Coverage for pygeodesy/albers.py: 97%
423 statements
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« prev ^ index » next coverage.py v7.2.2, created at 2023-08-12 12:31 -0400
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, isnear0, \
20 _0_0, _0_5, _1_0, _N_1_0, _2_0, _N_2_0, \
21 _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_
26from pygeodesy.interns import NN, _COMMASPACE_, _datum_, _gamma_, _k0_, \
27 _lat_, _lat1_, _lat2_, _lon_, _name_, _not_, \
28 _negative_, _scale_, _SPACE_, _x_, _y_
29from pygeodesy.karney import _diff182, _norm180, _signBit
30from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY
31from pygeodesy.named import _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 atand, atan2d, degrees360, sincos2, \
36 sincos2d, sincos2d_
38from math import atan, atan2, atanh, degrees, fabs, radians, sqrt
40__all__ = _ALL_LAZY.albers
41__version__ = '23.07.01'
43_k1_ = 'k1'
44_NUMIT = 8 # XXX 4?
45_NUMIT0 = 41 # XXX 21?
46_TERMS = 21 # XXX 16?
47_TOL0 = sqrt3(_TOL)
50def _Ks(**name_k):
51 '''(INTERNAL) Scale C{B{k} >= EPS0}.
52 '''
53 return Scalar_(Error=AlbersError, low=EPS0, **name_k) # > 0
56def _Lat(*lat, **Error_name_lat):
57 '''(INTERNAL) Latitude C{-90 <= B{lat} <= 90}.
58 '''
59 kwds = _xkwds(Error_name_lat, Error=AlbersError)
60 return Lat_(*lat, **kwds)
63def _qZx(alb):
64 '''(INTERNAL) Set C{alb._qZ} and C{alb._qx}.
65 '''
66 E = alb._datum.ellipsoid
67 alb._qZ = qZ = _1_0 + E.e21 * _atanhee(_1_0, E)
68 alb._qx = qZ / (_2_0 * E.e21)
69 return qZ
72class AlbersError(_ValueError):
73 '''An L{AlbersEqualArea}, L{AlbersEqualArea2}, L{AlbersEqualArea4},
74 L{AlbersEqualAreaCylindrical}, L{AlbersEqualAreaNorth},
75 L{AlbersEqualAreaSouth} or L{Albers7Tuple} issue.
76 '''
77 pass
80class _AlbersBase(_NamedBase):
81 '''(INTERNAL) Base class for C{AlbersEqualArea...} projections.
83 @see: I{Karney}'s C++ class U{AlbersEqualArea<https://GeographicLib.SourceForge.io/
84 html/classGeographicLib_1_1AlbersEqualArea.html>}, method C{Init}.
85 '''
86 _datum = _WGS84
87 _k = NN # or _k0_ or _k1_
88 _k0 = _Ks(k0=_1_0)
89 _k0n0 = None # (INTERNAL) k0 * no
90 _k02 = _1_0 # (INTERNAL) k0**2
91 _k02n0 = None # (INTERNAL) k02 * n0
92 _lat0 = None # lat origin
93 _lat1 = None # let 1st parallel
94 _lat2 = None # lat 2nd parallel
95 _m0 = _0_0 # if polar else sqrt(m02)
96 _m02 = None # (INTERNAL) cached
97 _n0 = None # (INTERNAL) cached
98 _nrho0 = _0_0 # if polar else m0 * E.a
99 _polar = False
100 _qx = None # (INTERNAL) see _qZx
101 _qZ = None # (INTERNAL) see _qZx
102 _scxi0_ = None # (INTERNAL) sec(xi) / (qZ * E.a2)
103 _sign = +1
104 _sxi0 = None # (INTERNAL) sin(xi)
105 _txi0 = None # (INTERNAL) tan(xi)
107 def __init__(self, sa1, ca1, sa2, ca2, k, datum, name):
108 '''(INTERNAL) New C{AlbersEqualArea...} instance.
