# space curve

## Kinematic definition.

A *parameterized space curve* is a parameterized curve taking
values in 3-dimensional Euclidean space. It may be interpreted as the
trajectory of a particle moving through space. Analytically, a smooth
space curve is represented by a sufficiently differentiable mapping
$\gamma :I\to {\mathbb{R}}^{3},$ of an interval^{} $I\subset \mathbb{R}$ into
3-dimensional Euclidean space ${\mathbb{R}}^{3}$. Equivalently, a
parameterized space curve can be considered a 3-vector of functions:

$$\gamma (t)=\left(\begin{array}{c}\hfill x(t)\hfill \\ \hfill y(t)\hfill \\ \hfill z(t)\hfill \end{array}\right),t\in I.$$ |

## Regularity hypotheses.

To preclude the possibility of kinks and corners, it is
necessary to add the hypothesis^{} that the mapping be regular^{} (http://planetmath.org/Curve), that is
to say that the derivative^{} ${\gamma}^{\prime}(t)$ never vanishes. Also, we say
that $\gamma (t)$ is a point of inflection if the first and second
derivatives ${\gamma}^{\prime}(t),{\gamma}^{\prime \prime}(t)$ are linearly dependent. Space curves
with points of inflection are beyond the scope of this entry.
Henceforth we make the assumption^{} that $\gamma (t)$ is both and
lacks points of inflection.

## Geometric definition.

A *space curve*, per se, needs to be conceived of as a subset of
${\mathbb{R}}^{3}$ rather than a mapping. Formally, we could define a space
curve to be the image of some parameterization $\gamma :I\to {\mathbb{R}}^{3}$. A
more useful concept, however, is the notion of an *oriented space
curve*, a space curve with a specified direction of motion.
Formally, an oriented space curve is an equivalence class^{} of
parameterized space curves; with ${\gamma}_{1}:{I}_{1}\to {\mathbb{R}}^{3}$ and
${\gamma}_{2}:{I}_{2}\to {\mathbb{R}}^{3}$ being judged equivalent^{} if there exists a
smooth, monotonically increasing reparameterization function $\sigma :{I}_{1}\to {I}_{2}$ such that

$${\gamma}_{1}(t)={\gamma}_{2}(\sigma (t)),t\in {I}_{1}.$$ |

## Arclength parameterization.

We say that $\gamma :I\to {\mathbb{R}}^{3}$ is an arclength parameterization of an oriented space curve if

$$\parallel {\gamma}^{\prime}(t)\parallel =1,t\in I.$$ |

With this hypothesis the length of the space curve between points $\gamma ({t}_{2})$ and $\gamma ({t}_{1})$ is just $|{t}_{2}-{t}_{1}|$. In other words, the parameter in such a parameterization measures the relative distance along the curve.

Starting with an arbitrary parameterization $\gamma :I\to {\mathbb{R}}^{3}$, one can obtain an arclength parameterization by fixing a ${t}_{0}\in I$, setting

$$\sigma (t)={\int}_{{t}_{0}}^{t}\parallel {\gamma}^{\prime}(x)\parallel \mathit{d}x,$$ |

and using the inverse function ${\sigma}^{-1}$ to reparameterize the curve. In other words,

$$\widehat{\gamma}(t)=\gamma ({\sigma}^{-1}(t))$$ |

is an arclength parameterization. Thus, every space curve possesses an arclength parameterization, unique up to a choice of additive constant in the arclength parameter.

Title | space curve |

Canonical name | SpaceCurve |

Date of creation | 2013-03-22 12:15:03 |

Last modified on | 2013-03-22 12:15:03 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 15 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 53A04 |

Synonym | oriented space curve |

Synonym | parameterized space curve |

Related topic | Torsion^{} |

Related topic | CurvatureOfACurve |

Related topic | MovingFrame |

Related topic | SerretFrenetFormulas |

Related topic | Helix |

Defines | point of inflection |

Defines | arclength parameterization |

Defines | reparameterization |