Some years ago, I wrote a program to compare distances and bearings on a spherical and an ellipsoid earth. In addition, it calculates some nice values with no direct utility. The formulae being not always self-explaining, I added remarks to remember what I did, as this one:

Geodesics vs. Great Circles >>>

The shortest path between two points on a curved surface is a Geodesic:

on a sphere, all Geodesics are part of a Great Circle,

on an oblate ellipsoid (the earth), only some Geodesics are part of a Great Circle, all meridians and the equator (the latter not for near antipodal distances).

A Great Circle always closes on itself without an intervening self-intersection, the sphere (and exceptionally the ellipsoid) is “cut” into two equal halves;

an oblique Geodesic on an ellipsoid self-intersects near its apex and nadir; it does not close on itself and therefore cannot contain a “cutting” surface.

A non-GC Geodesic around the ellipsoid earth returns to the original longitude higher or lower than the original latitude:

North or South difference depends on the initial bearing and on the geolatitude of the initial point.

Some sample values, calculated as ‘direct solution’ (given LatStart + InitBearing + Distance), by Vincenty and Karney (identical results) >>>

```
> LatStart InitBearing Distance Δ Latitude on Start-Longitude
> +47.57° 15° 40066.6 km 0.5270°N = 58.6 km N of Start
> +47.57° 30° 40061.0 km 0.4729°N 52.6 km N
> +47.57° 45° 40053.4 km 0.3857°N 42.9 km N
> +47.57° 60° 40045.9 km 0.2724°N 30.3 km N
> +47.57° 75° 40040.4 km 0.1398°N 15.5 km N
> +47.57° 90° 40038.5 km 0.0029°S 0.3 km S
> +47.57° 120° 40046.5 km 0.2796°S 31.1 km S
> +47.57° 150° 40062.2 km 0.4837°S 53.8 km S
> -47.57° 90° 40038.5 km 0.0029°N 0.3 km N
>
> +47.57° 89.69905° 40038.53628 km <1 millimeter off StartPos >>>
```

The last path returns quasi to its startpoint, its initial bearing is exactly at the limit between N and S deflection (analogously with all other start latitudes).

However, it is not a true Great Circle, the half-distance point is 81 meters South of the true geometric antipodal point and,

with the same initial bearing, but double distance (twice around the earth), the path returns to the initial longitude 645 meters South of the start-point.