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Encyclopedia > Curve

In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. Simple examples are the circle or the straight line. A large number of other curves have been studied in geometry. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... A circle, in Euclidean geometry, is the set of all points at a fixed distance, called the radius, from a fixed point, the centre. ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... This is a list of curves, by Wikipedia page. ... Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ...


This article is about the general theory. The term curve is also used in ways making it almost synonymous with mathematical function (as in learning curve), or graph of a function (Phillips curve). In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... The learning curve effect and the closely related experience curve effect express the relationship between experience and efficiency. ... In mathematics, the graph of a function f(x1, x2, ..., xn) is the collection of all tuples (x1, x2, ..., xn, f(x1, ..., xn)). In particular, graph means the graphical representation of this collection, in the form of a curve or surface, together with axes, etc. ... In macroeconomics, the Phillips curve is a supposed inverse relationship between inflation and unemployment. ...

Contents


Definitions

In mathematics, a (topological) curve is defined as follows. Let I be an interval of real numbers (i.e. a non-empty connected subset of mathbb{R}). Then a curve !,gamma is a continuous mapping ,!gamma : I rightarrow X, where X is a topological space. The curve !,gamma is said to be simple if it is injective, i.e. if for all x, y in I, we have ,!gamma(x) = gamma(y) rightarrow x = y. If I is a closed bounded interval ,![a, b], we also allow the possibility ,!gamma(a) = gamma(b) (this convention makes it possible to talk about closed simple curve). If γ(x) = γ(y) for some xne y (other than the extremities of I), then γ(x) is called a double (or: multiple) point of the curve. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... The word mapping has several senses: In mathematics and related technical fields, it is some kind of function: see map (mathematics). ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...


A curve !,gamma is said to be closed or a loop if ,!I = [a, b] and if !,gamma(a) = gamma(b). A closed curve is thus a continuous mapping of the circle S1; a simple closed curve is also called a Jordan curve.


A plane curve is a curve for which X is the mathematical plane — these are the examples first encountered — or in some cases the projective plane. A space curve is a curve for which X is of three dimensions, usually Euclidean space; a skew curve is a space curve which lies in no plane. These definitions also apply to algebraic curves (see below). In mathematics, a plane is the fundamental two-dimensional object. ... Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...


This definition of curve captures our intuitive notion of a curve as a connected, continuous geometric figure that is "like" a line, although it also includes figures that can be hardly called curves in common usage. For example, the image of a curve can cover a square in the plane (Peano curve). The image of simple plane curve can have Hausdorff dimension bigger than one (see Koch snowflake) and even positive Lebesgue measure (the last example can be obtained by small variation of the Peano curve construction). The dragon curve is yet another weird example. In plane geometry, a square is a polygon with four equal sides and equal angles. ... Intuitively, a continuous curve in the 2-dimensional plane or in the 3-dimensional space can be thought of as the path of a continuously moving point. To eliminate the inherent vagueness of this notion, Jordan in 1887 introduced the following rigorous definition, which has since been adopted as the... In mathematics, the Hausdorff dimension is an extended non-negative real number (that is a number in the closed infinite interval [0, ∞]) associated to any metric space . ... The first four iterations of the Koch snowflake The Koch curve is a mathematical curve, and one of the earliest fractal curves to have been described. ... A negative number is a number that is less than zero, such as −3. ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ... A dragon curve is the generic name for a member of a family of self similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. ...


Conventions and terminology

The distinction between a curve and its image is important. Two distinct curves may have the same image. For example, a line segment can be traced out at different speeds, or a circle can be traversed a different number of times. Many times, however, we are just interested in the image of the curve. It is important to pay attention to context and convention in reading. In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In mathematics, a line segment is a part of a line that is bounded by two end points. ...


Terminology is also not uniform. Often, topologists use the term "path" for what we are calling a curve, and "curve" for what we are calling the image of a curve. The term "curve" is more common in vector calculus and differential geometry. In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to X f : I → X. The initial point of the path is f(0) and the terminal point is f(1). ... Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...


Lengths of curves

If X is a metric space with metric d, then we can define the length of a curve !,gamma : [a, b] rightarrow X by In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

mbox{Length} (gamma)=sup left{ sum_{i=1}^n d(gamma(t_i),gamma(t_{i-1})) : n in mathbb{N} mbox{ and } a = t_0 < t_1 < dots < t_n = b right}.

