Curves in Euclidean Three-Space

Synonyms

Curves; Space curves

Related Concepts

Curvature; Differential Invariants; Euclidean Geometry; Osculating Paraboloids

Definition

Space curves are one-parameter manifolds immersed in Euclidean 3D space $\mathbf{r}(s) \subset {\mathbb{E}}^{3}$, where $s \in \mathbb{R}$. One requires differentiability to whatever order is necessary, and $\|{\frac{\partial \mathbf{r}(s)} {\partial s} \|}^{2}\neq 0$. It is convenient to require $\|{\frac{\partial \mathbf{r}(s)} {\partial s} \|}^{2} = 1$, though this can always be achieved through a reparameterization. Such curves are known as “rectified” or “parameterized by arc-length,” and one writes $\mathop{\mathbf{r}}\limits^{.}(s)$ for the partial derivative with respect to arc-length (r′(t) will be used if the parameter t is not arc-length). As discussed below, in addition one requires $\mathop{\mathbf{r}}\limits^{..}(s)\neq 0$for a generic space curve. The very notion of “rectifiable” is of course Euclidean. Curves in...