Of course there is no curvature. All the points on the horizon lie on the same plane level. It's like holding a hoola hoop horizontally and placing your head in the center. You don't see the curvature of the ring, you see a straight line.

Sorry, again as stated earlier, the horizon curves, but so faint, that's impossible to see it without additional references.

Your example:

Placing the hoola hoop exactly at your eye level, so that the observer is exactly at center of the circle, yes, what you see is a straight line. If you lower the hoola hoop, you see part of an ellipse!

And yes, the horizon is a circle, mathematically what you get if you intersect a cone (or easier a plane) with a sphere.

On a sphere the observer cannot be at the center of the circle of the horizon, he would have to dig a hole, but then could not see anything. The center of the horizon circle is below the observer.

The normal field of view of human eyes is about 110° (about 120°, 2*60°, cos(60°)=0.5).

The dip angle in arc minutes (1°/60) of the central point is about 2.1*sqrt(h) (h=observers hight in meters).

The formula is for nautical miles, which makes it easy, as 1 nm = 1 arc minute.

Not changing your view, the sides at the limits of you field of view are about cos(60°)*h lower. So the additional dip angle is about half of the dip angle at the center.

some examples

h=2m, distance to horizon 3 nm, dip 3 arc minutes, additional dip 1.5 arc minutes, almost imperceptible.

h=10m, distance to horizon 6.6 nm, dip 6.6 arc minutes, additional dip 3.3 arc minutes, even with a reference line hard to see.

h=90m, distance to horizon 20nm, dip 20 arc minutes 0.33°, additional dip 0.16°, could be seen with a reference line.

Larger distances would blur the horizon, so it cannot be measured precisely any more, due to atmospheric diffraction.

So to do the experiment right, you would need.

Excellent viewing conditions (20nm)

clear view of the horizon over the complete field of your vision.

a slight sea, no waves

moderate hight of observer

reference line - Try to align a water spirit level with the horizon line.

The observable dip and so the curvature, measured as viewing angle is only fractions of one degree, and conditions to see it are rare.

Mr. Rowbotham did a similar experiment in EnaG

http://www.sacred-texts.com/earth/za/za12.htmbut he didn't apply the necessary accuracy.

And overestimated the curve by far. The other figures he gives are ridiculous.