Curve

If ${\displaystyle X=\mathbb {R} ^{n}}$ is the ${\displaystyle n}$-dimensional Euclidean space, and if ${\displaystyle \gamma$ :[a,b]\to \mathbb {R} ^{n}} is an injective and continuously differentiable function, then the length of ${\displaystyle \gamma }$ is defined as the quantity

${\displaystyle \operatorname {Length} (\gamma )~{\stackrel {\text{def}}{=}}~\int _{a}^{b}|\gamma \,'(t)|~\mathrm {d} {t}.}$

The length of a curve is independent of the parametrization ${\displaystyle \gamma }$.

In particular, the length ${\displaystyle s}$ of the graph of a continuously differentiable function ${\displaystyle y=f(x)}$ defined on a closed interval ${\displaystyle [a,b]}$ is

${\displaystyle s=\int _{a}^{b}{\sqrt {1+[f'(x)]^{2}}}~\mathrm {d} {x}.}$

More generally, if ${\displaystyle X}$ is a metric space with metric ${\displaystyle d}$, then we can define the length of a curve ${\displaystyle \gamma$ :[a,b]\to X} by

${\displaystyle \operatorname {Length} (\gamma )~{\stackrel {\text{def}}{=}}~\sup \!\left(\left\{\sum _{i=1}^{n}d(\gamma (t_{i}),\gamma (t_{i-1}))~{\Bigg |}~n\in \mathbb {N} ~{\text{and}}~a=t_{0}<t_{1}<\ldots <t_{n}=b\right\}\right),}$

where the supremum is taken over all ${\displaystyle n\in \mathbb {N} }$ and all partitions ${\displaystyle t_{0}<t_{1}<\ldots <t_{n}}$ of ${\displaystyle [a,b]}$.

A rectifiable curve is a curve with finite length. A curve ${\displaystyle \gamma$ :[a,b]\to X} is called natural (or unit-speed or parametrized by arc length) if for any ${\displaystyle t_{1},t_{2}\in [a,b]}$ such that ${\displaystyle t_{1}\leq t_{2}}$, we have

${\displaystyle \operatorname {Length} \!\left(\gamma |_{[t_{1},t_{2}]}\right)=t_{2}-t_{1}.}$

If ${\displaystyle \gamma$ :[a,b]\to X} is a Lipschitz-continuous function, then it is automatically rectifiable. Moreover, in this case, one can define the speed (or metric derivative) of ${\displaystyle \gamma }$ at ${\displaystyle t\in [a,b]}$ as

${\displaystyle {\operatorname {Speed} _{\gamma }}(t)~{\stackrel {\text{def}}{=}}~\limsup _{[a,b]\ni s\to t}{\frac {d(\gamma (s),\gamma (t))}{|s-t|}}}$

and then show that

${\displaystyle \operatorname {Length} (\gamma )=\int _{a}^{b}{\operatorname {Speed} _{\gamma }}(t)~\mathrm {d} {t}.}$

While the first examples of curves that are met are mostly plane curves (that is, in everyday words, curved lines in two-dimensional space), there are obvious examples such as the helix which exist naturally in three dimensions. The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. In general relativity, a world line is a curve in spacetime.

If ${\displaystyle X}$ is a differentiable manifold, then we can define the notion of differentiable curve in ${\displaystyle X}$. This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take ${\displaystyle X}$ to be Euclidean space. On the other hand, it is useful to be more general, in that (for example) it is possible to define the tangent vectors to ${\displaystyle X}$ by means of this notion of curve.

If ${\displaystyle X}$ is a smooth manifold, a smooth curve in ${\displaystyle X}$ is a smooth map

${\displaystyle \gamma \colon I\rightarrow X}$.

This is a basic notion. There are less and more restricted ideas, too. If ${\displaystyle X}$ is a ${\displaystyle C^{k}}$ manifold (i.e., a manifold whose charts are ${\displaystyle k}$ times continuously differentiable), then a ${\displaystyle C^{k}}$ curve in ${\displaystyle X}$ is such a curve which is only assumed to be ${\displaystyle C^{k}}$ (i.e. ${\displaystyle k}$ times continuously differentiable). If ${\displaystyle X}$ is an analytic manifold (i.e. infinitely differentiable and charts are expressible as power series), and ${\displaystyle \gamma }$ is an analytic map, then ${\displaystyle \gamma }$ is said to be an analytic curve.

A differentiable curve is said to be regular if its derivative never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two ${\displaystyle C^{k}}$ differentiable curves

${\displaystyle \gamma _{1}\colon I\rightarrow X}$ and

${\displaystyle \gamma _{2}\colon J\rightarrow X}$

are said to be equivalent if there is a bijective ${\displaystyle C^{k}}$ map

${\displaystyle p\colon J\rightarrow I}$

such that the inverse map

${\displaystyle p^{-1}\colon I\rightarrow J}$

is also ${\displaystyle C^{k}}$, and

${\displaystyle \gamma _{2}(t)=\gamma _{1}(p(t))}$

for all ${\displaystyle t}$. The map ${\displaystyle \gamma _{2}}$ is called a reparametrization of ${\displaystyle \gamma _{1}}$; and this makes an equivalence relation on the set of all ${\displaystyle C^{k}}$ differentiable curves in ${\displaystyle X}$. A ${\displaystyle C^{k}}$ arc is an equivalence class of ${\displaystyle C^{k}}$ curves under the relation of reparametrization.