Editing Curve

{{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]].

==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]].

=={{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.

==Differentiable curve==
{{main|Differentiable curve}}
Roughly speaking a {{em|differentiable curve}} is a curve that is defined as being locally the image of an injective differentiable function <math>\gamma \colon I \rightarrow X</math> from an [[Interval (mathematics)|interval]] {{mvar|I}} of the [[real number]]s into a differentiable manifold {{mvar|X}}, often <math>\mathbb{R}^n.</math>

More precisely, a differentiable curve is a subset {{mvar|C}} of {{mvar|X}} where every point of {{mvar|C}} has a neighborhood {{mvar|U}} such that <math>C\cap U</math> is [[diffeomorphism|diffeomorphic]] to an interval of the real numbers.{{clarify|reason=This contradicts the definition given in [[Differential  geometry of curves]]|date=May 2019}} In other words, a differentiable curve is a differentiable manifold of dimension one.

===Differentiable arc===
{{redirect|Arc (geometry)|the use in finite projective geometry|Arc (projective geometry)|use in circles specifically|Circular arc}}

In [[Euclidean geometry]], an '''arc''' (symbol: '''⌒''') is a [[connected set|connected]] subset of a [[Differentiable function|differentiable]] curve. 

Arcs of [[line (geometry)|lines]] are called [[line segment|segments]], [[ray (geometry)|rays]], or [[line (geometry)|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===
{{main|Arc length}}
{{further|Differentiable curve#Length}}

If <math> X = \mathbb{R}^{n} </math> is the <math> n </math>-dimensional Euclidean space, and if <math> \gamma: [a,b] \to \mathbb{R}^{n} </math> is an injective and continuously differentiable function, then the length of <math> \gamma </math> is defined as the quantity
:<math>
\operatorname{Length}(\gamma) ~ \stackrel{\text{def}}{=} ~ \int_{a}^{b} |\gamma\,'(t)| ~ \mathrm{d}{t}.
</math>
The length of a curve is independent of the [[Parametrization (geometry)|parametrization]] <math> \gamma </math>.

In particular, the length <math> s </math> of the [[graph of a function|graph]] of a continuously differentiable function <math> y = f(x) </math> defined on a closed interval <math> [a,b] </math> is
:<math>
s = \int_{a}^{b} \sqrt{1 + [f'(x)]^{2}} ~ \mathrm{d}{x},
</math>
which can be thought of intuitively as using the [[Pythagorean theorem]] at the infinitesimal scale continuously over the full length of the curve.<ref>{{Cite book|url=https://books.google.com/books?id=OS4AAAAAYAAJ&dq=length+of+a+curve+formula+pythagorean&pg=RA2-PA108|title=The Calculus|last1=Davis|first1=Ellery W.|last2=Brenke|first2=William C.|date=1913|publisher=MacMillan Company|isbn=9781145891982|page=108|language=en}}</ref>

More generally, if <math> X </math> is a [[metric space]] with metric <math> d </math>, then we can define the length of a curve <math> \gamma: [a,b] \to X </math> by
:<math>
\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} < t_{1} < \ldots < t_{n} = b
\right\},
</math>
where the supremum is taken over all <math> n \in \mathbb{N} </math> and all partitions <math> t_{0} < t_{1} < \ldots < t_{n} </math> of <math> [a, b] </math>.

A rectifiable curve is a curve with [[wiktionary:finite|finite]] length. A curve <math> \gamma: [a,b] \to X </math> is called {{em|natural}} (or unit-speed or parametrized by arc length) if for any <math> t_{1},t_{2} \in [a,b] </math> such that <math> t_{1} \leq t_{2} </math>, we have
:<math>
\operatorname{Length} \! \left( \gamma|_{[t_{1},t_{2}]} \right) = t_{2} - t_{1}.
</math>

If <math> \gamma: [a,b] \to X </math> is a [[Lipschitz continuity|Lipschitz-continuous]] function, then it is automatically rectifiable. Moreover, in this case, one can define the speed (or [[metric derivative]]) of <math> \gamma </math> at <math> t \in [a,b] </math> as
:<math>
{\operatorname{Speed}_{\gamma}}(t) ~ \stackrel{\text{def}}{=} ~ \limsup_{s \to t} \frac{d(\gamma(s),\gamma(t))}{|s - t|}
</math>
and then show that
:<math>
\operatorname{Length}(\gamma) = \int_{a}^{b} {\operatorname{Speed}_{\gamma}}(t) ~ \mathrm{d}{t}.
</math>

