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# -*- coding: utf-8 -*- 

 

u'''A pure Python version of I{Karney}'s C++ class U{TransverseMercator 

<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1TransverseMercator.html>} 

based on I{Krüger} series. See also I{Karney}'s utility U{TransverseMercatorProj 

<https://GeographicLib.SourceForge.io/C++/doc/TransverseMercatorProj.1.html>}. 

 

Following and further below is a copy of I{Karney}'s U{TransverseMercator.hpp 

<https://GeographicLib.SourceForge.io/C++/doc/TransverseMercator_8hpp_source.html>} 

file C{Header}. 

 

This implementation follows closely JHS 154, ETRS89 - I{järjestelmään 

liittyvät karttaprojektiot, tasokoordinaatistot ja karttalehtijako} (Map 

projections, plane coordinates, and map sheet index for ETRS89), published 

by JUHTA, Finnish Geodetic Institute, and the National Land Survey of Finland 

(2006). The relevant section is available as the U{2008 PDF file 

<http://Docs.JHS-suositukset.FI/jhs-suositukset/JHS154/JHS154_liite1.pdf>}. 

 

This is a straight transcription of the formulas in this paper with the 

following exceptions: 

 

- Use of 6th order series instead of 4th order series. This reduces the 

error to about 5 nm for the UTM range of coordinates (instead of 200 nm), 

with a speed penalty of only 1%, 

 

- Use Newton's method instead of plain iteration to solve for latitude 

in terms of isometric latitude in the Reverse method, 

 

- Use of Horner's representation for evaluating polynomials and Clenshaw's 

method for summing trigonometric series, 

 

- Several modifications of the formulas to improve the numerical accuracy, 

 

- Evaluating the convergence and scale using the expression for the 

projection or its inverse. 

 

Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2008-2022) 

and licensed under the MIT/X11 License. For more information, see the 

U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation. 

''' 

# make sure int/int division yields float quotient 

from __future__ import division as _; del _ # PYCHOK semicolon 

 

from pygeodesy.basics import copysign0, isint, isodd, neg, neg_ 

from pygeodesy.constants import INF, _K0_UTM, NINF, PI, PI_2, _0_0s, \ 

_0_0, _1_0, _90_0, _180_0 

# from pygeodesy.datums import _spherical_datum # in KTransverseMercator.ellipsoid.setter 

from pygeodesy.errors import _or, _ValueError, _xkwds_get 

from pygeodesy.fmath import fsum1_, hypot, hypot1 

# from pygeodesy.fsums import fsum1_ # from .fmath 

from pygeodesy.interns import NN, _COMMASPACE_, _not_, _singular_ 

from pygeodesy.karney import _atan2d, _diff182, _EWGS84, _fix90, \ 

_NamedBase, _norm180, _polynomial, _unsigned2 

from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS, _pairs 

# from pygeodesy.named import _NamedBase # from .karney 

from pygeodesy.namedTuples import Forward4Tuple, Reverse4Tuple 

from pygeodesy.props import property_doc_, Property, Property_RO, \ 

_update_all 

# from pygeodesy.streprs import pairs as _pairs # from .lazily 

from pygeodesy.units import Degrees, Scalar_, _1mm as _TOL_10 # PYCHOK used! 

from pygeodesy.utily import atand, sincos2, sincos2d_ 

 

from cmath import phase 

from math import atan2, asinh, cos, cosh, degrees, sin, sinh, sqrt, tanh 

 

__all__ = _ALL_LAZY.ktm 

__version__ = '22.09.15' 

 

 

class KTMError(_ValueError): 

'''Error raised for L{KTransverseMercator} and L{KTransverseMercator.forward} issues. 

''' 

pass 

 

 

class KTransverseMercator(_NamedBase): 

'''Transverse Mercator projection based on Krüger's method which evaluates the 

projection and its inverse in terms of a series. 

 

There's a singularity in the projection at I{phi = 0, lam - lam0 = +/- (1 - e) 

90}, about +/- 82.6 degrees for WGS84, where I{e} is the eccentricity. Beyond 

this point, the series ceases to converge and the results from this method 

will be garbage. I{To be on the safe side, don't use this method if the 

angular distance from the central meridian exceeds (1 - 2e) x 90}, about 75 

degrees for the WGS84 ellipsoid. 

 

Class L{ExactTransverseMercator} is an alternative implementation of the 

projection using I{exact} formulas which yield accurate (to 8 nm) results 

over the entire ellipsoid. 

 

The ellipsoid parameters and the central scale are set in the constructor. 

The central meridian (which is a trivial shift of the longitude) is specified 

as the C{lon0} keyword argument of the L{KTransverseMercator.forward} and 

L{KTransverseMercator.reverse} methods. The latitude of origin is taken to 

be the equator. There is no provision in this class for specifying a false 

easting or false northing or a different latitude of origin. However these 

are can be simply included by the calling function. 

 

The L{KTransverseMercator.forward} and L{KTransverseMercator.reverse} methods 

also return the meridian convergence C{gamma} and scale C{k}. The meridian 

convergence is the bearing of grid North, the C{y axis}, measured clockwise 

from true North. 

''' 

_E = _EWGS84 

_k0 = _K0_UTM # central scale factor 

_lon0 = _0_0 # central meridian 

_mTM = 6 

_raiser = False # throw Error 

 

def __init__(self, a_earth=_EWGS84, f=None, lon0=0, k0=_K0_UTM, name=NN, 

raiser=False, **TMorder): 

'''New L{KTransverseMercator}. 

 

@kwarg a_earth: This rhumb's earth (L{Ellipsoid}, L{Ellipsoid2}, 

L{a_f2Tuple}, L{Datum}, 2-tuple (C{a, f})) or the 

equatorial radius (C{scalar}, C{meter}). 

@kwarg f: The ellipsoid's flattening (C{scalar}), iff B{C{a_earth}} is 

a C{scalar}, ignored otherwise. 

@kwarg lon0: The central meridian (C{degrees180}). 

@kwarg k0: Central scale factor (C{scalar}). 

@kwarg name: Optional name (C{str}). 

@kwarg raiser: If C{True}, throw a L{KTMError} for C{forward} 

singularities (C{bool}). 

@kwarg TMorder: Keyword argument B{C{TMorder}}, see property C{TMorder}. 

 

@raise KTMError: Invalid B{C{a_earth}}, B{C{f}} or B{C{TMorder}}. 

''' 

if f is not None: 

self.ellipsoid = a_earth, f 

elif a_earth not in (_EWGS84, None): 

self.ellipsoid = a_earth 

self.lon0 = lon0 

self.k0 = k0 

if name: # PYCHOK no cover 

self.name = name 

if raiser: 

self.raiser = True 

if TMorder: 

self.TMorder = _xkwds_get(TMorder, TMorder=self._mTM) 

 

@Property_RO 

def _Alp(self): 

return _Xs(_AlpCoeffs, self.TMorder, self.ellipsoid) 

 

@Property_RO 

def _b1(self): 

n = self.ellipsoid.n 

if n: # isEllipsoidal 

m = self.TMorder // 2 

B1 = _B1Coeffs[m] 

m += 1 

b1 = _polynomial(n**2, B1, 0, m) / (B1[m] * (n + _1_0)) 

else: # isSpherical 

b1 = _1_0 # B1[m - 1] / B1[m1] == 1, always 

return b1 

 

@Property_RO 

def _Bet(self): 

C = _Xs(_BetCoeffs, self.TMorder, self.ellipsoid) 

return tuple(map(neg, C)) if self.f else C # negated if isEllispoidal 

 

@Property 

def ellipsoid(self): 

'''Get the ellipsoid (L{Ellipsoid}). 

''' 

return self._E 

 

@ellipsoid.setter # PYCHOK setter! 

def ellipsoid(self, a_earth_f): 

'''Set this rhumb's ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum}, 

L{a_f2Tuple} or 2-tuple C{(a, f)}). 

