The Mercator projection will be the first under scrutiny. I chose to consider this projection because it is the most diffused globally, also used in popular software such as Google Maps.
It was invented in 1569 by Gerardus Mercator, Flemish philosopher and mathematician. It is one of the cylindrical map projection, that, as the name suggests, projects the globe on a cylinder, tangent on the equator.
This projection has three main properties:
As the image on the left shows, North-South direction is the vertical direction, and East-West the horizontal one;
The length of the equator is preserved;
“All paths of equal compass bearing on the sphere are straight lines”;
Thus, each point on the globe will have a unit to indicate the longitude, where , and a unit for the latitude, . All sources agreed that “it is more convenient to measure latitude and longitude in radians rather than degrees”
The plane to represent the map, instead, will be a Cartesian two-dimensional plane, with horizontal axis , and vertical axis
Since we know that the length of the equator is preserved, it can already be concluded that
Now, establishing how to calculate will be more challenging: it is here, indeed, that the distortion happens.
There are several ways to show how is derived. I chose the following one since it uses only elements of the syllabus.
The circle tangent to the cylinder (image on the left) has the same radius as the globe, to whom it can be assigned the value of 1. Then, is a certain latitude, that is the same as the latitude of P.
Imagine a parallel circle with latitude , tangent to P, which can be called . The radius of such circle is equal to . is projected on the cylinder onto a circle , which has radius 1, as the equator, where the cylinder is tangent to the globe.
Here it can be seen another distortion, because, by projecting onto , the values are distorted and one point on becomes many points on . This is why countries like Russia or Greenland are vertically stretched, and thus appear bigger than reality, on a Mercator projection.
Thus, the ratio between the two circumferences is: