, L{AlbersEqualArea4},
7L{AlbersEqualAreaCylindrical}, L{AlbersEqualAreaNorth}, L{AlbersEqualAreaSouth}
8and L{AlbersError}, transcoded from I{Charles Karney}'s C++ class U{AlbersEqualArea
9<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AlbersEqualArea.html>}.
11See also I{Albers Equal-Area Conic Projection} in U{John P. Snyder, "Map Projections
12-- A Working Manual", 1987<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 98-106
13and the Albers Conical Equal-Area examples on pp 291-294.
14'''
15# make sure int/int division yields float quotient, see .basics
16from __future__ import division as _; del _ # PYCHOK semicolon
18from pygeodesy.basics import neg, neg_
19from pygeodesy.constants import EPS0, EPS02, _EPSqrt as _TOL, \
20 _0_0, _0_5, _1_0, _N_1_0, _2_0, \
21 _N_2_0, _4_0, _6_0, _90_0, _N_90_0
22from pygeodesy.datums import _ellipsoidal_datum, _WGS84
23from pygeodesy.errors import _ValueError, _xkwds
24from pygeodesy.fmath import hypot, hypot1, sqrt3
25from pygeodesy.fsums import Fsum, _Fsum1f_, fsum1f_
26from pygeodesy.interns import NN, _COMMASPACE_, _datum_, _gamma_, _k0_, \
27 _lat_, _lat1_, _lat2_, _lon_, _negative_, \
28 _scale_, _SPACE_, _x_, _y_
29from pygeodesy.karney import _diff182, _norm180, _signBit
30from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY
31from pygeodesy.named import _name2__, _NamedBase, _NamedTuple, _Pass
32from pygeodesy.props import deprecated_Property_RO, Property_RO, _update_all
33from pygeodesy.streprs import Fmt, strs, unstr
34from pygeodesy.units import Bearing, Float_, Lat, Lat_, Lon, Meter, Scalar_
35from pygeodesy.utily import atan1, atan1d, atan2, degrees360, sincos2, \
36 sincos2d, sincos2d_
38from math import atanh, degrees, fabs, radians, sqrt
40__all__ = _ALL_LAZY.albers
41__version__ = '24.11.24'
43_k1_ = 'k1'
44_NUMIT = 8 # XXX 4?
45_NUMIT0 = 41 # XXX 21?
46_TERMS = 31 # XXX 16?
47_TOL0 = sqrt3(_TOL)
50def _ct2(s, c):
51 '''(INTERNAL) Avoid singularities at poles.
52 '''
53 c = max(EPS0, c)
54 return c, (s / c)
57def _Ks(**name_k):
58 '''(INTERNAL) Scale C{B{k} >= EPS0}.
59 '''
60 return Scalar_(Error=AlbersError, low=EPS0, **name_k) # > 0
63def _Lat(*lat, **Error_name_lat):
64 '''(INTERNAL) Latitude C{-90 <= B{lat} <= 90}.
65 '''
66 kwds = _xkwds(Error_name_lat, Error=AlbersError)
67 return Lat_(*lat, **kwds)
70def _qZx(albs):
71 '''(INTERNAL) Set C{albs._qZ} and C{albs._qx}.
72 '''
73 E = albs._datum.ellipsoid # _AlbersBase
74 albs._qZ = qZ = _1_0 + E.e21 * _atanheE(_1_0, E)
75 albs._qx = qZ / (_2_0 * E.e21)
76 return qZ
79class AlbersError(_ValueError):
80 '''An L{AlbersEqualArea}, L{AlbersEqualArea2}, L{AlbersEqualArea4},
81 L{AlbersEqualAreaCylindrical}, L{AlbersEqualAreaNorth},
82 L{AlbersEqualAreaSouth} or L{Albers7Tuple} issue.
83 '''
84 pass
87class _AlbersBase(_NamedBase):
88 '''(INTERNAL) Base class for C{AlbersEqualArea...} projections.
90 @see: I{Karney}'s C++ class U{AlbersEqualArea<https://GeographicLib.SourceForge.io/
91 C++/doc/classGeographicLib_1_1AlbersEqualArea.html>}, method C{Init}.
92 '''
93 _datum = _WGS84
94 _k = NN # or _k0_ or _k1_
95 _k0 = _Ks(k0=_1_0)
96# _k0n0 = None # (INTERNAL) k0 * no
97 _k02 = _1_0 # (INTERNAL) k0**2
98# _k02n0 = None # (INTERNAL) k02 * n0
99# _lat0 = None # lat origin
100 _lat1 = None # let 1st parallel
101 _lat2 = None # lat 2nd parallel
102 _m0 = _0_0 # if polar else sqrt(m02)
103# _m02 = None # (INTERNAL) cached
104# _n0 = None # (INTERNAL) cached
105 _nrho0 = _0_0 # if polar else m0 * E.a
106 _polar = False
107 _qx = None # (INTERNAL) see _qZx
108 _qZ = None # (INTERNAL) see _qZx
109# _scxi0_ = None # (INTERNAL) sec(xi) / (qZ * E.a2)
110 _sign = +1
111# _sxi0 = None # (INTERNAL) sin(xi)
112# _txi0 = None # (INTERNAL) tan(xi)
114 def __init__(self, sa1, ca1, sa2, ca2, k, datum, **name):
115 '''(INTERNAL) New C{AlbersEqualArea...} instance.
