ATCast,
what a brilliant conundrum!
Let me restate it in geometric terms.
Suppose the earth is a sphere, radius X. Suppose the aircraft is flying at constant altitude Y (in the same units). Then the aircraft trajectory is describing a circle of radius (X+Y). Let us suppose the aircraft is flying unaccelerated in Earth-fixed coordinates (1g force in the z-direction, no force in x or y, no rotational component in any axis). We may measure its progress by the constant angular velocity A of the CofG relative to the center of the earth (compatible units).
If the ground speed is the speed of the aircraft's projection on the ground, in Earth-axis coordinates, then the ground speed is (A x X). If it is the absolute speed, then it is (A x (X+Y)).
If it is the "horizontal projection of absolute speed", then we need to take into account the projection of the horizon on the aircraft's trajectory. The horizon is below the aircraft's trajectory by an angle Alpha, so the component of the aircraft's absolute velocity (in Earth-axis coordinates) on this would be (A x (X+Y)) x cosine(Alpha). Cosine(Alpha) appears from simple geometry to be X /(X+Y), so this yields (A x X) as ground speed.
This seems to me to be counterintuitive. I would prefer to take ground speed as the rate of change of the distance travelled by the aircraft in Earth-axis coordinates, which is (A x (X+Y)). This is also the "true airspeed corrected for wind" which is mentioned by some commentators.
I have a query out to see what reasons there might be for choosing the one over the other.
PBL