python-geographiclib 1.45-2 / usr / lib / python2.7 / dist-packages / geographiclib / geodesic.py

"""geodesic.py: transcription of GeographicLib::Geodesic class."""
# geodesic.py
#
# This is a rather literal translation of the GeographicLib::Geodesic class to
# python.  See the documentation for the C++ class for more information at
#
#    http://geographiclib.sourceforge.net/html/annotated.html
#
# The algorithms are derived in
#
#    Charles F. F. Karney,
#    Algorithms for geodesics, J. Geodesy 87, 43-55 (2013),
#    https://dx.doi.org/10.1007/s00190-012-0578-z
#    Addenda: http://geographiclib.sf.net/geod-addenda.html
#
# Copyright (c) Charles Karney (2011-2015) <charles@karney.com> and licensed
# under the MIT/X11 License.  For more information, see
# http://geographiclib.sourceforge.net/
######################################################################

import math
from geographiclib.geomath import Math
from geographiclib.constants import Constants
from geographiclib.geodesiccapability import GeodesicCapability

class Geodesic(object):
  """Solve geodesic problems.  The following illustrates its use

    import sys
    sys.path.append("/usr/local/lib/python/site-packages")
    from geographiclib.geodesic import Geodesic

    # The geodesic inverse problem
    Geodesic.WGS84.Inverse(-41.32, 174.81, 40.96, -5.50)

    # The geodesic direct problem
    Geodesic.WGS84.Direct(40.6, -73.8, 45, 10000e3)

    # How to obtain several points along a geodesic
    line = Geodesic.WGS84.Line(40.6, -73.8, 45)
    line.Position( 5000e3)
    line.Position(10000e3)

    # Computing the area of a geodesic polygon
    def p(lat,lon): return {'lat': lat, 'lon': lon}

    Geodesic.WGS84.Area([p(0, 0), p(0, 90), p(90, 0)])

  Documentation on these routines is available via

    help(Geodesic.__init__)
    help(Geodesic.Inverse)
    help(Geodesic.Direct)
    help(Geodesic.Line)
    help(line.Position)
    help(Geodesic.Area)

  All angles (latitudes, longitudes, azimuths, spherical arc lengths)
  are measured in degrees.  Latitudes must lie in [-90,90].  All lengths
  (distance, reduced length) are measured in meters.  Areas are measures
  in square meters.

  """

  GEOGRAPHICLIB_GEODESIC_ORDER = 6
  nA1_ = GEOGRAPHICLIB_GEODESIC_ORDER
  nC1_ = GEOGRAPHICLIB_GEODESIC_ORDER
  nC1p_ = GEOGRAPHICLIB_GEODESIC_ORDER
  nA2_ = GEOGRAPHICLIB_GEODESIC_ORDER
  nC2_ = GEOGRAPHICLIB_GEODESIC_ORDER
  nA3_ = GEOGRAPHICLIB_GEODESIC_ORDER
  nA3x_ = nA3_
  nC3_ = GEOGRAPHICLIB_GEODESIC_ORDER
  nC3x_ = (nC3_ * (nC3_ - 1)) // 2
  nC4_ = GEOGRAPHICLIB_GEODESIC_ORDER
  nC4x_ = (nC4_ * (nC4_ + 1)) // 2
  maxit1_ = 20
  maxit2_ = maxit1_ + Math.digits + 10

  tiny_ = math.sqrt(Math.minval)
  tol0_ = Math.epsilon
  tol1_ = 200 * tol0_
  tol2_ = math.sqrt(tol0_)
  tolb_ = tol0_ * tol2_
  xthresh_ = 1000 * tol2_

  CAP_NONE = GeodesicCapability.CAP_NONE
  CAP_C1   = GeodesicCapability.CAP_C1
  CAP_C1p  = GeodesicCapability.CAP_C1p
  CAP_C2   = GeodesicCapability.CAP_C2
  CAP_C3   = GeodesicCapability.CAP_C3
  CAP_C4   = GeodesicCapability.CAP_C4
  CAP_ALL  = GeodesicCapability.CAP_ALL
  CAP_MASK = GeodesicCapability.CAP_MASK
  OUT_ALL  = GeodesicCapability.OUT_ALL
  OUT_MASK = GeodesicCapability.OUT_MASK
  EMPTY         = GeodesicCapability.EMPTY
  LATITUDE      = GeodesicCapability.LATITUDE
  LONGITUDE     = GeodesicCapability.LONGITUDE
  AZIMUTH       = GeodesicCapability.AZIMUTH
  DISTANCE      = GeodesicCapability.DISTANCE
  STANDARD      = GeodesicCapability.STANDARD
  DISTANCE_IN   = GeodesicCapability.DISTANCE_IN
  REDUCEDLENGTH = GeodesicCapability.REDUCEDLENGTH
  GEODESICSCALE = GeodesicCapability.GEODESICSCALE
  AREA          = GeodesicCapability.AREA
  LONG_UNROLL   = GeodesicCapability.LONG_UNROLL
  LONG_NOWRAP   = GeodesicCapability.LONG_UNROLL
  ALL           = GeodesicCapability.ALL

  def SinCosSeries(sinp, sinx, cosx, c):
    """Private: Evaluate a trig series using Clenshaw summation."""
    # Evaluate
    # y = sinp ? sum(c[i] * sin( 2*i    * x), i, 1, n) :
    #            sum(c[i] * cos((2*i+1) * x), i, 0, n-1)
    # using Clenshaw summation.  N.B. c[0] is unused for sin series
    # Approx operation count = (n + 5) mult and (2 * n + 2) add
    k = len(c)                  # Point to one beyond last element
    n = k - sinp
    ar = 2 * (cosx - sinx) * (cosx + sinx) # 2 * cos(2 * x)
    y1 = 0                                 # accumulators for sum
    if n & 1:
      k -= 1; y0 = c[k]
    else:
      y0 = 0
    # Now n is even
    n = n // 2
    while n:                    # while n--:
      n -= 1
      # Unroll loop x 2, so accumulators return to their original role
      k -= 1; y1 = ar * y0 - y1 + c[k]
      k -= 1; y0 = ar * y1 - y0 + c[k]
    return ( 2 * sinx * cosx * y0 if sinp # sin(2 * x) * y0
             else cosx * (y0 - y1) )      # cos(x) * (y0 - y1)
  SinCosSeries = staticmethod(SinCosSeries)

  def Astroid(x, y):
    """Private: solve astroid equation."""
    # Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive root k.
    # This solution is adapted from Geocentric::Reverse.
    p = Math.sq(x)
    q = Math.sq(y)
    r = (p + q - 1) / 6
    if not(q == 0 and r <= 0):
      # Avoid possible division by zero when r = 0 by multiplying equations
      # for s and t by r^3 and r, resp.
      S = p * q / 4            # S = r^3 * s
      r2 = Math.sq(r)
      r3 = r * r2
      # The discriminant of the quadratic equation for T3.  This is zero on
      # the evolute curve p^(1/3)+q^(1/3) = 1
      disc = S * (S + 2 * r3)
      u = r
      if disc >= 0:
        T3 = S + r3
        # Pick the sign on the sqrt to maximize abs(T3).  This minimizes loss
        # of precision due to cancellation.  The result is unchanged because
        # of the way the T is used in definition of u.
        T3 += -math.sqrt(disc) if T3 < 0 else math.sqrt(disc) # T3 = (r * t)^3
        # N.B. cbrt always returns the real root.  cbrt(-8) = -2.
        T = Math.cbrt(T3)       # T = r * t
        # T can be zero; but then r2 / T -> 0.
        u += T + (r2 / T if T != 0 else 0)
      else:
        # T is complex, but the way u is defined the result is real.
        ang = math.atan2(math.sqrt(-disc), -(S + r3))
        # There are three possible cube roots.  We choose the root which
        # avoids cancellation.  Note that disc < 0 implies that r < 0.
        u += 2 * r * math.cos(ang / 3)
      v = math.sqrt(Math.sq(u) + q) # guaranteed positive
      # Avoid loss of accuracy when u < 0.
      uv = q / (v - u) if u < 0 else u + v # u+v, guaranteed positive
      w = (uv - q) / (2 * v)               # positive?
      # Rearrange expression for k to avoid loss of accuracy due to
      # subtraction.  Division by 0 not possible because uv > 0, w >= 0.
      k = uv / (math.sqrt(uv + Math.sq(w)) + w) # guaranteed positive
    else:                                       # q == 0 && r <= 0
      # y = 0 with |x| <= 1.  Handle this case directly.
      # for y small, positive root is k = abs(y)/sqrt(1-x^2)
      k = 0
    return k
  Astroid = staticmethod(Astroid)

  def A1m1f(eps):
    """Private: return A1-1."""
    coeff = [
      1, 4, 64, 0, 256,
    ]
    m = Geodesic.nA1_//2
    t = Math.polyval(m, coeff, 0, Math.sq(eps)) / coeff[m + 1]
    return (t + eps) / (1 - eps)
  A1m1f = staticmethod(A1m1f)

  def C1f(eps, c):
    """Private: return C1."""
    coeff = [
      -1, 6, -16, 32,
      -9, 64, -128, 2048,
      9, -16, 768,
      3, -5, 512,
      -7, 1280,
      -7, 2048,
    ]
    eps2 = Math.sq(eps)
    d = eps
    o = 0
    for l in range(1, Geodesic.nC1_ + 1): # l is index of C1p[l]
      m = (Geodesic.nC1_ - l) // 2        # order of polynomial in eps^2
      c[l] = d * Math.polyval(m, coeff, o, eps2) / coeff[o + m + 1]
      o += m + 2
      d *= eps
  C1f = staticmethod(C1f)

  def C1pf(eps, c):
    """Private: return C1'"""
    coeff = [
      205, -432, 768, 1536,
      4005, -4736, 3840, 12288,
      -225, 116, 384,
      -7173, 2695, 7680,
      3467, 7680,
      38081, 61440,
    ]
    eps2 = Math.sq(eps)
    d = eps
    o = 0
    for l in range(1, Geodesic.nC1p_ + 1): # l is index of C1p[l]
      m = (Geodesic.nC1p_ - l) // 2 # order of polynomial in eps^2
      c[l] = d * Math.polyval(m, coeff, o, eps2) / coeff[o + m + 1]
      o += m + 2
      d *= eps
  C1pf = staticmethod(C1pf)

  def A2m1f(eps):
    """Private: return A2-1"""
    coeff = [
      -11, -28, -192, 0, 256,
    ]
    m = Geodesic.nA2_//2
    t = Math.polyval(m, coeff, 0, Math.sq(eps)) / coeff[m + 1]
    return (t - eps) / (1 + eps)
  A2m1f = staticmethod(A2m1f)

  def C2f(eps, c):
    """Private: return C2"""
    coeff = [
      1, 2, 16, 32,
      35, 64, 384, 2048,
      15, 80, 768,
      7, 35, 512,
      63, 1280,
      77, 2048,
    ]
    eps2 = Math.sq(eps)
    d = eps
    o = 0
    for l in range(1, Geodesic.nC2_ + 1): # l is index of C2[l]
      m = (Geodesic.nC2_ - l) // 2        # order of polynomial in eps^2
      c[l] = d * Math.polyval(m, coeff, o, eps2) / coeff[o + m + 1]
      o += m + 2
      d *= eps
  C2f = staticmethod(C2f)

  def __init__(self, a, f):
    """Construct a Geodesic object for ellipsoid with major radius a and
    flattening f.

    """

    self._a = float(a)
    self._f = float(f)
    self._f1 = 1 - self._f
    self._e2 = self._f * (2 - self._f)
    self._ep2 = self._e2 / Math.sq(self._f1) # e2 / (1 - e2)
    self._n = self._f / ( 2 - self._f)
    self._b = self._a * self._f1
    # authalic radius squared
    self._c2 = (Math.sq(self._a) + Math.sq(self._b) *
                (1 if self._e2 == 0 else
                 (Math.atanh(math.sqrt(self._e2)) if self._e2 > 0 else
                  math.atan(math.sqrt(-self._e2))) /
                 math.sqrt(abs(self._e2))))/2
    # The sig12 threshold for "really short".  Using the auxiliary sphere
    # solution with dnm computed at (bet1 + bet2) / 2, the relative error in
    # the azimuth consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2.
    # (Error measured for 1/100 < b/a < 100 and abs(f) >= 1/1000.  For a given
    # f and sig12, the max error occurs for lines near the pole.  If the old
    # rule for computing dnm = (dn1 + dn2)/2 is used, then the error increases
    # by a factor of 2.)  Setting this equal to epsilon gives sig12 = etol2.
    # Here 0.1 is a safety factor (error decreased by 100) and max(0.001,
    # abs(f)) stops etol2 getting too large in the nearly spherical case.
    self._etol2 = 0.1 * Geodesic.tol2_ / math.sqrt( max(0.001, abs(self._f)) *
                                                    min(1.0, 1-self._f/2) / 2 )
    if not(Math.isfinite(self._a) and self._a > 0):
      raise ValueError("Major radius is not positive")
    if not(Math.isfinite(self._b) and self._b > 0):
      raise ValueError("Minor radius is not positive")
    self._A3x = list(range(Geodesic.nA3x_))
    self._C3x = list(range(Geodesic.nC3x_))
    self._C4x = list(range(Geodesic.nC4x_))
    self.A3coeff()
    self.C3coeff()
    self.C4coeff()

  def A3coeff(self):
    """Private: return coefficients for A3"""
    coeff = [
      -3, 128,
      -2, -3, 64,
      -1, -3, -1, 16,
      3, -1, -2, 8,
      1, -1, 2,
      1, 1,
    ]
    o = 0; k = 0
    for j in range(Geodesic.nA3_ - 1, -1, -1): # coeff of eps^j
      m = min(Geodesic.nA3_ - j - 1, j) # order of polynomial in n
      self._A3x[k] = Math.polyval(m, coeff, o, self._n) / coeff[o + m + 1]
      k += 1
      o += m + 2