116 '''
117 qZ = self._qZ
118 if datum not in (None, self._datum):
119 self._datum = _ellipsoidal_datum(datum, **name)
120 qZ = _qZx(self)
121 elif qZ is None:
122 qZ = _qZx(_AlbersBase)
123 if name:
124 self.name = name
126 E = self.ellipsoid
127 c = min(ca1, ca2)
128 if _signBit(c):
129 raise AlbersError(clat1=ca1, clat2=ca2, txt=_negative_)
130 polar = c < EPS0 # == 0
132 # determine hemisphere of tangent latitude
133 if sa1 < 0: # and sa2 < 0:
134 self._sign = -1
135 # internally, tangent latitude positive
136 sa1, sa2 = neg_(sa1, sa2)
137 if sa1 > sa2: # make phi1 < phi2
138 sa1, sa2 = sa2, sa1
139 ca1, ca2 = ca2, ca1
140 if sa1 < 0: # or sa2 < 0:
141 raise AlbersError(slat1=sa1, slat2=sa2, txt=_negative_)
142 ca1, ta1 = _ct2(sa1, ca1)
143 ca2, ta2 = _ct2(sa2, ca2)
145 par1 = fabs(ta1 - ta2) < EPS02 # ta1 == ta2
146 if par1 or polar:
147 ta0, C = ta2, _1_0
148 else:
149 ta0, C = self._ta0C2(ca1, sa1, ta1, ca2, sa2, ta2)
151 self._lat0 = _Lat(lat0=self._sign * atan1d(ta0))
152 self._m02 = m02 = _1_x21(E.f1 * ta0)
153 self._n0 = n0 = ta0 / hypot1(ta0)
154 if polar:
155 self._polar = True
156# self._nrho0 = self._m0 = _0_0
157 else: # m0 = nrho0 / E.a
158 self._m0 = sqrt(m02)
159 self._nrho0 = self._m0 * E.a
160 t = self._txi0 = self._txif(ta0)
161 h = hypot1(t)
162 s = self._sxi0 = t / h
163 if par1:
164 self._k0n0 = self._k02n0 = n0
165 else:
166 self._k0s(k * sqrt(C / (m02 + n0 * qZ * s)))
167 self._scxi0_ = h / (qZ * E.a2)
169 def _a_b_sxi3(self, *ca_sa_ta_scb_4s):
170 '''(INTERNAL) Sum of C{sm1} terms and C{sin(xi)}s for ._ta0C2.
171 '''
172 _1 = _1_0
173 a = b = s = _0_0
174 for ca, sa, ta, scb in ca_sa_ta_scb_4s:
175 cxi, sxi, _ = self._cstxif3(ta)
176 if sa > 0:
177 sa += _1
178 a += (cxi / ca)**2 * sa / (sxi + _1)
179 b += scb * ca**2 / sa
180 else:
181 sa = _1 - sa
182 a += (_1 - sxi) / sa
183 b += scb * sa
184 s += sxi
185 return a, b, s
187 def _azik(self, t, ta):
188 '''(INTERNAL) Compute the azimuthal scale C{_Ks(k=k)}.
189 '''
190 E = self.ellipsoid
191 return _Ks(k=hypot1(E.b_a * ta) * self._k0 * t / E.a)
193 def _cstxif3(self, ta):
194 '''(INTERNAL) Get 3-tuple C{(cos, sin, tan)} of M{xi(ta)}.
195 '''
196 t = self._txif(ta)
197 c = _1_0 / hypot1(t)
198 s = c * t
199 return c, s, t
201 @Property_RO
202 def datum(self):
203 '''Get the datum (L{Datum}).
204 '''
205 return self._datum
207 @Property_RO
208 def ellipsoid(self):
209 '''Get the datum's ellipsoid (L{Ellipsoid}).
210 '''
211 return self.datum.ellipsoid
213 @Property_RO
214 def equatoradius(self):
215 '''Get the C{ellipsoid}'s equatorial radius, semi-axis (C{meter}).
