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# -*- coding: utf-8 -*- 

 

u'''Classes L{AlbersEqualArea}, L{AlbersEqualArea2}, L{AlbersEqualArea4}, 

L{AlbersEqualAreaCylindrical}, L{AlbersEqualAreaNorth}, L{AlbersEqualAreaSouth} and 

L{AlbersError}, a transcription of I{Charles Karney}'s C++ class U{AlbersEqualArea 

<https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1AlbersEqualArea.html>}. 

 

See also I{Albers Equal-Area Conic Projection} in U{John P. Snyder, "Map Projections 

-- A Working Manual", 1987<https://pubs.er.USGS.gov/djvu/PP/PP_1395.pdf>}, pp 98-106 

and the Albers Conical Equal-Area examples on pp 291-294. 

 

@newfield example: Example, Examples 

''' 

 

from pygeodesy.basics import property_RO, _xkwds 

from pygeodesy.datums import Datums, _ellipsoidal_datum 

from pygeodesy.errors import _ValueError 

from pygeodesy.fmath import Fsum, fsum_, hypot, hypot1, sqrt3 

from pygeodesy.interns import EPS, NN, _datum_, _gamma_, _lat_, \ 

_lat0_, _lat1_, _lat2_, _lon_, _lon0_, \ 

_no_convergence_fmt_, _scale_, _x_, \ 

_y_, _0_0, _0_5, _1_0, _2_0, _3_0, _90_0 

from pygeodesy.karney import _diff182, _norm180 

from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY 

from pygeodesy.named import _NamedBase, _NamedTuple, _Pass 

from pygeodesy.units import Bearing, Float_, Lat, Lat_, Lon, Lon_, \ 

Meter, Scalar_ 

from pygeodesy.utily import atan2d, degrees360, sincos2, sincos2d 

 

from math import atan, atan2, atanh, degrees, radians, sqrt 

 

__all__ = _ALL_LAZY.albers 

__version__ = '20.10.02' 

 

_EPS__2 = EPS**2 

_EPS__4 = EPS**4 

_NUMIT = 8 # XXX 4? 

_NUMIT0 = 41 # XXX 21? 

_TERMS = 21 # XXX 9? 

_TOL = sqrt(EPS) 

_TOL0 = sqrt3(_TOL) 

 

 

class AlbersError(_ValueError): 

'''An L{AlbersEqualArea}, L{AlbersEqualArea2}, L{AlbersEqualArea4}, 

L{AlbersEqualAreaCylindrical}, L{AlbersEqualAreaNorth}, 

L{AlbersEqualAreaSouth} or L{Albers7Tuple} issue. 

''' 

pass 

 

 

def _Dsn(x, y, sx, sy): 

'''(INTERNAL) Divided differences, defined as 

M{Df(x, y) = (f(x) - f(y)) / (x - y)} with 

M{sn(x) = x / sqrt(1 + x^2)}: M{Dsn(x, y) = 

(x + y) / ((sn(x) + sn(y)) * (1 + x^2) * (1 + y^2))}. 

 

@see: U{W. M. Kahan and R. J. Fateman, "Sympbolic omputation of divided 

differences"<https://people.EECS.Berkeley.EDU/~fateman/papers/divdiff.pdf>}, 

U{ACM SIGSAM Bulletin 33(2), 7-28 (1999)<https://DOI.org/10.1145/334714.334716>} 

and U{AlbersEqualArea.hpp 

<https://geographiclib.sourceforge.io/html/AlbersEqualArea_8hpp_source.html>}. 

''' 

# sx = x / hypot1(x) 

d, t = _1_0, x * y 

if t > 0: 

s = sx + sy 

if s: 

d = (x + y) * ((sx * sy) / t)**2 / s 

else: 

t = x - y 

if t: 

d = (sx - sy) / t 

return d 

 

 

def _Ks(k, name, Error=AlbersError): 

'''(INTERNAL) Scale C{B{k} >= EPS}. 

''' 

return Scalar_(k, name=name, Error=Error, low=EPS) # > 0 

 

 

def _Lat(lat, name=_lat_, Error=AlbersError): 

'''(INTERNAL) Latitude C{-90 <= B{lat} <= 90}. 

