Do we see an ellipse when we view Earth's horizon? I wanted to try to determine mathematically exactly how much curvature one should see on the horizon when I viewed it from a given height, and it hit me that looking at the horizon is not the same as viewing a circle while standing perpendicularly to it. The horizon is a tilted circle, extremely so when one is viewing it from Earth's surface. Does that mean that, like with a smaller tilted circle, its apparent image is elliptical rather than circular?
To clarify what I mean when I say the horizon is a tilted circle, I mean first that it's a circle, with its center somewhere in Earth's interior directly beneath you. By tilted, I mean your line of sight isn't perpendicular to the plane it's in when you look out at it. You're looking at it from a tiny angle, like tilting a circle 89 degrees.
Am I wrong about this? Does anyone have a better idea of what shape we should expect to see on the horizon as our altitude increases?
 A: Imagine that your view is from a point located a height H above any arbitrary place on the globe.  Now draw all the lines from your point of view which are tangent to the globe.  Those lines form a cone, and the points where the lines are tangent to the globe form a circle.  That circle is the horizon.
That is a little different from what you might see.  What you see depends on where you look and what your brain does with the information it receives.  A person with certain kinds of distorted vision can see a straight horizontal line as a curve that rises on both ends. So I'm guessing that your question is more about the geometry of the point of view than about what a person might perceive.
A: The answer depends on what kind of eyes you are using (lol). We call them projections.
The first kind is perspective projection. Remember how the railways go closer and closer together as they extend farther and farther away. If they are infinitely long (neglecting the curvature of Earth), they will 'vanish' at a point. When you draw the railways on a paper, they will form 2 straight lines meeting at a point. This is because you are imagining a virtual 'canvas' in 3D space, and projecting the scene onto the canvas, i.e. connect each point you see in real space with your eye, the line will intersect with the canvas, and the intersection point is the corresponding point.

In this way, it is not an ellipse. Actually, it is a complicated shape because farther parts of the circle seems smaller, and breaks the symmetry.
The second type of projection is parallel projection. It is similar to how sunlight casts shadows on a flat ground. In this case, putting objects farther away does not affect the apparent size, and it is a linear transformation. Therefore, the shape is an ellipse.
