Radius of curvature formula derivation

[chandran]

for a curve defined by y=f(x) the radius of curvature is defined as
[f""(x)/(1+f"(x))] power 3/2. I need a good neat & understandable derivation for that. can anybody show a web.

 

[Mephistopheles]

Differential Geometry of Curves

I don't like the format of the forum reply, so click on the following link to view your derivative: DGC.

 

[amcavoy]

Well first understand that curvature for a vector function is given by:

\kappa=\frac{|\mathbf{r}'\times\mathbf{r}''|}{|\mathbf{r}'|^3}

Now, let r = xi + f(x)j and simplify. To prove the first formula \kappa, use the following fact and compute r' x r''. The answer should be clear from there.

\kappa=\frac{d\mathbf{T}}{ds}

\mathbf{r}'=\frac{ds}{dt}\mathbf{T}