''' 

return Lat_(lat, name=name, Error=Error) 

 

 

def _Lon(lon, name=_lon_, Error=AlbersError): 

'''(INTERNAL) Longitude C{-180 <= B{lon} <= 180}. 

''' 

return Lon_(lon, name=name, Error=Error) 

 

 

def _tol(tol, x): 

'''(INTERNAL) Converge tolerance. 

''' 

return tol * max(1, abs(x)) 

 

 

class _AlbersBase(_NamedBase): 

'''(INTERNAL) Base class for C{AlbersEqualArea...} projections. 

 

@see: I{Karney}'s C++ class U{AlbersEqualArea<https://GeographicLib.SourceForge.io/ 

html/classGeographicLib_1_1AlbersEqualArea.html>}, method C{Init}. 

''' 

_datum = Datums.WGS84 

_iteration = None 

_lat0 = None # lat origin 

_lat1 = None # let 1st parallel 

_lat2 = None # lat 2nd parallel 

_k0 = None # scale 

_k0n0 = None # (INTERNAL) k0 * no 

_k02 = None # (INTERNAL) k0**2 

_k02n0 = None # (INTERNAL) k02 * n0 

_m0 = _0_0 # if polar else sqrt(m02) 

_m02 = None # (INTERNAL) cached 

_n0 = None # (INTERNAL) cached 

_nrho0 = _0_0 # if polar else m0 * E.a 

_polar = False 

_qx = None # (INTERNAL) cached 

_qZ = None # (INTERNAL) cached 

_qZa2 = None # (INTERNAL) qZ * E.a**2 

_scxi0 = None # (INTERNAL) sec(xi) 

_sign = +1 

_sxi0 = None # (INTERNAL) sin(xi) 

_txi0 = None # (INTERNAL) tan(xi) 

 

def __init__(self, sa1, ca1, sa2, ca2, k, datum, name): 

'''(INTERNAL) New C{AlbersEqualArea...} instance. 

''' 

if datum not in (None, self._datum): 

self._datum = _ellipsoidal_datum(datum, name=name) 

if name: 

self.name = name 

 

E = self.datum.ellipsoid 

b_a = E.b_a # fm = 1 - E.f 

e2 = E.e2 

e12 = E.e12 # e2m = 1 - E.e2 

 

self._qZ = qZ = _1_0 + e12 * self._atanhee(1) 

self._qZa2 = qZ * E.a2 

self._qx = qZ / (_2_0 * e12) 

 

c = min(ca1, ca2) 

if c < 0: 

raise AlbersError(clat1=ca1, clat2=ca2) 

polar = c < _EPS__2 # == 0 

# determine hemisphere of tangent latitude 

if sa1 < 0: # and sa2 < 0: 

self._sign = -1 

# internally, tangent latitude positive 

sa1, sa2 = -sa1, -sa2 

if sa1 > sa2: # make phi1 < phi2 

sa1, sa2 = sa2, sa1 

ca1, ca2 = ca2, ca1 

if sa1 < 0: # or sa2 < 0: 

raise AlbersError(slat1=sa1, slat2=sa2) 

# avoid singularities at poles 

ca1, ca2 = max(_EPS__2, ca1), max(_EPS__2, ca2) 

ta1, ta2 = sa1 / ca1, sa2 / ca2 

 

par1 = abs(ta1 - ta2) < _EPS__4 # ta1 == ta2 

if par1 or polar: 

C, ta0 = _1_0, ta2 

else: 

s1_qZ, C = self._s1_qZ_C2(ca1, sa1, ta1, ca2, sa2, ta2) 

