What is a conformal and non conformal map?
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In a lot of contexts, you’re interested in translating coordinates in one space to coordinates in another. For example, you seem to be interested in geography, where we take the curved surface of the Earth and “flatten it out” into a map. You can think of latitude and longitude as “flat” coordinates that got “wrapped around” the curved Earth.
Formally speaking, latitude and longitude define a function f: I^2 -> S^(2), where I^2 is a square in the plane (in this case, the square of possible latitudes and longitudes) and S^2 is the sphere (i.e., the surface of the Earth). As is usually true in math, “map” here simply means “function”.
So what do we mean by *conformal*?
A conformal map (locally) *preserves angles*. That is, if you drew yourself a little triangle in the square of latitudes and longitudes, then put it through your map, you get a triangle with the the same angles on the sphere. If you look at the angle 45 degrees north of east in the space of latitudes and longitudes, it’ll remain 45 degrees north of east on the sphere. And so on. It may distort the *size* of objects, and it may distort *large* shapes, but small shapes remain the same.
A non-conformal map, on the other hand, doesn’t do this. A right angle in your coordinates might not map to a real right angle.
The latitude and longitude coordinates we were talking about above, for example, are **not** conformal. If you draw a triangle in your coordinates between, say, (89 latitude, 0 longitude), (89 latitude, 1 longitude), and (90 latitude, 0 longitude), you have a nice right triangle in your coordinates, with angles 45, 45, and 90 degrees. But on the globe, this triangle is extremely distorted, with one of its angles much smaller than the other two. Our nice isosceles triangle has been warped into an extremely long and narrow one.
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