36:
2324:. In general, such a projection is a curve whose singularities are self-crossing points and ordinary cusps. Self-crossing points appear when two different points of the curves have the same projection. Ordinary cusps appear when the tangent to the curve is parallel to the direction of projection (that is when the tangent projects on a single point). More complicated singularities occur when several phenomena occur simultaneously. For example, rhamphoid cusps occur for
101:
1256:
2301:
323:, in contrast to self-intersection points that involve more than one value. In some contexts, the condition on the directional derivative may be omitted, although, in this case, the singularity may look like a regular point.
501:
219:
2035:
1932:
434:
153:
2347:
of the restriction to a (smooth) spatial object of the projection. A cusp appears thus as a singularity of the contour of the image of the object (vision) or of its shadow (computer graphics).
311:
1707:
1569:
839:
889:
2232:
2187:
1162:
1062:
1618:
1357:
1199:
1110:
744:
1826:
927:
788:
1300:
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1005:
966:
374:
1236:
65:
2893:
2622:
429:
2985:
2582:
2463:
2427:
2401:
148:
1512: 'beak-like') denoted originally a cusp such that both branches are on the same side of the tangent, such as for the curve of equation
3062:
2673:
2572:
3052:
87:
2751:
2456:
1941:
2898:
2809:
2819:
2746:
2496:
2716:
1847:
2612:
2344:
2975:
2939:
3092:
2638:
2551:
420:
579:. In some contexts, the definition of a cusp is restricted to the case of cusps of order two—that is, the case where
2949:
2587:
2393:
700:
664:
416:
3087:
2995:
125:
48:
2908:
2888:
2824:
2741:
2602:
124:
where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of
2643:
2313:
1640:
58:
52:
44:
2607:
2799:
606:
598:
2592:
1515:
660:
625:
241:
69:
2970:
2706:
2506:
2350:
1630:
2668:
2617:
1934:
We can now make a diffeomorphic change of variable (in this case we simply substitute polynomials with
795:
251:
848:
3057:
2918:
2829:
2577:
2192:
2147:
1935:
1260:
1115:
1014:
680:
621:
523:
2883:
1574:
1313:
2761:
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2683:
2663:
1841:
1169:
1069:
714:
684:
668:
140:
1792:
900:
761:
2597:
2443:
2366:
2244:
2047:
1265:
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2784:
2756:
975:
936:
335:
2913:
2860:
2731:
2546:
2541:
2423:
2397:
2340:
692:
408:
394:
327:
136:
609:
of the point in the ambient space, which maps the curve onto one of the above-defined cusps.
2903:
2789:
2766:
2329:
2325:
2305:
1211:
384:
3029:
2834:
2776:
2678:
2501:
2480:
2411:
2336:
2321:
618:
594:
380:
2701:
841:
i.e. there exists a diffeomorphic change of coordinate in source and target which takes
589:
The definitions for plane curves and implicitly-defined curves have been generalized by
3003:
2526:
2511:
2488:
2416:
1637:. The rhamphoid cusp and the ordinary cusp are non-diffeomorphic. A parametric form is
1629:, the term has been extended to all such singularities. These cusps are non-generic as
1509:
892:
707:
688:
672:
602:
398:
3081:
3044:
2814:
2794:
2721:
2516:
1419:
397:; however, not all singular points that have this property are cusps. The theory of
2980:
2954:
2944:
2934:
2736:
2556:
2385:
2317:
545:
541:
590:
2855:
2693:
1571:
As such a singularity is in the same differential class as the cusp of equation
696:
675:
of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of
652:
573:
of the cusp, and is equal to the degree of the nonzero part of lowest degree of
513:
132:
109:
2850:
2357:
are other examples of curves having cusps that are visible in the real world.
