| 1 | # Geographical helper functions for nmea_info.py and friends |
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| 2 | # |
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| 3 | # Helps with geographic functions, including: |
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| 4 | # Lat+Long+Height -> XYZ |
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| 5 | # XYZ -> Lat+Long+Height |
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| 6 | # Lat+Long -> other Lat+Long (Helmert Transform) |
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| 7 | # Lat+Long -> easting/northing (OS Only) |
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| 8 | # easting/northing -> Lat+Long (OS Only) |
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| 9 | # OS easting/northing -> OS 6 figure ref |
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| 10 | # |
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| 11 | # See http://gagravarr.org/code/ for updates and information |
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| 12 | # |
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| 13 | # GPL |
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| 14 | # |
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| 15 | # Nick Burch - v0.03 (25/06/2006) |
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| 16 | |
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| 17 | import math |
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| 18 | |
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| 19 | # The earth's radius, in meters, at the equator (should be close enough) |
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| 20 | earths_radius = 6335.437 * 1000 |
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| 21 | |
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| 22 | # For each co-ordinate system we do, what are the A, B and E2 values? |
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| 23 | # List is A, B, E^2 (E^2 calculated after) |
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| 24 | abe_values = { |
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| 25 | 'wgs84': [ 6378137.0, 6356752.3141, -1 ], |
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| 26 | 'osgb' : [ 6377563.396, 6356256.91, -1 ], |
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| 27 | 'osie' : [ 6377340.189, 6356034.447, -1 ] |
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| 28 | } |
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| 29 | |
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| 30 | # Calculate the E2 values |
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| 31 | for system in abe_values.keys(): |
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| 32 | a = abe_values[system][0] |
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| 33 | b = abe_values[system][1] |
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| 34 | e2 = (a*a - b*b) / (a*a) |
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| 35 | abe_values[system][2] = e2 |
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| 36 | |
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| 37 | # For each co-ordinate system we can translate between, what are |
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| 38 | # the tx, ty, tz, s, rx, ry and rz values? |
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| 39 | # List is tx, ty, tz, s, rx, ry, rz |
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| 40 | transform_values = { |
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| 41 | 'wgs84_to_osgb' : [ -446.448, 125.157, -542.060, |
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| 42 | 20.4894 / 1000.0 / 1000.0, # given as ppm |
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| 43 | -0.1502 / 206265.0, # given as seconds of arc |
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| 44 | -0.2470 / 206265.0, # given as seconds of arc |
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| 45 | -0.8421 / 206265.0 # given as seconds of arc |
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| 46 | ], |
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| 47 | 'wgs84_to_osie' : [ -482.530, 130.596, -564.557, |
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| 48 | -8.1500 / 1000.0 / 1000.0, # given as ppm |
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| 49 | -1.0420 / 206265.0, # given as seconds of arc |
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| 50 | -0.2140 / 206265.0, # given as seconds of arc |
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| 51 | -0.6310 / 206265.0 # given as seconds of arc |
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| 52 | ], |
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| 53 | 'itrs2000_to_etrs89' : [ 0.054, 0.051, -0.048, 0, |
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| 54 | 0.000081 / 206265.0, # given as seconds of arc |
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| 55 | 0.00049 / 206265.0, # given as seconds of arc |
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| 56 | 0.000792 / 206265.0 # given as seconds of arc |
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| 57 | ] |
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| 58 | } |
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| 59 | |
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| 60 | # Calculate reverse transforms |
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| 61 | for systems in [('wgs84','osgb'), ('wgs84','osie'), ('itrs2000','etrs89')]: |
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| 62 | fs = systems[0] + "_to_" + systems[1] |
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| 63 | rs = systems[1] + "_to_" + systems[0] |
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| 64 | ra = [] |
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| 65 | for val in transform_values[fs]: |
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| 66 | ra.append(-1.0 * val) |
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| 67 | transform_values[rs] = ra |
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| 68 | |
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| 69 | # Easting and Northin system values, for the systems we work with. |
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| 70 | # List is n0, e0, F0, theta0 and landa0 |
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| 71 | en_values = { |
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| 72 | 'osgb' : [ -100000.0, 400000.0, 0.9996012717, |
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| 73 | 49.0 /360.0 *2.0*math.pi, |
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| 74 | -2.0 /360.0 *2.0*math.pi |
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| 75 | ], |
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| 76 | 'osie' : [ 250000.0, 200000.0, 1.000035, |
