How can one prove that the resulting projection onto a "canvas" of a conic is still a conic? In one of my recent exams from an analytical geometry course, we were asked to prove that the resulting image of the projection from a point onto a plane of an ellipse, is still an ellipse. I had no problem understanding a given proof that used analytical geometry arguments.
I was wondering, however, how one could prove that (or a more general) result using only arguments from pure/synthetic geometry, i.e., that the resulting image of the projection from a point onto a plane of a conic, is still a conic.
To be precise, I'm considering the projection from a point $\overrightarrow{P_0}$ onto a plane ($\alpha$) of a point ($\overrightarrow{P}$) in space as the point obtained when intersecting the line $\overrightarrow{P_0}+t(\overrightarrow{P_0}−\overrightarrow{P_0}$) and the plane $\alpha$.
Thanks in advance!
 A: I think the best approach would be to use the properties of an ellipse in terms of directrix and foci and eccentricity, and demonstrate that those properties remain the same or proportional.  For example, it should not be hard to show that a line projects onto a line, and a point projects onto a point.  So you could make an image of the directrix $D$, and an image of the focus $F$, and for an arbitrary point on the curve $Q$ show that its image $Q'$ shares the same ratio of distances $QF/QD$ as the original.  That would prove that the image is an ellipse, and the proof should generalize to all conic sections.
A: In the synthetic approach of Apollonios of Perga's Conics, a conic section is generated by cutting a conic surface with a plane, as in the figure below.
Thus it seems any conic section, such as ellipse $EFG$, can be  projected from the cone's vertex at $P$ either onto the plane of the cone's base to produce circle $ABC$, or onto a plane parallel to $PA$ to produce a parabola $DJK$, or onto any other plane intersecting the conic surface, to produce any other possible sections of the given cone.
A: I advise you to have a look here more precisely in the section "Two theorems" and the figure therein. It is based on the understanding of the so called "Dandelin spheres".
They allow a rather simple geometrical proof of the fact that the image of a circle on a certain base plane becomes an ellipse on a slant plane by using the characterization of an ellipse as the set of points such that the sum of distances to its foci is constant.
One can also, by similar means, obtain hyperbolas or parabolas.
