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Curve

From Wikipedia, the free encyclopedia
Mathematical idealization of the trace left by a moving point
For other uses, seeCurve (disambiguation).
Aparabola, one of the simplest curves, after (straight) lines

Inmathematics, acurve (also called acurved line in older texts) is an object similar to aline, but that does not have to bestraight.

Intuitively, a curve may be thought of as the trace left by a movingpoint. This is the definition that appeared more than 2000 years ago inEuclid'sElements: "The [curved] line[a] 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."[1]

This definition of a curve has been formalized in modern mathematics as:A curve is theimage of aninterval to atopological space by acontinuous function. In some contexts, the function that defines the curve is called aparametrization, and the curve is aparametric curve. In this article, these curves are sometimes calledtopological curves to distinguish them from more constrained curves such asdifferentiable curves. This definition encompasses most curves that are studied in mathematics; notable exceptions arelevel curves (which areunions of curves and isolated points), andalgebraic curves (see below). Level curves and algebraic curves are sometimes calledimplicit curves, since they are generally defined byimplicit 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 ofspace-filling curves andfractal curves. For ensuring more regularity, the function that defines a curve is often supposed to bedifferentiable, and the curve is then said to be adifferentiable curve.

Aplane algebraic curve is thezero set of apolynomial in twoindeterminates. More generally, analgebraic curve is the zero set of a finite set of polynomials, which satisfies the further condition of being analgebraic variety ofdimension one. If the coefficients of the polynomials belong to afieldk, the curve is said to bedefined overk. In the common case of areal algebraic curve, wherek is the field ofreal numbers, an algebraic curve is a finite union of topological curves. Whencomplex zeros are considered, one has acomplex algebraic curve, which, from thetopological point of view, is not a curve, but asurface, and is often called aRiemann surface. Although not being curves in the common sense, algebraic curves defined over other fields have been widely studied. In particular, algebraic curves over afinite field are widely used in moderncryptography.

History

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Megalithic art fromNewgrange showing an early interest in curves

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 prehistorictimes.[2] Curves, or at least their graphical representations, are simple to create, for example with a stick on the sand on a beach.

Historically, the termline was used in place of the more modern termcurve. Hence the termsstraight line andright line were used to distinguish what are today called lines from curved lines. For example, in Book I ofEuclid's Elements, a line is defined as a "breadthless length" (Def. 2), while astraight 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).[3] Later commentators further classified lines according to various schemes. For example:[4]

  • 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 curves created by slicing a cone (conic sections) were among the curves studied in ancientGreek mathematics.

The Greekgeometers had studied many other kinds of curves. One reason was their interest in solving geometrical problems that could not be solved using standardcompass and straightedge construction.These curves include:

Analytic geometry allowed curves, such as theFolium of Descartes, to be defined using equations instead of geometrical construction.

A fundamental advance in the theory of curves was the introduction ofanalytic geometry byRené 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 betweenalgebraic curves that can be defined usingpolynomial equations, andtranscendental curves that cannot. Previously, curves had been described as "geometrical" or "mechanical" according to how they were, or supposedly could be, generated.[2]

Conic sections were applied inastronomy byKepler.Newton also worked on an early example in thecalculus of variations. Solutions to variational problems, such as thebrachistochrone andtautochrone questions, introduced properties of curves in new ways (in this case, thecycloid). Thecatenary gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means ofdifferential calculus.

In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied thecubic curves, in the general description of the real points into 'ovals'. The statement ofBézout'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 ofmanifolds andalgebraic varieties. Nevertheless, many questions remain specific to curves, such asspace-filling curves,Jordan curve theorem andHilbert's sixteenth problem.

Topological curve

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Atopological curve can be specified by acontinuous functionγ:I→X{\displaystyle \gamma \colon I\rightarrow X} from anintervalI of thereal numbers into atopological spaceX. Properly speaking, thecurve is theimage ofγ.{\displaystyle \gamma .} However, in some contexts,γ{\displaystyle \gamma } itself is called a curve, especially when the image does not look like what is generally called a curve and does not characterize sufficientlyγ.{\displaystyle \gamma .}

For example, the image of thePeano curve or, more generally, aspace-filling curve completely fills a square, and therefore does not give any information on howγ{\displaystyle \gamma } is defined.

A curveγ{\displaystyle \gamma } isclosed[b] or is aloop ifI=[a,b]{\displaystyle I=[a,b]} andγ(a)=γ(b){\displaystyle \gamma (a)=\gamma (b)}. A closed curve is thus the image of a continuous mapping of acircle. A non-closed curve may also be called anopen curve.

If thedomain of a topological curve is a closed and bounded intervalI=[a,b]{\displaystyle I=[a,b]}, the curve is called apath, also known astopological arc (or justarc).

A curve issimple if it is the image of an interval or a circle by aninjective continuous function. In other words, if a curve is defined by a continuous functionγ{\displaystyle \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).[8]

Aplane curve is a curve for whichX{\displaystyle X} is theEuclidean plane—these are the examples first encountered—or in some cases theprojective plane.Aspace curve is a curve for whichX{\displaystyle X} is at least three-dimensional; askew curve is a space curve which lies in no plane. These definitions of plane, space and skew curves apply also toreal algebraic curves, although the above definition of a curve does not apply (a real algebraic curve may bedisconnected).

Adragon curve with a positive area

A plane simple closed curve is also called aJordan curve. It is also defined as a non-self-intersectingcontinuous loop in the plane.[9] TheJordan curve theorem states that theset complement in a plane of a Jordan curve consists of twoconnected components (that is the curve divides the plane in two non-intersectingregions that are both connected). The bounded region inside a Jordan curve is known asJordan domain.

The definition of a curve includes figures that can hardly be called curves in common usage. For example, the image of a curve can cover asquare in the plane (space-filling curve), and a simple curve may have a positive area.[10]Fractal curves can have properties that are strange for the common sense. For example, a fractal curve can have aHausdorff dimension bigger than one (seeKoch snowflake) and even a positive area. An example is thedragon curve, which has many other unusual properties.

Differentiable curve

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Main article:Differentiable curve

Roughly speaking adifferentiable curve is a curve that is defined as being locally the image of an injective differentiable functionγ:I→X{\displaystyle \gamma \colon I\rightarrow X} from anintervalI of thereal numbers into a differentiable manifoldX, oftenRn.{\displaystyle \mathbb {R} ^{n}.}

More precisely, a differentiable curve is a subsetC ofX where every point ofC has a neighborhoodU such thatC∩U{\displaystyle C\cap U} isdiffeomorphic to an interval of the real numbers.[clarification needed] In other words, a differentiable curve is a differentiable manifold of dimension one.

Differentiable arc

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"Arc (geometry)" redirects here. For the use in finite projective geometry, seeArc (projective geometry). For other uses, seeArc (disambiguation).

InEuclidean geometry, anarc (symbol:⌒) is aconnected subset of adifferentiable curve.

Arcs oflines are calledsegments,rays, orlines, depending on how they are bounded.

A common curved example is an arc of acircle, called acircular arc.

In asphere (or aspheroid), an arc of agreat circle (or agreat ellipse) is called agreat arc.

Length of a curve

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Main article:Arc length
Further information:Differentiable curve § Length

IfX=Rn{\displaystyle X=\mathbb {R} ^{n}} is then{\displaystyle n}-dimensional Euclidean space, and ifγ:[a,b]→Rn{\displaystyle \gamma :[a,b]\to \mathbb {R} ^{n}} is an injective and continuously differentiable function, then the length ofγ{\displaystyle \gamma } is defined as the quantity

Length⁡(γ) =def ∫ab|γ′(t)| dt.{\displaystyle \operatorname {Length} (\gamma )~{\stackrel {\text{def}}{=}}~\int _{a}^{b}|\gamma \,'(t)|~\mathrm {d} {t}.}

The length of a curve is independent of theparametrizationγ{\displaystyle \gamma }.

In particular, the lengths{\displaystyle s} of thegraph of a continuously differentiable functiony=f(x){\displaystyle y=f(x)} defined on a closed interval[a,b]{\displaystyle [a,b]} is

s=∫ab1+[f′(x)]2 dx,{\displaystyle s=\int _{a}^{b}{\sqrt {1+[f'(x)]^{2}}}~\mathrm {d} {x},}

which can be thought of intuitively as using thePythagorean theorem at the infinitesimal scale continuously over the full length of the curve.[11]

More generally, ifX{\displaystyle X} is ametric space with metricd{\displaystyle d}, then we can define the length of a curveγ:[a,b]→X{\displaystyle \gamma :[a,b]\to X} by

