# fundamental theorem of space curves

## Informal summary.

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

## Theorem.

Let $\boldsymbol{\gamma}:I\to\mathbb{R}$ be a regular, parameterized space curve, without points of inflection. Let $\kappa(t),\tau(t)$ be the corresponding curvature and torsion functions. Let $T:\mathbb{R}^{3}\to\mathbb{R}^{3}$ be a Euclidean isometry. The curvature and torsion of the transformed curve $T(\boldsymbol{\gamma}(t))$ are given by $\kappa(t)$ and $\tau(t)$, respectively.

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

Title fundamental theorem of space curves FundamentalTheoremOfSpaceCurves 2013-03-22 13:23:28 2013-03-22 13:23:28 rmilson (146) rmilson (146) 5 rmilson (146) Theorem msc 53A04 SpaceCurve