View source for Equation of an Ellipse

[[Image:ElipseAnimada.gif|thumb]]
[[Image:Ellipse axis.png|thumb]]

==Cartesian Equation of an Ellipse==
The general equation for an ellipse where its major, or longer, axis is horizontal is : <math>\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1</math>.<br />
Where the major axis is vertical, it is: <math>\frac{(y-k)^2}{a^2}+\frac{(x-h)^2}{b^2}=1</math>
* The center is located at <math>(h,k)</math>.
* The foci are found at a distance of <math>c</math> from the centre along the major axis, where <math>c = \sqrt{a^2 - b^2}</math>.
* The eccentricity of the ellipse can be found from the formula: <math>b^2 = a^2(1-e^2)</math> 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 <math>x = \pm \frac{a}{e}</math>
* The major axis has a length of <math>2a</math> and the minor one <math>2b</math>.
* The sum of the distance from each point to each of the foci is <math>2a</math>.

== Parametric representation ==
[[File:Elliko-sk.svg|thumb|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 <math>\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2} = 1</math> is:
: <math>(x,\, y) = (a \cos t,\, b \sin t),\ 0 \le t < 2\pi\ .</math>

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

===Rational representation===
With the substitution <math display="inline">u = \tan\left(\frac{t}{2}\right)</math> and trigonometric formulae one obtains
:<math>\cos t = \frac{1 - u^2}{u^2 + 1}\ ,\quad \sin t = \frac{2u}{u^2 + 1}</math>

and the ''rational'' parametric equation of an ellipse
: <math>\begin{align}
  x(u) &= a\frac{1 - u^2}{u^2 + 1} \\
  y(u) &= \frac{2bu}{u^2 + 1}
\end{align}\;,\quad -\infty < u < \infty\;,</math>

which covers any point of the ellipse <math>\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1</math> except the left vertex <math>(-a,\, 0)</math>.

For <math>u \in [0,\, 1],</math> this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing <math>u.</math> The left vertex is the limit <math display="inline">\lim_{u \to \pm \infty} (x(u),\, y(u)) = (-a,\, 0)\;.</math>

==Resources==
* [https://courses.lumenlearning.com/waymakercollegealgebra/chapter/equations-of-ellipses/ Equations of Ellipses], Lumen Learning

== Licensing == 
Content obtained and/or adapted from:
* [https://en.wikibooks.org/wiki/Conic_Sections/Ellipse Ellipse, Wikibooks: Conic Sections] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Ellipse Ellipse, Wikipedia] under a CC BY-SA license