1. 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. 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. 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. 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. 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. 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. 7 Coordinate Transformation: Spherical to Cartesian Coordinates  Spherical co-ordinates (R, , θ) : R: radius;  : latitude; Θ: longitude  Cartesian co-ordinates (x, y, z) :  x y z A: (R, , θ)

8. 8 To Find the Geodesic Distance  from Cartesian coordinates of points A and B   radian  AB = R  R

9. 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. 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 (= 1423.784 km) between OMHS and HLGS

11. 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 1416.15 km (www.movable-type.co.uk/scripts/latlong-vincenty.html)

12. 12 References  Wikipedia, definitions of longitude and latitude  《翰林數學天地》 Vol 30, 余文卿 〈 Characteristics of Earth’s Surface 〉  《翰林數學天地》 Vol 31, 簡茂祥 〈 Geodesic 〉  《數學傳播》 Serial 23, Vol 2, 徐正梅〈 On Distance between Two Points on Earth 〉  www.movable-type.co.uk/scripts/latlong-vincenty.html

13. 13 THE END Thank you