Point Scale For Normal Cylindrical Projections of The Sphere
The key to a quantitative understanding of scale is to consider an infinitesimal element on the sphere. The figure shows a point P at latitude and longitude on the sphere. The point Q is at latitude and longitude . The lines PK and MQ are arcs of meridians of length where is the radius of the sphere and is in radian measure. The lines PM and KQ are arcs of parallel circles of length with in radian measure. In deriving a point property of the projection at P it suffices to take an infinitesimal element PMQK of the surface: in the limit of Q approaching P such an element tends to an infinitesimally small planar rectangle.
Normal cylindrical projections of the sphere have and a function of latitude only. Therefore the infinitesimal element PMQK on the sphere projects to an infinitesimal element P'M'Q'K' which is an exact rectangle with a base and height . By comparing the elements on sphere and projection we can immediately deduce expressions for the scale factors on parallels and meridians. (We defer the treatment of the scale in a general direction to a mathematical addendum to this page.)
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- parallel scale factor
- meridian scale factor
Note that the parallel scale factor is independent of the definition of so it is the same for all normal cylindrical projections. It is useful to note that
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- at latitude 30 degrees the parallel scale is
- at latitude 45 degrees the parallel scale is
- at latitude 60 degrees the parallel scale is
- at latitude 80 degrees the parallel scale is
- at latitude 85 degrees the parallel scale is
The following examples illustrate three normal cylindrical projections and in each case the variation of scale with position and direction is illustrated by the use of Tissot's indicatrix.
Read more about this topic: Scale (map)
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