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geo_helper.py
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# Geographical helper functions for nmea_info.py and friends
#
# Helps with geographic functions, including:
# Lat+Long+Height -> XYZ
# XYZ -> Lat+Long+Height
# Lat+Long -> other Lat+Long (Helmert Transform)
# Lat+Long -> easting/northing (OS GB+IE Only)
# easting/northing -> Lat+Long (OS GB+IE Only)
# OS easting/northing -> OS 6 figure ref
#
# See http://gagravarr.org/code/ for updates and information
#
# GPL
#
# Nick Burch - v0.06 (30/05/2007)
import math
# For each co-ordinate system we do, what are the A, B and E2 values?
# List is A, B, E^2 (E^2 calculated after)
abe_values = {
'wgs84': [ 6378137.0, 6356752.3141, -1 ],
'osgb' : [ 6377563.396, 6356256.91, -1 ],
'osie' : [ 6377340.189, 6356034.447, -1 ]
}
# The earth's radius, in meters, as taken from an average of the WGS84
# a and b parameters (should be close enough)
earths_radius = (abe_values['wgs84'][0] + abe_values['wgs84'][1]) / 2.0
# Calculate the E2 values
for system in abe_values.keys():
a = abe_values[system][0]
b = abe_values[system][1]
e2 = (a*a - b*b) / (a*a)
abe_values[system][2] = e2
# For each co-ordinate system we can translate between, what are
# the tx, ty, tz, s, rx, ry and rz values?
# List is tx, ty, tz, s, rx, ry, rz
transform_values = {
'wgs84_to_osgb' : [ -446.448, 125.157, -542.060,
20.4894 / 1000.0 / 1000.0, # given as ppm
-0.1502 / 206265.0, # given as seconds of arc
-0.2470 / 206265.0, # given as seconds of arc
-0.8421 / 206265.0 # given as seconds of arc
],
'wgs84_to_osie' : [ -482.530, 130.596, -564.557,
-8.1500 / 1000.0 / 1000.0, # given as ppm
-1.0420 / 206265.0, # given as seconds of arc
-0.2140 / 206265.0, # given as seconds of arc
-0.6310 / 206265.0 # given as seconds of arc
],
'itrs2000_to_etrs89' : [ 0.054, 0.051, -0.048, 0,
0.000081 / 206265.0, # given as seconds of arc
0.00049 / 206265.0, # given as seconds of arc
0.000792 / 206265.0 # given as seconds of arc
]
}
# Calculate reverse transforms
for systems in [('wgs84','osgb'), ('wgs84','osie'), ('itrs2000','etrs89')]:
fs = systems[0] + "_to_" + systems[1]
rs = systems[1] + "_to_" + systems[0]
ra = []
for val in transform_values[fs]:
ra.append(-1.0 * val)
transform_values[rs] = ra
# Easting and Northin system values, for the systems we work with.
# List is n0, e0, F0, theta0 and landa0
en_values = {
'osgb' : [ -100000.0, 400000.0, 0.9996012717,
49.0 /360.0 *2.0*math.pi,
-2.0 /360.0 *2.0*math.pi
],
'osie' : [ 250000.0, 200000.0, 1.000035,
53.5 /360.0 *2.0*math.pi,
-8.0 /360.0 *2.0*math.pi
]
}
# Cassini Projection Origins
# List is lat (rad), long (rad), false easting, false northing
cassini_values = {
'osgb' : [ (53.0 + (13.0 / 60.0) + (17.274 / 3600.0)) /360.0 *2.0*math.pi,
-(2.0 + (41.0 / 60.0) + (3.562 / 3600.0)) /360.0 *2.0*math.pi,
0, 0 ]
}
# How many feet to the meter
feet_per_meter = 1.0 / 0.3048007491 # 3.28083
##############################################################
# OS GB Specific Helpers for Generic Methods #
##############################################################
def turn_wgs84_into_osgb36(lat_dec,long_dec,height):
"""See http://www.gps.gov.uk/guide6.asp#6.2 and http://www.gps.gov.uk/guide6.asp#6.6 for the calculations, and http://www.posc.org/Epicentre.2_2/DataModel/ExamplesofUsage/eu_cs34h.html for some background."""
