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I am studying about Rhumb lines and loxodromes following the information presented by the following link. The Rhumb line between two points using the Mercator projection looks like this:

enter image description here

My goal would be to find the mathematical formula to identify the coordinates of a generic point along the Rhumb line by knowing the distance from the point of origin (or of destination).

So if I know that Point 1 is my starting position, what are the lat/lon coordinates of the point positioned at the genric distance d from Point 1 along the Rhumb line connecting Point 1 and Point 2?

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You can get what you want from this interesting link: Constant Headings and Rhumb Lines

Some points you have to consider.

$\theta$ is the colatitude angle, so is, the equator is at $\pi/2$ and the North Pole at $0$, and $\phi$ the longitude one (easy to adapt to latitude), $s$ is the distance from the starting point, $R$ the earth's radius and $K$ a parameter that we can call "rhumb parameter" stated as follows:

$\dfrac{\Delta\theta}{\Delta\phi}=K$ and then $k=\dfrac{K}{R\sqrt{1+K^2}}$;$\Delta\theta=\theta_f-\theta_0,\Delta\phi=\phi_f-\phi_0$

$\theta(s)=\theta_0+ks$

$\phi(s)=\phi_0+\dfrac{1}{K}\ln\left(\dfrac{(1-\cos(ks+\theta_0))\sin\theta_0}{(1-\cos\theta_0)\sin(ks+\theta_0)}\right)$

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