Elliptic mountains and conic sections

Why is an ellipse actually a conic section? And what symmetries does a conic section have? The exhibits “Ellipse Mountains” and “Cone Sections” deal with these two questions. A conic section is defined as the intersection of a plane with a double cone. From your school lessons you know that parabolas, ellipses and hyperbolas can appear as such conic sections. But why is this true? And why do all these curves (except for the parabola) have two lines? These are the questions we want to explore in this in-depth text.

And now … the mathematics of it:

The standard double cone with apex in the coordinate origin is given by the equation x^2+y^2=z^2.

Dieses Bild hat ein leeres Alt-Attribut. Der Dateiname ist Ellipsengebirge-781x1024.jpg
Figure 1: The exhibit “Ellipse Mountains” shows the mirror symmetry of the ellipse