Construction of Conic Section

Theorem
Consider a right circular cone $\CC$ with opening angle $2 \alpha$ whose apex is at $O$.

Consider a plane $\PP$ which intersects $\CC$ at an angle $\beta$ to the axis of $\CC$.

Let the plane $OAA'$ through the axis of $\CC$ perpendicular to $\PP$ intersect $\PP$ in the line $AA'$.

Let $P$ be an arbitrary point on the intersection of $\PP$ with $\CC$.

Let $PM$ be constructed perpendicular to $AA'$.

$PM$ is then perpendicular to the plane $OAA'$.

The transverse section through $P$ then contains $PM$ and cuts $OAA'$ in $TT'$, which is a diameter of that transverse section.

The diagram below is the cross-section through $\CC$ corresponding to the plane $OAA'$.

The point $P$ is imagined to be perpendicular to the plane whose projection onto $OAA'$ coincides with $M$.


 * Conic-section-construction.png