408:
3686:
3073:
1807:
2063:
1295:
38:
2798:
489:. However, only three of these points may be real, so that the others cannot be seen in the real projective plane by drawing the curve. The nine inflection points of a non-singular cubic have the property that every line passing through two of them contains exactly three inflection points.
3663:
lies on the line of the circumcenter and the symmedian point (i.e., the
Brocard axis). The cubic passes through the centroid, symmedian point, both Fermat points, both isodynamic points, the Parry point, other triangle centers, and the vertices of the 2nd and 4th Brocard triangles.
2392:
1646:
2731:
500:; it appears as one or three infinite branches, containing the three real inflection points. The other oval, if it exists, does not contain any real inflection point and appears either as an oval or as two infinite branches. Like for
3933:
3348:
1459:
3050:
The Lucas cubic passes through the centroid, orthocenter, Gergonne point, Nagel point, de
Longchamps point, other triangle centers, the vertices of the anticomplementary triangle, and the foci of the Steiner circumellipse.
2202:
354:
3643:
4263:
3222:
3035:
2926:
1285:
3527:
3807:
1995:
1028:
2560:
4148:
1908:
797:
1109:
867:
496:. The real points of a non-singular projective cubic fall into one or two 'ovals'. One of these ovals crosses every real projective line, and thus is never bounded when the cubic is drawn in the
388:. Therefore, we can find some cubic curve through any nine given points, which may be degenerate, and may not be unique, but will be unique and non-degenerate if the points are in
2209:
1466:
4396:
2567:
4348:
116:
1210:
4303:
703:
662:
621:
164:
3408:
The 1st
Brocard cubic passes through the centroid, symmedian point, Steiner point, other triangle centers, and the vertices of the 1st and 3rd Brocard triangles.
2018:
The
Thomson cubic passes through the following points: incenter, centroid, circumcenter, orthocenter, symmedian point, other triangle centers, the vertices
3814:
3229:
2765:
The
Napoleon–Feuerbach cubic passes through the incenter, circumcenter, orthocenter, 1st and 2nd Napoleon points, other triangle centers, the vertices
1354:
2100:
4877:
198:
3534:
3983:
The 1st equal areas cubic passes through the incenter, Steiner point, other triangle centers, the 1st and 2nd
Brocard points, and the excenters.
4155:
3117:
2933:
2831:
1215:
3432:
3715:
2769:
the excenters, the projections of the centroid on the altitudes, and the centers of the 6 equilateral triangles erected on the sides of
5300:
1915:
878:
2479:
2439:
The
Darboux cubic passes through the incenter, circumcenter, orthocenter, de Longchamps point, other triangle centers, the vertices
4822:
4007:
450:
5392:
5029:
4989:
4870:
1842:
729:
717:
1039:
808:
558:. A reducible plane cubic curve is either a conic and a line or three lines, and accordingly have two double points or a
516:
5469:
5080:
4979:
5459:
4529:
712:, many named cubics pass through well-known points. Examples shown below use two kinds of homogeneous coordinates:
5158:
4863:
4420:
3977:
2759:
1686:
4655:
Cundy, H. M. & Parry, Cyril F. (2000), "Geometrical properties of some Euler and circular cubics (part 2)",
4626:
Cundy, H. M. & Parry, Cyril F. (1999), "Geometrical properties of some Euler and circular cubics (part 1)",
2424:
is the cevian triangle of some point (which lies on the Lucas cubic). Also, this cubic is the locus of a point
5305:
5226:
5216:
5153:
4903:
2387:{\displaystyle \sum _{\text{cyclic}}(2a^{2}(b^{2}+c^{2})+(b^{2}-c^{2})^{2}-3a^{4})x(c^{2}y^{2}-b^{2}z^{2})=0}
393:
1641:{\displaystyle \sum _{\text{cyclic}}(a^{2}(b^{2}+c^{2})+(b^{2}-c^{2})^{2}-2a^{4})x(c^{2}y^{2}-b^{2}z^{2})=0}
5123:
5019:
4503:
4481:
1655:
401:
5382:
5346:
5045:
4958:
17:
4827:
5356:
4994:
1033:
In the examples below, such equations are written more succinctly in "cyclic sum notation", like this:
5402:
2726:{\displaystyle \sum _{\text{cyclic}}(a^{2}(b^{2}+c^{2})+(b^{2}-c^{2})^{2})x(c^{2}y^{2}-b^{2}z^{2})=0}
5315:
5295:
5231:
5148:
5050:
5009:
4353:
5206:
5014:
4308:
124:
4419:
The 2nd equal areas cubic passes through the incenter, centroid, symmedian point, and points in
74:
4999:
5113:
1174:
515:
for which it has a point defined. Elliptic curves are now normally studied in some variant of
486:
5494:
5377:
5075:
5024:
4913:
4270:
713:
666:
625:
584:
131:
57:
4852:
lecture in July 2016, ICMS, Edinburgh at conference in honour of Dusa McDuff's 70th birthday
5464:
5325:
4984:
4597:
Cundy, H. M. & Parry, Cyril F. (1995), "Some cubic curves associated with a triangle",
539:
in
Weierstrass form. There are many cubic curves that have no such point, for example when
470:
397:
5236:
1788:
For a graphical representation and extensive list of properties of the
Neuberg cubic, see
8:
5290:
5168:
5133:
5090:
5070:
2413:
1659:
520:
454:
366:
4838:
Special
Isocubics in the Triangle Plane (pdf), by Jean-Pierre Ehrmann and Bernard Gibert
5431:
5211:
5191:
5004:
4789:
4672:
4643:
4614:
4585:
4556:
4518:
1116:
555:
30:"Cubic curve" redirects here. For information on polynomial functions of degree 3, see
5163:
4819:
1767:, the Euler infinity point, other triangle centers, the excenters, the reflections of
5320:
5267:
4953:
4948:
4793:
4741:
4676:
4647:
4618:
4589:
4560:
4525:
723:
To convert from trilinear to barycentric in a cubic equation, substitute as follows:
536:
524:
4844:"Real and Complex Cubic Curves - John Milnor, Stony Brook University [2016]"
5310:
5196:
5173:
4781:
4664:
4635:
4606:
4577:
4548:
1764:
563:
466:
389:
362:
169:
5436:
5241:
5183:
5085:
4908:
4887:
4802:
4498:
3928:{\displaystyle \sum _{\text{cyclic}}a^{2}(b^{2}-c^{2})x(c^{2}y^{2}-b^{2}z^{2})=0}
3343:{\displaystyle \sum _{\text{cyclic}}(a^{4}-b^{2}c^{2})x(c^{2}y^{2}+b^{2}z^{2})=0}
1454:{\displaystyle \sum _{\text{cyclic}}(\cos {A}-2\cos {B}\cos {C})x(y^{2}-z^{2})=0}
544:
497:
458:
4837:
2197:{\displaystyle \sum _{\text{cyclic}}(\cos {A}-\cos {B}\cos {C})x(y^{2}-z^{2})=0}
562:(if a conic and a line), or up to three double points or a single triple point (
5410:
5108:
4933:
4918:
4895:
4493:
550:
The singular points of an irreducible plane cubic curve are quite limited: one
532:
508:
478:
474:
349:{\displaystyle x^{3},y^{3},z^{3},x^{2}y,x^{2}z,y^{2}x,y^{2}z,z^{2}x,z^{2}y,xyz}
64:
31:
4843:
4463:
3987:
3668:
3412:
3055:
2780:
2459:
2045:
1789:
5488:
5451:
5221:
5201:
5128:
4923:
4855:
4684:
Ehrmann, Jean-Pierre & Gibert, Bernard (2001), "A Morley configuration",
4487:
1651:
501:
3638:{\displaystyle \sum _{\text{cyclic}}(b^{2}-c^{2})x(c^{2}y^{2}+b^{2}z^{2})=0}
1778:, and the vertices of the six equilateral triangles erected on the sides of
527:
made by extracting the square root of a cubic. This does depend on having a
5387:
5361:
5351:
5341:
5143:
4963:
2451:
on the cubic but not on a sideline of the cubic, the isogonal conjugate of
2037:
on the cubic but not on a sideline of the cubic, the isogonal conjugate of
1760:
1752:
551:
493:
173:
4258:{\displaystyle \sum _{\text{cyclic}}a(a^{2}-bc)x(c^{3}y^{2}-b^{3}z^{2})=0}
3047:
is the pedal triangle of some point; the point lies on the Darboux cubic.
