In short, two very good reasons, because if we measured from the center of the Earth:
- mountains that are easy climbs would be taller than mountains that are really really hard to climb because the Earth is bulged almost 22 km at the Equator
- if you say, “ok, Ciro, I will use an ellipsoid with 22 km more at the Equator as a reference”, as GPSs do, then you would measure height of rivers and some of those rivers would go uphill, depending on local gravity.
The second reason I give you is the reason why we use sea level as a reference.
As you can imagine, in common, normal life, we, regular folks, say that the sea is below the rivers.
Thus, we want to have heights measured around us that confirm the fact that water goes downhill until it reaches the sea.
Farmers would mock me if after using my really really expensive GPS that makes corrections based on the daily position of the Poles, measured by interferometry using a satellite that hovers 300.000 km above Earth and by also using atomic clocks on space in the US GPS network, Russian Glonass, European Galileos and Chinese Beidou, the ditch I designed flowed uphill.
I think this is crooked work, said the farmer
So, we use gravitational height.
This means we measure altitude:
- not with a sphere measured from the center of Earth
- nor with an elllipsoid taking in account Equatorial bulge but
- we do measure altitude using a reference surface called a geoid, which is expressed mathematically as sinusoidal waves on top of the ellipsoid that corrects height for measured gravitational variations which includes changes in direction of the vertical (nadir) and thus of horizon
Here you have the ellipsoid (including the Equatorial bulge) in red, the geoid in blue that takes in account gravitational anomalies and in black the real surface of Earth.
Notice that you can determine the geoid in blue by observing what the water would do.
Also notice that vertical distances in an ellipsoid when measured towards the center of Earth are not perpendicular to the ellipsoidal surface even if by a minute amount.
Finally I am not sure if you can imagine that, unlike on a sphere, latitude and longitude when projected over a geoid or even an ellipsoid vary with altitude.
It is a complicated world we live in when you can mesure things to the millimeter.
Notice that the heights given to you by a GPS are measured between the Earth surface in black and the ellipsoid in red, while the farmer that hired you wants to know geoid heights measured between the blue line of geoid and the black line of real surface and it can vary up to 85 meters…
In short, we use a complicated reference surface:
- because the Earth rotates and thus has an Equatorial bulge
- because the Earth is not uniformly dense and thus vertical directions and horizons change
- because we want to use altitude to predict what happens with water around us and with other things, like missiles, space rockets or long distance artillery fire.
If we followed the premise of the question we would be unable to trace agricultural ditches and kill people at long distances and we really, really like to eat and to kill people with minimal effort.
Next time you watch a thriller and you are eating popcorn and see the hero launch a missile, you can say “hey, this is the reason why we have geoids!! Popcorn and Schwarzenegger!!”
Geoids, geoids everywhere…