722:
707:
1592:
2084:
681:
669:
621:
693:
645:
1657:
633:
657:
2079:{\displaystyle {\begin{aligned}AC^{2}BC^{2}-CD^{2}AC^{2}-CD^{2}BC^{2}&=0\\AC^{2}BC^{2}&=CD^{2}BC^{2}+CD^{2}AC^{2}\\{\frac {1}{CD^{2}}}&={\frac {BC^{2}}{AC^{2}\cdot BC^{2}}}+{\frac {AC^{2}}{AC^{2}\cdot BC^{2}}}\\\therefore \;\;{\frac {1}{CD^{2}}}&={\frac {1}{AC^{2}}}+{\frac {1}{BC^{2}}}\end{aligned}}}
324:
not equal to zero. This equation has 15 constants. However, it can be multiplied by any non-zero constant without changing the curve; thus by the choice of an appropriate constant of multiplication, any one of the coefficients can be set to 1, leaving only 14 constants. Therefore, the space of
315:
1583:
2564:
2329:
884:
1144:
2672:
2453:
1008:
1353:
1662:
2222:
63:
1233:
2739:
363:
473:
444:
680:
668:
1449:
376:
2468:
2812:
368:
2233:
2978:
2822:
721:
706:
753:
1035:
2586:
2353:
911:
493:
Various combinations of coefficients in the above equation give rise to various important families of curves as listed below.
1153:
determines the size of the curve. The bicuspid has only the two cusps as singularities, and hence is a curve of genus one.
1268:
2133:
588:
1600:
310:{\displaystyle Ax^{4}+By^{4}+Cx^{3}y+Dx^{2}y^{2}+Exy^{3}+Fx^{3}+Gy^{3}+Hx^{2}y+Ixy^{2}+Jx^{2}+Ky^{2}+Lxy+Mx+Ny+P=0,}
2101:
1014:
2997:
2766:
556:
1370:
determining the shape of the curve. The cruciform curve is related by a standard quadratic transformation,
1178:
2699:
50:
333:
1405:
713:
326:
46:
561:
620:
2569:
where the two appearances of ± are independent of each other, giving up to four distinct values of
1588:
parametrizes the points on the curve outside of the exceptional cases where a denominator is zero.
890:
692:
644:
453:
424:
404:
2752:= 3. This curve has a triple point at the origin (0, 0) and has three double tangents.
537:
408:
38:
2974:
2946:
2921:
2896:
2871:
2846:
2818:
2787:
1591:
1435:
Because the curve is rational, it can be parametrized by rational functions. For instance, if
728:
573:
507:
371:, that there is exactly one quartic curve that passes through a set of 14 distinct points in
527:
482:
372:
54:
31:
2832:
2828:
2761:
1401:
632:
416:
412:
686:
Cruciform curve with parameters (b,a) being (1,1) in red; (2,1) in green; (3,1) in blue.
674:
Cruciform curve with parameters (b,a) being (1,1) in red; (2,2) in green; (3,3) in blue.
2095:
698:
578:
512:
1578:{\displaystyle x=-{\frac {t^{2}-2t+5}{t^{2}-2t-3}},\quad y={\frac {t^{2}-2t+5}{2t-2}}}
656:
2991:
2113:
595:
542:
522:
478:
2790:
2874:
2121:
517:
396:
2559:{\displaystyle y=\pm {\sqrt {\frac {-2x^{2}-3x\pm {\sqrt {16x^{3}+9x^{2}}}}{2}}},}
600:
2949:
2924:
2849:
2745:
568:
551:
2954:
2929:
2904:
2879:
2854:
2795:
2117:
1367:
1162:
532:
390:
2899:
583:
17:
2324:{\displaystyle r^{4}=dr^{2}\cos ^{2}\theta +er^{2}\sin ^{2}\theta +f.\,}
1595:
Illustration of the inverse
Pythagorean and regular Pythagorean theorems
2971:
Elementary
Geometry of Algebraic Curves, an Undergraduate Introduction
502:
1590:
1404:
of genus zero. The cruciform curve has three double points in the
893:
zero, with three ordinary double points, all in the real plane.
879:{\displaystyle \ (y^{2}-x^{2})(x-1)(2x-3)=4(x^{2}+y^{2}-2x)^{2}.}
1139:{\displaystyle (x^{2}-a^{2})(x-a)^{2}+(y^{2}-a^{2})^{2}=0\,}
496:
1246:=0, and consequently is a rational curve, with genus zero.