''' 

E = _MODS.datums._spherical_datum(a_earth_f, Error=KTMError).ellipsoid 

if self._E != E: 

_update_all(self) 

self._E = E 

 

@Property_RO 

def equatoradius(self): 

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

''' 

return self.ellipsoid.a 

 

a = equatoradius 

 

@Property_RO 

def flattening(self): 

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

''' 

return self.ellipsoid.f 

 

f = flattening 

 

def forward(self, lat, lon, lon0=None, name=NN): 

'''Forward projection, from geographic to transverse Mercator. 

 

@arg lat: Latitude of point (C{degrees90}). 

@arg lon: Longitude of point (C{degrees180}). 

@arg lon0: Central meridian of the projection (C{degrees180}). 

@kwarg name: Optional name (C{str}). 

 

@return: L{Forward4Tuple}C{(easting, northing, convergence, scale)} 

with C{easting} and C{northing} in C{meter}, unfalsed, the 

I{gamma} meridian C{convergence} at point in C{degrees180} 

and I{k} C{scale} of projection at point C{scalar}. Any 

value may be C{NAN}, C{NINF} or C{INF} for singularities. 

 

@raise KTMError: For singularities, iff property C{raiser} is 

C{True}. 

''' 

lat, _lat = _unsigned2(_fix90(lat)) 

lon, _ = _diff182((self.lon0 if lon0 is None else lon0), lon) 

lon, _lon = _unsigned2(lon) 

backside = lon > 90 

if backside: # PYCHOK no cover 

lon = _180_0 - lon 

if lat == 0: 

_lat = True 

 

sphi, cphi, slam, clam = sincos2d_(lat, lon) 

E = self.ellipsoid 

if cphi and lat != 90: 

t = sphi / cphi 

tp = E.es_taupf(t) 

h = hypot(tp, clam) 

if h: 

xip = atan2(tp, clam) 

etap = asinh(slam / h) # atanh(sin(lam) / cosh(psi)) 

g = _atan2d(slam * tp, clam * hypot1(tp)) # Krueger p 22 (44) 

k = sqrt(E.e21 + E.e2 * cphi**2) * hypot1(t) / h 

elif self.raiser: 

raise KTMError(lat=lat, lon=lon, lon0=lon0, txt=_singular_) 

else: # PYCHOK no cover 

xip, etap = _0_0, (NINF if slam < 0 else INF) 

g, k = copysign0(_90_0, slam), INF 

else: # PYCHOK no cover 

xip, etap = PI_2, _0_0 

g, k = lon, E.es_c 

y, x, t, z = self._yxgk4(xip, etap, self._Alp) 

g -= t 

k *= z * self._k0_b1 

 

if backside: # PYCHOK no cover 

y, g = (PI - y), (_180_0 - g) 

y *= self._k0_a1 

x *= self._k0_a1 

if _lat: 

y, g = neg_(y, g) 

if _lon: 

x, g = neg_(x, g) 

 

return Forward4Tuple(x, y, _norm180(g), k, name=name or self.name) 

 

@property_doc_(''' the central scale factor (C{float}).''') 

def k0(self): 

'''Get the central scale factor (C{float}), aka I{C{scale0}}. 

''' 

return self._k0 # aka scale0 

 

@k0.setter # PYCHOK setter! 

def k0(self, k0): 

'''Set the central scale factor (C{float}), aka I{C{scale0}}. 

 

@raise KTMError: Invalid B{C{k0}}. 

''' 

k0 = Scalar_(k0=k0, Error=KTMError, low=_TOL_10, high=_1_0) 

if self._k0 != k0: # PYCHOK no cover 

KTransverseMercator._k0_a1._update(self) # redo ._k0_a1 

KTransverseMercator._k0_b1._update(self) # redo ._k0_b1 

self._k0 = k0 

 

@Property_RO 

def _k0_a1(self): 

'''(INTERNAL) Cache C{k0 * _b1 * equatoradius}. 

''' 

return self._k0_b1 * self.equatoradius 

 

@Property_RO 

def _k0_b1(self): 

'''(INTERNAL) Cache C{k0 * _b1}. 

