Python Math.norm方法代码示例

# 需要导入模块: from geographiclib.geomath import Math [as 别名]
# 或者: from geographiclib.geomath.Math import norm [as 别名]
  def __init__(self, geod, lat1, lon1, azi1,
               caps = GeodesicCapability.STANDARD |
               GeodesicCapability.DISTANCE_IN,
               salp1 = Math.nan, calp1 = Math.nan):
    """Construct a GeodesicLine object

    :param geod: a :class:`~geographiclib.geodesic.Geodesic` 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>`

    This creates an object allowing points along a geodesic starting at
    (*lat1*, *lon1*), with azimuth *azi1* to be found.  The default
    value of *caps* is STANDARD | DISTANCE_IN.  The optional parameters
    *salp1* and *calp1* should not be supplied; they are part of the
    private interface.

    """

    from geographiclib.geodesic import Geodesic
    self.a = geod.a
    """The equatorial radius in meters (readonly)"""
    self.f = geod.f
    """The flattening (readonly)"""
    self._b = geod._b
    self._c2 = geod._c2
    self._f1 = geod._f1
    self.caps = (caps | Geodesic.LATITUDE | Geodesic.AZIMUTH |
                  Geodesic.LONG_UNROLL)
    """the capabilities (readonly)"""

    # Guard against underflow in salp0
    self.lat1 = Math.LatFix(lat1)
    """the latitude of the first point in degrees (readonly)"""
    self.lon1 = lon1
    """the longitude of the first point in degrees (readonly)"""
    if Math.isnan(salp1) or Math.isnan(calp1):
      self.azi1 = Math.AngNormalize(azi1)
      self.salp1, self.calp1 = Math.sincosd(Math.AngRound(azi1))
    else:
      self.azi1 = azi1
      """the azimuth at the first point in degrees (readonly)"""
      self.salp1 = salp1
      """the sine of the azimuth at the first point (readonly)"""
      self.calp1 = calp1
      """the cosine of the azimuth at the first point (readonly)"""

    # real cbet1, sbet1
    sbet1, cbet1 = Math.sincosd(Math.AngRound(lat1)); sbet1 *= self._f1
    # Ensure cbet1 = +epsilon at poles
    sbet1, cbet1 = Math.norm(sbet1, cbet1); cbet1 = max(Geodesic.tiny_, cbet1)
    self._dn1 = math.sqrt(1 + geod._ep2 * Math.sq(sbet1))

    # Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0),
    self._salp0 = self.salp1 * cbet1 # alp0 in [0, pi/2 - |bet1|]
    # Alt: calp0 = hypot(sbet1, calp1 * cbet1).  The following
    # is slightly better (consider the case salp1 = 0).
    self._calp0 = math.hypot(self.calp1, self.salp1 * sbet1)
    # Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
    # sig = 0 is nearest northward crossing of equator.
    # With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
    # With bet1 =  pi/2, alp1 = -pi, sig1 =  pi/2
    # With bet1 = -pi/2, alp1 =  0 , sig1 = -pi/2
    # Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
    # With alp0 in (0, pi/2], quadrants for sig and omg coincide.
    # No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
    # With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi.
    self._ssig1 = sbet1; self._somg1 = self._salp0 * sbet1
    self._csig1 = self._comg1 = (cbet1 * self.calp1
                                 if sbet1 != 0 or self.calp1 != 0 else 1)
    # sig1 in (-pi, pi]
    self._ssig1, self._csig1 = Math.norm(self._ssig1, self._csig1)
    # No need to normalize
    # self._somg1, self._comg1 = Math.norm(self._somg1, self._comg1)

    self._k2 = Math.sq(self._calp0) * geod._ep2
    eps = self._k2 / (2 * (1 + math.sqrt(1 + self._k2)) + self._k2)

    if self.caps & Geodesic.CAP_C1:
      self._A1m1 = Geodesic._A1m1f(eps)
      self._C1a = list(range(Geodesic.nC1_ + 1))
      Geodesic._C1f(eps, self._C1a)
      self._B11 = Geodesic._SinCosSeries(
        True, self._ssig1, self._csig1, self._C1a)
      s = math.sin(self._B11); c = math.cos(self._B11)
      # tau1 = sig1 + B11
      self._stau1 = self._ssig1 * c + self._csig1 * s
      self._ctau1 = self._csig1 * c - self._ssig1 * s
      # Not necessary because C1pa reverts C1a
      #    _B11 = -_SinCosSeries(true, _stau1, _ctau1, _C1pa)