109 '''
110 qZ = self._qZ
111 if datum not in (None, self._datum):
112 self._datum = _ellipsoidal_datum(datum, name=name)
113 qZ = _qZx(self)
114 elif qZ is None:
115 qZ = _qZx(_AlbersBase)
116 if name:
117 self.name = name
119 E = self.ellipsoid
120 c = min(ca1, ca2)
121 if _signBit(c):
122 raise AlbersError(clat1=ca1, clat2=ca2, txt=_negative_)
123 polar = c < EPS0 # == 0
125 # determine hemisphere of tangent latitude
126 if sa1 < 0: # and sa2 < 0:
127 self._sign = -1
128 # internally, tangent latitude positive
129 sa1, sa2 = neg_(sa1, sa2)
130 if sa1 > sa2: # make phi1 < phi2
131 sa1, sa2 = sa2, sa1
132 ca1, ca2 = ca2, ca1
133 if sa1 < 0: # or sa2 < 0:
134 raise AlbersError(slat1=sa1, slat2=sa2, txt=_negative_)
135 # avoid singularities at poles
136 ca1, ca2 = max(EPS0, ca1), max(EPS0, ca2)
137 ta1, ta2 = (sa1 / ca1), (sa2 / ca2)
139 par1 = fabs(ta1 - ta2) < EPS02 # ta1 == ta2
140 if par1 or polar:
141 ta0, C = ta2, _1_0
142 else:
143 ta0, C = self._ta0C2(ca1, sa1, ta1, ca2, sa2, ta2)
145 self._lat0 = _Lat(lat0=self._sign * atand(ta0))
146 self._m02 = m02 = _1_x21(E.f1 * ta0)
147 self._n0 = n0 = ta0 / hypot1(ta0)
148 if polar:
149 self._polar = True
150# self._nrho0 = self._m0 = _0_0
151 else: # m0 = nrho0 / E.a
152 self._m0 = sqrt(m02)
153 self._nrho0 = self._m0 * E.a
154 t = self._txi0 = self._txif(ta0)
155 h = hypot1(t)
156 s = self._sxi0 = t / h
157 if par1:
158 self._k0n0 = self._k02n0 = n0
159 else:
160 self._k0s(k * sqrt(C / (m02 + n0 * qZ * s)))
161 self._scxi0_ = h / (qZ * E.a2)
163 def _a_b_sxi3(self, *ca_sa_ta_scb_4s):
164 '''(INTERNAL) Sum of C{sm1} terms and C{sin(xi)}s for ._s1_qZ_C2.
165 '''
166 _1 = _1_0
167 a = b = s = _0_0
168 for ca, sa, ta, scb in ca_sa_ta_scb_4s:
169 cxi, sxi, _ = self._cstxif3(ta)
170 if sa > 0:
171 sa += _1
172 a += (cxi / ca)**2 * sa / (sxi + _1)
173 b += scb * ca**2 / sa
174 else:
175 sa = _1 - sa
176 a += (_1 - sxi) / sa
177 b += scb * sa
178 s += sxi
179 return a, b, s
181 def _azik(self, t, ta):
182 '''(INTERNAL) Compute the azimuthal scale C{_Ks(k=k)}.
183 '''
184 E = self.ellipsoid
185 t *= self._k0
186 return _Ks(k=t * hypot1(E.b_a * ta) / E.a)
188 def _cstxif3(self, ta):
189 '''(INTERNAL) Get 3-tuple C{(cos, sin, tan)} of M{xi(ta)}.
190 '''
191 t = self._txif(ta)
192 c = _1_0 / hypot1(t)
193 s = c * t
194 return c, s, t
196 @Property_RO
197 def datum(self):
198 '''Get the datum (L{Datum}).
199 '''
200 return self._datum
202 @Property_RO
203 def ellipsoid(self):
204 '''Get the datum's ellipsoid (L{Ellipsoid}).
205 '''
206 return self.datum.ellipsoid
208 @Property_RO
209 def equatoradius(self):
210 '''Get the geodesic's equatorial radius, semi-axis (C{meter}).
211 '''
212 return self.ellipsoid.a
214 @Property_RO
215 def flattening(self):
216 '''Get the geodesic's flattening (C{float}).
217 '''
218 return self.ellipsoid.f
220 def forward(self, lat, lon, lon0=0, name=NN):
221 '''Convert a geodetic location to east- and northing.
223 @arg lat: Latitude of the location (C{degrees}).
224 @arg lon: Longitude of the location (C{degrees}).
225 @kwarg lon0: Optional central meridian longitude (C{degrees}).
