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jk22
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 ?

Homework Helper
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:

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