geographiclib.geodesic — geographiclib 2.1 documentation

# Source code for geographiclib.geodesic

    """Define the :class:`~geographiclib.geodesic.Geodesic` class

    The ellipsoid parameters are defined by the constructor.  The direct and
    inverse geodesic problems are solved by

      * :meth:`~geographiclib.geodesic.Geodesic.Inverse` Solve the inverse
        geodesic problem
      * :meth:`~geographiclib.geodesic.Geodesic.Direct` Solve the direct
        geodesic problem
      * :meth:`~geographiclib.geodesic.Geodesic.ArcDirect` Solve the direct
        geodesic problem in terms of spherical arc length

    :class:`~geographiclib.geodesicline.GeodesicLine` objects can be created
    with

      * :meth:`~geographiclib.geodesic.Geodesic.Line`
      * :meth:`~geographiclib.geodesic.Geodesic.DirectLine`
      * :meth:`~geographiclib.geodesic.Geodesic.ArcDirectLine`
      * :meth:`~geographiclib.geodesic.Geodesic.InverseLine`

    :class:`~geographiclib.polygonarea.PolygonArea` objects can be created
    with

      * :meth:`~geographiclib.geodesic.Geodesic.Polygon`

    The public attributes for this class are

      * :attr:`~geographiclib.geodesic.Geodesic.a`
        :attr:`~geographiclib.geodesic.Geodesic.f`

    *outmask* and *caps* bit masks are

      * :const:`~geographiclib.geodesic.Geodesic.EMPTY`
      * :const:`~geographiclib.geodesic.Geodesic.LATITUDE`
      * :const:`~geographiclib.geodesic.Geodesic.LONGITUDE`
      * :const:`~geographiclib.geodesic.Geodesic.AZIMUTH`
      * :const:`~geographiclib.geodesic.Geodesic.DISTANCE`
      * :const:`~geographiclib.geodesic.Geodesic.STANDARD`
      * :const:`~geographiclib.geodesic.Geodesic.DISTANCE_IN`
      * :const:`~geographiclib.geodesic.Geodesic.REDUCEDLENGTH`
      * :const:`~geographiclib.geodesic.Geodesic.GEODESICSCALE`
      * :const:`~geographiclib.geodesic.Geodesic.AREA`
      * :const:`~geographiclib.geodesic.Geodesic.ALL`
      * :const:`~geographiclib.geodesic.Geodesic.LONG_UNROLL`

    :Example:

        >>> from geographiclib.geodesic import Geodesic
        >>> # The geodesic inverse problem
        ... Geodesic.WGS84.Inverse(-41.32, 174.81, 40.96, -5.50)
        {'lat1': -41.32,
         'a12': 179.6197069334283,
         's12': 19959679.26735382,
         'lat2': 40.96,
         'azi2': 18.825195123248392,
         'azi1': 161.06766998615882,
         'lon1': 174.81,
         'lon2': -5.5}

    """
    # 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
    #
    #    https://geographiclib.sourceforge.io/C++/doc/annotated.html
    #
    # The algorithms are derived in
    #
    #    Charles F. F. Karney,
    #    Algorithms for geodesics, J. Geodesy 87, 43-55 (2013),
    #    https://doi.org/10.1007/s00190-012-0578-z
    #    Addenda: https://geographiclib.sourceforge.io/geod-addenda.html
    #
    # Copyright (c) Charles Karney (2011-2022) <karney@alum.mit.edu> and licensed
    # under the MIT/X11 License.  For more information, see
    # https://geographiclib.sourceforge.io/
    ######################################################################

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

    class Geodesic:
      """Solve geodesic problems"""

      WGS84: "Geodesic"
      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_ + sys.float_info.mant_dig + 10

      tiny_ = math.sqrt(sys.float_info.min)
      tol0_ = sys.float_info.epsilon
      tol1_ = 200 * tol0_
      tol2_ = math.sqrt(tol0_)
      tolb_ = tol0_
      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

      @staticmethod
      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)

      @staticmethod
      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

      @staticmethod
      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)

      @staticmethod
      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

      @staticmethod
      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

      @staticmethod
      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)

      @staticmethod
      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

      def __init__(self, a, f):
        """Construct a Geodesic object

        :param a: the equatorial radius of the ellipsoid in meters
        :param f: the flattening of the ellipsoid

        An exception is thrown if *a* or the polar semi-axis *b* = *a* (1 -
        *f*) is not a finite positive quantity.

