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Navigational Mathematics

3. The great-circle track

The rhumb-line track is very convenient, because the ship keeps to the same course through the whole trip. Its disadvantage is that the straight line on a Mercator map may not be the shortest path between its endpoints, when measured back on the earth's surface.

The earth's surface is (to a good approximation) a sphere, and the path of shortest length between two points on a sphere is the great-circle arc between them. The great circle in question is the intersection of the spherical surface with the plane passing through the two points and the center of the earth.

The great-circle path is different from the rhumb line unless the two points are both on the equator or both on the same meridian. If the points are nearby, say within 50 miles of each other, the difference between the two paths is inconsequential. But for distant points the difference may be substantial.

Here is a classic navigation problem, taken from Benjamin Dutton's "Navigation and Nautical Astronomy," 7th Edition, a text once used at Annapolis.

  • Compute the distance and initial course by great circle sailing from a point in Lat. 37o-42' N., Long. 123o-04'W., near Farallon Island Lighthouse, to a point Lat. 34o-50' N., Long. 139o-53' E., near the entrance to the Bay of Tokio.
  • Answer: distance = 4461.7 nautical miles, course = N 57o46'15"W.

  • How is this answer calculated?