''' 

return self.k0 * self._b1 

 

@property_doc_(''' the central meridian (C{degrees180}).''') 

def lon0(self): 

'''Get the central meridian (C{degrees180}). 

''' 

return self._lon0 

 

@lon0.setter # PYCHOK setter! 

def lon0(self, lon0): 

'''Set the central meridian (C{degrees180}). 

 

@raise KTMError: Invalid B{C{lon0}}. 

''' 

self._lon0 = _norm180(Degrees(lon0=lon0, Error=KTMError)) 

 

@property_doc_(''' raise a L{KTMError} for C{forward} singularities (C{bool}).''') 

def raiser(self): 

'''Get the error setting (C{bool}). 

''' 

return self._raiser 

 

@raiser.setter # PYCHOK setter! 

def raiser(self, raiser): 

'''Set the error setting (C{bool}), to C{True} to throw a L{KTMError} 

for C{forward} singularities. 

''' 

self._raiser = bool(raiser) 

 

def reverse(self, x, y, lon0=None, name=NN): 

'''Reverse projection, from transverse Mercator to geographic. 

 

@arg x: Easting of point (C{meter}). 

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

@arg lon0: Central meridian of the projection (C{degrees180}). 

 

@return: L{Reverse4Tuple}C{(lat, lon, convergence, scale)} with 

C{lat}- and C{lon}gitude in C{degrees}, I{unfalsed}. 

''' 

eta, _lon = _unsigned2(x / self._k0_a1) 

xi, _lat = _unsigned2(y / self._k0_a1) 

backside = xi > PI_2 

if backside: # PYCHOK no cover 

xi = PI - xi 

 

xip, etap, g, k = self._yxgk4(xi, eta, self._Bet) 

t = self._k0_b1 

k = (t / k) if k else (NINF if t < 0 else INF) 

h, c = sinh(etap), cos(xip) 

if c > 0: 

r = hypot(h, c) 

else: # PYCHOK no cover 

r = abs(h) 

c = _0_0 

E = self.ellipsoid 

if r: 

lon = _atan2d(h, c) # Krueger p 17 (25) 

s = sin(xip) # Newton for tau 

t = E.es_tauf(s / r) 

lat = atand(t) 

g += _atan2d(s * tanh(etap), c) # Krueger p 19 (31) 

k *= sqrt(E.e21 + E.e2 / (t**2 + _1_0)) * hypot1(t) * r 

else: # PYCHOK no cover 

lat, lon = _90_0, _0_0 

k *= E.es_c 

 

if backside: # PYCHOK no cover 

lon, g = (_180_0 - lon), (_180_0 - g) 

if _lat: 

lat, g = neg_(lat, g) 

if _lon: 

lon, g = neg_(lon, g) 

 

lon += self.lon0 if lon0 is None else _norm180(lon0) 

return Reverse4Tuple(lat, _norm180(lon), _norm180(g), k, 

name=name or self.name) 

 

@Property 

def TMorder(self): 

'''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8). 

''' 

return self._mTM 

 

@TMorder.setter # PYCHOK setter! 

def TMorder(self, order): 

'''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8). 

''' 

m = _Xorder(_AlpCoeffs, KTMError, TMorder=order) 

if self._mTM != m: 

_update_all(self) 

self._mTM = m 

 

def toStr(self, **kwds): 

'''Return a C{str} representation. 

 

@arg kwds: Optional, overriding keyword arguments. 

''' 

d = dict(ellipsoid=self.ellipsoid, k0=self.k0, TMorder=self.TMorder) 

if self.name: # PYCHOK no cover 

d.update(name=self.name) 

return _COMMASPACE_.join(_pairs(d, **kwds)) 

 

def _yxgk4(self, xi_, eta_, C): 

'''(INTERNAL) Complex Clenshaw summation with 

C{B{C}=_Alp} or C{B{C}=-_Bet}, negated! 

