GIS, Mercator projection, calculate Y position

I am doing some stuff with GIS(or map), and I know how to calculate Y position from latitude in mercator projection. But I don't know how it comes?

Why it is ln(tan(x)+sec(x)) ? I couldn't 开发者_开发百科figure out why there is a 'ln'?

I know it is not directly related to programming, but I hope some one could help me understanding about this (better with some pictures to illustrate)

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It sounds as though you're asking about the last step in the derivation you've listed. It's the definition of arctanh combined with some trigonometry identities.