407:
5262:
5100:
3217:{\displaystyle \sum _{\text{cyclic}}bc(a^{4}-b^{2}c^{2})x(y^{2}+z^{2})=0}
1756:
49:
3030:{\displaystyle \sum _{\text{cyclic}}(b^{2}+c^{2}-a^{2})x(y^{2}-z^{2})=0}
396:. If two cubics pass through a given set of nine points, then in fact a
5257:
4785:
4712:
Gibert, Bernard (2003), "Orthocorrespondence and orthopivotal cubics",
4698:
Ehrmann, Jean-Pierre & Gibert, Bernard (2001), "The Simson cubic",
4668:
4639:
4610:
4581:
4552:
3685:
4832:
2921:{\displaystyle \sum _{\text{cyclic}}\cos(A)x(b^{2}y^{2}-c^{2}z^{2})=0}
1280:{\displaystyle X^{*}={\tfrac {1}{x}}:{\tfrac {1}{y}}:{\tfrac {1}{z}}.}
5118:
4726:
Kimberling, Clark (1998), "Triangle Centers and Central Triangles",
3522:{\displaystyle \sum _{\text{cyclic}}bc(b^{2}-c^{2})x(y^{2}+z^{2})=0}
3072:
4772:
Pinkernell, Guido M. (1996), "Cubic curves in the triangle plane",
4758:
Lang, Fred (2002), "Geometry and group structures of some cubics",
3802:{\displaystyle \sum _{\text{cyclic}}a(b^{2}-c^{2})x(y^{2}-z^{2})=0}
1748:
188:
400:
of cubics does, and the points satisfy additional properties; see
5441:
5426:
559:
1806:
5421:
4539:
Cerin, Zvonko (1998), "Locus properties of the Neuberg cubic",
2062:
1990:{\displaystyle \sum _{\text{cyclic}}x(c^{2}y^{2}-b^{2}z^{2})=0}
1294:
1023:{\displaystyle f(a,b,c,x,y,z)+f(b,c,a,y,z,x)+f(c,a,b,z,x,y)=0.}
4520:
Conics and Cubics: A Concrete Introduction to Algebraic Curves
2555:{\displaystyle \sum _{\text{cyclic}}\cos(B-C)x(y^{2}-z^{2})=0}
2797:
477:. This can be shown by taking the homogeneous version of the
37:
4143:{\displaystyle (bz+cx)(cx+ay)(ay+bz)=(bx+cy)(cy+az)(az+bx)}
2436:
are perspective; the perspector lies on the Thomson cubic.
361:
These are ten in number; therefore the cubic curves form a
569:
1747:
The Neuberg cubic passes through the following points:
481:, which defines again a cubic, and intersecting it with
1903:{\displaystyle \sum _{\text{cyclic}}bcx(y^{2}-z^{2})=0}
1167:
analogously. Then the three reflected lines concur in
1115:
The cubics listed below can be defined in terms of the
792:{\displaystyle x\to bcx,\quad y\to cay,\quad z\to abz;}
1814:
is on the cubic, such that the isogonal conjugate of
1263:
1248:
1233:
4356:
4311:
4273:
4158:
4010:
3817:
3718:
3537:
3435:
3386:
respectively. The 1st Brocard cubic is the locus of
3232:
3120:
2936:
2834:
2735:
The Napoleon–Feuerbach cubic is the locus of a point
2570:
2482:
2212:
2103:
1918:
1845:
1469:
1357:
1218:
1177:
1104:{\displaystyle \sum _{\text{cyclic}}f(x,y,z,a,b,c)=0}
1042:
881:
811:
732:
669:
628:
587:
201:
187:
is a non-zero linear combination of the third-degree
134:
77:
862:{\displaystyle x\to ax,\quad y\to by,\quad z\to cz.}
43:
Click the image to see information page for details.
392:; compare to two points determining a line and how
4807:(3rd ed.), Dublin: Hodges, Foster, and Figgis
4568:Cerin, Zvonko (1999), "On the cubic of Napoleon",
4517:
4390:
4342:
4297:
4257:
4142:
3937:The 1st equal areas cubic is the locus of a point
3927:
3801:
3637:
3521:
3359:be the 1st Brocard triangle. For arbitrary point
3342:
3216:
3029:
2920:
2725:
2554:
2386:
2196:
1989:
1902:
1640:
1453:
1279:
1204:
1103:
1022:
861:
791:
697:
656:
615:
348:
158:
110:
4742:"Cubics associated with triangles of equal areas"
504:, a line cuts this oval at, at most, two points.
5486:
492:The real points of cubic curves were studied by
4484:, on the intersection of two cubic plane curves
4885:
3647:The 2nd Brocard cubic is the locus of a point
802:to convert from barycentric to trilinear, use
4871:
4697:
4683:
4402:such that the area of the cevian triangle of
465:cubic curve is known to have nine points of
2471:
1149:about the internal angle bisector of angle
4878:
4864:
4771:
4739:
4725:
4654:
4625:
4596:
4409:equals the area of the cevian triangle of
4398:The 2nd equal areas cubic is the locus of
3945:equals the area of the cevian triangle of
3704:equals the area of the cevian triangle of
3095:is the first Brocard triangle of triangle
2396:The Darboux cubic is the locus of a point
1999:The Thomson cubic is the locus of a point
3941:such that area of the cevian triangle of
3700:such that area of the cevian triangle of
3999:
3684:
3680:
3110:are the first and second Brocard points.
3076:First Brocard Cubic: It is the locus of
3071:
3039:The Lucas cubic is the locus of a point
2796:
2061:
1810:Example of Thomson cubic (black curve).
1805:
1293:
872:Many equations for cubics have the form
485:; the intersections are then counted by
406:
36:
570:Cubic curves in the plane of a triangle
172:; or the inhomogeneous version for the
14:
5487:
5301:Clifford's theorem on special divisors
4850:. Graduate Mathematics. June 27, 2018.
4800:
4711:
2026:and the midpoints of the altitudes of
2022:the excenters, the midpoints of sides
507:A non-singular plane cubic defines an
4859:
4567:
4538:
2447:on the circumcircle. For each point
376:imposes a single linear condition on
4757:
3949:. Also, this cubic is the locus of
3424:
3067:
2816:is the pedal triangle of some point
2443:the excenters, and the antipodes of
2416:. Also, this cubic is the locus of
2081:are the feet of perpendiculars from
1689:). Also, this cubic is the locus of
4515:
4462:For a graphics and properties, see
3986:For a graphics and properties, see
3689:First equal area cubic of triangle
3667:For a graphics and properties, see
2779:For a graphics and properties, see
24:
5470:Vector bundles on algebraic curves
5393:Weber's theorem (Algebraic curves)
4990:Hasse's theorem on elliptic curves
4980:Counting points on elliptic curves
3378:be the intersections of the lines
25:
5506:
4813:
3411:For graphics and properties, see
3054:For graphics and properties, see
3043:such that the cevian triangle of
2812:such that the cevian triangle of
2458:For graphics and properties, see
4421:Encyclopedia of Triangle Centers
3978:Encyclopedia of Triangle Centers
2760:Encyclopedia of Triangle Centers
2428:such that the pedal triangle of
2420:such that the pedal triangle of
2057:
1801:
1687:Encyclopedia of Triangle Centers
1289:
1171:. In trilinear coordinates, if
517:Weierstrass's elliptic functions
426:. A parametrization is given by
5081:Hurwitz's automorphisms theorem
4820:A Catalog of Cubic Plane Curves
3651:for which the pole of the line
3102:, are collinear. In the figure
2432:and the anticevian triangle of
2044:For graphs and properties, see
843:
827:
770:
751:
581:is a triangle with sidelengths
365:of dimension 9, over any given
5306:Gonality of an algebraic curve
5217:Differential of the first kind
4246:
4200:
4194:
4172:
4137:
4119:
4116:
4098:
4095:
4077:
4071:
4053:
4050:
4032:
4029:
4011:
3916:
3870:
3864:
3838:
3790:
3764:
3758:
3732:
3626:
3580:
3574:
3548:
3510:
3484:
3478:
3452:
3331:
3285:
3279:
3243:
3205:
3179:
3173:
3137:
3018:
2992:
2986:
2947:
2909:
2863:
2857:
2851:
2792:
2714:
2668:
2662:
2653:
2626:
2620:
2594:
2581:
2543:
2517:
2511:
2499:
2375:
2329:
2323:
2298:
2271:
2265:
2239:
2223:
2185:
2159:
2153:
2114:
1978:
1932:
1891:
1865:
1629:
1583:
1577:
1552:
1525:
1519:
1493:
1480:
1442:
1416:
1410:
1368:
1092:
1056:
1011:
975:
966:
930:
921:
885:
847:
831:
815:
774:
755:
736:
688:
677:
647:
636:
606:
595:
153:
135:
99:
81:
13:
1:
5460:Birkhoff–Grothendieck theorem
5159:Nagata's conjecture on curves
5030:Schoof–Elkies–Atkin algorithm
4904:Five points determine a conic
4509:
1678:is the Euler infinity point (
394:five points determine a conic
41:A selection of cubic curves.