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| 77 | 53.5 /360.0 *2.0*math.pi, |
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| 78 | -8.0 /360.0 *2.0*math.pi |
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| 79 | ] |
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| 80 | } |
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| 81 | |
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| 82 | ############################################################## |
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| 83 | # OS Specific Helpers for Generic Methods # |
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| 84 | ############################################################## |
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| 85 | |
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| 86 | def turn_wgs84_into_osgb36(lat_dec,long_dec,height): |
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| 87 | """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.""" |
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| 88 | |
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| 89 | wgs84_xyz = turn_llh_into_xyz(lat_dec,long_dec,height,'wgs84') |
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| 90 | |
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| 91 | osgb_xyz = turn_xyz_into_other_xyz( |
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| 92 | wgs84_xyz[0],wgs84_xyz[1],wgs84_xyz[2],'wgs84','osgb') |
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| 93 | |
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| 94 | osgb_latlong = turn_xyz_into_llh( |
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| 95 | osgb_xyz[0],osgb_xyz[1],osgb_xyz[2],'osgb') |
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| 96 | |
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| 97 | return osgb_latlong |
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| 98 | def turn_osgb36_into_wgs84(lat_dec,long_dec,height): |
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| 99 | """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.""" |
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| 100 | |
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| 101 | osgb_xyz = turn_llh_into_xyz(lat_dec,long_dec,height,'osgb') |
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| 102 | |
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| 103 | wgs84_xyz = turn_xyz_into_other_xyz( |
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| 104 | osgb_xyz[0],osgb_xyz[1],osgb_xyz[2],'osgb','wgs84') |
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| 105 | |
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| 106 | wgs84_latlong = turn_xyz_into_llh( |
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| 107 | wgs84_xyz[0],wgs84_xyz[1],wgs84_xyz[2],'wgs84') |
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| 108 | |
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| 109 | return wgs84_latlong |
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| 110 | |
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| 111 | def turn_osgb36_into_eastingnorthing(lat_dec,long_dec): |
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| 112 | """Turn OSGB36 (decimal) lat/long values into OS easting and northing values.""" |
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| 113 | return turn_latlong_into_eastingnorthing(lat_dec,long_dec,'osgb') |
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| 114 | |
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| 115 | def turn_eastingnorthing_into_osgb36(easting,northing): |
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| 116 | """Turn OSGB36 easting and northing values into (decimal) lat/long values inOSGB36.""" |
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| 117 | return turn_eastingnorthing_into_latlong(easting,northing,'osgb') |
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| 118 | |
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| 119 | ############################################################## |
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| 120 | # Generic Transform Functions # |
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| 121 | ############################################################## |
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| 122 | |
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| 123 | def turn_llh_into_xyz(lat_dec,long_dec,height,system): |
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| 124 | """Convert Lat, Long and Height into 3D Cartesian x,y,z |
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| 125 | See http://www.ordnancesurvey.co.uk/gps/docs/convertingcoordinates3D.pdf""" |
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| 126 | |
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| 127 | a = abe_values[system][0] |
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| 128 | b = abe_values[system][1] |
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| 129 | e2 = abe_values[system][2] |
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| 130 | |
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| 131 | theta = float(lat_dec) / 360.0 * 2.0 * math.pi |
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| 132 | landa = float(long_dec) / 360.0 * 2.0 * math.pi |
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| 133 | height = float(height) |
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| 134 | |
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| 135 | v = a / math.sqrt( 1.0 - e2 * (math.sin(theta) * math.sin(theta)) ) |
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| 136 | x = (v + height) * math.cos(theta) * math.cos(landa) |
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| 137 | y = (v + height) * math.cos(theta) * math.sin(landa) |
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| 138 | z = ( (1.0 - e2) * v + height ) * math.sin(theta) |
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| 139 | |
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| 140 | return [x,y,z] |
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| 141 | |
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| 142 | def turn_xyz_into_llh(x,y,z,system): |
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| 143 | """Convert 3D Cartesian x,y,z into Lat, Long and Height |
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| 144 | See http://www.ordnancesurvey.co.uk/gps/docs/convertingcoordinates3D.pdf""" |
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| 145 | |
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| 146 | a = abe_values[system][0] |
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| 147 | b = abe_values[system][1] |
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| 148 | e2 = abe_values[system][2] |