2667:{\displaystyle x=\cos(3t)\cos t,\quad y=\cos(3t)\sin t.\,}
2448:{\displaystyle x^{4}+2x^{2}y^{2}+y^{4}-x^{3}+3xy^{2}=0.\,}
2124:. The name is from σπειρα meaning torus in ancient Greek.
2462:, the curve can be described by the following function:
1639:
are the endpoints of the hypotenuse of a right triangle
1003:{\displaystyle x^{4}+x^{2}y^{2}+y^{4}=x(x^{2}+y^{2}).\,}
27:
Plane algebraic curve defined by a 4th-degree polynomial
2104:
quartic curves that are symmetric with respect to the
1161:"Bow curve" redirects here. For the railway line, see
2817:(2nd ed.), Clarendon Press, Oxford, p. 72,
2702:
2589:
2471:
2356:
2236:
2136:
1660:
1452:
1271:
1181:
1038:
914:
756:
456:
427:
336:
66:
2112:
axes. Spiric sections are included in the family of
1651:, the vertex of the right angle, to the hypotenuse:
1603:
is obtained from the above equation by substituting
1348:{\displaystyle x^{2}y^{2}-b^{2}x^{2}-a^{2}y^{2}=0\,}
2733:
2666:
2558:
2447:
2323:
2217:{\displaystyle (x^{2}+y^{2})^{2}=dx^{2}+ey^{2}+f,}
2216:
2078:
1577:
1347:
1227:
1138:
1002:
878:
467:
438:
357:
309:
747:is a quartic plane curve given by the equation:
403:One may also consider quartic curves over other
2973:, Cambridge University Press, Cambridge, 2001,
1262:is a quartic plane curve given by the equation
481:. Additionally, one can look at curves in the
8:
1647:is the foot of a perpendicular dropped from
1172:is a quartic plane curve with the equation:
905:is a quartic plane curve with the equation:
1238:The bow curve has a single triple point at
1029:is a quartic plane curve with the equation
2811:Cundy, H. Martyn; Rollett, A. P. (1961) ,
1995:
1994:
1013:The bean curve has genus zero. It has one
325:quartic curves can be identified with the
2730:
2701:
2663:
2588:
2538:
2522:
2513:
2495:
2481:
2470:
2444:
2432:
2413:
2400:
2387:
2377:
2361:
2355:
2320:
2299:
2289:
2267:
2257:
2241:
2235:
2227:and the equation in polar coordinates as
2199:
2183:
2167:
2157:
2144:
2135:
2127:The Cartesian equation can be written as
2063:
2050:
2038:
2025:
2009:
1996:
1978:
1962:
1947:
1937:
1925:
1909:
1894:
1884:
1868:
1855:
1845:
1832:
1816:
1803:
1783:
1770:
1743:
1730:
1714:
1701:
1685:
1672:
1661:
1659:
1537:
1530:
1496:
1469:
1462:
1451:
1344:
1332:
1322:
1309:
1299:
1286:
1276:
1270:
1224:
1215:
1199:
1186:
1180:
1135:
1123:
1113:
1100:
1084:
1059:
1046:
1037:
1017:at the origin, an ordinary triple point.
999:
987:
974:
955:
942:
932:
919:
913:
867:
848:
835:
780:
767:
755:
458:
457:
455:
429:
428:
426:
419:, which are one-dimensional objects over
346:
342:
339:
338:
335:
256:
240:
224:
202:
186:
170:
154:
135:
125:
106:
90:
74:
65:
2778:
616:
382:A quartic curve can have a maximum of:
1228:{\displaystyle x^{4}=x^{2}y-y^{3}.\,}
7:
2734:{\displaystyle r=\cos(3\varphi ).\,}
2580:The parametric equation of curve is
485:, given by homogeneous polynomials.
