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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a curve (also called a curved line in older texts) is an object similar to a
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
, but that does not have to be
straight Straight may refer to: Slang * Straight, slang for heterosexual ** Straight-acting, an LGBT person who does not exhibit the appearance or mannerisms of the gay stereotype * Straight, a member of the straight edge subculture Sport and games * Str ...
. Intuitively, a curve may be thought of as the trace left by a moving
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
. 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 is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of an interval to a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
by a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a
parametric curve In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...
. 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 that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the sy ...
s. This definition encompasses most curves that are studied in mathematics; notable exceptions are
level curve In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is calle ...
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 c ...
s (see below). Level curves and algebraic curves are sometimes called
implicit curve In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly ''x'' and ''y''. For example, the unit circle is defined by the implicit equation x^2+y^2=1. In general, every implic ...
s, since they are generally defined by
implicit equation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit functi ...
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, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space ...
s and
fractal curve A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rec ...
s. For ensuring more regularity, the function that defines a curve is often supposed to be
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
, and the curve is then said to be a
differentiable curve Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the sy ...
. A
plane 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 c ...
is the
zero set In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equi ...
of a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
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 c ...
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, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
of
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
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 grass ...
, the curve is said to be ''defined over'' . In the common case of a
real algebraic curve In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial ...
, where is the field of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, an algebraic curve is a finite union of topological curves. When
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
zeros are considered, one has a ''complex algebraic curve'', which, from the
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
point of view, is not a curve, but a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
, and is often called a
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
. 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, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
are widely used in modern
cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
.


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 mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulat ...
, 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 In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
construction. These curves include: *The conic sections, studied in depth by
Apollonius of Perga Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribution ...
*The
cissoid of Diocles In geometry, the cissoid of Diocles (; named for Diocles) is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio. In particular, it can be used to double a cube. It can be de ...
, studied by Diocles and used as a method to double the cube. *The
conchoid of Nicomedes In geometry, a conchoid is a curve derived from a fixed point , another curve, and a length . It was invented by the ancient Greek mathematician Nicomedes. Description For every line through that intersects the given curve at the two points ...
, studied by
Nicomedes Nicomedes may refer to: *Nicomedes (mathematician), ancient Greek mathematician who discovered the conchoid *Nicomedes of Sparta, regent during the youth of King Pleistoanax, commanded the Spartan army at the Battle of Tanagra (457 BC) *Saint Nicom ...
as a method to both double the cube and to
trisect an angle Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge a ...
. *The
Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...
, studied by
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
as a method to trisect an angle and
square the circle Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficulty ...
. *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 bicircular quartic curves that are symme ...
s, sections of tori studied by
Perseus In Greek mythology, Perseus (Help:IPA/English, /ˈpɜːrsiəs, -sjuːs/; Greek language, Greek: Περσεύς, Romanization of Greek, translit. Perseús) is the legendary founder of Mycenae and of the Perseid dynasty. He was, alongside Cadmus ...
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 using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...
by
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
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 c ...
s that can be defined using
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation' ...
s, and
transcendental curve In analytical geometry , a transcendental curve is a curve that is not an algebraic curve.Newman, JA, ''The Universal Encyclopedia of Mathematics'', Pan Reference Books, 1976, , "Transcendental curves". Here for a curve, ''C'', what matters is the ...
s 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 () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
by
Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...
. Newton also worked on an early example in the
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. Solutions to variational problems, such as the
brachistochrone In physics and mathematics, a brachistochrone curve (), or curve of fastest descent, is the one lying on the plane between a point ''A'' and a lower point ''B'', where ''B'' is not directly below ''A'', on which a bead slides frictionlessly under ...
and
tautochrone A tautochrone or isochrone curve (from Greek prefixes tauto- meaning ''same'' or iso- ''equal'', and chrono ''time'') is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independe ...
questions, introduced properties of curves in new ways (in this case, the
cycloid In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve ...
). The
catenary In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field. The catenary curve has a U-like shape, superficia ...
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, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
. In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the
cubic curve In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an equ ...
s, in the general description of the real points into 'ovals'. The statement of
Bézout's theorem Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the degr ...
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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s and
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
. Nevertheless, many questions remain specific to curves, such as
space-filling curve In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space ...
s,
Jordan curve theorem In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior ...
and
Hilbert's sixteenth problem Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, as part of his list of 23 problems in mathematics. The original problem was posed as the ''Problem of the topolo ...
.