226 @kwarg name: Optional name for the location (C{str}).
228 @return: An L{Albers7Tuple}C{(x, y, lat, lon, gamma, scale, datum)}.
230 @note: The origin latitude is returned by C{property lat0}. No
231 false easting or northing is added. The value of B{C{lat}}
232 should be in the range C{[-90..90] degrees}. The returned
233 values C{x} and C{y} will be large but finite for points
234 projecting to infinity, i.e. one or both of the poles.
235 '''
236 a = self.ellipsoid.a
237 s = self._sign
239 k0 = self._k0
240 n0 = self._n0
241 nrho0 = self._nrho0
242 txi0 = self._txi0
244 sa, ca = sincos2d(s * _Lat(lat=lat))
245 ta = sa / max(EPS0, ca)
247 _, sxi, txi = self._cstxif3(ta)
248 dq = _Dsn(txi, txi0, sxi, self._sxi0) * \
249 (txi - txi0) * self._qZ
250 drho = a * dq / (sqrt(self._m02 - n0 * dq) + self._m0)
252 lon, _ = _diff182(lon0, lon)
253 x = radians(lon)
255 th = self._k02n0 * x
256 sth, cth = sincos2(th) # XXX sin, cos
257 if n0:
258 x = sth / n0
259 y = (_1_0 - cth) if cth < 0 else (sth**2 / (cth + _1_0))
260 y *= nrho0 / n0
261 else:
262 x *= self._k02
263 y = _0_0
264 t = nrho0 - n0 * drho
265 x = t * x / k0
266 y = s * (y + drho * cth) / k0
268 g = degrees360(s * th)
269 if t:
270 k0 = self._azik(t, ta)
271 return Albers7Tuple(x, y, lat, lon, g, k0, self.datum,
272 name=name or self.name)
274 @Property_RO
275 def ispolar(self):
276 '''Is this projection polar (C{bool})?
277 '''
278 return self._polar
280 isPolar = ispolar # synonym
282 def _k0s(self, k0):
283 '''(INTERNAL) Set C{._k0}, C{._k02}, etc.
284 '''
285 self._k0 = k = _Ks(k0=k0)
286 self._k02 = k2 = k**2
287 self._k0n0 = k * self._n0
288 self._k02n0 = k2 * self._n0
290 @Property_RO
291 def lat0(self):
292 '''Get the latitude of the projection origin (C{degrees}).
294 This is the latitude of minimum azimuthal scale and
295 equals the B{C{lat}} in the 1-parallel L{AlbersEqualArea}
296 and lies between B{C{lat1}} and B{C{lat2}} for the
297 2-parallel L{AlbersEqualArea2} and L{AlbersEqualArea4}
298 projections.
299 '''
300 return self._lat0
302 @Property_RO
303 def lat1(self):
304 '''Get the latitude of the first parallel (C{degrees}).
305 '''
306 return self._lat1
308 @Property_RO
309 def lat2(self):
310 '''Get the latitude of the second parallel (C{degrees}).
312 @note: The second and first parallel latitudes are the
313 same instance for 1-parallel C{AlbersEqualArea*}
314 projections.
315 '''
316 return self._lat2
318 @deprecated_Property_RO
319 def majoradius(self): # PYCHOK no cover
320 '''DEPRECATED, use property C{equatoradius}.'''
321 return self.equatoradius
323 def rescale0(self, lat, k=1): # PYCHOK no cover
324 '''Set the azimuthal scale for this projection.
326 @arg lat: Northern latitude (C{degrees}).
327 @arg k: Azimuthal scale at latitude B{C{lat}} (C{scalar}).
329 @raise AlbersError: Invalid B{C{lat}} or B{C{k}}.
331 @note: This allows a I{latitude of conformality} to be specified.
332 '''
333 k0 = _Ks(k=k) / self.forward(lat, _0_0).scale
334 if self._k0 != k0:
335 _update_all(self)
336 self._k0s(k0)
338 def reverse(self, x, y, lon0=0, name=NN, LatLon=None, **LatLon_kwds):
339 '''Convert an east- and northing location to geodetic lat- and longitude.
341 @arg x: Easting of the location (C{meter}).
342 @arg y: Northing of the location (C{meter}).
