What is the parametric equation of an ellipse?

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Answer 1 · 2018-04-21T00:39:33

Here is one example...

You can have $(n sin (t), m cos (t))$ $(nsin(t),mcos(t))$ when $n \neq m$ $n!=m$ , and $n$ $n$ and $m$ $m$ do not equal to $1$ $1$ .

This is essentially because:

$\Rightarrow x = n sin (t)$ $=>x=nsin(t)$

$\Rightarrow x^{2} = n^{2} {sin}^{2} (t)$ $=>x^2=n^2sin^2(t)$

$\Rightarrow \frac{x^{2}}{n^{2}} = {sin}^{2} (t)$ $=>x^2/n^2=sin^2(t)$

$\Rightarrow y = m cos (t)$ $=>y=mcos(t)$

$\Rightarrow \frac{y^{2}}{m^{2}} = {cos}^{2} (t)$ $=>y^2/m^2=cos^2(t)$

$\Rightarrow \frac{x^{2}}{n^{2}} + \frac{y^{2}}{m^{2}} = {sin}^{2} (t) + {cos}^{2} (t)$ $=>x^2/n^2+y^2/m^2=sin^2(t)+cos^2(t)$

Using the fact that ${sin}^{2} (x) + {cos}^{2} (x) = 1$ $sin^2(x)+cos^2(x)=1$ ...

$\Rightarrow \frac{x^{2}}{n^{2}} + \frac{y^{2}}{m^{2}} = 1$ $=>x^2/n^2+y^2/m^2=1$

This is essentially an ellipse!

Note that if you want a non-circle ellipse, you have to make sure that $n \neq m$ $n!=m$