Deriving calculation formulas for torsion and curvature With much blood, sweat, and tears, I have managed to derive the formulas
$$k = \frac{||a' \times a''||}{||a'||^3}, \quad \tau = \frac{\langle a' \times a'', a''' \rangle}{||a' \times a''||^2}$$
for the curvature and torsion of a smooth regular space curve $a$. In general, is there any elegant way of proving such formulas, or is it inevitably a mindless differentiation bash and application of cross product identities? I suppose these are what you'd call calculation formulas, so there might not be any meaning to assign to the right-hand sides, and hence my hope might be in vain. Nonetheless, thanks for any advice. Even any suggestions on how to cull the working would be good -- I spent at least two pages on the former and around three on the latter. 
 A: It shouldn't be "a mindless differentiation bash" if you write down the equation
$\alpha' = \upsilon T$, where $\upsilon=\|\alpha'\|$, use the Frenet equations and chain rule to differentiate twice. It helps to economize a bit by remembering that $w\times w=0$ for any vector $w$. In particular, $\alpha'\times\alpha''$ will point in the $T\times N= B$ direction, so you'll need only the $B$ component of $\alpha'''$. (P.S.@Jesse Madnick, I don't think the identity for  $\|v\times w\|$ is needed at all.)
A: Wolfram mathworld - http://mathworld.wolfram.com/Curvature.html
gives an easy proof of the first formula. The second one can be derived in a very similar way. The main idea is to write the derivatives of the curve in terms of the basis given by T, N and B. This should simplify the computations considerably.
A: The way I simplified the computations for myself at some point was by writing $\frac{d^2a}{ds^2}=\frac{d^2a}{dt^2}\left(\frac{dt}{ds}\right)^2+\frac{da}{dt}\frac{d^2t}{ds^2}$, figuring out the derivatives with respect to $s$ on the right hand side, and using the formula for orthogonal projection of $a''$ onto $a'.$ You can rewrite the curvature you wrote above similarly, using $\frac{dt}{ds}$, and writing it as the magnitude of the same projection.
(Note that $\|$orth$_{a'}(a'')\|=\frac{\|a'\times a''\|}{\|a'\|}.$)
