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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a curve (also called a curved line in older texts) is an object similar to a
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

line
, but that does not have to be
straight
straight
. Intuitively, a curve may be thought of as the trace left by a moving
point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...
. This is the definition that appeared more than 2000 years ago in Euclid's ''Elements'': "The urvedline 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." This definition of a curve has been formalized in modern mathematics as: ''A curve is the
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
of an interval to a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
by a
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a
parametric curve In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. In this article, these curves are sometimes called ''topological curves'' to distinguish them from more constrained curves such as
differentiable curve Differential geometry of curves is the branch of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematic ...
s. This definition encompasses most curves that are studied in mathematics; notable exceptions are
level curve In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...
s (which are unions of curves and isolated points), and
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
s (see below). Level curves and algebraic curves are sometimes called
implicit curve In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s, since they are generally defined by
implicit equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathemat ...
s and
fractal curve A fractal curve is, loosely, a mathematical curve 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 b ...
s. For ensuring more regularity, the function that defines a curve is often supposed to be
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

differentiable
, and the curve is then said to be a
differentiable curve Differential geometry of curves is the branch of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematic ...
. A
plane algebraic curve In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is the
zero set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...
of a
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
in two indeterminates. More generally, an
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
is the zero set of a finite set of polynomials, which satisfies the further condition of being an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematic ...
of
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
one. If the coefficients of the polynomials belong to a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
, the curve is said to be ''defined over'' . In the common case of a
real algebraic curveIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, where is the field of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, an algebraic curve is a finite union of topological curves. When
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

complex
zeros are considered, one has a ''complex algebraic curve'', which, from the
topological s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

topological
point of view, is not a curve, but a
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
, and is often called a
Riemann surface In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
are widely used in modern
cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia ''-logy'' is a suffix in the English language, used with words originally adapted from Ancient Greek ending in (''- ...

cryptography
.


History

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.Lockwood p. ix 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 was used in place of the more modern term . Hence the terms and were used to distinguish what are today called lines from curved lines. For example, in Book I of
Euclid's Elements The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematics, mathematical treatise consisting of 13 books attributed to the ancient Greek mathematics, Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a colle ...
, a line is defined as a "breadthless length" (Def. 2), while a 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). Later commentators further classified lines according to various schemes. For example: *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) The Greek
geometers A geometer is a mathematician whose area of study is geometry. Some notable geometers and their main fields of work, chronologically listed, are: 1000 BCE to 1 BCE * Baudhayana (fl. c. 800 BC) – Euclidean geometry, geometric algebra * M ...
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 Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandr ...
construction. These curves include: *The conic sections, studied in depth by
Apollonius of Perga Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος ''Apollonios o Pergeos''; la, Apollonius Pergaeus; ) was an Ancient Greece, Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the ...
*The
cissoid of Diocles In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

cissoid of Diocles
, studied by Diocles and used as a method to
double the cube Doubling the cube, also known as the Delian problem, is an ancient geometry, geometric problem. Given the Edge (geometry), edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. ...

double the cube
. *The
conchoid of Nicomedes
conchoid of Nicomedes
, studied by Nicomedes as a method to both double the cube and to
trisect an angle
trisect an angle
. *The
Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral In , a spiral is a which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the are:
Archimedean spiral
, studied by
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

Archimedes
as a method to trisect an angle and
square the circle
square the circle
. *The
spiric section In geometry, a spiric section, sometimes called a spiric of Perseus, is a quartic plane curve defined by equations of the form :(x^2+y^2)^2=dx^2+ey^2+f. \, Equivalently, spiric sections can be defined as Circular algebraic curve, bicircular quart ...

spiric section
s, sections of
tori Tori may refer to: Places * Tori (Georgia), an historic province of the nation of Georgia * Tori Parish, Pärnu County, Estonia ** Tori, Estonia * Tori, Ghana, a village in the kingdom of Chumburung, Ghana * Tori, Järva County, Estonia * Tori, ...

tori
studied by
Perseus In Greek mythology Greek mythology is the body of myth Myth is a folklore genre Folklore is the expressive body of culture shared by a particular group of people; it encompasses the tradition A tradition is a belief A b ...
as sections of cones had been studied by Apollonius. A fundamental advance in the theory of curves was the introduction of
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
by
René Descartes René Descartes ( or ; ; Latinized Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

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 In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
s that can be defined using
polynomial equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s, and transcendental curves that cannot. Previously, curves had been described as "geometrical" or "mechanical" according to how they were, or supposedly could be, generated. Conic sections were applied in
astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses mathematics, phys ...
by
Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German , , , and writer on music. He is a key figure in the 17th-century , best known for his , and his books ', ', and '. These works also provided one of the foundations for ...

