What is the relationship between the limit and curvature of a trajectory?

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The relationship between the limit and curvature of a trajectory is explained through the concept of how the tangent vector's direction changes per unit length along the curve. This change is quantified by ζ=dx/dl, indicating that a larger value signifies a greater directional change, which is why it is termed curvature. The curvature is invariant under different coordinate systems and can be derived from the curvature tensor. For any curve with defined curvature, it locally resembles a circle. This understanding can be further explored in resources like Wikipedia or standard textbooks.
rsaad
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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!
 

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

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