        """

        self.a = float(a)
        """The equatorial radius in meters (readonly)"""
        self.f = float(f)
        """The flattening (readonly)"""
        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("Equatorial radius is not positive")
        if not(math.isfinite(self._b) and self._b > 0):
          raise ValueError("Polar semi-axis 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 = m0x = J12 = 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, slam12, clam12,
                        # 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
        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 = lam12 / (self._f1 * dnm)
          somg12 = math.sin(omg12); comg12 = math.cos(omg12)
        else:
          somg12 = slam12; comg12 = clam12

        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
          lam12x = math.atan2(-slam12, -clam12)
          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 = lam12x / 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 = lam12x / 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,
                    slam120, clam120, 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, 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) + 0.0,
                                    csig1 * csig2 + ssig1 * ssig2)

        # omg12 = omg2 - omg1, limit to [0, pi]
        somg12 = max(0.0, comg1 * somg2 - somg1 * comg2) + 0.0
        comg12 =          comg1 * comg2 + somg1 * somg2
        # eta = omg12 - lam120
        eta = math.atan2(somg12 * clam120 - comg12 * slam120,
                         comg12 * clam120 + somg12 * slam120)

        # real B312
        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))
        domg12 =  -self.f * self._A3f(eps) * salp0 * (sig12 + B312)
        lam12 = eta + 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, salp1, calp1, salp2, calp2, m12, M12, M21, S12
      def _GenInverse(self, lat1, lon1, lat2, lon2, outmask):
        """Private: General version of the inverse problem"""
        a12 = s12 = 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.
        lon12, lon12s = Math.AngDiff(lon1, lon2)
        # Make longitude difference positive.
        lonsign = math.copysign(1, lon12)
        lon12 = lonsign * lon12; lon12s = lonsign * lon12s
        lam12 = math.radians(lon12)
        # Calculate sincos of lon12 + error (this applies AngRound internally).
        slam12, clam12 = Math.sincosde(lon12, lon12s)
        lon12s = (180 - lon12) - lon12s # the supplementary longitude difference

        # 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) or math.isnan(lat2) else 1
        if swapp < 0:
          lonsign *= -1
          lat2, lat1 = lat1, lat2
        # Make lat1 <= 0
        latsign = math.copysign(1, -lat1)
        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 = math.copysign(sbet1, sbet2)
        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))

        # 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 = 1.0; salp2 = 0.0       # 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) + 0.0,
                                      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 < Geodesic.tol2_ or m12x >= 0:
            if (sig12 < 3 * Geodesic.tiny_ or
                # Prevent negative s12 or m12 for short lines
                (sig12 < Geodesic.tol0_ and (s12x < 0 or m12x < 0))):
              sig12 = m12x = s12x = 0.0
            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:

        # somg12 == 2 marks that it needs to be calculated
        somg12 = 2.0; comg12 = 0.0; omg12 = 0.0
        if (not meridian and
            sbet1 == 0 and   # and sbet2 == 0
            # Mimic the way Lambda12 works with calp1 = 0
            (self.f <= 0 or lon12s >= self.f * 180)):

          # Geodesic runs along equator
          calp1 = calp2 = 0.0; salp1 = salp2 = 1.0
          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, slam12, clam12, 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 = 1.0
            salp1b = Geodesic.tiny_; calp1b = -1.0

