Ellipse

Definition: Ellipse

Let $\lambda$ be a plane and let $F_1$ and $F_2$ be points from $\lambda$.

The ellipse $E$ with foci $F_1$ and $F_2$ is the subset of $\lambda$ such that the sum of the distances between each point of $E$ and the points $F_1$ and $F_2$ is constant.

$$E=\{P\in \lambda \mid d(P;F_1)+d(P;F_2)=\text{const}\}$$

The midpoint $O$ of the line segment $F_1{F}_2$ is known as $E$‘s center.

The major axis of $E$ is the line segment joining the two points of $E$ which are the farthest away from $O$.

The minor axis of $E$ is the line segment connecting the two points of $E$ which are the closest to $O$.

Definition: Linear Eccentricity

The linear eccentricity of an Ellipse is the distance from its center to its foci.

NOTATION

The linear eccentricity is usually denoted by $c$.

Theorem: Calculating Linear Eccentricity

The linear eccentricity of an Ellipse can be calculated using the length $a$ of its semi-major axis and the length $b$ of its semi-minor axis:

$$c=\sqrt{a^2-b^2}$$

PROOF

TODO

Properties

Theorem: Distance in Ellipses

For each point $P$ of an Ellipse $E$, the sum of the distances from $P$ to the foci $F_1$ and $F_2$ is equal to the length of the major axis.

$$d(P,F_1)+d(P,F_2)=2a$$

PROOF

TODO

Standard Equation

Theorem: Standard Equation of an Ellipse

If $E$ is whose center has coordinates $(0,0)$, then the coordinates $(x,y)$ of each point of $E$ satisfy

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$$

where $a$ is half the length of $E$‘s major axis and $b$ is half the length of $E$‘s minor axis.

PROOF

TODO