 

ta0 = (ta2 + ta1) * _0_5 

Ta0 = Fsum(ta0) 

tol = _tol(_TOL0, ta0) 

for self._iteration in range(1, _NUMIT0): 

ta02 = ta0**2 

sca02 = ta02 + _1_0 

sca0 = sqrt(sca02) 

sa0 = ta0 / sca0 

sa01 = sa0 + _1_0 

sa02 = sa0**2 

# sa0m = 1 - sa0 = 1 / (sec(a0) * (tan(a0) + sec(a0))) 

sa0m = _1_0 / (sca0 * (ta0 + sca0)) # scb0^2 * sa0 

g = (_1_0 + (b_a * ta0)**2) * sa0 

dg = e12 * sca02 * (_1_0 + 2 * ta02) + e2 

D = sa0m * (_1_0 - e2 * (_1_0 + sa01 * 2 * sa0)) / (e12 * sa01) # dD/dsa0 

dD = -2 * (_1_0 - e2 * sa02 * (_3_0 + 2 * sa0)) / (e12 * sa01**2) 

sa02_ = _1_0 - e2 * sa02 

sa0m_ = sa0m / (_1_0 - e2 * sa0) 

BA = sa0m_ * (self._atanhx1(e2 * sa0m_**2) * e12 - e2 * sa0m) \ 

- sa0m**2 * e2 * (2 + (_1_0 + e2) * sa0) / (e12 * sa02_) # == B + A 

dAB = 2 * e2 * (2 - e2 * (_1_0 + sa02)) / (e12 * sa02_**2 * sca02) 

u_du = fsum_(s1_qZ * g, -D, g * BA) \ 

/ fsum_(s1_qZ * dg, -dD, dg * BA, g * dAB) # == u/du 

ta0, d = Ta0.fsum2_(-u_du * (sca0 * sca02)) 

if abs(d) < tol: 

break 

else: 

raise AlbersError(iteration=_NUMIT0, txt=_no_convergence_fmt_ % (tol,)) 

 

self._txi0 = txi0 = self._txif(ta0) 

self._scxi0 = hypot1(txi0) 

self._sxi0 = sxi0 = txi0 / self._scxi0 

self._m02 = m02 = _1_0 / (_1_0 + (b_a * ta0)**2) 

self._n0 = n0 = ta0 / hypot1(ta0) 

if polar: 

self._polar = True 

self._nrho0 = self._m0 = _0_0 

else: 

self._m0 = sqrt(m02) # == nrho0 / E.a 

self._nrho0 = E.a * self._m0 # == E.a * sqrt(m02) 

self._k0_(_1_0 if par1 else (k * sqrt(C / (m02 + n0 * qZ * sxi0)))) 

self._lat0 = _Lat(self._sign * degrees(atan(ta0)), name=_lat0_) 

 

def _s1_qZ_C2(self, ca1, sa1, ta1, ca2, sa2, ta2): 

'''(INTERNAL) Compute C{sm1 / (s / qZ)} and C{C} for .__init__. 

''' 

E = self.datum.ellipsoid 

b_a = E.b_a 

e2 = E.e2 

 

tb1 = b_a * ta1 

tb2 = b_a * ta2 

dtb12 = b_a * (tb1 + tb2) 

scb12 = _1_0 + tb1**2 

scb22 = _1_0 + tb2**2 

 

esa1_2 = (_1_0 - e2 * sa1**2) \ 

* (_1_0 - e2 * sa2**2) 

esa12 = _1_0 + e2 * sa1 * sa2 

 

dsn = _Dsn(ta2, ta1, sa2, sa1) 

axi, bxi, sxi = self._a_b_sxi3((ca1, sa1, ta1, scb12), 

(ca2, sa2, ta2, scb22)) 

 

dsxi = (esa12 / esa1_2 + self._Datanhee(sa2, sa1)) * dsn / (_2_0 * self._qx) 

C = fsum_(sxi * dtb12 / dsxi, scb22, scb12) / (_2_0 * scb22 * scb12) 

 

sa12 = fsum_(sa1, sa2, sa1 * sa2) 

axi *= (_1_0 + e2 * sa12) / (_1_0 + sa12) 

bxi *= e2 * fsum_(sa1, sa2, esa12) / esa1_2 + E.e12 * self._D2atanhee(sa1, sa2) 

s1_qZ = dsn * (axi * self._qZ - bxi) / (_2_0 * dtb12) 

return s1_qZ, C 

 

def _a_b_sxi3(self, *ca_sa_ta_scb): 

'''(INTERNAL) Sum of C{sm1} terms and C{sin(xi)}s for ._s1_qZ_C2. 