343 @kwarg lon0: Optional central meridian longitude (C{degrees}).
344 @kwarg name: Optional name for the location (C{str}).
345 @kwarg LatLon: Class to use (C{LatLon}) or C{None}.
346 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
347 arguments, ignored if C{B{LatLon} is None}.
349 @return: The geodetic (C{LatLon}) or if B{C{LatLon}} is C{None} an
350 L{Albers7Tuple}C{(x, y, lat, lon, gamma, scale, datum)}.
352 @note: The origin latitude is returned by C{property lat0}. No
353 false easting or northing is added. The returned value of
354 C{lon} is in the range C{[-180..180] degrees} and C{lat}
355 is in the range C{[-90..90] degrees}. If the given
356 B{C{x}} or B{C{y}} point is outside the valid projected
357 space the nearest pole is returned.
358 '''
359 k0 = self._k0
360 n0 = self._n0
361 k0n0 = self._k0n0
362 s = self._sign
363 txi = self._txi0
365 x = Meter(x=x)
366 nx = k0n0 * x
367 y = Meter(y=y)
368 y_ = s * y
369 ny = k0n0 * y_
370 t = nrho0 = self._nrho0
371 y1 = nrho0 - ny
373 den = drho = hypot(nx, y1) + nrho0 # 0 implies origin with polar aspect
374 if den:
375 drho = fsum1f_(x * nx, y_ * nrho0 * _N_2_0, y_ * ny) * k0 / den
376 # dsxia = scxi0 * dsxi
377 t += drho * n0
378 d_ = (nrho0 + t) * drho * self._scxi0_ # / (qZ * E.a2)
379 d_2 = (txi * _2_0 - d_) * d_ + _1_0
380 txi = (txi - d_) / (sqrt(d_2) if d_2 > EPS02 else EPS0)
382 ta = self._tanf(txi)
383 lat = atand(s * ta)
385 th = atan2(nx, y1)
386 lon = degrees((th / self._k02n0) if n0 else (x / (y1 * k0)))
387 if lon0:
388 lon += _norm180(lon0)
389 lon = _norm180(lon)
391 n = name or self.name
392 if LatLon is None:
393 g = degrees360(s * th)
394 if den:
395 k0 = self._azik(t, ta)
396 r = Albers7Tuple(x, y, lat, lon, g, k0, self.datum, name=n)
397 else: # PYCHOK no cover
398 kwds = _xkwds(LatLon_kwds, datum=self.datum, name=n)
399 r = LatLon(lat, lon, **kwds)
400 return r
402 @Property_RO
403 def scale0(self):
404 '''Get the central scale for the projection (C{float}).
406 This is the azimuthal scale on the latitude of origin
407 of the projection, see C{property lat0}.
408 '''
409 return self._k0
411 def _ta0(self, s1_qZ, ta0, E):
412 '''(INTERNAL) Refine C{ta0} for C{._ta0C2}.
413 '''
414 e2 = E.e2
415 e21 = E.e21
416 e22 = E.e22 # == e2 / e21
417 tol = _tol(_TOL0, ta0)
418 _Ta02 = Fsum(ta0).fsum2_
419 _fabs = fabs
420 _fsum1f_ = fsum1f_
421 _sqrt = sqrt
422 _1, _2 = _1_0, _2_0
423 _4, _6 = _4_0, _6_0
424 for self._iteration in range(1, _NUMIT0): # 4 trips
425 ta02 = ta0**2
426 sca02 = ta02 + _1
427 sca0 = _sqrt(sca02)
428 sa0 = ta0 / sca0
429 sa01 = sa0 + _1
430 sa02 = sa0**2
431 # sa0m = 1 - sa0 = 1 / (sec(a0) * (tan(a0) + sec(a0)))
432 sa0m = _1 / (sca0 * (ta0 + sca0)) # scb0^2 * sa0
433 sa0m1 = sa0m / (_1 - e2 * sa0)
434 sa021 = _1 - e2 * sa02
436 g = (_1 + ta02 * e21) * sa0
437 dg = (_1 + ta02 * _2) * sca02 * e21 + e2
438 D = (_1 - (_1 + sa0 * _2 * sa01) * e2) * sa0m / (e21 * sa01) # dD/dsa0
439 dD = (_2 - (_6 + sa0 * _4) * sa02 * e2) / (e21 * sa01**2)
440 BA = (_atanhs1(e2 * sa0m1**2) * e21 - e2 * sa0m) * sa0m1 \
441 - (_2 + (_1 + e2) * sa0) * sa0m**2 * e22 / sa021 # B + A
442 d = (_4 - (_1 + sa02) * e2 * _2) * e22 / (sa021**2 * sca02) # dAB
443 u = _fsum1f_(s1_qZ * g, -D, g * BA)
444 du = _fsum1f_(s1_qZ * dg, dD, dg * BA, g * d)
445 ta0, d = _Ta02(-u / du * (sca0 * sca02))
446 if _fabs(d) < tol:
447 return ta0
448 raise AlbersError(Fmt.no_convergence(d, tol), txt=repr(self))
450 def _ta0C2(self, ca1, sa1, ta1, ca2, sa2, ta2):
451 '''(INTERNAL) Compute C{ta0} and C{C} for C{.__init__}.
