# Radius of curvature formula derivation

1. Sep 4, 2005

### 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 any body show a web.

2. Sep 4, 2005

### 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.

3. Sep 4, 2005

### 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.

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

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