''' 

a = b = s = _0_0 

for ca, sa, ta, scb in ca_sa_ta_scb: 

cxi, sxi, _ = self._cstxif3(ta) 

if sa > 0: 

sa += _1_0 

a += (cxi / ca)**2 * sa / (_1_0 + sxi) 

b += scb * ca**2 / sa 

else: 

sa = _1_0 - sa 

a += (_1_0 - sxi) / sa 

b += scb * sa 

s += sxi 

return a, b, s 

 

@property_RO 

def datum(self): 

'''Get the datum (L{Datum}). 

''' 

return self._datum 

 

@property_RO 

def equatoradius(self): 

'''Get the geodesic's equatorial (major) radius, semi-axis (C{meter}). 

''' 

return self.datum.ellipsoid.a 

 

majoradius = equatoradius # for compatibility 

 

@property_RO 

def iteration(self): 

'''Get the iteration number (C{int}). 

''' 

return self._iteration 

 

@property_RO 

def flattening(self): 

'''Get the geodesic's flattening (C{float}). 

''' 

return self.datum.ellipsoid.f 

 

def forward(self, lat, lon, lon0=0, name=NN): 

'''Convert a geodetic location to east- and northing. 

 

@arg lat: Latitude of the location (C{degrees}). 

@arg lon: Longitude of the location (C{degrees}). 

@kwarg lon0: Optional central meridian longitude (C{degrees}). 

@kwarg name: Optional name for the location (C{str}). 

 

@return: An L{Albers7Tuple}C{(x, y, lat, lon, gamma, scale, datum)}. 

 

@note: The origin latitude is returned by C{property lat0}. No 

false easting or northing is added. The value of B{C{lat}} 

should be in the range C{[-90..90] degrees}. The returned 

values C{x} and C{y} will be large but finite for points 

projecting to infinity, i.e. one or both of the poles. 

''' 

E = self.datum.ellipsoid 

s = self._sign 

 

k0 = self._k0 

n0 = self._n0 

nrho0 = self._nrho0 

txi0 = self._txi0 

 

sa, ca = sincos2d(_Lat(lat) * s) 

ca = max(_EPS__2, ca) 

ta = sa / ca 

 

_, sxi, txi = self._cstxif3(ta) 

dq = self._qZ * _Dsn(txi, txi0, sxi, self._sxi0) * (txi - txi0) 

drho = -E.a * dq / (sqrt(self._m02 - n0 * dq) + self._m0) 

 

lon = _Lon(lon) 

if lon0: 

lon, _ = _diff182(_Lon(lon0, name=_lon0_), lon) 

b = radians(lon) 

 

th = self._k02n0 * b 

sth, cth = sincos2(th) # XXX sin, cos 

if n0: 

x = sth / n0 

y = (_1_0 - cth if cth < 0 else sth**2 / (_1_0 + cth)) / n0 

else: 

x = self._k02 * b 

y = _0_0 

t = nrho0 + n0 * drho 

x = t * x / k0 

y = s * (nrho0 * y - drho * cth) / k0 

 

g = degrees360(s * th) 

if t: 

k0 *= t * hypot1(E.b_a * ta) / E.a 

t = Albers7Tuple(x, y, lat, lon, g, k0, self.datum) 

return self._xnamed(t, name=name) 

 

@property_RO 

def ispolar(self): 

'''Is this projection polar (C{bool})? 

''' 

return self._polar 

 

isPolar = ispolar # synonym 

 

@property_RO 

def lat0(self): 

'''Get the latitude of the projection origin (C{degrees}). 

 

This is the latitude of minimum azimuthal scale and 

equals the B{C{lat}} in the 1-parallel L{AlbersEqualArea} 

and lies between B{C{lat1}} and B{C{lat2}} for the 

2-parallel L{AlbersEqualArea2} and L{AlbersEqualArea4} 

projections. 

''' 

return self._lat0 

 

@property_RO 

def lat1(self): 

'''Get the latitude of the first parallel (C{degrees}). 

''' 

return self._lat1 

 

@property_RO 

def lat2(self): 

'''Get the latitude of the second parallel (C{degrees}). 

 

@note: The second and first parallel latitudes are the 

same instance for 1-parallel C{AlbersEqualArea*} 

projections. 

