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Yikes, im really starting to spam this place up!

On the subject of curvature, it says that we can reparametrize the cure in terms of arc length instead of time. If we have time,t, and arc length s(t), we can write it as t=t(s).

It seems to me that this is NOT true in general, if you cannot factor t from s(t), then there is no way relate t in terms of t(s). So under what conditons is this true, and could you show a proof that shows that these conditions will insure reparametrization?

Secondly,

the curvature is defined as the rate of change of the unit tangent vector with respect to arc length.

I cannot see why they would choose this definition of curvature. What reasoning is behind using this as the definition?

Thanks,

Cyrus

ADDITION:Since not all functions can be parametrized in terms of arc length, that means that not all fuctions have cuvature, but all functions DO have arc length. Is that correct?

On the subject of curvature, it says that we can reparametrize the cure in terms of arc length instead of time. If we have time,t, and arc length s(t), we can write it as t=t(s).

It seems to me that this is NOT true in general, if you cannot factor t from s(t), then there is no way relate t in terms of t(s). So under what conditons is this true, and could you show a proof that shows that these conditions will insure reparametrization?

Secondly,

the curvature is defined as the rate of change of the unit tangent vector with respect to arc length.

I cannot see why they would choose this definition of curvature. What reasoning is behind using this as the definition?

Thanks,

Cyrus

ADDITION:Since not all functions can be parametrized in terms of arc length, that means that not all fuctions have cuvature, but all functions DO have arc length. Is that correct?

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