Definition:Conic Section/Eccentricity

This page is about Eccentricity of Conic Section. For other uses, see eccentricity.

Definition

Let $K$ be a conic section specified in terms of:

a given straight line $D$

a given point $F$

a given constant $e$

where $K$ is 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:

$q = e p$

The constant $e$ is known as the eccentricity of the conic section.

Also denoted as

Some sources use $\epsilon$ for the eccentricity of a conic section.

Also see

• Results about the eccentricity of a conic section can be found here.

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

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.21$: Newton's Law of Gravitation
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.8$: Pappus (fourth century A.D.): Appendix: The Focus-Directrix-Eccentricity Definitions of the Conic Sections
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): conic (conic section)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): eccentricity
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conic (conic section)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): eccentricity
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): eccentricity