What is the angle from (0,0) to (longitude, latitude)?

Question

Starting on the crossing between the equator and the Greenwich meridian (0,0), one can proceed to any point on Earth with a given longitude and latitude along a great circle.

Seen from the centre of the Earth, what is the angle from the point (0,0) to the point (longitude, latitude), along that great circle?

Answer 1

Let $l$ and $L$ be respectively the longitude and the latitude of a certain point on earth.

then Its coordinates will be

$$x=R\cos(L)\cos(l)$$ $$y=R\cos(L)\sin(l)$$ $$z=R\sin(L)$$

$R$ is the Earth radius. $L=\frac{\pi}{2}$ gives the northern pole with $(0,0,R)$ as coordinates.

Answer 2

Translating from spherical to Cartesian.

$x = 4000 \cos\theta\cos \phi\\ y = 4000 \sin\theta\cos \phi\\ z = 4000 \sin \phi$

$\phi$ is degrees latitude. South is negative degrees latitude. $\theta$ is degrees longitude. Lets make East positive, and West negative..

$4000$ is the radius of the earth in miles. (6,400 km if you prefer, or $r$, it won't really make a difference.)

$\cos \alpha = (\cos\theta\cos \phi, \sin\theta\cos \phi, \sin \phi)\cdot(1,0,0)\\ \alpha = \arccos (\cos\theta\cos\phi)$