2354:
1634:
1206:
The cusps are then given by the zero-level-sets of the representatives of the
676:
412:
225:
100:
2448:
2711:
248:, changes sign (the direction of the tangent is the direction of the slope
17:
1785:
divides the cubic terms. It follows that the third order taylor series of
2371:
496:{\displaystyle {\begin{aligned}x&=at^{m}\\y&=S(t),\end{aligned}}}
1771:
To see where these extra divisibility conditions come from, assume that
3034:
3019:
2123:
526:
245:
214:{\displaystyle {\begin{aligned}x&=f(t)\\y&=g(t),\end{aligned}}}
2300:
663:
from the plane to the line. The space of all such smooth functions is
3014:
2332:) for which the tangent is parallel to the direction of projection.
1418:
Having a degenerate quadratic part, i.e. the quadratic terms in the
2299:
1254:
121:
99:
2452:
1759:), and a final non-divisibility condition (giving type exactly
1255:
317:
in the sense that they involve only one value of the parameter
553:(degree of the nonzero term of the lowest degree) larger than
29:
383:, cusps are points where the terms of lowest degree of the
2030:{\displaystyle (L\pm Q/2)^{2}-Q^{4}/4\to x_{1}^{2}+P_{1}}
415:), a linear change of coordinates allows the curve to be
1484:
does not divide the cubic terms in the Taylor series of
2195:
2150:
1944:
1850:
1795:
1725:
to have a degenerate quadratic part (this gives type
1643:
1577:
1518:
1316:
1268:
1214:
1172:
1118:
1072:
1017:
978:
939:
903:
851:
798:
764:
717:
432:
338:
254:
151:
2292:, i.e. the zero-level-set will be a rhamphoid cusp.
3043:
2994:
2963:
2927:
2876:
2869:
2843:
2775:
2692:
2656:
2631:
2565:
2534:
2525:
2487:
1927:{\displaystyle L^{2}\pm LQ=(L\pm Q/2)^{2}-Q^{4}/4.}
2415:
2343:, the curve that is projected is the curve of the
2226:
2181:
2029:
1926:
1820:
1701:
1612:
1563:
1351:
1294:
1230:
1193:
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1056:
999:
960:
921:
883:
833:
782:
738:
495:
368:
305:
213:
27:Point on a curve where motion must move backwards
255:
57:but its sources remain unclear because it lacks
1750:), another divisibility condition (giving type
601:: a curve has a cusp at a point if there is a
2464:
1009:since the diffeomorphic change of coordinate
8:
845:into one of these forms. These simple forms
2873:
2531:
2471:
2457:
2449:
706:One such family of equivalence classes is
2444:Physicists See The Cosmos In A Coffee Cup
2218:
2205:
2200:
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1982:
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1943:
1916:
1910:
1897:
1885:
1855:
1849:
1800:
1794:
1702:{\displaystyle x=t^{2},\,y=ax^{4}+x^{5}.}
1690:
1677:
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1654:
1642:
1595:
1582:
1576:
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774:
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727:
722:
716:
454:
433:
431:
337:
278:
253:
152:
150:
88:Learn how and when to remove this message
2308:of light rays in the bottom of a teacup.
1741:divide the cubic terms (this gives type
2189:and change coordinates so that we have
613:Classification in differential geometry
2894:Clifford's theorem on special divisors
2068:. The divisibility condition for type
752:is a non-negative integer. A function
7:
1564:{\displaystyle x^{2}-x^{4}-y^{5}=0.}
3063:Vector bundles on algebraic curves
2986:Weber's theorem (Algebraic curves)
2583:Hasse's theorem on elliptic curves
2573:Counting points on elliptic curves
2304:An ordinary cusp occurring as the
1359:i.e. the zero-level-set of a type
25:
1391:and assume, for simplicity, that
834:{\displaystyle x^{2}\pm y^{k+1},}
306:{\displaystyle \lim(g'(t)/f'(t))}
2335:In many cases, and typically in
1775:has a degenerate quadratic part
931:-singularities. Notice that the
884:{\displaystyle x^{2}\pm y^{k+1}}
34:
2674:Hurwitz's automorphisms theorem
2227:{\displaystyle x_{2}^{2}+P_{2}}
2182:{\displaystyle x_{1}^{2}+P_{1}}
1620:which is a singularity of type
1157:{\displaystyle x^{2}-y^{2n+1}.}
1057:{\displaystyle (x,y)\to (x,-y)}
687:. This action splits the whole
2899:Gonality of an algebraic curve
2810:Differential of the first kind
2422:. Cambridge University Press.