A rectifiable curve is a curve with finite length. A parametrization of !,gamma is called natural (or unit speed or parametrised by arc length) if for any t1, t2 in [a,b], we have In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...

mbox{Length} (gamma|_{[t_1,t_2]})=|t_2-t_1|.

If !,gamma is Lipschitz then it is automatically rectifiable. Moreover, in this case, one can define speed of !,gamma at t0 as In mathematics, a function f : M → N between metric spaces M and N is called Lipschitz continuous (or is said to satisfy a Lipschitz condition) if there exists a constant K > 0 such that d(f(x), f(y)) ≤ K d(x, y) for all x and y in M...

mbox{Speed}(t_0)=limsup_{tto t_0} {d(gamma(t),gamma(t_0))over |t-t_0|}

and then

mbox{Length}(gamma)=int_a^b mbox{Speed}(t) , dt.

In particular, if X = mathbb{R}^n is Euclidean space and gamma : [a, b] rightarrow mathbb{R}^n is differentiable then In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

mbox{Length}(gamma)=int_a^b left| , {dgamma over dt} , right| , dt.

Differential geometry

Main article: differential geometry of curves In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ...


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. A helix (pl: helices), from the Greek word έλικας/έλιξ, is a twisted shape like a spring, screw or a spiral staircase. ... In physics, classical mechanics or Newtonian mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the set of physical laws governing and mathematically describing the motions of bodies and aggregates of bodies. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ...


If X is a differentiable manifold, 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. This article is in need of attention. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ...


If X is a smooth manifold, a smooth curve in X is a smooth map In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

!,gamma : I rightarrow X.

This is a basic notion. There are less and more restricted ideas, too. If X is a Ck manifold (i.e., a manifold whose charts are k times continuously differentiable), then a Ck curve in X is such a curve which is only assumed to be Ck (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. In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...


A differentiable curve is said to be regular if its derivative never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two Ck differentiable curves In mathematics, the derivative is one of the two central concepts of calculus. ...

!,gamma_1 :I rightarrow X and
!,gamma_2 : J rightarrow X

are said to be equivalent if there is a bijective Ck map In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...

!,p : J rightarrow I

such that the inverse map In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...

!,p^{-1} : I rightarrow J

is also Ck, and

!,gamma_{2}(t) = gamma_{1}(p(t))

for all t. The map !,gamma_2 is called a reparametrisation of !,gamma_1; and this makes an equivalence relation on the set of all Ck differentiable curves in X. A Ck arc is an equivalence class of Ck curves under the relation of reparametrisation. In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...


Algebraic curve

Main article: Algebraic curve In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ...


In the setting of algebraic geometry, a curve is usually defined to be an algebraic curve. These include, for example, elliptic curves, which are studied in number theory and which have important applications to cryptography. Algebraic curves are more akin to surfaces than curves. Non-singular complex projective algebraic curves are in fact compact Riemann surfaces. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... This article needs to be cleaned up to conform to a higher standard of quality. ... The Enigma machine, used by Germany in World War II, implemented a complex cipher to protect sensitive communications. ... An open surface with X-, Y-, and Z-contours shown. ... In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ...


History

A curve may be a locus, or a path. That is, it may be a graphical representation of some property of points; or it may be traced out, for example by a stick in the sand on a beach. Of course if one says curved in ordinary language, it means bent (not straight), so refers to a locus. This leads to the general idea of curvature. As we now understand, after Newtonian dynamics, to follow a curved path a body must experience acceleration. Before that, the application of current ideas to (for example) the physics of Aristotle is probably anachronistic. This is important because major examples of curves are the orbits of the planets. One reason for the use of the Ptolemaic system of epicycle and deferent was the special status accorded to the circle as curve. In mathematics, a locus (plural loci) is a collection of points which share a common property. ... Curvature is the amount by which a geometric object deviates from being flat. ... In the article vector quantities are written in bold whereas scalar ones are in italics. ... Acceleration is the time rate of change of velocity, and at any point on a v-t graph, it is given by the gradient of the tangent to that point In physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. ... A black hole concept drawing by NASA. Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... Aristotle, marble copy of bronze by Lysippos. ... In physics, an orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity. ... Mediaeval drawing of the Ptolemaic system. ... In the Ptolemaic system of astronomy, the epicycle (literally: on the cycle in Greek) was a geometric model to explain the variations in speed and direction of the apparent motion of the Moon, Sun, and planets. ... A circle, in Euclidean geometry, is the set of all points at a fixed distance, called the radius, from a fixed point, the centre. ...


