Definition:Conic Section

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Definition

Intersection with Cone

ConicSections-All.png

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:

$(1)$ the center is at the origin
$(2)$ the foci are at $\left({\pm c, 0}\right)$

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.


EllipseReducedForm.png


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.


HyperbolaReducedForm.png


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.


CircleReducedForm.png


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 $c \in \R_{> 0}$.


ParabolaReducedForm.png



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

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