Topological curve

A topological curve can be specified by a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
\gamma \colon I \rightarrow X from an interval of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s into a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
. Properly speaking, the ''curve'' is the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
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 In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective. ...
or, more generally, a
space-filling curve In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space ...
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 Loop or LOOP may refer to: Brands and enterprises * Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live * Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets * Loop Mobile, an ...
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 consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
. If the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
of a topological curve is a closed and bounded interval I = , b/math>, the curve is called a ''
path A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desire p ...
'', 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, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
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 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). A plane simple closed curve is also called a Jordan curve. It is also defined as a non-self-intersecting continuous loop in the plane. The
Jordan curve theorem In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior ...
states that the
set complement In set theory, the complement of a set , often denoted by (or ), is the set of elements not in . When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is the ...
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, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and the interaction of humanity and t ...
that are both connected). A ''
plane curve In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic pla ...
'' is a curve for which X is the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
—these are the examples first encountered—or in some cases the
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
. 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 curve In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial ...
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, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
in the plane (
space-filling curve In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space ...
) and thus have a positive area.
Fractal curve A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rec ...
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 is zero, of a ...
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 ...
) 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 repe ...
, 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, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
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 ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, an arc (symbol: ⌒) is a
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
subset of a
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
curve. Arcs of
lines Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
are called segments,
rays Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (gra ...
, or
lines Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
, depending on how they are bounded. A common curved example is an arc of a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
, called a
circular arc Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fallacy * Circular ...
. In a
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
(or a
spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...
), an arc of a
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
(or a
great ellipse 150px, A spheroid A great ellipse is an ellipse passing through two points on a spheroid and having the same center as that of the spheroid. Equivalently, it is an ellipse on the surface of a spheroid and centered on the origin, or the curve ...
) 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 discre ...
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, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
with metric d , then we can define the length of a curve \gamma: ,b\to X by : \operatorname(\gamma) ~ \stackrel ~ \sup \! \left\, where the supremum is taken over all n \in \mathbb and all partitions t_ < t_ < \ldots < t_ of
, b The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
. A rectifiable curve is a curve with
finite Finite is the opposite of 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 or marked ...
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 German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...
function, then it is automatically rectifiable. Moreover, in this case, one can define the speed (or
metric derivative In mathematics, the metric derivative is a notion of derivative appropriate to parametrized paths in metric spaces. It generalizes the notion of "speed" or "absolute velocity" to spaces which have a notion of distance (i.e. metric spaces) but not ...
) 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 () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, ...
which exist naturally in three dimensions. The needs of geometry, and also for example
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
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 and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, a
world line The world line (or worldline) of an object is the 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 distinguished from con ...
is a curve in
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
. If X is a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, 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 In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...
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 that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, a ''smooth curve'' in X is a
smooth map In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it ...
:\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 (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tabul ...
are k times
continuously differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
), 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 manifold In mathematics, an analytic manifold, also known as a C^\omega manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifolds are also analytic. In algebraic ge ...
(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 + \dots 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, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
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, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
C^k map :p \colon J \rightarrow I such that the
inverse map In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X\t ...
: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, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
on the set of all C^k differentiable curves in X. A C^k ''arc'' is an
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
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, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. A plane algebraic curve is the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
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, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
. 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, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, one normally considers points with
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
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, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
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 of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s, one simply talks of ''rational points''. For example,
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
may be restated as: ''For'' , ''every rational point of the
Fermat curve In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (''X'':''Y'':''Z'') by the Fermat equation :X^n + Y^n = Z^n.\ Therefore, in terms of the affine plane its equation is :x^n ...
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, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
of
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
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 ''V'' in projective space is a complete intersection if the ideal of ''V'' is generated by exactly ''codim V'' elements. That is, if ''V'' has dimension ''m'' and lies in projective space ''P'n'', there shou ...
. By eliminating variables (by any tool of
elimination theory Elimination may refer to: Science and medicine *Elimination reaction, an organic reaction in which two functional groups split to form an organic product *Bodily waste elimination, discharging feces, urine, or foreign substances from the body ...
), an algebraic curve may be projected onto a
plane 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 c ...
, which however may introduce new singularities such as
cusp A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifurca ...
s or
double point In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied. Algebraic curves in the plane Algebraic curv ...
s. A plane curve may also be completed to a curve in the
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
: if a curve is defined by a polynomial of total degree , then simplifies to a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
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 Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
, the simplest examples of algebraic curves are the
conics In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
, which are nonsingular curves of degree two and
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
zero.
Elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
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 pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, and have important applications to
cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
.


See also

*
Coordinate curve In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
*
Crinkled arc In mathematics, and in particular the study of Hilbert spaces, a crinkled arc is a type of continuous curve. The concept is usually credited to Paul Halmos. Specifically, consider f\colon ,1\to X, where X is a Hilbert space with inner product \ ...
*
Curve fitting Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is ...
*
Curve orientation In mathematics, an orientation of a curve is the choice of one of the two possible directions for travelling on the curve. For example, for Cartesian coordinates, the -axis is traditionally oriented toward the right, and the -axis is upward orie ...
*
Curve sketching In geometry, curve sketching (or curve tracing) are techniques for producing a rough idea of overall shape of a plane curve given its equation, without computing the large numbers of points required for a detailed plot. It is an application of t ...
*
Differential geometry of curves Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the sy ...
* Gallery of curves *
List of curves topics This is an alphabetical index of articles related to curves used in mathematics. * Acnode * Algebraic curve * Arc * Asymptote * Asymptotic curve * Barbier's theorem * Bézier curve * Bézout's theorem * Birch and Swinnerton-Dyer conjecture * Bi ...
*
List of curves This is a list of Wikipedia articles about curves used in different fields: mathematics (including geometry, statistics, and applied mathematics), physics, engineering, economics, medicine, biology, psychology, ecology, etc. Mathematics (Geometry) ...
*
Osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ''p'' on the curve has been traditionally defined as the circle passing through ''p'' and a pair of additional points on the curve i ...
*
Parametric surface A parametric surface is a surface in the Euclidean space \R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that oc ...
*
Path (topology) In mathematics, a path in a topological space X is a continuous function from the closed unit interval , 1/math> into X. Paths play an important role in the fields of topology and mathematical analysis. For example, a topological space for w ...
*
Polygonal curve In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments co ...
*
Position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or s ...
*
Vector-valued function A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...
** Infinite-dimensional vector function *
Winding number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turn ...


Notes


References

* * *
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
, commentary and trans. by
T. L. Heath Sir Thomas Little Heath (; 5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer. He was educated at Clifton College. Heath transla ...
''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

* ttp://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 Metric geometry Topology General topology