fundamental theorem of space curves

Informal summary.

The curvature and torsion of a spacecurve are invariant with respect to Euclidean motions. Conversely, agiven space curve is determined up to a Euclidean motion, by itscurvature and torsion, expressed as functions of the arclength.

Theorem.

Let $𝜸:I\to ℝ$ be a regular, parameterized space curve, withoutpoints of inflection. Let $\kappa (t),\tau (t)$ be thecorresponding curvature and torsion functions. Let$T:{ℝ}^3\to {ℝ}^3$ be a Euclidean isometry. The curvature andtorsion of the transformed curve$T(𝜸(t))$ are given by $\kappa (t)$ and $\tau (t)$, respectively.

Conversely, let $\kappa ,\tau :I\to ℝ$ be continuous functions,defined on an interval $I\subset ℝ$, and suppose that $\kappa (t)$never vanishes. Then, there exists an arclength parameterization$𝜸:I\to ℝ$ of a regular, oriented space curve, without points ofinflection, such that $\kappa (t)$ and $\tau (t)$ are the correspondingcurvature and torsion functions. If $\widehat{𝜸}:I\to ℝ$ is anothersuch space curve, then there exists a Euclidean isometry$T:{ℝ}^3\to {ℝ}^3$ such that $\widehat{𝜸}(t)=T(𝜸(t)).$