Lv = latitude of the vertex L1 = latitude of the point of departure (point 1) C1 = course angle at point 1 DLOv1 = difference in longitude from the vertex to point 1 Dv1 = distance from the vertex to point 1 (degrees) cos Lv = cos L1 * sin C1 sin DLOv1 = cos C1 * csc Lv or tan DLOv1 = ctn C1 * csc L1 sin Dv1 = cos L1 * sin DLOv1 or tan Dv1 = cos C1 * ctn L1 If a difference in longitude from the vertex is assumed (DLOvx), then the parameters for a point x (located anywhere along the great-circle path) can be determined. Lx = latitude at point x DLOvx = difference in longitude from the vertex to point x Cx = course angle at point x Dvx = distance from the vertex to point x (degrees) tan Lx = cos DLOvx * tan Lv cos Cx = sin DLOvx * sin Lv (usually not required) tan Dvx = tan DLOvx * cos Lv (optional)
When the above formulas are used, the distance between points will be a variable. To get equal distances the following method can be used. The incremental distance used is usually a multiple of the ship's cruising speed, so a very fast ship (super tanker) may pass a reference point every four hours. A slower ship would use six or more hours.
sin Lx = sin Lv * cos Dvx tan DLOvx = sec Lv * tan Dvx tan Cx = ctn Lv * csc Dvx (usually not required)
When sailing from San Francisco to Hilo Hawaii, the vertex will be near Lake Michigan. When sailing from San Francisco to Japan, the vertex will be just south of the Aleutian islands. This can be taken advantage of because the great-circle path will be symmetrical around the vertex and if the vertex longitude is used as a reference fewer calculations will be required. For example, 900 miles east or west of the vertex (along the great-circle path) will be at the identical latitude and the DLO from the vertex will be the same.
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