1 | #!/usr/bin/perl |
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2 | # |
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3 | # Geo/HelmertTransform.pm: |
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4 | # Perform "Helmert" (linear) transformations between coordinates referenced to |
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5 | # different datums. |
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6 | # |
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7 | # Reference: |
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8 | # http://www.gps.gov.uk/additionalInfo/images/A_guide_to_coord.pdf |
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9 | # |
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10 | # Copyright (c) 2005 UK Citizens Online Democracy. This module is free |
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11 | # software; you can redistribute it and/or modify it under the same terms as |
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12 | # Perl itself. |
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13 | # |
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14 | # Email: chris@mysociety.org; WWW: http://www.mysociety.org/ |
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15 | # |
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16 | # $Id: HelmertTransform.pm,v 1.6 2006/06/21 17:10:24 francis Exp $ |
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17 | # |
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18 | |
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19 | package Geo::HelmertTransform; |
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20 | |
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21 | use strict; |
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22 | |
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23 | =head1 NAME |
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24 | |
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25 | Geo::HelmertTransform |
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26 | |
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27 | =head1 SYNOPSIS |
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28 | |
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29 | use Geo::HelmertTransform; |
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30 | |
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31 | my ($lat, $lon, $height) = ...; # from OS map |
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32 | my $airy1830 = Geo::HelmertTransform::datum('Airy1830'); |
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33 | my $wgs84 = Geo::HelmertTransform::datum('WGS84'); |
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34 | |
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35 | ($lat, $lon, $height) |
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36 | = Geo::HelmertTransform::convert_datum($airy1830, $wgs84, |
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37 | $lat, $lon, $h); |
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38 | |
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39 | |
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40 | =head1 DESCRIPTION |
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41 | |
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42 | Perform transformations between geographical coordinates in different datums. |
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43 | |
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44 | It is usual to describe geographical points in terms of their polar coordinates |
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45 | (latitude, longitude and altitude) referenced to a "datum ellipsoid", which is |
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46 | used to approximate the Earth's geoid. The latitude, longitude and altitude of |
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47 | a given physical point vary depending on which datum ellipsoid is in use. |
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48 | Unfortunately, a number of ellipsoids are in everyday use, and so it is often |
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49 | necessary to transform geographical coordinates between different datum |
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50 | ellipsoids. |
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51 | |
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52 | Two different datum ellipsoids may differ in the locations of their centers, or |
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53 | in their shape; and there may be an angle between their equatorial planes or |
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54 | the meridians relative to which longitude is measured. The Helmert Transform, |
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55 | which this module implements, is a linear transformation of coordinates between |
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56 | pairs of datum ellipsoids in the limit of small angles of deviation between |
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57 | them. |
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58 | |
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59 | =head1 CONVENTIONS |
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60 | |
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61 | Latitude is expressed in degrees, positive-north; longitude in degrees, |
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62 | positive-east. Heights (ellipsoid) and cartesian coordinates are in meters. |
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63 | |
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64 | =head1 FUNCTIONS |
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65 | |
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66 | =over 4 |
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67 | |
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68 | =cut |
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69 | |
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70 | use constant M_PI => 3.141592654; |
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71 | |
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72 | =item rad_to_deg RADIANS |
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73 | |
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74 | Convert RADIANS to degrees. |
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75 | |
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76 | =cut |
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77 | sub rad_to_deg ($) { |
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78 | return 180. * $_[0] / M_PI; |
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79 | } |
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80 | |
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81 | =item deg_to_rad DEGREES |
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82 | |
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83 | Convert DEGREES to radians. |
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84 | |
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85 | =cut |
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86 | sub deg_to_rad ($) { |
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87 | return M_PI * $_[0] / 180.; |
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88 | } |
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89 | |
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90 | =item geo_to_xyz DATUM LAT LON H |
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91 | |
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92 | Return the Cartesian (X, Y, Z) coordinates for the geographical coordinates |
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93 | (LAT, LON, H) in the given DATUM. |
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94 | |
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95 | =cut |
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96 | sub geo_to_xyz ($$$$) { |
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97 | my ($datum, $lat, $lon, $h) = @_; |
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98 | $lat = deg_to_rad($lat); |
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99 | $lon = deg_to_rad($lon); |
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100 | |
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101 | my $v = $datum->a() / sqrt(1 - $datum->e2() * sin($lat) ** 2); |
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102 | return ( |
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103 | ($v + $h) * cos($lat) * cos($lon), |
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104 | ($v + $h) * cos($lat) * sin($lon), |
