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vincenty.py
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vincenty.py
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from math import radians as degrees_to_radians, degrees as radians_to_degrees, sin, cos, tan, atan, atan2, sqrt, acos
def lookup_ellipsoid_parameters(ellipsoid):
# parameters 'a' and 'f' of different ellipsoids
ellipsoids = {
"wgs84" : (6378137.0, 1000000000.0 / 298257223563.0),
"bessel" : (6377397.155, 10000000.0 / 2991528128.0),
"international" : (6378388.000, 1.0 / 297.0)
}
return ellipsoids[ellipsoid]
def normalize_degrees_signed(angle):
while (angle < -180.0):
angle += 360.0
while (angle >= 180.0):
angle -= 360.0
return angle
def normalize_degrees_unsigned(angle):
while (angle < 0.0):
angle += 360.0
while (angle >= 360.0):
angle -= 360.0
return angle
def square(x):
return x * x
#def coord(degs, mins, secs)
# return degs + mins / 60.0 + secs / 3600.0
#void print_dms(double dms)
#{
# double value = dms;
#
# bool negative = (value < 0.0);
#
# if (negative)
# {
# value = -value;
# }
#
# double deg = floor(value);
# value -= deg;
# value *= 60.0;
# double min = floor(value);
# value -= min;
# value *= 60.0;
# double sec = value;
#
# printf("%c %3.0f deg %02.0f min %20.17f sec [%+22.17f deg]", (negative ? '-' : '+'), deg, min, sec, dms)
#}
def vincenty_direct(latitude_departure_deg, longitude_departure_deg, azimuth_departure_deg, distance_meters, ellipsoid = "wgs84", ITERATION_LIMIT = 20, TOLERANCE = 1e-12):
# lookup ellipsoid parameters
(a, f) = lookup_ellipsoid_parameters(ellipsoid)
# massage the input parameters
phi1 = degrees_to_radians(latitude_departure_deg)
alpha1 = degrees_to_radians(azimuth_departure_deg)
s = distance_meters
# ellipsoid parameters
b = (1 - f) * a
# prepare iterations
tan_U1 = (1 - f) * tan(phi1)
U1 = atan(tan_U1)
sigma1 = atan2(tan_U1, cos(alpha1))
sin_alpha = cos(U1) * sin(alpha1)
cos_alpha_squared = 1.0 - square(sin_alpha)
u_squared = cos_alpha_squared * (square(a) - square(b)) / square(b)
A = 1.0 + u_squared/16384.0 * (4096.0 + u_squared * (-768.0 + u_squared * (320.0 - 175.0 * u_squared)))
B = u_squared/ 1024.0 * ( 256.0 + u_squared * (-128.0 + u_squared * ( 74.0 - 47.0 * u_squared)))
# iterate until sigma is stable
sigma = s/(b*A) # initial guess
for iteration in range(ITERATION_LIMIT):
cos_two_sigma_m = cos(2.0 * sigma1 + sigma)
delta_sigma = B * sin(sigma) * (cos_two_sigma_m + 1.0/4.0 * B * ( cos(sigma) * (-1.0 + 2.0 * square(cos_two_sigma_m))
- 1.0/6.0 * B * cos_two_sigma_m * (-3.0 + 4.0 * square(sin(sigma))) * (-3.0 + 4.0 * square(cos_two_sigma_m))))
old_sigma = sigma
sigma = s/(b*A) + delta_sigma
if (abs(sigma - old_sigma) < TOLERANCE):
break
else:
raise Exception("vincenty_direct failed to converge")
# done iterating; calculate where we are
phi2 = atan2(
sin(U1) * cos(sigma) + cos(U1) * sin(sigma) * cos(alpha1),
(1.0-f)*sqrt(square(sin_alpha)+square(sin(U1)*sin(sigma)-cos(U1)*cos(sigma)*cos(alpha1)))
)
lambda_ = atan2(sin(sigma)*sin(alpha1), cos(U1)*cos(sigma) - sin(U1)*sin(sigma)*cos(alpha1))
C = f / 16.0 * cos_alpha_squared * (4.0 + f * (4.0 - 3.0 * cos_alpha_squared))