===Differential geometry===
{{main|Differential geometry of curves}}
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 <math>X</math> is a [[differentiable manifold]], then we can define the notion of ''differentiable curve'' in <math>X</math>. This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take <math>X</math> 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 [[Differential geometry of curves|tangent vector]]s to <math>X</math> by means of this notion of curve.

If <math>X</math> is a [[smooth manifold]], a ''smooth curve'' in <math>X</math> is a [[smooth map]]

:<math>\gamma \colon I \rightarrow X</math>.

This is a basic notion. There are less and more restricted ideas, too. If <math>X</math> is a <math>C^k</math> manifold (i.e., a manifold whose [[chart (topology)|chart's]] [[Atlas (topology)#Transition_maps|transition maps]] are <math>k</math> times [[continuously differentiable]]), then a <math>C^k</math> curve in <math>X</math> is such a curve which is only assumed to be <math>C^k</math> (i.e. <math>k</math> times continuously differentiable).  If <math>X</math> is an [[manifold|analytic manifold]] (i.e. infinitely differentiable and charts are expressible as [[power series]]), and <math>\gamma</math> is an analytic map, then <math>\gamma</math> is said to be an ''analytic curve''.

A differentiable curve is said to be '''{{vanchor|regular}}''' if its [[derivative]] never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.)  Two <math>C^k</math> differentiable curves

:<math>\gamma_1 \colon I \rightarrow X</math> and

:<math>\gamma_2 \colon J \rightarrow X</math>

are said to be ''equivalent'' if there is a [[bijection|bijective]] <math>C^k</math> map

:<math>p \colon J \rightarrow I</math>

such that the [[inverse map]]

:<math>p^{-1} \colon I \rightarrow J</math>

is also <math>C^k</math>, and

:<math>\gamma_{2}(t) = \gamma_{1}(p(t))</math>

for all <math>t</math>.  The map <math>\gamma_2</math> is called a ''reparametrization'' of <math>\gamma_1</math>; and this makes an [[equivalence relation]] on the set of all <math>C^k</math> differentiable curves in <math>X</math>.  A <math>C^k</math> ''arc'' is an [[equivalence class]] of <math>C^k</math> curves under the relation of reparametrization.

==Algebraic curve==
{{main|Algebraic curve}}
Algebraic curves are the curves considered in [[algebraic geometry]]. A plane algebraic curve is the [[set (mathematics)|set]] of the points of coordinates {{math|''x'', ''y''}} such that {{math|1=''f''(''x'', ''y'') = 0}}, where {{math|''f''}} is a polynomial in two variables defined over some field {{math|''F''}}. One says that the curve is ''defined over'' {{math|''F''}}. Algebraic geometry normally considers not only points with coordinates in {{math|''F''}} but all the points with coordinates in an [[algebraically closed field]] {{math|''K''}}. 

If ''C'' is a curve defined by a polynomial ''f'' with coefficients in ''F'', the curve is said to be defined over ''F''. 

In the case of a curve defined over the [[real number]]s, one normally considers points with [[complex number|complex]] coordinates. In this case, a point with real coordinates is a ''real point'', and the set of all real points is the ''real part'' of the curve. It is therefore only the real part of an algebraic curve that can be a topological curve (this is not always the case, as the real part of an algebraic curve may be disconnected and contain isolated points). The whole curve, that is the set of its complex point is, from the topological point of view a surface. In particular, the nonsingular complex projective algebraic curves are called [[Riemann surface]]s.

The points of a curve {{math|''C''}} with coordinates in a field {{math|''G''}} are said to be rational over {{math|''G''}} and can be denoted {{math|''C''(''G'')}}. When {{math|''G''}} is the field of the [[rational number]]s, one simply talks of ''rational points''. For example, [[Fermat's Last Theorem]] may be restated as: ''For'' {{math|''n'' > 2}}, ''every rational point of the [[Fermat curve]] of degree {{mvar|n}} has a zero coordinate''.