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105 | ((1 - $datum->e2()) * $v + $h) * sin($lat) |
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106 | ); |
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107 | } |
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108 | |
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109 | =item xyz_to_geo DATUM X Y Z |
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110 | |
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111 | Return the geographical (LAT, LON, H) coordinates for the Cartesian coordinates |
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112 | (X, Y, Z) in the given DATUM. This is an iterative procedure. |
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113 | |
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114 | =cut |
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115 | sub xyz_to_geo ($$$$) { |
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116 | my ($datum, $x, $y, $z) = @_; |
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117 | my ($lat, $lat2, $lon, $h, $v, $p); |
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118 | $lon = atan2($y, $x); |
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119 | |
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120 | $p = sqrt($x**2 + $y**2); |
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121 | $lat2 = atan2($z, $p); |
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122 | |
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123 | my $niter = 0; |
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124 | do { |
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125 | $lat = $lat2; |
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126 | $v = $datum->a() / sqrt(1 - $datum->e2() * sin($lat) ** 2); |
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127 | $lat2 = atan2(($z + $datum->e2() * $v * sin($lat)), $p); |
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128 | die "exceeded 10000 iterations without converging in Geo::HelmertTransform::xyz_to_geo" |
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129 | if (++$niter > 10000); |
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130 | } while (abs($lat2 - $lat) > 2e-6); # about 1/10000 mile |
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131 | |
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132 | $h = $p / cos($lat) - $v; |
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133 | |
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134 | return (rad_to_deg($lat), rad_to_deg($lon), $h); |
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135 | } |
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136 | |
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137 | =item convert_datum D1 D2 LAT LON H |
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138 | |
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139 | Given geographical coordinates (LAT, LON, H) in datum D1, return the |
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140 | corresponding coordinates in datum D2. This assumes that the transformations |
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141 | are small, and always converts via WGS84. |
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142 | |
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143 | =cut |
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144 | sub convert_datum ($$$$$) { |
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145 | my ($d1, $d2, $lat, $lon, $h) = @_; |
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146 | my ($x1, $y1, $z1) = geo_to_xyz($d1, $lat, $lon, $h); |
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147 | my ($x, $y, $z) = ($x1, $y1, $z1); |
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148 | if (!$d1->is_wgs84()) { |
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149 | # Transform into WGS84. |
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150 | $x = $d1->tx() |
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151 | + (1 + $d1->s()) * $x1 |
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152 | - $d1->rz() * $y1 |
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153 | + $d1->ry() * $z1; |
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154 | $y = $d1->ty() |
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155 | + $d1->rz() * $x1 |
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156 | + (1 + $d1->s()) * $y1 |
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157 | - $d1->rx() * $z1; |
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158 | $z = $d1->tz() |
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159 | - $d1->ry() * $x1 |
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160 | + $d1->rx() * $y1 |
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161 | + (1 + $d1->s()) * $z1; |
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162 | } |
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163 | |
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164 | my ($x2, $y2, $z2) = ($x, $y, $z); |
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165 | if (!$d2->is_wgs84()) { |
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166 | $x2 = -$d2->tx() |
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167 | + (1 - $d2->s()) * $x |
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168 | + $d2->rz() * $y |
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169 | - $d2->ry() * $z; |
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170 | $y2 = -$d2->ty() |
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171 | - $d2->rz() * $x |
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172 | + (1 - $d2->s()) * $y |
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173 | + $d2->rx() * $z; |
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174 | $z2 = -$d2->tz() |
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175 | + $d2->ry() * $x |
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176 | - $d2->rx() * $y |
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177 | + (1 - $d2->s()) * $z; |
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178 | } |
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179 | |
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180 | return xyz_to_geo($d2, $x2, $y2, $z2); |
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181 | } |
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182 | |
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183 | =item datum NAME |
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184 | |
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185 | Return the datum of the given NAME. Currently implemented are: |
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186 | |
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187 | =over 4 |
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188 | |
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189 | =item Airy1830 |
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190 | |
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191 | The 1830 Airy ellipsoid to which the British Ordnance Survey's National Grid is |
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192 | referenced. |
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193 | |
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194 | =item Airy1830Modified |
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195 | |
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196 | The modified 1830 Airy ellipsoid to which the Irish Grid (as used by Ordnance |
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197 | Survey Ireland and Ordnance Survey Northern Ireland); also known as the Ireland |
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198 | 1975 datum. |
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199 | |
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200 | =item WGS84 |
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201 | |
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202 | The global datum used for GPS. |
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203 | |
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204 | =back |
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205 | |
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206 | =cut |
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207 | sub datum ($) { |
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208 | return new Geo::HelmertTransform::Datum(Name => $_[0]); |
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209 | } |
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210 | |
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211 | =back |
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212 | |
