# What is a conformal and non conformal map?

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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.

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.

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.

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.