L = lambda_ - (1.0 - C) * f * sin_alpha * (sigma + C * sin(sigma) * (cos_two_sigma_m + C * cos(sigma) * (-1.0 + 2.0 * square(cos_two_sigma_m))))
alpha2 = atan2(sin_alpha, -sin(U1) * sin(sigma) + cos(U1) * cos(sigma) * cos(alpha1))
# massage output
latitude_arrival_deg = radians_to_degrees(phi2)
longitude_arrival_deg = normalize_degrees_unsigned(longitude_departure_deg + radians_to_degrees(L))
azimuth_arrival_deg = normalize_degrees_unsigned(radians_to_degrees(alpha2))
return (latitude_arrival_deg, longitude_arrival_deg, azimuth_arrival_deg)
def vincenty_inverse(latitude_departure_deg, longitude_departure_deg, latitude_arrival_deg, longitude_arrival_deg, ellipsoid = "wgs84", ITERATION_LIMIT = 20, TOLERANCE = 1e-12):
# lookup ellipsoid parameters
(a, f) = lookup_ellipsoid_parameters(ellipsoid)
# massage the input parameters
phi1 = degrees_to_radians(latitude_departure_deg)
phi2 = degrees_to_radians(latitude_arrival_deg)
L = degrees_to_radians(longitude_arrival_deg - longitude_departure_deg)
# ellipsoid parameters
b = (1.0-f) * a
# prepare iterations
tan_U1 = (1.0-f) * tan(phi1)
U1 = atan(tan_U1)
tan_U2 = (1.0-f) * tan(phi2)
U2 = atan(tan_U2)
# iterate until lambda is stable
lambda_ = L # initial guess
for iteration in range(ITERATION_LIMIT):
sin_sigma_squared = square(cos(U2) * sin(lambda_)) + square(cos(U1)*sin(U2) - sin(U1)*cos(U2)*cos(lambda_))
sin_sigma = sqrt(sin_sigma_squared)
cos_sigma = sin(U1) * sin(U2) + cos(U1) * cos(U2) * cos(lambda_)
sigma = acos(cos_sigma)
sin_alpha = cos(U1) * cos(U2) * sin(lambda_) / sin_sigma
cos_alpha_squared = 1.0 - square(sin_alpha)
cos_two_sigma_m = cos_sigma - 2.0 * sin(U1) * sin(U2) / cos_alpha_squared
C = f / 16.0 * cos_alpha_squared * (4.0 + f * (4.0 - 3.0 * cos_alpha_squared))
old_lambda = lambda_
lambda_ = L + (1.0 - C) * f * sin_alpha * (sigma + C * sin_sigma * (cos_two_sigma_m + C * cos_sigma * (-1.0 + 2.0 * square(cos_two_sigma_m))))
if (abs(lambda_ - old_lambda) < TOLERANCE):
break
else:
raise Exception("vincenty_indirect failed to converge")
u_squared = cos_alpha_squared * (square(a) - square(b)) / square(b)
A = 1.0 + u_squared/16384.0 * (4096.0 + u_squared * (-768.0 + u_squared * (320.0 - 175.0 * u_squared)))
B = u_squared/ 1024.0 * ( 256.0 + u_squared * (-128.0 + u_squared * ( 74.0 - 47.0 * u_squared)))
delta_sigma = B * sin_sigma * (cos_two_sigma_m + 1.0/4.0 * B * ( cos_sigma * (-1.0 + 2.0 * square(cos_two_sigma_m))
- 1.0/6.0 * B * cos_two_sigma_m * (-3.0 + 4.0 * sin_sigma_squared) * (-3.0 + 4.0 * square(cos_two_sigma_m))))
s = b * A * (sigma - delta_sigma)
alpha1 = atan2(cos(U2) * sin(lambda_), cos(U1) * sin(U2) - sin(U1) * cos(U2) * cos(lambda_))
alpha2 = atan2(cos(U1) * sin(lambda_), - sin(U1) * cos(U2) + cos(U1) * sin(U2) * cos(lambda_))
# massage output
distance_meters = s
azimuth_departure_deg = normalize_degrees_unsigned(radians_to_degrees(alpha1))
azimuth_arrival_deg = normalize_degrees_unsigned(radians_to_degrees(alpha2))
return(distance_meters, azimuth_departure_deg, azimuth_arrival_deg)
if __name__ == "__main__":
forward = vincenty_direct(52.0, 4.0, 17.0, 100000.0)
print(forward)
inverse = vincenty_inverse(52.0, 4.0, forward[0], forward[1])
print(inverse)