Equation of an Ellipse

Cartesian Equation of an Ellipse

The general equation for an ellipse where its major, or longer, axis is horizontal is : ${\frac {(x-h)^{2}}{a^{2}}}+{\frac {(y-k)^{2}}{b^{2}}}=1$.
Where the major axis is vertical, it is: ${\frac {(y-k)^{2}}{a^{2}}}+{\frac {(x-h)^{2}}{b^{2}}}=1$

• The center is located at $(h,k)$.
• The foci are found at a distance of $c$ from the centre along the major axis, where $c={\sqrt {a^{2}-b^{2}}}$.
• The eccentricity of the ellipse can be found from the formula: $b^{2}=a^{2}(1-e^{2})$ 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 $x=\pm {\frac {a}{e}}$
• The major axis has a length of $2a$ and the minor one $2b$.
• The sum of the distance from each point to each of the foci is $2a$.

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 ${\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1$ is:

$(x,\,y)=(a\cos t,\,b\sin t),\ 0\leq t<2\pi \ .$

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

Rational representation

With the substitution ${\textstyle u=\tan \left({\frac {t}{2}}\right)}$ and trigonometric formulae one obtains

$\cos t={\frac {1-u^{2}}{u^{2}+1}}\ ,\quad \sin t={\frac {2u}{u^{2}+1}}$

and the rational parametric equation of an ellipse

${\begin{aligned}x(u)&=a{\frac {1-u^{2}}{u^{2}+1}}\\y(u)&={\frac {2bu}{u^{2}+1}}\end{aligned}}\;,\quad -\infty <u<\infty \;,$

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

For $u\in [0,\,1],$ this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing $u.$ The left vertex is the limit ${\textstyle \lim _{u\to \pm \infty }(x(u),\,y(u))=(-a,\,0)\;.}$

Resources

• Equations of Ellipses, Lumen Learning

Licensing

Content obtained and/or adapted from:

• Ellipse, Wikibooks: Conic Sections under a CC BY-SA license
• Ellipse, Wikipedia under a CC BY-SA license