452 '''
453 _1 = _1_0
454 _fsum1f_ = fsum1f_
455 E = self.ellipsoid
456 f1, e2 = E.f1, E.e2
458 tb1 = f1 * ta1
459 tb2 = f1 * ta2
460 dtb12 = f1 * (tb1 + tb2)
461 scb12 = _1 + tb1**2
462 scb22 = _1 + tb2**2
464 dsn_2 = _Dsn(ta2, ta1, sa2, sa1) * _0_5
465 sa12 = sa1 * sa2
467 esa1_2 = (_1 - e2 * sa1**2) \
468 * (_1 - e2 * sa2**2)
469 esa12 = _1 + e2 * sa12
471 axi, bxi, sxi = self._a_b_sxi3((ca1, sa1, ta1, scb12),
472 (ca2, sa2, ta2, scb22))
474 dsxi = (esa12 / esa1_2 + _Datanhee(sa2, sa1, E)) * dsn_2 / self._qx
475 C = _fsum1f_(sxi * dtb12 / dsxi, scb22, scb12) / (scb22 * scb12 * _2_0)
477 sa12 = _fsum1f_(sa1, sa2, sa12)
478 axi *= (sa12 * e2 + _1) / (sa12 + _1)
479 bxi *= _fsum1f_(sa1, sa2, esa12) * e2 / esa1_2 + E.e21 * _D2atanhee(sa1, sa2, E)
480 s1_qZ = (axi * self._qZ - bxi) * dsn_2 / dtb12
481 ta0 = self._ta0(s1_qZ, (ta1 + ta2) * _0_5, E)
482 return ta0, C
484 def _tanf(self, txi): # in .Ellipsoid.auxAuthalic
485 '''(INTERNAL) Function M{tan-phi from tan-xi}.
486 '''
487 tol = _tol(_TOL, txi)
488 e2 = self.ellipsoid.e2
489 qx = self._qx
491 ta = txi
492 _Ta2 = Fsum(ta).fsum2_
493 _fabs = fabs
494 _sqrt3 = sqrt3
495 _txif = self._txif
496 _1 = _1_0
497 for self._iteration in range(1, _NUMIT): # max 2, mean 1.99
498 # dtxi / dta = (scxi / sca)^3 * 2 * (1 - e^2)
499 # / (qZ * (1 - e^2 * sa^2)^2)
500 ta2 = ta**2
501 sca2 = _1 + ta2
502 txia = _txif(ta)
503 s3qx = _sqrt3(sca2 / (txia**2 + _1)) * qx # * _1_x21(txia)
504 eta2 = (_1 - e2 * ta2 / sca2)**2
505 ta, d = _Ta2((txi - txia) * s3qx * eta2)
506 if _fabs(d) < tol:
507 return ta
508 raise AlbersError(Fmt.no_convergence(d, tol), txt=repr(self))
510 def toRepr(self, prec=6, **unused): # PYCHOK expected
511 '''Return a string representation of this projection.
513 @kwarg prec: Number of (decimal) digits, unstripped (C{int}).
515 @return: This projection as C{"<classname>(lat1, lat2, ...)"}
516 (C{str}).
