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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
3956:
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
1860:
4409:
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,
890:
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
2726:
285:
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:
1537:
72:: "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."
2819:
420:
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
2087:{\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\},}
2471:
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1754:
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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
226:. Although not being curves in the common sense, algebraic curves defined over other fields have been widely studied. In particular, algebraic curves over a
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433:
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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4609:
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itself is called a curve, especially when the image does not look like what is generally called a curve and does not characterize sufficiently
325:
had studied many other kinds of curves. One reason was their interest in solving geometrical problems that could not be solved using standard
479:
showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions.
2721:{\displaystyle {\operatorname {Speed} _{\gamma }}(t)~{\stackrel {\text{def}}{=}}~\limsup _{s\to t}{\frac {d(\gamma (s),\gamma (t))}{|s-t|}}}
1014:
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
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1665:
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gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means of
2914:. This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take
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2935:
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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
1000:
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499:
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1532:{\displaystyle \operatorname {Length} (\gamma )~{\stackrel {\text{def}}{=}}~\int _{a}^{b}|\gamma \,'(t)|~\mathrm {d} {t}.}
1183:
to an interval of the real numbers. In other words, a differentiable curve is a differentiable manifold of dimension one.
471:
In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the
2128:
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2814:{\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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20:
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is a space curve which lies in no plane. These definitions of plane, space and skew curves apply also to
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In current mathematical usage, a line is straight. Previously lines could be either curved or straight.
3957:
point of view a surface. In particular, the nonsingular complex projective algebraic curves are called
2100:
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While the first examples of curves that are met are mostly plane curves (that is, in everyday words,
2524:
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
172:
is the zero set of a finite set of polynomials, which satisfies the further condition of being an
110:. 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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4233:. 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)
242:
223:
169:
141:. For ensuring more regularity, the function that defines a curve is often supposed to be
119:
51:
4036:
Algebraic curves can also be space curves, or curves in a space of higher dimension, say
964:, although the above definition of a curve does not apply (a real algebraic curve may be
806:
707:
4916:
4802:
4716:
Gallery of Bishop Curves and Other
Spherical Curves, includes animations by Peter Moses
4502:"Jordan arc definition at Dictionary.com. Dictionary.com Unabridged. Random House, Inc"
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3945:
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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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1191:"Arc (geometry)" redirects here. For the use in finite projective geometry, see
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203:
111:
39:
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that are both connected). The bounded region inside a Jordan curve is known as
475:, in the general description of the real points into 'ovals'. The statement of
5003:
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4734:
4728:
4387:
3304:
3010:
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2466:{\displaystyle \operatorname {Length} \!\left(\gamma |_{}\right)=t_{2}-t_{1}.}
2220:
1284:
651:
completely fills a square, and therefore does not give any information on how
374:
161:
4711:
Gallery of Space Curves Made from
Circles, includes animations by Peter Moses
4586:
1403:
is an injective and continuously differentiable function, then the length of
5024:
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4847:
4792:
3903:. Algebraic geometry normally considers not only points with coordinates in
3427:
2860:
867:
437:
322:
250:
55:
1763:
at the infinitesimal scale continuously over the full length of the curve.
924:—these are the examples first encountered—or in some cases the
4701:, School of Mathematics and Statistics, University of St Andrews, Scotland
3051:
This is a basic notion. There are less and more restricted ideas, too. If
456:
questions, introduced properties of curves in new ways (in this case, the
4980:
4896:
3241:
1260:
483:
461:
215:
157:
75:
This definition of a curve has been formalized in modern mathematics as:
31:
4594:
2851:
are to have a notion of curve in space of any number of dimensions. In
457:
296:
Determinate (lines that do not extend indefinitely, such as the circle)
412:
A fundamental advance in the theory of curves was the introduction of
4765:
4668:
4070:
polynomials are sufficient to define a curve in a space of dimension
1256:
1245:
787:
61:
Intuitively, a curve may be thought of as the trace left by a moving
4577:
4556:
408:, to be defined using equations instead of geometrical construction.
122:(see below). Level curves and algebraic curves are sometimes called
94:. In some contexts, the function that defines the curve is called a
26:
206:, an algebraic curve is a finite union of topological curves. When
65:. This is the definition that appeared more than 2000 years ago in
4773:
2844:
970:
490:. Nevertheless, many questions remain specific to curves, such as
399:
389:
25:
786:. A closed curve is thus the image of a continuous mapping of a
1749:{\displaystyle s=\int _{a}^{b}{\sqrt {1+^{2}}}~\mathrm {d} {x},}
281:, a line is defined as a "breadthless length" (Def. 2), while a
4738:
4050:
one. They may be obtained as the common solutions of at least
3948:
coordinates. In this case, a point with real coordinates is a
3244:(i.e. infinitely differentiable and charts are expressible as
1003:(that is the curve divides the plane in two non-intersecting
16:
Mathematical idealization of the trace left by a moving point
4252:, which are nonsingular curves of genus one, are studied in
4704:
4386:
This term my be ambiguous, as a non-closed curve may be a
3888:
is a polynomial in two variables defined over some field
2271:(or unit-speed or parametrized by arc length) if for any
106:
to distinguish them from more constrained curves such as
4088:, which however may introduce new singularities such as
803:
of a topological curve is a closed and bounded interval
180:
one. If the coefficients of the polynomials belong to a
4707:
A collection of 874 two-dimensional mathematical curves
4099:
A plane curve may also be completed to a curve in the
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102:. In this article, these curves are sometimes called
4240:, the simplest examples of algebraic curves are the
866:
if it is the image of an interval or a circle by an
396:
as sections of cones had been studied by
Apollonius.
34:, one of the simplest curves, after (straight) lines
4931:
4866:
4825:
4780:
4528:
Depth, Crossings and
Conflicts in Discrete Geometry
1137:More precisely, a differentiable curve is a subset
3827:
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2177:{\displaystyle t_{0}<t_{1}<\ldots <t_{n}}
2176:
2117:
2086:
1846:
1802:
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4561:Transactions of the American Mathematical Society
4244:, which are nonsingular curves of degree two and
2382:
1992:
1904:
1759:which can be thought of intuitively as using the
448:. Solutions to variational problems, such as the
3835:curves under the relation of reparametrization.
3635:{\displaystyle \gamma _{2}(t)=\gamma _{1}(p(t))}
3412:{\displaystyle \gamma _{2}\colon J\rightarrow X}
3369:{\displaystyle \gamma _{1}\colon I\rightarrow X}
2640:
1901:
1022:), and a simple curve may have a positive area.
987:. It is also defined as a non-self-intersecting
444:. Newton also worked on an early example in the
4084:), an algebraic curve may be projected onto a
3849:Algebraic curves are the curves considered in
1034:) and even a positive area. An example is the
404:Analytic geometry allowed curves, such as the
4750:
4608:Davis, Ellery W.; Brenke, William C. (1913).
999:in a plane of a Jordan curve consists of two
983:A plane simple closed curve is also called a
8:
4410:exempt de toute latitude." Pages 7 and 8 of
4208:is not zero. An example is the Fermat curve
3173:is such a curve which is only assumed to be
3040:{\displaystyle \gamma \colon I\rightarrow X}
1542:The length of a curve is independent of the
1396:{\displaystyle \gamma :\to \mathbb {R} ^{n}}
1081:{\displaystyle \gamma \colon I\rightarrow X}
542:{\displaystyle \gamma \colon I\rightarrow X}
4720:The Encyclopedia of Mathematics article on
4080:. By eliminating variables (by any tool of
3535:{\displaystyle p^{-1}\colon I\rightarrow J}
1810:, then we can define the length of a curve
1038:, which has many other unusual properties.
790:. A non-closed curve may also be called an
362:as a method to both double the cube and to
313:) were among the curves studied in ancient
4863:
4757:
4743:
4735:
4423:
4421:
3909:but all the points with coordinates in an
2843:), there are obvious examples such as the
1592:of a continuously differentiable function
265:was used in place of the more modern term
50:in older texts) is an object similar to a
4576:
4415:, by Pierre Mardele, Lyon, MDCXLV (1645).
3819:
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2014:
2007:
2006:
1991:
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905:
891:continuous non-self-intersecting curve).
875:
808:
747:
709:
679:
656:
623:
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522:
4103:: if a curve is defined by a polynomial
3952:, and the set of all real points is the
3940:In the case of a curve defined over the
3220:times continuously differentiable). If
333:The conic sections, studied in depth by
304:
290:Composite lines (lines forming an angle)
241:
4531:. Logos Verlag Berlin GmbH. p. 7.
4484:MacTutor History of Mathematics Archive
4402:
4370:
3933:, the curve is said to be defined over
1244:A common curved example is an arc of a
3484:{\displaystyle p\colon J\rightarrow I}
309:The curves created by slicing a cone (
126:, since they are generally defined by
4256:, and have important applications to
3944:, one normally considers points with
3295:A differentiable curve is said to be
2097:where the supremum is taken over all
1350:-dimensional Euclidean space, and if
1241:, depending on how they are bounded.
779:{\displaystyle \gamma (a)=\gamma (b)}
218:point of view, is not a curve, but a
145:, and the curve is then said to be a
7:
4351:Infinite-dimensional vector function
2219:A rectifiable curve is a curve with
377:as a method to trisect an angle and
329:construction. These curves include:
118:of curves and isolated points), and
3925:is a curve defined by a polynomial
2890:, then we can define the notion of
253:showing an early interest in curves
4614:. MacMillan Company. p. 108.
2958:by means of this notion of curve.
2799:
1734:
1517:
1323:{\displaystyle X=\mathbb {R} ^{n}}
1291:Differentiable curve § Length
14:
4557:"A Jordan Curve of Positive Area"
3853:. A plane algebraic curve is the
3098:manifold (i.e., a manifold whose
2118:{\displaystyle n\in \mathbb {N} }
1128:{\displaystyle \mathbb {R} ^{n}.}
954:is at least three-dimensional; a
4816:
258:a stick on the sand on a beach.
