space curve SpaceCurve 2002-02-02 00:59:16 2007-08-01 09:02:39 Definition point of inflection arclength parameterization reparameterization \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand{\reals}{\mathbb{R}} \paragraph{Kinematic definition.} A \emph{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 \reals^3, $ of an interval $I\subset \reals$ into 3-dimensional Euclidean space $\reals^3$. Equivalently, a parameterized space curve can be considered a 3-vector of functions: $$\gamma(t) = \begin{pmatrix} x(t)\\ y(t) \\ z(t) \end{pmatrix},\quad t\in I.$$ \paragraph{Regularity hypotheses.} To preclude the possibility of kinks and corners, it is necessary to add the hypothesis that the mapping be \PMlinkname{regular}{Curve}, 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 \PMlinkescapetext{regular} and lacks points of inflection. \paragraph{Geometric definition.} A \emph{space curve}, per se, needs to be conceived of as a subset of $\reals^3$ rather than a mapping. Formally, we could define a space curve to be the image of some parameterization $\gamma:I\to\reals^3$. A more useful concept, however, is the notion of an \emph{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 $\gamma_1:I_1\to\reals^3$ and $\gamma_2:I_2\to\reals^3$ being judged equivalent if there exists a smooth, monotonically increasing reparameterization function $\sigma: I_1\to I_2$ such that $$\gamma_1(t) = \gamma_2(\sigma(t)),\quad t\in I_1.$$ \paragraph{Arclength parameterization.} We say that $\gamma:I\to\reals^3$ is an arclength parameterization of an oriented space curve if $$\Vert \gamma'(t) \Vert = 1,\quad t\in I.$$ 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 $\gamma: I\to \reals^3$, one can obtain an arclength parameterization by fixing a $t_0\in I$, setting $$\sigma(t) = \int^t_{t_0} \Vert \gamma'(x)\Vert \, dx,$$ and using the inverse function $\sigma^{-1}$ to reparameterize the curve. In other words, $$\hat{\gamma}(t) = \gamma(\sigma^{-1}(t))$$ is an arclength parameterization. Thus, every space curve possesses an arclength parameterization, unique up to a choice of additive constant in the arclength parameter.