            while True:
              # the WGS84 test set: mean = 1.47, sd = 1.25, max = 16
              # WGS84 and random input: mean = 2.85, sd = 0.60
              (v, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
               eps, domg12, dv) = self._Lambda12(
                 sbet1, cbet1, dn1, sbet2, cbet2, dn2,
                 salp1, calp1, slam12, clam12, numit < Geodesic.maxit1_,
                 C1a, C2a, C3a)
              # Reversed test to allow escape with NaNs
              if (tripb or not (abs(v) >= (8 if tripn else 1) * Geodesic.tol0_) or
                  numit == Geodesic.maxit2_):
                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
                if abs(dalp1) < math.pi:
                  sdalp1 = math.sin(dalp1); cdalp1 = math.cos(dalp1)
                  nsalp1 = salp1 * cdalp1 + calp1 * sdalp1
                  if nsalp1 > 0:
                    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 positive 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)
            if outmask & Geodesic.AREA:
              # omg12 = lam12 - domg12
              sdomg12 = math.sin(domg12); cdomg12 = math.cos(domg12)
              somg12 = slam12 * cdomg12 - clam12 * sdomg12
              comg12 = clam12 * cdomg12 + slam12 * sdomg12

        # end elif not meridian

        if outmask & Geodesic.DISTANCE:
          s12 = 0.0 + s12x          # Convert -0 to 0

        if outmask & Geodesic.REDUCEDLENGTH:
          m12 = 0.0 + 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 = 0.0

          if not meridian and somg12 == 2.0:
            somg12 = math.sin(omg12); comg12 = math.cos(omg12)

          if (not meridian and
              # omg12 < 3/4 * pi
              comg12 > -0.7071 and   # Long difference not too big
              sbet2 - sbet1 < 1.75): # Lat difference not 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))
            domg12 = 1 + comg12; 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 = -1.0
            alp12 = math.atan2(salp12, calp12)
          S12 += self._c2 * alp12
          S12 *= swapp * lonsign * latsign
          # Convert -0 to 0
          S12 += 0.0

        # 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

        return a12, s12, salp1, calp1, salp2, calp2, m12, M12, M21, S12

      def Inverse(self, lat1, lon1, lat2, lon2,
                  outmask = GeodesicCapability.STANDARD):
        """Solve the inverse geodesic problem

        :param lat1: latitude of the first point in degrees
        :param lon1: longitude of the first point in degrees
        :param lat2: latitude of the second point in degrees
        :param lon2: longitude of the second point in degrees
        :param outmask: the :ref:`output mask <outmask>`
        :return: a :ref:`dict`

        Compute geodesic between (*lat1*, *lon1*) and (*lat2*, *lon2*).
        The default value of *outmask* is STANDARD, i.e., the *lat1*,
        *lon1*, *azi1*, *lat2*, *lon2*, *azi2*, *s12*, *a12* entries are
        returned.

        """

        a12, s12, salp1,calp1, salp2,calp2, m12, M12, M21, S12 = self._GenInverse(
          lat1, lon1, lat2, lon2, outmask)
        outmask &= Geodesic.OUT_MASK
        if outmask & Geodesic.LONG_UNROLL:
          lon12, e = Math.AngDiff(lon1, lon2)
          lon2 = (lon1 + lon12) + e
        else:
          lon2 = Math.AngNormalize(lon2)
        result = {'lat1': Math.LatFix(lat1),
                  'lon1': lon1 if outmask & Geodesic.LONG_UNROLL else
                  Math.AngNormalize(lon1),
                  'lat2': Math.LatFix(lat2),
                  'lon2': lon2}
        result['a12'] = a12
        if outmask & Geodesic.DISTANCE: result['s12'] = s12
        if outmask & Geodesic.AZIMUTH:
          result['azi1'] = Math.atan2d(salp1, calp1)
          result['azi2'] = Math.atan2d(salp2, calp2)
        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
        # Automatically supply DISTANCE_IN if necessary
        if not arcmode: outmask |= Geodesic.DISTANCE_IN
        line = GeodesicLine(self, lat1, lon1, azi1, outmask)
        return line._GenPosition(arcmode, s12_a12, outmask)

      def Direct(self, lat1, lon1, azi1, s12,
                 outmask = GeodesicCapability.STANDARD):
        """Solve the direct geodesic problem

        :param lat1: latitude of the first point in degrees
        :param lon1: longitude of the first point in degrees
        :param azi1: azimuth at the first point in degrees
        :param s12: the distance from the first point to the second in
          meters
        :param outmask: the :ref:`output mask <outmask>`
        :return: a :ref:`dict`

        Compute geodesic starting at (*lat1*, *lon1*) with azimuth *azi1*
        and length *s12*.  The default value of *outmask* is STANDARD, i.e.,
        the *lat1*, *lon1*, *azi1*, *lat2*, *lon2*, *azi2*, *s12*, *a12*
        entries are returned.