210:zeros are considered, one has a
4296:Differential geometry of curves
2831:Differential geometry of curves
2362:{\displaystyle t_{1}\leq t_{2}}
2322:{\displaystyle t_{1},t_{2}\in }
1099:into a differentiable manifold
436:Conic sections were applied in
54:, but that does not have to be
3629:
3626:
3620:
3614:
3598:
3592:
3526:
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2191:
1984:
1981:
1962:
1953:
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1934:
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1838:
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1823:
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1715:
1709:
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1608:
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1439:
1378:
1375:
1363:
1072:
828:
816:
773:
767:
758:
752:
729:
717:
643:For example, the image of the
533:
1:
4569:American Mathematical Society
3976:are said to be rational over
3894:. One says that the curve is
3857:of the points of coordinates
2513:{\displaystyle \gamma :\to X}
2260:{\displaystyle \gamma :\to X}
1847:{\displaystyle \gamma :\to X}
1627:defined on a closed interval
4076:, the curve is said to be a
4023:every rational point of the
3970:with coordinates in a field
4727:The Manifold Atlas page on
4671:, commentary and trans. by
4658:Encyclopedia of Mathematics
4640:Encyclopedia of Mathematics
4222:, which has an affine form
3716:{\displaystyle \gamma _{1}}
3685:{\displaystyle \gamma _{2}}
3124:continuously differentiable
1423:is defined as the quantity
598:However, in some contexts,
500:Hilbert's sixteenth problem
5085:
4633:A.S. Parkhomenko (2001) ,
4390:, as is a line in a plane.
3911:algebraically closed field
3842:
2828:
1568:In particular, the length
1288:
1282:
1190:
1045:
425:that can be defined using
230:are widely used in modern
194:. In the common case of a
187:, the curve is said to be
18:
4814:
3754:differentiable curves in
3268:is an analytic map, then
1193:Arc (projective geometry)
567:. Properly speaking, the
4678:Vol. 1 (1908 Cambridge)
4525:SulovskĂ˝, Marek (2012).
4506:Dictionary.reference.com
4489:University of St Andrews
4057:polynomial equations in
841:, the curve is called a
634:{\displaystyle \gamma .}
591:{\displaystyle \gamma .}
347:and used as a method to
327:compass and straightedge
222:, and is often called a
3281:{\displaystyle \gamma }
3261:{\displaystyle \gamma }
2888:differentiable manifold
2541:{\displaystyle \gamma }
1558:{\displaystyle \gamma }
1416:{\displaystyle \gamma }
1172:{\displaystyle C\cap U}
883:{\displaystyle \gamma }
687:{\displaystyle \gamma }
664:{\displaystyle \gamma }
611:{\displaystyle \gamma }
261:Historically, the term
212:complex algebraic curve
4651:B.I. Golubov (2001) ,
4479:"Spiral of Archimedes"
4346:Vector-valued function
4143:homogeneous polynomial
4042:. They are defined as
4003:, one simply talks of
3964:The points of a curve
3829:
3795:
3768:
3748:
3717:
3686:
3659:
3636:
3566:
3536:
3485:
3450:
3413:
3370:
3334:differentiable curves
3328:
3282:
3262:
3234:
3214:
3194:
3167:
3147:
3116:
3092:
3065:
3041:
3003:
2975:
2952:
2928:
2908:
2880:
2815:
2722:
2580:
2542:
2514:
2467:
2363:
2323:
2261:
2210:
2178:
2119:
2088:
1930:
1848:
1804:
1780:
1750:
1653:
1621:
1620:{\displaystyle y=f(x)}
1582:
1559:
1533:
1417:
1397:
1344:
1324:
1195:. For other uses, see
1173:
1129:
1082:
980:
948:
914:
884:
835:
780:
736:
688:
665:
647:or, more generally, a
635:
612:
592:
543:
514:can be specified by a
446:calculus of variations
409:
318:
254:
35:
21:Curve (disambiguation)
4311:List of curves topics
4086:plane algebraic curve
4078:complete intersection
4031:has a zero coordinate
4009:Fermat's Last Theorem
3929:with coefficients in
3830:
3828:{\displaystyle C^{k}}
3796:
3794:{\displaystyle C^{k}}
3769:
3749:
3747:{\displaystyle C^{k}}
3718:
3687:
3660:
3637:
3567:
3565:{\displaystyle C^{k}}
3537:
3486:
3451:
3449:{\displaystyle C^{k}}
3414:
3371:
3329:
3327:{\displaystyle C^{k}}
3283:
3263:
3235:
3215:
3195:
3193:{\displaystyle C^{k}}
3168:
3148:
3146:{\displaystyle C^{k}}
3117:
3093:
3091:{\displaystyle C^{k}}
3066:
3042:
3004:
2976:
2953:
2929:
2909:
2881:
2841:two-dimensional space
2825:Differential geometry
2816:
2723:
2581:
2579:{\displaystyle t\in }
2543:
2515:
2468:
2364:
2324:
2262:
2211:
2179:
2120:
2089:
1910:
1849:
1805:
1781:
1751:
1654:
1622:
1583:
1560:
1534:
1418:
1398:
1345:
1325:
1289:Further information:
1174:
1145:where every point of
1130:
1083:
1030:bigger than one (see
974:
962:real algebraic curves
949:
934:is a curve for which
915:
900:is a curve for which
885:
836:
781:
737:
689:
666:
636:
613:
593:
544:
466:differential calculus
431:transcendental curves
403:
356:conchoid of Nicomedes
308:
245:
168:. More generally, an
154:plane algebraic curve
108:differentiable curves
98:, and the curve is a
29:
4475:Robertson, Edmund F.
