 Solution

Parametrize the regular curve by arclength $s$. Its unit tangent is $T={\gamma }^{\prime }$, and its curvature is $\kappa =|T^{\prime }|$. Where $\kappa >0$, define the principal normal $N=T^{\prime }/\kappa$ and binormal $B=T\times N$. The torsion of a space curve is $\tau =-B^{\prime }\cdot N$. Orthonormality and differentiation give the Frenet-Serret formulasFor a general parameter, $\kappa =|\dot{\gamma }\times \ddot{\gamma }|/|\dot{\gamma }{|}^3$ and $\tau =\det (\dot{\gamma },\ddot{\gamma },\stackrel{...}{\gamma })/|\dot{\gamma }\times \ddot{\gamma }{|}^2$. The normal and torsion formulas require nonzero curvature; at zero-curvature points this particular Frenet frame need not be defined.