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.


Linguistic Note

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


Historical Note

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


Sources