5020:Supersingular elliptic curve
4833:Cubics in the Triangle Plane
4504:Catalogue of Triangle Cubics
4469:Cubics in the Triangle Plane
4391:{\displaystyle X_{Z}=z:x:y.}
3993:Cubics in the Triangle Plane
3674:Cubics in the Triangle Plane
3418:Cubics in the Triangle Plane
3061:Cubics in the Triangle Plane
2786:Cubics in the Triangle Plane
2465:Cubics in the Triangle Plane
2051:Cubics in the Triangle Plane
1795:Cubics in the Triangle Plane
1656:Joseph Jean Baptiste Neuberg
27:Type of a mathematical curve
7:
5227:Riemann's existence theorem
5154:Hilbert's sixteenth problem
5046:Elliptic curve cryptography
4959:Fundamental pair of periods
4736:. See Chapter 8 for cubics.
4475:
4343:{\displaystyle X_{Y}=y:z:x}
3655:in the circumconic through
2747:is the nine-point center, (
10:
5511:
5357:Moduli of algebraic curves
4740:Kimberling, Clark (2001),
3080:such the intersections of
2824:lies on the Darboux cubic.
2066:Darboux cubic of triangle
1298:Neuberg cubic of triangle
1145:be the reflection of line
183:in such an equation. Here
111:{\displaystyle F(x,y,z)=0}
29:
5450:
5401:
5370:
5334:
5283:
5276:
5250:
5182:
5099:
5063:
5038:
4972:
4941:
4932:
4894:
453:, in which case it has a
449:A cubic curve may have a
5124:Cayley–Bacharach theorem
5051:Elliptic curve primality
4482:Cayley–Bacharach theorem
2801:Lucas Cubic of triangle
2472:Napoleon–Feuerbach cubic
1205:{\displaystyle X=x:y:z,}
402:Cayley–Bacharach theorem
5383:Riemann–Hurwitz formula
5347:Gromov–Witten invariant
5207:Compact Riemann surface
4995:Mazur's torsion theorem
4801:Salmon, George (1879),
4298:{\displaystyle X=x:y:z}
3965:is the Steiner point. (
3696:: The locus of a point
2808:: The locus of a point
1693:such that the triangle
1324:are the reflections of
698:{\displaystyle c=|AB|.}
657:{\displaystyle b=|CA|,}
616:{\displaystyle a=|BC|,}
159:{\displaystyle (x:y:z)}
125:homogeneous coordinates
5000:Modular elliptic curve
4728:Congressus Numerantium
4524:, New York: Springer,
4392:
4344:
4299:
4259:
4144:
3929:
3811:Barycentric equation:
3803:
3709:
3639:
3531:Barycentric equation:
3523:
3344:
3226:Barycentric equation:
3218:
3111:
3031:
2930:Barycentric equation:
2922:
2825:
2727:
2564:Barycentric equation:
2556:
2455:is also on the cubic.
2388:
2206:Barycentric equation:
2198:
2094:
2041:is also on the cubic.
1991:
1912:Barycentric equation:
1904:
1836:
1642:
1463:Barycentric equation:
1455:
1348:
1281:
1206:
1105:
1024:
863:
793:
699:
658:
617:
535:, which serves as the
446:
350:
176:determined by setting
160:
112:
45:
4914:Rational normal curve
4490:, a cubic space curve
4393:
4345:
4300:
4260:
4152:Barycentric equation:
4145:
4000:2nd equal areas cubic
3930:
3804:
3688:
3681:1st equal areas cubic
3640:
3524:
3390:for which the points
3345:
3219:
3075:
3032:
2923:
2800:
2728:
2557:
2389:
2199:
2065:
1992:
1905:
1809:
1736:is the reflection of
1643:
1456:
1297:
1282:
1207:
1134:. A construction of
1127:not on a sideline of
1106:
1025:
864:
794:
700:
659:
618:
410:
351:
161:
113:
58:plane algebraic curve
40:
5465:Stable vector bundle
5326:Weil reciprocity law
5316:Riemann–Roch theorem
5296:Brill–Noether theory
5232:Riemann–Roch theorem
5149:Genus–degree formula
5010:Mordell–Weil theorem
4985:Division polynomials
4516:Bix, Robert (1998),
4459:(2053), and others.
4354:
4309:
4271:
4156:
4008:
4004:Trilinear equation:
3815:
3716:
3712:Trilinear equation:
3535:
3433:
3429:Trilinear equation:
3230:
3118:
2934:
2832:
2828:Trilinear equation:
2568:
2480:
2476:Trilinear equation:
2210:
2101:
1916:
1843:
1839:Trilinear equation:
1793:at Berhard Gibert's
1771:in the sidelines of
1467:
1355:
1351:Trilinear equation:
1216:
1175:
1040:
879:
809:
730:
667:
626:
585:
471:algebraically closed
199:
132:
75:
5277:Structure of curves
5169:Quartic plane curve
5091:Hyperelliptic curve
5071:De Franchis theorem
5015:Nagell–Lutz theorem
4804:Higher Plane Curves
4774:Journal of Geometry
4760:Forum Geometricorum
4746:Forum Geometricorum
4714:Forum Geometricorum
4700:Forum Geometricorum
4686:Forum Geometricorum
4657:Journal of Geometry
4628:Journal of Geometry
4599:Journal of Geometry
4570:Journal of Geometry
4541:Journal of Geometry
3382:with the sidelines
3114:Trilinear equation:
3084:with the sidelines
2414:de Longchamps point
2097:Trilinear equation:
521:quadratic extension
5284:Divisors on curves
5076:Faltings's theorem
5025:Schoof's algorithm
5005:Modularity theorem
4823:(archived version)
4786:10.1007/BF01223040
4669:10.1007/BF01221061
4640:10.1007/BF01225673
4611:10.1007/BF01224039
4582:10.1007/BF01225672
4553:10.1007/BF01221237
4388:
4340:
4305:(trilinears), let
4295:
4255:
4168:
4140:
3925:
3827:
3799:
3728:
3710:
3635:
3547:
3519:
3445:
3340:
3242:
3214:
3130:
3112:
3027:
2946:
2918:
2844:
2826:
2723:
2580:
2552:
2492:
2384:
2222:
2194:
2113:
2095:
2033:. For each point
1987:
1928:
1900:
1855:
1837:
1711:is perspective to
1638:
1479:
1451:
1367:
1349:
1277:
1272:
1257:
1242:
1202:
1117:isogonal conjugate
1101:
1052:
1020:
859:
789:
695:
654:
613:
566:) if three lines.
525:rational functions
473:field such as the
447:
346:
156:
108:
46:
5482:
5481:
5478:
5477:
5378:Hasse–Witt matrix
5321:Weierstrass point
5268:Smooth completion
5237:TeichmĂĽller space
5139:Cubic plane curve
5059:
5058:
4973:Arithmetic theory
4954:Elliptic integral
4949:Elliptic function
4166:
4159:
3825:
3818:
3726:
3719:
3545:
3538:
3443:
3436:
3425:2nd Brocard cubic
3240:
3233:
3128:
3121:
3068:1st Brocard cubic
2944:
2937:
2842:
2835:
2578:
2571:
2490:
2483:
2220:
2213:
2111:
2104:
2085:to the sidelines
2015:is the centroid.