  def C3coeff(self):
    """Private: return coefficients for C3"""
    coeff = [
      3, 128,
      2, 5, 128,
      -1, 3, 3, 64,
      -1, 0, 1, 8,
      -1, 1, 4,
      5, 256,
      1, 3, 128,
      -3, -2, 3, 64,
      1, -3, 2, 32,
      7, 512,
      -10, 9, 384,
      5, -9, 5, 192,
      7, 512,
      -14, 7, 512,
      21, 2560,
    ]
    o = 0; k = 0
    for l in range(1, Geodesic.nC3_): # l is index of C3[l]
      for j in range(Geodesic.nC3_ - 1, l - 1, -1): # coeff of eps^j
        m = min(Geodesic.nC3_ - j - 1, j) # order of polynomial in n
        self._C3x[k] = Math.polyval(m, coeff, o, self._n) / coeff[o + m + 1]
        k += 1
        o += m + 2

  def C4coeff(self):
    """Private: return coefficients for C4"""
    coeff = [
      97, 15015,
      1088, 156, 45045,
      -224, -4784, 1573, 45045,
      -10656, 14144, -4576, -858, 45045,
      64, 624, -4576, 6864, -3003, 15015,
      100, 208, 572, 3432, -12012, 30030, 45045,
      1, 9009,
      -2944, 468, 135135,
      5792, 1040, -1287, 135135,
      5952, -11648, 9152, -2574, 135135,
      -64, -624, 4576, -6864, 3003, 135135,
      8, 10725,
      1856, -936, 225225,
      -8448, 4992, -1144, 225225,
      -1440, 4160, -4576, 1716, 225225,
      -136, 63063,
      1024, -208, 105105,
      3584, -3328, 1144, 315315,
      -128, 135135,
      -2560, 832, 405405,
      128, 99099,
    ]
    o = 0; k = 0
    for l in range(Geodesic.nC4_): # l is index of C4[l]
      for j in range(Geodesic.nC4_ - 1, l - 1, -1): # coeff of eps^j
        m = Geodesic.nC4_ - j - 1 # order of polynomial in n
        self._C4x[k] = Math.polyval(m, coeff, o, self._n) / coeff[o + m + 1]
        k += 1
        o += m + 2

  def A3f(self, eps):
    """Private: return A3"""
    # Evaluate A3
    return Math.polyval(Geodesic.nA3_ - 1, self._A3x, 0, eps)

  def C3f(self, eps, c):
    """Private: return C3"""
    # Evaluate C3
    # Elements c[1] thru c[nC3_ - 1] are set
    mult = 1
    o = 0
    for l in range(1, Geodesic.nC3_): # l is index of C3[l]
      m = Geodesic.nC3_ - l - 1       # order of polynomial in eps
      mult *= eps
      c[l] = mult * Math.polyval(m, self._C3x, o, eps)
      o += m + 1

  def C4f(self, eps, c):
    """Private: return C4"""
    # Evaluate C4 coeffs by Horner's method
    # Elements c[0] thru c[nC4_ - 1] are set
    mult = 1
    o = 0
    for l in range(Geodesic.nC4_): # l is index of C4[l]
      m = Geodesic.nC4_ - l - 1    # order of polynomial in eps
      c[l] = mult * Math.polyval(m, self._C4x, o, eps)
      o += m + 1
      mult *= eps

  # return s12b, m12b, m0, M12, M21
  def Lengths(self, eps, sig12,
              ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2, outmask,
              # Scratch areas of the right size
              C1a, C2a):
    """Private: return a bunch of lengths"""
    # Return s12b, m12b, m0, M12, M21, where
    # m12b = (reduced length)/_b; s12b = distance/_b,
    # m0 = coefficient of secular term in expression for reduced length.
    outmask &= Geodesic.OUT_MASK
    # outmask & DISTANCE: set s12b
    # outmask & REDUCEDLENGTH: set m12b & m0
    # outmask & GEODESICSCALE: set M12 & M21

    s12b = m12b = m0 = M12 = M21 = Math.nan
    if outmask & (Geodesic.DISTANCE | Geodesic.REDUCEDLENGTH |
                  Geodesic.GEODESICSCALE):
      A1 = Geodesic.A1m1f(eps)
      Geodesic.C1f(eps, C1a)
      if outmask & (Geodesic.REDUCEDLENGTH | Geodesic.GEODESICSCALE):
        A2 = Geodesic.A2m1f(eps)
        Geodesic.C2f(eps, C2a)
        m0x = A1 - A2
        A2 = 1 + A2
      A1 = 1 + A1
    if outmask & Geodesic.DISTANCE:
      B1 = (Geodesic.SinCosSeries(True, ssig2, csig2, C1a) -
            Geodesic.SinCosSeries(True, ssig1, csig1, C1a))
      # Missing a factor of _b
      s12b = A1 * (sig12 + B1)
      if outmask & (Geodesic.REDUCEDLENGTH | Geodesic.GEODESICSCALE):
        B2 = (Geodesic.SinCosSeries(True, ssig2, csig2, C2a) -
              Geodesic.SinCosSeries(True, ssig1, csig1, C2a))
        J12 = m0x * sig12 + (A1 * B1 - A2 * B2)
    elif outmask & (Geodesic.REDUCEDLENGTH | Geodesic.GEODESICSCALE):
      # Assume here that nC1_ >= nC2_
      for l in range(1, Geodesic.nC2_):
        C2a[l] = A1 * C1a[l] - A2 * C2a[l]
      J12 = m0x * sig12 + (Geodesic.SinCosSeries(True, ssig2, csig2, C2a) -
                           Geodesic.SinCosSeries(True, ssig1, csig1, C2a))
    if outmask & Geodesic.REDUCEDLENGTH:
      m0 = m0x
      # Missing a factor of _b.
      # Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure
      # accurate cancellation in the case of coincident points.
      m12b = (dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) -
              csig1 * csig2 * J12)
    if outmask & Geodesic.GEODESICSCALE:
      csig12 = csig1 * csig2 + ssig1 * ssig2
      t = self._ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2)
      M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1
      M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2
    return s12b, m12b, m0, M12, M21

