Dandelin's Theorem/Directrices
Theorem
Let $\CC$ be a double napped right circular cone with apex $O$.
Let $\PP$ be a plane which intersects $\CC$ such that:
- $\PP$ does not pass through $O$
- $\PP$ is not perpendicular to the axis of $\CC$.
Let $\EE$ be the conic section arising as the intersection between $\PP$ and $\CC$.
Let $\SS$ and $\SS'$ be the Dandelin spheres with respect to $\PP$.
Let $\KK$ and $\KK'$ be the planes in which the ring-contacts of $\CC$ with $\SS$ and $\SS'$ are embedded respectively.
- The intersections of $\KK$ and $\KK'$ with $\PP$ form the directrices of $\EE$.
Proof
Let $\SS$ and $\SS'$ be the Dandelin spheres with respect to $\PP$.
Let $P$ be a point on $\EE$.
Let $F$ and $F'$ be the points at which $\SS$ and $\SS'$ are tangent to $\PP$ respectively.
Let the generatrix of $\CC$ which passes through $P$ touch $\SS$ and $\SS'$ at $E$ and $E'$ respectively.
Let $\theta$ be half the opening angle of $\CC$.
Let $\phi$ be the inclination of $\PP$ to the axis of $\CC$.
Let $\PP$ intersect $\KK$ in the straight line $NX$.
Let $PN$ be constructed perpendicular to $NX$.
Let $PK$ be constructed perpendicular to $\KK$.
Then:
- $PK = PN \cos \phi$
Also:
- $PK = PE \cos \theta = PF \cos \theta$
Hence:
- $\dfrac {PF} {PN} = \dfrac {\cos \phi} {\cos \theta}$
which is constant.
Hence $NX$ is a directrix of $\EE$ by definition.
A similar argument and construction applies with respect to $\SS'$ and $\KK$.
$\blacksquare$
Also see
Source of Name
This entry was named for Germinal Pierre Dandelin.
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $1 \text a$. Focal properties