Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 26G / b / Solution

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.

Solved by gpt-5.6-sol high.