Length⁡(γ) =def sup{∑i=1nd(γ(ti),γ(ti−1)) | n∈N and a=t0<t1<…<tn=b},{\displaystyle \operatorname {Length} (\gamma )~{\stackrel {\text{def}}{=}}~\sup \!\left\{\sum _{i=1}^{n}d(\gamma (t_{i}),\gamma (t_{i-1}))~{\Bigg |}~n\in \mathbb {N} ~{\text{and}}~a=t_{0}<t_{1}<\ldots <t_{n}=b\right\},}

where the supremum is taken over alln∈N{\displaystyle n\in \mathbb {N} } and all partitionst0<t1<…<tn{\displaystyle t_{0}<t_{1}<\ldots <t_{n}} of[a,b]{\displaystyle [a,b]}.

A rectifiable curve is a curve withfinite length. A curveγ:[a,b]→X{\displaystyle \gamma :[a,b]\to X} is callednatural (or unit-speed or parametrized by arc length) if for anyt1,t2∈[a,b]{\displaystyle t_{1},t_{2}\in [a,b]} such thatt1≤t2{\displaystyle t_{1}\leq t_{2}}, we have

Length(γ|[t1,t2])=t2−t1.{\displaystyle \operatorname {Length} \!\left(\gamma |_{[t_{1},t_{2}]}\right)=t_{2}-t_{1}.}

Ifγ:[a,b]→X{\displaystyle \gamma :[a,b]\to X} is aLipschitz-continuous function, then it is automatically rectifiable. Moreover, in this case, one can define the speed (ormetric derivative) ofγ{\displaystyle \gamma } att∈[a,b]{\displaystyle t\in [a,b]} as

Speedγ(t) =def lim sups→td(γ(s),γ(t))|s−t|{\displaystyle {\operatorname {Speed} _{\gamma }}(t)~{\stackrel {\text{def}}{=}}~\limsup _{s\to t}{\frac {d(\gamma (s),\gamma (t))}{|s-t|}}}

and then show that

Length⁡(γ)=∫abSpeedγ(t) dt.{\displaystyle \operatorname {Length} (\gamma )=\int _{a}^{b}{\operatorname {Speed} _{\gamma }}(t)~\mathrm {d} {t}.}

Differential geometry

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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 intwo-dimensional space), there are obvious examples such as thehelix which exist naturally in three dimensions. The needs of geometry, and also for exampleclassical mechanics are to have a notion of curve in space of any number of dimensions. Ingeneral relativity, aworld line is a curve inspacetime.

IfX{\displaystyle X} is adifferentiable manifold, then we can define the notion ofdifferentiable curve inX{\displaystyle X}. This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can takeX{\displaystyle 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 thetangent vectors toX{\displaystyle X} by means of this notion of curve.

IfX{\displaystyle X} is asmooth manifold, asmooth curve inX{\displaystyle X} is asmooth map

γ:I→X{\displaystyle \gamma \colon I\rightarrow X}.

This is a basic notion. There are less and more restricted ideas, too. IfX{\displaystyle X} is aCk{\displaystyle C^{k}} manifold (i.e., a manifold whosechart'stransition maps arek{\displaystyle k} timescontinuously differentiable), then aCk{\displaystyle C^{k}} curve inX{\displaystyle X} is such a curve which is only assumed to beCk{\displaystyle C^{k}} (i.e.k{\displaystyle k} times continuously differentiable). IfX{\displaystyle X} is ananalytic manifold (i.e. infinitely differentiable and charts are expressible aspower series), andγ{\displaystyle \gamma } is an analytic map, thenγ{\displaystyle \gamma } is said to be ananalytic curve.

A differentiable curve is said to beregular if itsderivative never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) TwoCk{\displaystyle C^{k}} differentiable curves

γ1:I→X{\displaystyle \gamma _{1}\colon I\rightarrow X} and
γ2:J→X{\displaystyle \gamma _{2}\colon J\rightarrow X}

are said to beequivalent if there is abijectiveCk{\displaystyle C^{k}} map

p:J→I{\displaystyle p\colon J\rightarrow I}

such that theinverse map

p−1:I→J{\displaystyle p^{-1}\colon I\rightarrow J}

is alsoCk{\displaystyle C^{k}}, and

γ2(t)=γ1(p(t)){\displaystyle \gamma _{2}(t)=\gamma _{1}(p(t))}

for allt{\displaystyle t}. The mapγ2{\displaystyle \gamma _{2}} is called areparametrization ofγ1{\displaystyle \gamma _{1}}; and this makes anequivalence relation on the set of allCk{\displaystyle C^{k}} differentiable curves inX{\displaystyle X}. ACk{\displaystyle C^{k}}arc is anequivalence class ofCk{\displaystyle C^{k}} curves under the relation of reparametrization.