216 '''
217 return self.ellipsoid.a
219 a = equatoradius
221 @Property_RO
222 def flattening(self):
223 '''Get the C{ellipsoid}'s flattening (C{scalar}).
224 '''
225 return self.ellipsoid.f
227 f = flattening
229 def forward(self, lat, lon, lon0=0, **name):
230 '''Convert a geodetic location to east- and northing.
232 @arg lat: Latitude of the location (C{degrees}).
233 @arg lon: Longitude of the location (C{degrees}).
234 @kwarg lon0: Optional central meridian longitude (C{degrees}).
235 @kwarg name: Optional C{B{name}=NN} for the location (C{str}).
237 @return: An L{Albers7Tuple}C{(x, y, lat, lon, gamma, scale, datum)},
238 with C{lon} offset by B{C{lon0}} and reduced C{[-180,180]}.
240 @note: The origin latitude is returned by C{property lat0}. No
241 false easting or northing is added. The value of B{C{lat}}
242 should be in the range C{[-90..90] degrees}. The returned
243 values C{x} and C{y} will be large but finite for points
244 projecting to infinity, i.e. one or both of the poles.
245 '''
246 a = self.ellipsoid.a
247 s = self._sign
249 k0 = self._k0
250 n0 = self._n0
251 nrho0 = self._nrho0
252 txi0 = self._txi0
254 _, ta = _ct2(*sincos2d(s * _Lat(lat=lat)))
256 _, sxi, txi = self._cstxif3(ta)
257 dq = _Dsn(txi, txi0, sxi, self._sxi0) * \
258 (txi - txi0) * self._qZ
259 drho = a * dq / (sqrt(self._m02 - n0 * dq) + self._m0)
261 lon, _ = _diff182(lon0, lon)
262 x = radians(lon)
263 th = self._k02n0 * x
264 sth, cth = sincos2(th) # XXX sin, cos
265 if n0:
266 x = sth / n0
267 y = (_1_0 - cth) if cth < 0 else (sth**2 / (cth + _1_0))
268 y *= nrho0 / n0
269 else:
270 x *= self._k02
271 y = _0_0
272 t = nrho0 - n0 * drho
273 x = t * x / k0
274 y = s * (y + drho * cth) / k0
276 g = degrees360(s * th)
277 if t:
278 k0 = self._azik(t, ta)
279 return Albers7Tuple(x, y, lat, lon, g, k0, self.datum,
280 name=self._name__(name))
282 @Property_RO
283 def ispolar(self):
284 '''Is this projection polar (C{bool})?
285 '''
286 return self._polar
288 isPolar = ispolar # synonym
290 def _k0s(self, k0):
291 '''(INTERNAL) Set C{._k0}, C{._k02}, etc.
292 '''
293 self._k0 = k0 = _Ks(k0=k0)
294 self._k02 = k02 = k0**2
295 self._k0n0 = k0 * self._n0
296 self._k02n0 = k02 * self._n0
298 @Property_RO
299 def lat0(self):
300 '''Get the latitude of the projection origin (C{degrees}).
302 This is the latitude of minimum azimuthal scale and
303 equals the B{C{lat}} in the 1-parallel L{AlbersEqualArea}
304 and lies between B{C{lat1}} and B{C{lat2}} for the
305 2-parallel L{AlbersEqualArea2} and L{AlbersEqualArea4}
306 projections.
307 '''
308 return self._lat0
310 @Property_RO
311 def lat1(self):
312 '''Get the latitude of the first parallel (C{degrees}).
313 '''
314 return self._lat1
316 @Property_RO
317 def lat2(self):
318 '''Get the latitude of the second parallel (C{degrees}).
320 @note: The second and first parallel latitudes are the
321 same instance for 1-parallel C{AlbersEqualArea*}
322 projections.
323 '''
324 return self._lat2
326 @deprecated_Property_RO
327 def majoradius(self): # PYCHOK no cover
328 '''DEPRECATED, use property C{equatoradius}.'''
329 return self.equatoradius
331 def rescale0(self, lat, k=1): # PYCHOK no cover
332 '''Set the azimuthal scale for this projection.
334 @arg lat: Northern latitude (C{degrees}).
335 @arg k: Azimuthal scale at latitude B{C{lat}} (C{scalar}).
337 @raise AlbersError: Invalid B{C{lat}} or B{C{k}}.
339 @note: This allows a I{latitude of conformality} to be specified.
340 '''
341 k0 = _Ks(k=k) / self.forward(lat, _0_0).scale
342 if self._k0 != k0:
343 _update_all(self)
344 self._k0s(k0)
346 def reverse(self, x, y, lon0=0, LatLon=None, **name_LatLon_kwds):
347 '''Convert an east- and northing location to geodetic lat- and longitude.
