TITLE: Mercator's Projection
QUESTION [2 upvotes]: Consider the two-sphere,
(a) Show that the following map does provide a chart for $S^{2}$,
$\phi^{-1}(u, v) = (sech(u) cos(v), sech(u)sin(v), tanh(u))$
Compute the inverse of this map. What values can $(u, v)$ take?
(b) Show that, in this chart, meridians and parallels (of common chart used for Earth Surface) correspond to perpendicular straight lines (in the $(u, v)$ plane).
I am done with the inverse of the map. I have found the values of $(u, v)$ in terms of $(x, y, z)$. I know that the meridians and parallels in this projection are orthogonal, but I don't know how to prove it mathematically. Can someone help me out?

REPLY [2 votes]: The parallels are $z = \mbox{const}_1,$ while the latitudes are $\frac{y}{x} = \mbox{const}_2.$ Are the images straight lines? Well, the image of a latitude is obviously a vertical straight line ($u=\mbox{const}$), so the image of a meridian (longtitude) better be $v=\mbox{const},$ if the claim is correct. That does not seem to be true.
{"set_name": "stack_exchange", "score": 2, "question_id": 4095726}