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geodetic.py
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geodetic.py
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#!/usr/bin/python
#
# ---------------------------------------------------------------------
# | |
# | geodetic.cc - a collection of geodetic functions |
# | Jim Leven - Dec 99 |
# | |
# | originally from: |
# | http://wegener.mechanik.tu-darmstadt.de/GMT-Help/Archiv/att-8710/Geodetic_py | |
# |ftp://pdsimage2.wr.usgs.gov/pub/pigpen/Python/Geodetic_py.py |
# | |
# ---------------------------------------------------------------------
#
#
# ----------------------------------------------------------------------
# | Algrothims from Geocentric Datum of Australia Technical Manual |
# | |
# | http://www.anzlic.org.au/icsm/gdatum/chapter4.html |
# | |
# | This page last updated 11 May 1999 |
# | |
# | Computations on the Ellipsoid |
# | |
# | There are a number of formulae that are available |
# | to calculate accurate geodetic positions, |
# | azimuths and distances on the ellipsoid. |
# | |
# | Vincenty's formulae (Vincenty, 1975) may be used |
# | for lines ranging from a few cm to nearly 20,000 km, |
# | with millimetre accuracy. |
# | The formulae have been extensively tested |
# | for the Australian region, by comparison with results |
# | from other formulae (Rainsford, 1955 & Sodano, 1965). |
# | |
# | * Inverse problem: azimuth and distance from known |
# | latitudes and longitudes |
# | * Direct problem: Latitude and longitude from known |
# | position, azimuth and distance. |
# | * Sample data |
# | * Excel spreadsheet |
# | |
# | Vincenty's Inverse formulae |
# | Given: latitude and longitude of two points |
# | (latitude1, longitude1 and latitude2, longitude2), |
# | Calculate: the ellipsoidal distance (s) and |
# | forward and reverse azimuths between the points (alpha12, alpha21). |
# | |
# ----------------------------------------------------------------------
import math
# def vinc_dist( f, a, latitude1, longitude1, latitude2, longitude2 ) :
def vinc_dist(latitude1, longitude1, latitude2, longitude2 ) :
"""
Returns the distance between two geographic points on the ellipsoid
and the forward and reverse azimuths between these points.
lats, longs and azimuths are in decimal degrees, distance in metres
Returns ( s, alpha12, alpha21 ) as a tuple
"""
f = 1.0 / 298.257223563 # WGS84
a = 6378137.0 # metres
if (abs( latitude2 - latitude1 ) < 1e-8) and ( abs( longitude2 - longitude1) < 1e-8 ) :
return 0.0, 0.0, 0.0
piD4 = math.atan( 1.0 )
two_pi = piD4 * 8.0
latitude1 = latitude1 * piD4 / 45.0
longitude1 = longitude1 * piD4 / 45.0 # unfortunately lambda is a key word!
latitude2 = latitude2 * piD4 / 45.0
longitude2 = longitude2 * piD4 / 45.0
b = a * (1.0 - f)
TanU1 = (1-f) * math.tan( latitude1 )
TanU2 = (1-f) * math.tan( latitude2 )
U1 = math.atan(TanU1)
U2 = math.atan(TanU2)
lembda = longitude2 - longitude1
last_lembda = -4000000.0 # an impossibe value
omega = lembda
# Iterate the following equations,
# until there is no significant change in lembda
while ( last_lembda < -3000000.0 or lembda != 0 and abs( (last_lembda - lembda)/lembda) > 1.0e-9 ) :
sqr_sin_sigma = pow( math.cos(U2) * math.sin(lembda), 2) + \
pow( (math.cos(U1) * math.sin(U2) - \
math.sin(U1) * math.cos(U2) * math.cos(lembda) ), 2 )
Sin_sigma = math.sqrt( sqr_sin_sigma )
Cos_sigma = math.sin(U1) * math.sin(U2) + math.cos(U1) * math.cos(U2) * math.cos(lembda)
sigma = math.atan2( Sin_sigma, Cos_sigma )
Sin_alpha = math.cos(U1) * math.cos(U2) * math.sin(lembda) / math.sin(sigma)
alpha = math.asin( Sin_alpha )
Cos2sigma_m = math.cos(sigma) - (2 * math.sin(U1) * math.sin(U2) / pow(math.cos(alpha), 2) )
C = (f/16) * pow(math.cos(alpha), 2) * (4 + f * (4 - 3 * pow(math.cos(alpha), 2)))
last_lembda = lembda
lembda = omega + (1-C) * f * math.sin(alpha) * (sigma + C * math.sin(sigma) * \
