1
$\begingroup$

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

  • 1
    It seems you need to address the question of how $N'$ relates to $B$ and $T$. When I derive things I often find it helpful to write $V = c_1T+c_2N+c_3B$ and use dot-products to select the values of $c_1,c_2,c_3$. For example, $c_1 = V \cdot T$. We can use the orthonormality of the Frenet frame to make nice calculations.2012-12-08

1 Answers 1

1

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)

  • 0
    no problem. thx again2012-12-11