In
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 ...
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
However, in some contexts,
itself is called a curve, especially when the image does not look like what is generally called a curve and does not characterize sufficiently
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
is defined.
A curve
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