''' 

return self._lat2 

 

def rescale0(self, lat, k=1): 

'''Set the azimuthal scale for this projection. 

 

@arg lat: Northern latitude (C{degrees}). 

@arg k: Azimuthal scale at latitude B{C{lat}} (C{scalar}). 

 

@raise AlbersError: Invalid B{C{lat}} or B{C{k}}. 

 

@note: This allows a I{latitude of conformality} to be specified. 

''' 

self._k0_(_Ks(k, 'k') / self.forward(lat, _0_0).scale) 

 

def reverse(self, x, y, lon0=0, name=NN, LatLon=None, **LatLon_kwds): 

'''Convert an east- and northing location to geodetic lat- and longitude. 

 

@arg x: Easting of the location (C{meter}). 

@arg y: Northing of the location (C{meter}). 

@kwarg lon0: Optional central meridian longitude (C{degrees}). 

@kwarg name: Optional name for the location (C{str}). 

@kwarg LatLon: Class to use (C{LatLon}) or C{None}. 

@kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword 

arguments, ignored if C{B{LatLon}=None}. 

 

@return: The geodetic (C{LatLon}) or if B{C{LatLon}} is C{None} an 

L{Albers7Tuple}C{(x, y, lat, lon, gamma, scale, datum)}. 

 

@note: The origin latitude is returned by C{property lat0}. No 

false easting or northing is added. The returned value of 

C{lon} is in the range C{[-180..180] degrees} and C{lat} 

is in the range C{[-90..90] degrees}. If the given 

B{C{x}} or B{C{y}} point is outside the valid projected 

space the nearest pole is returned. 

''' 

x = Meter(x=x) 

y = Meter(y=y) 

E = self.datum.ellipsoid 

 

k0 = self._k0 

n0 = self._n0 

k0n0 = self._k0n0 

nrho0 = self._nrho0 

txi0 = self._txi0 

 

y_ = self._sign * y 

nx = k0n0 * x 

ny = k0n0 * y_ 

y1 = nrho0 - ny 

 

drho = den = nrho0 + hypot(nx, y1) # 0 implies origin with polar aspect 

if den: 

drho = fsum_(x * nx, -_2_0 * y_ * nrho0, y_ * ny) * k0 / den 

# dsxia = scxi0 * dsxi 

dsxia = -self._scxi0 * (_2_0 * nrho0 + n0 * drho) * drho / self._qZa2 

t = _1_0 - dsxia * (_2_0 * txi0 + dsxia) 

txi = (txi0 + dsxia) / (sqrt(t) if t > _EPS__4 else _EPS__2) 

 

ta = self._tanf(txi) 

lat = degrees(atan(ta * self._sign)) 

 

b = atan2(nx, y1) 

lon = degrees(b / self._k02n0 if n0 else x / (y1 * k0)) 

if lon0: 

lon += _norm180(_Lon(lon0, name=_lon0_)) 

lon = _norm180(lon) 

 

if LatLon is None: 

g = degrees360(self._sign * b) 

if den: 

k0 *= (nrho0 + n0 * drho) * hypot1(E.b_a * ta) / E.a 

r = Albers7Tuple(x, y, lat, lon, g, k0, self.datum) 

else: 

kwds = _xkwds(LatLon_kwds, datum=self.datum) 

r = LatLon(lat, lon, **kwds) 

return self._xnamed(r, name=name) 

 

@property_RO 

def scale0(self): 

'''Get the central scale for the projection (C{float}). 

 

This is the azimuthal scale on the latitude of origin 

of the projection, see C{property lat0}. 

''' 

return self._k0 

 

def _atanhee(self, x): 

'''(INTERNAL) Function M{atanhee(x)}, defined as ... 

atanh( E.e * x) / E.e if f > 0 # oblate 

atan (sqrt(-E.e2) * x) / sqrt(-E.e2) if f < 0 # prolate 

x if f = 0. 

''' 

E = self.datum.ellipsoid 

if E.isOblate: 

x = atanh(x * E.e) / E.e 

elif E.isProlate: 

x = atan2(abs(x) * E.e, -_1_0 if x < 0 else _1_0) / E.e 

return x 

 

def _atanhx1(self, x): 

'''(INTERNAL) Function M{atanh(sqrt(x)) / sqrt(x) - 1}. 