517 '''
518 t = self.toStr(prec=prec, sep=_COMMASPACE_)
519 return Fmt.PAREN(self.classname, t)
521 def toStr(self, prec=6, sep=_SPACE_, **unused): # PYCHOK expected
522 '''Return a string representation of this projection.
524 @kwarg prec: Number of (decimal) digits, unstripped (C{int}).
525 @kwarg sep: Separator to join (C{str}).
527 @return: This projection as C{"lat1 lat2"} (C{str}).
528 '''
529 k = self._k
530 t = (self.lat1, self.lat2, self._k0) if k is _k1_ else (
531 (self.lat1, self._k0) if k is _k0_ else
532 (self.lat1,))
533 t = strs(t, prec=prec)
534 if k:
535 t = t[:-1] + (Fmt.EQUAL(k, t[-1]),)
536 if self.datum != _WGS84:
537 t += (Fmt.EQUAL(_datum_, self.datum),)
538 if self.name:
539 t += (Fmt.EQUAL(_name_, repr(self.name)),)
540 return t if sep is None else sep.join(t)
542 def _txif(self, ta): # in .Ellipsoid.auxAuthalic
543 '''(INTERNAL) Function M{tan-xi from tan-phi}.
544 '''
545 E = self.ellipsoid
547 ca2 = _1_x21(ta)
548 sa = sqrt(ca2) * fabs(ta) # enforce odd parity
549 sa1 = _1_0 + sa
551 es1 = sa * E.e2
552 es1m1 = sa1 * (_1_0 - es1)
553 es1p1 = sa1 / (_1_0 + es1)
554 es2m1 = _1_0 - sa * es1
555 es2m1a = es2m1 * E.e21 # e2m
556 s = sqrt((ca2 / (es1p1 * es2m1a) + _atanhee(ca2 / es1m1, E))
557 * (es1m1 / es2m1a + _atanhee(es1p1, E)))
558 t = (sa / es2m1 + _atanhee(sa, E)) / s
559 return neg(t) if ta < 0 else t
562class AlbersEqualArea(_AlbersBase):
563 '''An Albers equal-area (authalic) projection with a single standard parallel.
565 @see: L{AlbersEqualArea2} and L{AlbersEqualArea4}.
566 '''
567 _k = _k0_
569 def __init__(self, lat, k0=1, datum=_WGS84, name=NN):
570 '''New L{AlbersEqualArea} projection.
572 @arg lat: Standard parallel (C{degrees}).
573 @kwarg k0: Azimuthal scale on the standard parallel (C{scalar}).
574 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid},
575 L{Ellipsoid2} or L{a_f2Tuple}).
576 @kwarg name: Optional name for the projection (C{str}).
578 @raise AlbersError: Invalid B{C{lat}}, B{C{k0}} or no convergence.
579 '''
580 self._lat1 = self._lat2 = lat = _Lat(lat1=lat)
581 args = tuple(sincos2d(lat)) * 2 + (_Ks(k0=k0), datum, name)
582 _AlbersBase.__init__(self, *args)
585class AlbersEqualArea2(_AlbersBase):
586 '''An Albers equal-area (authalic) projection with two standard parallels.
588 @see: L{AlbersEqualArea} and L{AlbersEqualArea4}.
589 '''
590 _k = _k1_
592 def __init__(self, lat1, lat2, k1=1, datum=_WGS84, name=NN):
593 '''New L{AlbersEqualArea2} projection.
595 @arg lat1: First standard parallel (C{degrees}).
596 @arg lat2: Second standard parallel (C{degrees}).
597 @kwarg k1: Azimuthal scale on the standard parallels (C{scalar}).
598 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid},
599 L{Ellipsoid2} or L{a_f2Tuple}).
600 @kwarg name: Optional name for the projection (C{str}).
602 @raise AlbersError: Invalid B{C{lat1}}m B{C{lat2}}, B{C{k1}}
603 or no convergence.
604 '''
605 self._lat1, self._lat2 = lats = _Lat(lat1=lat1), _Lat(lat2=lat2)
606 args = tuple(sincos2d_(*lats)) + (_Ks(k1=k1), datum, name)
607 _AlbersBase.__init__(self, *args)
610class AlbersEqualArea4(_AlbersBase):
611 '''An Albers equal-area (authalic) projection specified by the C{sin}
612 and C{cos} of both standard parallels.