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213 | =cut |
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214 | |
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215 | # Datum class for internal use (alternative spelling: "I can't be bothered to |
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216 | # document it now"). |
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217 | package Geo::HelmertTransform::Datum; |
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218 | |
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219 | use fields qw(name a b e2 tx ty tz s rx ry rz is_wgs84); |
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220 | |
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221 | # Fields are: semi-major and -minor axes; and the x-, y-, and z-displacements, |
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222 | # scale change, and rotations to transform from this datum into WGS84. |
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223 | # |
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224 | # a (m) b tx ty tz s (ppm) rx (sec) ry rz |
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225 | # -------------- --------------- --------- --------- --------- --------- -------- -------- ------- |
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226 | my %known_datums = ( |
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227 | # from OS article above |
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228 | Airy1830 => [6_377_563.396, 6_356_256.910, +446.448, -125.157, +542.060, -20.4894, +0.1502, +0.2470, +0.8421], |
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229 | # from http://www.osni.gov.uk/downloads/Making%20maps%20GPS%20compatible.pdf |
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230 | Airy1830Modified => [6_377_340.189, 6_356_034.447, +482.530, -130.596, +564.557, +8.150, +1.042, +0.214, +0.631], |
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231 | # International1924 => [6_378_388.000, 6_356_911.946, ??? ], |
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232 | WGS84 => [6_378_137.000, 6_356_752.3141, 0.000, 0.000, 0.000, 0.0000, 0.0000, 0.0000, 0.0000] |
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233 | ); |
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234 | |
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235 | sub new ($%) { |
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236 | my ($class, %p) = @_; |
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237 | if (exists($p{Name})) { |
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238 | die "datum \"$p{Name}\" not known" |
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239 | if (!exists($known_datums{$p{Name}})); |
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240 | |
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241 | # Take a copy of the datum data |
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242 | my $reald = $known_datums{$p{Name}}; |
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243 | my $d; |
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244 | foreach my $df (@$reald) { |
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245 | push @$d,$df; |
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246 | } |
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247 | |
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248 | my $s = fields::new($class); |
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249 | foreach (qw(a b tx ty tz)) { |
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250 | $s->{$_} = shift(@$d); |
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251 | } |
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252 | $s->{s} = shift(@$d) / 1_000_000; # ppm |
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253 | foreach (qw(rx ry rz)) { |
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254 | $s->{$_} = Geo::HelmertTransform::deg_to_rad(shift(@$d) / 3600.); # seconds |
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255 | } |
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256 | $s->{is_wgs84} = ($p{Name} eq 'WGS84'); |
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257 | return $s; |
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258 | } elsif (!exists($p{a}) || !exists($p{b})) { |
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259 | die "must specify semi-major axis a and semi-minor axis b"; |
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260 | } else { |
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261 | my $s = fields::new($class); |
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262 | foreach (qw(a b tx ty tz s rx ry rz)) { |
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263 | $s->{$_} = 0; |
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264 | $s->{$_} = $p{$_} if (exists($p{$_})); |
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265 | } |
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266 | $s->{is_wgs84} = 0; |
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267 | return $s; |
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268 | } |
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269 | } |
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270 | |
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271 | foreach (qw(a b tx ty tz s rx ry rz is_wgs84)) { |
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272 | eval <<EOF; |
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273 | sub $_ (\$) { |
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274 | return \$_[0]->{$_}; |
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275 | } |
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276 | EOF |
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277 | } |
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278 | |
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279 | sub e2 ($) { |
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280 | my $s = shift; |
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281 | if (!exists($_[0]->{e2})) { |
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282 | if($s->a() == 0) { |
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283 | $s->{e2} = 1; |
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284 | } else { |
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285 | $s->{e2} = 1 - ($s->b() / $s->a()) ** 2; |
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286 | } |
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287 | } |
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288 | return $s->{e2} |
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289 | } |
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290 | |
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291 | =head1 SEE ALSO |
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292 | |
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293 | I<A guide to coordinate systems in Great Britain>, |
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294 | http://www.gps.gov.uk/guidecontents.asp |
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295 | |
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296 | I<Making maps compatible with GPS>, |
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297 | http://www.osni.gov.uk/downloads/Making%20maps%20GPS%20compatible.pdf |
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298 | |
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299 | =head1 AUTHOR AND COPYRIGHT |
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300 | |
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301 | Written by Chris Lightfoot, chris@mysociety.org |
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302 | |
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303 | Copyright (c) UK Citizens Online Democracy. |
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304 | |
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305 | This module is free software; you can redistribute it and/or modify it under |
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306 | the same terms as Perl itself. |
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307 | |
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308 | =head1 VERSION |
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309 | |
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310 | $Id: HelmertTransform.pm,v 1.6 2006/06/21 17:10:24 francis Exp $ |
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311 | |
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312 | =cut |
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313 | |
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314 | 1; |
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