wgs84_xyz = turn_llh_into_xyz(lat_dec,long_dec,height,'wgs84')
osgb_xyz = turn_xyz_into_other_xyz(
wgs84_xyz[0],wgs84_xyz[1],wgs84_xyz[2],'wgs84','osgb')
osgb_latlong = turn_xyz_into_llh(
osgb_xyz[0],osgb_xyz[1],osgb_xyz[2],'osgb')
return osgb_latlong
def turn_osgb36_into_wgs84(lat_dec,long_dec,height):
"""See http://www.gps.gov.uk/guide6.asp#6.2 and http://www.gps.gov.uk/guide6.asp#6.6 for the calculations, and http://www.posc.org/Epicentre.2_2/DataModel/ExamplesofUsage/eu_cs34h.html for some background."""
osgb_xyz = turn_llh_into_xyz(lat_dec,long_dec,height,'osgb')
wgs84_xyz = turn_xyz_into_other_xyz(
osgb_xyz[0],osgb_xyz[1],osgb_xyz[2],'osgb','wgs84')
wgs84_latlong = turn_xyz_into_llh(
wgs84_xyz[0],wgs84_xyz[1],wgs84_xyz[2],'wgs84')
return wgs84_latlong
def turn_osgb36_into_eastingnorthing(lat_dec,long_dec):
"""Turn OSGB36 (decimal) lat/long values into OS easting and northing values."""
return turn_latlong_into_eastingnorthing(lat_dec,long_dec,'osgb')
def turn_eastingnorthing_into_osgb36(easting,northing):
"""Turn OSGB36 easting and northing values into (decimal) lat/long values inOSGB36."""
return turn_eastingnorthing_into_latlong(easting,northing,'osgb')
##############################################################
# OS IE Specific Helpers for Generic Methods #
##############################################################
def turn_wgs84_into_osie36(lat_dec,long_dec,height):
"""As per turn_wgs84_into_osgb36, but for Irish grid"""
wgs84_xyz = turn_llh_into_xyz(lat_dec,long_dec,height,'wgs84')
osie_xyz = turn_xyz_into_other_xyz(
wgs84_xyz[0],wgs84_xyz[1],wgs84_xyz[2],'wgs84','osie')
osie_latlong = turn_xyz_into_llh(
osie_xyz[0],osie_xyz[1],osie_xyz[2],'osie')
return osie_latlong
def turn_osie36_into_wgs84(lat_dec,long_dec,height):
"""As per turn_osgb36_into_wgs84, but for Irish grid"""
osie_xyz = turn_llh_into_xyz(lat_dec,long_dec,height,'osie')
wgs84_xyz = turn_xyz_into_other_xyz(
osie_xyz[0],osie_xyz[1],osie_xyz[2],'osie','wgs84')
wgs84_latlong = turn_xyz_into_llh(
wgs84_xyz[0],wgs84_xyz[1],wgs84_xyz[2],'wgs84')
return wgs84_latlong
def turn_osie36_into_eastingnorthing(lat_dec,long_dec):
"""Turn OSIE36 (decimal) lat/long values into OS IE easting and northing values."""
return turn_latlong_into_eastingnorthing(lat_dec,long_dec,'osie')
def turn_eastingnorthing_into_osie36(easting,northing):
"""Turn OSIE36 easting and northing values into (decimal) lat/long values inOSIE36."""
