Editing Curve (section)

{{short description|Mathematical idealization of the trace left by a moving point}}
{{other uses}}
[[File:Parabola.svg|right|thumb|A [[parabola]], one of the simplest curves, after (straight) lines]]

In [[mathematics]], a '''curve''' (also called a '''curved line''' in older texts) is an object similar to a [[line (geometry)|line]], but that does not have to be [[Linearity|straight]].

Intuitively, a curve may be thought of as the trace left by a moving [[point (geometry)|point]]. This is the definition that appeared more than 2000 years ago in [[Euclid's Elements|Euclid's ''Elements'']]: "The [curved] line{{efn|In current mathematical usage, a line is straight. Previously lines could be either curved or straight.}} is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width."<ref>In (rather old) French: "La ligne est la première espece de quantité, laquelle a tant seulement une dimension à sçavoir longitude, sans aucune latitude ni profondité, & n'est autre chose que le flux ou coulement du poinct, lequel […] laissera de son mouvement imaginaire quelque vestige en long, exempt de toute latitude." Pages 7 and 8 of ''Les quinze livres des éléments géométriques d'Euclide Megarien, traduits de Grec en François, & augmentez de plusieurs figures & demonstrations, avec la corrections des erreurs commises és autres traductions'', by Pierre Mardele, Lyon, MDCXLV (1645).</ref>

This definition of a curve has been formalized in modern mathematics as: ''A curve is the [[image (mathematics)|image]] of an [[interval (mathematics)|interval]] to a [[topological space]] by a [[continuous function]]''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a [[parametric curve]]. In this article, these curves are sometimes called ''topological curves'' to distinguish them from more constrained curves such as [[differentiable curve]]s. This definition encompasses most curves that are studied in mathematics; notable exceptions are [[level curve]]s (which are [[union (set theory)|unions]] of curves and isolated points), and [[algebraic curve]]s (see below). Level curves and algebraic curves are sometimes called [[implicit curve]]s, since they are generally defined by [[implicit equation]]s.

Nevertheless, the class of topological curves is very broad, and contains some curves that do not look as one may expect for a curve, or even cannot be drawn. This is the case of [[space-filling curve]]s and [[fractal curve]]s. For ensuring more regularity, the function that defines a curve is often supposed to be [[differentiable function|differentiable]], and the curve is then said to be a [[differentiable curve]].

A [[plane algebraic curve]] is the [[zero set]] of a [[polynomial]] in two [[indeterminate (variable)|indeterminate]]s. More generally, an [[algebraic curve]] is the zero set of a finite set of polynomials, which satisfies the further condition of being an [[algebraic variety]] of [[dimension of an algebraic variety|dimension]] one. If the coefficients of the polynomials belong to a [[field (mathematics)|field]] {{mvar|k}}, the curve is said to be ''defined over'' {{mvar|k}}. In the common case of a [[real algebraic curve]], where {{mvar|k}} is the field of [[real number]]s, an algebraic curve is a finite union of topological curves. When [[complex number|complex]] zeros are considered, one has a ''complex algebraic curve'', which, from the [[topology|topological]] point of view, is not a curve, but a [[surface (mathematics)|surface]], and is often called a [[Riemann surface]]. Although not being curves in the common sense, algebraic curves defined over other fields have been widely studied. In particular, algebraic curves over a [[finite field]] are widely used in modern [[cryptography]].