4011:may be restated as:
3999:is the field of the
3812:
3778:
3758:
3731:
3725:equivalence relation
3723:; and this makes an
3700:
3669:
3649:
3579:
3549:
3504:
3463:
3433:
3384:
3341:
3311:
3272:
3252:
3224:
3204:
3177:
3157:
3130:
3106:
3075:
3055:
3019:
2993:
2965:
2942:
2918:
2898:
2892:differentiable curve
2870:
2738:
2593:
2552:
2532:
2522:Lipschitz-continuous
2480:
2376:
2333:
2275:
2227:
2188:
2129:
2101:
1861:
1814:
1794:
1770:
1666:
1631:
1596:
1572:
1549:
1430:
1407:
1354:
1334:
1299:
1197:Arc (disambiguation)
1157:
1107:
1060:
1054:differentiable curve
1048:Differentiable curve
1042:Differentiable curve
1001:connected components
993:Jordan curve theorem
979:with a positive area
938:
904:
874:
807:
746:
708:
678:
655:
622:
602:
579:
521:
496:Jordan curve theorem
492:space-filling curves
427:polynomial equations
196:real algebraic curve
147:differentiable curve
135:space-filling curves
19:For other uses, see
4705:Mathematical curves
4699:Famous Curves Index
4653:"Rectifiable curve"
4473:O'Connor, John J.;
4044:algebraic varieties
3982:and can be denoted
2849:classical mechanics
2773:
2731:and then show that
2125:and all partitions
1766:More generally, if
1761:Pythagorean theorem
1689:
1484:
1149:has a neighborhood
1052:Roughly speaking a
1028:Hausdorff dimension
1020:space-filling curve
649:space-filling curve
516:continuous function
488:algebraic varieties
406:Folium of Descartes
335:Apollonius of Perga
91:continuous function
4553:Osgood, William F.
4326:Parametric surface
4306:Index of the curve
4082:elimination theory
3851:algebraic geometry
3825:
3791:
3764:
3744:
3727:on the set of all
3713:
3682:
3655:
3632:
3562:
3532:
3481:
3446:
3409:
3366:
3324:
3278:
3258:
3230:
3210:
3190:
3163:
3143:
3112:
3088:
3061:
3037:
2999:
2971:
2948:
2924:
2904:
2876:
2853:general relativity
2811:
2759:
2718:
2654:
2576:
2538:
2510:
2463:
2359:
2319:
2257:
2206:
2174:
2115:
2084:
1844:
1800:
1776:
1746:
1675:
1649:
1617:
1578:
1555:
1529:
1470:
1413:
1393:
1340:
1320:
1204:Euclidean geometry
1187:Differentiable arc
1169:
1125:
1078:
991:in the plane. The
981:
944:
910:
880:
834:{\displaystyle I=}
831:
776:
735:{\displaystyle I=}
732:
684:
661:
631:
608:
588:
539:
410:
371:Archimedean spiral
341:cissoid of Diocles
319:
293:Incomposite lines
269:. Hence the terms
255:
214:, which, from the
128:implicit equations
104:topological curves
36:
5038:
5037:
4927:
4926:
4321:Osculating circle
4301:Gallery of curves
4286:Curve orientation
3806:equivalence class
3767:{\displaystyle X}
3694:reparametrization
3658:{\displaystyle t}
3288:is said to be an
3242:analytic manifold
3233:{\displaystyle X}
3213:{\displaystyle k}
3166:{\displaystyle X}
3115:{\displaystyle k}
3064:{\displaystyle X}
3002:{\displaystyle X}
2974:{\displaystyle X}
2951:{\displaystyle X}
2927:{\displaystyle X}
2907:{\displaystyle X}
2879:{\displaystyle X}
2797:
2716:
2639:
2638:
2633:
2631:
2619:
2526:metric derivative
2021:
2017:
2013:
1999:
1989:
1900:
1895:
1893:
1881:
1803:{\displaystyle d}
1779:{\displaystyle X}
1732:
1728:
1581:{\displaystyle s}
1515:
1469:
1464:
1462:
1450:
1343:{\displaystyle n}
1279:Length of a curve
947:{\displaystyle X}
913:{\displaystyle X}
562:topological space
512:topological curve
506:Topological curve
414:analytic geometry
379:square the circle
315:Greek mathematics
279:Euclid's Elements
174:algebraic variety
87:topological space
5076:
5069:General topology
4864:
4843:Boerdijk–Coxeter
4820:
4819:
4759:
4752:
4745:
4736:
4688:(1961 Cambridge)
4686:A Book of Curves
4665:
4647:
4626:
4625:
4605:
4599:
4598:
4580:
4555:(January 1903).
