def measure(self, a, b):
a, b = Point(a), Point(b)
lat1, lng1 = radians(degrees=a.latitude), radians(degrees=a.longitude)
lat2, lng2 = radians(degrees=b.latitude), radians(degrees=b.longitude)
sin_lat1, cos_lat1 = sin(lat1), cos(lat1)
sin_lat2, cos_lat2 = sin(lat2), cos(lat2)
delta_lng = lng2 - lng1
cos_delta_lng, sin_delta_lng = cos(delta_lng), sin(delta_lng)
central_angle = acos(
# We're correcting from floating point rounding errors on very-near and exact points here
min(1.0, sin_lat1 * sin_lat2 +
cos_lat1 * cos_lat2 * cos_delta_lng))
# From http://en.wikipedia.org/wiki/Great_circle_distance:
# Historically, the use of this formula was simplified by the
# availability of tables for the haversine function. Although this
# formula is accurate for most distances, it too suffers from
# rounding errors for the special (and somewhat unusual) case of
# antipodal points (on opposite ends of the sphere). A more
# complicated formula that is accurate for all distances is: (below)
d = atan2(sqrt((cos_lat2 * sin_delta_lng) ** 2 +
(cos_lat1 * sin_lat2 -
sin_lat1 * cos_lat2 * cos_delta_lng) ** 2),
sin_lat1 * sin_lat2 + cos_lat1 * cos_lat2 * cos_delta_lng)
return self.RADIUS * d
def destination(self, point, bearing, distance=None):
point = Point(point)
lat1 = units.radians(degrees=point.latitude)
lng1 = units.radians(degrees=point.longitude)
bearing = units.radians(degrees=bearing)
if distance is None:
distance = self
if isinstance(distance, Distance):
distance = distance.kilometers
d_div_r = float(distance) / self.RADIUS
lat2 = asin(
sin(lat1) * cos(d_div_r) +
cos(lat1) * sin(d_div_r) * cos(bearing)
)
lng2 = lng1 + atan2(
sin(bearing) * sin(d_div_r) * cos(lat1),
cos(d_div_r) - sin(lat1) * sin(lat2)
)
return Point(units.degrees(radians=lat2), units.degrees(radians=lng2))
def destination(self, point, bearing, distance=None):
point = Point(point)
lat1 = units.radians(degrees=point.latitude)
lng1 = units.radians(degrees=point.longitude)
bearing = units.radians(degrees=bearing)
if distance is None:
distance = self
if isinstance(distance, Distance):
distance = distance.kilometers
ellipsoid = self.ELLIPSOID
if isinstance(ellipsoid, basestring):
ellipsoid = ELLIPSOIDS[ellipsoid]
major, minor, f = ellipsoid
tan_reduced1 = (1 - f) * tan(lat1)
cos_reduced1 = 1 / sqrt(1 + tan_reduced1 ** 2)
sin_reduced1 = tan_reduced1 * cos_reduced1
sin_bearing, cos_bearing = sin(bearing), cos(bearing)
sigma1 = atan2(tan_reduced1, cos_bearing)
sin_alpha = cos_reduced1 * sin_bearing
cos_sq_alpha = 1 - sin_alpha ** 2
u_sq = cos_sq_alpha * (major ** 2 - minor ** 2) / minor ** 2
A = 1 + u_sq / 16384. * (
4096 + u_sq * (-768 + u_sq * (320 - 175 * u_sq))
)
B = u_sq / 1024. * (256 + u_sq * (-128 + u_sq * (74 - 47 * u_sq)))
sigma = distance / (minor * A)
sigma_prime = 2 * pi
while abs(sigma - sigma_prime) > 10e-12:
cos2_sigma_m = cos(2 * sigma1 + sigma)
sin_sigma, cos_sigma = sin(sigma), cos(sigma)
delta_sigma = B * sin_sigma * (
cos2_sigma_m + B / 4. * (
cos_sigma * (
-1 + 2 * cos2_sigma_m