    if self.caps & Geodesic.CAP_C1p:
      self._C1pa = list(range(Geodesic.nC1p_ + 1))
      Geodesic._C1pf(eps, self._C1pa)

    if self.caps & Geodesic.CAP_C2:
      self._A2m1 = Geodesic._A2m1f(eps)
      self._C2a = list(range(Geodesic.nC2_ + 1))
      Geodesic._C2f(eps, self._C2a)
#.........这里部分代码省略.........

# 需要导入模块: from geographiclib.geomath import Math [as 别名]
# 或者: from geographiclib.geomath.Math import norm [as 别名]
  def __init__(self, geod, lat1, lon1, azi1, caps = GeodesicCapability.ALL):
    from geographiclib.geodesic import Geodesic
    self._a = geod._a
    self._f = geod._f
    self._b = geod._b
    self._c2 = geod._c2
    self._f1 = geod._f1
    self._caps = (caps | Geodesic.LATITUDE | Geodesic.AZIMUTH |
                  Geodesic.LONG_UNROLL)

    # Guard against underflow in salp0
    azi1 = Math.AngRound(Math.AngNormalize(azi1))
    self._lat1 = lat1
    self._lon1 = lon1
    self._azi1 = azi1
    # alp1 is in [0, pi]
    alp1 = azi1 * Math.degree
    # Enforce sin(pi) == 0 and cos(pi/2) == 0.  Better to face the ensuing
    # problems directly than to skirt them.
    self._salp1 = 0 if     azi1  == -180 else math.sin(alp1)
    self._calp1 = 0 if abs(azi1) ==   90 else math.cos(alp1)
    # real cbet1, sbet1, phi
    phi = lat1 * Math.degree
    # Ensure cbet1 = +epsilon at poles
    sbet1 = self._f1 * math.sin(phi)
    cbet1 = Geodesic.tiny_ if abs(lat1) == 90 else math.cos(phi)
    sbet1, cbet1 = Math.norm(sbet1, cbet1)
    self._dn1 = math.sqrt(1 + geod._ep2 * Math.sq(sbet1))

    # Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0),
    self._salp0 = self._salp1 * cbet1 # alp0 in [0, pi/2 - |bet1|]
    # Alt: calp0 = hypot(sbet1, calp1 * cbet1).  The following
    # is slightly better (consider the case salp1 = 0).
    self._calp0 = math.hypot(self._calp1, self._salp1 * sbet1)
    # Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
    # sig = 0 is nearest northward crossing of equator.
    # With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
    # With bet1 =  pi/2, alp1 = -pi, sig1 =  pi/2
    # With bet1 = -pi/2, alp1 =  0 , sig1 = -pi/2
    # Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
    # With alp0 in (0, pi/2], quadrants for sig and omg coincide.
    # No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
    # With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi.
    self._ssig1 = sbet1; self._somg1 = self._salp0 * sbet1
    self._csig1 = self._comg1 = (cbet1 * self._calp1
                                 if sbet1 != 0 or self._calp1 != 0 else 1)
    # sig1 in (-pi, pi]
    self._ssig1, self._csig1 = Math.norm(self._ssig1, self._csig1)
    # No need to normalize
    # self._somg1, self._comg1 = Math.norm(self._somg1, self._comg1)

    self._k2 = Math.sq(self._calp0) * geod._ep2
    eps = self._k2 / (2 * (1 + math.sqrt(1 + self._k2)) + self._k2)

    if self._caps & Geodesic.CAP_C1:
      self._A1m1 = Geodesic.A1m1f(eps)
      self._C1a = list(range(Geodesic.nC1_ + 1))
      Geodesic.C1f(eps, self._C1a)
      self._B11 = Geodesic.SinCosSeries(
        True, self._ssig1, self._csig1, self._C1a)
      s = math.sin(self._B11); c = math.cos(self._B11)
      # tau1 = sig1 + B11
      self._stau1 = self._ssig1 * c + self._csig1 * s
      self._ctau1 = self._csig1 * c - self._ssig1 * s
      # Not necessary because C1pa reverts C1a
      #    _B11 = -SinCosSeries(true, _stau1, _ctau1, _C1pa)

