space curve

Kinematic definition.

A parameterized space curve is a parameterized curve takingvalues in 3-dimensional Euclidean space. It may be interpreted as thetrajectory of a particle moving through space. Analytically, a smoothspace curve is represented by a sufficiently differentiable mapping$\gamma :I\to {ℝ}^3,$ of an interval $I\subset ℝ$ into3-dimensional Euclidean space ${ℝ}^3$. Equivalently, aparameterized 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 isnecessary to add the hypothesis that the mapping be regular (http://planetmath.org/Curve), that isto say that the derivative ${\gamma }^{\prime }(t)$ never vanishes. Also, we saythat $\gamma (t)$ is a point of inflection if the first and secondderivatives ${\gamma }^{\prime }(t),{\gamma }^{\prime \prime }(t)$ are linearly dependent. Space curveswith points of inflection are beyond the scope of this entry.Henceforth we make the assumption that $\gamma (t)$ is both andlacks points of inflection.

Geometric definition.

A space curve, per se, needs to be conceived of as a subset of${ℝ}^3$ rather than a mapping. Formally, we could define a spacecurve to be the image of some parameterization $\gamma :I\to {ℝ}^3$. Amore useful concept, however, is the notion of an oriented spacecurve, a space curve with a specified direction of motion.Formally, an oriented space curve is an equivalence class ofparameterized space curves; with ${\gamma }_1:I_1\to {ℝ}^3$ and${\gamma }_2:I_2\to {ℝ}^3$ being judged equivalent if there exists asmooth, 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 $\gamma :I\to {ℝ}^3$ is an arclength parameterization of anoriented space curve if

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

With this hypothesis thelength of the space curve between points $\gamma (t_2)$ and $\gamma (t_1)$ isjust $|t_2-t_1|$. In other words, the parameter in such aparameterization measures the relative distance along thecurve.

Starting with an arbitrary parameterization $\gamma :I\to {ℝ}^3$,one can obtain an arclength parameterization by fixing a $t_0\in I$,setting

$$\sigma (t)={\int }_{t_0}^t\parallel {\gamma }^{\prime }(x)\parallel 𝑑x,$$

and using theinverse function ${\sigma }^{-1}$ to reparameterize the curve. In otherwords,

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

is an arclengthparameterization. Thus, every space curve possesses an arclengthparameterization, unique up to a choice of additive constant in thearclength 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