What are the ranges of $x$ and $y$ in the Mercator projection?

Question

What are the ranges of $x$ and $y$ in the Mercator projection ?

I searched about it but I couldn't find anything.

Any ideas?

Thanks in advance

Answer 1

As a mathematician, I'd say $\mathbb R$ for both $x$ and $y$. The Wikipedia article has many formulas describing this projection. A Mercator map is periodic in the East/West direction, reflecting the fact that you can circle the earth as many times by going East or West. That's what's behind the range $\mathbb R$ for $x$. The range $\mathbb R$ for $y$ is caused by the fact that the North and South poles are not represented by a Mercator map, even though they are actual places on the Earth.

As a practical matter, one prints a bounded rectangular portion of a Mercator map to put in an atlas or to put on a wall map. Such a truncated map will not include some neighborhood of the N and S poles, but (because of the E/W periodicity) show the full extent of the Equator, etc. Often one sees (on American or European-made) Mercator maps two copies of the International Date Line; one can imagine maps printed in (say New Zeeland) as having two copies of the Greenwich meridian and one of the IDL.

Most software packages deliver an $x,y$ pair in the ranges $-\pi/2 \le x \le \pi/2$ (or $-180\le x\le180$) and $-\infty<y<\infty$. It is not clear to me from your comments how this matches up with what you want.