4549:
4543:
4542:
4522:
4516:
4515:
4513:
4512:
4498:
4492:
4491:
4470:
4464:
4461:
4455:
4452:
4446:
4443:
4437:
4434:
4428:
4425:
4416:
4407:
4391:
4384:
4378:
4375:
4271:Coordinate curve
4232:
4221:
4207:
4201:
4182:
4169:. The values of
4168:
4162:
4141:simplifies to a
4140:
4114:
4109:of total degree
4108:
4101:projective plane
4075:
4069:
4062:
4056:
4041:
4030:
4020:
4001:rational numbers
3998:
3992:
3981:
3975:
3969:
3959:Riemann surfaces
3917:
3908:
3902:
3893:
3887:
3881:
3866:
3834:
3832:
3831:
3826:
3824:
3823:
3800:
3798:
3797:
3792:
3790:
3789:
3773:
3771:
3770:
3765:
3753:
3751:
3750:
3745:
3743:
3742:
3722:
3720:
3719:
3714:
3712:
3711:
3691:
3689:
3688:
3683:
3681:
3680:
3664:
3662:
3661:
3656:
3641:
3639:
3638:
3633:
3613:
3612:
3591:
3590:
3571:
3569:
3568:
3563:
3561:
3560:
3541:
3539:
3538:
3533:
3519:
3518:
3490:
3488:
3487:
3482:
3455:
3453:
3452:
3447:
3445:
3444:
3418:
3416:
3415:
3410:
3396:
3395:
3375:
3373:
3372:
3367:
3353:
3352:
3333:
3331:
3330:
3325:
3323:
3322:
3301:
3300:
3287:
3285:
3284:
3279:
3267:
3265:
3264:
3259:
3239:
3237:
3236:
3231:
3219:
3217:
3216:
3211:
3199:
3197:
3196:
3191:
3189:
3188:
3172:
3170:
3169:
3164:
3152:
3150:
3149:
3144:
3142:
3141:
3121:
3119:
3118:
3113:
3097:
3095:
3094:
3089:
3087:
3086:
3070:
3068:
3067:
3062:
3046:
3044:
3043:
3038:
3008:
3006:
3005:
3000:
2980:
2978:
2977:
2972:
2957:
2955:
2954:
2949:
2933:
2931:
2930:
2925:
2913:
2911:
2910:
2905:
2885:
2883:
2882:
2877:
2820:
2818:
2817:
2812:
2807:
2802:
2795:
2785:
2784:
2783:
2772:
2767:
2727:
2725:
2724:
2719:
2717:
2715:
2714:
2700:
2694:
2656:
2653:
2636:
2635:
2634:
2632:
2629:
2627:
2622:
2617:
2607:
2606:
2605:
2585:
2583:
2582:
2577:
2547:
2545:
2544:
2539:
2519:
2517:
2516:
2511:
2472:
2470:
2469:
2464:
2459:
2458:
2446:
2445:
2433:
2429:
2428:
2427:
2423:
2422:
2410:
2409:
2396:
2368:
2366:
2365:
2360:
2358:
2357:
2345:
2344:
2328:
2326:
2325:
2320:
2300:
2299:
2287:
2286:
2266:
2264:
2263:
2258:
2223:length. A curve
2215:
2213:
2212:
2209:{\displaystyle }
2207:
2183:
2181:
2180:
2175:
2173:
2172:
2154:
2153:
2141:
2140:
2124:
2122:
2121:
2116:
2114:
2093:
2091:
2090:
2085:
2080:
2076:
2069:
2068:
2050:
2049:
2037:
2036:
2019:
2018:
2015:
2011:
2010:
1997:
1996:
1995:
1987:
1980:
1979:
1952:
1951:
1929:
1924:
1898:
1897:
1896:
1894:
1891:
1889:
1884:
1879:
1853:
1851:
1850:
1845:
1809:
1807:
1806:
1801:
1785:
1783:
1782:
1777:
1755:
1753:
1752:
1747:
1742:
1737:
1730:
1729:
1727:
1726:
1708:
1691:
1688:
1683:
1658:
1656:
1655:
1652:{\displaystyle }
1650:
1626:
1624:
1623:
1618:
1587:
1585:
1584:
1579:
1564:
1562:
1561:
1556:
1538:
1536:
1535:
1530:
1525:
1520:
1513:
1512:
1498:
1489:
1483:
1478:
1467:
1466:
1465:
1463:
1460:
1458:
1453:
1448:
1422:
1420:
1419:
1414:
1402:
1400:
1399:
1394:
1392:
1391:
1386:
1349:
1347:
1346:
1341:
1329:
1327:
1326:
1321:
1319:
1318:
1313:
1178:
1176:
1175:
1170:
1152:
1148:
1144:
1140:
1134:
1132:
1131:
1126:
1121:
1120:
1115:
1102:
1094:
1087:
1085:
1084:
1079:
995:states that the
953:
951:
950:
945:
926:projective plane
919:
917:
916:
911:
889:
887:
886:
881:
857:
856:
847:, also known as
840:
838:
837:
832:
785:
783:
782:
777:
741:
739:
738:
733:
693:
691:
690:
685:
670:
668:
667:
662:
640:
638:
637:
632:
617:
615:
614:
609:
597:
595:
594:
589:
566:
555:
548:
546:
545:
540:
477:BĂ©zout's theorem
423:algebraic curves
364:trisect an angle
202:is the field of
201:
193:
186:
120:algebraic curves
100:parametric curve
5084:
5083:
5079:
5078:
5077:
5075:
5074:
5073:
5059:Metric geometry
5044:
5043:
5041:
5039:
5034:
4923:
4877:
4862:
4821:
4817:
4812:
4776:
4763:
4695:
4684:E. H. Lockwood
4650:
4632:
4629:
4622:
4607:
4606:
4602:
4578:10.2307/1986455
4551:
4550:
4546:
4539:
4524:
4523:
4519:
4510:
4508:
4500:
4499:
4495:
4472:
4471:
4467:
4463:Lockwood p. 129
4462:
4458:
4454:Lockwood p. 132
4453:
4449:
4444:
4440:
4435:
4431:
4426:
4419:
4408:
4404:
4400:
4395:
4394:
4385:
4381:
4376:
4372:
4367:
4362:
4341:Position vector
4336:Polygonal curve
4331:Path (topology)
4291:Curve sketching
4266:
4250:Elliptic curves
4223:
4209:
4203:
4184:
4170:
4164:
4145:
4116:
4110:
4104:
4071:
4064:
4058:
4051:
4037:
4028:
4015:
4007:. For example,
4005:rational points
3994:
3983:
3977:
3971:
3965:
3913:
3904:
3898:
3889:
3883:
3868:
3858:
3847:
3845:Algebraic curve
3841:
3839:Algebraic curve
3815:
3810:
3809:
3781:
3776:
3775:
3756:
3755:
3734:
3729:
3728:
3703:
3698:
3697:
3672:
3667:
3666:
3647:
3646:
3604:
3582:
3577:
3576:
3552:
3547:
3546:
3507:
3502:
3501:
3461:
3460:
3436:
3431:
3430:
3422:are said to be
3387:
3382:
3381:
3344:
3339:
3338:
3314:
3309:
3308:
3298:
3297:
3270:
3269:
3250:
3249:
3222:
3221:
3202:
3201:
3180:
3175:
3174:
3155:
3154:
3133:
3128:
3127:
3104:
3103:
3078:
3073:
3072:
3053:
3052:
3017:
3016:
2991:
2990:
2983:smooth manifold
2963:
2962:
2940:
2939:
2936:tangent vectors
2916:
2915:
2896:
2895:
2868:
2867:
2833:
2827:
2775:
2736:
2735:
2695:
2657:
2597:
2591:
2590:
2550:
2549:
2530:
2529:
2478:
2477:
2450:
2437:
2414:
2401:
2391:
2387:
2383:
2374:
2373:
2349:
2336:
2331:
2330:
2291:
2278:
2273:
2272:
2225:
2224:
2186:
2185:
2164:
2145:
2132:
2127:
2126:
2099:
2098:
2060:
2041:
2028:
1965:
1943:
1909:
1905:
1859:
1858:
1812:
1811:
1792:
1791:
1768:
1767:
1718:
1701:
1664:
1663:
1629:
1628:
1594:
1593:
1570:
1569:
1547:
1546:
1544:parametrization
1493:
1428:
1427:
1405:
1404:
1381:
1352:
1351:
1332:
1331:
1308:
1297:
1296:
1293:
1287:
1281:
1263:), an arc of a
1200:
1189:
1155:
1154:
1150:
1146:
1142:
1138:
1110:
1105:
1104:
1100:
1092:
1058:
1057:
1050:
1044:
989:continuous loop
936:
935:
922:Euclidean plane
902:
901:
872:
871:
854:
853:
849:topological arc
805:
804:
744:
743:
706:
705:
676:
675:
653:
652:
620:
619:
600:
599:
577:
576:
564:
553:
519:
518:
508:
450:brachistochrone
386:spiric sections
349:double the cube
240:
224:Riemann surface
199:
191:
184:
170:algebraic curve
124:implicit curves
96:parametrization
77:A curve is the
46:(also called a
24:
17:
12:
11:
5:
5082:
5080:
5072:
5071:
5066:
5061:
5056:
5046:
5045:
5036:
5035:
5033:
5032:
5027:
5022:
5017:
5012:
5007:
5000:
4999:
4998:
4988:
4983:
4978:
4973:
4968:
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4693:External links
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4635:"Line (curve)"
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4358:Winding number
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4316:List of curves
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4063:variables. If
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1046:Main article:
1043:
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1032:Koch snowflake
1024:Fractal curves
1018:in the plane (
997:set complement
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418:René Descartes
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311:conic sections
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247:Megalithic art
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166:indeterminates
143:differentiable
139:fractal curves
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4281:Curve fitting
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4254:number theory
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1269:great ellipse
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1181:diffeomorphic
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1021:
1017:
1012:
1010:
1009:Jordan domain
1006:
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986:
978:
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428:
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415:
407:
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395:
391:
387:
383:
380:
376:
373:, studied by
372:
368:
365:
361:
358:, studied by
357:
353:
350:
346:
343:, studied by
342:
338:
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332:
331:
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316:
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271:straight line
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33:
28:
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5040:
5002:
4867:Biochemistry
4769:
4685:
4680:Google Books
4675:
4656:
4638:
4611:The Calculus
4610:
4603:
4564:
4560:
4547:
4527:
4520:
4509:. Retrieved
4496:
4482:
4468:
4459:
4450:
4445:Heath p. 160
4441:
4436:Heath p. 153
4432:
4411:
4405:
4382:
4373:
4276:Crinkled arc
4258:cryptography
4235:
4228:
4224:
4218:
4214:
4210:
4204:
4197:
4193:
4189:
4185:
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4120:
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4098:
4072:
4065:
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4052:
4038:
4035:
4025:Fermat curve
4022:
4016:
4012:
4004:
3995:
3988:
3984:
3978:
3972:
3966:
3963:
3953:
3949:
3942:real numbers
3939:
3934:
3930:
3926:
3922:
3920:
3914:
3905:
3899:
3896:defined over
3895:
3890:
3884:
3877:
3873:
3869:
3863:
3859:
3848:
3801:
3693:
3692:is called a
3644:
3544:
3493:
3423:
3421:
3296:
3294:
3289:
3246:power series
3050:
2987:smooth curve
2986:
2960:
2891:
2865:
2840:
2837:curved lines
2836:
2834:
2730:
2475:
2218:
2096:
1790:with metric
1788:metric space
1765:
1758:
1567:
1541:
1294:
1272:
1265:great circle
1254:
1250:circular arc
1243:
1224:
1218:subset of a
1211:
1207:
1201:
1136:
1097:real numbers
1051:
1036:dragon curve
1013:
1008:
985:Jordan curve
984:
982:
977:dragon curve
966:disconnected
955:
929:
895:
893:
863:
861:
852:
848:
842:
798:
792:
791:
699:
695:
673:
671:is defined.
642:
568:
558:real numbers
511:
509:
481:
473:cubic curves
470:
435:
411:
320:
260:
256:
232:cryptography
228:finite field
211:
204:real numbers
189:defined over
188:
151:
132:
112:level curves
103:
95:
76:
74:
68:
60:
47:
43:
37:
5015:Pitch angle
4991:Logarithmic
4939:Archimedean
4902:Polyproline
4729:1-manifolds
4673:T. L. Heath
4571:: 107–112.
4413:traductions
4236:Except for
3665:. The map
3496:inverse map
1248:, called a
1229:are called
931:space curve
897:plane curve
862:A curve is
645:Peano curve
454:tautochrone
392:studied by
216:topological
114:(which are
48:curved line
40:mathematics
5048:Categories
5004:On Spirals
4954:Hyperbolic
4511:2012-03-14
4398:References
4388:closed set
4183:such that
4163:of degree
4027:of degree
3950:real point
3867:such that
3424:equivalent
3305:derivative
3126:), then a
3011:smooth map
2857:world line
2369:, we have
2329:such that
2267:is called
1285:Arc length
1153:such that
957:skew curve
793:open curve
375:Archimedes
321:The Greek
275:right line
162:polynomial
5025:Spirangle
5020:Theodorus
4959:Poinsot's
4949:Epispiral
4793:Curvature
4788:Algebraic
4663:EMS Press
4645:EMS Press
4587:0002-9947
4048:dimension
3954:real part
3705:γ
3674:γ
3606:γ
3584:γ
3527:→
3521::
3513:−
3476:→
3470::
3428:bijective
3404:→
3398::
3389:γ
3361:→
3355::
3346:γ
3276:γ
3256:γ
3153:curve in
3032:→
3026::
3023:γ
2861:spacetime
2781:γ
2761:∫
2751:γ
2745:
2705:−
2680:γ
2665:γ
2648:→
2603:γ
2559:∈
2536:γ
2505:→
2484:γ
2448:−
2389:γ
2347:≤
2302:∈
2252:→
2231:γ
2159:…
2108:∈
2055:…
2004:∈
1974:−
1960:γ
1938:γ
1912:∑
1874:γ
1868:
1839:→
1818:γ
1677:∫
1553:γ
1491:γ
1472:∫
1443:γ
1437:
1411:γ
1379:→
1358:γ
1273:great arc
1216:connected
1210:(symbol:
1164:∩
1073:→
1067::
1064:γ
878:γ
868:injective
851:(or just
765:γ
750:γ
682:γ
659:γ
626:γ
606:γ
583:γ
534:→
528::
525:γ
484:manifolds
438:astronomy
360:Nicomedes
323:geometers
251:Newgrange
178:dimension
67:Euclid's
5064:Topology
4981:Involute
4976:Fermat's
4917:Collagen
4853:Symmetry
4676:Elements
4264:See also
3882:, where
3645:for all
3545:is also
1706:′
1496:′
1261:spheroid
1231:segments
1225:Arcs of
1222:curve.
