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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. Curves, or at least their graphical representations, are simple to create, for example with
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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
1871:
4420:
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,
901:
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
2737:
296:
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:
1548:
83:: "The line 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."
2830:
431:
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
2098:{\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\},}
2482:
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1765:
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Les 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
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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
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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
237:. Although not being curves in the common sense, algebraic curves defined over other fields have been widely studied. In particular, algebraic curves over a
2386:
444:
that cannot. Previously, curves had been described as "geometrical" or "mechanical" according to how they were, or supposedly could be, generated.
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629:
itself is called a curve, especially when the image does not look like what is generally called a curve and does not characterize sufficiently
336:
had studied many other kinds of curves. One reason was their interest in solving geometrical problems that could not be solved using standard
490:
showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions.
2732:{\displaystyle {\operatorname {Speed} _{\gamma }}(t)~{\stackrel {\text{def}}{=}}~\limsup _{s\to t}{\frac {d(\gamma (s),\gamma (t))}{|s-t|}}}
1025:
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 a
4709:
1676:
475:
gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means of
2925:. This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take
4494:
2945:
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
4361:
1543:{\displaystyle \operatorname {Length} (\gamma )~{\stackrel {\text{def}}{=}}~\int _{a}^{b}|\gamma \,'(t)|~\mathrm {d} {t}.}
1194:
to an interval of the real numbers. In other words, a differentiable curve is a differentiable manifold of dimension one.
482:
In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the
2139:
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2825:{\displaystyle \operatorname {Length} (\gamma )=\int _{a}^{b}{\operatorname {Speed} _{\gamma }}(t)~\mathrm {d} {t}.}
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Since the nineteenth century, curve theory is viewed as the special case of dimension one of the theory of
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is a space curve which lies in no plane. These definitions of plane, space and skew curves apply also to
1309:
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In current mathematical usage, a line is straight. Previously lines could be either curved or straight.
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point of view a surface. In particular, the nonsingular complex projective algebraic curves are called
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While the first examples of curves that are met are mostly plane curves (that is, in everyday words,
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function, then it is automatically rectifiable. Moreover, in this case, one can define the speed (or
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can have properties that are strange for the common sense. For example, a fractal curve can have a
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were used to distinguish what are today called lines from curved lines. For example, in Book I of
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is the zero set of a finite set of polynomials, which satisfies the further condition of being an
121:. This definition encompasses most curves that are studied in mathematics; notable exceptions are
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never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two
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93:
31:
4721:
4244:. A similar process of homogenization may be defined for curves in higher dimensional spaces.
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is a curve that is defined as being locally the image of an injective differentiable function
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Indeterminate (lines that extend indefinitely, such as the straight line and the parabola)
152:. For ensuring more regularity, the function that defines a curve is often supposed to be
8:
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Algebraic curves can also be space curves, or curves in a space of higher dimension, say
2859:
1771:
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1038:
1030:
975:, although the above definition of a curve does not apply (a real algebraic curve may be
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Gallery of Bishop Curves and Other
Spherical Curves, includes animations by Peter Moses
4601:
4513:"Jordan arc definition at Dictionary.com. Dictionary.com Unabridged. Random House, Inc"
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which exist naturally in three dimensions. The needs of geometry, and also for example
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continuous function. In other words, if a curve is defined by a continuous function
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1202:"Arc (geometry)" redirects here. For the use in finite projective geometry, see
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that are both connected). The bounded region inside a Jordan curve is known as
1007:
486:, in the general description of the real points into 'ovals'. The statement of
396:
257:
218:
134:
4745:
4739:
2477:{\displaystyle \operatorname {Length} \!\left(\gamma |_{}\right)=t_{2}-t_{1}.}
2231:
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completely fills a square, and therefore does not give any information on how
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4722:
Gallery of Space Curves Made from
Circles, includes animations by Peter Moses
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4264:
4252:
1414:
is an injective and continuously differentiable function, then the length of
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1226:
1034:
321:
149:
3914:. Algebraic geometry normally considers not only points with coordinates in
316:
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at the infinitesimal scale continuously over the full length of the curve.
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1241:
1046:
987:
935:—these are the examples first encountered—or in some cases the
253:
242:
238:
4712:, School of Mathematics and Statistics, University of St Andrews, Scotland
3062:
This is a basic notion. There are less and more restricted ideas, too. If
467:
questions, introduced properties of curves in new ways (in this case, the
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This definition of a curve has been formalized in modern mathematics as:
50:
5014:
4917:
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4398:
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3021:
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are to have a notion of curve in space of any number of dimensions. In
1295:
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385:
307:
Determinate (lines that do not extend indefinitely, such as the circle)
172:
423:
A fundamental advance in the theory of curves was the introduction of
411:
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4858:
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4081:
polynomials are sufficient to define a curve in a space of dimension
3438:
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448:
333:
261:
72:
Intuitively, a curve may be thought of as the trace left by a moving
66:
4588:
4567:
419:, to be defined using equations instead of geometrical construction.
