def _rhumb3(self, other):
'''(INTERNAL) Rhumb_ helper function.
@param other: The other point (spherical C{LatLon}).
'''
self.others(other)
a1, b1 = self.to2ab()
a2, b2 = other.to2ab()
# if |db| > 180 take shorter rhumb
# line across the anti-meridian
db = wrapPI(b2 - b1)
dp = log(tanPI_2_2(a2) / tanPI_2_2(a1))
return (a2 - a1), db, dp
def rhumbMidpointTo(self, other, height=None):
'''Return the (loxodromic) midpoint between this and
an other point.
@param other: The other point (spherical LatLon).
@keyword height: Optional height, overriding the mean height
(C{meter}).
@return: The midpoint (spherical C{LatLon}).
@raise TypeError: The I{other} point is not spherical.
@example:
>>> p = LatLon(51.127, 1.338)
>>> q = LatLon(50.964, 1.853)
>>> m = p.rhumb_midpointTo(q)
>>> m.toStr() # '51.0455°N, 001.5957°E'
'''
self.others(other)
# see <https://MathForum.org/library/drmath/view/51822.html>
a1, b1 = self.to2ab()
a2, b2 = other.to2ab()
if abs(b2 - b1) > PI:
b1 += PI2 # crossing anti-meridian
a3 = favg(a1, a2)
b3 = favg(b1, b2)
f1 = tanPI_2_2(a1)
if abs(f1) > EPS:
f2 = tanPI_2_2(a2)
f = f2 / f1
if abs(f) > EPS:
f = log(f)
if abs(f) > EPS:
f3 = tanPI_2_2(a3)
b3 = fsum_(b1 * log(f2), -b2 * log(f1),
(b2 - b1) * log(f3)) / f
h = self._havg(other) if height is None else height
return self.classof(degrees90(a3), degrees180(b3), height=h)
def rhumbDestination(self, distance, bearing, radius=R_M, height=None):
'''Return the destination point having travelled along a rhumb
(loxodrome) line from this point the given distance on the
given bearing.
@param distance: Distance travelled (C{meter}, same units as
I{radius}).
@param bearing: Bearing from this point (compass C{degrees360}).
@keyword radius: Optional, mean earth radius (C{meter}).
@keyword height: Optional height, overriding the default
height (C{meter}, same unit as I{radius}).
@return: The destination point (spherical C{LatLon}).
@example:
>>> p = LatLon(51.127, 1.338)
>>> q = p.rhumbDestination(40300, 116.7) # 50.9642°N, 001.8530°E
@JSname: I{rhumbDestinationPoint}
'''
a1, b1 = self.to2ab()
r = float(distance) / float(radius) # angular distance in radians
sb, cb = sincos2d(bearing)
da = r * cb
a2 = a1 + da
# normalize latitude if past pole
if a2 > PI_2:
a2 = PI - a2
elif a2 < -PI_2:
a2 = -PI - a2
dp = log(tanPI_2_2(a2) / tanPI_2_2(a1))
# E-W course becomes ill-conditioned with 0/0
q = (da / dp) if abs(dp) > EPS else cos(a1)
b2 = (b1 + r * sb / q) if abs(q) > EPS else b1
h = self.height if height is None else height
return self.classof(degrees90(a2), degrees180(b2), height=h)
def _tdef(self, lat):
'''(INTERNAL) Compute t(lat).
'''
return max(0, tanPI_2_2(-lat) / self._pdef(lat))