        """

        a12, lat2, lon2, azi2, s12, m12, M12, M21, S12 = self._GenDirect(
          lat1, lon1, azi1, False, s12, outmask)
        outmask &= Geodesic.OUT_MASK
        result = {'lat1': Math.LatFix(lat1),
                  'lon1': lon1 if outmask & Geodesic.LONG_UNROLL else
                  Math.AngNormalize(lon1),
                  'azi1': Math.AngNormalize(azi1),
                  's12': s12}
        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 = GeodesicCapability.STANDARD):
        """Solve the direct geodesic problem in terms of spherical arc length

        :param lat1: latitude of the first point in degrees
        :param lon1: longitude of the first point in degrees
        :param azi1: azimuth at the first point in degrees
        :param a12: spherical arc length from the first point to the second
          in degrees
        :param outmask: the :ref:`output mask <outmask>`
        :return: a :ref:`dict`

        Compute geodesic starting at (*lat1*, *lon1*) with azimuth *azi1*
        and arc length *a12*.  The default value of *outmask* is STANDARD,
        i.e., the *lat1*, *lon1*, *azi1*, *lat2*, *lon2*, *azi2*, *s12*,
        *a12* entries are returned.

        """

        a12, lat2, lon2, azi2, s12, m12, M12, M21, S12 = self._GenDirect(
          lat1, lon1, azi1, True, a12, outmask)
        outmask &= Geodesic.OUT_MASK
        result = {'lat1': Math.LatFix(lat1),
                  'lon1': lon1 if outmask & Geodesic.LONG_UNROLL else
                  Math.AngNormalize(lon1),
                  'azi1': Math.AngNormalize(azi1),
                  'a12': a12}
        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 = GeodesicCapability.STANDARD |
               GeodesicCapability.DISTANCE_IN):
        """Return a GeodesicLine object

        :param lat1: latitude of the first point in degrees
        :param lon1: longitude of the first point in degrees
        :param azi1: azimuth at the first point in degrees
        :param caps: the :ref:`capabilities <outmask>`
        :return: a :class:`~geographiclib.geodesicline.GeodesicLine`

        This allows points along a geodesic starting at (*lat1*, *lon1*),
        with azimuth *azi1* to be found.  The default value of *caps* is
        STANDARD | DISTANCE_IN, allowing direct geodesic problem to be
        solved.

        """

        from geographiclib.geodesicline import GeodesicLine
        return GeodesicLine(self, lat1, lon1, azi1, caps)

      def _GenDirectLine(self, lat1, lon1, azi1, arcmode, s12_a12,
                         caps = GeodesicCapability.STANDARD |
                         GeodesicCapability.DISTANCE_IN):
        """Private: general form of DirectLine"""
        from geographiclib.geodesicline import GeodesicLine
        # Automatically supply DISTANCE_IN if necessary
        if not arcmode: caps |= Geodesic.DISTANCE_IN
        line = GeodesicLine(self, lat1, lon1, azi1, caps)
        if arcmode:
          line.SetArc(s12_a12)
        else:
          line.SetDistance(s12_a12)
        return line

      def DirectLine(self, lat1, lon1, azi1, s12,
                     caps = GeodesicCapability.STANDARD |
                     GeodesicCapability.DISTANCE_IN):
        """Define a GeodesicLine object in terms of the direct geodesic
        problem specified in terms of spherical arc length

        :param lat1: latitude of the first point in degrees
        :param lon1: longitude of the first point in degrees
        :param azi1: azimuth at the first point in degrees
        :param s12: the distance from the first point to the second in
          meters
        :param caps: the :ref:`capabilities <outmask>`
        :return: a :class:`~geographiclib.geodesicline.GeodesicLine`

        This function sets point 3 of the GeodesicLine to correspond to
        point 2 of the direct geodesic problem.  The default value of *caps*
        is STANDARD | DISTANCE_IN, allowing direct geodesic problem to be
        solved.

