Originally Posted by

**Kiwi_Dave** Show that when the equation of a curve is given in parametric form x=x(t),y=y(t) that the curvature given by:

$\displaystyle \frac{\dot x \ddot y - \dot y \ddot x}{(\dot x^2 + \dot y^2)^{-3/2}}$ (**)

remains invariant under the change of parameter $\displaystyle t=t(\bar t)$

I think I have to make the substitutions:

$\displaystyle \dot x = \frac{dx}{d \bar t}\cdot \frac{d \bar t}{dt}$

and

$\displaystyle \ddot x = \frac{d^2x}{d \bar t^2} (\frac{d \bar t}{dt})^2+\frac{dx}{d \bar t} \frac{d^2 \bar t}{dt^2}$

and expect to find that the terms like $\displaystyle \frac {d \bar t}{dt} $disappear.

Unfortunately I have not been able to make that work.