(Cos2sigma_m + C * math.cos(sigma) * (-1 + 2 * pow(Cos2sigma_m, 2) )))
u2 = pow(math.cos(alpha),2) * (a*a-b*b) / (b*b)
A = 1 + (u2/16384) * (4096 + u2 * (-768 + u2 * (320 - 175 * u2)))
B = (u2/1024) * (256 + u2 * (-128+ u2 * (74 - 47 * u2)))
delta_sigma = B * Sin_sigma * (Cos2sigma_m + (B/4) * \
(Cos_sigma * (-1 + 2 * pow(Cos2sigma_m, 2) ) - \
(B/6) * Cos2sigma_m * (-3 + 4 * sqr_sin_sigma) * \
(-3 + 4 * pow(Cos2sigma_m,2 ) )))
s = b * A * (sigma - delta_sigma)
alpha12 = math.atan2( (math.cos(U2) * math.sin(lembda)), \
(math.cos(U1) * math.sin(U2) - math.sin(U1) * math.cos(U2) * math.cos(lembda)))
alpha21 = math.atan2( (math.cos(U1) * math.sin(lembda)), \
(-math.sin(U1) * math.cos(U2) + math.cos(U1) * math.sin(U2) * math.cos(lembda)))
if ( alpha12 < 0.0 ) :
alpha12 = alpha12 + two_pi
if ( alpha12 > two_pi ) :
alpha12 = alpha12 - two_pi
alpha21 = alpha21 + two_pi / 2.0
if ( alpha21 < 0.0 ) :
alpha21 = alpha21 + two_pi
if ( alpha21 > two_pi ) :
alpha21 = alpha21 - two_pi
alpha12 = alpha12 * 45.0 / piD4
alpha21 = alpha21 * 45.0 / piD4
return s, alpha12, alpha21
# END of Vincenty's Inverse formulae
#-------------------------------------------------------------------------------
# Vincenty's Direct formulae |
# Given: latitude and longitude of a point (latitude1, longitude1) and |
# the geodetic azimuth (alpha12) |
# and ellipsoidal distance in metres (s) to a second point, |
# |
# Calculate: the latitude and longitude of the second point (latitude2, longitude2) |
# and the reverse azimuth (alpha21). |
# |
#-------------------------------------------------------------------------------
def vincentyDirect(latitude1, longitude1, alpha12, s ) :
"""
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 ( latitude2, lambda2, alpha21 ) as a tuple
"""
f = 1.0 / 298.257223563 # WGS84
a = 6378137.0 # metres
piD4 = math.atan( 1.0 )
two_pi = piD4 * 8.0
latitude1 = latitude1 * piD4 / 45.0
longitude1 = longitude1 * 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(latitude1)
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 approximation
sigma = (s / (b * A))
last_sigma = 2.0 * sigma + 2.0 # something impossible
# Iterate the following three equations
# until there is no significant 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
latitude2 = 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) )))
longitude2 = longitude1 + 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
latitude2 = latitude2 * 45.0 / piD4
longitude2 = longitude2 * 45.0 / piD4
alpha21 = alpha21 * 45.0 / piD4
return latitude2, longitude2, alpha21
# END of Vincenty's Direct formulae
#--------------------------------------------------------------------------
# Notes:
#
# * "The inverse formulae may give no solution over a line
# between two nearly antipodal points. This will occur when
# lembda ... is greater than pi in absolute value". (Vincenty, 1975)
#
# * In Vincenty (1975) L is used for the difference in longitude,
# however for consistency with other formulae in this Manual,
# omega is used here.
#
# * Variables specific to Vincenty's formulae are shown below,
# others common throughout the manual are shown in the Glossary.
#
#
# alpha = Azimuth of the geodesic at the equator
# U = Reduced latitude
# lembda = Difference in longitude on an auxiliary sphere (longitude1 & longitude2
# are the geodetic longitudes of points 1 & 2)
# sigma = Angular distance on a sphere, from point 1 to point 2
# sigma1 = Angular distance on a sphere, from the equator to point 1
# sigma2 = Angular distance on a sphere, from the equator to point 2
# sigma_m = Angular distance on a sphere, from the equator to the
# midpoint of the line from point 1 to point 2
# u, A, B, C = Internal variables
#
#
# Sample Data
#
# Flinders Peak
# -37 57'03.72030"
# 144 25'29.52440"
# Buninyong
# -37 39'10.15610"
# 143 55'35.38390"
# Ellipsoidal Distance
# 54,972.271 m
#
# Forward Azimuth
# 306 52'05.37"
#
# Reverse Azimuth
# 127 10'25.07"
#
#
#*******************************************************************