        """

        return self._GenDirectLine(lat1, lon1, azi1, False, s12, caps)

      def ArcDirectLine(self, lat1, lon1, azi1, a12,
                     caps = GeodesicCapability.STANDARD |
                     GeodesicCapability.DISTANCE_IN):
        """Define a GeodesicLine object in terms of the direct geodesic
        problem specified in terms of spherical arc length

        :param lat1: latitude of the first point in degrees
        :param lon1: longitude of the first point in degrees
        :param azi1: azimuth at the first point in degrees
        :param a12: spherical arc length from the first point to the second
          in degrees
        :param caps: the :ref:`capabilities <outmask>`
        :return: a :class:`~geographiclib.geodesicline.GeodesicLine`

        This function sets point 3 of the GeodesicLine to correspond to
        point 2 of the direct geodesic problem.  The default value of *caps*
        is STANDARD | DISTANCE_IN, allowing direct geodesic problem to be
        solved.

        """

        return self._GenDirectLine(lat1, lon1, azi1, True, a12, caps)

      def InverseLine(self, lat1, lon1, lat2, lon2,
                      caps = GeodesicCapability.STANDARD |
                      GeodesicCapability.DISTANCE_IN):
        """Define a GeodesicLine object in terms of the invese geodesic problem

        :param lat1: latitude of the first point in degrees
        :param lon1: longitude of the first point in degrees
        :param lat2: latitude of the second point in degrees
        :param lon2: longitude of the second point in degrees
        :param caps: the :ref:`capabilities <outmask>`
        :return: a :class:`~geographiclib.geodesicline.GeodesicLine`

        This function sets point 3 of the GeodesicLine to correspond to
        point 2 of the inverse geodesic problem.  The default value of *caps*
        is STANDARD | DISTANCE_IN, allowing direct geodesic problem to be
        solved.

        """

        from geographiclib.geodesicline import GeodesicLine
        a12, _, salp1, calp1, _, _, _, _, _, _ = self._GenInverse(
          lat1, lon1, lat2, lon2, 0)
        azi1 = Math.atan2d(salp1, calp1)
        if caps & (Geodesic.OUT_MASK & Geodesic.DISTANCE_IN):
          caps |= Geodesic.DISTANCE
        line = GeodesicLine(self, lat1, lon1, azi1, caps, salp1, calp1)
        line.SetArc(a12)
        return line

      def Polygon(self, polyline = False):
        """Return a PolygonArea object

        :param polyline: if True then the object describes a polyline
          instead of a polygon
        :return: a :class:`~geographiclib.polygonarea.PolygonArea`

        """

        from geographiclib.polygonarea import PolygonArea
        return PolygonArea(self, polyline)

      EMPTY         = GeodesicCapability.EMPTY
      """No capabilities, no output."""
      LATITUDE      = GeodesicCapability.LATITUDE
      """Calculate latitude *lat2*."""
      LONGITUDE     = GeodesicCapability.LONGITUDE
      """Calculate longitude *lon2*."""
      AZIMUTH       = GeodesicCapability.AZIMUTH
      """Calculate azimuths *azi1* and *azi2*."""
      DISTANCE      = GeodesicCapability.DISTANCE
      """Calculate distance *s12*."""
      STANDARD      = GeodesicCapability.STANDARD
      """All of the above."""
      DISTANCE_IN   = GeodesicCapability.DISTANCE_IN
      """Allow distance *s12* to be used as input in the direct geodesic
      problem."""
      REDUCEDLENGTH = GeodesicCapability.REDUCEDLENGTH
      """Calculate reduced length *m12*."""
      GEODESICSCALE = GeodesicCapability.GEODESICSCALE
      """Calculate geodesic scales *M12* and *M21*."""
      AREA          = GeodesicCapability.AREA
      """Calculate area *S12*."""
      ALL           = GeodesicCapability.ALL
      """All of the above."""
      LONG_UNROLL   = GeodesicCapability.LONG_UNROLL
      """Unroll longitudes, rather than reducing them to the range
      [-180d,180d].

      """

    Geodesic.WGS84 = Geodesic(Constants.WGS84_a, Constants.WGS84_f)
    """Instantiation for the WGS84 ellipsoid"""