What is a conformal and non conformal map?
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A conformal map is a type of function, or a set of closely related functions (I.e. a vector-valued function or a “coordinate transform function.”) See: the definition of a function.
In the sense of basic cartesian x-y coordinates, a conformal mapping function takes every point in the plane and moves it to a different location on the plane. This is a specific example but you got the general idea. But such a mapping function would need to be carefully concocted otherwise it probably won’t satisfy the conformal property.
A conformal map, locally preserves the value of notions like angles, and measures similar to it such as the concept of a “solid angle” in 3D space. That means for example, you can measure angles between 3 points that are quite close together and those aren’t changed. However in the case of points quite far away, angles may not be preserved on a macroscopic scale.
A conformal map doesn’t preserve measures of *distance*, and analogous concepts, such a the vector norm or magnitude, even on a small scale. It also doesn’t preserve parallelism.
A simple example of this is an actual map, like the commonly seen Mercator Projection of the globe. You can use such a map to navigate in the arctic, the angles between destinations will be accurate. However in a Mercator projection map, Greenland, the northern Russian and canadian coast and the various arctic islands (Svalbard, anyone?) are shown far larger than they actually are. Therefore the distance scale varies depending on how far from the equator.
Another important property in this example is “great circle” routes of flying across the globe such as Calgary canada to Warsaw Poland The two are similar latitude so on a Mercator projection map the shortest distance would be to seemingly fly due east along the latitude lines. However in reality the shortest distance on the globe is to fly northeast over greenland. This is because the only latitude line that’s straight is the equator. All others are curved in a globe. Hence in a conformal map large scale angles aren’t preserved.
However the notion of conformal mapping is quite general and can apply to other kinds of spaces besides 2D or 3D space. Anything where you can come up with a notion allowing something like angle measurement. That in itself is a very complex topic.
A nonconformal map is one that doesn’t preserve notions of angle measurement. A simple example is printing a square grid on a square sheet of rubber, then pulling in two opposite corners, causing the sheet to become a parallelogram. The angles between the gridlines will change in this square-to-parallelogram mapping. In this case it doesn’t matter how closely you zoom in on the gridlines, the angles between gridlines won’t be 90° anymore.
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