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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.

draw a perfect ellipseIf 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.

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.

valid pointsPoints 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.

minor and major axisSegments \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.

semi-major and semi-minor axesSince 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.

linear eccentricityFrom 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.

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