Equation of an Ellipse

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Cartesian Equation of an Ellipse

The general equation for an ellipse where its major, or longer, axis is horizontal is : .
Where the major axis is vertical, it is:

  • The center is located at .
  • The foci are found at a distance of from the centre along the major axis, where .
  • The eccentricity of the ellipse can be found from the formula: where e is eccentricity. A higher eccentricity makes the curve appear more 'squashed', whereas an eccentricity of 0 makes the ellipse a circle.
  • The directrices are the lines
  • The major axis has a length of and the minor one .
  • The sum of the distance from each point to each of the foci is .

Parametric representation

The construction of points based on the parametric equation and the interpretation of parameter t, which is due to de la Hire

Standard parametric representation

Using trigonometric functions, a parametric representation of the standard ellipse is:

The parameter t (called the eccentric anomaly in astronomy) is not the angle of with the x-axis, but has a geometric meaning due to Philippe de La Hire.

Rational representation

With the substitution and trigonometric formulae one obtains

and the rational parametric equation of an ellipse

which covers any point of the ellipse except the left vertex .

For this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing The left vertex is the limit

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