Equation of an Ellipse

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