On a sphere the shortest path between two points is an arc of the great circle — the circle whose centre is the centre of the earth. Sinnott's haversine form computes its length without losing precision on short legs: $d = 2R\arcsin\sqrt{\operatorname{hav}\Delta\varphi + \cos\varphi_1\cos\varphi_2\operatorname{hav}\Delta\lambda}$.

The letters, once, plainly. $\varphi$ — phi — is latitude, north positive and south negative. $\lambda$ — lambda — is longitude, east positive and west negative. Subscript 1 is where you are; subscript 2 is where you are going, and $\Delta\varphi$ means $\varphi_2 - \varphi_1$. $R$ is the earth's mean radius, 6,371 km. The haversine itself, $\operatorname{hav}x = \sin^{2}(x/2)$, is just a compact way of writing a very small angle without losing it to rounding.

Look at what $\cos\varphi_1\cos\varphi_2$ is doing: it shrinks the longitude difference as you go poleward, because meridians converge. A degree of latitude is 111 km everywhere. A degree of longitude is 111 km at the equator, 71 km at 50° N, and nothing at all at the pole. Forget that cosine and you have drawn a flat earth.

The initial bearing, $\theta = \operatorname{atan2}\!\left(\sin\Delta\lambda\cos\varphi_2,\; \cos\varphi_1\sin\varphi_2 - \sin\varphi_1\cos\varphi_2\cos\Delta\lambda\right)$, gives the course at the moment of departure, clockwise from true north. The word INITIAL is load-bearing. A great circle crosses every meridian at a different angle — that is exactly what makes it short — so steering one bearing the whole way puts you on a different track altogether. Great circle sailing in practice means breaking the route into legs, commonly every 5° of longitude, and recomputing.

The alternative is the rhumb line, or loxodrome: the track that crosses every meridian at the SAME angle, which is the route you get when you set one course and hold it. $d = R\sqrt{\Delta\varphi^{2} + q^{2}\Delta\lambda^{2}}$, where $q$ is the ratio of the true latitude difference to the stretched Mercator latitude difference — the same stretching that makes Greenland look continental.

The trade: the rhumb line is always longer, and the surplus grows with latitude and with east-west distance. It is a rounding error on a coastal hop and a serious fuel figure on a North Atlantic crossing. What you buy for it is one course to steer, which is worth a great deal on a small boat and very little on a ship with a plotter. One more caution: all of this is a sphere. Against the real ellipsoid the answer can be half a percent out, which is noise for a passage plan and money for a cable schedule — that is when Vincenty's method earns its keep.