614 @see: L{AlbersEqualArea} and L{AlbersEqualArea2}.
615 '''
616 _k = _k1_
618 def __init__(self, slat1, clat1, slat2, clat2, k1=1, datum=_WGS84, name=NN):
619 '''New L{AlbersEqualArea4} projection.
621 @arg slat1: Sine of first standard parallel (C{scalar}).
622 @arg clat1: Cosine of first standard parallel (non-negative C{scalar}).
623 @arg slat2: Sine of second standard parallel (C{scalar}).
624 @arg clat2: Cosine of second standard parallel (non-negative C{scalar}).
625 @kwarg k1: Azimuthal scale on the standard parallels (C{scalar}).
626 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid},
627 L{Ellipsoid2} or L{a_f2Tuple}).
628 @kwarg name: Optional name for the projection (C{str}).
630 @raise AlbersError: Negative B{C{clat1}} or B{C{clat2}}, B{C{slat1}}
631 and B{C{slat2}} have opposite signs (hemispheres),
632 invalid B{C{k1}} or no convergence.
633 '''
634 def _Lat_s_c3(name, s, c):
635 r = Float_(hypot(s, c), name=name, Error=AlbersError)
636 L = _Lat(atan2d(s, c), name=name)
637 return L, (s / r), (c / r)
639 self._lat1, sa1, ca1 = _Lat_s_c3(_lat1_, slat1, clat1)
640 self._lat2, sa2, ca2 = _Lat_s_c3(_lat2_, slat2, clat2)
641 _AlbersBase.__init__(self, sa1, ca1, sa2, ca2, _Ks(k1=k1), datum, name)
644class AlbersEqualAreaCylindrical(_AlbersBase):
645 '''An L{AlbersEqualArea} projection at C{lat=0} and C{k0=1} degenerating
646 to the cylindrical-equal-area projection.
647 '''
648 _lat1 = _lat2 = _Lat(lat1=_0_0)
650 def __init__(self, lat=_0_0, datum=_WGS84, name=NN):
651 '''New L{AlbersEqualAreaCylindrical} projection.
653 @kwarg lat: Standard parallel (C{0 degrees} I{fixed}).
654 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid},
655 L{Ellipsoid2} or L{a_f2Tuple}).
656 @kwarg name: Optional name for the projection (C{str}).
657 '''
658 _xlat(lat, _0_0, AlbersEqualAreaCylindrical)
659 _AlbersBase.__init__(self, _0_0, _1_0, _0_0, _1_0, 1, datum, name)
662class AlbersEqualAreaNorth(_AlbersBase):
663 '''An azimuthal L{AlbersEqualArea} projection at C{lat=90} and C{k0=1}
664 degenerating to the L{azimuthal} L{LambertEqualArea} projection.
665 '''
666 _lat1 = _lat2 = _Lat(lat1=_90_0)
668 def __init__(self, lat=_90_0, datum=_WGS84, name=NN):
669 '''New L{AlbersEqualAreaNorth} projection.
671 @kwarg lat: Standard parallel (C{90 degrees} I{fixed}).
672 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid},
673 L{Ellipsoid2} or L{a_f2Tuple}).
674 @kwarg name: Optional name for the projection (C{str}).
675 '''
676 _xlat(lat, _90_0, AlbersEqualAreaNorth)
677 _AlbersBase.__init__(self, _1_0, _0_0, _1_0, _0_0, 1, datum, name)
680class AlbersEqualAreaSouth(_AlbersBase):
681 '''An azimuthal L{AlbersEqualArea} projection at C{lat=-90} and C{k0=1}
682 degenerating to the L{azimuthal} L{LambertEqualArea} projection.
683 '''
684 _lat1 = _lat2 = _Lat(lat1=_N_90_0)
686 def __init__(self, lat=_N_90_0, datum=_WGS84, name=NN):
687 '''New L{AlbersEqualAreaSouth} projection.
689 @kwarg lat: Standard parallel (C{-90 degrees} I{fixed}).
690 @kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid},
691 L{Ellipsoid2} or L{a_f2Tuple}).
692 @kwarg name: Optional name for the projection (C{str}).