  # return sig12, salp1, calp1, salp2, calp2, dnm
  def InverseStart(self, sbet1, cbet1, dn1, sbet2, cbet2, dn2, lam12,
                   # Scratch areas of the right size
                   C1a, C2a):
    """Private: Find a starting value for Newton's method."""
    # Return a starting point for Newton's method in salp1 and calp1 (function
    # value is -1).  If Newton's method doesn't need to be used, return also
    # salp2 and calp2 and function value is sig12.
    sig12 = -1; salp2 = calp2 = dnm = Math.nan # Return values
    # bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0]
    sbet12 = sbet2 * cbet1 - cbet2 * sbet1
    cbet12 = cbet2 * cbet1 + sbet2 * sbet1
    # Volatile declaration needed to fix inverse cases
    # 88.202499451857 0 -88.202499451857 179.981022032992859592
    # 89.262080389218 0 -89.262080389218 179.992207982775375662
    # 89.333123580033 0 -89.333123580032997687 179.99295812360148422
    # which otherwise fail with g++ 4.4.4 x86 -O3
    sbet12a = sbet2 * cbet1
    sbet12a += cbet2 * sbet1

    shortline = cbet12 >= 0 and sbet12 < 0.5 and cbet2 * lam12 < 0.5
    omg12 = lam12
    if shortline:
      sbetm2 = Math.sq(sbet1 + sbet2)
      # sin((bet1+bet2)/2)^2
      # =  (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2)
      sbetm2 /= sbetm2 + Math.sq(cbet1 + cbet2)
      dnm = math.sqrt(1 + self._ep2 * sbetm2)
      omg12 /= self._f1 * dnm
    somg12 = math.sin(omg12); comg12 = math.cos(omg12)

    salp1 = cbet2 * somg12
    calp1 = (
      sbet12 + cbet2 * sbet1 * Math.sq(somg12) / (1 + comg12) if comg12 >= 0
      else sbet12a - cbet2 * sbet1 * Math.sq(somg12) / (1 - comg12))

    ssig12 = math.hypot(salp1, calp1)
    csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12

    if shortline and ssig12 < self._etol2:
      # really short lines
      salp2 = cbet1 * somg12
      calp2 = sbet12 - cbet1 * sbet2 * (Math.sq(somg12) / (1 + comg12)
                                        if comg12 >= 0 else 1 - comg12)
      salp2, calp2 = Math.norm(salp2, calp2)
      # Set return value
      sig12 = math.atan2(ssig12, csig12)
    elif (abs(self._n) >= 0.1 or # Skip astroid calc if too eccentric
          csig12 >= 0 or
          ssig12 >= 6 * abs(self._n) * math.pi * Math.sq(cbet1)):
      # Nothing to do, zeroth order spherical approximation is OK
      pass
    else:
      # Scale lam12 and bet2 to x, y coordinate system where antipodal point
      # is at origin and singular point is at y = 0, x = -1.
      # real y, lamscale, betscale
      # Volatile declaration needed to fix inverse case
      # 56.320923501171 0 -56.320923501171 179.664747671772880215
      # which otherwise fails with g++ 4.4.4 x86 -O3
      # volatile real x
      if self._f >= 0:            # In fact f == 0 does not get here
        # x = dlong, y = dlat
        k2 = Math.sq(sbet1) * self._ep2
        eps = k2 / (2 * (1 + math.sqrt(1 + k2)) + k2)
        lamscale = self._f * cbet1 * self.A3f(eps) * math.pi
        betscale = lamscale * cbet1
        x = (lam12 - math.pi) / lamscale
        y = sbet12a / betscale
      else:                     # _f < 0
        # x = dlat, y = dlong
        cbet12a = cbet2 * cbet1 - sbet2 * sbet1
        bet12a = math.atan2(sbet12a, cbet12a)
        # real m12b, m0, dummy
        # In the case of lon12 = 180, this repeats a calculation made in
        # Inverse.
        dummy, m12b, m0, dummy, dummy = self.Lengths(
          self._n, math.pi + bet12a, sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
          cbet1, cbet2, Geodesic.REDUCEDLENGTH, C1a, C2a)
        x = -1 + m12b / (cbet1 * cbet2 * m0 * math.pi)
        betscale = (sbet12a / x if x < -0.01
                    else -self._f * Math.sq(cbet1) * math.pi)
        lamscale = betscale / cbet1
        y = (lam12 - math.pi) / lamscale

      if y > -Geodesic.tol1_ and x > -1 - Geodesic.xthresh_:
        # strip near cut
        if self._f >= 0:
          salp1 = min(1.0, -x); calp1 = - math.sqrt(1 - Math.sq(salp1))
        else:
          calp1 = max((0.0 if x > -Geodesic.tol1_ else -1.0), x)
          salp1 = math.sqrt(1 - Math.sq(calp1))
      else:
        # Estimate alp1, by solving the astroid problem.
        #
        # Could estimate alpha1 = theta + pi/2, directly, i.e.,
        #   calp1 = y/k; salp1 = -x/(1+k);  for _f >= 0
        #   calp1 = x/(1+k); salp1 = -y/k;  for _f < 0 (need to check)
        #
        # However, it's better to estimate omg12 from astroid and use
        # spherical formula to compute alp1.  This reduces the mean number of
        # Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12
        # (min 0 max 5).  The changes in the number of iterations are as
        # follows:
        #
        # change percent
        #    1       5
        #    0      78
        #   -1      16
        #   -2       0.6
        #   -3       0.04
        #   -4       0.002
        #
        # The histogram of iterations is (m = number of iterations estimating
        # alp1 directly, n = number of iterations estimating via omg12, total
        # number of trials = 148605):
        #
        #  iter    m      n
        #    0   148    186
        #    1 13046  13845
        #    2 93315 102225
        #    3 36189  32341
        #    4  5396      7
        #    5   455      1
        #    6    56      0
        #
        # Because omg12 is near pi, estimate work with omg12a = pi - omg12
        k = Geodesic.Astroid(x, y)
        omg12a = lamscale * ( -x * k/(1 + k) if self._f >= 0
                              else -y * (1 + k)/k )
        somg12 = math.sin(omg12a); comg12 = -math.cos(omg12a)
        # Update spherical estimate of alp1 using omg12 instead of lam12
        salp1 = cbet2 * somg12
        calp1 = sbet12a - cbet2 * sbet1 * Math.sq(somg12) / (1 - comg12)
    # Sanity check on starting guess.  Backwards check allows NaN through.
    if not (salp1 <= 0):
      salp1, calp1 = Math.norm(salp1, calp1)
    else:
      salp1 = 1; calp1 = 0
    return sig12, salp1, calp1, salp2, calp2, dnm

  # return lam12, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2, eps,
  # domg12, dlam12
  def Lambda12(self, sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1, diffp,
               # Scratch areas of the right size
               C1a, C2a, C3a):
    """Private: Solve hybrid problem"""
    if sbet1 == 0 and calp1 == 0:
      # Break degeneracy of equatorial line.  This case has already been
      # handled.
      calp1 = -Geodesic.tiny_

    # sin(alp1) * cos(bet1) = sin(alp0)
    salp0 = salp1 * cbet1
    calp0 = math.hypot(calp1, salp1 * sbet1) # calp0 > 0

    # real somg1, comg1, somg2, comg2, omg12, lam12
    # tan(bet1) = tan(sig1) * cos(alp1)
    # tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1)
    ssig1 = sbet1; somg1 = salp0 * sbet1
    csig1 = comg1 = calp1 * cbet1
    ssig1, csig1 = Math.norm(ssig1, csig1)
    # Math.norm(somg1, comg1); -- don't need to normalize!

