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On other parallels

On any other parallel the scale factor is sec φ so that

For the above model 1 cm corresponds to 1,500 km at a latitude of 60°.

This is not the shortest distance between the chosen endpoints on the parallel because a parallel is not a great circle. The difference is small for short distances but increases as λ, the longitudinal separation, increases. For two points, A and B, separated by 10° of longitude on the parallel at 60° the distance along the parallel is approximately 0.5 km greater than the great circle distance. (The distance AB along the parallel is (a cos φ) λ. The length of the chord AB is 2(a cos φ) sin λ / 2. This chord subtends an angle at the centre equal to 2arcsin(cos φ sin λ / 2) and the great circle distance between A and B is 2 a arcsin(cos φ sin λ / 2).) In the extreme case where the longitudinal separation is 180°, the distance along the parallel is one half of the circumference of that parallel; i.e., 10,007.5 km. On the other hand, the geodesic between these points is a great circle arc through the pole subtending an angle of 60° at the center: the length of this arc is one sixth of the great circle circumference, about 6,672 km. The difference is 3,338 km so the ruler distance measured from the map is quite misleading even after correcting for the latitude variation of the scale factor.