return turn_eastingnorthing_into_latlong(easting,northing,'osie')
##############################################################
# Generic Transform Functions #
##############################################################
def turn_llh_into_xyz(lat_dec,long_dec,height,system):
"""Convert Lat, Long and Height into 3D Cartesian x,y,z
See http://www.ordnancesurvey.co.uk/gps/docs/convertingcoordinates3D.pdf"""
a = abe_values[system][0]
b = abe_values[system][1]
e2 = abe_values[system][2]
theta = float(lat_dec) / 360.0 * 2.0 * math.pi
landa = float(long_dec) / 360.0 * 2.0 * math.pi
height = float(height)
v = a / math.sqrt( 1.0 - e2 * (math.sin(theta) * math.sin(theta)) )
x = (v + height) * math.cos(theta) * math.cos(landa)
y = (v + height) * math.cos(theta) * math.sin(landa)
z = ( (1.0 - e2) * v + height ) * math.sin(theta)
return [x,y,z]
def turn_xyz_into_llh(x,y,z,system):
"""Convert 3D Cartesian x,y,z into Lat, Long and Height
See http://www.ordnancesurvey.co.uk/gps/docs/convertingcoordinates3D.pdf"""
a = abe_values[system][0]
b = abe_values[system][1]
e2 = abe_values[system][2]
p = math.sqrt(x*x + y*y)
long = math.atan(y/x)
lat_init = math.atan( z / (p * (1.0 - e2)) )
v = a / math.sqrt( 1.0 - e2 * (math.sin(lat_init) * math.sin(lat_init)) )
lat = math.atan( (z + e2*v*math.sin(lat_init)) / p )
height = (p / math.cos(lat)) - v # Ignore if a bit out
# Turn from radians back into degrees
long = long / 2 / math.pi * 360
lat = lat / 2 / math.pi * 360
return [lat,long,height]
def turn_xyz_into_other_xyz(old_x,old_y,old_z,from_scheme,to_scheme):
"""Helmert Transformation between one lat+long system and another
See http://www.ordnancesurvey.co.uk/oswebsite/gps/information/coordinatesystemsinfo/guidecontents/guide6.html for the calculations, and http://www.movable-type.co.uk/scripts/LatLongConvertCoords.html for a friendlier version with examples"""
transform = from_scheme + "_to_" + to_scheme
tx = transform_values[transform][0]
ty = transform_values[transform][1]
tz = transform_values[transform][2]
s = transform_values[transform][3]
rx = transform_values[transform][4]
ry = transform_values[transform][5]
rz = transform_values[transform][6]
# Do the transform
new_x = tx + ((1.0+s) * old_x) + (-rz * old_y) + (ry * old_z)
new_y = ty + (rz * old_x) + ((1.0+s) * old_y) + (-rx * old_z)
new_z = tz + (-ry * old_x) + (rx * old_y) + ((1.0+s) * old_z)
return [new_x,new_y,new_z]
def calculate_distance_and_bearing(from_lat_dec,from_long_dec,to_lat_dec,to_long_dec):
"""Uses the spherical law of cosines to calculate the distance and bearing between two positions"""
# Turn them all into radians
from_theta = float(from_lat_dec) / 360.0 * 2.0 * math.pi
from_landa = float(from_long_dec) / 360.0 * 2.0 * math.pi
to_theta = float(to_lat_dec) / 360.0 * 2.0 * math.pi
to_landa = float(to_long_dec) / 360.0 * 2.0 * math.pi
d = math.acos(
math.sin(from_theta) * math.sin(to_theta) +
math.cos(from_theta) * math.cos(to_theta) * math.cos(to_landa-from_landa)
) * earths_radius
bearing = math.atan2(
math.sin(to_landa-from_landa) * math.cos(to_theta),
math.cos(from_theta) * math.sin(to_theta) -
math.sin(from_theta) * math.cos(to_theta) * math.cos(to_landa-from_landa)
)
bearing = bearing / 2.0 / math.pi * 360.0
return [d,bearing]
##############################################################
# Easting/Northing Transform Methods #
##############################################################
def turn_latlong_into_eastingnorthing(lat_dec,long_dec,scheme):
"""Turn OSGB36 or OSIE36 (decimal) lat/long values into OS easting and northing values. See http://www.ordnancesurvey.co.uk/oswebsite/gps/information/coordinatesystemsinfo/guidecontents/guide7.html for the calculations, and http://www.posc.org/Epicentre.2_2/DataModel/ExamplesofUsage/eu_cs34h.html for some background."""