    # Enforce symmetries in the case abs(bet2) = -bet1.  Need to be careful
    # about this case, since this can yield singularities in the Newton
    # iteration.
    # sin(alp2) * cos(bet2) = sin(alp0)
    salp2 = salp0 / cbet2 if cbet2 != cbet1 else salp1
    # calp2 = sqrt(1 - sq(salp2))
    #       = sqrt(sq(calp0) - sq(sbet2)) / cbet2
    # and subst for calp0 and rearrange to give (choose positive sqrt
    # to give alp2 in [0, pi/2]).
    calp2 = (math.sqrt(Math.sq(calp1 * cbet1) +
                       ((cbet2 - cbet1) * (cbet1 + cbet2) if cbet1 < -sbet1
                        else (sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2
             if cbet2 != cbet1 or abs(sbet2) != -sbet1 else abs(calp1))
    # tan(bet2) = tan(sig2) * cos(alp2)
    # tan(omg2) = sin(alp0) * tan(sig2).
    ssig2 = sbet2; somg2 = salp0 * sbet2
    csig2 = comg2 = calp2 * cbet2
    ssig2, csig2 = Math.norm(ssig2, csig2)
    # Math.norm(somg2, comg2); -- don't need to normalize!

    # sig12 = sig2 - sig1, limit to [0, pi]
    sig12 = math.atan2(max(0.0, csig1 * ssig2 - ssig1 * csig2),
                       csig1 * csig2 + ssig1 * ssig2)

    # omg12 = omg2 - omg1, limit to [0, pi]
    omg12 = math.atan2(max(0.0, comg1 * somg2 - somg1 * comg2),
                       comg1 * comg2 + somg1 * somg2)
    # real B312, h0
    k2 = Math.sq(calp0) * self._ep2
    eps = k2 / (2 * (1 + math.sqrt(1 + k2)) + k2)
    self.C3f(eps, C3a)
    B312 = (Geodesic.SinCosSeries(True, ssig2, csig2, C3a) -
            Geodesic.SinCosSeries(True, ssig1, csig1, C3a))
    h0 = -self._f * self.A3f(eps)
    domg12 = salp0 * h0 * (sig12 + B312)
    lam12 = omg12 + domg12

    if diffp:
      if calp2 == 0:
        dlam12 = - 2 * self._f1 * dn1 / sbet1
      else:
        dummy, dlam12, dummy, dummy, dummy = self.Lengths(
          eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2,
          Geodesic.REDUCEDLENGTH, C1a, C2a)
        dlam12 *= self._f1 / (calp2 * cbet2)
    else:
      dlam12 = Math.nan

    return (lam12, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2, eps,
            domg12, dlam12)

  # return a12, s12, azi1, azi2, m12, M12, M21, S12
  def GenInverse(self, lat1, lon1, lat2, lon2, outmask):
    """Private: General version of the inverse problem"""
    a12 = s12 = azi1 = azi2 = m12 = M12 = M21 = S12 = Math.nan # return vals

    outmask &= Geodesic.OUT_MASK
    # Compute longitude difference (AngDiff does this carefully).  Result is
    # in [-180, 180] but -180 is only for west-going geodesics.  180 is for
    # east-going and meridional geodesics.
    # If very close to being on the same half-meridian, then make it so.
    lon12 = Math.AngRound(Math.AngDiff(lon1, lon2))
    # Make longitude difference positive.
    lonsign = 1 if lon12 >= 0 else -1
    lon12 *= lonsign
    # If really close to the equator, treat as on equator.
    lat1 = Math.AngRound(Math.LatFix(lat1))
    lat2 = Math.AngRound(Math.LatFix(lat2))
    # Swap points so that point with higher (abs) latitude is point 1
    # If one latitude is a nan, then it becomes lat1.
    swapp = -1 if abs(lat1) < abs(lat2) else 1
    if swapp < 0:
      lonsign *= -1
      lat2, lat1 = lat1, lat2
    # Make lat1 <= 0
    latsign = 1 if lat1 < 0 else -1
    lat1 *= latsign
    lat2 *= latsign
    # Now we have
    #
    #     0 <= lon12 <= 180
    #     -90 <= lat1 <= 0
    #     lat1 <= lat2 <= -lat1
    #
    # longsign, swapp, latsign register the transformation to bring the
    # coordinates to this canonical form.  In all cases, 1 means no change was
    # made.  We make these transformations so that there are few cases to
    # check, e.g., on verifying quadrants in atan2.  In addition, this
    # enforces some symmetries in the results returned.

    # real phi, sbet1, cbet1, sbet2, cbet2, s12x, m12x

    sbet1, cbet1 = Math.sincosd(lat1); sbet1 *= self._f1
    # Ensure cbet1 = +epsilon at poles
    sbet1, cbet1 = Math.norm(sbet1, cbet1); cbet1 = max(Geodesic.tiny_, cbet1)

    sbet2, cbet2 = Math.sincosd(lat2); sbet2 *= self._f1
    # Ensure cbet2 = +epsilon at poles
    sbet2, cbet2 = Math.norm(sbet2, cbet2); cbet2 = max(Geodesic.tiny_, cbet2)

    # If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the
    # |bet1| - |bet2|.  Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is
    # a better measure.  This logic is used in assigning calp2 in Lambda12.
    # Sometimes these quantities vanish and in that case we force bet2 = +/-
    # bet1 exactly.  An example where is is necessary is the inverse problem
    # 48.522876735459 0 -48.52287673545898293 179.599720456223079643
    # which failed with Visual Studio 10 (Release and Debug)

    if cbet1 < -sbet1:
      if cbet2 == cbet1:
        sbet2 = sbet1 if sbet2 < 0 else -sbet1
    else:
      if abs(sbet2) == -sbet1:
        cbet2 = cbet1

    dn1 = math.sqrt(1 + self._ep2 * Math.sq(sbet1))
    dn2 = math.sqrt(1 + self._ep2 * Math.sq(sbet2))

    lam12 = math.radians(lon12)
    slam12, clam12 = Math.sincosd(lon12)

    # real a12, sig12, calp1, salp1, calp2, salp2
    # index zero elements of these arrays are unused
    C1a = list(range(Geodesic.nC1_ + 1))
    C2a = list(range(Geodesic.nC2_ + 1))
    C3a = list(range(Geodesic.nC3_))

    meridian = lat1 == -90 or slam12 == 0

    if meridian:

      # Endpoints are on a single full meridian, so the geodesic might lie on
      # a meridian.

      calp1 = clam12; salp1 = slam12 # Head to the target longitude
      calp2 = 1; salp2 = 0           # At the target we're heading north

      # tan(bet) = tan(sig) * cos(alp)
      ssig1 = sbet1; csig1 = calp1 * cbet1
      ssig2 = sbet2; csig2 = calp2 * cbet2

      # sig12 = sig2 - sig1
      sig12 = math.atan2(max(0.0, csig1 * ssig2 - ssig1 * csig2),
                         csig1 * csig2 + ssig1 * ssig2)

      s12x, m12x, dummy, M12, M21 = self.Lengths(
        self._n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2,
        outmask | Geodesic.DISTANCE | Geodesic.REDUCEDLENGTH, C1a, C2a)

