Evolute of parabola and other

Hi,

I am trying to solve the following question:

For the plane curve $\displaystyle r(t) = (t, t^2)$ , find a parametric equation of its evolute.

I looked at this website: Parabola Evolute -- from Wolfram MathWorld

From there I get the cartesian equation? Is there a general formula for evolute? I am not sure which equation to use from Evolute - Wikipedia, the free encyclopedia . Can someone point me in the right direction? Thanks

Re: Evolute of parabola and other

Hey M.R.

The parametricform of the evolute is given on that page in the second pair of formulas since you have the same form as your question gives.

Re: Evolute of parabola and other

So is the answer:

$\displaystyle x_{e}=\frac{1}{2}(1+6t^2)$

and

$\displaystyle y_{e}=-4t^3$ ???

But how would I find the evolute of other curves? Is there a general formula to use?

Re: Evolute of parabola and other

For that one yes, (since a = 1) since it has the same functional form.

There is a link on that page that takes you to this:

Evolute -- from Wolfram MathWorld

It outlines how to get the non-parametric form for the evolute, but be aware that if it's only y in terms of x, all you have to do is let x = t and y = f(t) to get a parametric form of the equation (in other words, x = t and y = whatever the final function is given by that formula which are just functions of t).

When I say the above I don't mean the final x value will be t: only the intermediate value within the functions. For example if the original curve is y = x^3 - x^2 + 5x^4, then x = t and y = f(x) = f(t) but for the final involute you will have combinations of f(t), g(t) and its derivatives which will give you a complex parametrization for your evolute x(t) and y(t).