349 @arg x: Easting of the location (C{meter}).
350 @arg y: Northing of the location (C{meter}).
351 @kwarg lon0: Optional central meridian longitude (C{degrees}).
352 @kwarg LatLon: Class to use (C{LatLon}) or C{None}.
353 @kwarg name_LatLon_kwds: Optional C{B{name}=NN} for the location
354 and optional, additional B{C{LatLon}} keyword
355 arguments, ignored if C{B{LatLon} is None}.
357 @return: The geodetic (C{LatLon}) or if C{B{LatLon} is None} an
358 L{Albers7Tuple}C{(x, y, lat, lon, gamma, scale, datum)}.
360 @note: The origin latitude is returned by C{property lat0}. No
361 false easting or northing is added. The returned value of
362 C{lon} is in the range C{[-180..180] degrees} and C{lat}
363 is in the range C{[-90..90] degrees}. If the given
364 B{C{x}} or B{C{y}} point is outside the valid projected
365 space the nearest pole is returned.
366 '''
367 k0 = self._k0
368 n0 = self._n0
369 k0n0 = self._k0n0
370 s = self._sign
371 txi = self._txi0
373 x = Meter(x=x)
374 nx = k0n0 * x
375 y = Meter(y=y)
376 y_ = s * y
377 ny = k0n0 * y_
378 t = nrho0 = self._nrho0
379 y1 = nrho0 - ny
381 den = hypot(nx, y1) + nrho0 # 0 implies origin with polar aspect
382 if den:
383 drho = _Fsum1f_(x * nx, y_ * nrho0 * _N_2_0, y_ * ny).fover(den / k0)
384 # dsxia = scxi0 * dsxi
385 t += drho * n0 # k0 below
386 d_ = (nrho0 + t) * drho * self._scxi0_ # / (qZ * E.a2)
387 t_ = txi - d_
388 d_ = (txi + t_) * d_ + _1_0
389 txi = t_ / (sqrt(d_) if d_ > EPS02 else EPS0)
391 ta = self._tanf(txi)
392 lat = atan1d(s * ta)
394 th = atan2(nx, y1)
395 lon = degrees((th / self._k02n0) if n0 else (x / (y1 * k0)))
396 if lon0:
397 lon += _norm180(lon0)
398 lon = _norm180(lon)
400 n, LatLon_kwds = _name2__(name_LatLon_kwds, _or_nameof=self)
401 if LatLon is None:
402 g = degrees360(s * th)
403 if den:
404 k0 = self._azik(t, ta)
405 r = Albers7Tuple(x, y, lat, lon, g, k0, self.datum, name=n)
406 else: # PYCHOK no cover
407 kwds = _xkwds(LatLon_kwds, datum=self.datum, name=n)
408 r = LatLon(lat, lon, **kwds)
409 return r
411 @Property_RO
412 def scale0(self):
413 '''Get the central scale for the projection (C{float}).
415 This is the azimuthal scale on the latitude of origin
416 of the projection, see C{property lat0}.
417 '''
418 return self._k0
420 def _ta0(self, s1_qZ, ta0, E):
421 '''(INTERNAL) Refine C{ta0} for C{._ta0C2}.
422 '''
423 e2 = E.e2
424 e21 = E.e21
425 e22 = E.e22 # == e2 / e21
426 tol = _tol(_TOL0, ta0)
427 _Ta02 = Fsum(ta0).fsum2f_
428 _1, _2 = _1_0, _2_0
429 _4, _6 = _4_0, _6_0
430 for self._iteration in range(1, _NUMIT0): # 4 trips
431 ta02 = ta0**2
432 sca02 = ta02 + _1
433 sca0 = sqrt(sca02)
434 sa0 = ta0 / sca0
435 sa01 = sa0 + _1
436 sa02 = sa0**2
437 # sa0m = 1 - sa0 = 1 / (sec(a0) * (tan(a0) + sec(a0)))
438 sa0m = _1 / (sca0 * (ta0 + sca0)) # scb0^2 * sa0
439 sa0m1 = sa0m / (_1 - e2 * sa0)
440 sa021 = _1 - e2 * sa02
442 g = (_1 + ta02 * e21) * sa0
443 dg = (_1 + ta02 * _2) * sca02 * e21 + e2
444 D = (_1 - (_1 + sa0 * _2 * sa01) * e2) * sa0m / (e21 * sa01) # dD/dsa0
445 dD = (_2 - (_6 + sa0 * _4) * sa02 * e2) / (e21 * sa01**2)
446 BA = (_atanh1(e2 * sa0m1**2) * e21 - e2 * sa0m) * sa0m1 \
447 - (_2 + (_1 + e2) * sa0) * sa0m**2 * e22 / sa021 # B + A
448 d = (_4 - (_1 + sa02) * e2 * _2) * e22 / (sa021**2 * sca02) # dAB
449 u = fsum1f_(s1_qZ * g, -D, g * BA)
450 du = fsum1f_(s1_qZ * dg, dD, dg * BA, g * d)
451 ta0, d = _Ta02(-u / du * (sca0 * sca02))
452 if fabs(d) < tol:
453 return ta0
454 raise AlbersError(Fmt.no_convergence(d, tol), txt=repr(self))
456 def _ta0C2(self, ca1, sa1, ta1, ca2, sa2, ta2):
457 '''(INTERNAL) Compute C{ta0} and C{C} for C{.__init__}.
