Conic Sections (the cross-sections of a double cone)

Think of the double cone as something physical you could actually cut — a cone-shaped vegetable like a carrot, or two ice-cream cones joined tip to tip. Slicing it with a flat plane, at different angles relative to the axis, produces every one of the conic sections:

1. Circle — if the cutting plane is perpendicular to the axis and does not pass through the vertex, the cross-section is a circle. (This is the special case you have already studied in an earlier chapter.)
2. Parabola — if the plane is parallel to exactly one generator of the cone (i.e. tilted so that it never meets that one generator again) and does not pass through the vertex, the cross-section is a parabola.
3. Ellipse — if the plane is oblique to the axis (neither perpendicular to it nor parallel to any generator) and cuts only one nappe all the way around, the cross-section is a closed oval curve, an ellipse.
4. Hyperbola — if the plane is parallel to the axis itself and is positioned so that it cuts through both nappes of the double cone, you get two separate open curves, one in each nappe — together these form a hyperbola.
5. Pair of straight lines (a degenerate case) — if the plane contains a generator of the cone and is tangent to the cone along that generator, the "cross-section" collapses to that single generator; more generally, a plane through the vertex tilted like the ellipse/hyperbola/parabola case gives a pair of straight lines crossing at the vertex, rather than a curved conic. …