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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]].
=={{anchor|Definitions|Topology|In topology}}Topological curve==
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{{anchor|Jordan}}A plane simple closed curve is also called a '''Jordan curve'''. It is also defined as a non-self-intersecting [[loop (topology)|continuous loop]] in the plane.<ref>{{Cite book|url=https://books.google.com/books?id=0Q9mbXCQRyoC&pg=PA7|title=Depth, Crossings and Conflicts in Discrete Geometry|last=Sulovský|first=Marek|date=2012|publisher=Logos Verlag Berlin GmbH| isbn=9783832531195|page=7|language=en}}</ref> The [[Jordan curve theorem]] states that the [[set complement]] in a plane of a Jordan curve consists of two [[connected component (topology)|connected component]]s (that is the curve divides the plane in two non-intersecting [[region (mathematics)|regions]] that are both connected). The bounded region inside a Jordan curve is known as '''Jordan domain'''.
The definition of a curve includes figures that can hardly be called curves in common usage. For example, the image of a curve can cover a [[Square (geometry)|square]] in the plane ([[space-filling curve]]), and a simple curve may have a positive area.<ref>{{cite journal|last=Osgood|first=William F.|date=January 1903|title=A Jordan Curve of Positive Area|journal=Transactions of the American Mathematical Society|publisher=[[American Mathematical Society]]|volume=4|issue=1|pages=107–112|doi=10.2307/1986455|issn=0002-9947|jstor=1986455|author-link1=William Fogg Osgood|doi-access=free}}<!--|access-date=2008-06-04--></ref> [[Fractal curve]]s can have properties that are strange for the common sense. For example, a fractal curve can have a [[Hausdorff dimension]] bigger than one (see [[Koch snowflake]]) and even a positive area. An example is the [[dragon curve]], which has many other unusual properties.
==Differentiable curve==
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