    if self._caps & Geodesic.CAP_C1p:
      self._C1pa = list(range(Geodesic.nC1p_ + 1))
      Geodesic.C1pf(eps, self._C1pa)

    if self._caps & Geodesic.CAP_C2:
      self._A2m1 = Geodesic.A2m1f(eps)
      self._C2a = list(range(Geodesic.nC2_ + 1))
      Geodesic.C2f(eps, self._C2a)
      self._B21 = Geodesic.SinCosSeries(
        True, self._ssig1, self._csig1, self._C2a)

    if self._caps & Geodesic.CAP_C3:
      self._C3a = list(range(Geodesic.nC3_))
      geod.C3f(eps, self._C3a)
      self._A3c = -self._f * self._salp0 * geod.A3f(eps)
      self._B31 = Geodesic.SinCosSeries(
        True, self._ssig1, self._csig1, self._C3a)

    if self._caps & Geodesic.CAP_C4:
      self._C4a = list(range(Geodesic.nC4_))
      geod.C4f(eps, self._C4a)
      # Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
      self._A4 = Math.sq(self._a) * self._calp0 * self._salp0 * geod._e2
      self._B41 = Geodesic.SinCosSeries(
        False, self._ssig1, self._csig1, self._C4a)

# 需要导入模块: from geographiclib.geomath import Math [as 别名]
# 或者: from geographiclib.geomath.Math import norm [as 别名]
  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.
    lon12 = Math.AngDiff(Math.AngNormalize(lon1), Math.AngNormalize(lon2))
    # If very close to being on the same half-meridian, then make it so.
    lon12 = Math.AngRound(lon12)
    # 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(lat1)
    lat2 = Math.AngRound(lat2)
    # Swap points so that point with higher (abs) latitude is point 1
    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

    phi = lat1 * Math.degree
    # Ensure cbet1 = +epsilon at poles
    sbet1 = self._f1 * math.sin(phi)
    cbet1 = Geodesic.tiny_ if lat1 == -90 else math.cos(phi)
    sbet1, cbet1 = Math.norm(sbet1, cbet1)

    phi = lat2 * Math.degree
    # Ensure cbet2 = +epsilon at poles
    sbet2 = self._f1 * math.sin(phi)
    cbet2 = Geodesic.tiny_ if abs(lat2) == 90 else math.cos(phi)
    sbet2, cbet2 = Math.norm(sbet2, 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 = lon12 * Math.degree
    slam12 = 0 if lon12 == 180 else math.sin(lam12)
    clam12 = math.cos(lam12)      # lon12 == 90 isn't interesting

    # 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(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,
#.........这里部分代码省略.........

# 需要导入模块: from geographiclib.geomath import Math [as 别名]
# 或者: from geographiclib.geomath.Math import norm [as 别名]
  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(csig1 * ssig2 - ssig1 * csig2, 0.0),
                       csig1 * csig2 + ssig1 * ssig2)

    # omg12 = omg2 - omg1, limit to [0, pi]
    omg12 = math.atan2(max(comg1 * somg2 - somg1 * comg2, 0.0),
                       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,
          False, C1a, C2a)
        dlam12 *= self._f1 / (calp2 * cbet2)
    else:
      dlam12 = Math.nan

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

# 需要导入模块: from geographiclib.geomath import Math [as 别名]
# 或者: from geographiclib.geomath.Math import norm [as 别名]
  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, False, 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
#.........这里部分代码省略.........

# 需要导入模块: from geographiclib.geomath import Math [as 别名]
# 或者: from geographiclib.geomath.Math import norm [as 别名]
  def __init__(self, geod, lat1, lon1, azi1, caps = GeodesicCapability.ALL):
    """Construct a GeodesicLine object describing a geodesic line
    starting at (lat1, lon1) with azimuth azi1.  geod is a Geodesic
    object (which embodies the ellipsoid parameters).  caps is 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.REDUCEDLENGTH
      Geodesic.GEODESICSCALE
      Geodesic.AREA
      Geodesic.DISTANCE_IN
      Geodesic.ALL (all of the above)

    The default value of caps is ALL.