133:(see below). Level curves and algebraic curves are sometimes called
105:. In some contexts, the function that defines the curve is called a
4991:
4907:
3252:
1271:
494:
472:
226:
217:, an algebraic curve is a finite union of topological curves. When
168:
76:. This is the definition that appeared more than 2000 years ago in
42:
501:. Nevertheless, many questions remain specific to curves, such as
797:. A closed curve is thus the image of a continuous mapping of a
468:
1760:{\displaystyle s=\int _{a}^{b}{\sqrt {1+^{2}}}~\mathrm {d} {x},}
292:, a line is defined as a "breadthless length" (Def. 2), while a
4776:
4679:
1267:
1256:
798:
4061:
one. They may be obtained as the common solutions of at least
3959:
coordinates. In this case, a point with real coordinates is a
3255:(i.e. infinitely differentiable and charts are expressible as
4784:
2855:
1014:(that is the curve divides the plane in two non-intersecting
400:
27:
Mathematical idealization of the trace left by a moving point
4263:, which are nonsingular curves of genus one, are studied in
4715:
37:
4397:
This term my be ambiguous, as a non-closed curve may be a
3899:
is a polynomial in two variables defined over some field
2282:(or unit-speed or parametrized by arc length) if for any
117:
to distinguish them from more constrained curves such as
4099:, which however may introduce new singularities such as
814:
of a topological curve is a closed and bounded interval
191:
one. If the coefficients of the polynomials belong to a
4718:
A collection of 874 two-dimensional mathematical curves
4110:
A plane curve may also be completed to a curve in the
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113:. In this article, these curves are sometimes called
4251:, the simplest examples of algebraic curves are the
877:
if it is the image of an interval or a circle by an
407:
as sections of cones had been studied by
Apollonius.
45:, one of the simplest curves, after (straight) lines
4539:
Depth, Crossings and
Conflicts in Discrete Geometry
1148:More precisely, a differentiable curve is a subset
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2188:{\displaystyle t_{0}<t_{1}<\ldots <t_{n}}
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4572:Transactions of the American Mathematical Society
4483:
4255:, which are nonsingular curves of degree two and
2393:
2003:
1915:
1770:which can be thought of intuitively as using the
459:. Solutions to variational problems, such as the
5056:
4643:
3846:curves under the relation of reparametrization.
3646:{\displaystyle \gamma _{2}(t)=\gamma _{1}(p(t))}
3423:{\displaystyle \gamma _{2}\colon J\rightarrow X}
3380:{\displaystyle \gamma _{1}\colon I\rightarrow X}
2651:
1912:
1033:), and a simple curve may have a positive area.
998:. It is also defined as a non-self-intersecting
455:. Newton also worked on an early example in the
4775:
4095:), an algebraic curve may be projected onto a
3860:Algebraic curves are the curves considered in
1045:) and even a positive area. An example is the
415:Analytic geometry allowed curves, such as the
4761:
4619:Davis, Ellery W.; Brenke, William C. (1913).
1010:in a plane of a Jordan curve consists of two
994:A plane simple closed curve is also called a
4661:
4421:exempt de toute latitude." Pages 7 and 8 of
4219:is not zero. An example is the Fermat curve
3184:is such a curve which is only assumed to be
3051:{\displaystyle \gamma \colon I\rightarrow X}
1553:The length of a curve is independent of the
1407:{\displaystyle \gamma :\to \mathbb {R} ^{n}}
1092:{\displaystyle \gamma \colon I\rightarrow X}
553:{\displaystyle \gamma \colon I\rightarrow X}
4877:
4731:The Encyclopedia of Mathematics article on
4618:
4091:. By eliminating variables (by any tool of
3546:{\displaystyle p^{-1}\colon I\rightarrow J}
1821:, then we can define the length of a curve
1049:, which has many other unusual properties.
801:. A non-closed curve may also be called an
373:as a method to both double the cube and to
324:) were among the curves studied in ancient
4768:
4754:
4434:
4432:
3920:but all the points with coordinates in an
2854:), there are obvious examples such as the
1603:of a continuously differentiable function
276:was used in place of the more modern term
61:in older texts) is an object similar to a
4587:
4426:, by Pierre Mardele, Lyon, MDCXLV (1645).
2122:
2018:
1505:
1394:
1321:
1123:
902:continuous non-self-intersecting curve).
4535:
4114:: if a curve is defined by a polynomial
3963:, and the set of all real points is the
3951:In the case of a curve defined over the
3231:times continuously differentiable). If
2835:
981:
410:
344:The conic sections, studied in depth by
315:
301:Composite lines (lines forming an angle)
252:
36:
4542:. Logos Verlag Berlin GmbH. p. 7.
4495:MacTutor History of Mathematics Archive
4429:
3944:, the curve is said to be defined over
1255:A common curved example is an arc of a
1052:
14:
5057:
4562:
3495:{\displaystyle p\colon J\rightarrow I}
320:The curves created by slicing a cone (
137:, since they are generally defined by
4749:
4267:, and have important applications to
3955:, one normally considers points with
3306:A differentiable curve is said to be
2108:where the supremum is taken over all
1361:-dimensional Euclidean space, and if
1252:, depending on how they are bounded.
1197:
790:{\displaystyle \gamma (a)=\gamma (b)}
229:point of view, is not a curve, but a
156:, and the curve is then said to be a
4362:Infinite-dimensional vector function
2230:A rectifiable curve is a curve with
1289:
516:
388:as a method to trisect an angle and
340:construction. These curves include:
129:of curves and isolated points), and
3936:is a curve defined by a polynomial
2901:, then we can define the notion of
264:showing an early interest in curves
24:
4625:. MacMillan Company. p. 108.
3849:
2969:by means of this notion of curve.
