Let me see if I can draw this as an ascii picture

So heres the earth (hypothetically) as a globe.

..,,,----'''''''''----,,,

(...................)

..'''----.....----''''

Ok so we want to know what the "curvature" is over a distance such as 100 miles

But that definition does not represent what we think it does.

At a glance we think curvature measures top-most first comma (,) to the top-most last apostrophe (')

..,,,----'''''''''----,,,

(...................)

..'''----.....----''''

It isnt.

So I will draw a line to show what curvature actually measures:

..,,,----'''''''|---,,,

(...................)

..'''----.....----''''

Curvature is the distance between the same first comma (,) to the bottom point of the pipe (|)

The bottom point of the pipe digs deep within the surface at a 90 degree pie.

Edit

I will attempt to illustrate this again but written

Imagine a pumpkin pie.

The outer crust is round and completes a circle of crust around the pie.

Now cut a slice of the pie thin or thick.

You make two cuts and are left with 1 side round and 2 sides triangular on a straight line

What we want to measure is the point of one side of the crust to the other side of the crust on the rounded side.

However, we have accidentally measured the "curvature" which is the distance the cut crust starts on a 90 angle deep below the crust towards the pointed side.

We actually want to measure what ( I think) is defined as "horizon dip"

Any considerstions on what it would take to disprove refraction?