1926:
1919:
1853:
1846:
1765:isodynamic points
1477:
1470:
1365:
1358:
1332:, then the lines
1328:in the sidelines
1271:
1256:
1241:
1050:
1043:
537:point at infinity
511:, over any field
380:, if we ask that
54:cubic plane curve
16:(Redirected from
5502:
5311:Jacobian variety
5281:
5280:
5184:Riemann surfaces
5174:Real plane curve
5134:Cramer's paradox
5114:BĂ©zout's theorem
4939:
4938:
4888:algebraic curves
4880:
4873:
4866:
4857:
4856:
4851:
4828:Points on Cubics
4808:
4796:
4780:(1–2): 142–161,
4767:
4753:
4735:
4721:
4707:
4693:
4679:
4650:
4621:
4592:
4563:
4534:
4523:
4415:
4408:
4401:
4397:
4395:
4394:
4389:
4366:
4365:
4349:
4347:
4346:
4341:
4321:
4320:
4304:
4302:
4301:
4296:
4264:
4262:
4261:
4256:
4245:
4244:
4235:
4234:
4222:
4221:
4212:
4211:
4184:
4183:
4167:
4164:
4149:
4147:
4146:
4141:
3975:
3964:
3960:
3956:
3952:
3948:
3944:
3940:
3934:
3932:
3931:
3926:
3915:
3914:
3905:
3904:
3892:
3891:
3882:
3881:
3863:
3862:
3850:
3849:
3837:
3836:
3826:
3823:
3808:
3806:
3805:
3800:
3789:
3788:
3776:
3775:
3757:
3756:
3744:
3743:
3727:
3724:
3707:
3703:
3699:
3695:
3662:
3658:
3654:
3650:
3644:
3642:
3641:
3636:
3625:
3624:
3615:
3614:
3602:
3601:
3592:
3591:
3573:
3572:
3560:
3559:
3546:
3543:
3528:
3526:
3525:
3520:
3509:
3508:
3496:
3495:
3477:
3476:
3464:
3463:
3444:
3441:
3404:
3389:
3385:
3381:
3377:
3362:
3358:
3349:
3347:
3346:
3341:
3330:
3329:
3320:
3319:
3307:
3306:
3297:
3296:
3278:
3277:
3268:
3267:
3255:
3254:
3241:
3238:
3223:
3221:
3220:
3215:
3204:
3203:
3191:
3190:
3172:
3171:
3162:
3161:
3149:
3148:
3129:
3126:
3109:
3105:
3101:
3094:
3087:
3083:
3079:
3046:
3042:
3036:
3034:
3033:
3028:
3017:
3016:
3004:
3003:
2985:
2984:
2972:
2971:
2959:
2958:
2945:
2942:
2927:
2925:
2924:
2919:
2908:
2907:
2898:
2897:
2885:
2884:
2875:
2874:
2843:
2840:
2823:
2819:
2815:
2811:
2807:
2775:
2768:
2757:
2746:
2742:
2738:
2732:
2730:
2729:
2724:
2713:
2712:
2703:
2702:
2690:
2689:
2680:
2679:
2661:
2660:
2651:
2650:
2638:
2637:
2619:
2618:
2606:
2605:
2593:
2592:
2579:
2576:
2561:
2559:
2558:
2553:
2542:
2541:
2529:
2528:
2491:
2488:
2454:
2450:
2446:
2442:
2435:
2431:
2427:
2423:
2419:
2411:
2407:
2403:
2399:
2393:
2391:
2390:
2385:
2374:
2373:
2364:
2363:
2351:
2350:
2341:
2340:
2322:
2321:
2306:
2305:
2296:
2295:
2283:
2282:
2264:
2263:
2251:
2250:
2238:
2237:
2221:
2218:
2203:
2201:
2200:
2195:
2184:
2183:
2171:
2170:
2152:
2141:
2127:
2112:
2109:
2092:
2088:
2084:
2080:
2076:
2072:
2040:
2036:
2032:
2025:
2021:
2014:
2010:
2006:
2002:
1996:
1994:
1993:
1988:
1977:
1976:
1967:
1966:
1954:
1953:
1944:
1943:
1927:
1924:
1909:
1907:
1906:
1901:
1890:
1889:
1877:
1876:
1854:
1851:
1834:
1824:
1813:
1784:
1777:
1770:
1743:
1739:
1735:
1717:
1710:
1692:
1684:
1677:
1673:
1669:
1665:
1647:
1645:
1644:
1639:
1628:
1627:
1618:
1617:
1605:
1604:
1595:
1594:
1576:
1575:
1560:
1559:
1550:
1549:
1537:
1536:
1518:
1517:
1505:
1504:
1492:
1491:
1478:
1475:
1460:
1458:
1457:
1452:
1441:
1440:
1428:
1427:
1409:
1398:
1381:
1366:
1363:
1346:
1331:
1327:
1323:
1308:
1304:
1286:
1284:
1283:
1278:
1273:
1264:
1258:
1249:
1243:
1234:
1228:
1227:
1211:
1209:
1208:
1203:
1170:
1166:
1159:
1152:
1148:
1144:
1137:
1133:
1126:
1122:
1110:
1108:
1107:
1102:
1051:
1048:
1029:
1027:
1026:
1021:
868:
866:
865:
860:
798:
796:
795:
790:
711:
704:
702:
701:
696:
691:
680:
663:
661:
660:
655:
650:
639:
622:
620:
619:
614:
609:
598:
580:
564:concurrent lines
542:
530:
523:of the field of
514:
487:BĂ©zout's theorem
484:
444:
425:
390:general position
387:
383:
379:
375:
371:
363:projective space
357:
355:
353:
352:
347:
330:
329:
314:
313:
298:
297:
282:
281:
266:
265:
250:
249:
237:
236:
224:
223:
211:
210:
186:
182:
170:projective plane
167:
165:
163:
162:
157:
119:
117:
115:
114:
109:
62:
44:
21:
5510:
5509:
5505:
5504:
5503:
5501:
5500:
5499:
5485:
5484:
5483:
5474:
5446:
5437:Delta invariant
5415:
5397:
5366:
5330:
5291:Abel–Jacobi map
5272:
5246:
5242:Torelli theorem
5212:Dessin d'enfant
5192:Belyi's theorem
5178:
5164:PlĂĽcker formula
5095:
5086:Hurwitz surface
5055:
5034:
4968:
4942:Analytic theory
4934:Elliptic curves
4928:
4909:Projective line
4896:Rational curves
4890:
4884:
4842:
4816:
4634:(1–2): 72–103,
4532:
4512:
4499:Witch of Agnesi
4478:
4414:
4410:
4407:
4403:
4399:
4361:
4357:
4355:
4352:
4351:
4316:
4312:
4310:
4307:
4306:
4272:
4269:
4268:
4240:
4236:
4230:
4226:
4217:
4213:
4207:
4203:
4179:
4175:
4163:
4157:
4154:
4153:
4009:
4006:
4005:
4002:
3966:
3962:
3958:
3957:is on the line
3954:
3950:
3946:
3942:
3938:
3910:
3906:
3900:
3896:
3887:
3883:
3877:
3873:
3858:
3854:
3845:
3841:
3832:
3828:
3822:
3816:
3813:
3812:
3784:
3780:
3771:
3767:
3752:
3748:
3739:
3735:
3723:
3717:
3714:
3713:
3705:
3701:
3697:
3690:
3683:
3660:
3656:
3652:
3648:
3620:
3616:
3610:
3606:
3597:
3593:
3587:
3583:
3568:
3564:
3555:
3551:
3542:
3536:
3533:
3532:
3504:
3500:
3491:
3487:
3472:
3468:
3459:
3455:
3440:
3434:
3431:
3430:
3427:
3405:are collinear.
3403:
3399:
3395:
3391:
3387:
3383:
3379:
3376:
3372:
3368:
3364:
3360:
3353:
3325:
3321:
3315:
3311:
3302:
3298:
3292:
3288:
3273:
3269:
3263:
3259:
3250:
3246:
3237:
3231:
3228:
3227:
3199:
3195:
3186:
3182:
3167:
3163:
3157:
3153:
3144:
3140:
3125:
3119:
3116:
3115:
3107:
3103:
3096:
3089:
3085:
3081:
3077:
3070:
3044:
3040:
3012:
3008:
2999:
2995:
2980:
2976:
2967:
2963:
2954:
2950:
2941:
2935:
2932:
2931:
2903:
2899:
2893:
2889:
2880:
2876:
2870:
2866:
2839:
2833:
2830:
2829:
2821:
2817:
2813:
2809:
2802:
2795:
2770:
2766:
2748:
2744:
2740:
2739:is on the line
2736:
2708:
2704:
2698:
2694:
2685:
2681:
2675:
2671:
2656:
2652:
2646:
2642:
2633:
2629:
2614:
2610:
2601:
2597:
2588:
2584:
2575:
2569:
2566:
2565:
2537:
2533:
2524:
2520:
2487:
2481:
2478:
2477:
2474:
2452:
2448:
2444:
2440:
2433:
2429:
2425:
2421:
2417:
2409:
2405:
2404:is on the line
2401:
2397:
2369:
2365:
2359:
2355:
2346:
2342:
2336:
2332:
2317:
2313:
2301:
2297:
2291:
2287:
2278:
2274:
2259:
2255:
2246:
2242:
2233:
2229:
2217:
2211:
2208:
2207:
2179:
2175:
2166:
2162:
2148:
2137:
2123:
2108:
2102:
2099:
2098:
2093:are concurrent.
2090:
2089:then the lines
2086:
2082:
2078:
2074:
2073:: The locus of
2067:
2060:
2038:
2034:
2027:
2023:
2019:
2012:
2008:
2007:is on the line
2004:
2000:
1972:
1968:
1962:
1958:
1949:
1945:
1939:
1935:
1923:
1917:
1914:
1913:
1885:
1881:
1872:
1868:
1850:
1844:
1841:
1840:
1826:
1825:is on the line
1815:
1811:
1804:
1779:
1772:
1768:
1741:
1737:
1733:
1729:
1725:
1719:
1712:
1708:
1704:
1700:
1694:
1690:
1679:
1675:
1671:
1670:is on the line
1667:
1663:
1623:
1619:
1613:
1609:
1600:
1596:
1590:
1586:
1571:
1567:
1555:
1551:
1545:
1541:
1532:
1528:
1513:
1509:
1500:
1496:
1487:
1483:
1474:
1468:
1465:
1464:
1436:
1432:
1423:
1419:
1405:
1394:
1377:
1362:
1356:
1353:
1352:
1347:are concurrent.
