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The curvature of a space curve measures turning of its unit tangent vector; the torsion of a curve measures twisting of its osculating plane. For a regular curve $r(t)$ in Euclidean space ${\mathbb{R}}^3$,The first formula requires two derivatives and $r^{\prime }\ne 0$. The second requires three derivatives and $r^{\prime }\times r^{\prime \prime }\ne 0$: torsion of a curve is undefined by this formula at zero curvature of a space curve.

With arc length $s$ and Frenet frame $(T,N,B)$, the formulas become $\kappa =\Vert dT/ds\Vert$ and $\tau =-⟨dB/ds,N⟩$. A circle of radius $a>0$ has $\kappa =1/a$ and $\tau =0$. The circular helix $r(t)=(a\cos t,a\sin t,bt)$ has $\kappa =a/(a^2+b^2)$ and $\tau =b/(a^2+b^2)$. These data, expressed in arc length, determine the curve up to a proper Euclidean motion of Euclidean three-space by the fundamental theorem of regular space curves.^[ref]