369:Cramer's theorem on algebraic curves
358:{\displaystyle \mathbb {RP} ^{14}.}
53:. It can be defined by a bivariate
2693: sin φ) the equation is
2100:Spiric sections can be defined as
25:
720:
705:
691:
679:
667:
655:
643:
631:
619:
2626:
2335:Three-leaved clover (trifolium)
1523:
2724:
2715:
2648:
2639:
2611:
2602:
2164:
2137:
1402:rational plane algebraic curve
1120:
1093:
1081:
1068:
1065:
1039:
993:
967:
864:
828:
819:
804:
801:
789:
786:
760:
1:
468:{\displaystyle \mathbb {R} .}
448:but are two-dimensional over
439:{\displaystyle \mathbb {C} ,}
30:For the univariate case, see
2347:is the quartic plane curve
1601:inverse Pythagorean theorem
3014:
2116:and include the family of
2093:
1160:
29:
386:Four connected components
375:, since a quartic has 14
2744:It is a special case of
1400:= 1, and is therefore a
415:. In this way, one gets
2767:Bitangents of a quartic
727:Three-leaved clover in
712:Three-leaved clover in
557:Lemniscate of Bernoulli
2735:
2677:In polar coordinates (
2668:
2560:
2449:
2325:
2218:
2080:
1596:
1579:
1349:
1229:
1140:
1004:
880:
589:Lamé's special quartic
469:
440:
367:It also follows, from
359:
311:
2736:
2669:
2561:
2450:
2326:
2219:
2081:
1594:
1580:
1406:real projective plane
1350:
1230:
1141:
1005:
881:
714:Cartesian coordinates
470:
441:
360:
327:real projective space
320:with at least one of
312:
47:plane algebraic curve
2700:
2587:
2469:
2354:
2234:
2134:
1658:
1450:
1269:
1179:
1036:
912:
754:
562:Lemniscate of Gerono
454:
425:
411:), for instance the
334:
64:
2814:Mathematical models
2341:three-leaved clover
43:quartic plane curve
2981:. Pages 12 and 78.
2947:Weisstein, Eric W.
2922:Weisstein, Eric W.
2897:Weisstein, Eric W.
2872:Weisstein, Eric W.
2847:Weisstein, Eric W.
2788:Weisstein, Eric W.
2731:
2685: cos φ,
2664:
2556:
2445:
2321:
2214:
2120:and the family of
2076:
2074:
1597:
1575:
1345:
1225:
1136:
1000:
876:
538:Kampyle of Eudoxus
477:An example is the
465:
436:
377:degrees of freedom
355:
307:
39:algebraic geometry
2979:978-0-521-64641-3
2925:"Cruciform curve"
2824:978-0-906212-20-2
2791:"Ampersand Curve"
2551:
2550:
2544:
2070:
2045:
2016:
1985:
1932:
1875:
1573:
1518:
759:
729:polar coordinates
611:
610:
508:Bullet-nose curve
16:(Redirected from
3005:
2982:
2967:
2961:
2960:
2959:
2942:
2936:
2935:
2934:
2917:
2911:
2910:
2909:
2892:
2886:
2885:
2884:
2875:"Bicuspid Curve"
2867:
2861:
2860:
2859:
2842:
2836:
2835:
2808:
2802:
2801:
2800:
2783:
2740:
2738:
2737:
2732:
2673:
2671:
2670:
2665:
2565:
2563:
2562:
2557:
2552:
2546:
2545:
2543:
2542:
2527:
2526:
2514:
2500:
2499:
2483:
2482:
2454:
2452:
2451:
2446:
2437:
2436:
2418:
2417:
2405:
2404:
2392:
2391:
2382:
2381:
2366:
2365:
2330:
2328:
2327:
2322:
2304:
2303:
2294:
2293:
2272:
2271:
2262:
2261:
2246:
2245:
2223:
2221:
2220:
2215:
2204:
2203:
2188:
2187:
2172:
2171:
2162:
2161:
2149:
2148:
2085:
2083:
2082:
2077:
2075:
2071:
2069:
2068:
2067:
2051:
2046:
2044:
2043:
2042:
2026:
2017:
2015:
2014:
2013:
1997:
1986:
1984:
1983:
1982:
1967:
1966:
1953:
1952:
1951:
1938:
1933:
1931:
1930:
1929:
1914:
1913:
1900:
1899:
1898:
1885:
1876:
1874:
1873:
1872:
1856:
1850:
1849:
1837:
1836:
1821:
1820:
1808:
1807:
1788:
1787:
1775:
1774:
1748:
1747:
1735:
1734:
1719:
1718:
1706:
1705:
1690:
1689:
1677:
1676:
1584:
1582:
1581:
1576:
1574:
1572:
1558:
1542:
1541:
1531:
1519:
1517:
1501:
1500:
1490:
1474:
1473:
1463:
1354:
1352:
1351:
1346:
1337:
1336:
1327:
1326:
1314:
1313:
1304:
1303:
1291:
1290:
1281:
1280:
1234:
1232:
1231:
1226:
1220:
1219:
1204:
1203:
1191:
1190:
1145:
1143:
1142:
1137:
1128:
1127:
1118:
1117:
1105:
1104:
1089:
1088:
1064:
1063:
1051:
1050:
1009:
1007:
1006:
1001:
992:
991:
979:
978:
960:
959:
947:
946:
937:
936:
924:
923:
885:
883:
882:
877:
872:
871:
853:
852:
840:
839:
785:
784:
772:
771:
757:
724:
709:
695:
683:
671:
659:
647:
635:
623:
497:
483:projective plane
476:
474:
472:
471:
466:
461:
447:
445:
443:
442:
437:
432:
417:Riemann surfaces
373:general position
366:
364:
362:
361:
356:
351:
350:
345:
323:
316:
314:
313:
308:
261:
260:
245:
244:
229:
228:
207:
206:
191:
190:
175:
174:
159:
158:
140:
139:
130:
129:
111:
110:
95:
94:
79:
78:
55:quartic equation
32:Quartic function
21:
3013:
3012:
3008:
3007:
3006:
3004:
3003:
3002:
2988:
2987:
2986:
2985:
2969:Gibson, C. G.,
2968:
2964:
2945:
2944:
2943:
2939:
2920:
2919:
2918:
2914:
2895:
2894:
2893:
2889:
2870:
2869:
2868:
2864:
2845:
2844:
2843:
2839:
2825:
2810:
2809:
2805:
2786:
2785:
2784:
2780:
2775:
2762:Ternary quartic
2758:
2698:
2697:
2585:
2584:
2534:
2518:
2491:
2484:
2467:
2466:
2458:By solving for
2428:
2409:
2396:
2383:
2373:
2357:
2352:
2351:
2337:
2295:
2285:
2263:
2253:
2237:
2232:
2231:
2195:
2179:
2163:
2153:
2140:
2132:
2131:
2098:
2092:
2073:
2072:
2059:
2055:
2034:
2030:
2018:
2005:
2001:
1988:
1987:
1974:
1958:
1954:
1943:
1939:
1921:
1905:
1901:
1890:
1886:
1877:
1864:
1860:
1852:
1851:
1841:
1828:
1812:
1799:
1789:
1779:
1766:
1760:
1759:
1749:
1739:
1726:
1710:
1697:
1681:
1668:
1656:
1655:
1559:
1533:
1532:
1492:
1491:
1465:
1464:
1448:
1447:
1386:to the ellipse
1328:
1318:
1305:
1295:
1282:
1272:
1267:
1266:
1256:cruciform curve
1252:
1250:Cruciform curve
1211:
1195:
1182:
1177:
1176:
1166:
1159:
1119:
1109:
1096:
1080:
1055:
1042:
1034:
1033:
1023:
983:
970:
951:
938:
928:
915:
910:
909:
899:
863:
844:
831:
776:
763:
752:
751:
745:ampersand curve
741:
739:Ampersand curve
736:
735:
734:
731:
725:
716:
710:
701:
696:
687:
684:
675:
672:
663:
660:
651:
648:
639:
636:
627:
626:Ampersand curve
624:
612:
491:
452:
451:
449:
423:
422:
420:
413:complex numbers
395:Three ordinary
337:
332:
331:
329:
321:
252:
236:
220:
198:
182:
166:
150:
131:
121:
102:
86:
70:
62:
61:
35:
28:
23:
22:
15:
12:
11:
5:
3011:
3009:
3001:
3000:
2998:Quartic curves
2990:
2989:
2984:
2983:
2962:
2937:
2912:
2887:
2862:
2837:
2823:
2803:
2777:
2776:
2774:
2771:
2770:
2769:
2764:
2757:
2754:
2742:
2741:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2675:
2674:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2625:
2622:
2619:
2616:
2613:
2610:
2607:
2604:
2601:
2598:
2595:
2592:
2567:
2566:
2555:
2549:
2541:
2537:
2533:
2530:
2525:
2521:
2517:
2512:
2509:
2506:
2503:
2498:
2494:
2490:
2487:
2480:
2477:
2474:
2456:
2455:
2443:
2440:
2435:
2431:
2427:
2424:
2421:
2416:
2412:
2408:
2403:
2399:
2395:
2390:
2386:
2380:
2376:
2372:
2369:
2364:
2360:
2336:
2333:
2332:
2331:
2319:
2316:
2313:
2310:
2307:
2302:
2298:
2292:
2288:
2284:
2281:
2278:
2275:
2270:
2266:
2260:
2256:
2252:
2249:
2244:
2240:
2225:
2224:
2213:
2210:
2207:
2202:
2198:
2194:
2191:
2186:
2182:
2178:
2175:
2170:
2166:
2160:
2156:
2152:
2147:
2143:
2139:
2114:toric sections
2096:Spiric section
2094:Main article:
2091:
2090:Spiric section
2088:
2087:
2086:
2066:
2062:
2058:
2054:
2049:
2041:
2037:
2033:
2029:
2024:
2021:
2019:
2012:
2008:
2004:
2000:
1993:
1990:
1989:
1981:
1977:
1973:
1970:
1965:
1961:
1957:
1950:
1946:
1942:
1936:
1928:
1924:
1920:
1917:
1912:
1908:
1904:
1897:
1893:
1889:
1883:
1880:
1878:
1871:
1867:
1863:
1859:
1854:
1853:
1848:
1844:
1840:
1835:
1831:
1827:
1824:
1819:
1815:
1811:
1806:
1802:
1798:
1795:
1792:
1790:
1786:
1782:
1778:
1773:
1769:
1765:
1762:
1761:
1758:
1755:
1752:
1750:
1746:
1742:
1738:
1733:
1729:
1725:
1722:
1717:
1713:
1709:
1704:
1700:
1696:
1693:
1688:
1684:
1680:
1675:
1671:
1667:
1664:
1663:
1586:
1585:
1571:
1568:
1565:
1562:
1557:
1554:
1551:
1548:
1545:
1540:
1536:
1529:
1526:
1522:
1516:
1513:
1510:
1507:
1504:
1499:
1495:
1489:
1486:
1483:
1480:
1477:
1472:
1468:
1461:
1458:
1455:
1356:
1355:
1343:
1340:
1335:
1331:
1325:
1321:
1317:
1312:
1308:
1302:
1298:
1294:
1289:
1285:
1279:
1275:
1251:
1248:
1236:
1235:
1223:
1218:
1214:
1210:
1207:
1202:
1198:
1194:
1189:
1185:
1158:
1155:
1147:
1146:
1134:
1131:
1126:
1122:
1116:
1112:
1108:
1103:
1099:
1095:
1092:
1087:
1083:
1079:
1076:
1073:
1070:
1067:
1062:
1058:
1054:
1049:
1045:
1041:
1022:
1021:Bicuspid curve
1019:
1011:
1010:
998:
995:
990:
986:
982:
977:
973:
969:
966:
963:
958:
954:
950:
945:
941:
935:
931:
927:
922:
918:
898:
895:
887:
886:
875:
870:
866:
862:
859:
856:
851:
847:
843:
838:
834:
830:
827:
824:
821:
818:
815:
812:
809:
806:
803:
800:
797:
794:
791:
788:
783:
779:
775:
770:
766:
762:
740:
737:
733:
732:
726:
719:
717:
711:
704:
702:
699:Spiric section
697:
690:
688:
685:
678:
676:
673:
666:
664:
661:
654:
652:
650:Bicuspid curve
649:
642:
640:
637:
630:
628:
625:
618:
615:
614:
613:
609:
608:
604:
603:
598:
593:
592:
591:
581:
579:Spiric section
576:
574:Lüroth quartic
571:
566:
565:
564:
559:
547:
546:
545:
540:
535:
530:
525:
520:
515:
513:Cartesian oval
510:
505:
495:
490:
487:
464:
460:
435:
431:
401:
400:
393:
387:
354:
349:
344:
341:
318:
317:
306:
303:
300:
297:
294:
291:
288:
285:
282:
279:
276:
273:
270:
267:
264:
259:
255:
251:
248:
243:
239:
235:
232:
227:
223:
219:
216:
213:
210:
205:
201:
197:
194:
189:
185:
181:
178:
173:
169:
165:
162:
157:
153:
149:
146:
143:
138:
134:
128:
124:
120:
117:
114:
109:
105:
101:
98:
93:
89:
85:
82:
77:
73:
69:
49:of the fourth
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3010:
2999:
2996:
2995:
2993:
2980:
2976:
2972:
2966:
2963:
2957:
2956:
2951:
2948:
2941:
2938:
2932:
2931:
2926:
2923:
2916:
2913:
2907:
2906:
2901:
2898:
2891:
2888:
2882:
2881:
2876:
2873:
2866:
2863:
2857:
2856:
2851:
2848:
2841:
2838:
2834:
2830:
2826:
2820:
2816:
2815:
2807:
2804:
2798:
2797:
2792:
2789:
2782:
2779:
2772:
2768:
2765:
2763:
2760:
2759:
2755:
2753:
2751:
2747:
2727:
2721:
2718:
2712:
2709:
2706:
2703:
2696:
2695:
2694:
2692:
2688:
2684:
2680:
2660:
2657:
2654:
2651:
2645:
2642:
2636:
2633:
2630:
2627:
2623:
2620:
2617:
2614:
2608:
2605:
2599:
2596:
2593:
2590:
2583:
2582:
2581:
2578:
2576:
2572:
2553:
2547:
2539:
2535:
2531:
2528:
2523:
2519:
2515:
2510:
2507:
2504:
2501:
2496:
2492:
2488:
2485:
2478:
2475:
2472:
2465:
2464:
2463:
2461:
2441:
2438:
2433:
2429:
2425:
2422:
2419:
2414:
2410:
2406:
2401:
2397:
2393:
2388:
2384:
2378:
2374:
2370:
2367:
2362:
2358:
2350:
2349:
2348:
2346:
2342:
2334:
2317:
2314:
2311:
2308:
2305:
2300:
2296:
2290:
2286:
2282:
2279:
2276:
2273:
2268:
2264:
2258:
2254:
2250:
2247:
2242:
2238:
2230:
2229:
2228:
2211:
2208:
2205:
2200:
2196:
2192:
2189:
2184:
2180:
2176:
2173:
2168:
2158:
2154:
2150:
2145:
2141:
2130:
2129:
2128:
2125:
2123:
2122:Cassini ovals
2119:
2115:
2111:
2107:
2103:
2097:
2089:
2064:
2060:
2056:
2052:
2047:
2039:
2035:
2031:
2027:
2022:
2020:
2010:
2006:
2002:
1998:
1991:
1979:
1975:
1971:
1968:
1963:
1959:
1955:
1948:
1944:
1940:
1934:
1926:
1922:
1918:
1915:
1910:
1906:
1902:
1895:
1891:
1887:
1881:
1879:
1869:
1865:
1861:
1857:
1846:
1842:
1838:
1833:
1829:
1825:
1822:
1817:
1813:
1809:
1804:
1800:
1796:
1793:
1791:
1784:
1780:
1776:
1771:
1767:
1763:
1756:
1753:
1751:
1744:
1740:
1736:
1731:
1727:
1723:
1720:
1715:
1711:
1707:
1702:
1698:
1694:
1691:
1686:
1682:
1678:
1673:
1669:
1665:
1654:
1653:
1652:
1650:
1646:
1642:
1638:
1634:
1630:
1626:
1622:
1618:
1614:
1610:
1606:
1602:
1593:
1589:
1569:
1566:
1563:
1560:
1555:
1552:
1549:
1546:
1543:
1538:
1534:
1527:
1524:
1520:
1514:
1511:
1508:
1505:
1502:
1497:
1493:
1487:
1484:
1481:
1478:
1475:
1470:
1466:
1459:
1456:
1453:
1446:
1445:
1444:
1442:
1438:
1433:
1431:
1427:
1423:
1419:
1415:
1411:
1407:
1403:
1399:
1396:
1392:
1389:
1385:
1381:
1377:
1373:
1369:
1365:
1361:
1341:
1338:
1333:
1329:
1323:
1319:
1315:
1310:
1306:
1300:
1296:
1292:
1287:
1283:
1277:
1273:
1265:
1264:
1263:
1261:
1257:
1249:
1247:
1245:
1241:
1221:
1216:
1212:
1208:
1205:
1200:
1196:
1192:
1187:
1183:
1175:
1174:
1173:
1171:
1164:
1156:
1154:
1152:
1132:
1129:
1124:
1114:
1110:
1106:
1101:
1097:
1090:
1085:
1077:
1074:
1071:
1060:
1056:
1052:
1047:
1043:
1032:
1031:
1030:
1028:
1020:
1018:
1016:
996:
988:
984:
980:
975:
971:
964:
961:
956:
952:
948:
943:
939:
933:
929:
925:
920:
916:
908:
907:
906:
904:
896:
894:
892:
873:
868:
860:
857:
854:
849:
845:
841:
836:
832:
825:
822:
816:
813:
810:
807:
798:
795:
792:
781:
777:
773:
768:
764:
750:
749:
748:
746:
738:
730:
723:
718:
715:
708:
703:
700:
694:
689:
682:
677:
670:
665:
658:
653:
646:
641:
634:
629:
622:
617:
607:
602:
599:
597:
596:Toric section
594:
590:
587:
586:
585:
582:
580:
577:
