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Curvature of a trajectory 
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#1
Feb813, 09:55 AM

P: 77

Hi!
Can someone please explain how does the limit in the attachment equals the curvature of a trajectory? I do not understand it. Why is it defined this way? ζ=dx/dl and it is in the direction of T. Thank you! 


#2
Feb813, 01:06 PM

Sci Advisor
Thanks
P: 2,326

It tells you, how much the direction of the tangent vector changes per unit length, when you walk along the curve. The larger this number (of dimension 1/length), the more your direction changes per unit path length. Thus it makes sense to call this the curvature. One can also show that it is an invariant wrt. to the choice of coordinates and parametrizations of the curve, because it can be derived from the curvature tensor.



#3
Feb813, 01:09 PM

Mentor
P: 11,576

You can check how ##\tau## and ##l## look for a circle  and all curves (with a welldefined curvature) locally look like circles.
Alternatively, see Wikipedia, or any textbook. 


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