The conic sections had been deeply studied by Apollonius of Perga. They were applied in astronomy by Kepler. The Greek geometers had studied many other kinds of curves. One reason was their interest in geometric constructions, going beyond ruler-and-compass constructions. In that way, the intersection of curves could be used to solve some polynomial equations, such as that involved in trisecting an angle. In mathematics, a conic section (or just conic) is a curve formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Lunar astronomy: the large crater is Daedalus, photographed by the crew of Apollo 11 as they circled the Moon in 1969. ... Johannes Kepler Johannes Kepler (December 27, 1571 – November 15, 1630), a key figure in the scientific revolution, was a German astronomer, mathematician and astrologer. ... A geometer is a mathematician whose area of study is geometry. ... Creating a regular hexagon with a ruler and compass A number of ancient problems in plane geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... A number of ancient problems in geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass. ...


Newton also worked on an early example in the calculus of variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways (in this case, the cycloid). The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means of differential calculus. Calculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ... A Brachistochrone curve, or curve of fastest descent, is the curve between two points that is covered in the least time by a body that starts at the first point with zero speed and passes down along the curve to the second point, under the action of constant gravity and... A tautochrone curve is the curve for which the time taken by a particle sliding down it under uniform gravity to its lowest point is independent of its starting point. ... Cycloid (red) generated by a rolling circle A cycloid is the curve defined by a fixed point on a wheel as it rolls, or, more precisely, the locus of a point on the rim of a circle rolling along a straight line. ... In mathematics, the catenary is the shape of a hanging flexible chain or cable when supported at its ends and acted upon by a uniform gravitational force (its own weight). ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ...


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. In mathematics, a cubic curve is a plane curve C defined by a cubic equation F(X,Y,Z) = 0 applied to homogeneous coordinates [X:Y:Z] for the projective plane; or the inhomogeneous version for the affine space determined by setting Z = 1 in such an equation. ... This article refers to Bézouts theorem in algebraic geometry. ...


From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry; and later topology, when for example the Jordan curve theorem was understood to lie quite deep, as well as being required in complex analysis. The era of the space-filling curves finally provoked the modern definitions of curve. Projective geometry can be thought of informally as the geometry which arises from placing ones eye at a point. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... In topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane divides the plane into an inside and an outside. It was proved by Oswald Veblen in 1905. ... Complex analysis is the branch of mathematics investigating functions of complex numbers. ... Space-filling curves or Peano curves are curves, first described by Giuseppe Peano, whose ranges contain the entire 2-dimensional unit square (or the 3-dimensional unit cube). ...


See also

Curvature is the amount by which a geometric object deviates from being flat. ... In differential geometry, the osculating circle of a curve at a point shares a common tangent line and a common radius of curvature with the curve at that point. ... In mathematics, a positively oriented curve is a planar simple closed curve (that is, a curve in the plane whose starting point is also the end point and which has no other self-intersections) such that when travelling on it one always has the curve interior to the left (and... This is a list of curves, by Wikipedia page. ... This is a list of curve topics in mathematics, by Wikipedia page. ...

External links

  • List of famous curves

  Results from FactBites:
 
Curve - Wikipedia, the free encyclopedia (1231 words)
A space curve is a curve for which X is of three dimensions, usually Euclidean space; a skew curve is a space curve which lies in no plane.
A rectifiable curve is a curve with finite length.
From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry; and later topology, when for example the Jordan curve theorem was understood to lie quite deep, as well as being required in complex analysis.
PlanetMath: curve (450 words)
The second notion is geometric; in this sense a curve is an arc, a 1-dimensional subset of an ambient space.
The two notions are related: the image of a parameterized curve describes the trajectory of a moving particle.
In algebraic geometry, the term curve is used to describe a 1-dimensional variety relative to the complex numbers or some other ground field.
  More results at FactBites »

 
 

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