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In a projective plane I choose a line at infinity and a conic section not intersecting that line, i.e. an ellipse.

On the ellipse I choose a direction or orientation.

I move the ellipse to the line at infinity and pass through it.
Visually on the paper the orientation has changed its direction.

How do I prove that without using formulas and without using a third dimension?

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    If you don't want formulas, can you use a [picture](https://upload.wikimedia.org/wikipedia/commons/thumb/9/9b/ProjectivePlaneAsSquare.svg/240px-ProjectivePlaneAsSquare.svg.png)? – 2017-01-21
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    Ok I understand the picture and it indeed shows how orientation is reversed. But is it possible to use that picture with its border at infinity? The square form is irrelevant it may as well be just a circle. – 2017-01-22
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    But when I translate an ellipse to infinity how do I see that the one side of the circle returns on the other side after passing infinity? – 2017-01-22

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The short answer is to say that since antipodal points at infinity are identified, anything passing through infinity gets orientation-reversed. But if you'd count this as a proof, you'd probably not be asking. So I'll try to stick closer to what you have in your situation.

In between the ellipse this side of infinity and the ellipse on the other side of infinity, there likely is a situation where the “ellipse” does cross the line at infinity. In that case, it's not an ellipse any more, but a hyperbola. Think of the hyperbola $xy=1$ for a moment. It crosses the line at infinity in the points at infinity in the $x$ direction and in the $y$ direction, corresponding to the asymptotict directions of the hyperbola. Suppose you pick a counter-clockwise orientation of the hyperbola in the first quadrant. That means traveling along the conic in the chosen direction you'll come from “up”, and move first “down” then to “right”. Passing through infinity in the “right” direction, you then traverse the hyperbola branch in the third quadrant startig “left” and going “right” then “down”. So you have a clockwise change of direction there. So the cyclic orientation observed in one branch of the hyperbola is opposite that in the other branch. Moving your ellipse through infinity, you are enlarging one branch and reducing the other, so the whole process will swap the orientation.

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    This will do. Thank you, I don't know why I didn't come up with that myself ... – 2017-02-02