Curve

For other uses, see Curve (disambiguation).

In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A simple example is the circle. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In everyday use of the term "curve", a straight line is not curved, but in mathematical parlance curves include straight lines and line segments. A large number of other curves have been studied in geometry. This is a list of Curves, by Wikipedia page See also List of curve topics, List of surfaces, Riemann surface. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position

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). The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function The term learning curve refers to the graphical relation between the amount of Learning and the time it takes to learn In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x) The Phillips curve is a historical inverse relation between the rate of Unemployment and the rate of Inflation in an Economy.

An example of a (simple, closed) curve: a hypotrochoid. A hypotrochoid is a roulette traced by a point attached to a Circle of Radius r rolling around the inside of a fixed circle of radius R

1 Definitions
2 Conventions and terminology
3 Lengths of curves
4 Differential geometry
5 Algebraic curve
6 History
7 See also
8 References

Definitions

In mathematics, a (topological) curve is defined as follows. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Let $I$ be an interval of real numbers (i. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Mathematics, the real numbers may be described informally in several different ways e. a non-empty connected subset of $\mathbb{R}$ ). In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of Then a curve $\!\,\gamma$ is a continuous mapping $\,\!\gamma : I \rightarrow X$ , where $X$ is a topological space. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. The curve $\!\,\gamma$ is said to be simple if it is injective, i. e. if for all $x$ , $y$ in $I$ , we have $\,\!\gamma(x) = \gamma(y) \implies x = y$ . 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 $x\ne y$ (other than the extremities of $I$ ), then $γ(x)$ is called a double (or multiple) point of the curve.

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 $S 1$ ; a simple closed curve is also called a Jordan curve or a Jordan arc.

A plane curve is a curve for which X is the Euclidean plane — these are the examples first encountered — or in some cases the projective plane. In mathematics a plane curve is a Curve in a Euclidian plane (cf Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics 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 Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one However, in the case of algebraic curves it is very common not to restrict the curve to having points only defined over the real numbers.

This definition of curve captures our intuitive notion of a curve as a connected, continuous geometric figure that is "like" a line, without thickness and drawn without interruption, although it also includes figures that can hardly be called curves in common usage. For example, the image of a curve can cover a square in the plane (space-filling curve). Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides Space-filling curves or Peano curves are Curves first described by Giuseppe Peano (1858–1932 whose ranges contain the entire 2-dimensional Unit The image of simple plane curve can have Hausdorff dimension bigger than one (see Koch snowflake) and even positive Lebesgue measure^[1] (the last example can be obtained by small variation of the Peano curve construction). In Mathematics, the Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative Real number associated The Koch snowflake (or Koch star) 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 −2 In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to The dragon curve is another unusual example. A dragon curve is the generic name for any member of a family of self similar Fractal curves which can be approximated by recursive methods such as

Conventions and terminology

The distinction between a curve and its image is important. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage 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. In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points 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.

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. In Mathematics, a path in a Topological space X is a continuous map f from the Unit interval I = to The term "curve" is more common in vector calculus and differential geometry. Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry

Lengths of curves

Main article: Arc length

If $X$ is a metric space with metric $d$ , then we can define the length of a curve $\!\,\gamma : [a, b] \rightarrow X$ by

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

A rectifiable curve is a curve with finite length. Determining the length of an irregular arc segment — also called Rectification of a Curve — was historically difficult In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined A parametrization of $\!\,\gamma$ is called natural (or unit speed or parametrised by arc length) if for any $t 1$ , $t 2$ in $[a, b]$ , we have

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

If $\!\,\gamma$ is a Lipschitz-continuous function, then it is automatically rectifiable. In Mathematics, parametric equations are a method of defining a curve In Mathematics, more specifically in Real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions Moreover, in this case, one can define speed of $\!\,\gamma$ at $t 0$ as

$\mbox{speed}(t_0)=\limsup_{t\to t_0} {d(\gamma(t),\gamma(t_0))\over |t-t_0|}$

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

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

Differential geometry

Main article: Differential geometry of curves

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. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change This article only considers curves in Euclidean space Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian A helix (pl helixes or helices) from the Greek word έλιξ, is a special kind of Space curve, i The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects In general relativity, a world line is a curve in spacetime. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 In physics the world line of an object is the unique path of that object as it travels through 4- Dimensional Spacetime. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS

If $X$ is a differentiable manifold, then we can define the notion of differentiable curve in $X$ . A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. 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

$\!\,\gamma : I \rightarrow X.$

This is a basic notion. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability There are less and more restricted ideas, too. If $X$ is a $C k$ manifold (i. e. , a manifold whose charts are $k$ times continuously differentiable), then a $C k$ curve in $X$ is such a curve which is only assumed to be $C k$ (i. For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability e. $k$ times continuously differentiable). If $X$ is an analytic manifold (i. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be 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 Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 +

A differentiable curve is said to be regular if its derivative never vanishes. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change (In words, a regular curve never slows to a stop or backtracks on itself. ) Two $C k$ differentiable curves

$\!\,\gamma_1 :I \rightarrow X$ and

$\!\,\gamma_2 : J \rightarrow X$

are said to be equivalent if there is a bijective $C k$ map

$\!\,p : J \rightarrow I$

such that the inverse map

$\!\,p^{-1} : I \rightarrow J$

is also $C k$ , and

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

for all $t$ . In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B The map $\!\,\gamma_2$ is called a reparametrisation of $\!\,\gamma_1$ ; and this makes an equivalence relation on the set of all $C k$ differentiable curves in $X$ . In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" A $C k$ arc is an equivalence class of $C k$ curves under the relation of reparametrisation. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X

Algebraic curve

Main article: Algebraic curve

Algebraic curves are the curves considered in algebraic geometry. In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with A plane algebraic curve is the locus of points f(x, y) = 0, where f(x, y) is a polynomial in two variables defined over some field F. Algebraic geometry normally looks at such curves in the context of algebraically closed fields. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients If K is the algebraic closure of F, and C is a curve defined by a polynomial f(x, y) defined over F, the points of the curve defined over F, consisting of pairs (a, b) with a and b in F, can be denoted C(F); the full curve itself being C(K). In Mathematics, particularly Abstract algebra, an algebraic closure of a field K is an Algebraic extension of K that is

Algebraic curves can also be space curves, or curves in even higher dimensions, obtained as the intersection (common solution set) of more than one polynomial equation in more than two variables. By eliminating variables by means of the resultant, these can be reduced to plane algebraic curves, which however may introduce singularities such as cusps or double points. In Mathematics, the resultant of two Monic polynomials P and Q over a field k is defined as the product In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one We may also consider these curves to have points defined in the projective plane; if f(x, y) = 0 then if x = u/w and y = v/w, and n is the total degree of f, then by expanding out wⁿf(u/w, v/w) = 0 we obtain g(u, v, w) = 0, where g is homogeneous of degree n. See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics In Mathematics, a homogeneous polynomial is a Polynomial whose terms are Monomials all having the same total degree; or are elements of the same An example is the Fermat curve uⁿ + vⁿ = wⁿ, which has an affine form xⁿ + yⁿ = 1. In Mathematics, the Fermat curve is the Algebraic curve in the Complex projective plane defined in Homogeneous coordinates ( X:

Important examples of algebraic curves are the conics, which are nonsingular curves of degree two and genus zero, and elliptic curves, which are nonsingular curves of genus one studied in number theory and which have important applications to cryptography. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface In Mathematics, genus has a few different but closely related meanings Topology Orientable surface In Mathematics, an elliptic curve is a smooth, projective Algebraic curve of genus one on which there is a specified point O Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" Because algebraic curves in fields of characteristic zero are most often studied over the complex numbers, algbebraic curves in algebraic geometry look like real surfaces. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the real numbers may be described informally in several different ways Looking at them projectively, if we have a nonsingular curve in n dimensions, we obtain a picture in the complex projective space of dimension n, which corresponds to a real manifold of dimension 2n, in which the curve is an embedded smooth and compact surface with a certain number of holes in it, the genus. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In fact, non-singular complex projective algebraic curves are compact Riemann surfaces. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional

History

A curve may be a locus, or a path. In Mathematics, a locus ( Latin for "place" plural loci) is a collection of points which share a property 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. In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry As we now understand, after Newtonian dynamics, to follow a curved path a body must experience acceleration. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Before that, the application of current ideas to (for example) the physics of Aristotle is probably anachronistic. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. This is important because major examples of curves are the orbits of the planets. In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star One reason for the use of the Ptolemaic system of epicycle and deferent was the special status accorded to the circle as curve. In Astronomy, the geocentric model of the Universe is the superseded theory that the Earth is the center of the universe and other In the Ptolemaic system of Astronomy, the epicycle (literally on the circle in Greek) was a geometric model used to explain the variations in Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the

The conic sections had been deeply studied by Apollonius of Perga. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface They were applied in astronomy by Kepler. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study Johannes Kepler (ˈkɛplɚ ( December 27 1571 &ndash November 15 1630) was a German Mathematician, Astronomer The Greek geometers had studied many other kinds of curves. A geometer is a Mathematician whose area of study is Geometry. One reason was their interest in geometric constructions, going beyond compass and straightedge. Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles In that way, the intersection of curves could be used to solve some polynomial equations, such as that involved in trisecting an angle. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with The problem of trisecting the angle is a classic problem of Compass and straightedge constructions of ancient Greek mathematics.

Newton also worked on an early example in the calculus of variations. Calculus of variations is a field of Mathematics that deals with functionals, as opposed to ordinary Calculus which deals with functions. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways (in this case, the cycloid). A Brachistochrone curve, (Greek - "brachistos" shortest "chronos" time or curve of fastest descent is the curve between two points that is covered in the least time A tautochrone or isochrone curve is the curve for which the time taken by an object sliding without friction in uniform Gravity to its lowest point is independent A cycloid is the curve defined by the path of a point on the edge of circular wheel as the wheel rolls along a straight line 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 Physics and Geometry, the catenary is the theoretical Shape of a hanging flexible Chain or Cable when supported at its ends and Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change

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'. In Mathematics, a cubic plane curve is a Plane algebraic curve C defined by a cubic equation F ( x, y, 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. Bézout's theorem is a statement in Algebraic geometry concerning the number of common points or intersection points of two plane Algebraic curves The theorem claims

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. Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of 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" Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex The era of the space-filling curves finally provoked the modern definitions of curve. Space-filling curves or Peano curves are Curves first described by Giuseppe Peano (1858–1932 whose ranges contain the entire 2-dimensional Unit

References

^ Osgood, William F. This article only considers curves in Euclidean space Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian 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 This page addresses the Mathematical topic of Curves in Differential geometry. This is a list of Curves, by Wikipedia page See also List of curve topics, List of surfaces, Riemann surface. This is a list of curve topics in Mathematics. See also Curve, List of curves, and List of differential geometry topics. In Differential geometry of curves, the osculating circle of a sufficiently smooth plane Curve at a given point on the curve is the Circle whose center A parametric surface is a Surface in the Euclidean space R 3 which is defined by a Parametric equation with two parameters In Mathematics, a path in a Topological space X is a continuous map f from the Unit interval I = to A position, location or radius vector is a vector which represents the position of an object in space in relation to an arbitrary reference A vector-valued function is a mathematical function that maps Real numbers onto vectors Vector-valued functions can be defined as \mathbf{r}(t=f(t\mathbf+g(t\mathbf (January 1903). "A Jordan Curve of Positive Area" (in English). Transactions of the American Mathematical Society 4 (1): 107-112. American Mathematical Society.

B. I. Golubov (2001), “Rectifiable curve”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
Famous Curves Index, School of Mathematics and Statistics, University of St Andrews, Scotland

The Encyclopaedia of Mathematics is a large reference work in Mathematics.

Dictionary

-adjective

(obsolete) Bent without angles; crooked; curved.

-noun

A gentle bend, such as in a road.
A simple figure containing no straight portions and no angles; a curved line.
(analytic geometry) A continuous map from a one-dimensional space to a multidimensional space.
(geometry) A one-dimensional figure of non-zero length; the graph of a continuous map from a one-dimensional space.
(algebraic geometry) An algebraic curve; a polynomial relation of the planar coordinates.
(topology) A one-dimensional continuum.
(informal, usually in plural curves) The attractive shape of a woman's body.

-verb

(transitive) To bend; to crook.
(transitive) To cause to swerve from a straight course.
(intransitive) To bend or turn gradually from a given direction.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents