Radius of curvature formula derivation

  • Thread starter chandran
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  • #1
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Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
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
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Well first understand that curvature for a vector function is given by:

[tex]\kappa=\frac{|\mathbf{r}'\times\mathbf{r}''|}{|\mathbf{r}'|^3}[/tex]

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

[tex]\kappa=\frac{d\mathbf{T}}{ds}[/tex]

[tex]\mathbf{r}'=\frac{ds}{dt}\mathbf{T}[/tex]
 

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