2810:
1745:
1528:
1334:{\displaystyle X=\mathbb {R} ^{n}}
1302:Differentiable curve § Length
25:
5091:
4703:
4568:"A Jordan Curve of Positive Area"
3864:. A plane algebraic curve is the
3109:manifold (i.e., a manifold whose
2129:{\displaystyle n\in \mathbb {N} }
1139:{\displaystyle \mathbb {R} ^{n}.}
965:is at least three-dimensional; a
4827:
269:a stick on the sand on a beach.
221:zeros are considered, one has a
4612:
4556:
4529:
4505:
4307:Differential geometry of curves
2842:Differential geometry of curves
2373:{\displaystyle t_{1}\leq t_{2}}
2333:{\displaystyle t_{1},t_{2}\in }
1110:into a differentiable manifold
447:Conic sections were applied in
65:, but that does not have to be
4477:
4468:
4459:
4450:
4441:
4414:
4391:
4382:
3640:
3637:
3631:
3625:
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3603:
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3414:
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3042:
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2797:
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2722:
2708:
2702:
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2619:
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2515:
2512:
2500:
2435:
2409:
2404:
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2315:
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2214:
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1995:
1992:
1973:
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1945:
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728:
654:For example, the image of the
544:
13:
1:
4580:American Mathematical Society
4408:
3987:are said to be rational over
3905:. One says that the curve is
3868:of the points of coordinates
2524:{\displaystyle \gamma :\to X}
2271:{\displaystyle \gamma :\to X}
1858:{\displaystyle \gamma :\to X}
1638:defined on a closed interval
4087:, the curve is said to be a
4034:every rational point of the
3981:with coordinates in a field
7:
4738:The Manifold Atlas page on
4682:, commentary and trans. by
4669:Encyclopedia of Mathematics
4651:Encyclopedia of Mathematics
4274:
4233:, which has an affine form
3727:{\displaystyle \gamma _{1}}
3696:{\displaystyle \gamma _{2}}
3135:continuously differentiable
1434:is defined as the quantity
609:However, in some contexts,
511:Hilbert's sixteenth problem
10:
5096:
4644:A.S. Parkhomenko (2001) ,
4401:, as is a line in a plane.
3922:algebraically closed field
3853:
2839:
1579:In particular, the length
1299:
1293:
1201:
1056:
436:that can be defined using
248:
241:are widely used in modern
205:. In the common case of a
198:, the curve is said to be
29:
4942:
4836:
4825:
4791:
3765:differentiable curves in
3279:is an analytic map, then
1204:Arc (projective geometry)
578:. Properly speaking, the
4689:Vol. 1 (1908 Cambridge)
4536:SulovskĂ˝, Marek (2012).
4517:Dictionary.reference.com
4500:University of St Andrews
4375:
4068:polynomial equations in
852:, the curve is called a
645:{\displaystyle \gamma .}
602:{\displaystyle \gamma .}
358:and used as a method to
338:compass and straightedge
233:, and is often called a
3292:{\displaystyle \gamma }
3272:{\displaystyle \gamma }
2899:differentiable manifold
2552:{\displaystyle \gamma }
1569:{\displaystyle \gamma }
1427:{\displaystyle \gamma }
1183:{\displaystyle C\cap U}
894:{\displaystyle \gamma }
698:{\displaystyle \gamma }
675:{\displaystyle \gamma }
622:{\displaystyle \gamma }
272:Historically, the term
223:complex algebraic curve
4662:B.I. Golubov (2001) ,
4490:"Spiral of Archimedes"
4357:Vector-valued function
4154:homogeneous polynomial
4053:. They are defined as
4014:, one simply talks of
3975:The points of a curve
3840:
3806:
3779:
3759:
3728:
3697:
3670:
3647:
3577:
3547:
3496:
3461:
3424:
3381:
3345:differentiable curves
3339:
3293:
3273:
3245:
3225:
3205:
3178:
3158:
3127:
3103:
3076:
3052:
3014:
2986:
2963:
2939:
2919:
2891:
2826:
2733:
2591:
2553:
2525:
2478:
2374:
2334:
2272:
2221:
2189:
2130:
2099:
1941:
1859:
1815:
1791:
1761:
1664:
1632:
1631:{\displaystyle y=f(x)}
1593:
1570:
1544:
1428:
1408:
1355:
1335:
1206:. For other uses, see
1184:
1140:
1093:
991:
959:
925:
895:
846:
791:
747:
699:
676:
658:or, more generally, a
646:
623:
603:
554:
525:can be specified by a
457:calculus of variations
420:
329:
265:
46:
32:Curve (disambiguation)
4322:List of curves topics
4097:plane algebraic curve
4089:complete intersection
4042:has a zero coordinate
4020:Fermat's Last Theorem
3940:with coefficients in
3841:
3839:{\displaystyle C^{k}}
3807:
3805:{\displaystyle C^{k}}
3780:
3760:
3758:{\displaystyle C^{k}}
3729:
3698:
3671:
3648:
3578:
3576:{\displaystyle C^{k}}
3548:
3497:
3462:
3460:{\displaystyle C^{k}}
3425:
3382:
3340:
3338:{\displaystyle C^{k}}
3294:
3274:
3246:
3226:
3206:
3204:{\displaystyle C^{k}}
3179:
3159:
3157:{\displaystyle C^{k}}
3128:
3104:
3102:{\displaystyle C^{k}}
3077:
3053:
3015:
2987:
2964:
2940:
2920:
2892:
2852:two-dimensional space
2836:Differential geometry
2827:
2734:
2592:
2590:{\displaystyle t\in }
2554:
2526:
2479:
2375:
2335:
2273:
2222:
2190:
2131:
2100:
1921:
1860:
1816:
1792:
1762:
1665:
1633:
1594:
1571:
1545:
1429:
1409:
1356:
1336:
1300:Further information:
1185:
1156:where every point of
1141:
1094:
1041:bigger than one (see
985:
973:real algebraic curves
960:
945:is a curve for which
926:
911:is a curve for which
896:
847:
792:
748:
700:
677:
647:
624:
604:
555:
477:differential calculus
442:transcendental curves
414:
367:conchoid of Nicomedes
319:
256:
179:. More generally, an
165:plane algebraic curve
119:differentiable curves
109:, and the curve is a
40:
4486:Robertson, Edmund F.
