1 Geodesic Distance Between Otia Maizuru High School and National Hualien Girls Senior High School by Rui-Ling Hsu Serena Wan December 15, 2010 2 How Far Are We Apart? (131.61E, 33.23N) OMHS (121.61E, 23.97N) HLGS (x A, y A ) OMHS (x B, y B ) HLGS Shortest distance between points A and B on a plane = 3 Latitude and Longitude 60 Latitude and Longitude being angular distances Cartesian co-ordinates Longitude: ang. dist. from the Prime Meridian Latitude: ang. dist. from the Equator Distance between two points on and along the earth surface geodesic distance 4 Research Objectives and Methodology Research Objectives To understand the concept of geodesic distance To calculate the geodesic distance between OMHS and HLGS Methodology Prove that a Great Circle gives the shortest arc length between two locations Find the Cartesian coordinates of a location from its latitude and longitude through its spherical coordinates Find the included angle at the center of the Earth for two locations Calculate the arc length along a great circle from the included angle and the radius of the Earth Assumption: the Earth is a sphere 5 Geodesic Distance many arcs passing through points A and B geodesic distance: arc length between two points along a great circle great circle Radius = radius of the earth Center = center of the earth small circle Radius < radius of the earth Center center of the earth 6 x Geodesic Distance intuitively clear that the arc length along the great circle is shorter in (0, ), for < R(2 ) < r(2 ) A B A B R r O o1o1 7 Coordinate Transformation: Spherical to Cartesian Coordinates Spherical co-ordinates (R, , ) : R: radius; : latitude; : longitude Cartesian co-ordinates (x, y, z) : x y z A: (R, , ) 8 To Find the Geodesic Distance from Cartesian coordinates of points A and B radian AB = R R 9 Spherical & Cartesian Co-ordinates of OMHS and HLGS Cartesian co-ordinates of OMHS Cartesian co-ordinates of HLGS Cartesian co-ordinates (x, y, z) : radian 10 Conclusions we have proven that the shortest distance between two points on a sphere is along the great circle found the spherical coordinates of OMHS and HLGS in terms of their latitudes and longitudes transformed the spherical coordinates of OMHS and HLGS to their Cartesian co-ordinates found the geodesic distance (= km) between OMHS and HLGS 11 Conclusions refinement the Earth is not exactly spherical in shape the data is not exact from web, the exact geodesic distance between OMHS and HLGS is km (www.movable-type.co.uk/scripts/latlong-vincenty.html) 12 References Wikipedia, definitions of longitude and latitude Vol 30, Characteristics of Earths Surface Vol 31, Geodesic Serial 23, Vol 2, On Distance between Two Points on Earth 13 THE END Thank you