Editing Curve (section)

=={{anchor|Definitions|Topology|In topology}}Topological curve==
A '''topological curve''' can be specified by a [[continuous function (topology)|continuous function]] <math>\gamma \colon I \rightarrow X</math> from an [[Interval (mathematics)|interval]] {{mvar|I}} of the [[real number]]s into a [[topological space]] {{mvar|X}}. Properly speaking, the ''curve'' is the [[image (mathematics)|image]] of <math>\gamma.</math> However, in some contexts, <math>\gamma</math> itself is called a curve, especially when the image does not look like what is generally called a curve and does not characterize sufficiently <math>\gamma.</math>

For example, the image of the [[Peano curve]] or, more generally, a [[space-filling curve]] completely fills a square, and therefore does not give any information on how <math>\gamma</math> is defined.

A curve <math>\gamma</math> is '''closed'''{{efn|This term my be ambiguous, as a non-closed curve may be a [[closed set]], as is a line in a plane.}} or is a ''[[loop (topology)|loop]]'' if <math>I = [a,
b]</math> and <math>\gamma(a) = \gamma(b)</math>.  A closed curve is thus the image of a continuous mapping of a [[circle]]. A non-closed curve may also be called an '''''open curve'''''.

If the [[domain of a function|domain]] of a topological curve is a closed and bounded interval <math>I = [a,
b]</math>, the curve is called a ''[[path (topology)|path]]'', also known as ''topological arc'' (or just '''{{vanchor|arc}}''').

A curve is '''simple''' if it is the image of an interval or a circle by an [[injective]] continuous function. In other words, if a curve is defined by a continuous function <math>\gamma</math> with an interval as a domain, the curve is simple if and only if any two different points of the interval have different images, except, possibly, if the points are the endpoints of the interval. Intuitively, a simple curve is a curve that "does not cross itself and has no missing points" (a continuous non-self-intersecting curve).<ref>{{cite web|url=http://dictionary.reference.com/browse/jordan%20arc |title=Jordan arc definition at Dictionary.com. Dictionary.com Unabridged. Random House, Inc |publisher=[[Dictionary.reference.com]] |access-date=2012-03-14}}</ref>

A ''[[plane curve]]'' is a curve for which <math>X</math> is the [[Euclidean plane]]&mdash;these are the examples first encountered&mdash;or in some cases the [[projective plane]]. {{anchor|Space curve}}A '''{{em|space curve}}''' is a curve for which <math>X</math> is at least three-dimensional; a '''{{em|skew curve}}''' {{anchor|skew curve}} is a space curve which lies in no plane. These definitions of plane, space and skew curves apply also to [[real algebraic geometry|real algebraic curve]]s, although the above definition of a curve does not apply (a real algebraic curve may be [[connected space|disconnected]]).

[[File:Fractal dragon curve.jpg|thumb|A [[dragon curve]] with a positive area]]
{{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.