What would influence an alien race to map their planet in a way other than the traditional map of the Earth
I was looking through a bunch of different ways we can shape a map of the Earth and it got me thinking. We map out planet as a stretched rectangle with NSEW as Up Down Right Left, with the Equator right in the middle. But what would cause a species to prefer a different way of mapping?
Tidal Locking may cause a species to develop a map for each hemisphere. Maybe they would focus more on the positions of extreme weather conditions. What would cause other unique ways of mapping a planet?
Edit: I'm specifically looking for 2D maps not 3D maps like globes.
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The aliens know some mathematics that we don't.
In the question, there is mention of a particular rectangular projection, best known as the Mercator projection. However, there are many other ways we Earth people have figured out for projecting a sphere onto something flat, including conic, conformal, gnomonic. There is a decent article at Wikipedia.
Each of these projection techniques requires an understanding of some type of geometry, trigonometry, or related areas of mathematics, and there could easily be other possible projections we have yet to discover, but the aliens have known for a long time. Their maps won't be radically different as a result, but perhaps just different enough to handwave it into your story.
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Maps don't have to be geographical, they can also be diagrammatical, like the London Tube map. For instance, ancient Roman maps, the itinerarium, were graphs showing points of interest along a road rather than geographical relations. A branch in the road was a branch in the graph line.
Another approach is symbolic maps, like the medieval T-O maps and mappa mundis. The simplest of such are extremely abstract, but at the other extreme, you could have more detailed ones as well, like the one at Hereford Cathedral. These maps are constructed the way they are to fit the prevailing religious worldview of the time.
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