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Python implementation of Vincenty's direct formula
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def vinc_pt(f, a, phi1, lembda1, alpha12, s ) : | |
import math | |
""" | |
Returns the lat and long of projected point and reverse azimuth | |
given a reference point and a distance and azimuth to project. | |
lats, longs and azimuths are passed in decimal degrees | |
Returns ( phi2, lambda2, alpha21 ) as a tuple | |
Parameters: | |
=========== | |
f: flattening of the ellipsoid | |
a: radius of the ellipsoid, meteres | |
phil: latitude of the start point, decimal degrees | |
lembda1: longitude of the start point, decimal degrees | |
alpha12: bearing, decimal degrees | |
s: Distance to endpoint, meters | |
NOTE: This code could have some license issues. It has been obtained | |
from a forum and its license is not clear. I'll reimplement with | |
GPL3 as soon as possible. | |
The code has been taken from | |
https://isis.astrogeology.usgs.gov/IsisSupport/index.php?topic=408.0 | |
and refers to (broken link) | |
http://wegener.mechanik.tu-darmstadt.de/GMT-Help/Archiv/att-8710/Geodetic_py | |
""" | |
piD4 = math.atan( 1.0 ) | |
two_pi = piD4 * 8.0 | |
phi1 = phi1 * piD4 / 45.0 | |
lembda1 = lembda1 * piD4 / 45.0 | |
alpha12 = alpha12 * piD4 / 45.0 | |
if ( alpha12 < 0.0 ) : | |
alpha12 = alpha12 + two_pi | |
if ( alpha12 > two_pi ) : | |
alpha12 = alpha12 - two_pi | |
b = a * (1.0 - f) | |
TanU1 = (1-f) * math.tan(phi1) | |
U1 = math.atan( TanU1 ) | |
sigma1 = math.atan2( TanU1, math.cos(alpha12) ) | |
Sinalpha = math.cos(U1) * math.sin(alpha12) | |
cosalpha_sq = 1.0 - Sinalpha * Sinalpha | |
u2 = cosalpha_sq * (a * a - b * b ) / (b * b) | |
A = 1.0 + (u2 / 16384) * (4096 + u2 * (-768 + u2 * \ | |
(320 - 175 * u2) ) ) | |
B = (u2 / 1024) * (256 + u2 * (-128 + u2 * (74 - 47 * u2) ) ) | |
# Starting with the approx | |
sigma = (s / (b * A)) | |
last_sigma = 2.0 * sigma + 2.0 # something impossible | |
# Iterate the following 3 eqs unitl no sig change in sigma | |
# two_sigma_m , delta_sigma | |
while ( abs( (last_sigma - sigma) / sigma) > 1.0e-9 ): | |
two_sigma_m = 2 * sigma1 + sigma | |
delta_sigma = B * math.sin(sigma) * ( math.cos(two_sigma_m) \ | |
+ (B/4) * (math.cos(sigma) * \ | |
(-1 + 2 * math.pow( math.cos(two_sigma_m), 2 ) - \ | |
(B/6) * math.cos(two_sigma_m) * \ | |
(-3 + 4 * math.pow(math.sin(sigma), 2 )) * \ | |
(-3 + 4 * math.pow( math.cos (two_sigma_m), 2 ))))) | |
last_sigma = sigma | |
sigma = (s / (b * A)) + delta_sigma | |
phi2 = math.atan2 ( (math.sin(U1) * math.cos(sigma) +\ | |
math.cos(U1) * math.sin(sigma) * math.cos(alpha12) ), \ | |
((1-f) * math.sqrt( math.pow(Sinalpha, 2) + \ | |
pow(math.sin(U1) * math.sin(sigma) - math.cos(U1) * \ | |
math.cos(sigma) * math.cos(alpha12), 2)))) | |
lembda = math.atan2( (math.sin(sigma) * math.sin(alpha12 )),\ | |
(math.cos(U1) * math.cos(sigma) - \ | |
math.sin(U1) * math.sin(sigma) * math.cos(alpha12))) | |
C = (f/16) * cosalpha_sq * (4 + f * (4 - 3 * cosalpha_sq )) | |
omega = lembda - (1-C) * f * Sinalpha * \ | |
(sigma + C * math.sin(sigma) * (math.cos(two_sigma_m) + \ | |
C * math.cos(sigma) * (-1 + 2 *\ | |
math.pow(math.cos(two_sigma_m), 2) ))) | |
lembda2 = lembda1 + omega | |
alpha21 = math.atan2 ( Sinalpha, (-math.sin(U1) * \ | |
math.sin(sigma) + | |
math.cos(U1) * math.cos(sigma) * math.cos(alpha12))) | |
alpha21 = alpha21 + two_pi / 2.0 | |
if ( alpha21 < 0.0 ) : | |
alpha21 = alpha21 + two_pi | |
if ( alpha21 > two_pi ) : | |
alpha21 = alpha21 - two_pi | |
phi2 = phi2 * 45.0 / piD4 | |
lembda2 = lembda2 * 45.0 / piD4 | |
alpha21 = alpha21 * 45.0 / piD4 | |
return phi2, lembda2, alpha21 |
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