n0 = en_values[scheme][0]
e0 = en_values[scheme][1]
f0 = en_values[scheme][2]
theta0 = en_values[scheme][3]
landa0 = en_values[scheme][4]
a = abe_values[scheme][0]
b = abe_values[scheme][1]
e2 = abe_values[scheme][2]
theta = float(lat_dec) /360.0 *2.0*math.pi
landa = float(long_dec) /360.0 *2.0*math.pi
n = (a-b) / (a+b)
v = a * f0 * math.pow( (1 - e2 * math.sin(theta)*math.sin(theta)), -0.5 )
ro = a * f0 * (1 - e2) * math.pow( (1 - e2 * math.sin(theta)*math.sin(theta)), -1.5 )
nu2 = v/ro - 1
M = b * f0 * ( \
(1.0 + n + 5.0/4.0 *n*n + 5.0/4.0 *n*n*n) * (theta-theta0) - \
(3.0*n + 3.0*n*n + 21.0/8.0 *n*n*n) *math.sin(theta-theta0) *math.cos(theta+theta0) + \
(15.0/8.0*n*n + 15.0/8.0*n*n*n) *math.sin(2.0*(theta-theta0)) *math.cos(2.0*(theta+theta0)) - \
35.0/24.0*n*n*n *math.sin(3.0*(theta-theta0)) *math.cos(3.0*(theta+theta0)) \
)
I = M + n0
II = v/2.0 * math.sin(theta) * math.cos(theta)
III = v/24.0 * math.sin(theta) * math.pow( math.cos(theta),3 ) * \
(5.0 - math.pow(math.tan(theta),2) + 9.0*nu2)
IIIa = v/720.0 * math.sin(theta) * math.pow( math.cos(theta),5 ) * \
( 61.0 - 58.0 *math.pow(math.tan(theta),2) + math.pow(math.tan(theta),4) )
IV = v * math.cos(theta)
V = v/6.0 * math.pow( math.cos(theta),3 ) * \
( v/ro - math.pow(math.tan(theta),2) )
VI = v/120.0 * math.pow(math.cos(theta),5) * \
( 5.0 - 18.0 *math.pow(math.tan(theta),2) + \
math.pow(math.tan(theta),4) + 14.0*nu2 - \
58.0 * math.pow(math.tan(theta),2)*nu2 )
northing = I + II*math.pow(landa-landa0,2) + \
III*math.pow(landa-landa0,4) + \
IIIa*math.pow(landa-landa0,6)
easting = e0 + IV*(landa-landa0) + V*math.pow(landa-landa0,3) + \
VI*math.pow(landa-landa0,5)
return (easting,northing)
def turn_eastingnorthing_into_latlong(easting,northing,scheme):
"""Turn OSGB36 or OSIE36 easting and northing values into (decimal) lat/long values in OSGB36 / OSIE36. See http://www.ordnancesurvey.co.uk/oswebsite/gps/information/coordinatesystemsinfo/guidecontents/guide7.html for the calculations, and http://www.posc.org/Epicentre.2_2/DataModel/ExamplesofUsage/eu_cs34h.html for some background."""
n0 = en_values[scheme][0]
e0 = en_values[scheme][1]
f0 = en_values[scheme][2]
theta0 = en_values[scheme][3]
landa0 = en_values[scheme][4]
a = abe_values[scheme][0]
b = abe_values[scheme][1]
e2 = abe_values[scheme][2]
n = (a-b) / (a+b)
# Prepare to iterate
M = 0
theta = theta0
# Iterate, 4 times should be enough
for i in range(4):
theta = ((northing - n0 - M) / (a * f0)) + theta
M = b * f0 * ( \
(1.0 + n + 5.0/4.0 *n*n + 5.0/4.0 *n*n*n) * (theta-theta0) - \
(3.0*n + 3.0*n*n + 21.0/8.0 *n*n*n) *math.sin(theta-theta0) *math.cos(theta+theta0) + \
(15.0/8.0*n*n + 15.0/8.0*n*n*n) *math.sin(2.0*(theta-theta0)) *math.cos(2.0*(theta+theta0)) - \
35.0/24.0*n*n*n *math.sin(3.0*(theta-theta0)) *math.cos(3.0*(theta+theta0)) \
)
# Compute intermediate values
v = a * f0 * math.pow( (1 - e2 * math.sin(theta)*math.sin(theta)), -0.5 )
