What is an ellipse?

Image source: (http://www.qrg.northwestern.edu/projects/vss/docs/space-environment/2-how-ellipse-is-different.html)

Ellipse Definition: On a plane, ellipse is defined as follows – If two special points (called the foci) are picked on a plane and if we collect all the points around those foci such that the sum of the distances between any point in that collection and the two foci are a constant, then the locus of all these points form a curve called Ellipse.

Though this definition is for ellipse as a plane curve, this definition can be extended to define ellipse on non-planar surfaces, like for example on Earth.

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Ellipses are symmetric about exactly two axes that are perpendicular to each other. If we align those two axes along the two cartesian axes ##X## and ##Y## and have the point of intersection coincide with the coordinate origin then the ellipse can be described by the following simple equation,

Cartesian Equation of an Ellipse: ##frac{x^2}{a^2}+frac{y^2}{b^2}=1##.

Here ##a## is called the semi-major axis and ##b## is called the semi-minor axis.

Ellipses are characterised by a parameter called eccentricity (##e##) which is related to the semi-major and semi-minor axes as follows, ##e=sqrt{1-frac{b^2}{a^2}}##.

A circle is a special ellipse with eccentricity zero (##e=0##).

If one of the focus is placed at the coordinate origin and measure the angle ( ##theta## ) from the semi-major axis in the counter-clockwise direction, the ellipse of eccentricity ##e##, can be described by the following simple polar equation,

##r(theta) = frac{a(1-e^2)}{1+ecostheta}##

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