I want to find the distance between two points A and B on a sphere. The coordinates (x,y,z) of my points can be given by
$x=R\cos(θ)\sin(ϕ)$
$y=-R\cos(θ)\sin(ϕ)$
$z=R\sin(ϕ)$
Where $\theta$ is the angle between the point and the positive z-axis and $\phi$ is the angle subtended by the positive x-axis. The final equation for the geodesic I got (similar to this video) is
$θ=\arccos(b\cotθ )+c$
To go ahead and calculate the distance between two points, I'm assuming I'm going to have to use the arc length formula
$d=\int_a^b\sqrt{1+f'(\theta )} d\theta$
But in order to do that, I need to determine the constants $b$ and $c$. I tried using simultaneous equations but I'm getting an expression for $b$ in terms of $c$, whereas I need them separately. How do I go ahead with this?
WLOG, the two points can be on the equator, at angular distance $\Delta\theta=\alpha$, and the curvilinear distance is just $R\alpha$.
If the points are located anywhere, you can find the subtended angle by a dot product,
$$\cos\alpha=(\cos\theta_A\sin\phi_A, \sin\theta_A\sin\phi_A,\cos\phi_B)\cdot (\cos\theta_B\sin\phi_B, \sin\theta_B\sin\phi_B,\cos\phi_B)\\ =\sin(\theta_A+\theta_B)\sin\phi_A\sin\phi_B+\cos\phi_B\cos\phi_B.$$