Question about Mercator's projection

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In summary, in the mathematics of the Mercator projection, in order for a map to be conformal, the local latitude scale must match the local longitude scale. This is achieved by integrating a constant towards the variable of integration, resulting in the equations x(λ,ϕ)=Ccos(ϕ)λ and y(λ,ϕ)=Cϕ. This means that x (longitude) will always be proportional to the longitude and y (latitude) will have a scale equal to the x scale. The full derivation can be found in the Wikipedia article on the Mercator projection."
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jk22
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I read the Wikipedia article : https://en.m.wikipedia.org/wiki/Mercator_projection

Section : Mathematics of the Mercator projection.

For a map to be conformal should not it be $$k(\phi)=C,h(\phi)=C$$, or the shrinking coefficient shall be not only equal but homogeneous, in order to be conformal ? We then get two partial differential equations and their solution is Simply obtained by integrating a constant towards the variable of integration :

$$x(\lambda,\phi)=Ccos(\phi)\lambda$$
$$y(\lambda,\phi)=C\phi$$

Does this anyhow makes sense ?
 
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To be conformal, it is sufficient that the local latitude scale match the local longitude scale.
Here is the legend (as I understand it) for you equations:
x,y: The map coordinates.
ϕ: Latitude
λ: Longitude I am not clear on your k and h - perhaps they indicate x,y scale. If they do, then you are glazing over the fact that there are many C's. I would say you have a ##C_\phi##, not a C.

For Mercator, x (the horizontal, longitude direction) will always be proportional to the Longitude.
So x(λ,ϕ)=R ϕ

That leaves y to be a function that always has a scale (differential) equal to the x scale.

The full derivation is in the wiki article you cited:
{\displaystyle x=R(\lambda -\lambda _{0}),\qquad y=R\ln \left[\tan \left({\frac {\pi }{4}}+{\frac {\varphi }{2}}\right)\right].}
 
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Related to Question about Mercator's projection

1. What is Mercator's projection?

Mercator's projection is a map projection developed by Gerardus Mercator in 1569. It is a cylindrical projection that preserves angles and shapes, but distorts the size of objects as they get farther away from the equator.

2. What are the advantages of Mercator's projection?

Mercator's projection is advantageous for navigation and route planning, as it accurately represents the direction and bearing between two points. It also provides a rectangular grid that makes it easy to measure distances and angles.

3. What are the drawbacks of Mercator's projection?

The main drawback of Mercator's projection is the significant distortion of size and shape towards the poles. This makes it unsuitable for accurately representing the size of countries and continents, particularly near the poles.

4. How does Mercator's projection affect our perception of the world?

Mercator's projection has been criticized for perpetuating a Eurocentric view of the world, as it distorts the size and importance of countries located closer to the equator. This can lead to a skewed perception of the world and its geography.

5. Are there alternative map projections to Mercator's projection?

Yes, there are many alternative map projections that attempt to address the drawbacks of Mercator's projection. Some examples include the Robinson projection, the Gall-Peters projection, and the Winkel Tripel projection. Each projection has its own strengths and weaknesses, and the choice of projection often depends on the purpose of the map.

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