      # Add the check for sig12 since zero length geodesics might yield m12 <
      # 0.  Test case was
      #
      #    echo 20.001 0 20.001 0 | GeodSolve -i
      #
      # In fact, we will have sig12 > pi/2 for meridional geodesic which is
      # not a shortest path.
      if sig12 < 1 or m12x >= 0:
        if sig12 < 3 * Geodesic.tiny_:
          sig12 = m12x = s12x = 0
        m12x *= self._b
        s12x *= self._b
        a12 = math.degrees(sig12)
      else:
        # m12 < 0, i.e., prolate and too close to anti-podal
        meridian = False
    # end if meridian:

    #real omg12
    if (not meridian and
        sbet1 == 0 and   # and sbet2 == 0
        # Mimic the way Lambda12 works with calp1 = 0
        (self._f <= 0 or lam12 <= math.pi - self._f * math.pi)):

      # Geodesic runs along equator
      calp1 = calp2 = 0; salp1 = salp2 = 1
      s12x = self._a * lam12
      sig12 = omg12 = lam12 / self._f1
      m12x = self._b * math.sin(sig12)
      if outmask & Geodesic.GEODESICSCALE:
        M12 = M21 = math.cos(sig12)
      a12 = lon12 / self._f1

    elif not meridian:

      # Now point1 and point2 belong within a hemisphere bounded by a
      # meridian and geodesic is neither meridional or equatorial.

      # Figure a starting point for Newton's method
      sig12, salp1, calp1, salp2, calp2, dnm = self.InverseStart(
        sbet1, cbet1, dn1, sbet2, cbet2, dn2, lam12, C1a, C2a)

      if sig12 >= 0:
        # Short lines (InverseStart sets salp2, calp2, dnm)
        s12x = sig12 * self._b * dnm
        m12x = (Math.sq(dnm) * self._b * math.sin(sig12 / dnm))
        if outmask & Geodesic.GEODESICSCALE:
          M12 = M21 = math.cos(sig12 / dnm)
        a12 = math.degrees(sig12)
        omg12 = lam12 / (self._f1 * dnm)
      else:

        # Newton's method.  This is a straightforward solution of f(alp1) =
        # lambda12(alp1) - lam12 = 0 with one wrinkle.  f(alp) has exactly one
        # root in the interval (0, pi) and its derivative is positive at the
        # root.  Thus f(alp) is positive for alp > alp1 and negative for alp <
        # alp1.  During the course of the iteration, a range (alp1a, alp1b) is
        # maintained which brackets the root and with each evaluation of f(alp)
        # the range is shrunk if possible.  Newton's method is restarted
        # whenever the derivative of f is negative (because the new value of
        # alp1 is then further from the solution) or if the new estimate of
        # alp1 lies outside (0,pi); in this case, the new starting guess is
        # taken to be (alp1a + alp1b) / 2.
        # real ssig1, csig1, ssig2, csig2, eps
        numit = 0
        tripn = tripb = False
        # Bracketing range
        salp1a = Geodesic.tiny_; calp1a = 1
        salp1b = Geodesic.tiny_; calp1b = -1

        while numit < Geodesic.maxit2_:
          # the WGS84 test set: mean = 1.47, sd = 1.25, max = 16
          # WGS84 and random input: mean = 2.85, sd = 0.60
          (nlam12, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
           eps, omg12, dv) = self.Lambda12(
             sbet1, cbet1, dn1, sbet2, cbet2, dn2,
             salp1, calp1, numit < Geodesic.maxit1_, C1a, C2a, C3a)
          v = nlam12 - lam12
          # 2 * tol0 is approximately 1 ulp for a number in [0, pi].
          # Reversed test to allow escape with NaNs
          if tripb or not (abs(v) >= (8 if tripn else 2) * Geodesic.tol0_):
            break
          # Update bracketing values
          if v > 0 and (numit > Geodesic.maxit1_ or
                        calp1/salp1 > calp1b/salp1b):
            salp1b = salp1; calp1b = calp1
          elif v < 0 and (numit > Geodesic.maxit1_ or
                          calp1/salp1 < calp1a/salp1a):
            salp1a = salp1; calp1a = calp1

          numit += 1
          if numit < Geodesic.maxit1_ and dv > 0:
            dalp1 = -v/dv
            sdalp1 = math.sin(dalp1); cdalp1 = math.cos(dalp1)
            nsalp1 = salp1 * cdalp1 + calp1 * sdalp1
            if nsalp1 > 0 and abs(dalp1) < math.pi:
              calp1 = calp1 * cdalp1 - salp1 * sdalp1
              salp1 = nsalp1
              salp1, calp1 = Math.norm(salp1, calp1)
              # In some regimes we don't get quadratic convergence because
              # slope -> 0.  So use convergence conditions based on epsilon
              # instead of sqrt(epsilon).
              tripn = abs(v) <= 16 * Geodesic.tol0_
              continue
          # Either dv was not postive or updated value was outside legal range.
          # Use the midpoint of the bracket as the next estimate.  This
          # mechanism is not needed for the WGS84 ellipsoid, but it does catch
          # problems with more eccentric ellipsoids.  Its efficacy is such for
          # the WGS84 test set with the starting guess set to alp1 = 90deg:
          # the WGS84 test set: mean = 5.21, sd = 3.93, max = 24
          # WGS84 and random input: mean = 4.74, sd = 0.99
          salp1 = (salp1a + salp1b)/2
          calp1 = (calp1a + calp1b)/2
          salp1, calp1 = Math.norm(salp1, calp1)
          tripn = False
          tripb = (abs(salp1a - salp1) + (calp1a - calp1) < Geodesic.tolb_ or
                   abs(salp1 - salp1b) + (calp1 - calp1b) < Geodesic.tolb_)

        lengthmask = (outmask |
                      (Geodesic.DISTANCE
                       if (outmask & (Geodesic.REDUCEDLENGTH |
                                      Geodesic.GEODESICSCALE))
                       else Geodesic.EMPTY))
        s12x, m12x, dummy, M12, M21 = self.Lengths(
          eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2,
          lengthmask, C1a, C2a)

        m12x *= self._b
        s12x *= self._b
        a12 = math.degrees(sig12)
        omg12 = lam12 - omg12
    # end elif not meridian

    if outmask & Geodesic.DISTANCE:
      s12 = 0 + s12x           # Convert -0 to 0

    if outmask & Geodesic.REDUCEDLENGTH:
      m12 = 0 + m12x           # Convert -0 to 0

    if outmask & Geodesic.AREA:
      # From Lambda12: sin(alp1) * cos(bet1) = sin(alp0)
      salp0 = salp1 * cbet1
      calp0 = math.hypot(calp1, salp1 * sbet1) # calp0 > 0
      # real alp12
      if calp0 != 0 and salp0 != 0:
        # From Lambda12: tan(bet) = tan(sig) * cos(alp)
        ssig1 = sbet1; csig1 = calp1 * cbet1
        ssig2 = sbet2; csig2 = calp2 * cbet2
        k2 = Math.sq(calp0) * self._ep2
        eps = k2 / (2 * (1 + math.sqrt(1 + k2)) + k2)
        # Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0).
        A4 = Math.sq(self._a) * calp0 * salp0 * self._e2
        ssig1, csig1 = Math.norm(ssig1, csig1)
        ssig2, csig2 = Math.norm(ssig2, csig2)
        C4a = list(range(Geodesic.nC4_))
        self.C4f(eps, C4a)
        B41 = Geodesic.SinCosSeries(False, ssig1, csig1, C4a)
        B42 = Geodesic.SinCosSeries(False, ssig2, csig2, C4a)
        S12 = A4 * (B42 - B41)
      else:
        # Avoid problems with indeterminate sig1, sig2 on equator
        S12 = 0
      if (not meridian and
          omg12 < 0.75 * math.pi and # Long difference too big
          sbet2 - sbet1 < 1.75):     # Lat difference too big
        # Use tan(Gamma/2) = tan(omg12/2)
        # * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2))
        # with tan(x/2) = sin(x)/(1+cos(x))
        somg12 = math.sin(omg12); domg12 = 1 + math.cos(omg12)
        dbet1 = 1 + cbet1; dbet2 = 1 + cbet2
        alp12 = 2 * math.atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
                                domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) )
      else:
        # alp12 = alp2 - alp1, used in atan2 so no need to normalize
        salp12 = salp2 * calp1 - calp2 * salp1
        calp12 = calp2 * calp1 + salp2 * salp1
        # The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
        # salp12 = -0 and alp12 = -180.  However this depends on the sign
        # being attached to 0 correctly.  The following ensures the correct
        # behavior.
        if salp12 == 0 and calp12 < 0:
          salp12 = Geodesic.tiny_ * calp1
          calp12 = -1
        alp12 = math.atan2(salp12, calp12)
      S12 += self._c2 * alp12
      S12 *= swapp * lonsign * latsign
      # Convert -0 to 0
      S12 += 0

    # Convert calp, salp to azimuth accounting for lonsign, swapp, latsign.
    if swapp < 0:
      salp2, salp1 = salp1, salp2
      calp2, calp1 = calp1, calp2
      if outmask & Geodesic.GEODESICSCALE:
        M21, M12 = M12, M21

    salp1 *= swapp * lonsign; calp1 *= swapp * latsign
    salp2 *= swapp * lonsign; calp2 *= swapp * latsign

    if outmask & Geodesic.AZIMUTH:
      # minus signs give range [-180, 180). 0- converts -0 to +0.
      azi1 = Math.atan2d(salp1, calp1)
      azi2 = Math.atan2d(salp2, calp2)

    # Returned value in [0, 180]
    return a12, s12, azi1, azi2, m12, M12, M21, S12

  def CheckPosition(lat, lon):
    """Check that lat and lon are legal and return normalized lon"""
    if abs(lat) > 90:
      raise ValueError("latitude " + str(lat) + " not in [-90, 90]")
    # if not Math.isfinite(lon):
    #   raise ValueError("longitude " + str(lon) + " not a finite number")
    return Math.AngNormalize(lon)
  CheckPosition = staticmethod(CheckPosition)

  def CheckAzimuth(azi):
    """Check that azi is legal and return normalized value"""
    # if not Math.isfinite(azi):
    #   raise ValueError("azimuth " + str(azi) + " not a finite number")
    return Math.AngNormalize(azi)
  CheckAzimuth = staticmethod(CheckAzimuth)

  def CheckDistance(s):
    """Check that s is a legal distance"""
    # if not Math.isfinite(s):
    #   raise ValueError("distance " + str(s) + " not a finite number")
  CheckDistance = staticmethod(CheckDistance)

  def Inverse(self, lat1, lon1, lat2, lon2, outmask = STANDARD):
    """Solve the inverse geodesic problem.  Compute geodesic between (lat1,
    lon1) and (lat2, lon2).  Return a dictionary with (some) of the
    following entries:

      lat1 latitude of point 1
      lon1 longitude of point 1
      azi1 azimuth of line at point 1
      lat2 latitude of point 2
      lon2 longitude of point 2
      azi2 azimuth of line at point 2
      s12 distance from 1 to 2
      a12 arc length on auxiliary sphere from 1 to 2
      m12 reduced length of geodesic
      M12 geodesic scale 2 relative to 1
      M21 geodesic scale 1 relative to 2
      S12 area between geodesic and equator

    outmask determines which fields get included and if outmask is
    omitted, then only the basic geodesic fields are computed.  The mask
    is an or'ed combination of the following values

      Geodesic.AZIMUTH
      Geodesic.DISTANCE
      Geodesic.STANDARD (all of the above)
      Geodesic.REDUCEDLENGTH
      Geodesic.GEODESICSCALE
      Geodesic.AREA
      Geodesic.ALL (all of the above)
      Geodesic.LONG_UNROLL

    If Geodesic.LONG_UNROLL is set, then lon1 is unchanged and lon2 -
    lon1 indicates whether the geodesic is east going or west going.
    Otherwise lon1 and lon2 are both reduced to the range [-180,180).

    The default value of outmask is STANDARD.

    """

    lon1a = Geodesic.CheckPosition(lat1, lon1)
    lon2a = Geodesic.CheckPosition(lat2, lon2)
    if outmask & Geodesic.LONG_UNROLL:
      lon2 = lon1 + Math.AngDiff(lon1a, lon2a)
    else:
      lon1 = lon1a; lon2 = lon2a

    result = {'lat1': lat1, 'lon1': lon1, 'lat2': lat2, 'lon2': lon2}
    a12, s12, azi1, azi2, m12, M12, M21, S12 = self.GenInverse(
      lat1, lon1a, lat2, lon2a, outmask)
    outmask &= Geodesic.OUT_MASK
    result['a12'] = a12
    if outmask & Geodesic.DISTANCE: result['s12'] = s12
    if outmask & Geodesic.AZIMUTH:
      result['azi1'] = azi1; result['azi2'] = azi2
    if outmask & Geodesic.REDUCEDLENGTH: result['m12'] = m12
    if outmask & Geodesic.GEODESICSCALE:
      result['M12'] = M12; result['M21'] = M21
    if outmask & Geodesic.AREA: result['S12'] = S12
    return result

  # return a12, lat2, lon2, azi2, s12, m12, M12, M21, S12
  def GenDirect(self, lat1, lon1, azi1, arcmode, s12_a12, outmask):
    """Private: General version of direct problem"""
    from geographiclib.geodesicline import GeodesicLine
    line = GeodesicLine(
      self, lat1, lon1, azi1,
      # Automatically supply DISTANCE_IN if necessary
      outmask | (Geodesic.EMPTY if arcmode else Geodesic.DISTANCE_IN))
    return line.GenPosition(arcmode, s12_a12, outmask)

  def Direct(self, lat1, lon1, azi1, s12, outmask = STANDARD):
    """Solve the direct geodesic problem.  Compute geodesic starting at
    (lat1, lon1) with azimuth azi1 and length s12.  Return a dictionary
    with (some) of the following entries:

      lat1 latitude of point 1
      lon1 longitude of point 1
      azi1 azimuth of line at point 1
      lat2 latitude of point 2
      lon2 longitude of point 2
      azi2 azimuth of line at point 2
      s12 distance from 1 to 2
      a12 arc length on auxiliary sphere from 1 to 2
      m12 reduced length of geodesic
      M12 geodesic scale 2 relative to 1
      M21 geodesic scale 1 relative to 2
      S12 area between geodesic and equator

    outmask determines which fields get included and if outmask is
    omitted, then only the basic geodesic fields are computed.  The mask
    is an or'ed combination of the following values

      Geodesic.LATITUDE
      Geodesic.LONGITUDE
      Geodesic.AZIMUTH
      Geodesic.STANDARD (all of the above)
      Geodesic.REDUCEDLENGTH
      Geodesic.GEODESICSCALE
      Geodesic.AREA
      Geodesic.ALL (all of the above)
      Geodesic.LONG_UNROLL

    The LONG_UNROLL bit unrolls the longitudes (instead of reducing them
    to the range [-180,180)); the quantity lon2 - lon1 then indicates
    how many times and in what sense the geodesic encircles the
    ellipsoid.

    The default value of outmask is STANDARD.

    """

    if outmask & Geodesic.LONG_UNROLL:
      Geodesic.CheckPosition(lat1, lon1)
    else:
      lon1 = Geodesic.CheckPosition(lat1, lon1)
    azi1 = Geodesic.CheckAzimuth(azi1)
    Geodesic.CheckDistance(s12)

    result = {'lat1': lat1, 'lon1': lon1, 'azi1': azi1, 's12': s12}
    a12, lat2, lon2, azi2, s12, m12, M12, M21, S12 = self.GenDirect(
      lat1, lon1, azi1, False, s12, outmask)
    outmask &= Geodesic.OUT_MASK
    result['a12'] = a12
    if outmask & Geodesic.LATITUDE: result['lat2'] = lat2
    if outmask & Geodesic.LONGITUDE: result['lon2'] = lon2
    if outmask & Geodesic.AZIMUTH: result['azi2'] = azi2
    if outmask & Geodesic.REDUCEDLENGTH: result['m12'] = m12
    if outmask & Geodesic.GEODESICSCALE:
      result['M12'] = M12; result['M21'] = M21
    if outmask & Geodesic.AREA: result['S12'] = S12
    return result

  def ArcDirect(self, lat1, lon1, azi1, a12, outmask = STANDARD):
    """Solve the direct geodesic problem.  Compute geodesic starting at
    (lat1, lon1) with azimuth azi1 and spherical arc length a12.  Return
    a dictionary with (some) of the following entries:

      lat1 latitude of point 1
      lon1 longitude of point 1
      azi1 azimuth of line at point 1
      lat2 latitude of point 2
      lon2 longitude of point 2
      azi2 azimuth of line at point 2
      s12 distance from 1 to 2
      a12 arc length on auxiliary sphere from 1 to 2
      m12 reduced length of geodesic
      M12 geodesic scale 2 relative to 1
      M21 geodesic scale 1 relative to 2
      S12 area between geodesic and equator

    outmask determines which fields get included and if outmask is
    omitted, then only the basic geodesic fields are computed.  The
    LONG_UNROLL bit unrolls the longitudes (instead of reducing them to
    the range [-180,180)).  The mask is an or'ed combination of the
    following values

      Geodesic.LATITUDE
      Geodesic.LONGITUDE
      Geodesic.AZIMUTH
      Geodesic.DISTANCE
      Geodesic.STANDARD (all of the above)
      Geodesic.REDUCEDLENGTH
      Geodesic.GEODESICSCALE
      Geodesic.AREA
      Geodesic.ALL (all of the above)
      Geodesic.LONG_UNROLL

    The default value of outmask is STANDARD.

    """

    if outmask & Geodesic.LONG_UNROLL:
      Geodesic.CheckPosition(lat1, lon1)
    else:
      lon1 = Geodesic.CheckPosition(lat1, lon1)
    azi1 = Geodesic.CheckAzimuth(azi1)
    Geodesic.CheckDistance(a12)

    result = {'lat1': lat1, 'lon1': lon1, 'azi1': azi1, 'a12': a12}
    a12, lat2, lon2, azi2, s12, m12, M12, M21, S12 = self.GenDirect(
      lat1, lon1, azi1, True, a12, outmask)
    outmask &= Geodesic.OUT_MASK
    if outmask & Geodesic.DISTANCE: result['s12'] = s12
    if outmask & Geodesic.LATITUDE: result['lat2'] = lat2
    if outmask & Geodesic.LONGITUDE: result['lon2'] = lon2
    if outmask & Geodesic.AZIMUTH: result['azi2'] = azi2
    if outmask & Geodesic.REDUCEDLENGTH: result['m12'] = m12
    if outmask & Geodesic.GEODESICSCALE:
      result['M12'] = M12; result['M21'] = M21
    if outmask & Geodesic.AREA: result['S12'] = S12
    return result

  def Line(self, lat1, lon1, azi1, caps = ALL):
    """Return a GeodesicLine object to compute points along a geodesic
    starting at lat1, lon1, with azimuth azi1.  caps is an or'ed
    combination of bit the following values indicating the capabilities
    of the returned object

      Geodesic.LATITUDE
      Geodesic.LONGITUDE
      Geodesic.AZIMUTH
      Geodesic.DISTANCE
      Geodesic.STANDARD (all of the above)
      Geodesic.REDUCEDLENGTH
      Geodesic.GEODESICSCALE
      Geodesic.AREA
      Geodesic.DISTANCE_IN
      Geodesic.ALL (all of the above)

    The default value of caps is ALL.

    """

    from geographiclib.geodesicline import GeodesicLine
    lon1 = Geodesic.CheckPosition(lat1, lon1)
    azi1 = Geodesic.CheckAzimuth(azi1)
    return GeodesicLine(
      self, lat1, lon1, azi1,
      # Automatically supply DISTANCE_IN
      caps | Geodesic.DISTANCE_IN)

  def Area(self, points, polyline = False):
    """Compute the area of a geodesic polygon given by points, an array of
    dictionaries with entries lat and lon.  Return a dictionary with
    entries

      number the number of verices
      perimeter the perimeter
      area the area (counter-clockwise traversal positive)

    There is no need to "close" the polygon.  If polyline is set to
    True, then the points define a polyline instead of a polygon, the
    length is returned as the perimeter, and the area is not returned.

    """

    from geographiclib.polygonarea import PolygonArea
    for p in points:
      Geodesic.CheckPosition(p['lat'], p['lon'])
    num, perimeter, area = PolygonArea.Area(self, points, polyline)
    result = {'number': num, 'perimeter': perimeter}
    if not polyline: result['area'] = area
    return result

Geodesic.WGS84 = Geodesic(Constants.WGS84_a, Constants.WGS84_f)