Why does the sea horizon line always seems to be at the same height as one's eyes? I wonder why, when I look at a sea/ocean, the horizon line always seems to be at the same height as my eyes, no matter how many meters I am above the sea level. 
This is something I noticed when I trek, and it happened in many places. It's particularly noticeable when I climb down a mountain, from the peek to the beach. 
However, this isn't the case when landing in an airplane (although maybe in the last seconds, I haven't had the occasion to test), it seems one need to "touch the earth" to see this effect. 
Thanks a lot for your help!
Edit:
Here is my understanding of the triangle thing. But it doesn't make any sense as the higher I go, the less far I see. To make it right I should increase the lenght of the two opposite side (cathetus) of the hypothenus but to what extend should I increase them? 
Thanks a lot for your help! 

Second try! Where the earth look s like a "line"... I still don't get it!

 A: The picture drawn by docscience is basically correct, but the math that goes along with it was wrong - as a result, the "height of 370000 km" part of the answer is completely off.
The effect you describe is well known by navigators: when you measure the height of the sun or star with a sextant, you actually have to take into account the height of the eye above the water because even a few meters height difference is enough to affect your position calculation. 

Now 
$$\cos\theta=\frac{r}{r+h}\\
1-\cos\theta = \frac{h}{r+h}$$
And for $h<<r$ and small values of $\theta$, we obtain the approximation
$$h = \frac12 r\theta^2$$
So the angular error as a function of height can be given by
$$\theta = \sqrt{\frac{2h}{r}}$$
A plot of this value looks like this:

This graph demonstrates why this correction is so important for sextant observations: the initial slope is very steep, and at sea an error of one degree leads to a position error of 60 nautical miles... By comparison, your body's ability to tell "level" from "not quite level" is probably a couple of degrees - so you really need the altitude of a plane (or a high mountain next to the sea) to convince your brain that the horizon you are seeing is not "level".  According to my graph, 1 degree happens at about 1000 meters, and 2 degrees at 4000 meters. That seems consistent with your own observations.
A: The construction and calculations below show that if the altitude is very small compared to Earth's radius, the line of sight as measured from local vertical is very near 90 degrees. The Earth's radius is about 6371 km. For the line of sight to fall 1 degree you would have to elevate your point of view by about 370000 km!

A: The reason it may not happen in a plane is probably because you are tilted with respect to the horizon (even if it may not feel like it due to fictitious forces). "Touching the Earth", you are "forced" by gravity to be straight up, and thus the triangle described in the comment above works.
Also, these kinds of strange properties usually come from an infinity involved. In this case, it's the fact that the horizon is so far away it's almost like standing in an infinite plane.
In the situation of an infinite plane, it's easy to see that you get to the same situation if you climb up to twice your height, than if you scaled all distances by two (because twice infinity is infinity). And it's clear that an overall scaling wouldn't change the direction in which you see the horizon. Or, as they say, if you scaled everything you wouldn't notice it.
Finally, Earth isn't infinite, so you will eventually notice the horizon dropping if you go high enough, but close to the surface (including a mountain) it's very close to the infinite case.
