Suppose I have a thing such as an ellipse:
$$\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1$$
now we can define it so that $\frac{x}{a}=cos(\theta)$ and $\frac{y}{b}=sin(\theta)$. I know the perimeter formula
$$\mu(S)=\int\sqrt{1+\left(f'(x)\right)^{2}} dx.$$
It is easy to paramerize the ellipse but how can I parametrize the perimeter formula so that I can easily calculate the perimeter?
I find that I am doing things the hard way like this:
$$y=\pm b \sqrt{1-\left(\frac{x}{a}\right)^{2}}$$
now if I plug in the y into the formula of perimeter, it is messy. Can I do it elegantly with parametric form somehow?
Best Answer
I would write the following
$$ds=\sqrt{\left(\frac{dx}{d\theta}\right)^2+\left(\frac{dy}{d\theta}\right)^2}d\theta$$
and so
$$P=\int_0^{2\pi}\sqrt{a^2\sin^2\theta+b^2\cos^2\theta}d\theta$$
and this is just an elliptic integral. The final result takes the form ($b>a$)
$$P=4bE(e)$$
being
$$E(e)=\int_0^{\frac{\pi}{2}}\sqrt{1-e^2\sin^2\theta}d\theta$$
and $e^2=1-\frac{a^2}{b^2}$ the eccentricity.