Curvature in terms of the tangent vector

[jaejoon89]

My teacher wrote an alternative equation on the board for curvature, and I am wondering how it is true:

k = | dT/dt / |dR/dt| |

where T is the unit tangent vector.

I know k = |R' x R''| / |R'|^3 = |dT/ds|
but I am not sure about the formula in question. How is it true/derived?

 

[kof9595995]

i think these are equivalent, i think the basic definition of curvature is |dT|/ds,meaning the angle turned over curve length traveled, indeed you can find |dT| is angle turned if T is unit tangent vector,and ds=|dR| where R is the position vector, and all the mentioned formulas can be derived.

 

[chrisk]

Check out this website for clarification.

http://www.usd.edu/~jflores/MultiCalc02/WebBook/Chapter_14/Graphics/Chapter14_3/Html14_3/14.3%20Arc%20Length%20and%20Curvature.htm