575:
572:
570:
567:
563:
560:
558:
555:
554:
553:
550:
549:
548:
544:
543:Klein quartic
541:
539:
536:
534:
531:
529:
528:Devil's curve
526:
524:
523:Deltoid curve
521:
519:
516:
514:
511:
509:
506:
504:
501:
500:
499:
498:
494:
488:
486:
484:
480:
479:Klein quartic
462:
433:
418:
414:
410:
406:
398:
397:double points
394:
392:
389:Twenty-eight
388:
385:
384:
383:
380:
378:
374:
370:
352:
347:
328:
322:A, B, C, D, E
304:
301:
298:
295:
292:
289:
286:
283:
280:
277:
274:
271:
268:
265:
262:
257:
253:
249:
246:
241:
237:
233:
230:
225:
221:
217:
214:
211:
208:
203:
199:
195:
192:
187:
183:
179:
176:
171:
167:
163:
160:
155:
151:
147:
144:
141:
136:
132:
126:
122:
118:
115:
112:
107:
103:
99:
96:
91:
87:
83:
80:
75:
71:
67:
60:
59:
58:
56:
52:
48:
44:
40:
33:
19:
2970:
2965:
2953:
2940:
2928:
2915:
2903:
2890:
2878:
2865:
2853:
2850:"Bean Curve"
2840:
2813:
2806:
2794:
2781:
2749:
2743:
2690:
2686:
2682:
2678:
2676:
2579:
2574:
2570:
2568:
2459:
2457:
2344:
2340:
2338:
2226:
2126:
2109:
2105:
2099:
1648:
1644:
1640:
1636:
1632:
1628:
1624:
1620:
1616:
1612:
1608:
1604:
1598:
1587:
1440:
1436:
1434:
1429:
1425:
1421:
1417:
1413:
1409:
1397:
1394:
1390:
1387:
1383:
1379:
1375:
1371:
1363:
1359:
1357:
1259:
1255:
1253:
1243:
1239:
1237:
1169:
1167:
1150:
1148:
1026:
1024:
1012:
902:
900:
888:
744:
742:
605:
518:Cassini oval
492:
402:
381:
319:
42:
36:
2950:"Trifolium"
1619:, and each
1260:cross curve
1015:singularity
601:Trott curve
391:bi-tangents
2773:References
2746:rose curve
2118:hippopedes
2102:bicircular
1368:parameters
903:bean curve
897:Bean curve
638:Bean curve
552:Lemniscate
2955:MathWorld
2930:MathWorld
2905:MathWorld
2880:MathWorld
2855:MathWorld
2796:MathWorld
2722:φ
2713:
2655:
2637:
2618:
2600:
2573:for each
2511:±
2502:−
2486:−
2479:±
2407:−
2345:trifolium
2309:θ
2306:
2277:θ
2274:
1992:∴
1969:⋅
1916:⋅
1721:−
1692:−
1567:−
1544:−
1512:−
1503:−
1476:−
1460:−
1443:=2, then
1316:−
1293:−
1209:−
1170:bow curve
1163:Bow Curve
1157:Bow curve
1107:−
1075:−
1053:−
855:−
814:−
796:−
774:−
662:Bow curve
533:Hippopede
407:(or even
18:Bow curve
2992:Category
2756:See also
1631:, where
1424:=0, and
1366:are two
1027:bicuspid
584:Squircle
489:Examples
2833:0124167
1439:=1 and
1428:=0 and
1420:=0 and
1412:=0 and
889:It has
569:Limaçon
475:
450:
446:
421:
365:
330:
2977:
2831:
2821:
1643:, and
1358:where
1149:where
758:
606:
503:Bicorn
405:fields
51:degree
2900:"Bow"
2748:with
1627:with
1615:with
1607:with
1432:=0.
1408:, at
1258:, or
891:genus
409:rings
45:is a
2975:ISBN
2819:ISBN
2339:The
2108:and
1623:and
1599:The
1416:=0,
1382:↦ 1/
1374:↦ 1/
1362:and
1254:The
1242:=0,
1168:The
1025:The
901:The
743:The
41:, a
2710:cos
2652:sin
2634:cos
2615:cos
2597:cos
2343:or
2297:sin
2265:cos
1641:ABC
37:In
2994::
2952:.
2927:.
2902:.
2877:.
2852:.
2829:MR
2827:,
2793:.
2689:=
2681:=
2577:.
2516:16
2442:0.
1635:,
1629:CD
1617:BC
1611:,
1609:AC
1393:+
1378:,
379:.
348:14
57::
2958:.
2933:.
2908:.
2883:.
2858:.
2799:.
2750:k
2728:.
2725:)
2719:3
2716:(
2707:=
2704:r
2691:r
2687:y
2683:r
2679:x
2661:.
