Definition:Ellipse/Focus-Directrix
Definition
Let $D$ be a straight line.
Let $F$ be a point.
Let $e \in \R: 0 < e < 1$.
Let $K$ be the locus of points $P$ such that the distance $p$ from $P$ to $D$ and the distance $q$ from $P$ to $F$ are related by the condition:
- $e p = q$
Then $K$ is an ellipse.
Directrix
Let $K$ be the ellipse specified as the locus of points $P$ to a straight line $D$ and a point $F$ such that $PD$ and $PF$ are related by the condition:
- $\size {PF} = e \size {PD}$
where $0 < e < 1$.
The line $D$ is known as the directrix of the ellipse.
Focus
The point $F$ is known as the focus of the ellipse.
Eccentricity
Let $K$ be the ellipse specified as the locus of points $P$ to a straight line $D$ and a point $F$ such that $PD$ and $PF$ are related by the condition:
- $\size {PF} = e \size {PD}$
where $0 < e < 1$.
The constant $e$ is known as the eccentricity of the ellipse.
Also see
Historical Note
The focus-directrix definition of a conic section was first documented by Pappus of Alexandria.
It appears in his Collection.
As he was scrupulous in documenting his sources, and he gives none for this construction, it can be supposed that it originated with him.
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $1 \text a$. Focal properties
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ellipse
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ellipse