4022:may be restated as:
4010:is the field of the
3823:
3789:
3769:
3742:
3736:equivalence relation
3734:; and this makes an
3711:
3680:
3660:
3590:
3560:
3515:
3474:
3444:
3395:
3352:
3322:
3283:
3263:
3235:
3215:
3188:
3168:
3141:
3117:
3086:
3066:
3030:
3004:
2976:
2953:
2929:
2909:
2903:differentiable curve
2881:
2749:
2604:
2563:
2543:
2533:Lipschitz-continuous
2491:
2387:
2344:
2286:
2238:
2199:
2140:
2112:
1872:
1825:
1805:
1781:
1677:
1642:
1607:
1583:
1560:
1441:
1418:
1365:
1345:
1310:
1208:Arc (disambiguation)
1168:
1118:
1071:
1065:differentiable curve
1059:Differentiable curve
1053:Differentiable curve
1012:connected components
1004:Jordan curve theorem
990:with a positive area
949:
915:
885:
818:
757:
719:
689:
666:
633:
613:
590:
532:
507:Jordan curve theorem
503:space-filling curves
438:polynomial equations
207:real algebraic curve
158:differentiable curve
146:space-filling curves
30:For other uses, see
4716:Mathematical curves
4710:Famous Curves Index
4664:"Rectifiable curve"
4484:O'Connor, John J.;
4055:algebraic varieties
3993:and can be denoted
2860:classical mechanics
2784:
2742:and then show that
2136:and all partitions
1777:More generally, if
1772:Pythagorean theorem
1700:
1495:
1160:has a neighborhood
1063:Roughly speaking a
1039:Hausdorff dimension
1031:space-filling curve
660:space-filling curve
527:continuous function
499:algebraic varieties
417:Folium of Descartes
346:Apollonius of Perga
102:continuous function
4564:Osgood, William F.
4337:Parametric surface
4317:Index of the curve
4093:elimination theory
3862:algebraic geometry
3836:
3802:
3775:
3755:
3738:on the set of all
3724:
3693:
3666:
3643:
3573:
3543:
3492:
3457:
3420:
3377:
3335:
3289:
3269:
3241:
3221:
3201:
3174:
3154:
3123:
3099:
3072:
3048:
3010:
2982:
2959:
2935:
2915:
2887:
2864:general relativity
2822:
2770:
2729:
2665:
2587:
2549:
2521:
2474:
2370:
2330:
2268:
2217:
2185:
2126:
2095:
1855:
1811:
1787:
1757:
1686:
1660:
1628:
1589:
1566:
1540:
1481:
1424:
1404:
1351:
1331:
1215:Euclidean geometry
1198:Differentiable arc
1180:
1136:
1089:
1002:in the plane. The
992:
955:
921:
891:
845:{\displaystyle I=}
842:
787:
746:{\displaystyle I=}
743:
695:
672:
642:
619:
599:
550:
421:
382:Archimedean spiral
352:cissoid of Diocles
330:
304:Incomposite lines
280:. Hence the terms
266:
225:, which, from the
139:implicit equations
115:topological curves
47:
5049:
5048:
4938:
4937:
4332:Osculating circle
4312:Gallery of curves
4297:Curve orientation
3817:equivalence class
3778:{\displaystyle X}
3705:reparametrization
3669:{\displaystyle t}
3299:is said to be an
3253:analytic manifold
3244:{\displaystyle X}
3224:{\displaystyle k}
3177:{\displaystyle X}
3126:{\displaystyle k}
3075:{\displaystyle X}
3013:{\displaystyle X}
2985:{\displaystyle X}
2962:{\displaystyle X}
2938:{\displaystyle X}
2918:{\displaystyle X}
2890:{\displaystyle X}
2808:
2727:
2650:
2649:
2644:
2642:
2630:
2537:metric derivative
2032:
2028:
2024:
2010:
2000:
1911:
1906:
1904:
1892:
1814:{\displaystyle d}
1790:{\displaystyle X}
1743:
1739:
1592:{\displaystyle s}
1526:
1480:
1475:
1473:
1461:
1354:{\displaystyle n}
1290:Length of a curve
958:{\displaystyle X}
924:{\displaystyle X}
573:topological space
523:topological curve
517:Topological curve
425:analytic geometry
390:square the circle
326:Greek mathematics
290:Euclid's Elements
185:algebraic variety
98:topological space
16:(Redirected from
5087:
5080:General topology
4875:
4874:
4854:Boerdijk–Coxeter
4831:
4830:
4770:
4763:
4756:
4747:
4746:
4699:(1961 Cambridge)
4697:A Book of Curves
4676:
4658:
4637:
4636:
4616:
4610:
4609:
4591:
4566:(January 1903).
