Why does the Transverse Mercator projection have a scale factor in ArcGIS and the standard Mercator projection not?
In other words what is the difference between these two projections beside being a horizontal respectively vertical cylinder?
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But it does--it's just implicitly defined. Instead, a standard parallel (+/-) is used to make the cylinder secant. For the transverse case, there's no easily defined line (you can't use a meridian), so a scale factor at the central meridian is used instead.
I've written tiny programs to go between the two variants.
At some point, we'll add the scale-factor version of the Mercator projection, but it hasn't been scheduled yet (as of 24 Feb 2014).
Disclosure: I work at Esri.
They are not so different, but being horizontal instead of vertical has some implications. Indeed, the Earth is not a perfect sphere, as you know. So when you use a cylindrical projection, you must take this into account.
Another difference, that is not linked with Transverse or not Transverse, is that the cylindrical projection can be tangent or secant. When it is secant, the scale between the two secant lines is reduced. UTM projections are secant, the standard world Mercator is not.