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Projective space can, in principle, be thought of as Euclidean space supplemented with points at infinity. Say, we have a hyperbola on projective plane. It has two asymptotes, one for each branch. And those asymptotes are not parallel lines (in projective geometry they themselves may cross at any angle, but zero). It means that they can not cross at the same point of infinity, which would give us a nice closed curve. Instead, they both cross the same (and the only) line at infinity at different points. That makes a closed hyperbola consisting of two parts: the curved segment and the line segment at infinity. How can it be considered as a qualitative equivalent of ellipse that is curved everywhere?

The same is true for parabola, there are no asymptotes here, but tangent lines are not parallel anywhere, which means a parabola can not be closed without some kind of similar linear construction, can it?