4560:
4554:
4553:
4533:
4527:
4526:
4524:
4523:
4509:
4503:
4502:
4481:
4475:
4472:
4466:
4463:
4457:
4454:
4448:
4445:
4439:
4436:
4427:
4418:
4402:
4395:
4389:
4386:
4282:Coordinate curve
4243:
4232:
4218:
4212:
4193:
4180:. The values of
4179:
4173:
4152:simplifies to a
4151:
4125:
4120:of total degree
4119:
4112:projective plane
4086:
4080:
4073:
4067:
4052:
4041:
4031:
4012:rational numbers
4009:
4003:
3992:
3986:
3980:
3970:Riemann surfaces
3928:
3919:
3913:
3904:
3898:
3892:
3877:
3845:
3843:
3842:
3837:
3835:
3834:
3811:
3809:
3808:
3803:
3801:
3800:
3784:
3782:
3781:
3776:
3764:
3762:
3761:
3756:
3754:
3753:
3733:
3731:
3730:
3725:
3723:
3722:
3702:
3700:
3699:
3694:
3692:
3691:
3675:
3673:
3672:
3667:
3652:
3650:
3649:
3644:
3624:
3623:
3602:
3601:
3582:
3580:
3579:
3574:
3572:
3571:
3552:
3550:
3549:
3544:
3530:
3529:
3501:
3499:
3498:
3493:
3466:
3464:
3463:
3458:
3456:
3455:
3429:
3427:
3426:
3421:
3407:
3406:
3386:
3384:
3383:
3378:
3364:
3363:
3344:
3342:
3341:
3336:
3334:
3333:
3312:
3311:
3298:
3296:
3295:
3290:
3278:
3276:
3275:
3270:
3250:
3248:
3247:
3242:
3230:
3228:
3227:
3222:
3210:
3208:
3207:
3202:
3200:
3199:
3183:
3181:
3180:
3175:
3163:
3161:
3160:
3155:
3153:
3152:
3132:
3130:
3129:
3124:
3108:
3106:
3105:
3100:
3098:
3097:
3081:
3079:
3078:
3073:
3057:
3055:
3054:
3049:
3019:
3017:
3016:
3011:
2991:
2989:
2988:
2983:
2968:
2966:
2965:
2960:
2944:
2942:
2941:
2936:
2924:
2922:
2921:
2916:
2896:
2894:
2893:
2888:
2831:
2829:
2828:
2823:
2818:
2813:
2806:
2796:
2795:
2794:
2783:
2778:
2738:
2736:
2735:
2730:
2728:
2726:
2725:
2711:
2705:
2667:
2664:
2647:
2646:
2645:
2643:
2640:
2638:
2633:
2628:
2618:
2617:
2616:
2596:
2594:
2593:
2588:
2558:
2556:
2555:
2550:
2530:
2528:
2527:
2522:
2483:
2481:
2480:
2475:
2470:
2469:
2457:
2456:
2444:
2440:
2439:
2438:
2434:
2433:
2421:
2420:
2407:
2379:
2377:
2376:
2371:
2369:
2368:
2356:
2355:
2339:
2337:
2336:
2331:
2311:
2310:
2298:
2297:
2277:
2275:
2274:
2269:
2234:length. A curve
2226:
2224:
2223:
2220:{\displaystyle }
2218:
2194:
2192:
2191:
2186:
2184:
2183:
2165:
2164:
2152:
2151:
2135:
2133:
2132:
2127:
2125:
2104:
2102:
2101:
2096:
2091:
2087:
2080:
2079:
2061:
2060:
2048:
2047:
2030:
2029:
2026:
2022:
2021:
2008:
2007:
2006:
1998:
1991:
1990:
1963:
1962:
1940:
1935:
1909:
1908:
1907:
1905:
1902:
1900:
1895:
1890:
1864:
1862:
1861:
1856:
1820:
1818:
1817:
1812:
1796:
1794:
1793:
1788:
1766:
1764:
1763:
1758:
1753:
1748:
1741:
1740:
1738:
1737:
1719:
1702:
1699:
1694:
1669:
1667:
1666:
1663:{\displaystyle }
1661:
1637:
1635:
1634:
1629:
1598:
1596:
1595:
1590:
1575:
1573:
1572:
1567:
1549:
1547:
1546:
1541:
1536:
1531:
1524:
1523:
1509:
1500:
1494:
1489:
1478:
1477:
1476:
1474:
1471:
1469:
1464:
1459:
1433:
1431:
1430:
1425:
1413:
1411:
1410:
1405:
1403:
1402:
1397:
1360:
1358:
1357:
1352:
1340:
1338:
1337:
1332:
1330:
1329:
1324:
1189:
1187:
1186:
1181:
1163:
1159:
1155:
1151:
1145:
1143:
1142:
1137:
1132:
1131:
1126:
1113:
1105:
1098:
1096:
1095:
1090:
1006:states that the
964:
962:
961:
956:
937:projective plane
930:
928:
927:
922:
900:
898:
897:
892:
868:
867:
858:, also known as
851:
849:
848:
843:
796:
794:
793:
788:
752:
750:
749:
744:
704:
702:
701:
696:
681:
679:
678:
673:
651:
649:
648:
643:
628:
626:
625:
620:
608:
606:
605:
600:
577:
566:
559:
557:
556:
551:
488:BĂ©zout's theorem
434:algebraic curves
375:trisect an angle
213:is the field of
212:
204:
197:
131:algebraic curves
111:parametric curve
21:
5095:
5094:
5090:
5089:
5088:
5086:
5085:
5084:
5070:Metric geometry
5055:
5054:
5052:
5050:
5045:
4934:
4888:
4873:
4832:
4828:
4823:
4787:
4774:
4706:
4695:E. H. Lockwood
4640:
4633:
4617:
4613:
4589:10.2307/1986455
4561:
4557:
4550:
4534:
4530:
4521:
4519:
4511:
4510:
4506:
4482:
4478:
4474:Lockwood p. 129
4473:
4469:
4465:Lockwood p. 132
4464:
4460:
4455:
4451:
4446:
4442:
4437:
4430:
4419:
4415:
4411:
4406:
4405:
4396:
4392:
4387:
4383:
4378:
4373:
4352:Position vector
4347:Polygonal curve
4342:Path (topology)
4302:Curve sketching
