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For a unit-speed curve with nonzero curvature of a space curve $\kappa$, define the Frenet frameThe torsion of a space curve iswith this choice of convention.

For the requested example, take the unit-speed circleand the unit-speed circular helixBoth have the constant nonzero curvature $\kappa =1$, but their torsions are respectively $0$ and $1$. A proper Euclidean motion of Euclidean three-space preserves torsion, so no such rigid motion relates the two curves.

If $\kappa \equiv 0$, then $T^{\prime }=0$, so the unit tangent is constant andThus both curves are straight line segments. Equal arc length makes their parameter intervals equally long, and a rotation in $SO(3)$ followed by a translation maps one to the other. Hence they are related by a proper Euclidean motion of Euclidean three-space.

The conclusion fails for $\kappa \equiv 1$, because curvature alone does not determine a space curve. A unit circle has curvature one and torsion zero. The circular helix with $a=b=1/2$,is unit speed and has curvature one but torsion one. Proper Euclidean motions preserve torsion, so equal-length pieces of these two curves cannot be related by one.