Arc length parameterized curve

[odyssey]

Show that for any arc length parameterized curve there is a vector \(\displaystyle \delta(s)\) that satisfies:

\(\displaystyle T'(s) = \delta(s) \times T(s)\)

\(\displaystyle N'(s) = \delta(s) \times N(s)\)

\(\displaystyle B'(s) = \delta(s) \times B(s).\)

I know I can write \(\displaystyle \delta(s)\) as a linear combination thus \(\displaystyle \delta(s) = a(s)T(s) + b(s)N(s) + c(s)B(s)\) but I don't know how to find the coefficients and apply them to the proof above?

 

[HallsofIvy]

It would help if you would tell us what those symbols mean! T is the unit tangent, N is the unit normal, and B is the unit binormal? Take the dot product of each successively with aT+ bN+ cB.

 

[odyssey]

Yes, T is the unit tangent, N is the unit normal, and B is the unit binormal.

 

[daon2]

Yes, T is the unit tangent, N is the unit normal, and B is the unit binormal.

Maybe a little time saver for you:

\(\displaystyle ||T'||N = T'\)

So if there is such a delta it must satisfy

\(\displaystyle \delta\times T = ||T'||N\), i.e. delta is orthogonal to N, so that the coefficient of N in your linear combination is zero.