458 '''
459 E = self.ellipsoid
460 f1, e2 = E.f1, E.e2
461 _1 = _1_0
463 tb1 = f1 * ta1
464 tb2 = f1 * ta2
465 dtb12 = f1 * (tb1 + tb2)
466 scb12 = _1 + tb1**2
467 scb22 = _1 + tb2**2
469 dsn_2 = _Dsn(ta2, ta1, sa2, sa1) * _0_5
470 sa12 = sa1 * sa2
472 esa1_2 = (_1 - e2 * sa1**2) \
473 * (_1 - e2 * sa2**2)
474 esa12 = _1 + e2 * sa12
476 axi, bxi, sxi = self._a_b_sxi3((ca1, sa1, ta1, scb12),
477 (ca2, sa2, ta2, scb22))
479 dsxi = ((esa12 / esa1_2) + _DatanheE(sa2, sa1, E)) * dsn_2 / self._qx
480 C = _Fsum1f_(sxi * dtb12 / dsxi, scb22, scb12).fover(scb22 * scb12 * _2_0)
482 S = _Fsum1f_(sa1, sa2, sa12)
483 axi *= (S * e2 + _1).fover(S + _1, raiser=False)
484 bxi *= _Fsum1f_(sa1, sa2, esa12).fover(esa1_2) * e2 + _D2atanheE(sa1, sa2, E) * E.e21
485 s1_qZ = (axi * self._qZ - bxi) * dsn_2 / dtb12
486 ta0 = self._ta0(s1_qZ, (ta1 + ta2) * _0_5, E)
487 return ta0, C
489 def _tanf(self, txi): # in .Ellipsoid.auxAuthalic
490 '''(INTERNAL) Function M{tan-phi from tan-xi}.
491 '''
492 tol = _tol(_TOL, txi)
493 e2 = self.ellipsoid.e2
494 qx = self._qx
496 ta = txi
497 _Ta2 = Fsum(ta).fsum2f_
498 _txif = self._txif
499 _1 = _1_0
500 for self._iteration in range(1, _NUMIT): # max 2, mean 1.99
501 # dtxi / dta = (scxi / sca)^3 * 2 * (1 - e^2)
502 # / (qZ * (1 - e^2 * sa^2)^2)
503 ta2 = ta**2
504 sca2 = _1 + ta2
505 txia = _txif(ta)
506 s3qx = sqrt3(sca2 / (txia**2 + _1)) * qx # * _1_x21(txia)
507 eta2 = (_1 - e2 * ta2 / sca2)**2
508 ta, d = _Ta2((txi - txia) * s3qx * eta2)
509 if fabs(d) < tol:
510 return ta
511 raise AlbersError(Fmt.no_convergence(d, tol), txt=repr(self))
513 def toRepr(self, prec=6, **unused): # PYCHOK expected
514 '''Return a string representation of this projection.
516 @kwarg prec: Number of (decimal) digits, unstripped (C{int}).
518 @return: This projection as C{"<classname>(lat1, lat2, ...)"}
519 (C{str}).
520 '''
521 t = self.toStr(prec=prec, sep=_COMMASPACE_)
522 return Fmt.PAREN(self.classname, t)
524 def toStr(self, prec=6, sep=_SPACE_, **unused): # PYCHOK expected
525 '''Return a string representation of this projection.
527 @kwarg prec: Number of (decimal) digits, unstripped (C{int}).
528 @kwarg sep: Separator to join (C{str}).
530 @return: This projection as C{"lat1 lat2"} (C{str}).
531 '''
532 k = self._k
533 t = (self.lat1, self.lat2, self._k0) if k is _k1_ else (
534 (self.lat1, self._k0) if k is _k0_ else
535 (self.lat1,))
536 t = strs(t, prec=prec)
537 if k:
538 t = t[:-1] + (Fmt.EQUAL(k, t[-1]),)
539 if self.datum != _WGS84:
540 t += Fmt.EQUAL(datum=self.datum),
541 if self.name:
542 t += Fmt.EQUAL(name=repr(self.name)),
543 return t if sep is None else sep.join(t)
545 def _txif(self, ta): # in .Ellipsoid.auxAuthalic
546 '''(INTERNAL) Function M{tan-xi from tan-phi}.