Kepler
. Newton also worked on an early example in the
calculus of variations The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathematic ...
. Solutions to variational problems, such as the
brachistochrone In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

brachistochrone
and
tautochroneImage:Tautochrone curve.gif, 300px, Four balls slide down a cycloid curve from different positions, but they arrive at the bottom at the same time. The blue arrows show the points' acceleration along the curve. On the top is the time-position diagram ...
questions, introduced properties of curves in new ways (in this case, the
cycloid In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

cycloid
). The
catenary forming multiple Elastic deformation, elastic catenaries. In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or wire rope, cable assumes under its own weight when supported only at its ends. The catenary cu ...

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 mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the
cubic curve In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

manifold
s and
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures ...
. Nevertheless, many questions remain specific to curves, such as
space-filling curve In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathemat ...
s,
Jordan curve theorem In topology, a Jordan curve, sometimes called a plane simple closed curve, is a non-self-intersecting loop (topology), continuous loop in the plane. The Jordan curve theorem asserts that every Jordan curve divides the plane into an "interior" reg ...

Jordan curve theorem
and
Hilbert's sixteenth problemHilbert's 16th problem was posed by David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and de ...
.


Topological curve

A topological curve can be specified by a
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
\gamma \colon I \rightarrow X from an interval of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s into a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
. Properly speaking, the ''curve'' is the
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
of \gamma. However, in some contexts, \gamma itself is called a curve, especially when the image does not look like what is generally called a curve and does not characterize sufficiently \gamma. For example, the image of the
Peano curve 400px, Three iterations of a Peano curve construction, whose limit is a space-filling curve. In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arit ...

Peano curve
or, more generally, a
space-filling curve In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathemat ...
completely fills a square, and therefore does not give any information on how \gamma is defined. A curve \gamma is closed or is a loop if I = , b/math> and \gamma(a) = \gamma(b). A closed curve is thus the image of a continuous mapping of a
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

circle
. If the
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
of a topological curve is a closed and bounded interval I = , b/math>, it is called a '' path'', also known as ''topological arc'' (or just ). A curve is simple if it is the image of an interval or a circle by an
injective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
continuous function. In other words, if a curve is defined by a continuous function \gamma with an interval as a domain, the curve is simple if and only if two different points of the interval have different images, except, possibly, if the points are the end points of the interval. Intuitively, a simple curve is a curve that "does not cross itself and has no missing points". A simple closed curve is also called a
Jordan curve Jordan ( ar, الأردن; tr. ' ), officially the Hashemite Kingdom of Jordan ( ar, المملكة الأردنية الهاشمية; tr. '), is an Arab country in the Levant The Levant () is an approximate historical geographical ...
. The
Jordan curve theorem In topology, a Jordan curve, sometimes called a plane simple closed curve, is a non-self-intersecting loop (topology), continuous loop in the plane. The Jordan curve theorem asserts that every Jordan curve divides the plane into an "interior" reg ...

Jordan curve theorem
states that the
set complement In , the complement of a , often denoted by (or ), are the not in . When all sets under consideration are considered to be s of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of ...
in a plane of a Jordan curve consists of two connected components (that is the curve divides the plane in two non-intersecting
regions In geography Geography (from Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and phenomena of the Earth and planets. The first person to use the wor ...
that are both connected). A ''
plane curve In mathematics, a plane curve is a curve 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 o ...
'' is a curve for which X is the
Euclidean plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
—these are the examples first encountered—or in some cases the
projective plane In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. A is a curve for which X is at least three-dimensional; a is a space curve which lies in no plane. These definitions of plane, space and skew curves apply also to
real algebraic curveIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s, although the above definition of a curve does not apply (a real algebraic curve may be disconnected). The definition of a curve includes figures that can hardly be called curves in common usage. For example, the image of a simple curve can cover a
square In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

square
in the plane (
space-filling curve In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathemat ...
) and thus have a positive area.
Fractal curve A fractal curve is, loosely, a mathematical curve 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 b ...
s can have properties that are strange for the common sense. For example, a fractal curve can have a
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point (geometry), ...
bigger than one (see
Koch snowflake The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...

