示例#1
0
  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
示例#2
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 def _transit(lon1, lon2):
     """Count crossings of prime meridian for AddPoint."""
     # Return 1 or -1 if crossing prime meridian in east or west direction.
     # Otherwise return zero.
     # Compute lon12 the same way as Geodesic::Inverse.
     lon1 = Math.AngNormalize(lon1)
     lon2 = Math.AngNormalize(lon2)
     lon12, _ = Math.AngDiff(lon1, lon2)
     cross = (1 if lon1 <= 0 and lon2 > 0 and lon12 > 0 else
              (-1 if lon2 <= 0 and lon1 > 0 and lon12 < 0 else 0))
     return cross
示例#3
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 def transit(lon1, lon2):
   # Return 1 or -1 if crossing prime meridian in east or west direction.
   # Otherwise return zero.
   from geographiclib.geodesic import Geodesic
   # Compute lon12 the same way as Geodesic::Inverse.
   lon1 = Math.AngNormalize(lon1);
   lon2 = Math.AngNormalize(lon2);
   lon12 = Math.AngDiff(lon1, lon2);
   cross = (1 if lon1 < 0 and lon2 >= 0 and lon12 > 0
            else (-1 if lon2 < 0 and lon1 >= 0 and lon12 < 0 else 0))
   return cross
示例#4
0
  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
示例#5
0
  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 = 1 if lon12 >= 0 else -1
    # If very close to being on the same half-meridian, then make it so.
    lon12 = lonsign * Math.AngRound(lon12)
    lon12s = Math.AngRound((180 - lon12) - lonsign * lon12s)
    lam12 = math.radians(lon12)
    if lon12 > 90:
      slam12, clam12 = Math.sincosd(lon12s); clam12 = -clam12
    else:
      slam12, clam12 = Math.sincosd(lon12)

    # 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))

    # 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),
                                  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 = 0.0
        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 > 1 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 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
          (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)
          # 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 1) * 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 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 > 1:
        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