''' 

def _sinhcosh2(x): 

return sinh(x), cosh(x) 

 

x = complex(xi_, eta_) 

if self.f: # isEllipsoidal 

s, c = sincos2( xi_ * 2) 

sh, ch = _sinhcosh2(eta_ * 2) 

n = -s 

s = complex(s * ch, c * sh) # sin(zeta * 2) 

c = complex(c * ch, n * sh) # cos(zeta * 2) 

 

y0 = y1 = z0 = z1 = complex(0) # 0+j0 

n = self.TMorder # == len(C) - 1 

if isodd(n): 

Cn = C[n] 

y0 = complex(Cn) # +j0 

z0 = complex(Cn * (n * 2)) 

n -= 1 

a = c * 2 # cos(zeta * 2) * 2 

while n > 0: 

Cn = C[n] 

y1 = _c(a, y0, y1, Cn) 

z1 = _c(a, z0, z1, Cn * (n * 2)) 

n -= 1 

Cn = C[n] 

y0 = _c(a, y1, y0, Cn) 

z0 = _c(a, z1, z0, Cn * (n * 2)) 

n -= 1 

# assert n == 0 

x = _c(s, y0, -x, _0_0) 

c = _c(c, z0, z1, _1_0) 

 

# Gauss-Schreiber to Gauss-Krueger TM 

# C{cmath.phase} handles INF, NAN, etc. 

g, k = degrees(phase(c)), abs(c) 

else: # isSpherical 

g, k = _0_0, _1_0 

 

return x.real, x.imag, g, k 

 

 

def _c(a, b0, b1, Cn): 

'''(INTERNAL) Accurately compute complex M{a * b0 - b1 + Cn} 

with complex args C{a}, C{b0} and C{b1} and scalar C{Cn}. 

 

@see: CPython function U{_Py_c_prod<https://GitHub.com/python/ 

cpython/blob/main/Objects/complexobject.c>}. 

 

@note: Python function C{cmath.fsum} is no longer available, 

but stil mentioned in Note 4 of the comments before 

CPython function U{math_fsum<https://GitHub.com/python/ 

cpython/blob/main/Modules/mathmodule.c>} 

''' 

r = fsum1_(a.real * b0.real, -a.imag * b0.imag, -b1.real, Cn, floats=True) 

j = fsum1_(a.real * b0.imag, a.imag * b0.real, -b1.imag, floats=True) 

return complex(r, j) 

 

 

def _Xorder(_Coeffs, Error, **Xorder): # in .rhumbx 

'''(INTERNAL) Validate C{RAorder} or C{TMorder}. 

''' 

X, m = Xorder.popitem() 

if m in _Coeffs and isint(m): 

return m 

t = sorted(map(str, _Coeffs.keys())) 

raise Error(X, m, txt=_not_(_or(*t))) 

 

 

def _Xs(_Coeffs, m, E, RA=False): # in .rhumbx 

'''(INTERNAL) Compute the C{A}, C{B} or C{RA} terms of order 

B{C{m}} for I{Krüger} series and I{rhumbx._sincosSeries}, 

return a tuple with C{B{m} + 1} terms C{X}, C{X[0]==0}. 

''' 

Cs = _Coeffs[m] 

assert len(Cs) == (((m + 1) * (m + 4)) if RA else 

((m + 3) * m)) // 2 

n = n_ = E.n 

if n: # isEllipsoidal 

X = [0] # X[0] never used, it's just an integration 

# constant, it cancels when evaluating a definite 

# integral. Don't bother computing it, it is not 

# used in C{KTransverseMercator._yxgk4} above nor 

# in C{rhumbx._sincosSeries}. 

i = (m + 2) if RA else 0 

for r in range(m - 1, -1, -1): # [m-1 ... 0] 

j = i + r + 1 

X.append(_polynomial(n, Cs, i, j) * n_ / Cs[j]) 

i = j + 1 

n_ *= n 

X = tuple(X) 

else: # isSpherical 

X = _0_0s(m + 1) 

return X 

 

 