1103:, often
1090:interval
1088:from an
698:or is a
674:A curve
551:interval
549:from an
462:catenary
283:straight
198:, where
158:zero set
83:interval
69:Elements
56:straight
32:parabola
5010:Padovan
4944:Cotes's
4932:Spirals
4838:Antenna
4826:Helices
4798:Gallery
4774:helices
4766:Spirals
4595:1986455
4115:, then
3993:. When
3946:complex
3303:if its
3299:regular
3248:), and
2269:natural
1588:of the
1330:is the
1214:) is a
1095:of the
1005:regions
920:is the
799:If the
571:is the
560:into a
556:of the
460:). The
458:cycloid
394:Perseus
345:Diocles
238:History
220:surface
208:complex
164:in two
156:is the
5054:Curves
4996:Golden
4912:Triple
4892:Double
4858:Triple
4808:Topics
4781:Curves
4770:curves
4669:Euclid
4618:
4593:
4585:
4535:
4248:zero.
4242:conics
4019:> 2
3804:is an
3572:, and
3240:is an
3200:(i.e.
3122:times
3100:charts
2796:
2742:Length
2637:
2618:
2380:Length
2221:finite
2020:
2012:
1998:
1988:
1899:
1880:
1865:Length
1731:
1514:
1468:
1449:
1434:Length
1267:(or a
1259:(or a
1257:sphere
1246:circle
1016:square
864:simple
801:domain
788:circle
696:closed
442:Kepler
429:, and
116:unions
81:of an
4971:Euler
4966:Doyle
4907:Super
4882:Alpha
4833:Angle
4722:lines
4591:JSTOR
4567:(1).
4365:Notes
4246:genus
4238:lines
4200:) = 0
4090:cusps
3880:) = 0
3774:. A
3071:is a
3009:is a
2981:is a
2886:is a
2845:helix
2777:Speed
2599:Speed
2528:) of
2520:is a
1786:is a
1590:graph
1255:In a
1239:lines
1237:, or
1227:lines
1206:, an
573:image
569:curve
267:curve
249:from
182:field
160:of a
89:by a
85:to a
79:image
63:point
44:curve
5030:Ulam
4986:List
4887:Beta
4848:Hemi
4803:List
4772:and
4616:ISBN
4583:ISSN
4533:ISBN
3456:map
3102:are
2985:, a
2855:, a
2162:<
2156:<
2143:<
2058:<
2052:<
2039:<
1235:rays
928:. A
844:path
742:and
701:loop
498:and
486:and
452:and
390:tori
384:The
369:The
354:The
339:The
273:and
263:line
137:and
52:line
42:, a
4573:doi
4231:= 1
4092:or
4046:of
4013:For
3937:.
3921:If
3918:.
3855:set
3808:of
3802:arc
3696:of
3376:and
2989:in
2961:If
2938:to
2894:in
2866:If
2839:in
2630:def
2586:as
2548:at
2476:If
2184:of
2016:and
1902:sup
1892:def
1854:by
1659:is
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1295:If
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1202:In
1179:is
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1011:.
968:).
859:).
855:arc
704:if
694:is
575:of
440:by
416:by
176:of
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5050::
4897:Pi
4876:10
4768:,
4661:,
4655:,
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4637:,
4589:.
4581:.
4563:.
4559:.
4504:.
4487:,
4481:,
4477:,
4420:^
4260:.
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4217:=
4213:+
4196:,
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4096:.
4068:–1
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4758:e
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4112:d
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4073:n
4066:n
4060:n
4053:n
4039:n
4029:n
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3996:G
3991:)
3989:G
3987:(
3985:C
3979:G
3973:G
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3891:F
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2493:a
2490:[
2487::
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2308:a
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2246:b
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2240:a
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2204:]
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2192:[
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2112:N
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2082:,
2078:}
2074:b
2071:=
2066:n
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2034:0
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2026:=
2023:a
2008:N
2001:n
1993:|
1985:)
1982:)
1977:1
1971:i
1967:t
1963:(
1957:,
1954:)
1949:i
1945:t
1941:(
1935:(
1932:d
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1916:i
1907:{
1887:=
1877:)
1871:(
1842:X
1836:]
1833:b
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1824:[
1821::
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1600:y
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1199:.
1167:U
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1101:X
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820:a
817:[
814:=
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771:b
768:(
762:=
759:)
756:a
753:(
730:]
727:b
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586:.
565:X
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381:.
366:.
351:.
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200:k
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