2658:t
2649:)
2646:t
2643:3
2640:(
2631:=
2628:y
2624:,
2621:t
2612:)
2609:t
2606:3
2603:(
2594:=
2591:x
2575:x
2571:y
2554:,
2548:2
2540:2
2536:x
2532:9
2529:+
2524:3
2520:x
2508:x
2505:3
2497:2
2493:x
2489:2
2476:=
2473:y
2460:y
2439:=
2434:2
2430:y
2426:x
2423:3
2420:+
2415:3
2411:x
2402:4
2398:y
2394:+
2389:2
2385:y
2379:2
2375:x
2371:2
2368:+
2363:4
2359:x
2318:.
2315:f
2312:+
2301:2
2291:2
2287:r
2283:e
2280:+
2269:2
2259:2
2255:r
2251:d
2248:=
2243:4
2239:r
2212:,
2209:f
2206:+
2201:2
2197:y
2193:e
2190:+
2185:2
2181:x
2177:d
2174:=
2169:2
2165:)
2159:2
2155:y
2151:+
2146:2
2142:x
2138:(
2110:y
2106:x
2065:2
2061:C
2057:B
2053:1
2048:+
2040:2
2036:C
2032:A
2028:1
2023:=
2011:2
2007:D
2003:C
1999:1
1980:2
1976:C
1972:B
1964:2
1960:C
1956:A
1949:2
1945:C
1941:A
1935:+
1927:2
1923:C
1919:B
1911:2
1907:C
1903:A
1896:2
1892:C
1888:B
1882:=
1870:2
1866:D
1862:C
1858:1
1847:2
1843:C
1839:A
1834:2
1830:D
1826:C
1823:+
1818:2
1814:C
1810:B
1805:2
1801:D
1797:C
1794:=
1785:2
1781:C
1777:B
1772:2
1768:C
1764:A
1757:0
1754:=
1745:2
1741:C
1737:B
1732:2
1728:D
1724:C
1716:2
1712:C
1708:A
1703:2
1699:D
1695:C
1687:2
1683:C
1679:B
1674:2
1670:C
1666:A
1649:C
1645:D
1637:B
1633:A
1625:b
1621:a
1613:y
1605:x
1570:2
1564:t
1561:2
1556:5
1553:+
1550:t
1547:2
1539:2
1535:t
1528:=
1525:y
1521:,
1515:3
1509:t
1506:2
1498:2
1494:t
1488:5
1485:+
1482:t
1479:2
1471:2
1467:t
1457:=
1454:x
1441:b
1437:a
1430:z
1426:y
1422:z
1418:x
1414:y
1410:x
1398:y
1395:b
1391:x
1388:a
1384:y
1380:y
1376:x
1372:x
1364:b
1360:a
1342:0
1339:=
1334:2
1330:y
1324:2
1320:a
1311:2
1307:x
1301:2
1297:b
1288:2
1284:y
1278:2
1274:x
1244:y
1240:x
1222:.
1217:3
1213:y
1206:y
1201:2
1197:x
1193:=
1188:4
1184:x
1165:.
1151:a
1133:0
1130:=
1125:2
1121:)
1115:2
1111:a
1102:2
1098:y
1094:(
1091:+
1086:2
1082:)
1078:a
1072:x
1069:(
1066:)
1061:2
1057:a
1048:2
1044:x
1040:(
997:.
994:)
989:2
985:y
981:+
976:2
972:x
968:(
965:x
962:=
957:4
953:y
949:+
944:2
940:y
934:2
930:x
926:+
921:4
917:x
874:.
869:2
865:)
861:x
858:2
850:2
846:y
842:+
837:2
833:x
829:(
826:4
823:=
820:)
817:3
811:x
808:2
805:(
802:)
799:1
793:x
790:(
787:)
782:2
778:x
769:2
765:y
761:(
463:.
459:R
434:,
430:C
399:.
353:.
343:P
340:R
305:,
302:0
299:=
296:P
293:+
290:y
287:N
284:+
281:x
278:M
275:+
272:y
269:x
266:L
263:+
258:2
254:y
250:K
247:+
242:2
238:x
234:J
231:+
226:2
222:y
218:x
215:I
212:+
209:y
204:2
200:x
196:H
193:+
188:3
184:y
180:G
177:+
172:3
168:x
164:F
161:+
156:3
152:y
148:x
145:E
142:+
137:2
133:y
127:2
123:x
119:D
116:+
113:y
108:3
104:x
100:C
97:+
92:4
88:y
84:B
81:+
76:4
72:x
68:A
34:.
20:)
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