''' 

s = abs(x) 

if s < _0_5: # for typical ... 

# x < E.e^2 = 2 * E.f use ... 

# x / 3 + x^2 / 5 + x^3 / 7 + ... 

y, k = x, 3 

S = Fsum(y / k) 

for _ in range(_TERMS): 

y *= x # x**n 

k += 2 # 2*n + 1 

s, d = S.fsum2_(y / k) 

if not d: 

break 

else: 

s = sqrt(s) 

s = (atanh(s) if x > 0 else atan(s)) / s - _1_0 

return s 

 

def _cstxif3(self, ta): 

'''(INTERNAL) Get 3-tuple C{(cos, sin, tan)} of M{xi(ta)}. 

''' 

t = self._txif(ta) 

c = _1_0 / hypot1(t) 

s = c * t 

return c, s, t 

 

def _Datanhee(self, x, y): 

'''(INTERNAL) Function M{Datanhee(x, y)}, defined as 

M{atanhee((x - y) / (1 - E.e^2 * x * y)) / (x - y)}. 

''' 

E = self.datum.ellipsoid 

d = _1_0 - E.e2 * x * y 

t = x - y 

if t: 

s = self._atanhee(t / d) / t 

elif d: 

s = _1_0 / d 

else: 

raise AlbersError(x=x, y=y, d=d, txt=_AlbersBase._Datanhee.__name__) 

return s 

 

def _D2atanhee(self, x, y): 

'''(INTERNAL) Function M{D2atanhee(x, y)}, defined as 

M{(Datanhee(1, y) - Datanhee(1, x)) / (y - x)}. 

''' 

e2 = self.datum.ellipsoid.e2 

 

if (abs(x) + abs(y)) * e2 < _0_5: 

e = z = _1_0 

k = 1 

C = Fsum() 

S = Fsum() 

T = Fsum() # Taylor expansion 

for _ in range(_TERMS): 

T *= y; p = T.fsum_(z); z *= x # PYCHOK ; 

T *= y; t = T.fsum_(z); z *= x # PYCHOK ; 

e *= e2 

k += 2 

s, d = S.fsum2_(e * C.fsum_(p, t) / k) 

if not d: 

break 

elif (_1_0 - x): 

s = (self._Datanhee(_1_0, y) - self._Datanhee(x, y)) / (_1_0 - x) 

else: 

raise AlbersError(x=x, y=y, txt=_AlbersBase._D2atanhee.__name__) 

return s 

 

def _k0_(self, k): 

'''(INTERNAL) Set C{._k0}, C{._k02}, etc. 

''' 

self._k0 = k = _Ks(k, 'k0') 

self._k02 = k2 = k**2 

self._k0n0 = k * self._n0 

self._k02n0 = k2 * self._n0 

 

def _tanf(self, txi): # called from .Ellipsoid.auxAuthalic 

'''(INTERNAL) Function M{tan-phi from tan-xi}. 

''' 

tol = _tol(_TOL, txi) 

 

e2 = self.datum.ellipsoid.e2 

qx = self._qx 

 

ta = txi 

Ta = Fsum(ta) 

for self._iteration in range(1, _NUMIT): # max 2, mean 1.99 

# dtxi/dta = (scxi / sca)^3 * 2 * (1 - e^2) / (qZ * (1 - e^2 * sa^2)^2) 

ta2 = ta**2 

sca2 = ta2 + _1_0 

txia = self._txif(ta) 

s3qx = sqrt3(sca2 / (_1_0 + txia**2)) * qx 

ta, d = Ta.fsum2_((txi - txia) * s3qx * (_1_0 - e2 * ta2 / sca2)**2) 

if abs(d) < tol: 

return ta 

raise AlbersError(iteration=_NUMIT, txt=_no_convergence_fmt_ % (tol,)) 

 

def _txif(self, ta): # called from .Ellipsoid.auxAuthalic 

'''(INTERNAL) Function M{tan-xi from tan-phi}. 