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| 149 | |
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| 150 | p = math.sqrt(x*x + y*y) |
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| 151 | |
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| 152 | long = math.atan(y/x) |
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| 153 | lat_init = math.atan( z / (p * (1.0 - e2)) ) |
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| 154 | v = a / math.sqrt( 1.0 - e2 * (math.sin(lat_init) * math.sin(lat_init)) ) |
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| 155 | lat = math.atan( (z + e2*v*math.sin(lat_init)) / p ) |
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| 156 | |
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| 157 | height = (p / math.cos(lat)) - v # Ignore if a bit out |
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| 158 | |
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| 159 | # Turn from radians back into degrees |
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| 160 | long = long / 2 / math.pi * 360 |
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| 161 | lat = lat / 2 / math.pi * 360 |
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| 162 | |
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| 163 | return [lat,long,height] |
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| 164 | |
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| 165 | def turn_xyz_into_other_xyz(old_x,old_y,old_z,from_scheme,to_scheme): |
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| 166 | """Helmert Transformation between one lat+long system and another |
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| 167 | 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""" |
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| 168 | |
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| 169 | transform = from_scheme + "_to_" + to_scheme |
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| 170 | tx = transform_values[transform][0] |
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| 171 | ty = transform_values[transform][1] |
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| 172 | tz = transform_values[transform][2] |
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| 173 | s = transform_values[transform][3] |
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| 174 | rx = transform_values[transform][4] |
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| 175 | ry = transform_values[transform][5] |
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| 176 | rz = transform_values[transform][6] |
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| 177 | |
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| 178 | # Do the transform |
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| 179 | new_x = tx + ((1.0+s) * old_x) + (-rz * old_y) + (ry * old_z) |
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| 180 | new_y = ty + (rz * old_x) + ((1.0+s) * old_y) + (-rx * old_z) |
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| 181 | new_z = tz + (-ry * old_x) + (rx * old_y) + ((1.0+s) * old_z) |
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| 182 | |
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| 183 | return [new_x,new_y,new_z] |
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| 184 | |
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| 185 | def calculate_distance_and_bearing(from_lat_dec,from_long_dec,to_lat_dec,to_long_dec): |
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| 186 | """Uses the spherical law of cosines to calculate the distance and bearing between two positions""" |
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| 187 | |
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| 188 | # Turn them all into radians |
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| 189 | from_theta = float(from_lat_dec) / 360.0 * 2.0 * math.pi |
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| 190 | from_landa = float(from_long_dec) / 360.0 * 2.0 * math.pi |
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| 191 | to_theta = float(to_lat_dec) / 360.0 * 2.0 * math.pi |
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| 192 | to_landa = float(to_long_dec) / 360.0 * 2.0 * math.pi |
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| 193 | |
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| 194 | d = math.acos( |
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| 195 | math.sin(from_theta) * math.sin(to_theta) + |
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| 196 | math.cos(from_theta) * math.cos(to_theta) * math.cos(to_landa-from_landa) |
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| 197 | ) * earths_radius |
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| 198 | |
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| 199 | bearing = math.atan2( |
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| 200 | math.sin(to_landa-from_landa) * math.cos(to_theta), |
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| 201 | math.cos(from_theta) * math.sin(to_theta) - |
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| 202 | math.sin(from_theta) * math.cos(to_theta) * math.cos(to_landa-from_landa) |
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| 203 | ) |
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| 204 | bearing = bearing / 2.0 / math.pi * 360.0 |
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| 205 | |
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| 206 | return [d,bearing] |
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| 207 | |
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| 208 | ############################################################## |
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| 209 | # Easting/Northing Transform Methods # |
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| 210 | ############################################################## |
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| 211 | |
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| 212 | def turn_latlong_into_eastingnorthing(lat_dec,long_dec,scheme): |
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| 213 | """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.""" |
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| 214 | |
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| 215 | n0 = en_values[scheme][0] |
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| 216 | e0 = en_values[scheme][1] |
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| 217 | f0 = en_values[scheme][2] |
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| 218 | |
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| 219 | theta0 = en_values[scheme][3] |
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| 220 | landa0 = en_values[scheme][4] |
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| 221 | |