Koch snowflake
) and even a positive area. An example is the
dragon curve A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. The dragon curve is probably most commonly thought of as the shape that is generated from re ...

dragon curve
, which has many other unusual properties.


Differentiable curve

Roughly speaking a is a curve that is defined as being locally the image of an injective differentiable function \gamma \colon I \rightarrow X from an interval of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s into a differentiable manifold , often \mathbb^n. More precisely, a differentiable curve is a subset of where every point of has a neighborhood such that C\cap U is
diffeomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
to an interval of the real numbers. In other words, a differentiable curve is a differentiable manifold of dimension one.


Differentiable arc

In
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
, an arc (symbol: ⌒) is a connected subset of a
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

differentiable
curve. Arcs of
lines Long interspersed nuclear elements (LINEs) (also known as long interspersed nucleotide elements or long interspersed elements) are a group of non-LTR (long terminal repeat A long terminal repeat (LTR) is a pair of identical sequences of DNA ...

lines
are called or rays, depending whether they are bounded or not. A common curved example is an arc of a
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

circle
, called a
circular arc Circular may refer to: * The shape of a circle * Circular (album), ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fal ...
. In a
sphere A sphere (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

sphere
(or a
spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipse In math ...

spheroid
), an arc of a
great circle A great circle, also known as an orthodrome, of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathem ...

great circle
(or a
great ellipse image:OblateSpheroid.PNG, 150px, A spheroid A great ellipse is an ellipse passing through two point (geometry), points on a spheroid and having the same center (geometry), center as that of the spheroid. Equivalently, it is an ellipse on the surfa ...
) is called a great arc.


Length of a curve

If X = \mathbb^ is the n -dimensional Euclidean space, and if \gamma: ,b\to \mathbb^ is an injective and continuously differentiable function, then the length of \gamma is defined as the quantity : \operatorname(\gamma) ~ \stackrel ~ \int_^ , \gamma\,'(t), ~ \mathrm. The length of a curve is independent of the parametrization \gamma . In particular, the length s of the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

graph
of a continuously differentiable function y = f(x) defined on a closed interval ,b is : s = \int_^ \sqrt ~ \mathrm. More generally, if X is a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
with metric d , then we can define the length of a curve \gamma: ,b\to X by : \operatorname(\gamma) ~ \stackrel ~ \sup \! \left( \left\ \right), where the supremum is taken over all n \in \mathbb and all partitions t_ < t_ < \ldots < t_ of
, b The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
. A rectifiable curve is a curve with
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
length. A curve \gamma: ,b\to X is called (or unit-speed or parametrized by arc length) if for any t_,t_ \in ,b such that t_ \leq t_ , we have : \operatorname \! \left( \gamma, _ \right) = t_ - t_. If \gamma: ,b\to X is a
Lipschitz-continuous In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for function (mathematics), functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there ex ...

Lipschitz-continuous
function, then it is automatically rectifiable. Moreover, in this case, one can define the speed (or metric derivative) of \gamma at t \in ,b as : (t) ~ \stackrel ~ \limsup_ \frac and then show that : \operatorname(\gamma) = \int_^ (t) ~ \mathrm.


Differential geometry

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 A helix (), plural helixes or helices (), is a shape like a corkscrew or spiral staircase. It is a type of smooth Smooth may refer to: Mathematics * Smooth function is a smooth function with compact support. In mathematical analysis, 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 General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
, a
world line The world line (or worldline) of an object is the path (topology), path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is disti ...

world line
is a curve in
spacetime In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
. If X is a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
, then we can define the notion of ''differentiable curve'' in X. This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take X 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
tangent vector :''For a more general — but much more technical — treatment of tangent vectors, see tangent space.'' In mathematics, a tangent vector is a Vector (geometry), vector that is tangent to a curve or Surface (mathematics), surface at a given point. T ...
s to X by means of this notion of curve. If X is a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
, a ''smooth curve'' in X is a
smooth map In mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
:\gamma \colon I \rightarrow X. This is a basic notion. There are less and more restricted ideas, too. If X is a C^k manifold (i.e., a manifold whose
charts A chart is a graphical representation Graphic communication as the name suggests is communication using graphic elements. These elements include symbols such as glyphs and icon (computing), icons, images such as drawings and photographs, and c ...
are k times
continuously differentiable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
), then a C^k curve in X is such a curve which is only assumed to be C^k (i.e. k times continuously differentiable). If X is an
analytic manifoldIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

