Editing Curve (section)

==Differentiable curve==
{{main|Differentiable curve}}
Roughly speaking a {{em|differentiable curve}} is a curve that is defined as being locally the image of an injective differentiable function <math>\gamma \colon I \rightarrow X</math> from an [[Interval (mathematics)|interval]] {{mvar|I}} of the [[real number]]s into a differentiable manifold {{mvar|X}}, often <math>\mathbb{R}^n.</math>

More precisely, a differentiable curve is a subset {{mvar|C}} of {{mvar|X}} where every point of {{mvar|C}} has a neighborhood {{mvar|U}} such that <math>C\cap U</math> is [[diffeomorphism|diffeomorphic]] to an interval of the real numbers.{{clarify|reason=This contradicts the definition given in [[Differential  geometry of curves]]|date=May 2019}} In other words, a differentiable curve is a differentiable manifold of dimension one.

===Differentiable arc===
{{redirect|Arc (geometry)|the use in finite projective geometry|Arc (projective geometry)|use in circles specifically|Circular arc}}

In [[Euclidean geometry]], an '''arc''' (symbol: '''⌒''') is a [[connected set|connected]] subset of a [[Differentiable function|differentiable]] curve. 

Arcs of [[line (geometry)|lines]] are called [[line segment|segments]], [[ray (geometry)|rays]], or [[line (geometry)|lines]], depending on how they are bounded.

A common curved example is an arc of a [[circle]], called a [[circular arc]]. 

In a [[sphere]] (or a [[spheroid]]), an arc of a [[great circle]] (or a [[great ellipse]]) is called a '''great arc'''.

===Length of a curve===
{{main|Arc length}}
{{further|Differentiable curve#Length}}

If <math> X = \mathbb{R}^{n} </math> is the <math> n </math>-dimensional Euclidean space, and if <math> \gamma: [a,b] \to \mathbb{R}^{n} </math> is an injective and continuously differentiable function, then the length of <math> \gamma </math> is defined as the quantity
:<math>
\operatorname{Length}(\gamma) ~ \stackrel{\text{def}}{=} ~ \int_{a}^{b} |\gamma\,'(t)| ~ \mathrm{d}{t}.
</math>
The length of a curve is independent of the [[Parametrization (geometry)|parametrization]] <math> \gamma </math>.

In particular, the length <math> s </math> of the [[graph of a function|graph]] of a continuously differentiable function <math> y = f(x) </math> defined on a closed interval <math> [a,b] </math> is
:<math>
s = \int_{a}^{b} \sqrt{1 + [f'(x)]^{2}} ~ \mathrm{d}{x},
</math>
which can be thought of intuitively as using the [[Pythagorean theorem]] at the infinitesimal scale continuously over the full length of the curve.<ref>{{Cite book|url=https://books.google.com/books?id=OS4AAAAAYAAJ&dq=length+of+a+curve+formula+pythagorean&pg=RA2-PA108|title=The Calculus|last1=Davis|first1=Ellery W.|last2=Brenke|first2=William C.|date=1913|publisher=MacMillan Company|isbn=9781145891982|page=108|language=en}}</ref>

More generally, if <math> X </math> is a [[metric space]] with metric <math> d </math>, then we can define the length of a curve <math> \gamma: [a,b] \to X </math> by
:<math>
\operatorname{Length}(\gamma)
~ \stackrel{\text{def}}{=} ~
\sup \!
\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\},
</math>
where the supremum is taken over all <math> n \in \mathbb{N} </math> and all partitions <math> t_{0} < t_{1} < \ldots < t_{n} </math> of <math> [a, b] </math>.

A rectifiable curve is a curve with [[wiktionary:finite|finite]] length. A curve <math> \gamma: [a,b] \to X </math> is called {{em|natural}} (or unit-speed or parametrized by arc length) if for any <math> t_{1},t_{2} \in [a,b] </math> such that <math> t_{1} \leq t_{2} </math>, we have
:<math>
\operatorname{Length} \! \left( \gamma|_{[t_{1},t_{2}]} \right) = t_{2} - t_{1}.
</math>

If <math> \gamma: [a,b] \to X </math> is a [[Lipschitz continuity|Lipschitz-continuous]] function, then it is automatically rectifiable. Moreover, in this case, one can define the speed (or [[metric derivative]]) of <math> \gamma </math> at <math> t \in [a,b] </math> as
:<math>
{\operatorname{Speed}_{\gamma}}(t) ~ \stackrel{\text{def}}{=} ~ \limsup_{s \to t} \frac{d(\gamma(s),\gamma(t))}{|s - t|}
</math>
and then show that
:<math>
\operatorname{Length}(\gamma) = \int_{a}^{b} {\operatorname{Speed}_{\gamma}}(t) ~ \mathrm{d}{t}.
</math>

===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.