# Ellipse

Just like a circle has one focal point – center, ellipse has two. We are using those two foci and one positive number to define it… But, lets start from the beginning.

Through history people were always interested in the universe, especially planet movement. In the very beginnings of astronomy, it was considered that the Earth was in the center of the universe and that all planets revolve around it in concentric circles. Later on, when new technology was introduced, telescopes, better lenses and more collected data people started noticing some irregularities.

These irregularities were explained later on, when a man named Tycho Brahe set a theory where planets revolve around the Sun in elliptic paths. People used to think that those elliptic paths are nothing but irregular circles, but studying their properties they found out that ellipses were everything but irregular.

If $F_1, F_2$ are two fixed points of the plain M, and a positive real number greater than $\frac{1}{2} \mid F_1 F_2 \mid$, ellipse is the set that contains all points of the plain for which the sum of the distances from points $F_1$ and $F_2$ is a constant and equal to $2a$.

To put it in a mathematical form, the definition states:

$E = (T \in M : \mid F_1 T \mid + \mid F_2 T \mid = 2a, a \in \mathbb{R})$

Let’s explain this definition  using the drawing of the ellipse.

$\mid F_1 I \mid + \mid F_2 I \mid = \mid F_1 L \mid + \mid F_2 L \mid \rightarrow e + f = i + j$

This will be valid for any point on the ellipse.

Points $F_1$ and $F_2$ are called foci. The midpoint $O$ of the segment $\bar{F_1F_2}$ is called the center of an ellipse.

The line $F_1F_2$ cuts the ellipse in two points – A and B. Those points are called vertexes. Just the same, the center line of the segment $F_1F_2$ cuts the ellipse in points C and D. Those points are also called vertexes.

Segment $\bar{AB}$ is called the major, and segment $\bar{CD}$ is called the minor axis.

Segments $\bar{OA}$ and $\bar{OB}$ are called semi-major axes whose length is usually marked with $a$. Segments $\bar{OC}$ and $\bar{OD}$ are called semi-minor axes whose length is usually marked with $b$.

Since the two most important points in the ellipse are the foci, their distance from the center is also very important. Number $e = \mid F_1O \mid = \mid F_2O \mid = \frac{1}{2} \mid F_1 F_2 \mid$ is called linear eccentricity of the ellipse.

From here we can see an link between major axis, minor axis and linear eccentricity:

$b^2 = a^2 – e^2$

Coefficient $\epsilon = \frac{e}{a}$ is called numerical eccentricity of an ellipse.