693 '''
694 _xlat(lat, _N_90_0, AlbersEqualAreaSouth)
695 _AlbersBase.__init__(self, _N_1_0, _0_0, _N_1_0, _0_0, 1, datum, name)
698class Albers7Tuple(_NamedTuple):
699 '''7-Tuple C{(x, y, lat, lon, gamma, scale, datum)}, in C{meter},
700 C{meter}, C{degrees90}, C{degrees180}, C{degrees360}, C{scalar} and
701 C{Datum} where C{(x, y)} is the projected, C{(lat, lon)} the geodetic
702 location, C{gamma} the meridian convergence at point, the bearing of
703 the y-axis measured clockwise from true North and C{scale} is the
704 azimuthal scale of the projection at point. The radial scale is
705 the reciprocal C{1 / scale}.
706 '''
707 _Names_ = (_x_, _y_, _lat_, _lon_, _gamma_, _scale_, _datum_)
708 _Units_ = ( Meter, Meter, Lat, Lon, Bearing, _Pass, _Pass)
711def _atanhee(x, E): # see Ellipsoid._es_atanh, .AuxLat._atanhee
712 '''(INTERNAL) Function M{atanhee(x)}, defined as ...
713 atanh( E.e * x) / E.e if f > 0 # oblate
714 atan (sqrt(-E.e2) * x) / sqrt(-E.e2) if f < 0 # prolate
715 x if f = 0.
716 '''
717 e = E.e
718 if E.f > 0: # .isOblate
719 x = atanh( x * e) / e
720 elif E.f < 0: # .isProlate
721 x = (atan2(-x * e, _N_1_0) if x < 0 else
722 atan2( x * e, _1_0)) / e
723 return x
726def _atanhs1(x):
727 '''(INTERNAL) Function M{atanh(sqrt(x)) / sqrt(x) - 1}.
728 '''
729 s = fabs(x)
730 if s < _0_5: # for typical ...
731 # x < E.e^2 == 2 * E.f use ...
732 # x / 3 + x^2 / 5 + x^3 / 7 + ...
733 y, k = x, 3
734 _S2 = Fsum(y / k).fsum2_
735 for _ in range(_TERMS): # 9 terms
736 y *= x # x**n
737 k += 2 # 2*n + 1
738 s, d = _S2(y / k)
739 if not d:
740 break
741 else:
742 s = sqrt(s)
743 s = (atanh(s) if x > 0 else atan(s)) / s - _1_0
744 return s
747def _Datanhee(x, y, E): # see .rhumbx._DeatanhE
748 '''(INTERNAL) Function M{Datanhee(x, y)}, defined as
749 M{atanhee((x - y) / (1 - E.e^2 * x * y)) / (x - y)}.
750 '''
751 d = _1_0 - E.e2 * x * y
752 if d:
753 d = _1_0 / d
754 t = x - y
755 if t:
756 d = _atanhee(t * d, E) / t
757 else:
758 raise AlbersError(x=x, y=y, txt=_Datanhee.__name__)
759 return d
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 e2 = E.e2
767 if ((fabs(x) + fabs(y)) * e2) < _0_5:
768 e = z = _1_0
769 k = 1
770 T = Fsum() # Taylor expansion
771 _T = T.fsum_
772 _C = Fsum().fsum_
773 _S2 = Fsum().fsum2_
774 for _ in range(_TERMS): # 15 terms
775 T *= y; p = _T(z); z *= x # PYCHOK ;
776 T *= y; t = _T(z); z *= x # PYCHOK ;
777 e *= e2
778 k += 2
779 s, d = _S2(e * _C(p, t) / k)
780 if not d:
781 break
782 else: # PYCHOK no cover
783 d = _1_0 - x
784 if isnear0(d):
785 raise AlbersError(x=x, y=y, txt=_D2atanhee.__name__)
786 s = (_Datanhee(_1_0, y, E) - _Datanhee(x, y, E)) / d
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.__name__, lat=lat)
830 raise AlbersError(t, txt=_not_(f))
833__all__ += _ALL_DOCS(_AlbersBase)
835# **) MIT License
836#
837# Copyright (C) 2016-2023 -- 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
847# in all copies or substantial portions of the Software.
848#
849# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
850# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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852# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
853# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
854# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
855# OTHER DEALINGS IN THE SOFTWARE.