Roughly speaking a is a curve that is defined as being locally the image of an injective differentiable function \gamma \colon I \rightarrow X from an
interval of the
real numbers into a differentiable manifold , often \mathbb{R}^n. More precisely, a differentiable curve is a subset of where every point of has a neighborhood such that C\cap U is
diffeomorphic to an interval of the real numbers. In other words, a differentiable curve is a differentiable manifold of dimension one.
Differentiable arc In
Euclidean geometry, an
arc (symbol:
⌒) is a
connected subset of a
differentiable curve. Arcs of
lines are called
segments,
rays, or
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 If X = \mathbb{R}^{n} is the n -dimensional Euclidean space, and if \gamma: [a,b] \to \mathbb{R}^{n} is an injective and continuously differentiable function, then the length of \gamma is defined as the quantity : \operatorname{Length}(\gamma) ~ \stackrel{\text{def}}{=} ~ \int_{a}^{b} |\gamma\,'(t)| ~ \mathrm{d}{t}. The length of a curve is independent of the
parametrization \gamma . In particular, the length s of the
graph of a continuously differentiable function y = f(x) defined on a closed interval [a,b] is : s = \int_{a}^{b} \sqrt{1 + [f'(x)]^{2}} ~ \mathrm{d}{x}, which can be thought of intuitively as using the
Pythagorean theorem at the infinitesimal scale continuously over the full length of the curve. More generally, if X is a
metric space with metric d , then we can define the length of a curve \gamma: [a,b] \to X by : \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} where the supremum is taken over all n \in \mathbb{N} and all partitions t_{0} of [a, b] . A rectifiable curve is a curve with
finite length. A curve \gamma: [a,b] \to X is called (or unit-speed or parametrized by arc length) if for any t_{1},t_{2} \in [a,b] such that t_{1} \leq t_{2} , we have : \operatorname{Length} \! \left( \gamma|_{[t_{1},t_{2}]} \right) = t_{2} - t_{1}. If \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 \gamma at t \in [a,b] as : {\operatorname{Speed}_{\gamma}}(t) ~ \stackrel{\text{def}}{=} ~ \limsup_{s \to t} \frac{d(\gamma(s),\gamma(t))} and then show that : \operatorname{Length}(\gamma) = \int_{a}^{b} {\operatorname{Speed}_{\gamma}}(t) ~ \mathrm{d}{t}.
Differential geometry 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 X is a
differentiable manifold, then we can define the notion of
differentiable curve in 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 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 X by means of this notion of curve. If X is a
smooth manifold, a
smooth curve in X is a
smooth map :\gamma \colon I \rightarrow X. This is a basic notion. There are less and more restricted ideas, too. If X is a C^k manifold (i.e., a manifold whose
chart's transition maps are k times
continuously differentiable), then a C^k curve in X is such a curve which is only assumed to be C^k (i.e. k times continuously differentiable). If X is an
analytic manifold (i.e. infinitely differentiable and charts are expressible as
power series), and \gamma is an analytic map, then \gamma is said to be an
analytic curve. A differentiable curve is said to be '''''' if its
derivative never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two C^k differentiable curves :\gamma_1 \colon I \rightarrow X and :\gamma_2 \colon J \rightarrow X are said to be
equivalent if there is a
bijective C^k map :p \colon J \rightarrow I such that the
inverse map :p^{-1} \colon I \rightarrow J is also C^k, and :\gamma_{2}(t) = \gamma_{1}(p(t)) for all t. The map \gamma_2 is called a
reparametrization of \gamma_1; and this makes an
equivalence relation on the set of all C^k differentiable curves in X. A C^k
arc is an
equivalence class of C^k curves under the relation of reparametrization. ==Algebraic curve==