ro = a * f0 * (1 - e2) * math.pow( (1 - e2 * math.sin(theta)*math.sin(theta)), -1.5 )
nu2 = v/ro - 1
tantheta2 = math.pow(math.tan(theta),2)
VII = math.tan(theta) / (2 * ro * v)
VIII = math.tan(theta) / (24 * ro * math.pow(v,3)) \
* (5 + 3 * tantheta2 + nu2 - \
9 * tantheta2 * nu2 )
IX = math.tan(theta) / (720 * ro * math.pow(v,5)) \
* (61 + 90 * tantheta2 + 45 * tantheta2 * tantheta2)
X = 1 / (math.cos(theta) * v)
XI = 1 / (math.cos(theta) * 6 * math.pow(v,3)) \
* (v/ro + 2*tantheta2)
XII = 1 / (math.cos(theta) * 120 * math.pow(v,5)) \
* (5 + 28 * tantheta2 + 24 * tantheta2 * tantheta2)
XIIa = 1 / (math.cos(theta) * 5040 * math.pow(v,7)) \
* (61 + 662 * tantheta2 + 1320 * tantheta2 * tantheta2 \
+ 720 * tantheta2 * tantheta2 * tantheta2)
lat_rad = theta - VII * math.pow((easting-e0),2) \
+ VIII * math.pow((easting-e0),4) \
- IX * math.pow((easting-e0),6)
long_rad = landa0 + X * (easting-e0) \
- XI * math.pow((easting-e0),3) \
+ XII * math.pow((easting-e0),5) \
- XIIa * math.pow((easting-e0),7)
lat = lat_rad / 2.0 / math.pi * 360.0
long = long_rad / 2.0 / math.pi * 360.0
return (lat,long)
##############################################################
# Cassini Easting/Northing Transform Methods #
##############################################################
def turn_latlong_into_cassini_en(lat_dec,long_dec,scheme):
"""Latitude and Longitude, into Cassini-Soldner easting and northing co-ordinates, in the given scheme. See http://www.posc.org/Epicentre.2_2/DataModel/ExamplesofUsage/eu_cs34g.html for details of the calculation used"""
a = abe_values[scheme][0]
b = abe_values[scheme][1]
e2 = abe_values[scheme][2]
e4 = e2 * e2
e6 = e2 * e2 * e2
theta = float(lat_dec) /360.0 *2.0*math.pi
landa = float(long_dec) /360.0 *2.0*math.pi
theta0 = cassini_values[scheme][0]
landa0 = cassini_values[scheme][1]
false_easting = cassini_values[scheme][2]
false_northing = cassini_values[scheme][3]
# Compute intermediate values
A = (landa - landa0) * math.cos(theta)
T = math.tan(theta) * math.tan(theta)
C = e2 / (1.0 - e2) * math.cos(theta) * math.cos(theta)
v = a / math.sqrt( 1 - (e2 * math.sin(theta) * math.sin(theta)) )
A2 = A ** 2
A3 = A ** 3
A4 = A ** 4
A5 = A ** 5
# And M, which is how far along the meridian our latitude is from the origin
def makeM(picked_theta):
return a * (
(1.0 - e2/4.0 - 3.0*e4/64.0 - 5.0*e6/256.0) * picked_theta
- (3.0*e2/8.0 + 3.0*e4/32.0 + 45.0*e6/1024.0) * math.sin(2.0*picked_theta)
+ (15.0*e4/256.0 + 45.0*e6/1024.0) * math.sin(4.0*picked_theta)
- (35.0*e6/3072.0) * math.sin(6.0*picked_theta)
)
M = makeM(theta)
M0 = makeM(theta0)
# Now calculate
easting = false_easting + v * (
A - T * A3 / 6.0 - (8.0 - T + 8.0*C) * T * A5 / 120.0 )
northing = false_northing + M - M0 + v * math.tan(theta) * (
A2 / 2.0 + (5.0 - T + 6.0*C) * A4 / 24.0 )
return (easting,northing)
def turn_cassini_en_into_latlong(easting,northing,scheme):
