I'm trying to show that (1) $T'\times T'' = k^2(kB +\tau T)$
$T' = \kappa N$, from Frenet Serret
$T'' = \kappa'N + N'\kappa$, but the algebra didn't follow when I tried to substitute this on the Left hand side, of (1) above
I'm trying to show that (1) $T'\times T'' = k^2(kB +\tau T)$
$T' = \kappa N$, from Frenet Serret
$T'' = \kappa'N + N'\kappa$, but the algebra didn't follow when I tried to substitute this on the Left hand side, of (1) above
First using the frenet serret equation N' = -kT + τB, substitute it into T" to get
T" = k'N - k2T + kτB
so T' x T" = kN x (-k'N - k2T + kτB) = -kk'(N x N) - k3(N x T) + k2τ(N x B)
If you unsure how I got to this point, here's a link to the properties of the cross product
Since {T,N,B} is an orthonormal basis (meaning each vector is of unit length and each vector is orthogonal to one another), N x N = 0, T x N = B which implies N x T = -B, and N x B = T.
Going back to our equation we get
T' x T" = k3B + k2τT = k2(kB + τT)