how to find the parametric equation of an ellipse?
x^2/a^2 + y^2/b^2 =1
thank you very much.
we know that $\displaystyle \cos^2 t + \sin^2 t = 1$
and we also know from our knowledge of polar coordinates that it is more appropriate to relate the x to the cosine and the y to the sine, so
$\displaystyle \frac {x^2}{a^2} + \frac {y^2}{b^2} = \left( \frac xa \right)^2 + \left( \frac yb \right)^2 = 1$
thus we can let $\displaystyle \cos t = \frac xa$ and $\displaystyle \sin t = \frac yb$ (and so we would obtain $\displaystyle \cos^2 t + \sin^2 t = 1$)
solving for $\displaystyle x$ and $\displaystyle y$ we get: $\displaystyle x = a \cos t$ and $\displaystyle y = b \sin t$ and we restrict $\displaystyle t$ to $\displaystyle 0 \le t \le 2 \pi$ since sine and cosine are periodic for that interval