1. ## Arc Length Parametrisation

Hi all,

I need to find the arc length parametrisation of alpha = ( (1-t^2)/(1+t^2) , (2t)/(1+t^2) )

So first of all I differentiated to find alpha' = ( (-4t) / (1+t^2)^2 , (2-2t^2) / (1+t^2)^2 )

and then tried sqrt(x^2 + y^2) but didn't get anything near what the answer is supposed to be ( |alpha'| = 2 / 1 + t^2 )

Many thanks

2. ## Re: Arc Length Parametrisation

$\displaystyle \sqrt{\left(\frac{-4t}{(1+t^2)^2}\right)^2 + \left(\frac{2-2t^2}{(1+t^2)^2}\right)^2}$

$\displaystyle \sqrt{\frac{16t^2}{(1+t^2)^4} + \frac{4-8t^2+4t^4}{(1+t^2)^4}}$

$\displaystyle \sqrt{\frac{4t^4+8t^2+4}{(1+t^2)^4}}$

$\displaystyle \sqrt{\frac{4(t^4+2t^2+1)}{(1+t^2)^4}}$

$\displaystyle \sqrt{\frac{4(t^2+1)^2}{(1+t^2)^4}}$

$\displaystyle \sqrt{\frac{4}{(1+t^2)^2}}$

$\displaystyle \frac{2}{1+t^2}$