Conics in real projective plane and application to Poncelet's theorem

Question

I know that all conics (ellipse, hyperbola and parabola) are equivalent in real projective plane. But when we are drawing things from projective plane on piece of paper, are we still allowed to draw hyperbolas and parabolas as we know them in $\mathbb{R}^2$?

And if so. I am trying to figure out Poncelet's theorem. It states: "Let K and C be nondegenerate conics in general position (two tangents from intersection points to both conics are different for all intersection points). Suppose there is an n-sided polygon inscribed in K and circumscribed about C such that none of its vertices belongs to C (in the case when K and C intersect or meet). Further suppose there is an (n − 1)-sided polygonal chain with vertices on K such that all its sides are tangent to C and none of its vertices belongs to C. Then the side, which closes the polygonal chain, is also tangent to C."

If I draw hyperbola $x^2 - y^2 = 1$ and ellipse $x^2+3y^2 =4$, then we can find 4-gon inscribed in ellipse and circumscribed in hyperbola. This can not be done if we take one of intersections as vertex of 4-gon or some other points on ellipse, but the theorems assumptions take care of that. Since this can be done for hyperbolla and ellipse, it sholud be possible for two ellipses that have 4 common points, lets say $4x^2+y^2=4$ and $x^2+4y^2=4$. But I dont see how is this possible.

Another thing. Let's say we do not have to have conics in general case. So we can have hyperbola $x^2 - y^2 = 1$ and ellipse $(x+1)^2+4y^2 =4$ (they have same tangent in (1,0)). It seems obvious (in this case), that we can not find such polygon and polygonal chain as Poncelet's theorem states. But is this an exception or a rule? How can we be sure, that if conics are not in general case, Poncelet's theorem can not be applied? (Therefore it is not necessary to say they are in general position.)

Answer 1

Poncelet's closure theorem (porism) states that if there is an $n$-sided polygon of a certain kind, then there are an infinite number of them. For any given pair of conics and integer $n\ge 3$ there may not be an $n$-sided Poncelet polygon. For example, for two non-intersecting circles in the plane there is a certain condition on the radii and distance between the centers that must be satisfied for a triangle to be inscribed in one circle and circumscribed around the other.

The theorem does not state that it is always possible to draw an $n$-sided polygon that is inscribed in one of the conics and circumscribed around the other conic. It states that if it is possible, then it is possible to do so with any starting point. There may be some obvious exceptions if the conics intersect each other. Also, the theorem may not always hold in the real plane.

The Wikipedia article Conic section states:

... That is, there is a projective transformation that will map any non-degenerate conic to any other non-degenerate conic.

The three types of conic sections will reappear in the affine plane obtained by choosing a line of the projective space to be the line at infinity. The three types are then determined by how this line at infinity intersects the conic in the projective space. In the corresponding affine space, one obtains an ellipse if the conic does not intersect the line at infinity, a parabola if the conic intersects the line at infinity in one double point corresponding to the axis, and a hyperbola if the conic intersects the line at infinity in two points corresponding to the asymptotes

Thus, in the real projective plane there is only type of non-degenerate conic, but when passing back to the affine or Euclidean plane by an arbitrary choice of a line "at infinity", the three different types appear. Also, in the Euclidean plane, the circle is a special case of an ellipse.