Saturday, April 19, 2014

Geodesics on a Sphere

  As the bounds of a map increase the curvature of the Earth starts to deviate from a plane and this will distort the positions on the map even if we use a projection onto a plane tangent to the surface at the center of the map. The distances, angles from north and the direction of lines of sight are altered from those of plane geometry. The simplest solution is to use a sphere to represent the surface of the Earth and work with spherical geometry.

  The first question that we need to ask is how do we measure distances on the surface of a sphere? If we stretch a string between two points on the surface it becomes something like a straight line and we can used the string length as a measure of the distance. The string's arc is the arc common to the sphere's surface and a plane passing through the two points and the center of the sphere. This is a section of a great circle. The equator and meridian arcs that are used in spherical coordinates are great circles. The other lines of latitude are not but are arcs drawn at equal distances from the pole of the coordinate system.

  To show that the great circle is the minimum distance between two points we can compare the great circle length  with the lengths of other arcs connecting them. These arcs will have centers on a great circle passing through the center of the great circle connecting the two points and normal to it. To compare the arcs we need their lengths, Δs, and the maximum distance, β, of the arc normal to the great circle connecting the points.

When the lengths are computed and plotted we see that β = 0 corresponds to the minimum length which is that of the great circle.

So we see that great circles play the role of straight lines in spherical geometry. The great circle is a geodesic.

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