# 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 $\bm{\gamma}:I\to \mathbb{R}$ 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
$T(\bm{\gamma}(t))$ 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
$\bm{\gamma}:I\to \mathbb{R}$ 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 $\widehat{\bm{\gamma}}:I\to \mathbb{R}$ is another
such space curve, then there exists a Euclidean isometry
$T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ such that $\widehat{\bm{\gamma}}(t)=T(\bm{\gamma}(t)).$

Title | fundamental theorem of space curves^{} |
---|---|

Canonical name | FundamentalTheoremOfSpaceCurves |

Date of creation | 2013-03-22 13:23:28 |

Last modified on | 2013-03-22 13:23:28 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 5 |

Author | rmilson (146) |

Entry type | Theorem |

Classification | msc 53A04 |

Related topic | SpaceCurve |