4277:
4261:Elliptic curves
4234:
4220:
4214:
4195:
4181:
4175:
4156:
4127:
4121:
4115:
4082:
4075:
4069:
4062:
4048:
4039:
4026:
4018:. For example,
4016:rational points
4005:
3994:
3988:
3982:
3976:
3924:
3915:
3909:
3900:
3894:
3879:
3869:
3858:
3856:Algebraic curve
3852:
3850:Algebraic curve
3830:
3826:
3824:
3821:
3820:
3796:
3792:
3790:
3787:
3786:
3770:
3767:
3766:
3749:
3745:
3743:
3740:
3739:
3718:
3714:
3712:
3709:
3708:
3687:
3683:
3681:
3678:
3677:
3661:
3658:
3657:
3619:
3615:
3597:
3593:
3591:
3588:
3587:
3567:
3563:
3561:
3558:
3557:
3522:
3518:
3516:
3513:
3512:
3475:
3472:
3471:
3451:
3447:
3445:
3442:
3441:
3433:are said to be
3402:
3398:
3396:
3393:
3392:
3359:
3355:
3353:
3350:
3349:
3329:
3325:
3323:
3320:
3319:
3309:
3308:
3284:
3281:
3280:
3264:
3261:
3260:
3236:
3233:
3232:
3216:
3213:
3212:
3195:
3191:
3189:
3186:
3185:
3169:
3166:
3165:
3148:
3144:
3142:
3139:
3138:
3118:
3115:
3114:
3093:
3089:
3087:
3084:
3083:
3067:
3064:
3063:
3031:
3028:
3027:
3005:
3002:
3001:
2994:smooth manifold
2977:
2974:
2973:
2954:
2951:
2950:
2947:tangent vectors
2930:
2927:
2926:
2910:
2907:
2906:
2882:
2879:
2878:
2844:
2838:
2814:
2809:
2790:
2786:
2785:
2779:
2774:
2750:
2747:
2746:
2721:
2707:
2706:
2668:
2666:
2654:
2639:
2634:
2632:
2631:
2612:
2608:
2607:
2605:
2602:
2601:
2564:
2561:
2560:
2544:
2541:
2540:
2492:
2489:
2488:
2465:
2461:
2452:
2448:
2429:
2425:
2416:
2412:
2408:
2403:
2402:
2398:
2394:
2388:
2385:
2384:
2364:
2360:
2351:
2347:
2345:
2342:
2341:
2306:
2302:
2293:
2289:
2287:
2284:
2283:
2239:
2236:
2235:
2200:
2197:
2196:
2179:
2175:
2160:
2156:
2147:
2143:
2141:
2138:
2137:
2121:
2113:
2110:
2109:
2075:
2071:
2056:
2052:
2043:
2039:
2025:
2017:
2002:
2001:
1980:
1976:
1958:
1954:
1936:
1925:
1920:
1916:
1901:
1896:
1894:
1893:
1873:
1870:
1869:
1826:
1823:
1822:
1806:
1803:
1802:
1782:
1779:
1778:
1749:
1744:
1733:
1729:
1712:
1701:
1695:
1690:
1678:
1675:
1674:
1643:
1640:
1639:
1608:
1605:
1604:
1584:
1581:
1580:
1561:
1558:
1557:
1555:parametrization
1532:
1527:
1519:
1504:
1496:
1490:
1485:
1470:
1465:
1463:
1462:
1442:
1439:
1438:
1419:
1416:
1415:
1398:
1393:
1392:
1366:
1363:
1362:
1346:
1343:
1342:
1325:
1320:
1319:
1311:
1308:
1307:
1304:
1298:
1292:
1274:), an arc of a
1211:
1200:
1169:
1166:
1165:
1161:
1157:
1153:
1149:
1127:
1122:
1121:
1119:
1116:
1115:
1111:
1103:
1072:
1069:
1068:
1061:
1055:
1000:continuous loop
950:
947:
946:
933:Euclidean plane
916:
913:
912:
886:
883:
882:
865:
864:
860:topological arc
819:
816:
815:
758:
755:
754:
720:
717:
716:
690:
687:
686:
667:
664:
663:
634:
631:
630:
614:
611:
610:
591:
588:
587:
575:
564:
533:
530:
529:
519:
461:brachistochrone
397:spiric sections
360:double the cube
251:
235:Riemann surface
210:
202:
195:
181:algebraic curve
135:implicit curves
107:parametrization
88:A curve is the
57:(also called a
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4704:External links
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4646:"Line (curve)"
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4327:List of curves
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1043:Koch snowflake
1035:Fractal curves
1029:in the plane (
1008:set complement
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258:Megalithic art
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177:indeterminates
154:differentiable
150:fractal curves
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1020:Jordan domain
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384:, studied by
383:
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369:, studied by
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354:, studied by
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4878:Biochemistry
4780:
4696:
4691:Google Books
4686:
4667:
4649:
4622:The Calculus
4621:
4614:
4575:
4571:
4558:
4538:
4531:
4520:. Retrieved
4507:
4493:
4479:
4470:
4461:
4456:Heath p. 160
4452:
4447:Heath p. 153
4443:
4422:
4416:
4393:
4384:
4287:Crinkled arc