"""Cassini-Soldner easting and northing, into Latitude and Longitude, in the given scheme. See http://www.posc.org/Epicentre.2_2/DataModel/ExamplesofUsage/eu_cs34g.html for details of the calculation used"""
a = abe_values[scheme][0]
b = abe_values[scheme][1]
e2 = abe_values[scheme][2]
e4 = e2 * e2
e6 = e2 * e2 * e2
theta0 = cassini_values[scheme][0]
landa0 = cassini_values[scheme][1]
false_easting = cassini_values[scheme][2]
false_northing = cassini_values[scheme][3]
def makeM(picked_theta):
return a * (
(1.0 - e2/4.0 - 3.0*e4/64.0 - 5.0*e6/256.0) * picked_theta
- (3.0*e2/8.0 + 3.0*e4/32.0 + 45.0*e6/1024.0) * math.sin(2.0*picked_theta)
+ (15.0*e4/256.0 + 45.0*e6/1024.0) * math.sin(4.0*picked_theta)
- (35.0*e6/3072.0) * math.sin(6.0*picked_theta)
)
# Compute first batch of intermediate values
M1 = makeM(theta0) + (northing - false_northing)
mu1 = M1 / (a * (1.0 - e2/4.0 - 3.0*e4/64.0 - 5.0*e6/256.0) )
e1 = (1 - ((1-e2) ** 0.5)) / (1 + ((1-e2) ** 0.5))
e1_2 = e1 ** 2
e1_3 = e1 ** 3
e1_4 = e1 ** 4
# Now compute theta1 at T1
theta1 = mu1 + (
+ (3.0*e1 / 2.0 - 27.0*e1_3 / 32.0) * math.sin(2.0*mu1)
+ (21.0*e1_2 / 16.0 - 55.0*e1_4 / 32.0) * math.sin(4.0*mu1)
+ (151.0*e1_3 / 96.0) * math.sin(6.0*mu1)
+ (1097.0*e1_4 / 512.0) * math.sin(8.0*mu1)
)
T1 = (math.tan(theta1)) ** 2
# Now we can find v1, ro1 and D
v1 = a / math.sqrt( 1.0 - (e2 * math.sin(theta1) * math.sin(theta1)) )
ro1 = a * (1 - e2) / ((1 - e2 * math.sin(theta1) * math.sin(theta1)) ** 1.5)
D = (easting - false_easting) / v1
# And finally the lat and long
lat = theta1 - (v1 * math.tan(theta1)) / ro1 * (
D*D/2.0 - (1.0 + 3.0 * T1) * ( (D**4) / 24.0 ) )
long = landa0 + (
D - T1 * (D**3) / 3.0 + (1 + 3.0 * T1) * T1 * (D**5) / 15.0
) / math.cos(theta1)
# Now make decimal versions
lat_dec = lat * 360.0 / 2.0 / math.pi
long_dec = long * 360.0 / 2.0 / math.pi
return (lat_dec,long_dec)
##############################################################
# OS Specific Methods Follow #
##############################################################
def turn_easting_northing_into_six_fig(easting,northing):
"""Turn OS easting and northing values into the six figure OS grid refecence. See http://www.jstott.me.uk/jscoord/"""
first_letter = ""
second_letter = ""
easting = int(easting)
northing = int(northing)
# Get the 100 km part
hkm_east = int( math.floor(easting / 100000.0) )
hkm_north = int( math.floor(northing / 100000.0) )
if hkm_north < 5:
if hkm_east < 5:
first_letter = "S"
else:
first_letter = "T"
elif hkm_north < 10:
if hkm_east < 5:
first_letter = "N"
else:
first_letter = "O"
else:
first_letter = "H"
# Get the 10km part
index = 65 + ((4 - (hkm_north % 5)) * 5) + (hkm_east % 5)
ti = index
if index >= 73:
index += 1
second_letter = chr(index)
# Get digits 2-4 on easting and northing
e = math.floor( (easting - (100000.0 * hkm_east)) / 100.0)
n = math.floor( (northing - (100000.0 * hkm_north)) / 100.0)
e = "%03d" % e
n = "%03d" % n
return first_letter + second_letter + e + n