analytic manifold
(i.e. infinitely differentiable and charts are expressible as
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
), and \gamma is an analytic map, then \gamma is said to be an ''analytic curve''. A differentiable curve is said to be if its
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two C^k differentiable curves :\gamma_1 \colon I \rightarrow X and :\gamma_2 \colon J \rightarrow X are said to be ''equivalent'' if there is a
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

bijective
C^k map :p \colon J \rightarrow I such that the inverse map :p^ \colon I \rightarrow J is also C^k, and :\gamma_(t) = \gamma_(p(t)) for all t. The map \gamma_2 is called a ''reparametrization'' of \gamma_1; and this makes an
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
on the set of all C^k differentiable curves in X. A C^k ''arc'' is an
equivalence class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of C^k curves under the relation of reparametrization.


Algebraic curve

Algebraic curves are the curves considered in
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

algebraic geometry
. A plane algebraic curve is the set of the points of coordinates such that , where is a polynomial in two variables defined over some field . One says that the curve is ''defined over'' . Algebraic geometry normally considers not only points with coordinates in but all the points with coordinates in an
algebraically closed field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, one normally considers points with
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s. The points of a curve with coordinates in a field are said to be rational over and can be denoted . When is the field of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s, one simply talks of ''rational points''. For example,
Fermat's Last Theorem In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
may be restated as: ''For'' , ''every rational point of the Fermat curve of degree has a zero coordinate''. Algebraic curves can also be space curves, or curves in a space of higher dimension, say . They are defined as
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures ...
of
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
one. They may be obtained as the common solutions of at least polynomial equations in variables. If polynomials are sufficient to define a curve in a space of dimension , the curve is said to be a
complete intersection In mathematics, an algebraic variety Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern ...
. By eliminating variables (by any tool of
elimination theory In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating some variables between polynomials of several variables, in order to solve systems of polynomial equations. The classi ...
), an algebraic curve may be projected onto a
plane algebraic curve In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, which however may introduce new singularities such as
cusp Cusp may refer to: Mathematics *Cusp (singularity) In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
s or
double pointIn geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

double point
s. A plane curve may also be completed to a curve in the
projective plane In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
: if a curve is defined by a polynomial of total degree , then simplifies to a
homogeneous polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
of degree . The values of such that 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 is not zero. An example is the Fermat curve , which has an affine form . A similar process of homogenization may be defined for curves in higher dimensional spaces. Except for
lines Long interspersed nuclear elements (LINEs) (also known as long interspersed nucleotide elements or long interspersed elements) are a group of non-LTR (long terminal repeat A long terminal repeat (LTR) is a pair of identical sequences of DNA ...

lines
, the simplest examples of algebraic curves are the
conics In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the Conical surface, surface of a cone (geometry), cone with a plane (mathematics), plane. The three types of conic section are the hyperbola, the para ...
, which are nonsingular curves of degree two and
genus Genus /ˈdʒiː.nəs/ (plural genera /ˈdʒen.ər.ə/) is a taxonomic rank In biological classification In biology, taxonomy () is the scientific study of naming, defining (Circumscription (taxonomy), circumscribing) and classifying gr ...
zero.
Elliptic curve In , an elliptic curve is a , , of one, on which there is a specified point ''O''. An elliptic curve is defined over a ''K'' and describes points in ''K''2, the of ''K'' with itself. If the field's is different from 2 and 3, then the curv ...
s, which are nonsingular curves of genus one, are studied in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
, and have important applications to
cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia ''-logy'' is a suffix in the English language, used with words originally adapted from Ancient Greek ending in (''- ...

cryptography
.


See also

* Coordinate curve *Crinkled arc *Curve fitting *Curve orientation *Curve sketching *Differential geometry of curves *Gallery of curves *List of curves topics *List of curves *Osculating circle *Parametric surface *Path (topology) *Position vector *Vector-valued function **Infinite–dimensional vector function *Winding number


Notes


References

* * * Euclid, commentary and trans. by T. L. Heath ''Elements'' Vol. 1 (1908 Cambridge
Google Books
* E. H. Lockwood ''A Book of Curves'' (1961 Cambridge)


External links



School of Mathematics and Statistics, University of St Andrews, Scotland
Mathematical curves
A collection of 874 two-dimensional mathematical curves

*[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 o
lines
* The Manifold Atlas page o
1-manifolds
{{Authority control Curves, Metric geometry Topology General topology