Algebraic curves can also be space curves, or curves in a space of higher dimension, say {{math|''n''}}. They are defined as [[algebraic varieties]] of [[dimension of an algebraic variety|dimension]] one. They may be obtained as the common solutions of at least {{math|''n''–1}} polynomial equations in {{math|''n''}} variables. If {{math|''n''–1}} polynomials are sufficient to define a curve in a space of dimension {{math|''n''}}, the curve is said to be a [[complete intersection]]. By eliminating variables (by any tool of [[elimination theory]]), an algebraic curve may be projected onto a [[plane algebraic curve]], which however may introduce new singularities such as [[cusp (singularity)|cusp]]s or [[double point]]s.

A plane curve may also be completed to a curve in the [[projective plane]]: if a curve is defined by a polynomial {{math|''f''}} of total degree {{math|''d''}}, then {{math|''w''<sup>''d''</sup>''f''(''u''/''w'', ''v''/''w'')}} simplifies to a [[homogeneous polynomial]] {{math|''g''(''u'', ''v'', ''w'')}} of degree {{math|''d''}}. The values of {{math|''u'', ''v'', ''w''}} such that {{math|1=''g''(''u'', ''v'', ''w'') = 0}} are the homogeneous coordinates of the points of the completion of the curve in the projective plane and the points of the initial curve are those such that {{math|''w''}} is not zero. An example is the Fermat curve {{math|1=''u''<sup>''n''</sup> + ''v''<sup>''n''</sup> = ''w''<sup>''n''</sup>}}, which has an affine form {{math|1=''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> = 1}}. A similar process of homogenization may be defined for curves in higher dimensional spaces.

Except for [[line (geometry)|lines]], the simplest examples of algebraic curves are the [[conic section|conics]], which are nonsingular curves of degree two and [[genus (mathematics)|genus]] zero. [[Elliptic curve]]s, which are nonsingular curves of genus one, are studied in [[number theory]], and have important applications to [[cryptography]].

==See also==
{{Div col|colwidth=20em}}
*[[Coordinate curve]]
*[[Crinkled arc]]
*[[Curve fitting]]
*[[Curve orientation]]
*[[Curve sketching]]
*[[Differential geometry of curves]]
*[[Gallery of curves]]
*[[Index of the curve]]
*[[List of curves topics]]
*[[List of curves]]
*[[Osculating circle]]
*[[Parametric surface]]
*[[Path (topology)]]
*[[Polygonal curve]]
*[[Position vector]]
*[[Vector-valued function]]
**[[Infinite-dimensional vector function]]
*[[Winding number]]
{{div col end}}

==Notes==
{{notelist}}

==References==
{{reflist|2}}

* {{springer|author=A.S. Parkhomenko|id=l/l059020|title=Line (curve)}}
* {{springer|author=B.I. Golubov|id=r/r080130|title=Rectifiable curve}}
* [[Euclid]], commentary and trans. by [[T. L. Heath]] ''Elements'' Vol. 1 (1908 Cambridge) [https://books.google.com/books?id=UhgPAAAAIAAJ Google Books]
* E. H. Lockwood ''A Book of Curves'' (1961 Cambridge)

==External links==
{{Commons category|Curves}}
*[https://web.archive.org/web/20040609111105/http://www-gap.dcs.st-and.ac.uk/~history/Curves/Curves.html Famous Curves Index], School of Mathematics and Statistics, University of St Andrews, Scotland
*[http://www.2dcurves.com/ Mathematical curves] A collection of 874 two-dimensional mathematical curves
*[http://faculty.evansville.edu/ck6/Gallery/Introduction.html Gallery of Space Curves Made from Circles, includes animations by Peter Moses]
*[http://faculty.evansville.edu/ck6/GalleryTwo/Introduction2.html Gallery of Bishop Curves and Other Spherical Curves, includes animations by Peter Moses]
* The Encyclopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/Line_(curve) lines].
* The Manifold Atlas page on [http://www.map.mpim-bonn.mpg.de/1-manifolds 1-manifolds].

{{Curves}}
{{Authority control}}

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