Equivalence of Definitions of Conic Section

Theorem
The following definitions of a conic section are equivalent:

Proof

 * ApolloniusPappusEquivalence.png

Consider the cone in which $\alpha$ is half the opening angle.

Let the tilting angle of the slicing plane to the horizontal be $\beta$.

The conic section thus defined is given here as an ellipse, but the argument holds for the other cases also.

Inside the cone, let a sphere be inscribed which is tangent to the slicing plane at the point $F$.

Let the sphere be tangent to the cone along the circle $C$.

Let $d$ be the line in which the slicing plane intersects the plane of the circle $C$.

It is to be demonstrated that the conic section induced by the slicing plane has $F$ as its focus and $d$ as its directrix.

Let $P$ be a point on the conic section.

Let $PQ$ be constructed parallel to the axis of the cone such that $Q$ lies on the plane of the circle $C$.

Let $R$ be the point where the generatrix of the cone through $P$ intersects $C$.

Let a perpendicular be constructed from $P$ to $d$, intersecting $d$ at $D$.

Hence:
 * $\beta = \angle PDQ$

We have that $PR$ and $PF$ are both tangents to the sphere from the same point $P$.

Thus $PF$ and $PR$ are the same length:
 * $(1): \quad PR = PF$

From the right triangle $\triangle PQR$:
 * $PQ = PR \cos \alpha$

From the right triangle $\triangle PQD$:
 * $PQ = PD \sin \beta$

Thus:

Let us define:
 * $e = \dfrac {\cos \gamma} {\cos \alpha}$

For a given slicing plane and cone this is constant.

It is noted that:


 * $(1): \quad$ when $\gamma < \alpha$, that is, the tilting angle is less than half the opening angle of the cone, the conic section is an ellipse


 * $(2): \quad$ when $\gamma = \alpha$, that is, the tilting angle equals half the opening angle of the cone, the conic section is a parabola


 * $(3): \quad$ when $\gamma > \alpha$, that is, the tilting angle is greater than half the opening angle of the cone, the conic section is a hyperbola


 * $(4): \quad$ when $\gamma = 0$, that is, the slicing plane is parallel to the plane of the circle $C$., the conic section is a circle and there is no directrix

The result follows.