Algebraic curve

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Main article:Algebraic curve

Algebraic curves are the curves considered inalgebraic geometry. A plane algebraic curve is theset of the points of coordinatesx,y such thatf(x,y) = 0, wheref is a polynomial in two variables defined over some fieldF. One says that the curve isdefined overF. Algebraic geometry normally considers not only points with coordinates inF but all the points with coordinates in analgebraically closed fieldK.

IfC is a curve defined by a polynomialf with coefficients inF, the curve is said to be defined overF.

In the case of a curve defined over thereal numbers, one normally considers points withcomplex coordinates. In this case, a point with real coordinates is areal point, and the set of all real points is thereal 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 calledRiemann surfaces.

The points of a curveC with coordinates in a fieldG are said to be rational overG and can be denotedC(G). WhenG is the field of therational numbers, one simply talks ofrational points. For example,Fermat's Last Theorem may be restated as:Forn > 2,every rational point of theFermat curve of degreen has a zero coordinate.

Algebraic curves can also be space curves, or curves in a space of higher dimension, sayn. They are defined asalgebraic varieties ofdimension one. They may be obtained as the common solutions of at leastn–1 polynomial equations inn variables. Ifn–1 polynomials are sufficient to define a curve in a space of dimensionn, the curve is said to be acomplete intersection. By eliminating variables (by any tool ofelimination theory), an algebraic curve may be projected onto aplane algebraic curve, which however may introduce new singularities such ascusps ordouble points.

A plane curve may also be completed to a curve in theprojective plane: if a curve is defined by a polynomialf of total degreed, thenwdf(u/w,v/w) simplifies to ahomogeneous polynomialg(u,v,w) of degreed. The values ofu,v,w such thatg(u,v,w) = 0 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 thatw is not zero. An example is the Fermat curveun +vn =wn, which has an affine formxn +yn = 1. A similar process of homogenization may be defined for curves in higher dimensional spaces.

Except forlines, the simplest examples of algebraic curves are theconics, which are nonsingular curves of degree two andgenus zero.Elliptic curves, which are nonsingular curves of genus one, are studied innumber theory, and have important applications tocryptography.

See also

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Notes

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  1. ^In current mathematical usage, a line is straight. Previously lines could be either curved or straight.
  2. ^This term my be ambiguous, as a non-closed curve may be aclosed set, as is a line in a plane.

References

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  1. ^In (rather old) French: "La ligne est la première espece de quantité, laquelle a tant seulement une dimension à sçavoir longitude, sans aucune latitude ni profondité, & n'est autre chose que le flux ou coulement du poinct, lequel […] laissera de son mouvement imaginaire quelque vestige en long, exempt de toute latitude." Pages 7 and 8 ofLes quinze livres des éléments géométriques d'Euclide Megarien, traduits de Grec en François, & augmentez de plusieurs figures & demonstrations, avec la corrections des erreurs commises és autres traductions, by Pierre Mardele, Lyon, MDCXLV (1645).
  2. ^abLockwood p. ix
  3. ^Heath p. 153
  4. ^Heath p. 160
  5. ^Lockwood p. 132
  6. ^Lockwood p. 129
  7. ^O'Connor, John J.;Robertson, Edmund F.,"Spiral of Archimedes",MacTutor History of Mathematics Archive,University of St Andrews
  8. ^"Jordan arc definition at Dictionary.com. Dictionary.com Unabridged. Random House, Inc".Dictionary.reference.com. Retrieved2012-03-14.
  9. ^Sulovský, Marek (2012).Depth, Crossings and Conflicts in Discrete Geometry. Logos Verlag Berlin GmbH. p. 7.ISBN 9783832531195.
  10. ^Osgood, William F. (January 1903)."A Jordan Curve of Positive Area".Transactions of the American Mathematical Society.4 (1).American Mathematical Society:107–112.doi:10.2307/1986455.ISSN 0002-9947.JSTOR 1986455.
  11. ^Davis, Ellery W.; Brenke, William C. (1913).The Calculus. MacMillan Company. p. 108.ISBN 9781145891982.

External links

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