    """

    from geographiclib.geodesic import Geodesic
    self._a = geod._a
    self._f = geod._f
    self._b = geod._b
    self._c2 = geod._c2
    self._f1 = geod._f1
    self._caps = (caps | Geodesic.LATITUDE | Geodesic.AZIMUTH |
                  Geodesic.LONG_UNROLL)

    # Guard against underflow in salp0
    self._lat1 = Math.LatFix(lat1)
    self._lon1 = lon1
    self._azi1 = Math.AngNormalize(azi1)
    self._salp1, self._calp1 = Math.sincosd(Math.AngRound(azi1))

    # real cbet1, sbet1
    sbet1, cbet1 = Math.sincosd(Math.AngRound(lat1)); sbet1 *= self._f1
    # Ensure cbet1 = +epsilon at poles
    sbet1, cbet1 = Math.norm(sbet1, cbet1); cbet1 = max(Geodesic.tiny_, cbet1)
    self._dn1 = math.sqrt(1 + geod._ep2 * Math.sq(sbet1))

    # Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0),
    self._salp0 = self._salp1 * cbet1 # alp0 in [0, pi/2 - |bet1|]
    # Alt: calp0 = hypot(sbet1, calp1 * cbet1).  The following
    # is slightly better (consider the case salp1 = 0).
    self._calp0 = math.hypot(self._calp1, self._salp1 * sbet1)
    # Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
    # sig = 0 is nearest northward crossing of equator.
    # With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
    # With bet1 =  pi/2, alp1 = -pi, sig1 =  pi/2
    # With bet1 = -pi/2, alp1 =  0 , sig1 = -pi/2
    # Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
    # With alp0 in (0, pi/2], quadrants for sig and omg coincide.
    # No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
    # With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi.
    self._ssig1 = sbet1; self._somg1 = self._salp0 * sbet1
    self._csig1 = self._comg1 = (cbet1 * self._calp1
                                 if sbet1 != 0 or self._calp1 != 0 else 1)
    # sig1 in (-pi, pi]
    self._ssig1, self._csig1 = Math.norm(self._ssig1, self._csig1)
    # No need to normalize
    # self._somg1, self._comg1 = Math.norm(self._somg1, self._comg1)

    self._k2 = Math.sq(self._calp0) * geod._ep2
    eps = self._k2 / (2 * (1 + math.sqrt(1 + self._k2)) + self._k2)

    if self._caps & Geodesic.CAP_C1:
      self._A1m1 = Geodesic.A1m1f(eps)
      self._C1a = list(range(Geodesic.nC1_ + 1))
      Geodesic.C1f(eps, self._C1a)
      self._B11 = Geodesic.SinCosSeries(
        True, self._ssig1, self._csig1, self._C1a)
      s = math.sin(self._B11); c = math.cos(self._B11)
      # tau1 = sig1 + B11
      self._stau1 = self._ssig1 * c + self._csig1 * s
      self._ctau1 = self._csig1 * c - self._ssig1 * s
      # Not necessary because C1pa reverts C1a
      #    _B11 = -SinCosSeries(true, _stau1, _ctau1, _C1pa)

    if self._caps & Geodesic.CAP_C1p:
      self._C1pa = list(range(Geodesic.nC1p_ + 1))
      Geodesic.C1pf(eps, self._C1pa)

    if self._caps & Geodesic.CAP_C2:
      self._A2m1 = Geodesic.A2m1f(eps)
      self._C2a = list(range(Geodesic.nC2_ + 1))
      Geodesic.C2f(eps, self._C2a)
      self._B21 = Geodesic.SinCosSeries(
        True, self._ssig1, self._csig1, self._C2a)

    if self._caps & Geodesic.CAP_C3:
      self._C3a = list(range(Geodesic.nC3_))
      geod.C3f(eps, self._C3a)
      self._A3c = -self._f * self._salp0 * geod.A3f(eps)
      self._B31 = Geodesic.SinCosSeries(
        True, self._ssig1, self._csig1, self._C3a)

    if self._caps & Geodesic.CAP_C4:
      self._C4a = list(range(Geodesic.nC4_))
#.........这里部分代码省略.........