1345:
1341:
1337:
1333:
1329:
1325:
1322:
1318:
1314:
1310:
1306:
1305:: The locus of
1299:
1292:
1262:
1247:
1232:
1223:
1219:
1217:
1214:
1213:
1176:
1173:
1172:
1168:
1165:
1161:
1158:
1154:
1150:
1146:
1143:
1139:
1135:
1128:
1124:
1120:
1047:
1041:
1038:
1037:
880:
877:
876:
810:
807:
806:
731:
728:
727:
706:
687:
676:
668:
665:
664:
646:
635:
627:
624:
623:
605:
594:
586:
583:
582:
575:
572:
545:rational number
540:
528:
512:
498:Euclidean plane
482:
475:complex numbers
459:projective line
455:parametrization
427:
412:
411:Singular cubic
385:
381:
377:
373:
369:
325:
321:
309:
305:
293:
289:
277:
273:
261:
257:
245:
241:
232:
228:
219:
215:
206:
202:
200:
197:
196:
194:
184:
177:
133:
130:
129:
127:
76:
73:
72:
70:
60:
42:
35:
28:
23:
22:
15:
12:
11:
5:
5508:
5498:
5497:
5480:
5479:
5476:
5475:
5473:
5472:
5467:
5462:
5456:
5454:
5452:Vector bundles
5448:
5447:
5445:
5444:
5439:
5434:
5429:
5424:
5419:
5413:
5407:
5405:
5399:
5398:
5396:
5395:
5390:
5385:
5380:
5374:
5372:
5368:
5367:
5365:
5364:
5359:
5354:
5349:
5344:
5338:
5336:
5332:
5331:
5329:
5328:
5323:
5318:
5313:
5308:
5303:
5298:
5293:
5287:
5285:
5278:
5274:
5273:
5271:
5270:
5265:
5260:
5254:
5252:
5248:
5247:
5245:
5244:
5239:
5234:
5229:
5224:
5219:
5214:
5209:
5204:
5199:
5194:
5188:
5186:
5180:
5179:
5177:
5176:
5171:
5166:
5161:
5156:
5151:
5146:
5141:
5136:
5131:
5126:
5121:
5116:
5111:
5105:
5103:
5097:
5096:
5094:
5093:
5088:
5083:
5078:
5073:
5067:
5065:
5061:
5060:
5057:
5056:
5054:
5053:
5048:
5042:
5040:
5036:
5035:
5033:
5032:
5027:
5022:
5017:
5012:
5007:
5002:
4997:
4992:
4987:
4982:
4976:
4974:
4970:
4969:
4967:
4966:
4961:
4956:
4951:
4945:
4943:
4936:
4930:
4929:
4927:
4926:
4921:
4919:Riemann sphere
4916:
4911:
4906:
4900:
4898:
4892:
4891:
4883:
4882:
4875:
4868:
4860:
4854:
4853:
4840:
4835:
4830:
4825:
4815:
4814:External links
4812:
4811:
4810:
4798:
4769:
4755:
4737:
4723:
4709:
4695:
4681:
4663:(1–2): 58–75,
4652:
4623:
4605:(1–2): 41–66,
4594:
4576:(1–2): 55–71,
4565:
4547:(1–2): 39–56,
4536:
4530:
4511:
4508:
4507:
4506:
4501:
4496:
4494:Elliptic curve
4491:
4485:
4477:
4474:
4412:
4405:
4387:
4384:
4381:
4378:
4375:
4372:
4369:
4364:
4360:
4339:
4336:
4333:
4330:
4327:
4324:
4319:
4315:
4294:
4291:
4288:
4285:
4282:
4279:
4276:
4267:For any point
4254:
4251:
4248:
4243:
4239:
4233:
4229:
4225:
4220:
4216:
4210:
4206:
4202:
4199:
4196:
4193:
4190:
4187:
4182:
4178:
4174:
4171:
4162:
4139:
4136:
4133:
4130:
4127:
4124:
4121:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4028:
4025:
4022:
4019:
4016:
4013:
4001:
3998:
3924:
3921:
3918:
3913:
3909:
3903:
3899:
3895:
3890:
3886:
3880:
3876:
3872:
3869:
3866:
3861:
3857:
3853:
3848:
3844:
3840:
3835:
3831:
3821:
3798:
3795:
3792:
3787:
3783:
3779:
3774:
3770:
3766:
3763:
3760:
3755:
3751:
3747:
3742:
3738:
3734:
3731:
3722:
3682:
3679:
3634:
3631:
3628:
3623:
3619:
3613:
3609:
3605:
3600:
3596:
3590:
3586:
3582:
3579:
3576:
3571:
3567:
3563:
3558:
3554:
3550:
3541:
3518:
3515:
3512:
3507:
3503:
3499:
3494:
3490:
3486:
3483:
3480:
3475:
3471:
3467:
3462:
3458:
3454:
3451:
3448:
3439:
3426:
3423:
3401:
3397:
3393:
3374:
3370:
3366:
3339:
3336:
3333:
3328:
3324:
3318:
3314:
3310:
3305:
3301:
3295:
3291:
3287:
3284:
3281:
3276:
3272:
3266:
3262:
3258:
3253:
3249:
3245:
3236:
3213:
3210:
3207:
3202:
3198:
3194:
3189:
3185:
3181:
3178:
3175:
3170:
3166:
3160:
3156:
3152:
3147:
3143:
3139:
3136:
3133:
3124:
3069:
3066:
3026:
3023:
3020:
3015:
3011:
3007:
3002:
2998:
2994:
2991:
2988:
2983:
2979:
2975:
2970:
2966:
2962:
2957:
2953:
2949:
2940:
2917:
2914:
2911:
2906:
2902:
2896:
2892:
2888:
2883:
2879:
2873:
2869:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2838:
2794:
2791:
2722:
2719:
2716:
2711:
2707:
2701:
2697:
2693:
2688:
2684:
2678:
2674:
2670:
2667:
2664:
2659:
2655:
2649:
2645:
2641:
2636:
2632:
2628:
2625:
2622:
2617:
2613:
2609:
2604:
2600:
2596:
2591:
2587:
2583:
2574:
2551:
2548:
2545:
2540:
2536:
2532:
2527:
2523:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2486:
2473:
2470:
2383:
2380:
2377:
2372:
2368:
2362:
2358:
2354:
2349:
2345:
2339:
2335:
2331:
2328:
2325:
2320:
2316:
2312:
2309:
2304:
2300:
2294:
2290:
2286:
2281:
2277:
2273:
2270:
2267:
2262:
2258:
2254:
2249:
2245:
2241:
2236:
2232:
2228:
2225:
2216:
2193:
2190:
2187:
2182:
2178:
2174:
2169:
2165:
2161:
2158:
2155:
2151:
2147:
2144:
2140:
2136:
2133:
2130:
2126:
2122:
2119:
2116:
2107:
2059:
2056:
1986:
1983:
1980:
1975:
1971:
1965:
1961:
1957:
1952:
1948:
1942:
1938:
1934:
1931:
1922:
1899:
1896:
1893:
1888:
1884:
1880:
1875:
1871:
1867:
1864:
1861:
1858:
1849:
1803:
1800:
1731:
1727:
1723:
1706:
1702:
1698:
1637:
1634:
1631:
1626:
1622:
1616:
1612:
1608:
1603:
1599:
1593:
1589:
1585:
1582:
1579:
1574:
1570:
1566:
1563:
1558:
1554:
1548:
1544:
1540:
1535:
1531:
1527:
1524:
1521:
1516:
1512:
1508:
1503:
1499:
1495:
1490:
1486:
1482:
1473:
1450:
1447:
1444:
1439:
1435:
1431:
1426:
1422:
1418:
1415:
1412:
1408:
1404:
1401:
1397:
1393:
1390:
1387:
1384:
1380:
1376:
1373:
1370:
1361:
1343:
1339:
1335:
1320:
1316:
1312:
1309:such that, if
1291:
1288:
1276:
1270:
1267:
1261:
1255:
1252:
1246:
1240:
1237:
1231:
1226:
1222:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1163:
1156:
1141:
1138:follows. Let
1113:
1112:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1046:
1031:
1030:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
929:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
893:
890:
887:
884:
870:
869:
858:
855:
852:
849:
846:
842:
839:
836:
833:
830:
826:
823:
820:
817:
814:
800:
799:
788:
785:
782:
779:
776:
773:
769:
766:
763:
760:
757:
754:
750:
747:
744:
741:
738:
735:
694:
690:
686:
683:
679:
675:
672:
653:
649:
645:
642:
638:
634:
631:
612:
608:
604:
601:
597:
593:
590:
571:
568:
533:rational point
509:elliptic curve
502:conic sections
479:Hessian matrix
