# 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 (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

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

## 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.

 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