2122:(the zero-level-set here is a
1996:
1966:
1945:
1894:
1873:
1613:{\displaystyle x^{2}-y^{5}=0,}
1352:{\displaystyle x^{2}-y^{3}=0,}
1051:
1036:
1033:
1030:
1018:
483:
477:
354:
342:
300:
297:
291:
275:
269:
258:
201:
195:
175:
169:
120:in old texts, is a point on a
1:
3053:Birkhoff–Grothendieck theorem
2752:Nagata's conjecture on curves
2623:Schoof–Elkies–Atkin algorithm
2497:Five points determine a conic
1194:{\displaystyle A_{2n}^{\pm }}
1105:{\displaystyle x^{2}+y^{k+1}}
739:{\displaystyle A_{k}^{\pm },}
224:a cusp is a point where both
2613:Supersingular elliptic curve
2312:Cusps appear naturally when
1821:{\displaystyle L^{2}\pm LQ,}
922:{\displaystyle A_{k}^{\pm }}
783:{\displaystyle A_{k}^{\pm }}
2820:Riemann's existence theorem
2747:Hilbert's sixteenth problem
2639:Elliptic curve cryptography
2552:Fundamental pair of periods
1426:form a perfect square, say
1295:{\displaystyle y^{2}=x^{3}}
1240:equivalence classes, where
792:if it lies in the orbit of
3109:
2950:Moduli of algebraic curves
2394:Cambridge University Press
2283:then we have exactly type
2144:we complete the square on
2113:then we have type exactly
1164:So we can drop the ± from
1000:{\displaystyle A_{2n}^{-}}
961:{\displaystyle A_{2n}^{+}}
326:For a curve defined by an
244:, in the direction of the
2418:Geometric Differentiation
1415:can be characterised by:
369:{\displaystyle F(x,y)=0,}
126:singular point of a curve
2717:Cayley–Bacharach theorem
2644:Elliptic curve primality
2390:Curves and Singularities
1383:be a smooth function of
599:differentiable functions
565:is sometimes called the
43:This article includes a
2976:Riemann–Hurwitz formula
2940:Gromov–Witten invariant
2800:Compact Riemann surface
2588:Mazur's torsion theorem
72:more precise citations.
2593:Modular elliptic curve
2309:
2228:
2183:
2031:
1938:linear parts) so that
1928:
1822:
1703:
1614:
1565:
1353:
1302:
1296:
1232:
1231:{\displaystyle A_{2n}}
1195:
1158:
1106:
1058:
1001:
962:
923:
885:
835:
784:
756:is said to be of type
740:
497:
370:
307:
242:directional derivative
215:
105:
2507:Rational normal curve
2320:in three-dimensional
2303:
2229:
2184:
2032:
1929:
1823:
1721:-singularity we need
1704:
1615:
1566:
1354:
1297:
1258:
1233:
1196:
1159:
1107:
1059:
1002:
963:
924:
886:
836:
785:
741:
597:to curves defined by
498:
371:
308:
216:
103:
3058:Stable vector bundle
2919:Weil reciprocity law
2909:Riemann–Roch theorem
2889:Brill–Noether theory
2825:Riemann–Roch theorem
2742:Genus–degree formula
2603:Mordell–Weil theorem
2578:Division polynomials
2193:
2148:
1942:
1936:linearly independent
1848:
1793:
1641:
1575:
1516:
1314:
1266:
1261:semicubical parabola
1212:
1170:
1116:
1070:
1066:in the source takes
1015:
976:
970:are the same as the
937:
901:
849:
796:
762:
715:
622:real-valued function
430:
336:
252:
149:
2870:Structure of curves
2762:Quartic plane curve
2684:Hyperelliptic curve
2664:De Franchis theorem
2608:Nagell–Lutz theorem
2210:
2165:
2013:
1842:complete the square
1190:
996:
957:
918:
779:
732:
693:equivalence classes
315:local singularities
141:parametric equation
116:, sometimes called
104:A cusp at (0, 1/2)
3093:Singularity theory
2877:Divisors on curves
2669:Faltings's theorem
2618:Schoof's algorithm
2598:Modularity theorem
2310:
2224:
2196:
2179:
2151:
2027:
1999:
1924:
1818:
1699:
1610:
1561:
1368:-singularity. Let
1349:
1303:
1292:
1228:
1191:
1173:
1154:
1102:
1054:
997:
979:
958:
940:
919:
904:
881:
831:
780:
765:
736:
718:
493:
491:
366:
303:
240:are zero, and the
211:
209:
106:
45:list of references
3075:
3074:
3071:
3070:
2971:Hasse–Witt matrix
2914:Weierstrass point
2861:Smooth completion
2830:TeichmĂĽller space
2732:Cubic plane curve
2652:
2651:
2566:Arithmetic theory
2547:Elliptic integral
2542:Elliptic function
2429:978-0-521-39063-7
2403:978-0-521-42999-3
2341:computer graphics
2330:undulation points
2326:inflection points
891:are said to give
409:analytic function
401:implies that, if
395:linear polynomial
393:are a power of a
328:implicit equation
98:
97:
90:
16:(Redirected from
3100:
3088:Algebraic curves
2904:Jacobian variety
2874:
2777:Riemann surfaces
2767:Real plane curve
2727:Cramer's paradox
2707:BĂ©zout's theorem
2532:
2481:algebraic curves
2473:
2466:
2459:
2450:
2433:
2421:
2407:
2367:Cusp catastrophe
2291:
2282:
2274:does not divide
2273:
2264:
2255:
2247:(order five) in
2242:
2233:
2231:
2230:
2225:
2223:
2222:
2209:
2204:
2188:
2186:
2185:
2180:
2178:
2177:
2164:
2159:
2143:
2134:
2121:
2112:
2104:does not divide
2103:
2094:
2085:
2076:
2067:
2058:
2050:(order four) in
2045:
2036:
2034:
2033:
2028:
2026:
2025:
2012:
2007:
1992:
1987:
1986:
1974:
1973:
1961:
1933:
1931:
1930:
1925:
1920:
1915:
1914:
1902:
1901:
1889:
1860:
1859:
1839:
1835:
1832:is quadratic in
1831:
1827:
1825:
1824:
1819:
1805:
1804:
1788:
1784:
1780:
1774:
1767:
1758:
1749:
1737:
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1700:
1695:
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1616:
1611:
1600:
1599:
1587:
1586:
1570:
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1562:
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1553:
1541:
1540:
1528:
1527:
1498:
1483:
1463:
1459:
1455:
1440:
1425:
1414:
1410:
1407:-singularity of
1406:
1397:
1390:
1386:
1382:
1367:
1358:
1356:
1355:
1350:
1339:
1338:
1326:
1325:
1301:
1299:
1298:
1293:
1291:
1290:
1278:
1277:
1246:
1239:
1237:
1235:
1234:
1229:
1227:
1226:
1202:
1200:
1198:
1197:
1192:
1189:
1184:
1163:
1161:
1160:
1155:
1150:
1149:
1128:
1127:
1111:
1109:
1108:
1103:
1101:
1100:
1082:
1081:
1065:
1063:
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1055:
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1006:
1004:
1003:
998:
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990:
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928:
926:
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917:
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888:
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861:
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844:
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832:
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789:
787:
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778:
773:
755:
751:
747:
745:
743:
742:
737:
731:
726:
658:
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646:
642:
585:
578:
564:
558:
552:
539:
521:
511:
502:
500:
499:
494:
492:
459:
458:
423:of the cusp, as
406:
392:
385:Taylor expansion
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367:
322:
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309:
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82:
79:
73:
68:this article by
59:inline citations
38:
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3039:
3030:Delta invariant
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2959:
2923:
2884:Abel–Jacobi map
2865:
2839:
2835:Torelli theorem
2805:Dessin d'enfant
2785:Belyi's theorem
2771:
2757:PlĂĽcker formula
2688:
2679:Hurwitz surface
2648:
2627:
2561:
2535:Analytic theory
2527:Elliptic curves
2521:
2502:Projective line
2489:Rational curves
2483:
2477:
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2410:
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2363:
2345:critical points
2337:computer vision
2322:Euclidean space
2316:into a plane a
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1253:
1247:is an integer.