461:. Otherwise a
457:in terms of a
451:singular point
359:
358:
345:
342:
339:
336:
333:
328:
324:
320:
317:
312:
308:
304:
301:
296:
292:
288:
285:
280:
276:
272:
269:
264:
260:
256:
253:
248:
244:
240:
235:
231:
227:
222:
218:
214:
209:
205:
155:
152:
149:
146:
143:
140:
137:
121:
120:
107:
104:
101:
98:
95:
92:
89:
86:
83:
80:
65:cubic equation
32:Cubic function
26:
9:
6:
4:
3:
2:
5507:
5496:
5493:
5492:
5490:
5471:
5468:
5466:
5463:
5461:
5458:
5457:
5455:
5453:
5449:
5443:
5440:
5438:
5435:
5433:
5430:
5428:
5425:
5423:
5420:
5418:
5416:
5409:
5408:
5406:
5404:
5403:Singularities
5400:
5394:
5391:
5389:
5386:
5384:
5381:
5379:
5376:
5375:
5373:
5369:
5363:
5360:
5358:
5355:
5353:
5350:
5348:
5345:
5343:
5340:
5339:
5337:
5333:
5327:
5324:
5322:
5319:
5317:
5314:
5312:
5309:
5307:
5304:
5302:
5299:
5297:
5294:
5292:
5289:
5288:
5286:
5282:
5279:
5275:
5269:
5266:
5264:
5261:
5259:
5256:
5255:
5253:
5251:Constructions
5249:
5243:
5240:
5238:
5235:
5233:
5230:
5228:
5225:
5223:
5222:Klein quartic
5220:
5218:
5215:
5213:
5210:
5208:
5205:
5203:
5202:Bolza surface
5200:
5198:
5197:Bring's curve
5195:
5193:
5190:
5189:
5187:
5185:
5181:
5175:
5172:
5170:
5167:
5165:
5162:
5160:
5157:
5155:
5152:
5150:
5147:
5145:
5142:
5140:
5137:
5135:
5132:
5130:
5129:Conic section
5127:
5125:
5122:
5120:
5117:
5115:
5112:
5110:
5109:AF+BG theorem
5107:
5106:
5104:
5102:
5098:
5092:
5089:
5087:
5084:
5082:
5079:
5077:
5074:
5072:
5069:
5068:
5066:
5062:
5052:
5049:
5047:
5044:
5043:
5041:
5037:
5031:
5028:
5026:
5023:
5021:
5018:
5016:
5013:
5011:
5008:
5006:
5003:
5001:
4998:
4996:
4993:
4991:
4988:
4986:
4983:
4981:
4978:
4977:
4975:
4971:
4965:
4962:
4960:
4957:
4955:
4952:
4950:
4947:
4946:
4944:
4940:
4937:
4935:
4931:
4925:
4924:Twisted cubic
4922:
4920:
4917:
4915:
4912:
4910:
4907:
4905:
4902:
4901:
4899:
4897:
4893:
4889:
4881:
4876:
4874:
4869:
4867:
4862:
4861:
4858:
4849:
4845:
4841:
4839:
4836:
4834:
4831:
4829:
4826:
4824:
4821:
4818:
4817:
4806:
4805:
4799:
4795:
4791:
4787:
4783:
4779:
4775:
4770:
4765:
4761:
4756:
4751:
4747:
4743:
4738:
4733:
4729:
4724:
4719:
4715:
4710:
4705:
4701:
4696:
4691:
4687:
4682:
4678:
4674:
4670:
4666:
4662:
4658:
4653:
4649:
4645:
4641:
4637:
4633:
4629:
4624:
4620:
4616:
4612:
4608:
4604:
4600:
4595:
4591:
4587:
4583:
4579:
4575:
4571:
4566:
4562:
4558:
4554:
4550:
4546:
4542:
4537:
4533:
4531:0-387-98401-1
4527:
4522:
4521:
4514:
4513:
4505:
4502:
4500:
4497:
4495:
4492:
4489:
4488:Twisted cubic
4486:
4483:
4480:
4479:
4473:
4471:
4470:
4466:
4460:
4458:
4454:
4450:
4446:
4442:
4438:
4434:
4430:
4426:
4422:
4417:
4385:
4382:
4379:
4376:
4373:
4370:
4367:
4362:
4358:
4337:
4334:
4331:
4328:
4325:
4322:
4317:
4313:
4292:
4289:
4286:
4283:
4280:
4277:
4274:
4265:
4252:
4249:
4241:
4237:
4231:
4227:
4223:
4218:
4214:
4208:
4204:
4197:
4191:
4188:
4185:
4180:
4176:
4169:
4160:
4150:
4134:
4131:
4128:
4125:
4122:
4113:
4110:
4107:
4104:
4101:
4092:
4089:
4086:
4083:
4080:
4074:
4068:
4065:
4062:
4059:
4056:
4047:
4044:
4041:
4038:
4035:
4026:
4023:
4020:
4017:
4014:
3997:
3995:
3994:
3990:
3984:
3981:
3979:
3973:
3969:
3935:
3922:
3919:
3911:
3907:
3901:
3897:
3893:
3888:
3884:
3878:
3874:
3867:
3859:
3855:
3851:
3846:
3842:
3833:
3829:
3819:
3809:
3796:
3793:
3785:
3781:
3777:
3772:
3768:
3761:
3753:
3749:
3745:
3740:
3736:
3729:
3720:
3694:
3687:
3678:
3676:
3675:
3671:
3665:
3645:
3632:
3629:
3621:
3617:
3611:
3607:
3603:
3598:
3594:
3588:
3584:
3577:
3569:
3565:
3561:
3556:
3552:
3539:
3529:
3516:
3513:
3505:
3501:
3497:
3492:
3488:
3481:
3473:
3469:
3465:
3460:
3456:
3449:
3446:
3437:
3422:
3420:
3419:
3415:
3409:
3406:
3380:XA′, XB′, XC′
3357:
3350:
3337:
3334:
3326:
3322:
3316:
3312:
3308:
3303:
3299:
3293:
3289:
3282:
3274:
3270:
3264:
3260:
3256:
3251:
3247:
3234:
3224:
3211:
3208:
3200:
3196:
3192:
3187:
3183:
3176:
3168:
3164:
3158:
3154:
3150:
3145:
3141:
3134:
3131:
3122:
3100:
3093:
3082:XA', XB', XC'
3074:
3065:
3063:
3062:
3058:
3052:
3048:
3037:
3024:
3021:
3013:
3009:
3005:
3000:
2996:
2989:
2981:
2977:
2973:
2968:
2964:
2960:
2955:
2951:
2938:
2928:
2915:
2912:
2904:
2900:
2894:
2890:
2886:
2881:
2877:
2871:
2867:
2860:
2854:
2848:
2845:
2836:
2806:
2799:
2790:
2788:
2787:
2783:
2777:
2774:
2763:
2761:
2755:
2751:
2733:
2720:
2717:
2709:
2705:
2699:
2695:
2691:
2686:
2682:
2676:
2672:
2665:
2657:
2647:
2643:
2639:
2634:
2630:
2623:
2615:
2611:
2607:
2602:
2598:
2589:
2585:
2572:
2562:
2549:
2546:
2538:
2534:
2530:
2525:
2521:
2514:
2508:
2505:
2502:
2496:
2493:
2484:
2469:
2467:
2466:
2462:
2456:
2437:
2415:
2394:
2381:
2378:
2370:
2366:
2360:
2356:
2352:
2347:
2343:
2337:
2333:
2326:
2318:
2314:
2310:
2307:
2302:
2292:
2288:
2284:
2279:
2275:
2268:
2260:
2256:
2252:
2247:
2243:
2234:
2230:
2226:
2214:
2204:
2191:
2188:
2180:
2176:
2172:
2167:
2163:
2156:
2149:
2145:
2142:
2138:
2134:
2131:
2128:
2124:
2120:
2117:
2105:
2077:such that if
2071:
2064:
2058:Darboux cubic
2055:
2053:
2052:
2048:
2042:
2031:
2016:
1997:
1984:
1981:
1973:
1969:
1963:
1959:
1955:
1950:
1946:
1940:
1936:
1929:
1920:
1910:
1897:
1894:
1886:
1882:
1878:
1873:
1869:
1862:
1859:
1856:
1847:
1833:
1829:
1822:
1818:
1808:
1802:Thomson cubic
1799:
1797:
1796:
1792:
1786:
1783:
1776:
1766:
1762:
1761:Fermat points
1758:
1754:
1750:
1745:
1744:respectively
1740:in the lines
1734:
1716:
1709:
1688:
1682:
1661:
1657:
1654:(named after
1653:
1652:Neuberg cubic
1648:
1635:
1632:
1624:
1620:
1614:
1610:
1606:
1601:
1597:
1591:
1587:
1580:
1572:
1568:
1564:
1561:
1556:
1546:
1542:
1538:
1533:
1529:
1522:
1514:
1510:
1506:
1501:
1497:
1488:
1484:
1471:
1461:
1448:
1445:
1437:
1433:
1429:
1424:
1420:
1413:
1406:
1402:
1399:
1395:
1391:
1388:
1385:
1382:
1378:
1374:
1371:
1359:
1303:
1296:
1290:Neuberg cubic
1287:
1274:
1268:
1265:
1259:
1253:
1250:
1244:
1238:
1235:
1229:
1224:
