PlanetMath: space curve

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space curve

(Definition)

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:

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

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 $\gamma'(t)$ never vanishes. Also, we say that $\gamma(t)$ is a point of inflection if the first and second derivatives $\gamma'(t), \gamma''(t)$ 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

$\displaystyle \gamma_1(t) = \gamma_2(\sigma(t)),\quad t\in I_1.$

Arclength parameterization.

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

$\displaystyle \Vert \gamma'(t) \Vert = 1,\quad t\in I.$

With this hypothesis the length of the space curve between points $\gamma(t_2)$ and $\gamma(t_1)$ is just $\vert t_2-t_1 \vert$ . 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

$\displaystyle \sigma(t) = \int^t_{t_0} \Vert \gamma'(x)\Vert \, dx,$

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

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

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

"space curve" is owned by Mathprof. [ full author list (3) | owner history (2) ]

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Other names:

oriented space curve, parameterized space curve

Also defines:

point of inflection, arclength parameterization, reparameterization

Attachments:: line in space (Topic) by pahio

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Cross-references: additive, inverse function, distance, measures, parameter, points, length, monotonically increasing, equivalent, equivalence class, image, subset, scope, linearly dependent, second derivatives, vanishes, derivative, mapping, hypothesis, necessary, functions, interval, differentiable mapping, smooth, Euclidean space, parameterized curve
There are 15 references to this entry.

This is version 11 of space curve, born on 2002-02-02, modified 2007-08-01.
Object id is 1633, canonical name is SpaceCurve.
Accessed 14918 times total.

Classification:

AMS MSC:

53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)

Pending Errata and Addenda

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