The projection of Mercator is a cylindrical projection terrestrial sphere on a plane chart named by Gerardus Mercator in 1569. The parallel S and the Méridien S are straight lines and the inevitable East-West stretching apart from the equator is accompanied by a North-South stretching corresponding, so that the East-West scale is similar on a North-South scale everywhere. A chart of Mercator cannot cover the pole S: they would be infinitely hauts.

It is about a projection conforms, i.e. it preserves the angles. Any straight line on a chart of Mercator is a line of constant Azimut. This makes it marine particularly useful for the S, even if the way thus defined is generally not on a Grand circle and is thus not the shortest way.
At the time of large the sailing, the duration of the voyage was subjected to the elements, and thus the distance from the way was less important than the direction, especially because the Longitude was difficult to calculate précisément.

The traditional charts inspired of the work of Mercator intended for navigation have for principal defect to give us an erroneous idea of the surfaces occupied by the various areas of the world, and thus of the relationship between the peuples.
Some examples:
The South America seems smaller than the Greenland; actually, it is nine times larger: 17,8 million km ² against 2,1 million. The India (3,3 million km ²) seems smaller than the Scandinavia (1,1 million km ²). Europe (9,7 million km ²) seems wider than South America, however close to twice larger.

The choice of projection can thus be a tool of centering or even of Propagande. On the first projections, the chart is européocentrée, i.e. the Europe appears in the center. A few centuries afterwards, the the United States employ a chart which places the American continent in the center of the representation. The continent of Asia being divided into two.

Formulas

The following equations determine the Coordonnée S X and there of a point on a chart of Mercator starting from its Latitude φ and of its Longitude λ (with λ₀ in the center of the chart)

$$

\begin{matrix} X &=& \ lambda - \ lambda_0 \ \ there &=& \ ln \ left \ tan \ left (\ frac {1} {4} \ pi + \ frac {1} {2} \ varphi \ right) \ right \ \ \ & =& \ frac {1} {2} \ ln \ left (\ frac {1 + \ sin \ varphi} {1 - \ sin \ varphi} \ right) \ \ \ & =& \ sinh^ {- 1} \ left (\ tan \ varphi \ right) \ \ \ & =& \ tanh^ {- 1} \ left (\ sin \ varphi \ right) \ \ \ & =& \ ln \ left (\ tan \ varphi + \ dry \ varphi \ right) \end{matrix}

And here the function known as function of Gudermann reverses:

$$

\begin{matrix} \ varphi &=& 2 \ tan^ {- 1} \ left (e^y \ right) - \ frac {1} {2} \ pi \ \ \ &=& \ tan^ {- 1} \ left (\ sinh there \ right) \ \ \ lambda &=& X + \ lambda_0 \end{matrix}

Calculation

Since a cylindrical projection is used, X depends only on λ and there depends only on φ. The North-South scale (in φ) must be everywhere equal on a East-West scale (in λ), but a radian of longitude does not make the same size with the poles as at the equator. The report/ratio of derived must thus be equal to the report/ratio length of the parallel compared to the length of the meridian line.

$\backslash \; forall\; \backslash \; varphi,\; \backslash \; lambda\; \backslash :\; \backslash \; frac\; \{\backslash \; frac\; \{\backslash \; partial\; X\}\; \{\backslash \; partial\; \backslash \; lambda\}\}\; \{\backslash \; frac\; \{\backslash \; partial\; there\}\; \{\backslash \; partial\; \backslash \; varphi\}\}\; =\; \backslash \; frac\; \{2\; \backslash \; pi\; R\; \backslash \; cos\; (\backslash \; varphi)\}\{2\; \backslash \; pi\; R\}$

And since one chooses $\backslash \; frac\; \{\backslash \; partial\; X\}\; \{\backslash \; partial\; \backslash \; lambda\}\; =\; 1$
One finds

$\backslash \; frac\; \{\backslash \; partial\; there\}\; \{\backslash \; partial\; \backslash \; varphi\}\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; cos\; (\backslash \; varphi)\}\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{2\}\; +\; \backslash \; varphi)\}\; =\; \backslash \; frac\; \{1\}\; \{2\; \backslash \; sin\; (\backslash \; frac\; \{\backslash \; pi\}\; \{4\}\; +\; \backslash \; frac\; \{\backslash \; varphi\}\; \{2\})\; \backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{4\}\; +\; \backslash \; frac\; \{\backslash \; varphi\}\; \{2\})\}\; =\; \backslash \; frac\; \{\backslash \; frac\; \{1\}\; \{2\}\; \backslash \; frac\; \{1\}\; \{\backslash \; cos\; (\backslash \; frac\; \{\backslash \; pi\}\; \{4\}\; +\; \backslash \; frac\; \{\backslash \; varphi\}\; \{2\})\; ^2\}\}\; \{\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{4\}\; +\; \backslash \; frac\; \{\backslash \; varphi\}\; \{2\})\}\; =\; \backslash \; frac\; \{\backslash \; frac\; \{\backslash \; partial\; (\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{4\}\; +\; \backslash \; frac\; \{\backslash \; varphi\}\; \{2\})\}\{\backslash \; partial\; \backslash \; varphi\}\}\; \{\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{4\}\; +\; \backslash \; frac\; \{\backslash \; varphi\}\; \{2\})\}$

then while integrating $y\; =\; \backslash \; ln\; \backslash \; left\; (\backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{4\}\; +\; \backslash \; frac\; \{\backslash \; varphi\}\; \{2\})\; \backslash \; right)$

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