''' 

E = self.datum.ellipsoid 

 

ca2 = _1_0 / (_1_0 + ta**2) 

sa = sqrt(ca2) * abs(ta) # enforce odd parity 

 

es1 = sa * E.e2 

es2m1 = _1_0 - sa * es1 

sp1 = _1_0 + sa 

es1p1 = sp1 / (_1_0 + es1) 

es1m1 = sp1 * (_1_0 - es1) 

es2m1a = es2m1 * E.e12 # e2m 

s = sqrt((ca2 / (es1p1 * es2m1a) + self._atanhee(ca2 / es1m1)) * 

(es1m1 / es2m1a + self._atanhee(es1p1))) 

t = (sa / es2m1 + self._atanhee(sa)) / s 

if ta < 0: 

t = -t 

return t 

 

 

class AlbersEqualArea(_AlbersBase): 

'''An Albers equal-area (authalic) projection with a single standard parallel. 

 

@see: L{AlbersEqualArea4} and L{AlbersEqualArea}. 

''' 

def __init__(self, lat, k0=1, datum=Datums.WGS84, name=NN): 

'''New L{AlbersEqualArea} projection. 

 

@arg lat: Standard parallel (C{degrees}). 

@kwarg k0: Azimuthal scale on the standard parallel (C{scalar}). 

@kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, 

L{Ellipsoid2} or L{a_f2Tuple}). 

@kwarg name: Optional name for the projection (C{str}). 

 

@raise AlbertError: Invalid B{C{lat}}, B{C{k0}} or no convergence. 

''' 

self._lat1 = self._lat2 = lat = _Lat(lat, name=_lat1_) 

args = tuple(sincos2d(lat)) * 2 + (_Ks(k0, 'k0'), datum, name) 

_AlbersBase.__init__(self, *args) 

 

 

class AlbersEqualArea2(_AlbersBase): 

'''An Albers equal-area (authalic) projection with two standard parallels. 

 

@see: L{AlbersEqualArea4} and L{AlbersEqualArea}. 

''' 

def __init__(self, lat1, lat2, k1=1, datum=Datums.WGS84, name=NN): 

'''New L{AlbersEqualArea2} projection. 

 

@arg lat1: First standard parallel (C{degrees}). 

@arg lat2: Second standard parallel (C{degrees}). 

@kwarg k1: Azimuthal scale on the standard parallels (C{scalar}). 

@kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, 

L{Ellipsoid2} or L{a_f2Tuple}). 

@kwarg name: Optional name for the projection (C{str}). 

 

@raise AlbertError: Invalid B{C{lat1}}m B{C{lat2}}, B{C{k1}} 

or no convergence. 

''' 

self._lat1, self._lat2 = lats = _Lat(lat1, name=_lat1_), \ 

_Lat(lat2, name=_lat2_) 

args = tuple(sincos2d(*lats)) + (_Ks(k1, 'k1'), datum, name) 

_AlbersBase.__init__(self, *args) 

 

 

class AlbersEqualArea4(_AlbersBase): 

'''An Albers equal-area (authalic) projection specified by the C{sin} 

and C{cos} of both standard parallels. 

 

@see: L{AlbersEqualArea2} and L{AlbersEqualArea}. 

''' 

def __init__(self, slat1, clat1, slat2, clat2, k1=1, datum=Datums.WGS84, name=NN): 

'''New L{AlbersEqualArea4} projection. 

 

@arg slat1: Sine of first standard parallel (C{scalar}). 

@arg clat1: Cosine of first standard parallel (non-negative C{scalar}). 

@arg slat2: Sine of second standard parallel (C{scalar}). 

@arg clat2: Cosine of second standard parallel (non-negative C{scalar}). 

@kwarg k1: Azimuthal scale on the standard parallels (C{scalar}). 

@kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, 

L{Ellipsoid2} or L{a_f2Tuple}). 

@kwarg name: Optional name for the projection (C{str}). 

 

@raise AlbertError: Negative B{C{clat1}} or B{C{clat2}}, B{C{slat1}} 

and B{C{slat2}} have opposite signs (hemispheres), 

invalid B{C{k1}} or no convergence. 

''' 

def _Lat_s_c3(s, c, name): 

L = _Lat(atan2d(s, c), name=name) 

r = _1_0 / Float_(hypot(s, c), Error=AlbersError, name=name) 

return L, s * r, c * r 

 

self._lat1, sa1, ca1 = _Lat_s_c3(slat1, clat1, _lat1_) 

self._lat2, sa2, ca2 = _Lat_s_c3(slat2, clat2, _lat2_) 

_AlbersBase.__init__(self, sa1, ca1, sa2, ca2, _Ks(k1, 'k1'), datum, name) 

 

 

class AlbersEqualAreaCylindrical(_AlbersBase): 

'''An L{AlbersEqualArea} projection at C{lat=0} and C{k0=1} degenerating 

to the cylindrical-equal-area projection. 

''' 

_lat1 = _lat2 = _Lat(_0_0, name=_lat1_) 

 

def __init__(self, datum=Datums.WGS84, name=NN): 

'''New L{AlbersEqualAreaCylindrical} projection. 

 

@kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, 

L{Ellipsoid2} or L{a_f2Tuple}). 

@kwarg name: Optional name for the projection (C{str}). 

''' 

_AlbersBase.__init__(self, _0_0, _1_0, _0_0, _1_0, _1_0, datum, name) 

 

 

class AlbersEqualAreaNorth(_AlbersBase): 

'''An azimuthal L{AlbersEqualArea} projection at C{lat=90} and C{k0=1} 

degenerating to the L{azimuthal} L{LambertEqualArea} projection. 

''' 

_lat1 = _lat2 = _Lat(_90_0, name=_lat1_) 

 

def __init__(self, datum=Datums.WGS84, name=NN): 

'''New L{AlbersEqualAreaNorth} projection. 

 

@kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, 

L{Ellipsoid2} or L{a_f2Tuple}). 

@kwarg name: Optional name for the projection (C{str}). 

''' 

_AlbersBase.__init__(self, _1_0, _0_0, _1_0, _0_0, _1_0, datum, name) 

 

 

class AlbersEqualAreaSouth(_AlbersBase): 

'''An azimuthal L{AlbersEqualArea} projection at C{lat=-90} and C{k0=1} 

degenerating to the L{azimuthal} L{LambertEqualArea} projection. 

''' 

_lat1 = _lat2 = _Lat(-_90_0, name=_lat1_) 

 

def __init__(self, datum=Datums.WGS84, name=NN): 

'''New L{AlbersEqualAreaSouth} projection. 

 

@kwarg datum: Optional datum or ellipsoid (L{Datum}, L{Ellipsoid}, 

L{Ellipsoid2} or L{a_f2Tuple}). 

@kwarg name: Optional name for the projection (C{str}). 

''' 

_AlbersBase.__init__(self, -_1_0, _0_0, -_1_0, _0_0, _1_0, datum, name) 

 

 

class Albers7Tuple(_NamedTuple): 

'''7-Tuple C{(x, y, lat, lon, gamma, scale, datum)}, in C{meter}, 

C{meter}, C{degrees90}, C{degrees180}, C{degrees360}, C{scalar} and 

C{Datum} where C{(x, y)} is the projected, C{(lat, lon)} the geodetic 

location, C{gamma} the meridian convergence at point, the bearing of 

the y-axis measured clockwise from true North and C{scale} is the 

azimuthal scale of the projection at point. The radial scale is 

the reciprocal C{1 / scale}. 

''' 

_Names_ = (_x_, _y_, _lat_, _lon_, _gamma_, _scale_, _datum_) 

_Units_ = ( Meter, Meter, Lat, Lon, Bearing, _Ks, _Pass) 

 

 

__all__ += _ALL_DOCS(Albers7Tuple, _AlbersBase) 

 

# **) MIT License 

# 

# Copyright (C) 2016-2020 -- mrJean1 at Gmail -- All Rights Reserved. 

# 

# Permission is hereby granted, free of charge, to any person obtaining a 

# copy of this software and associated documentation files (the "Software"), 

# to deal in the Software without restriction, including without limitation 

# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

# and/or sell copies of the Software, and to permit persons to whom the 

# Software is furnished to do so, subject to the following conditions: 

# 

# The above copyright notice and this permission notice shall be included 

# in all copies or substantial portions of the Software. 

# 

# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

# OTHER DEALINGS IN THE SOFTWARE.