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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. 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 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 this article, these curves are sometimes called ''topological curves'' to distinguish them from more constrained curves such as differentiable curves. This definition encompasses most curves that are studied in mathematics; notable exceptions are level curves (which are unions of curves and isolated points), and algebraic curves (see below). Level curves and algebraic curves are sometimes called implicit curves, since they are generally defined by implicit equations. 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 curves and fractal curves. For ensuring more regularity, the function that defines a curve is often supposed to be differentiable, and the curve is then said to be a differentiable curve. A plane algebraic curve is the zero set of a polynomial in two
indeterminate Indeterminate may refer to: In mathematics * Indeterminate (variable), a symbol that is treated as a variable * Indeterminate system, a system of simultaneous equations that has more than one solution * Indeterminate equation, an equation that ha ...
s. More generally, an algebraic curve is the zero set of a finite set of polynomials, which satisfies the further condition of being an algebraic variety of dimension one. If the coefficients of the polynomials belong to a field , the curve is said to be ''defined over'' . In the common case of a real algebraic curve, where is the field of real numbers, an algebraic curve is a finite union of topological curves. When complex zeros are considered, one has a ''complex algebraic curve'', which, from the topological point of view, is not a curve, but a surface, and is often called a Riemann surface. Although not being curves in the common sense, algebraic curves defined over other fields have been widely studied. In particular, algebraic curves over a finite field are widely used in modern cryptography.


History

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, 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 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, studied by Diocles and used as a method to
double the cube Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related pro ...
. *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 sections, 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 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 curves that can be defined using polynomial equations, 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 by Kepler. 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 and tautochrone questions, introduced properties of curves in new ways (in this case, the cycloid). The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means of
differential calculus In 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 curves, 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 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. Nevertheless, many questions remain specific to curves, such as space-filling curves, Jordan curve theorem 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 numbers into a topological space . 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 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. 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'', 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 states that the set complement in a plane of a Jordan curve consists of two connected components (that is the curve divides the plane in two non-intersecting regions that are both connected). A '' plane curve'' 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. 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 curves, 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 the plane ( space-filling curve) and thus have a positive area. Fractal curves can have properties that are strange for the common sense. For example, a fractal curve can have a Hausdorff dimension bigger than one (see Koch snowflake) and even a positive area. An example is the
dragon curve 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 numbers 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 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, an arc (symbol: ⌒) is a connected subset of a differentiable 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, 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, called a circular arc. In a sphere (or a spheroid), 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) 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 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 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 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) 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 which exist naturally in three dimensions. The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. In general relativity, a world line is a curve in spacetime. If 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 vectors to X by means of this notion of curve. If X is a smooth manifold, a ''smooth curve'' in X is a smooth map :\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 (i.e. infinitely differentiable and charts are expressible as power series), 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 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 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, 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 numbers, one normally considers points with 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 surfaces. 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 numbers, one simply talks of ''rational points''. For example, Fermat's Last Theorem 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 of dimension 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), an algebraic curve may be projected onto a plane algebraic curve, which however may introduce new singularities such as cusps 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: if a curve is defined by a polynomial of total degree , then simplifies to a homogeneous polynomial 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 zero. Elliptic curves, which are nonsingular curves of genus one, are studied in number theory, and have important applications to cryptography.


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 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 * Differential geometry of curves *
Gallery of curves This is a gallery of curves used in mathematics, by Wikipedia page. See also list of curves. Algebraic curves Rational curves Degree 1 File:FuncionLineal01.svg, Line Degree 2 File:Circle-withsegments.svg, Circle File:Ellipse Properties of D ...
*
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 *
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) * Polygonal curve *
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 **
Infinite-dimensional vector function An infinite-dimensional vector function is a function whose values lie in an infinite-dimensional topological vector space, such as a Hilbert space or a Banach space. Such functions are applied in most sciences including physics. Example Set f_k( ...
* Winding number


Notes


References

* * * Euclid, 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