1220:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1153:, and define
1132:
1123:, of a point
1119:, denoted by
1118:
1098:
1095:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1053:
1044:
1036:
1035:
1034:
1017:
1014:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
972:
969:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
927:
924:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
882:
875:
874:
873:
856:
853:
850:
844:
840:
837:
834:
828:
824:
821:
818:
812:
805:
804:
803:
786:
783:
780:
777:
771:
767:
764:
761:
758:
752:
748:
745:
742:
739:
733:
726:
725:
724:
721:
719:
715:
710:
692:
684:
681:
673:
670:
651:
643:
640:
632:
629:
610:
602:
599:
591:
588:
579:
574:Suppose that
567:
565:
561:
557:
553:
548:
546:
538:
534:
526:
522:
519:, defining a
518:
510:
505:
503:
499:
495:
490:
488:
480:
476:
472:
468:
464:
460:
456:
452:
442:
438:
434:
430:
423:
419:
415:
409:
405:
403:
399:
395:
391:
384:pass through
372:. Each point
368:
364:
343:
340:
337:
334:
331:
326:
322:
318:
315:
310:
306:
302:
299:
294:
290:
286:
283:
278:
274:
270:
267:
262:
258:
254:
251:
246:
242:
238:
233:
229:
225:
220:
216:
212:
207:
203:
193:
192:
191:
190:
180:
175:
171:
150:
147:
144:
141:
138:
126:
105:
102:
96:
93:
90:
87:
84:
78:
69:
68:
67:
66:
63:defined by a
59:
55:
51:
39:
33:
19:
5495:Cubic curves
5411:
5388:Prym variety
5362:Stable curve
5352:Hodge bundle
5342:ELSV formula
5144:Fermat curve
5138:
5101:Plane curves
5064:Higher genus
5039:Applications
4964:Modular form
4847:
4803:
4777:
4773:
4763:
4759:
4749:
4745:
4731:
4727:
4717:
4713:
4703:
4699:
4689:
4685:
4660:
4656:
4631:
4627:
4602:
4598:
4573:
4569:
4544:
4540:
4519:
4468:
4464:
4461:
4456:
4452:
4448:
4444:
4440:
4436:
4432:
4428:
4424:
4418:
4266:
4151:
4003:
3992:
3988:
3985:
3982:
3971:
3967:
3936:
3810:
3711:
3692:
3673:
3669:
3666:
3646:
3530:
3428:
3417:
3413:
3410:
3407:
3355:
3351:
3225:
3113:
3098:
3091:
3060:
3056:
3053:
3049:
3038:
2929:
2827:
2820:; the point
2804:
2785:
2781:
2778:
2772:
2764:
2753:
2749:
2734:
2563:
2475:
2464:
2460:
2457:
2438:
2395:
2205:
2096:
2069:
2050:
2046:
2043:
2029:
2017:
1998:
1911:
1838:
1831:
1827:
1820:
1816:
1794:
1790:
1787:
1781:
1774:
1753:circumcenter
1746:
1721:
1714:
1696:
1680:
1649:
1462:
1350:
1301:
1130:
1114:
1032:
871:
801:
722:
708:
705:Relative to
577:
573:
552:double point
549:
506:
494:Isaac Newton
491:
463:non-singular
462:
448:
440:
436:
432:
428:
421:
417:
413:
360:
178:
174:affine space
122:
53:
47:
5417:singularity
5263:Polar curve
4423:indexed as
3384:BC, CA, AB,
3086:BC, CA, CB,
2793:Lucas cubic
2024:BC, CA, AB,
1757:orthocenter
1742:BC, CA, AB,
1662:of a point
718:barycentric
123:applied to
50:mathematics
18:Cubic curve
5258:Dual curve
4886:Topics in
4510:References
3953:for which
2400:such that
2091:AD, BE, CF
2087:BC, CA, AB
2003:such that
1666:such that
1330:BC, CA, AB
469:, over an
467:inflection
5371:Morphisms
5119:Bitangent
4794:123411561
4766:: 135–146
4752:: 161–171
4706:: 107–114
4677:126542269
4648:119886462
4619:122633134
4590:120174967
4561:116778499
4224:−
4186:−
4161:∑
3894:−
3852:−
3820:∑
3778:−
3746:−
3721:∑
3562:−
3540:∑
3466:−
3438:∑
3257:−
3235:∑
3151:−
3123:∑
3006:−
2974:−
2939:∑
2887:−
2849:
2837:∑
2692:−
2640:−
2573:∑
2531:−
2506:−
2497:
2485:∑
2353:−
2308:−
2285:−
2215:∑
2173:−
2146:
2135:
2129:−
2121:
2106:∑
1956:−
1921:∑
1879:−
1848:∑
1658:) is the
1607:−
1562:−
1539:−
1472:∑
1430:−
1403:
1392:
1383:−
1375:
1360:∑
1225:∗
1045:∑
848:→
832:→
816:→
775:→
756:→
737:→
714:trilinear
554:, or one
189:monomials
5489:Category
4476:See also
4455:(1931),
4451:(1453),
3961:, where
2767:A, B, C,
2743:, where
2441:A, B, C,
2408:, where
2020:A, B, C,
2011:, where
1749:incenter
1718:, where
1674:, where
168:for the
5442:Tacnode
5427:Crunode
4848:YouTube
4734:: 1–295
4692:: 51–58
4447:(672),
4443:(365),
4439:(292),
4435:(238),
4431:(105),
3976:in the
2758:in the
2445:A, B, C
2412:is the
2079:D, E, F
1769:A, B, C
1763:, both
1759:, both
1685:in the
560:tacnode
547:field.
543:is the
356:
195:
166:
128:
118:
71:
5422:Acnode
5335:Moduli
4792:
4720:: 1–27
4675:
4646:
4617:
4588:
4559:
4528:
4427:(31),
4165:cyclic
3824:cyclic
3725:cyclic
3544:cyclic
3442:cyclic
3363:, let
3356:A'B'C'
3239:cyclic
3127:cyclic
3092:A'B'C'
3088:where
2943:cyclic
2841:cyclic
2577:cyclic
2489:cyclic
2219:cyclic
2110:cyclic
1925:cyclic
1852:cyclic
1830:(2) –
1476:cyclic
1364:cyclic
1049:cyclic
398:pencil
4790:S2CID
4673:S2CID
4644:S2CID
4615:S2CID
4586:S2CID
4557:S2CID
1660:locus
1212:then
443:– 1))
435:– 1,
367:field
56:is a
5432:Cusp
4526:ISBN
4465:K155
4350:and
3989:K021
3974:(99)
3670:K018
3659:and
3414:K017
3352:Let
3106:and
3057:K007
2782:K005
2461:K004
2047:K002
1791:K001
1683:(30)
1650:The
1342:, CX
1338:, BX
1160:and
716:and
556:cusp
424:+ 1)
52:, a
4782:doi
4732:129
4665:doi
4636:doi
4607:doi
4578:doi
4549:doi
4467:at
3991:at
3980:).
3959:S*X
3693:ABC
3672:at
3653:XX*
3416:at
3400:, X
3396:, X
3373:, X
3369:, X
3099:ABC
3059:at
2846:cos
2805:ABC
2784:at
2773:ABC
2762:).
2756:(5)
2494:cos
2463:at
2143:cos
2132:cos
2118:cos
2070:ABC
2049:at
2030:ABC
1782:ABC
1775:ABC
1715:ABC
1400:cos
1389:cos
1372:cos
1319:, X
1315:, X
1302:ABC
1131:ABC
709:ABC
578:ABC
439:â‹… (
431:↦ (
420:â‹… (
181:= 1
48:In
5491::
4846:.
4788:,
4778:55
4776:,
4762:,
4748:,
4744:,
4730:,
4716:,
4702:,
4688:,
4671:,
4661:68
4659:,
4642:,
4632:66
4630:,
4613:,
4603:53
4601:,
4584:,
4574:66
4572:,
4555:,
4545:63
4543:,
4472:.
4416:.
3996:.
3970:=
3955:X*
3947:X*
3706:X*
3677:.
3661:X*
3421:.
3108:Ω′
3064:.
2822:X'
2818:X'
2789:.
2776:.
2752:=
2741:NX
2737:X*
2468:.
2406:LX
2402:X*
2054:.
2009:GX
2005:X*
1823:′)
1798:.
1785:.
1755:,
1751:,
1672:EX
1668:X*
1334:AX
1169:X*
1147:XA
1136:X*
1121:X*
1018:0.
720:.
416:=
404:.
5414:k
5412:A
4879:e
4872:t
4865:v
4809:.
4797:.
4784::
4768:.
4764:2
4754:.
4750:1
4722:.
4718:3
4708:.
