Editing Curve (section)

===Differential geometry===
{{main|Differential geometry of curves}}
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 <math>X</math> is a [[differentiable manifold]], then we can define the notion of ''differentiable curve'' in <math>X</math>. This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take <math>X</math> 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 [[Differential geometry of curves|tangent vector]]s to <math>X</math> by means of this notion of curve.

If <math>X</math> is a [[smooth manifold]], a ''smooth curve'' in <math>X</math> is a [[smooth map]]

:<math>\gamma \colon I \rightarrow X</math>.

This is a basic notion. There are less and more restricted ideas, too. If <math>X</math> is a <math>C^k</math> manifold (i.e., a manifold whose [[chart (topology)|chart's]] [[Atlas (topology)#Transition_maps|transition maps]] are <math>k</math> times [[continuously differentiable]]), then a <math>C^k</math> curve in <math>X</math> is such a curve which is only assumed to be <math>C^k</math> (i.e. <math>k</math> times continuously differentiable).  If <math>X</math> is an [[manifold|analytic manifold]] (i.e. infinitely differentiable and charts are expressible as [[power series]]), and <math>\gamma</math> is an analytic map, then <math>\gamma</math> is said to be an ''analytic curve''.

A differentiable curve is said to be '''{{vanchor|regular}}''' if its [[derivative]] never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.)  Two <math>C^k</math> differentiable curves

:<math>\gamma_1 \colon I \rightarrow X</math> and

:<math>\gamma_2 \colon J \rightarrow X</math>

are said to be ''equivalent'' if there is a [[bijection|bijective]] <math>C^k</math> map

:<math>p \colon J \rightarrow I</math>

such that the [[inverse map]]

:<math>p^{-1} \colon I \rightarrow J</math>

is also <math>C^k</math>, and

:<math>\gamma_{2}(t) = \gamma_{1}(p(t))</math>

for all <math>t</math>.  The map <math>\gamma_2</math> is called a ''reparametrization'' of <math>\gamma_1</math>; and this makes an [[equivalence relation]] on the set of all <math>C^k</math> differentiable curves in <math>X</math>.  A <math>C^k</math> ''arc'' is an [[equivalence class]] of <math>C^k</math> curves under the relation of reparametrization.