package GIS::Distance::Formula::Vincenty; =head1 NAME GIS::Distance::Formula::Vincenty - Thaddeus Vincenty distance calculations. =head1 DESCRIPTION For the benefit of the terminally obsessive (as well as the genuinely needy), Thaddeus Vincenty devised formulae for calculating geodesic distances between a pair of latitude/longitude points on the earth's surface, using an accurate ellipsoidal model of the earth. Vincenty's formula is accurate to within 0.5mm, or 0.000015", on the ellipsoid being used. Calculations based on a spherical model, such as the (much simpler) Haversine, are accurate to around 0.3% (which is still good enough for most purposes, of course). Note: the accuracy quoted by Vincenty applies to the theoretical ellipsoid being used, which will differ (to varying degree) from the real earth geoid. If you happen to be located in Colorado, 2km above msl, distances will be 0.03% greater. In the UK, if you measure the distance from Land's End to John O' Groats using WGS-84, it will be 28m - 0.003% - greater than using the Airy ellipsoid, which provides a better fit for the UK. NOTE: This formula is still considered alpha quality in GIS::Distance. It has not been tested enough to be used in production. Normally this module is not used directly. Instead L<GIS::Distance> is used which in turn interfaces with the various formula classes. =head1 FORMULA a, b = major & minor semiaxes of the ellipsoid f = flattening (a-b)/a L = lon2 - lon1 u1 = atan((1-f) * tan(lat1)) u2 = atan((1-f) * tan(lat2)) sin_u1 = sin(u1) cos_u1 = cos(u1) sin_u2 = sin(u2) cos_u2 = cos(u2) lambda = L lambda_pi = 2PI while abs(lambda-lambda_pi) > 1e-12 sin_lambda = sin(lambda) cos_lambda = cos(lambda) sin_sigma = sqrt((cos_u2 * sin_lambda) * (cos_u2*sin_lambda) + (cos_u1*sin_u2-sin_u1*cos_u2*cos_lambda) * (cos_u1*sin_u2-sin_u1*cos_u2*cos_lambda)) cos_sigma = sin_u1*sin_u2 + cos_u1*cos_u2*cos_lambda sigma = atan2(sin_sigma, cos_sigma) alpha = asin(cos_u1 * cos_u2 * sin_lambda / sin_sigma) cos_sq_alpha = cos(alpha) * cos(alpha) cos2sigma_m = cos_sigma - 2*sin_u1*sin_u2/cos_sq_alpha cc = f/16*cos_sq_alpha*(4+f*(4-3*cos_sq_alpha)) lambda_pi = lambda lambda = L + (1-cc) * f * sin(alpha) * (sigma + cc*sin_sigma*(cos2sigma_m+cc*cos_sigma*(-1+2*cos2sigma_m*cos2sigma_m))) } usq = cos_sq_alpha*(a*a-b*b)/(b*b); aa = 1 + usq/16384*(4096+usq*(-768+usq*(320-175*usq))) bb = usq/1024 * (256+usq*(-128+usq*(74-47*usq))) delta_sigma = bb*sin_sigma*(cos2sigma_m+bb/4*(cos_sigma*(-1+2*cos2sigma_m*cos2sigma_m)- bb/6*cos2sigma_m*(-3+4*sin_sigma*sin_sigma)*(-3+4*cos2sigma_m*cos2sigma_m))) c = b*aa*(sigma-delta_sigma) =cut use Moose; extends 'GIS::Distance::Formula'; use Class::Measure::Length qw( length ); use Math::Trig qw( deg2rad pi tan atan asin ); =head1 METHODS =head2 distance This method is called by L<GIS::Distance>'s distance() method. =cut sub distance { my($self,$lat1,$lon1,$lat2,$lon2) = @_; return length(0,'km') if (($lon1==$lon2) and ($lat1==$lat2)); $lon1 = deg2rad($lon1); $lat1 = deg2rad($lat1); $lon2 = deg2rad($lon2); $lat2 = deg2rad($lat2); my($a,$b,$f) = (6378137,6356752.3142,1/298.257223563); my $l = $lon2 - $lon1; my $u1 = atan((1-$f) * tan($lat1)); my $u2 = atan((1-$f) * tan($lat2)); my $sin_u1 = sin($u1); my $cos_u1 = cos($u1); my $sin_u2 = sin($u2); my $cos_u2 = cos($u2); my $lambda = $l; my $lambda_pi = 2 * pi; my $iter_limit = 20; my($cos_sq_alpha,$sin_sigma,$cos2sigma_m,$cos_sigma,$sigma) = (0,0,0,0,0); while( abs($lambda-$lambda_pi) > 1e-12 && --$iter_limit>0 ){ my $sin_lambda = sin($lambda); my $cos_lambda = cos($lambda); $sin_sigma = sqrt(($cos_u2*$sin_lambda) * ($cos_u2*$sin_lambda) + ($cos_u1*$sin_u2-$sin_u1*$cos_u2*$cos_lambda) * ($cos_u1*$sin_u2-$sin_u1*$cos_u2*$cos_lambda)); $cos_sigma = $sin_u1*$sin_u2 + $cos_u1*$cos_u2*$cos_lambda; $sigma = atan2($sin_sigma, $cos_sigma); my $alpha = asin($cos_u1 * $cos_u2 * $sin_lambda / $sin_sigma); $cos_sq_alpha = cos($alpha) * cos($alpha); $cos2sigma_m = $cos_sigma - 2*$sin_u1*$sin_u2/$cos_sq_alpha; my $cc = $f/16*$cos_sq_alpha*(4+$f*(4-3*$cos_sq_alpha)); $lambda_pi = $lambda; $lambda = $l + (1-$cc) * $f * sin($alpha) * ($sigma + $cc*$sin_sigma*($cos2sigma_m+$cc*$cos_sigma*(-1+2*$cos2sigma_m*$cos2sigma_m))); } my $usq = $cos_sq_alpha*($a*$a-$b*$b)/($b*$b); my $aa = 1 + $usq/16384*(4096+$usq*(-768+$usq*(320-175*$usq))); my $bb = $usq/1024 * (256+$usq*(-128+$usq*(74-47*$usq))); my $delta_sigma = $bb*$sin_sigma*($cos2sigma_m+$bb/4*($cos_sigma*(-1+2*$cos2sigma_m*$cos2sigma_m)- $bb/6*$cos2sigma_m*(-3+4*$sin_sigma*$sin_sigma)*(-3+4*$cos2sigma_m*$cos2sigma_m))); my $c = $b*$aa*($sigma-$delta_sigma); return length( $c, 'm' ); } 1; __END__ =head1 SEE ALSO L<GIS::Distanc> L<GIS::Distance::Formula::Vincenty::Fast> =head1 RESOURCES L<http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf> L<http://www.movable-type.co.uk/scripts/LatLongVincenty.html> =head1 AUTHOR Aran Clary Deltac <bluefeet@cpan.org> =head1 LICENSE This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself.