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Robin and all please consider the following argument, which I think goes a long way towards covering some of the points we’ve been trying to clarify.

1.  First change the title to this thread to ‘To show that the projection of the surface of a sphere from a point at its centre onto the surface of a cylinder encircling it is not orthomorphic and therefore should not be used as a simple way of depicting a Mercator Projection’

2.  The Mercator projection was designed mathematically to be orthomorphic and has been shown by use over the years to be so.

3.  The test for a chart to be orthomorphic is that at any point on a chart the scale should be the same in every direction.

4.  In the diagram below, A is a plan view of the Earth.  B is a side view of the Earth.  Both are surrounded by a cylinder whose axis is that of the Earth and whose radius r is also that of the Earth.

5.  At the Equator, the length of a ten degree section of longitude is 2 pi r x 10/360 on both the sphere and the cylinder so the scale is one.  The length of a ten degree section of latitude is also 2 pi r x 10/360 on the sphere and almost the same on the cylinder, so the scale is also one.

6.  At latitude 60N, the length of a ten degree section of longitude has halved on the sphere, but it is still 2 pi r x 10/360 on the cylinder, so the scale E/W has increased to two.  The length of 10 degrees of latitude on the sphere is still 2 pi r x 10/360, but on the cylinder it isn’t constrained to quite the same extent as longitude and appears to be about four times this amount, so the scale N/S has increased to about four.

7.  At 60N, the scale on the cylinder is no longer the same in every direction (despite the meridians and parallels crossing at right angles), so the projection is not orthomorphic, and it therefore should not be used to depict a Mercator chart.

Note: The ten degree square could be reduced to a tiny square and the same argument followed using calculus, but the above is enough to satisfy me. DaveP

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