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| 222 | a = abe_values[scheme][0] |
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| 223 | b = abe_values[scheme][1] |
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| 224 | e2 = abe_values[scheme][2] |
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| 225 | |
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| 226 | theta = float(lat_dec) /360.0 *2.0*math.pi |
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| 227 | landa = float(long_dec) /360.0 *2.0*math.pi |
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| 228 | |
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| 229 | n = (a-b) / (a+b) |
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| 230 | v = a * f0 * math.pow( (1 - e2 * math.sin(theta)*math.sin(theta)), -0.5 ) |
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| 231 | ro = a * f0 * (1 - e2) * math.pow( (1 - e2 * math.sin(theta)*math.sin(theta)), -1.5 ) |
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| 232 | nu2 = v/ro - 1 |
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| 233 | |
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| 234 | M = b * f0 * ( \ |
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| 235 | (1.0 + n + 5.0/4.0 *n*n + 5.0/4.0 *n*n*n) * (theta-theta0) - \ |
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| 236 | (3.0*n + 3.0*n*n + 21.0/8.0 *n*n*n) *math.sin(theta-theta0) *math.cos(theta+theta0) + \ |
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| 237 | (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)) - \ |
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| 238 | 35.0/24.0*n*n*n *math.sin(3.0*(theta-theta0)) *math.cos(3.0*(theta+theta0)) \ |
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| 239 | ) |
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| 240 | |
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| 241 | I = M + n0 |
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| 242 | II = v/2.0 * math.sin(theta) * math.cos(theta) |
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| 243 | III = v/24.0 * math.sin(theta) * math.pow( math.cos(theta),3 ) * \ |
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| 244 | (5.0 - math.pow(math.tan(theta),2) + 9.0*nu2) |
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| 245 | IIIa = v/720.0 * math.sin(theta) * math.pow( math.cos(theta),5 ) * \ |
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| 246 | ( 61.0 - 58.0 *math.pow(math.tan(theta),2) + math.pow(math.tan(theta),4) ) |
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| 247 | IV = v * math.cos(theta) |
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| 248 | V = v/6.0 * math.pow( math.cos(theta),3 ) * \ |
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| 249 | ( v/ro - math.pow(math.tan(theta),2) ) |
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| 250 | VI = v/120.0 * math.pow(math.cos(theta),5) * \ |
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| 251 | ( 5.0 - 18.0 *math.pow(math.tan(theta),2) + \ |
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| 252 | math.pow(math.tan(theta),4) + 14.0*nu2 - \ |
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| 253 | 58.0 * math.pow(math.tan(theta),2)*nu2 ) |
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| 254 | |
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| 255 | northing = I + II*math.pow(landa-landa0,2) + \ |
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| 256 | III*math.pow(landa-landa0,4) + \ |
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| 257 | IIIa*math.pow(landa-landa0,6) |
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| 258 | easting = e0 + IV*(landa-landa0) + V*math.pow(landa-landa0,3) + \ |
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| 259 | VI*math.pow(landa-landa0,5) |
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| 260 | |
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| 261 | return (easting,northing) |
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| 262 | |
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| 263 | def turn_eastingnorthing_into_latlong(easting,northing,scheme): |
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| 264 | """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.""" |
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| 265 | |
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| 266 | n0 = en_values[scheme][0] |
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| 267 | e0 = en_values[scheme][1] |
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| 268 | f0 = en_values[scheme][2] |
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| 269 | |
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| 270 | theta0 = en_values[scheme][3] |
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| 271 | landa0 = en_values[scheme][4] |
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| 272 | |
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| 273 | a = abe_values[scheme][0] |
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| 274 | b = abe_values[scheme][1] |
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| 275 | e2 = abe_values[scheme][2] |
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| 276 | |
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| 277 | n = (a-b) / (a+b) |
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| 278 | |
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| 279 | # Prepare to iterate |
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| 280 | M = 0 |
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| 281 | theta = theta0 |
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| 282 | # Iterate, 4 times should be enough |
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| 283 | for i in range(4): |
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| 284 | theta = ((northing - n0 - M) / (a * f0)) + theta |
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| 285 | M = b * f0 * ( \ |
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| 286 | (1.0 + n + 5.0/4.0 *n*n + 5.0/4.0 *n*n*n) * (theta-theta0) - \ |
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| 287 | (3.0*n + 3.0*n*n + 21.0/8.0 *n*n*n) *math.sin(theta-theta0) *math.cos(theta+theta0) + \ |
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| 288 | (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)) - \ |
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| 289 | 35.0/24.0*n*n*n *math.sin(3.0*(theta-theta0)) *math.cos(3.0*(theta+theta0)) \ |
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| 290 | ) |
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| 291 | |
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| 292 | # Compute intermediate values |
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| 293 | v = a * f0 * math.pow( (1 - e2 * math.sin(theta)*math.sin(theta)), -0.5 ) |
