I am new to working with coordinate data and figured out the equation I am looking for is the Rhumb Line. I went to go research it and found a lot of equations and I still have no idea where to start.
The data I DO have is my heading, my distance, and my starting coordinate pair. How would I use this equation to find my new coordinate pair?
I'll use spherical coordinates ($\theta =$ longitude, $\phi =$ latitude, all angles measured in radians) on a sphere of radius $R$. Mercator projection maps $(\theta, \phi)$ to $\theta, \ln(\tan(\phi/2 + \pi/4))$, and a rhumb line on the sphere corresponds to a straight line on the Mercator projection. Thus the rhumb line through $(\theta_0, \phi_0)$ at heading $\alpha$ (measured clockwise from north) is given parametrically by $\theta = \theta_0 - t \sin(\alpha)$, $\ln(\tan(\phi/2 + \pi/4)) = \ln(\tan(\phi_0/2 + \pi/4)) + t \cos(\alpha)$ or $\phi = 2 \arctan \left( e^{t \cos(\alpha)} \tan(\phi_0/2 + \pi/4) \right)$. Now the distance between two points $(\theta_0, \phi_0)$ and $\theta_1, \phi_1)$ along the rhumb line is $R \sec(\alpha) (\phi_1 - \phi_0)$. So if that distance is $d$, $\phi_1 = \phi_0 + \frac{d}{R} \cos(\alpha)$, corresponding to $t = \sec(\alpha) \ln \left( \frac{\tan(\phi_1/2 + \pi/4)}{\tan(\phi_0/2 + \pi/4)}\right)$, and $\theta_1 = \theta_0 - \tan(\alpha) \ln \left( \frac{\tan(\phi_1/2 + \pi/4)}{\tan(\phi_0/2 + \pi/4)}\right)$.