# Calculate position of point along Rhumb line given the distance from starting point

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:

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?

$$\theta$$ 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 $$k=\dfrac{K}{R\sqrt{1+K^2}}$$;$$\Delta\theta=\theta_f-\theta_0,\Delta\phi=\phi_f-\phi_0$$
$$\theta(s)=\theta_0+ks$$
$$\phi(s)=\phi_0+\dfrac{1}{K}\ln\left(\dfrac{(1-\cos(ks+\theta_0))\sin\theta_0}{(1-\cos\theta_0)\sin(ks+\theta_0)}\right)$$