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# 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:

$\gamma(t)=\begin{pmatrix}x(t)\\ y(t)\\ z(t)\end{pmatrix},\quad t\in I.$ |

# Regularity hypotheses.

To preclude the possibility of kinks and corners, it is necessary to add the hypothesis that the mapping be regular, 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 regular 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

$\gamma_{1}(t)=\gamma_{2}(\sigma(t)),\quad t\in I_{1}.$ |

# Arclength parameterization.

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

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

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

$\sigma(t)=\int^{t}_{{t_{0}}}\|\gamma^{{\prime}}(x)\|\,dx,$ |

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

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

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

## Mathematics Subject Classification

53A04*no label found*

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## Attached Articles

## Corrections

typo in formula by stevecheng ✓

plane curve by pahio ✘

revise the order a bit? by Mathprof ✓