Definition:Ellipse

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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 $\phi$.


ConicSections-Ellipse.png


Let $\theta < \phi < \dfrac \pi 2 - \theta$.

That is, the angle between $D$ and the axis of $C$ is between that for which $K$ is a circle and that which $K$ is a parabola.

Then $K$ is an ellipse.


Focus-Directrix Property

EllipseFocusDirectrix.png


Let $D$ be a straight line.

Let $F$ be a point.

Let $\epsilon \in \R: 0 < \epsilon < 1$.


Let $K$ be the locus of points $b$ such that the distance $p$ from $P$ to $D$ and the distance $q$ from $P$ to $F$ are related by the condition:

$\epsilon \, p = q$


Then $K$ is an ellipse.


Equidistance Property

EllipseEquidistance.png

Let $F_1$ and $F_2$ be two points in the plane.

Let $d$ be a length greater than the distance between $F_1$ and $F_2$.

Let $K$ be the locus of points $P$ which are subject to the condition:

$d_1 + d_2 = d$

where:

$d_1$ is the distance from $P$ to $F_1$
$d_2$ is the distance from $P$ to $F_2$.

Then $K$ is an ellipse.


Parts of Ellipse

EllipseParts.png

Consider an ellipse $K$ whose foci are $F_1$ and $F_2$.


Center

The center of $K$ is the point midway between the foci.

By definition of the major axis and minor axis, this is the point where the major axis and minor axis of $K$ cross.


Major Axis

The major axis of $K$ is the line segment passing through both $F_1$ and $F_2$ whose endpoints are where it intersects $K$.


Minor Axis

The minor axis of $K$ is the line segment through the center of $K$ perpendicular to the major axis of $K$ such that its endpoints are the points of intersection with $K$.


Vertex

A vertex of $K$ is either one of the endpoints of the major axis of $K$.


Covertex

A covertex of $K$ is either one of the endpoints of the minor axis of $K$.


Reduced Form

Let $K$ be an ellipse embedded in a cartesian coordinate 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


Also see

  • Results about ellipses can be found here.


Historical Note

The word ellipse was provided by Apollonius of Perga, who did considerable work on establishing its properties.


Linguistic Note

The word ellipse is pronounced with the stress on the second syllable: el-lipse.


Sources