Editing Curve (section)

==History==
[[File:Newgrange Entrance Stone.jpg|thumb|225px|[[Megalithic art]] from [[Newgrange]] showing an early interest in curves]]
Interest in curves began long before they were the subject of mathematical study.  This can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric
times.<ref name="Lockwood">Lockwood p. ix</ref> Curves, or at least their graphical representations, are simple to create, for example with a stick on the sand on a beach.

Historically, the term {{em|line}} was used in place of the more modern term {{em|curve}}. Hence the terms {{em|straight line}} and {{em|right line}} were used to distinguish what are today called lines from curved lines. For example, in Book I of [[Euclid's Elements]], a line is defined as a "breadthless length" (Def. 2), while a {{em|straight}} line is defined as "a line that lies evenly with the points on itself" (Def. 4). Euclid's idea of a line is perhaps clarified by the statement "The extremities of a line are points," (Def. 3).<ref>Heath p. 153</ref> Later commentators further classified lines according to various schemes. For example:<ref>Heath p. 160</ref>
*Composite lines (lines forming an angle)
*Incomposite lines
**Determinate (lines that do not extend indefinitely, such as the circle)
**Indeterminate (lines that extend indefinitely, such as the straight line and the parabola)

[[File:Conic sections with plane.svg|thumb|225px|The curves created by slicing a cone ([[conic section]]s) were among the curves studied in ancient [[Greek mathematics]].]]
The Greek [[geometers]] had studied many other kinds of curves. One reason was their interest in solving geometrical problems that could not be solved using standard [[compass and straightedge]] construction.
These curves include:
*The conic sections, studied in depth by [[Apollonius of Perga]]
*The [[cissoid of Diocles]], studied by [[Diocles (mathematician)|Diocles]] and used as a method to [[doubling the cube|double the cube]].<ref>Lockwood p. 132</ref>
*The [[conchoid of Nicomedes]], studied by [[Nicomedes (mathematician)|Nicomedes]] as a method to both double the cube and to [[angle trisection|trisect an angle]].<ref>Lockwood p. 129</ref>
*The [[Archimedean spiral]], studied by [[Archimedes]] as a method to trisect an angle and [[Squaring the circle|square the circle]].<ref>{{MacTutor|class=Curves|id=Spiral|title=Spiral of Archimedes}}</ref>
*The [[spiric section]]s, sections of [[torus|tori]] studied by [[Perseus (geometer)|Perseus]] as sections of cones had been studied by Apollonius.

[[File:Folium Of Descartes.svg|thumb|225px|left|Analytic geometry allowed curves, such as the [[Folium of Descartes]], to be defined using equations instead of geometrical construction.]]
A fundamental advance in the theory of curves was the introduction of [[analytic geometry]] by [[René Descartes]] in the seventeenth century. This enabled a curve to be described using an equation rather than an elaborate geometrical construction. This not only allowed new curves to be defined and studied, but it enabled a formal distinction to be made between [[algebraic curve]]s that can be defined using [[polynomial equation]]s, and [[transcendental curve]]s that cannot. Previously, curves had been described as "geometrical" or "mechanical" according to how they were, or supposedly could be, generated.<ref name="Lockwood" />

Conic sections were applied in [[astronomy]] by [[Johannes Kepler|Kepler]].
Newton also worked on an early example in the [[calculus of variations]]. Solutions to variational problems, such as the [[brachistochrone]] and [[tautochrone]] questions, introduced properties of curves in new ways (in this case, the [[cycloid]]). The [[catenary]] gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means of [[differential calculus]].

In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the [[cubic curve]]s, in the general description of the real points into 'ovals'. The statement of [[Bézout's theorem]] showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions.

Since the nineteenth century, curve theory is viewed as the special case of dimension one of the theory of [[manifold]]s and [[algebraic varieties]]. Nevertheless, many questions remain specific to curves, such as [[space-filling curve]]s, [[Jordan curve theorem]] and [[Hilbert's sixteenth problem]].