4269:cryptography
4246:
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4036:Fermat curve
4033:
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4023:
4015:
4006:
3999:
3995:
3989:
3983:
3977:
3974:
3964:
3960:
3953:real numbers
3950:
3945:
3941:
3937:
3933:
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3925:
3916:
3910:
3907:defined over
3906:
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3870:
3859:
3812:
3704:
3703:is called a
3655:
3555:
3504:
3434:
3432:
3307:
3305:
3300:
3257:power series
3061:
2998:smooth curve
2997:
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2902:
2876:
2851:
2848:curved lines
2847:
2845:
2741:
2486:
2229:
2107:
1801:with metric
1799:metric space
1776:
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1578:
1552:
1305:
1283:
1276:great circle
1265:
1261:circular arc
1254:
1235:
1229:subset of a
1222:
1218:
1212:
1147:
1108:real numbers
1062:
1047:dragon curve
1024:
1019:
996:Jordan curve
995:
993:
988:dragon curve
977:disconnected
966:
940:
906:
904:
874:
872:
863:
859:
853:
809:
803:
802:
710:
706:
684:
682:is defined.
653:
579:
569:real numbers
522:
520:
492:
484:cubic curves
481:
446:
422:
331:
271:
267:
243:cryptography
239:finite field
222:
215:real numbers
200:defined over
199:
162:
143:
123:level curves
114:
106:
87:
85:
79:
71:
58:
54:
48:
18:Jordan curve
5026:Pitch angle
5002:Logarithmic
4950:Archimedean
4913:Polyproline
4740:1-manifolds
4684:T. L. Heath
4582:: 107–112.
4424:traductions
4247:Except for
3676:. The map
3507:inverse map
1259:, called a
1240:are called
942:space curve
908:plane curve
873:A curve is
656:Peano curve
465:tautochrone
403:studied by
227:topological
125:(which are
59:curved line
51:mathematics
5059:Categories
5015:On Spirals
4965:Hyperbolic
4522:2012-03-14
4409:References
4399:closed set
4194:such that
4174:of degree
4038:of degree
3961:real point
3878:such that
3435:equivalent
3316:derivative
3137:), then a
3022:smooth map
2868:world line
2380:, we have
2340:such that
2278:is called
1296:Arc length
1164:such that
968:skew curve
804:open curve
386:Archimedes
332:The Greek
286:right line
173:polynomial
5036:Spirangle
5031:Theodorus
4970:Poinsot's
4960:Epispiral
4804:Curvature
4799:Algebraic
4674:EMS Press
4656:EMS Press
4598:0002-9947
4059:dimension
3965:real part
3716:γ
3685:γ
3617:γ
3595:γ
3538:→
3532::
3524:−
3487:→
3481::
3439:bijective
3415:→
3409::
3400:γ
3372:→
3366::
3357:γ
3287:γ
3267:γ
3164:curve in
3043:→
3037::
3034:γ
2872:spacetime
2792:γ
2772:∫
2762:γ
2756:
2716:−
2691:γ
2676:γ
2659:→
2614:γ
2570:∈
2547:γ
2516:→
2495:γ
2459:−
2400:γ
2358:≤
2313:∈
2263:→
2242:γ
2170:…
2119:∈
2066:…
2015:∈
1985:−
1971:γ
1949:γ
1923:∑
1885:γ
1879:
1850:→
1829:γ
1688:∫
1564:γ
1502:γ
1483:∫
1454:γ
1448:
1422:γ
1390:→
1369:γ
1284:great arc
1227:connected
1221:(symbol:
1175:∩
1084:→
1078::
1075:γ
889:γ
879:injective
862:(or just
776:γ
761:γ
693:γ
670:γ
637:γ
617:γ
594:γ
545:→
539::
536:γ
495:manifolds
449:astronomy
371:Nicomedes
334:geometers
262:Newgrange
189:dimension
78:Euclid's
5075:Topology
4992:Involute
4987:Fermat's
4928:Collagen
4864:Symmetry
4687:Elements
4275:See also
3893:, where
3656:for all
3556:is also
1717:′
1507:′
1272:spheroid
1242:segments
1236:Arcs of
1233:curve.
1114:, often
1101:interval
1099:from an
709:or is a
685:A curve
562:interval
560:from an
473:catenary
294:straight
209:, where
169:zero set
94:interval
80:Elements
67:straight
43:parabola
5021:Padovan
4955:Cotes's
4943:Spirals
4849:Antenna
4837:Helices
4809:Gallery
4785:helices
4777:Spirals
4606:1986455
4126:, then
4004:. When
3957:complex
3314:if its
3310:regular
3259:), and
2280:natural
1599:of the
1341:is the
1225:) is a
1106:of the
1016:regions
931:is the
810:If the
582:is the
571:into a
567:of the
471:). The
469:cycloid
405:Perseus
356:Diocles
249:History
231:surface
219:complex
175:in two
167:is the
5065:Curves
5007:Golden
4923:Triple
4903:Double
4869:Triple
4819:Topics
4792:Curves
4781:curves
4680:Euclid
4629:
4604:
4596:
4546:
4259:zero.
