I have some questions about the TNB frame.(adsbygoogle = window.adsbygoogle || []).push({});

The T unit vector is defined this way:

[tex]\hat{T} = \frac{dr(t)/dt}{ds/dt} = \frac{dr(s(t))}{ds}[/tex]

So, it is parametrized by arc length.Why can't t be left as the parameter?Is this just for definition-of-curvature-sake? If so is there any reason why the TNB frame couldn't be parametrized all in time, not by arc length?

Ok, next question. I see that the N unit vector is defined this way:

[tex]\hat{N} = \frac{d\hat{T}/ds}{|d\hat{T}/ds|} = \frac{(d\hat{T}/dt)/(dt/ds)}{|d\hat{T}/dt|/|dt/ds|} = \frac{d\hat{T}/dt}{|d\hat{T}/dt|}[/tex]

Ok, confusion. The T unit vector is defined by parameter s, but then let's compute the N unit vector and for fun make it parametrized by t?! What is going on?

Another thing about the N unit vector. It seems to me that if this TNB frame is so great it should be able to deal with a straight line.

However if the path is a straight line:

[tex]\frac{d\hat{T}}{ds} = \frac{d\hat{T}}{dt} = 0[/tex]

So, there would be no way to find an N unit vector.

For this reason I am assuming that it must be a curve in order to be defined. Is this a reasonable assumption?

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# TNB Frenet Frames

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