Definition of a Conic

Definition Let a line, the axis, and a second line, the generator, meet at a point $V$, the vertex, at an angle $\alpha$ with $0 < \alpha < \frac{\pi}{2}$. Rotating the generator about the axis sweeps out a right circular double cone. With vertex at the origin and the $z$ axis as axis, it is the surface

The two parts with $z \ge 0$ and $z \le 0$ are the nappes, and they meet only at the vertex.

Definition Let a plane not through the vertex meet the double cone with half angle $\alpha$, and let $\beta$ be the angle between the plane and the axis, $0 \le \beta \le \frac{\pi}{2}$. The intersection is

• an ellipse if $\beta > \alpha$, and a circle when $\beta = \frac{\pi}{2}$,
• a parabola if $\beta = \alpha$, that is when the plane is parallel to a generator,
• a hyperbola if $\beta < \alpha$, which cuts both nappes.

Definition The ellipses, parabolas and hyperbolas from planes that miss the vertex are the nondegenerate conics. A plane through the vertex gives a degenerate conic, which is a single point, a single line (when the plane touches the cone along a generator), or a pair of lines crossing at the vertex.

Definition Fix a point $F$, the focus, a line $L$ not through $F$, the directrix, and a number $e > 0$. The set of points $P$ in the plane with

is a conic. It is an ellipse if $0 < e < 1$, a parabola if $e = 1$, and a hyperbola if $e > 1$.

Definition The number $e$ in the focus directrix definition is the eccentricity of the conic. A circle is given eccentricity $e = 0$, the limit of ellipses as $e \to 0$, although a circle has no directrix.

Definition In a coordinate plane every conic, nondegenerate or degenerate, is the set of solutions of a second degree equation

Conversely, the solution set of every such equation is an ellipse (possibly a circle), a parabola, a hyperbola, a single point, a single line, a pair of crossing lines, a pair of parallel lines, or the empty set. Only the last two cannot be cut from a double cone by a plane.