Definition:Conic Section
Definition
Intersection with Cone
Let $C$ be a double napped right circular cone whose base is $B$.
Let $\theta$ be half the opening angle of $C$.
That is, let $\theta$ be the angle between the axis of $C$ and a generatrix of $C$.
Let a plane $D$ intersect $C$.
Let $\phi$ be the inclination of $D$ to the axis of $C$.
Let $K$ be the set of points which forms the intersection of $C$ with $D$.
Then $K$ is a conic section, whose nature depends on $\phi$.
Focus-Directrix Property
A conic section is a plane curve which can be specified in terms of:
- a given straight line $D$ known as the directrix
- a given point $F$ known as a focus
- a given constant $\epsilon$ known as the eccentricity.
Let $K$ be the locus of points $b$ such that the distance $p$ from $b$ to $D$ and the distance $q$ from $b$ to $F$ are related by the condition:
- $(1): \quad q = \epsilon \, p$
Then $K$ is a conic section.
Equation $(1)$ is known as the focus-directrix property of $K$.
Reduced Form
Let $K$ be a conic section.
Let $K$ be embedded in a cartesian plane such that:
for some $c \in \R_{\ge 0}$.
This can be interpreted in the contexts of the specific classes of conic section as follows:
Reduced Form of Ellipse
Let $K$ be an ellipse embedded in a cartesian plane.
$K$ is in reduced form if and only if:
- $(1)$ its major axis is aligned with the $x$-axis
- $(2)$ its minor axis is aligned with the $y$-axis.
Reduced Form of Hyperbola
Let $K$ be a hyperbola embedded in a cartesian plane.
$K$ is in reduced form if and only if:
- $(1)$ its major axis is aligned with the $x$-axis
- $(2)$ its minor axis is aligned with the $y$-axis.
Reduced Form of Circle
Let $K$ be a circle embedded in a cartesian plane.
$K$ is in reduced form if and only if its center is located at the origin.
Reduced Form of Parabola
Let $K$ be a parabola embedded in a cartesian plane.
As a Parabola has no Center, it is not possible to define the reduced form of a parabola in the same way as for the other classes of conic section.
Instead, $K$ is in reduced form if and only if:
- $(1)$ its focus is at the point $\tuple {c, 0}$
- $(2)$ its directrix is aligned with the line $x = -c$
for some $c \in \R_{> 0}$.
Also known as
A conic section is also known just as a conic.
Also see
- Results about conic sections can be found here.
Historical Note
The conic sections were first studied, by Menaechmus in around $\text {350 BCE}$.
He was the first to identify them as different types of slices through a right circular cone.
Other early investigations were made by Conon of Samos at around $\text {245 BCE}$.
Apollonius of Perga made an extensive study of them in around $\text {225 BCE}$, the results of which he published his book Conics.
He demonstrated rigorously that they can all be generated by different sections of the surface of a right circular cone.
Apollonius of Perga was also the first to recognise that a double napped cone is used to generate the hyperbola.
In the $17$th century, conic sections were initially investigated using the techniques of analytic geometry.
This was mainly initiated by Jan de Witt, who introduced the focus-directrix property in around $\text {1659}$ – $\text {61}$.
This was also done independently by John Wallis in $\text {1655}$.
Sources
- 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies
- 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): conic section
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): section (plane section)
- 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): conic section
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): section (plane section)
- Weisstein, Eric W. "Conic Section." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConicSection.html