) - B / 6. * cos2_sigma_m * (
-3 + 4 * sin_sigma ** 2) * (
-3 + 4 * cos2_sigma_m ** 2
)
)
)
sigma_prime = sigma
sigma = distance / (minor * A) + delta_sigma
sin_sigma, cos_sigma = sin(sigma), cos(sigma)
lat2 = atan2(
sin_reduced1 * cos_sigma + cos_reduced1 * sin_sigma * cos_bearing,
(1 - f) * sqrt(
sin_alpha ** 2 + (
sin_reduced1 * sin_sigma -
cos_reduced1 * cos_sigma * cos_bearing
) ** 2
)
)
lambda_lng = atan2(
sin_sigma * sin_bearing,
cos_reduced1 * cos_sigma - sin_reduced1 * sin_sigma * cos_bearing
)
C = f / 16. * cos_sq_alpha * (4 + f * (4 - 3 * cos_sq_alpha))
delta_lng = (
lambda_lng - (1 - C) * f * sin_alpha * (
sigma + C * sin_sigma * (
cos2_sigma_m + C * cos_sigma * (
-1 + 2 * cos2_sigma_m ** 2
)
)
)
)
final_bearing = atan2(
sin_alpha,
cos_reduced1 * cos_sigma * cos_bearing - sin_reduced1 * sin_sigma
)
lng2 = lng1 + delta_lng
return Point(units.degrees(radians=lat2), units.degrees(radians=lng2))
def measure(self, a, b):
a, b = Point(a), Point(b)
lat1, lng1 = radians(degrees=a.latitude), radians(degrees=a.longitude)
lat2, lng2 = radians(degrees=b.latitude), radians(degrees=b.longitude)
if isinstance(self.ELLIPSOID, basestring):
major, minor, f = ELLIPSOIDS[self.ELLIPSOID]
else:
major, minor, f = self.ELLIPSOID
delta_lng = lng2 - lng1
reduced_lat1 = atan((1 - f) * tan(lat1))
reduced_lat2 = atan((1 - f) * tan(lat2))
sin_reduced1, cos_reduced1 = sin(reduced_lat1), cos(reduced_lat1)
sin_reduced2, cos_reduced2 = sin(reduced_lat2), cos(reduced_lat2)
lambda_lng = delta_lng
lambda_prime = 2 * pi
iter_limit = 20
while abs(lambda_lng - lambda_prime) > 10e-12 and iter_limit > 0:
sin_lambda_lng, cos_lambda_lng = sin(lambda_lng), cos(lambda_lng)
sin_sigma = sqrt(
(cos_reduced2 * sin_lambda_lng) ** 2 +
(cos_reduced1 * sin_reduced2 -
sin_reduced1 * cos_reduced2 * cos_lambda_lng) ** 2
)
if sin_sigma == 0:
return 0 # Coincident points
cos_sigma = (
sin_reduced1 * sin_reduced2 +
cos_reduced1 * cos_reduced2 * cos_lambda_lng
)
sigma = atan2(sin_sigma, cos_sigma)
sin_alpha = (
cos_reduced1 * cos_reduced2 * sin_lambda_lng / sin_sigma
)
cos_sq_alpha = 1 - sin_alpha ** 2
if cos_sq_alpha != 0:
cos2_sigma_m = cos_sigma - 2 * (
sin_reduced1 * sin_reduced2 / cos_sq_alpha
)
else:
cos2_sigma_m = 0.0 # Equatorial line
C = f / 16. * cos_sq_alpha * (4 + f * (4 - 3 * cos_sq_alpha))
lambda_prime = lambda_lng
lambda_lng = (
delta_lng + (1 - C) * f * sin_alpha * (
sigma + C * sin_sigma * (
cos2_sigma_m + C * cos_sigma * (
-1 + 2 * cos2_sigma_m ** 2
)
)
)
)
iter_limit -= 1
if iter_limit == 0:
raise ValueError("Vincenty formula failed to converge!")
u_sq = cos_sq_alpha * (major ** 2 - minor ** 2) / minor ** 2
A = 1 + u_sq / 16384. * (
4096 + u_sq * (-768 + u_sq * (320 - 175 * u_sq))
)
B = u_sq / 1024. * (256 + u_sq * (-128 + u_sq * (74 - 47 * u_sq)))
delta_sigma = (
B * sin_sigma * (
cos2_sigma_m + B / 4. * (
cos_sigma * (
-1 + 2 * cos2_sigma_m ** 2
) - B / 6. * cos2_sigma_m * (
-3 + 4 * sin_sigma ** 2
) * (
-3 + 4 * cos2_sigma_m ** 2
)
)
)
)
s = minor * A * (sigma - delta_sigma)
return s