# _Alp- and _BetCoeffs in .rhumbx 

_AlpCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00 

4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4 

164, 225, -480, 360, 720, # Alp[1]/n^1, polynomial(n), order 3 

557, -864, 390, 1440, # Alp[2]/n^2, polynomial(n), order 2 

-1236, 427, 1680, # PYCHOK Alp[3]/n^3, polynomial(n), order 1 

49561, 161280), # Alp[4]/n^4, polynomial(n), order 0, count = 14 

5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5 

-635, 328, 450, -960, 720, 1440, # Alp[1]/n^1, polynomial(n), order 4 

4496, 3899, -6048, 2730, 10080, # PYCHOK Alp[2]/n^2, polynomial(n), order 3 

15061, -19776, 6832, 26880, # PYCHOK Alp[3]/n^3, polynomial(n), order 2 

-171840, 49561, 161280, # Alp[4]/n^4, polynomial(n), order 1 

34729, 80640), # PYCHOK Alp[5]/n^5, polynomial(n), order 0, count = 20 

6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6 

31564, -66675, 34440, 47250, -100800, 75600, 151200, # Alp[1]/n^1, polynomial(n), order 5 

-1983433, 863232, 748608, -1161216, 524160, 1935360, # PYCHOK Alp[2]/n^2, polynomial(n), order 4 

670412, 406647, -533952, 184464, 725760, # Alp[3]/n^3, polynomial(n), order 3 

6601661, -7732800, 2230245, 7257600, # Alp[4]/n^4, polynomial(n), order 2 

-13675556, 3438171, 7983360, # PYCHOK Alp[5]/n^5, polynomial(n), order 1 

212378941, 319334400), # Alp[6]/n^6, polynomial(n), order 0, count = 27 

7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7 

1804025, 2020096, -4267200, 2204160, 3024000, -6451200, 4838400, 9676800, # Alp[1]/n^1, polynomial(n), order 6 

4626384, -9917165, 4316160, 3743040, -5806080, 2620800, 9676800, # Alp[2]/n^2, polynomial(n), order 5 

-67102379, 26816480, 16265880, -21358080, 7378560, 29030400, # PYCHOK Alp[3]/n^3, polynomial(n), order 4 

155912000, 72618271, -85060800, 24532695, 79833600, # Alp[4]/n^4, polynomial(n), order 3 

102508609, -109404448, 27505368, 63866880, # Alp[5]/n^5, polynomial(n), order 2 

-12282192400, 2760926233, 4151347200, # PYCHOK Alp[6]/n^6, polynomial(n), order 1 

1522256789, 1383782400), # Alp[7]/n^7, polynomial(n), order 0, count = 35 

8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8 

-75900428, 37884525, 42422016, -89611200, 46287360, 63504000, -135475200, 101606400, 203212800, # Alp[1]/n^1, polynomial(n), order 7 

148003883, 83274912, -178508970, 77690880, 67374720, -104509440, 47174400, 174182400, # PYCHOK Alp[2]/n^2, polynomial(n), order 6 

318729724, -738126169, 294981280, 178924680, -234938880, 81164160, 319334400, # PYCHOK Alp[3]/n^3, polynomial(n), order 5 

-40176129013, 14967552000, 6971354016, -8165836800, 2355138720, 7664025600, # Alp[4]/n^4, polynomial(n), order 4 

10421654396, 3997835751, -4266773472, 1072709352, 2490808320, # PYCHOK Alp[5]/n^5, polynomial(n), order 3 

175214326799, -171950693600, 38652967262, 58118860800, # PYCHOK Alp[6]/n^6, polynomial(n), order 2 

-67039739596, 13700311101, 12454041600, # PYCHOK Alp[7]/n^7, polynomial(n), order 1 

1424729850961, 743921418240) # PYCHOK Alp[8]/n^8, polynomial(n), order 0, count = 44 