# Test driver
if __name__ == "__main__" :
f = 1.0 / 298.257223563 # WGS84
a = 6378137.0 # metres
print ("\n Ellipsoidal major axis = %12.3f metres\n" % ( a ))
print ("\n Inverse flattening = %15.9f\n" % ( 1.0/f ))
print ("\n Test Flinders Peak to Buninyon")
print ("\n ****************************** \n")
latitude1 = -(( 3.7203 / 60. + 57) / 60. + 37 )
longitude1 = ( 29.5244 / 60. + 25) / 60. + 144
print ("Flinders Peak = %12.6f, %13.6f \n" % ( latitude1, longitude1 ))
deg = int(latitude1)
min = int(abs( ( latitude1 - deg) * 60.0 ))
sec = abs(latitude1 * 3600 - deg * 3600) - min * 60
print (" Flinders Peak = %3i\xF8%3i\' %6.3f\", " % ( deg, min, sec ),)
deg = int(longitude1)
min = int(abs( ( longitude1 - deg) * 60.0 ))
sec = abs(longitude1 * 3600 - deg * 3600) - min * 60
print (" %3i\xF8%3i\' %6.3f\" \n" % ( deg, min, sec ))
latitude2 = -(( 10.1561 / 60. + 39) / 60. + 37 )
longitude2 = ( 35.3839 / 60. + 55) / 60. + 143
print ("\n Buninyon = %12.6f, %13.6f \n" % ( latitude2, longitude2 ))
deg = int(latitude2)
min = int(abs( ( latitude2 - deg) * 60.0 ))
sec = abs(latitude2 * 3600 - deg * 3600) - min * 60
print (" Buninyon = %3i\xF8%3i\' %6.3f\", " % ( deg, min, sec ),)
deg = int(longitude2)
min = int(abs( ( longitude2 - deg) * 60.0 ))
sec = abs(longitude2 * 3600 - deg * 3600) - min * 60
print (" %3i\xF8%3i\' %6.3f\" \n" % ( deg, min, sec ))
dist, alpha12, alpha21 = vinc_dist ( f, a, latitude1, longitude1, latitude2, longitude2 )
print ("\n Ellipsoidal Distance = %15.3f metres\n should be 54972.271 m\n" % ( dist ))
print ("\n Forward and back azimuths = %15.6f, %15.6f \n" % ( alpha12, alpha21 ))
deg = int(alpha12)
min = int( abs(( alpha12 - deg) * 60.0 ) )
sec = abs(alpha12 * 3600 - deg * 3600) - min * 60
print (" Forward azimuth = %3i\xF8%3i\' %6.3f\"\n" % ( deg, min, sec ))
deg = int(alpha21)
min = int(abs( ( alpha21 - deg) * 60.0 ))
sec = abs(alpha21 * 3600 - deg * 3600) - min * 60
print (" Reverse azimuth = %3i\xF8%3i\' %6.3f\"\n" % ( deg, min, sec ))
# Test the direct function */
latitude1 = -(( 3.7203 / 60. + 57) / 60. + 37 )
longitude1 = ( 29.5244 / 60. + 25) / 60. + 144
dist = 54972.271
alpha12 = ( 5.37 / 60. + 52) / 60. + 306
latitude2 = longitude2 = 0.0
alpha21 = 0.0
latitude2, longitude2, alpha21 = vincentyDirect (latitude1, longitude1, alpha12, dist )
print ("\n Projected point =%11.6f, %13.6f \n" % ( latitude2, longitude2 ))
deg = int(latitude2)
min = int(abs( ( latitude2 - deg) * 60.0 ))
sec = abs( latitude2 * 3600 - deg * 3600) - min * 60
print (" Projected Point = %3i\xF8%3i\' %6.3f\", " % ( deg, min, sec ),)
deg = int(longitude2)
min = int(abs( ( longitude2 - deg) * 60.0 ))
sec = abs(longitude2 * 3600 - deg * 3600) - min * 60
print (" %3i\xF8%3i\' %6.3f\"\n" % ( deg, min, sec ))
print (" Should be Buninyon \n" )
print ("\n Reverse azimuth = %10.6f \n" % ( alpha21 ))
deg = int(alpha21)
min = int(abs( ( alpha21 - deg) * 60.0 ))
sec = abs(alpha21 * 3600 - deg * 3600) - min * 60
print (" Reverse azimuth = %3i\xF8%3i\' %6.3f\"\n\n" % ( deg, min, sec ))
#*******************************************************************
def est_dist( latitude1, longitude1, latitude2, longitude2 ) :
"""
Returns an estimate of the distance between two geographic points
This is a quick and dirty vinc_dist
which will generally estimate the distance to within 1%
Returns distance in metres
"""
f = 1.0 / 298.257223563 # WGS84
a = 6378137.0 # metres
piD4 = 0.785398163397
latitude1 = latitude1 * piD4 / 45.0
longitude1 = longitude1 * piD4 / 45.0
latitude2 = latitude2 * piD4 / 45.0
longitude2 = longitude2 * piD4 / 45.0
c = math.cos((latitude2+latitude1)/2.0)
return math.sqrt( pow(math.fabs(latitude2-latitude1), 2) + \
pow(math.fabs(longitude2-longitude1)*c, 2) ) * a * ( 1.0 - f + f * c )
# END of rough estimate of the distance.