4704:1
4694:.
4690:1
4680:.
4667::
4651:.
4638::
4622:.
4609::
4593:.
4580::
4564:.
4551::
4535:.
4457:X
4453:X
4449:X
4445:X
4441:X
4437:X
4433:X
4429:X
4425:X
4413:Z
4411:X
4406:Y
4404:X
4400:X
4386:.
4383:y
4380::
4377:x
4374::
4371:z
4368:=
4363:Z
4359:X
4338:x
4335::
4332:z
4329::
4326:y
4323:=
4318:Y
4314:X
4293:z
4290::
4287:y
4284::
4281:x
4278:=
4275:X
4253:0
4250:=
4247:)
4242:2
4238:z
4232:3
4228:b
4219:2
4215:y
4209:3
4205:c
4201:(
4198:x
4195:)
4192:c
4189:b
4181:2
4177:a
4173:(
4170:a
4138:)
4135:x
4132:b
4129:+
4126:z
4123:a
4120:(
4117:)
4114:z
4111:a
4108:+
4105:y
4102:c
4099:(
4096:)
4093:y
4090:c
4087:+
4084:x
4081:b
4078:(
4075:=
4072:)
4069:z
4066:b
4063:+
4060:y
4057:a
4054:(
4051:)
4048:y
4045:a
4042:+
4039:x
4036:c
4033:(
4030:)
4027:x
4024:c
4021:+
4018:z
4015:b
4012:(
3972:X
3968:S
3963:S
3951:X
3943:X
3939:X
3923:0
3920:=
3917:)
3912:2
3908:z
3902:2
3898:b
3889:2
3885:y
3879:2
3875:c
3871:(
3868:x
3865:)
3860:2
3856:c
3847:2
3843:b
3839:(
3834:2
3830:a
3797:0
3794:=
3791:)
3786:2
3782:z
3773:2
3769:y
3765:(
3762:x
3759:)
3754:2
3750:c
3741:2
3737:b
3733:(
3730:a
3708:.
3702:X
3698:X
3691:â–ł
3657:X
3649:X
3633:0
3630:=
3627:)
3622:2
3618:z
3612:2
3608:b
3604:+
3599:2
3595:y
3589:2
3585:c
3581:(
3578:x
3575:)
3570:2
3566:c
3557:2
3553:b
3549:(
3517:0
3514:=
3511:)
3506:2
3502:z
3498:+
3493:2
3489:y
3485:(
3482:x
3479:)
3474:2
3470:c
3461:2
3457:b
3453:(
3450:c
3447:b
3402:C
3398:B
3394:A
3392:X
3388:X
3375:C
3371:B
3367:A
3365:X
3361:X
3354:â–ł
3338:0
3335:=
3332:)
3327:2
3323:z
3317:2
3313:b
3309:+
3304:2
3300:y
3294:2
3290:c
3286:(
3283:x
3280:)
3275:2
3271:c
3265:2
3261:b
3252:4
3248:a
3244:(
3212:0
3209:=
3206:)
3201:2
3197:z
3193:+
3188:2
3184:y
3180:(
3177:x
3174:)
3169:2
3165:c
3159:2
3155:b
3146:4
3142:a
3138:(
3135:c
3132:b
3104:Ω
3097:â–ł
3090:â–ł
3078:X
3045:X
3041:X
3025:0
3022:=
3019:)
3014:2
3010:z
3001:2
2997:y
2993:(
2990:x
2987:)
2982:2
2978:a
2969:2
2965:c
2961:+
2956:2
2952:b
2948:(
2916:0
2913:=
2910:)
2905:2
2901:z
2895:2
2891:c
2882:2
2878:y
2872:2
2868:b
2864:(
2861:x
2858:)
2855:A
2852:(
2814:X
2810:X
2803:â–ł
2771:â–ł
2754:X
2750:N
2745:N
2721:0
2718:=
2715:)
2710:2
2706:z
2700:2
2696:b
2687:2
2683:y
2677:2
2673:c
2669:(
2666:x
2663:)
2658:2
2654:)
2648:2
2644:c
2635:2
2631:b
2627:(
2624:+
2621:)
2616:2
2612:c
2608:+
2603:2
2599:b
2595:(
2590:2
2586:a
2582:(
2550:0
2547:=
2544:)
2539:2
2535:z
2526:2
2522:y
2518:(
2515:x
2512:)
2509:C
2503:B
2500:(
2453:P
2449:P
2434:X
2430:X
2426:X
2422:X
2418:X
2410:L
2398:X
2382:0
2379:=
2376:)
2371:2
2367:z
2361:2
2357:b
2348:2
2344:y
2338:2
2334:c
2330:(
2327:x
2324:)
2319:4
2315:a
2311:3
2303:2
2299:)
2293:2
2289:c
2280:2
2276:b
2272:(
2269:+
2266:)
2261:2
2257:c
2253:+
2248:2
2244:b
2240:(
2235:2
2231:a
2227:2
2224:(
2192:0
2189:=
2186:)
2181:2
2177:z
2168:2
2164:y
2160:(
2157:x
2154:)
2150:C
2139:B
2125:A
2115:(
2083:X
2075:X
2068:â–ł
2039:P
2035:P
2028:â–ł
2013:G
2001:X
1985:0
1982:=
1979:)
1974:2
1970:z
1964:2
1960:b
1951:2
1947:y
1941:2
1937:c
1933:(
1930:x
1898:0
1895:=
1892:)
1887:2
1883:z
1874:2
1870:y
1866:(
1863:x
1860:c
1857:b
1835:.
1832:X
1828:X
1821:X
1819:(
1817:X
1812:X
1780:â–ł
1773:â–ł
1738:X
1732:C
1730:X
1728:B
1726:X
1724:A
1722:X
1720:â–ł
1713:â–ł
1707:C
1705:X
1703:B
1701:X
1699:A
1697:X
1695:â–ł
1691:X
1681:X
1676:E
1664:X
1636:0
1633:=
1630:)
1625:2
1621:z
1615:2
1611:b
1602:2
1598:y
1592:2
1588:c
1584:(
1581:x
1578:)
1573:4
1569:a
1565:2
1557:2
1553:)
1547:2
1543:c
1534:2
1530:b
1526:(
1523:+
1520:)
1515:2
1511:c
1507:+
1502:2
1498:b
1494:(
1489:2
1485:a
1481:(
1449:0
1446:=
1443:)
1438:2
1434:z
1425:2
1421:y
1417:(
1414:x
1411:)
1407:C
1396:B
1386:2
1379:A
1369:(
1344:C
1340:B
1336:A
1326:X
1321:C
1317:B
1313:A
1311:X
1307:X
1300:â–ł
1275:.
1269:z
1266:1
1260::
1254:y
1251:1
1245::
1239:x
1236:1
1230:=
1221:X
1200:,
1197:z
1194::
1191:y
1188::
1185:x
1182:=
1179:X
1164:C
1162:L
1157:B
1155:L
1151:A
1142:A
1140:L
1129:â–ł
1125:X
1111:.
1099:0
1096:=
1093:)
1090:c
1087:,
1084:b
1081:,
1078:a
1075:,
1072:z
1069:,
1066:y
1063:,
1060:x
1057:(
1054:f
1015:=
1012:)
1009:y
1006:,
1003:x
1000:,
997:z
994:,
991:b
988:,
985:a
982:,
979:c
976:(
973:f
970:+
967:)
964:x
961:,
958:z
955:,
952:y
949:,
946:a
943:,
940:c
937:,
934:b
931:(
928:f
925:+
922:)
919:z
916:,
913:y
910:,
907:x
904:,
901:c
898:,
895:b
892:,
889:a
886:(
883:f
857:.
854:z
851:c
845:z
841:,
838:y
835:b
829:y
825:,
822:x
819:a
813:x
787:;
784:z
781:b
778:a
772:z
768:,
765:y
762:a
759:c
753:y
749:,
746:x
743:c
740:b
734:x
707:â–ł
693:.
689:|
685:B
682:A
678:|
674:=
671:c
652:,
648:|
644:A
641:C
637:|
633:=
630:b
611:,
607:|
603:C
600:B
596:|
592:=
589:a
576:â–ł
541:K
531:-
529:K
513:K
483:C
445:.
441:t
437:t
433:t
429:t
422:x
418:x
414:y
386:P
382:C
378:F
374:P
370:K
344:z
341:y
338:x
335:,
332:y
327:2
323:z
319:,
316:x
311:2
307:z
303:,
300:z
295:2
291:y
287:,
284:x
279:2
275:y
271:,
268:z
263:2
259:x
255:,
252:y
247:2
243:x
239:,
234:3
230:z
226:,
221:3
217:y
213:,
208:3
204:x
185:F
179:z
154:)
151:z
148::
145:y
142::
139:x
136:(
106:0
103:=
100:)
97:z
94:,
91:y
88:,
85:x
82:(
79:F
61:C
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