} 

_B1Coeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00 

2: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 2 

1, 16, 64, 64), # b1 * (n + 1), polynomial(n2), order 2 

3: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 3 

1, 4, 64, 256, 256), # b1 * (n + 1), polynomial(n2), order 3 

4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 4 

25, 64, 256, 4096, 16384, 16384) # PYCHOK b1 * (n + 1), polynomial(n2), order 4 

} 

_BetCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00 

4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4 

-4, 555, -960, 720, 1440, # Bet[1]/n^1, polynomial(n), order 3 

-437, 96, 30, 1440, # Bet[2]/n^2, polynomial(n), order 2 

-148, 119, 3360, # Bet[3]/n^3, polynomial(n), order 1 

4397, 161280), # PYCHOK Bet[4]/n^4, polynomial(n), order 0, count = 14 

5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5 

-3645, -64, 8880, -15360, 11520, 23040, # Bet[1]/n^1, polynomial(n), order 4 

4416, -3059, 672, 210, 10080, # PYCHOK Bet[2]/n^2, polynomial(n), order 3 

-627, -592, 476, 13440, # Bet[3]/n^3, polynomial(n), order 2 

-3520, 4397, 161280, # Bet[4]/n^4, polynomial(n), order 1 

4583, 161280), # PYCHOK Bet[5]/n^5, polynomial(n), order 0, count = 20 

6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6 

384796, -382725, -6720, 932400, -1612800, 1209600, 2419200, # Bet[1]/n^1, polynomial(n), order 5 

-1118711, 1695744, -1174656, 258048, 80640, 3870720, # PYCHOK Bet[2]/n^2, polynomial(n), order 4 

22276, -16929, -15984, 12852, 362880, # Bet[3]/n^3, polynomial(n), order 3 

-830251, -158400, 197865, 7257600, # PYCHOK Bet[4]/n^4, polynomial(n), order 2 

-435388, 453717, 15966720, # PYCHOK Bet[5]/n^5, polynomial(n), order 1 

20648693, 638668800), # Bet[6]/n^6, polynomial(n), order 0, count = 27 

7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7 

-5406467, 6156736, -6123600, -107520, 14918400, -25804800, 19353600, 38707200, # Bet[1]/n^1, polynomial(n), order 6 

829456, -5593555, 8478720, -5873280, 1290240, 403200, 19353600, # PYCHOK Bet[2]/n^2, polynomial(n), order 5 

9261899, 3564160, -2708640, -2557440, 2056320, 58060800, # PYCHOK Bet[3]/n^3, polynomial(n), order 4 

14928352, -9132761, -1742400, 2176515, 79833600, # PYCHOK Bet[4]/n^4, polynomial(n), order 3 

-8005831, -1741552, 1814868, 63866880, # Bet[5]/n^5, polynomial(n), order 2 

-261810608, 268433009, 8302694400, # Bet[6]/n^6, polynomial(n), order 1 

219941297, 5535129600), # PYCHOK Bet[7]/n^7, polynomial(n), order 0, count = 35 

8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8 

31777436, -37845269, 43097152, -42865200, -752640, 104428800, -180633600, 135475200, 270950400, # Bet[1]/n^1, polynomial(n), order 7 

24749483, 14930208, -100683990, 152616960, -105719040, 23224320, 7257600, 348364800, # Bet[2]/n^2, polynomial(n), order 6 

-232468668, 101880889, 39205760, -29795040, -28131840, 22619520, 638668800, # PYCHOK Bet[3]/n^3, polynomial(n), order 5 

324154477, 1433121792, -876745056, -167270400, 208945440, 7664025600, # Bet[4]/n^4, polynomial(n), order 4 

457888660, -312227409, -67920528, 70779852, 2490808320, # Bet[5]/n^5, polynomial(n), order 3 

-19841813847, -3665348512, 3758062126, 116237721600, # PYCHOK Bet[6]/n^6, polynomial(n), order 2 

-1989295244, 1979471673, 49816166400, # PYCHOK Bet[7]/n^7, polynomial(n), order 1 

191773887257, 3719607091200) # Bet[8]/n^8, polynomial(n), order 0, count = 44 

} 

 

assert set(_AlpCoeffs.keys()) == set(_BetCoeffs.keys()) 

 

if __name__ == '__main__': 

 

from pygeodesy.interns import _usage 

from sys import argv, exit as _exit 

 

_exit(_usage(*argv).replace('.ktm', '.etm -series')) 

 

# **) MIT License 

# 

# Copyright (C) 2022-2022 -- 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.