def _sigmaInv0(self, xi, eta):
'''Starting point for C{sigmaInv}.
@return: 3-Tuple C{(u, v, trip)}.
@see: C{bool TMExact::sigmainv0(real xi, real eta,
real &u, real &v)}.
'''
trip = False
if eta > self._Ev_KE_5_4 or xi < min(-self._Eu_E_1_4,
eta - self._Ev_KE):
# sigma as a simple pole at
# w = w0 = Eu.K() + i * Ev.K()
# and sigma is approximated by
# sigma = (Eu.E() + i * Ev.KE()) + 1/(w - w0)
x = xi - self._Eu_E
y = eta - self._Ev_KE
d = x**2 + y**2
u = self._Eu_K + x / d
v = self._Ev_K - y / d
elif eta > self._Ev_KE or (eta > self._Ev_KE_3_4
and xi < self._Eu_E_1_4):
# At w = w0 = i * Ev.K(), we have
# sigma = sigma0 = i * Ev.KE()
# sigma' = sigma'' = 0
#
# including the next term in the Taylor series gives:
# sigma = sigma0 - _mv / 3 * (w - w0)^3
#
# When inverting this, we map arg(w - w0) = [-pi/2, -pi/6]
# to arg(sigma - sigma0) = [-pi/2, pi/2]
# mapping arg = [-pi/2, -pi/6] to [-pi/2, pi/2]
d = eta - self._Ev_KE
r = hypot(xi, d)
# Error using this guess is about 0.068 * rad^(5/3)
trip = r < _TAYTOL2
# Map the range [-90, 180] in sigma space to [-90, 0] in
# w space. See discussion in zetainv0 on the cut for ang.
r = cbrt(r * self._3_mv)
a = atan2(d - xi, xi + d) / 3.0 - PI_4
s, c = sincos2(a)
u = r * c
v = r * s + self._Ev_K
else: # use w = sigma * Eu.K/Eu.E (correct in the limit _e -> 0)
r = self._Eu_K / self._Eu_E
u = xi * r
v = eta * r
return u, v, trip
def toNvector(self, datum=Datums.WGS84):
'''Convert this cartesian to an (ellipsoidal) n-vector.
@keyword datum: Optional datum to use (L{Datum}).
@return: The ellipsoidal n-vector (L{Nvector}).
@raise ValueError: The I{Cartesian} at origin.
@example:
>>> from ellipsoidalNvector import LatLon
>>> c = Cartesian(3980581, 97, 4966825)
>>> n = c.toNvector() # (0.62282, 0.000002, 0.78237, +0.24)
'''
if self._Nv is None or datum != self._Nv.datum:
E = datum.ellipsoid
x, y, z = self.to3xyz()
# Kenneth Gade eqn 23
p = (x**2 + y**2) * E.a2_
q = (z**2 * E.e12) * E.a2_
r = fsum_(p, q, -E.e4) / 6
s = (p * q * E.e4) / (4 * r**3)
t = cbrt(fsum_(1, s, sqrt(s * (2 + s))))
u = r * fsum_(1, t, 1 / t)
v = sqrt(u**2 + E.e4 * q)
w = E.e2 * fsum_(u, v, -q) / (2 * v)
k = sqrt(fsum_(u, v, w**2)) - w
if abs(k) < EPS:
raise ValueError('%s: %r' % ('origin', self))
e = k / (k + E.e2)
d = e * hypot(x, y)
t = hypot(d, z)
if t < EPS:
raise ValueError('%s: %r' % ('origin', self))
h = fsum_(k, E.e2, -1) / k * t
s = e / t
self._Nv = Nvector(x * s,
y * s,
z / t,
h=h,
datum=datum,
name=self.name)
return self._Nv
def _zetaInv0(self, psi, lam):
'''Starting point for C{zetaInv}.
@return: 3-Tuple C{(u, v, trip)}.
@see: C{bool TMExact::zetainv0(real psi, real lam, # radians
real &u, real &v)}.
'''
trip = False
if (psi < -self._e_PI_4 and lam > self._1_e2_PI_2
and psi < lam - self._1_e_PI_2):
# N.B. this branch is normally not taken because psi < 0
# is converted psi > 0 by Forward.
#
# There's a log singularity at w = w0 = Eu.K() + i * Ev.K(),
# corresponding to the south pole, where we have, approximately
#
# psi = _e + i * pi/2 - _e * atanh(cos(i * (w - w0)/(1 + _mu/2)))
#
# Inverting this gives:
h = sinh(1 - psi / self._e)
a = (PI_2 - lam) / self._e
s, c = sincos2(a)
u = self._Eu_K - asinh(s / hypot(c, h)) * self._mu_2_1
v = self._Ev_K - atan2(c, h) * self._mu_2_1
elif (psi < self._e_PI_2 and lam > self._1_e2_PI_2):
# At w = w0 = i * Ev.K(), we have
#
# zeta = zeta0 = i * (1 - _e) * pi/2
# zeta' = zeta'' = 0
#
# including the next term in the Taylor series gives:
#
# zeta = zeta0 - (_mv * _e) / 3 * (w - w0)^3
#
# When inverting this, we map arg(w - w0) = [-90, 0] to
# arg(zeta - zeta0) = [-90, 180]
d = lam - self._1_e_PI_2
r = hypot(psi, d)
# Error using this guess is about 0.21 * (rad/e)^(5/3)
trip = r < self._e_taytol_
# atan2(dlam-psi, psi+dlam) + 45d gives arg(zeta - zeta0)
# in range [-135, 225). Subtracting 180 (since multiplier
# is negative) makes range [-315, 45). Multiplying by 1/3
# (for cube root) gives range [-105, 15). In particular
# the range [-90, 180] in zeta space maps to [-90, 0] in
# w space as required.
r = cbrt(r * self._3_mv_e)
a = atan2(d - psi, psi + d) / 3.0 - PI_4
s, c = sincos2(a)
u = r * c
v = r * s + self._Ev_K
else:
# Use spherical TM, Lee 12.6 -- writing C{atanh(sin(lam) /
# cosh(psi)) = asinh(sin(lam) / hypot(cos(lam), sinh(psi)))}.
# This takes care of the log singularity at C{zeta = Eu.K()},
# corresponding to the north pole.
s, c = sincos2(lam)
h, r = sinh(psi), self._Eu_K / PI_2
# But scale to put 90, 0 on the right place
u = r * atan2(h, c)
v = r * asinh(s / hypot(c, h))
return u, v, trip