I have the following map in ArcGIS desktop. It has a geographic coordinate system of GCS_WGS_1984.
I want to project it to Mercator so that the countries don't look as squished (like so: https://www.google.ca/search?q=mercator+projection&biw=1366&bih=643&source=lnms&tbm=isch&sa=X&ved=0C... )
I projected it to World Mercator and WGS_1984_World_Mercator but it still looks the same
If you have a file with a geographic coordinate system of GCS_WGS_1984 you need to do one of 2 things to make it look different in whatever coordinate system you want.
Under scenario one, when a single file is first added to a data frame, the coordinate system of the data frame is set to the coordinate system of file. If you project it (using the Project tool) to a new coordinate system and add it to the same data frame...nothing happens...because the file is projected-on-the-fly back to the coordinate system of the data frame, that is why it must be added to a new data frame for its true appearance to reveal itself.
Under scenario two, if you set the coordinate system of the data frame to something other that its coordinate system, the files are projected-on-the-fly to the new coordinate system...which is ok, you can now 'see' what it would look like if you physically projected it.
Under scenario three, you just ruined a perfectly good file, telling it that it is something that it isn't and it will never display properly.
So my suggestion is
Thank you! I did #1 and it worked