547 '''
548 E = self.ellipsoid
549 _1 = _1_0
551 ca2 = _1_x21(ta)
552 sa = sqrt(ca2) * fabs(ta) # enforce odd parity
553 sa1 = _1 + sa
555 es1 = sa * E.e2
556 es1m1 = sa1 * (_1 - es1)
557 es1p1 = sa1 / (_1 + es1)
558 es2m1 = _1 - sa * es1
559 es2m1a = es2m1 * E.e21 # e2m
560 s = sqrt((ca2 / (es1p1 * es2m1a) + _atanheE(ca2 / es1m1, E))
561 * (es1m1 / es2m1a + _atanheE(es1p1, E)))
562 t = _Fsum1f_(sa / es2m1, _atanheE(sa, E)).fover(s)
563 return neg(t) if ta < 0 else t
566class AlbersEqualArea(_AlbersBase):
567 '''An Albers equal-area (authalic) projection with a single standard parallel.
569 @see: L{AlbersEqualArea2} and L{AlbersEqualArea4}.
570 '''
571 _k = _k0_
573 def __init__(self, lat, k0=1, datum=_WGS84, **name):
574 '''New L{AlbersEqualArea} projection.
576 @arg lat: Standard parallel (C{degrees}).
577 @kwarg k0: Azimuthal scale on the standard parallel (C{scalar}).
578 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid},
579 L{Ellipsoid2} or L{a_f2Tuple}).
580 @kwarg name: Optional C{B{name}=NN} for the projection (C{str}).
582 @raise AlbersError: Invalid B{C{lat}}, B{C{k0}} or no convergence.
583 '''
584 self._lat1 = self._lat2 = lat = _Lat(lat1=lat)
585 args = tuple(sincos2d(lat)) * 2 + (_Ks(k0=k0), datum)
586 _AlbersBase.__init__(self, *args, **name)
589class AlbersEqualArea2(_AlbersBase):
590 '''An Albers equal-area (authalic) projection with two standard parallels.
592 @see: L{AlbersEqualArea} and L{AlbersEqualArea4}.
593 '''
594 _k = _k1_
596 def __init__(self, lat1, lat2, k1=1, datum=_WGS84, **name):
597 '''New L{AlbersEqualArea2} projection.
599 @arg lat1: First standard parallel (C{degrees}).
600 @arg lat2: Second standard parallel (C{degrees}).
601 @kwarg k1: Azimuthal scale on the standard parallels (C{scalar}).
602 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid},
603 L{Ellipsoid2} or L{a_f2Tuple}).
604 @kwarg name: Optional C{B{name}=NN} for the projection (C{str}).
606 @raise AlbersError: Invalid B{C{lat1}}m B{C{lat2}}, B{C{k1}}
607 or no convergence.
608 '''
609 self._lat1, self._lat2 = lats = _Lat(lat1=lat1), _Lat(lat2=lat2)
610 args = tuple(sincos2d_(*lats)) + (_Ks(k1=k1), datum)
611 _AlbersBase.__init__(self, *args, **name)
614class AlbersEqualArea4(_AlbersBase):
615 '''An Albers equal-area (authalic) projection specified by the C{sin}
616 and C{cos} of both standard parallels.
618 @see: L{AlbersEqualArea} and L{AlbersEqualArea2}.
619 '''
620 _k = _k1_
622 def __init__(self, slat1, clat1, slat2, clat2, k1=1, datum=_WGS84, **name):
623 '''New L{AlbersEqualArea4} projection.
625 @arg slat1: Sine of first standard parallel (C{scalar}).
626 @arg clat1: Cosine of first standard parallel (non-negative C{scalar}).
627 @arg slat2: Sine of second standard parallel (C{scalar}).
628 @arg clat2: Cosine of second standard parallel (non-negative C{scalar}).
629 @kwarg k1: Azimuthal scale on the standard parallels (C{scalar}).
630 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid},
631 L{Ellipsoid2} or L{a_f2Tuple}).
632 @kwarg name: Optional C{B{name}=NN} for the projection (C{str}).
634 @raise AlbersError: Negative B{C{clat1}} or B{C{clat2}}, B{C{slat1}}
635 and B{C{slat2}} have opposite signs (hemispheres),
636 invalid B{C{k1}} or no convergence.