4253:conics
4030:> 2
3815:is an
3583:, and
3251:is an
3211:(i.e.
3133:times
3111:charts
2807:
2753:Length
2648:
2629:
2391:Length
2232:finite
2031:
2023:
2009:
1999:
1910:
1891:
1876:Length
1742:
1525:
1479:
1460:
1445:Length
1278:(or a
1270:(or a
1268:sphere
1257:circle
1027:square
875:simple
812:domain
799:circle
707:closed
453:Kepler
440:, and
127:unions
92:of an
4982:Euler
4977:Doyle
4918:Super
4893:Alpha
4844:Angle
4733:lines
4602:JSTOR
4578:(1).
4376:Notes
4257:genus
4249:lines
4211:) = 0
4101:cusps
3891:) = 0
3785:. A
3082:is a
3020:is a
2992:is a
2897:is a
2856:helix
2788:Speed
2610:Speed
2539:) of
2531:is a
1797:is a
1601:graph
1266:In a
1250:lines
1248:, or
1238:lines
1217:, an
584:image
580:curve
278:curve
260:from
193:field
171:of a
100:by a
96:to a
90:image
74:point
55:curve
5041:Ulam
4997:List
4898:Beta
4859:Hemi
4814:List
4783:and
4627:ISBN
4594:ISSN
4544:ISBN
3467:map
3113:are
2996:, a
2866:, a
2173:<
2167:<
2154:<
2069:<
2063:<
2050:<
1246:rays
939:. A
855:path
753:and
712:loop
509:and
497:and
463:and
401:tori
395:The
380:The
365:The
350:The
284:and
274:line
148:and
63:line
53:, a
4584:doi
4242:= 1
4103:or
4057:of
4024:For
3948:.
3932:If
3929:.
3866:set
3819:of
3813:arc
3707:of
3387:and
3000:in
2972:If
2949:to
2905:in
2877:If
2850:in
2641:def
2597:as
2559:at
2487:If
2195:of
2027:and
1913:sup
1903:def
1865:by
1670:is
1472:def
1306:If
1263:.
1219:arc
1213:In
1190:is
1152:of
1022:.
979:).
870:).
866:arc
715:if
705:is
586:of
451:by
427:by
187:of
49:In
5061::
4908:Pi
4887:10
4779:,
4672:,
4666:,
4654:,
4648:,
4600:.
4592:.
4574:.
4570:.
4515:.
4498:,
4492:,
4488:,
4431:^
4271:.
4238:+
4228:=
4224:+
4207:,
4203:,
4189:,
4185:,
4168:,
4164:,
4142:,
4107:.
4079:–1
4066:–1
4044:.
4032:,
3972:.
3887:,
3873:,
3303:.
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1286:.
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986:A
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807:.
521:A
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505:,
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163:A
160:.
141:.
69:.
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4148:w
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4144:v
4140:w
4138:/
4136:u
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4132:f
4129:w
4123:d
4117:f
4084:n
4077:n
4071:n
4064:n
4050:n
4040:n
4028:n
4007:G
4002:)
4000:G
3998:(
3996:C
3990:G
3984:G
3978:C
3946:F
3942:F
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3934:C
3926:K
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3896:f
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3883:(
3881:f
3875:y
3871:x
3832:k
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3641:)
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3635:t
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3629:p
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3607:t
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2768:=
2765:)
2759:(
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2697:t
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2688:,
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2636:=
2626:)
2623:t
2620:(
2585:]
2582:b
2579:,
2576:a
2573:[
2567:t
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2513:]
2510:b
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2504:a
2501:[
2498::
2472:.
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2442:)
2436:]
2431:2
2427:t
2423:,
2418:1
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2410:[
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2396:(
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2328:]
2325:b
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2308:2
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2260:]
2257:b
2254:,
2251:a
2248:[
2245::
2215:]
2212:b
2209:,
2206:a
2203:[
2181:n
2177:t
2162:1
2158:t
2149:0
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2123:N
2116:n
2093:,
2089:}
2085:b
2082:=
2077:n
2073:t
2058:1
2054:t
2045:0
2041:t
2037:=
2034:a
2019:N
2012:n
2004:|
1996:)
1993:)
1988:1
1982:i
1978:t
1974:(
1968:,
1965:)
1960:i
1956:t
1952:(
1946:(
1943:d
1938:n
1933:1
1930:=
1927:i
1918:{
1898:=
1888:)
1882:(
1853:X
1847:]
1844:b
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1838:a
1835:[
1832::
1809:d
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1755:,
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1731:]
1727:)
1724:x
1721:(
1714:f
1710:[
1707:+
1704:1
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1684:=
1681:s
1658:]
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1614:=
1611:y
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1467:=
1457:)
1451:(
1400:n
1395:R
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1384:b
1381:,
1378:a
1375:[
1372::
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1317:=
1314:X
1223:⌒
1210:.
1178:U
1172:C
1162:U
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1129:n
1124:R
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840:]
837:b
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828:[
825:=
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785:)
782:b
779:(
773:=
770:)
767:a
764:(
741:]
738:b
735:,
732:a
729:[
726:=
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640:.
597:.
576:X
565:I
548:X
542:I
392:.
377:.
362:.
328:.
211:k
203:k
196:k
34:.
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