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from math import atan, tan, sin, cos, pi, sqrt, atan2, acos, asin
from geopy.units import radians
from geopy import units, util
from geopy.point import Point
# Average great-circle radius in kilometers, from Wikipedia.
# Using a sphere with this radius results in an error of up to about 0.5%.
EARTH_RADIUS = 6372.795
# From http://www.movable-type.co.uk/scripts/LatLongVincenty.html:
# The most accurate and widely used globally-applicable model for the earth
# ellipsoid is WGS-84, used in this script. Other ellipsoids offering a
# better fit to the local geoid include Airy (1830) in the UK, International
# 1924 in much of Europe, Clarke (1880) in Africa, and GRS-67 in South
# America. America (NAD83) and Australia (GDA) use GRS-80, functionally
# equivalent to the WGS-84 ellipsoid.
ELLIPSOIDS = {
# model major (km) minor (km) flattening
'WGS-84': (6378.137, 6356.7523142, 1 / 298.257223563),
'GRS-80': (6378.137, 6356.7523141, 1 / 298.257222101),
'Airy (1830)': (6377.563396, 6356.256909, 1 / 299.3249646),
'Intl 1924': (6378.388, 6356.911946, 1 / 297.0),
'Clarke (1880)': (6378.249145, 6356.51486955, 1 / 293.465),
'GRS-67': (6378.1600, 6356.774719, 1 / 298.25)
}
class Distance(object):
def __init__(self, *args, **kwargs):
kilometers = kwargs.pop('kilometers', 0)
if len(args) == 1:
# if we only get one argument we assume
# it's a known distance instead of
# calculating it first
kilometers += args[0]
elif len(args) > 1:
for a, b in util.pairwise(args):
kilometers += self.measure(a, b)
kilometers += units.kilometers(**kwargs)
self.__kilometers = kilometers
def __add__(self, other):
if isinstance(other, Distance):
return self.__class__(self.kilometers + other.kilometers)
else:
raise TypeError(
"Distance instance must be added with Distance instance."
)
def __neg__(self):
return self.__class__(-self.kilometers)
def __sub__(self, other):
return self + -other
def __mul__(self, other):
return self.__class__(self.kilometers * other)
def __div__(self, other):
if isinstance(other, Distance):
return self.kilometers / other.kilometers
else:
return self.__class__(self.kilometers / other)
def __abs__(self):
return self.__class__(abs(self.kilometers))
def __nonzero__(self):
return bool(self.kilometers)
def measure(self, a, b):
raise NotImplementedError
def __repr__(self):
return 'Distance(%s)' % self.kilometers
def __str__(self):
return '%s km' % self.__kilometers
def __cmp__(self, other):
if isinstance(other, Distance):
return cmp(self.kilometers, other.kilometers)
else:
return cmp(self.kilometers, other)
@property
def kilometers(self):
return self.__kilometers
@property
def km(self):
return self.kilometers
@property
def meters(self):
return units.meters(kilometers=self.kilometers)
@property
def m(self):
return self.meters
@property
def miles(self):
return units.miles(kilometers=self.kilometers)
@property
def mi(self):
return self.miles
@property
def feet(self):
return units.feet(kilometers=self.kilometers)
@property
def ft(self):
return self.feet
@property
def nautical(self):
return units.nautical(kilometers=self.kilometers)
@property
def nm(self):
return self.nautical
class GreatCircleDistance(Distance):
"""
Use spherical geometry to calculate the surface distance between two
geodesic points. This formula can be written many different ways,
including just the use of the spherical law of cosines or the haversine
formula.
The class attribute `RADIUS` indicates which radius of the earth to use,
in kilometers. The default is to use the module constant `EARTH_RADIUS`,
which uses the average great-circle radius.
"""
RADIUS = EARTH_RADIUS
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))
class VincentyDistance(Distance):
"""
Calculate the geodesic distance between two points using the formula
devised by Thaddeus Vincenty, with an accurate ellipsoidal model of the
earth.
The class attribute `ELLIPSOID` indicates which ellipsoidal model of the
earth to use. If it is a string, it is looked up in the `ELLIPSOIDS`
dictionary to obtain the major and minor semiaxes and the flattening.
Otherwise, it should be a tuple with those values. The most globally
accurate model is WGS-84. See the comments above the `ELLIPSOIDS`
dictionary for more information.
"""
ELLIPSOID = 'WGS-84'
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
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))
# Set the default distance formula to the most generally accurate.
distance = VincentyDistance
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