637 '''
638 def _Lat_s_c3(n, s, c):
639 r = Float_(hypot(s, c), name=n, Error=AlbersError)
640 L = _Lat( atan1d(s, c), name=n)
641 return L, (s / r), (c / r)
643 self._lat1, sa1, ca1 = _Lat_s_c3(_lat1_, slat1, clat1)
644 self._lat2, sa2, ca2 = _Lat_s_c3(_lat2_, slat2, clat2)
645 _AlbersBase.__init__(self, sa1, ca1, sa2, ca2, _Ks(k1=k1), datum, **name)
648class AlbersEqualAreaCylindrical(_AlbersBase):
649 '''An L{AlbersEqualArea} projection at C{lat=0} and C{k0=1} degenerating
650 to the cylindrical-equal-area projection.
651 '''
652 _lat1 = _lat2 = _Lat(lat1=_0_0)
654 def __init__(self, lat=_0_0, datum=_WGS84, **name):
655 '''New L{AlbersEqualAreaCylindrical} projection.
657 @kwarg lat: Standard parallel (C{0 degrees} I{fixed}).
658 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid},
659 L{Ellipsoid2} or L{a_f2Tuple}).
660 @kwarg name: Optional C{B{name}=NN} for the projection (C{str}).
661 '''
662 _xlat(lat, _0_0, AlbersEqualAreaCylindrical)
663 _AlbersBase.__init__(self, _0_0, _1_0, _0_0, _1_0, 1, datum, **name)
666class AlbersEqualAreaNorth(_AlbersBase):
667 '''An azimuthal L{AlbersEqualArea} projection at C{lat=90} and C{k0=1}
668 degenerating to the L{azimuthal} L{LambertEqualArea} projection.
669 '''
670 _lat1 = _lat2 = _Lat(lat1=_90_0)
672 def __init__(self, lat=_90_0, datum=_WGS84, **name):
673 '''New L{AlbersEqualAreaNorth} projection.
675 @kwarg lat: Standard parallel (C{90 degrees} I{fixed}).
676 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid},
677 L{Ellipsoid2} or L{a_f2Tuple}).
678 @kwarg name: Optional C{B{name}=NN} for the projection (C{str}).
679 '''
680 _xlat(lat, _90_0, AlbersEqualAreaNorth)
681 _AlbersBase.__init__(self, _1_0, _0_0, _1_0, _0_0, 1, datum, **name)
684class AlbersEqualAreaSouth(_AlbersBase):
685 '''An azimuthal L{AlbersEqualArea} projection at C{lat=-90} and C{k0=1}
686 degenerating to the L{azimuthal} L{LambertEqualArea} projection.
687 '''
688 _lat1 = _lat2 = _Lat(lat1=_N_90_0)
690 def __init__(self, lat=_N_90_0, datum=_WGS84, **name):
691 '''New L{AlbersEqualAreaSouth} projection.
693 @kwarg lat: Standard parallel (C{-90 degrees} I{fixed}).
694 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid},
695 L{Ellipsoid2} or L{a_f2Tuple}).
696 @kwarg name: Optional C{B{name}=NN} for the projection (C{str}).
697 '''
698 _xlat(lat, _N_90_0, AlbersEqualAreaSouth)
699 _AlbersBase.__init__(self, _N_1_0, _0_0, _N_1_0, _0_0, 1, datum, **name)
702class Albers7Tuple(_NamedTuple):
703 '''7-Tuple C{(x, y, lat, lon, gamma, scale, datum)}, in C{meter},
704 C{meter}, C{degrees90}, C{degrees180}, C{degrees360}, C{scalar} and
705 C{Datum} where C{(x, y)} is the projected, C{(lat, lon)} the geodetic
706 location, C{gamma} the meridian convergence at point, the bearing of
707 the y-axis measured clockwise from true North and C{scale} is the
708 azimuthal scale of the projection at point. The radial scale is
709 the reciprocal C{1 / scale}.
710 '''
711 _Names_ = (_x_, _y_, _lat_, _lon_, _gamma_, _scale_, _datum_)
712 _Units_ = ( Meter, Meter, Lat, Lon, Bearing, _Pass, _Pass)
715def _atanh1(x):
716 '''(INTERNAL) Function M{atanh(sqrt(x)) / sqrt(x) - 1}.
717 '''
718 s = fabs(x)
719 if 0 < s < _0_5: # for typical ...
720 # x < E.e^2 == 2 * E.f use ...
721 # x / 3 + x^2 / 5 + x^3 / 7 + ...
722 y, k = x, 3
723 _S2 = Fsum(y / k).fsum2f_
724 for _ in range(_TERMS): # 9 terms
725 y *= x # x**n
726 k += 2 # 2*n + 1
727 s, d = _S2(y / k)
728 if not d:
729 break
730 elif s:
731 s = sqrt(s)
732 s = (atanh(s) if x > 0 else atan1(s)) / s - _1_0
733 return s
736def _atanheE(x, E): # see Ellipsoid._es_atanh, .AuxLat._atanhee
737 '''(INTERNAL) Function M{atanhee(x)}, defined as ...
738 atanh( E.e * x) / E.e if f > 0 # oblate
739 atan (sqrt(-E.e2) * x) / sqrt(-E.e2) if f < 0 # prolate
740 x if f = 0.
741 '''
742 e = E.e # == sqrt(E.e2abs)
743 if e and x:
744 if E.f > 0: # .isOblate
745 x = atanh(x * e) / e
746 elif E.f < 0: # .isProlate
747 x = atan1(x * e) / e
748 return x
751def _DatanheE(x, y, E): # see .rhumb.ekx._DeatanhE
752 '''(INTERNAL) Function M{Datanhee(x, y)}, defined as
753 M{atanhee((x - y) / (1 - E.e^2 * x * y)) / (x - y)}.
754 '''
755 e = _1_0 - E.e2 * x * y
756 if e:
757 d = x - y
758 e = (_atanheE(d / e, E) / d) if d else (_1_0 / e)
759 return e
762def _D2atanheE(x, y, E):
763 '''(INTERNAL) Function M{D2atanhee(x, y)}, defined as
764 M{(Datanhee(1, y) - Datanhee(1, x)) / (y - x)}.
765 '''
766 s, e2 = _0_0, E.e2
767 if e2:
768 if ((fabs(x) + fabs(y)) * e2) < _0_5:
769 e = z = _1_0
770 k = 1
771 T = Fsum() # Taylor expansion
772 _T = T.Fsumf_
773 _C = Fsum().Fsum_
774 _S2 = Fsum().fsum2_
775 for _ in range(_TERMS): # 15 terms
776 T *= y; P = _T(z); z *= x # PYCHOK ;
777 T *= y; Q = _T(z); z *= x # PYCHOK ;
778 e *= e2
779 k += 2
780 s, d = _S2(_C(P, Q) * e / k)
781 if not d:
782 break
783 else: # PYCHOK no cover
784 s = _1_0 - x
785 if s:
786 s = (_DatanheE(_1_0, y, E) - _DatanheE(x, y, E)) / s
787 return s
790def _Dsn(x, y, sx, sy):
791 '''(INTERNAL) Divided differences, defined as M{Df(x, y) = (f(x) - f(y)) / (x - y)}
792 with M{sn(x) = x / sqrt(1 + x^2)}: M{Dsn(x, y) = (x + y) / ((sn(x) + sn(y)) *
793 (1 + x^2) * (1 + y^2))}.
795 @see: U{W. M. Kahan and R. J. Fateman, "Sympbolic Computation of Divided
796 Differences"<https://People.EECS.Berkeley.EDU/~fateman/papers/divdiff.pdf>},
797 U{ACM SIGSAM Bulletin 33(2), 7-28 (1999)<https://DOI.org/10.1145/334714.334716>}
798 and U{AlbersEqualArea.hpp
799 <https://GeographicLib.SourceForge.io/C++/doc/AlbersEqualArea_8hpp_source.html>}.
800 '''
801 # sx = x / hypot1(x)
802 d, t = _1_0, (x * y)
803 if t > 0:
804 s = sx + sy
805 if s:
806 t = sx * sy / t
807 d = t**2 * (x + y) / s
808 elif x != y:
809 d = (sx - sy) / (x - y)
810 return d
813def _tol(tol, x):
814 '''(INTERNAL) Converge tolerance.
815 '''
816 return tol * max(_1_0, fabs(x))
819def _1_x21(x):
820 '''(INTERNAL) Return M{1 / (x**2 + 1)}.
821 '''
822 return _1_0 / (x**2 + _1_0)
825def _xlat(lat, f, where):
826 '''(INTERNAL) check fixed C{lat}.
827 '''
828 if lat is not f and _Lat(lat=lat) != f:
829 t = unstr(where, lat=lat)
830 raise AlbersError(t, txt_not_=f)
833__all__ += _ALL_DOCS(_AlbersBase)
835# **) MIT License
836#
837# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.
838#
839# Permission is hereby granted, free of charge, to any person obtaining a
840# copy of this software and associated documentation files (the "Software"),
841# to deal in the Software without restriction, including without limitation
842# the rights to use, copy, modify, merge, publish, distribute, sublicense,
843# and/or sell copies of the Software, and to permit persons to whom the
844# Software is furnished to do so, subject to the following conditions:
845#
846# The above copyright notice and this permission notice shall be included
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848#
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850# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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854# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
855# OTHER DEALINGS IN THE SOFTWARE.