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| 294 | ro = a * f0 * (1 - e2) * math.pow( (1 - e2 * math.sin(theta)*math.sin(theta)), -1.5 ) |
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| 295 | nu2 = v/ro - 1 |
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| 296 | tantheta2 = math.pow(math.tan(theta),2) |
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| 297 | |
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| 298 | VII = math.tan(theta) / (2 * ro * v) |
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| 299 | VIII = math.tan(theta) / (24 * ro * math.pow(v,3)) \ |
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| 300 | * (5 + 3 * tantheta2 + nu2 - \ |
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| 301 | 9 * tantheta2 * nu2 ) |
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| 302 | IX = math.tan(theta) / (720 * ro * math.pow(v,5)) \ |
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| 303 | * (61 + 90 * tantheta2 + 45 * tantheta2 * tantheta2) |
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| 304 | X = 1 / (math.cos(theta) * v) |
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| 305 | XI = 1 / (math.cos(theta) * 6 * math.pow(v,3)) \ |
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| 306 | * (v/ro + 2*tantheta2) |
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| 307 | XII = 1 / (math.cos(theta) * 120 * math.pow(v,5)) \ |
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| 308 | * (5 + 28 * tantheta2 + 24 * tantheta2 * tantheta2) |
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| 309 | XIIa = 1 / (math.cos(theta) * 5040 * math.pow(v,7)) \ |
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| 310 | * (61 + 662 * tantheta2 + 1320 * tantheta2 * tantheta2 \ |
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| 311 | + 720 * tantheta2 * tantheta2 * tantheta2) |
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| 312 | |
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| 313 | lat_rad = theta - VII * math.pow((easting-e0),2) \ |
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| 314 | + VIII * math.pow((easting-e0),4) \ |
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| 315 | - IX * math.pow((easting-e0),6) |
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| 316 | long_rad = landa0 + X * (easting-e0) \ |
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| 317 | - XI * math.pow((easting-e0),3) \ |
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| 318 | + XII * math.pow((easting-e0),5) \ |
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| 319 | - XIIa * math.pow((easting-e0),7) |
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| 320 | |
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| 321 | lat = lat_rad / 2.0 / math.pi * 360.0 |
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| 322 | long = long_rad / 2.0 / math.pi * 360.0 |
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| 323 | |
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| 324 | return (lat,long) |
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| 325 | |
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| 326 | ############################################################## |
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| 327 | # OS Specific Methods Follow # |
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| 328 | ############################################################## |
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| 329 | |
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| 330 | def turn_easting_northing_into_six_fig(easting,northing): |
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| 331 | """Turn OS easting and northing values into the six figure OS grid refecence. See http://www.jstott.me.uk/jscoord/""" |
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| 332 | first_letter = "" |
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| 333 | second_letter = "" |
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| 334 | |
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| 335 | easting = int(easting) |
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| 336 | northing = int(northing) |
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| 337 | |
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| 338 | # Get the 100 km part |
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| 339 | hkm_east = int( math.floor(easting / 100000.0) ) |
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| 340 | hkm_north = int( math.floor(northing / 100000.0) ) |
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| 341 | if hkm_north < 5: |
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| 342 | if hkm_east < 5: |
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| 343 | first_letter = "S" |
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| 344 | else: |
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| 345 | first_letter = "T" |
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| 346 | elif hkm_north < 10: |
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| 347 | if hkm_east < 5: |
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| 348 | first_letter = "N" |
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| 349 | else: |
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| 350 | first_letter = "O" |
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| 351 | else: |
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| 352 | first_letter = "H" |
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| 353 | |
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| 354 | # Get the 10km part |
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| 355 | index = 65 + ((4 - (hkm_north % 5)) * 5) + (hkm_east % 5) |
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| 356 | ti = index |
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| 357 | if index >= 73: |
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| 358 | index += 1 |
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| 359 | second_letter = chr(index) |
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| 360 | |
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| 361 | # Get digits 2-4 on easting and northing |
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| 362 | e = math.floor( (easting - (100000.0 * hkm_east)) / 100.0) |
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| 363 | n = math.floor( (northing - (100000.0 * hkm_north)) / 100.0) |
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| 364 | e = "%03d" % e |
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| 365 | n = "%03d" % n |
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| 366 | |
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| 367 | return first_letter + second_letter + e + n |
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