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1132:
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1114:
1113:
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673:diffeomorphisms
656:
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595:Vladimir Arnold
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554:
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411:(for example a
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49:related reading
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3045:Vector bundles
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2522:
2520:
2519:
2514:
2512:Riemann sphere
2509:
2504:
2499:
2493:
2491:
2485:
2484:
2478:
2476:
2475:
2468:
2461:
2453:
2447:
2446:
2439:
2438:External links
2436:
2435:
2434:
2428:
2408:
2402:
2384:Bruce, J. W.;
2379:
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2369:
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2109:
2100:
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1560:
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1548:
1544:
1539:
1535:
1531:
1526:
1522:
1506:rhamphoid cusp
1502:
1501:
1500:
1468:
1403:
1398:. Then a type
1364:
1348:
1345:
1342:
1337:
1333:
1329:
1324:
1320:
1289:
1285:
1281:
1276:
1272:
1259:A cusp in the
1252:
1249:
1225:
1222:
1218:
1188:
1183:
1180:
1176:
1153:
1148:
1145:
1142:
1139:
1135:
1131:
1126:
1122:
1099:
1096:
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1089:
1085:
1080:
1076:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
994:
989:
986:
982:
955:
950:
947:
943:
916:
911:
907:
878:
875:
872:
868:
864:
859:
855:
830:
825:
822:
819:
815:
811:
806:
802:
777:
772:
768:
735:
730:
725:
721:
689:function space
614:
611:
603:diffeomorphism
522:is a positive
504:
503:
488:
485:
482:
479:
476:
473:
470:
468:
466:
463:
462:
457:
453:
449:
446:
443:
441:
439:
436:
435:
399:Puiseux series
377:
376:
365:
362:
359:
356:
353:
350:
347:
344:
341:
313:). Cusps are
302:
299:
296:
293:
289:
286:
281:
277:
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222:
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186:
184:
181:
180:
177:
174:
171:
168:
165:
162:
160:
158:
155:
154:
135:defined by an
96:
95:
53:external links
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3105:
3094:
3091:
3089:
3086:
3085:
3083:
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3061:
3059:
3056:
3054:
3051:
3050:
3048:
3046:
3042:
3036:
3033:
3031:
3028:
3026:
3023:
3021:
3018:
3016:
3013:
3011:
3009:
3002:
3001:
2999:
2997:
2996:Singularities
2993:
2987:
2984:
2982:
2979:
2977:
2974:
2972:
2969:
2968:
2966:
2962:
2956:
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2926:
2920:
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2915:
2912:
2910:
2907:
2905:
2902:
2900:
2897:
2895:
2892:
2890:
2887:
2885:
2882:
2881:
2879:
2875:
2872:
2868:
2862:
2859:
2857:
2854:
2852:
2849:
2848:
2846:
2844:Constructions
2842:
2836:
2833:
2831:
2828:
2826:
2823:
2821:
2818:
2816:
2815:Klein quartic
2813:
2811:
2808:
2806:
2803:
2801:
2798:
2796:
2795:Bolza surface
2793:
2791:
2790:Bring's curve
2788:
2786:
2783:
2782:
2780:
2778:
2774:
2768:
2765:
2763:
2760:
2758:
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2750:
2748:
2745:
2743:
2740:
2738:
2735:
2733:
2730:
2728:
2725:
2723:
2722:Conic section
2720:
2718:
2715:
2713:
2710:
2708:
2705:
2703:
2702:AF+BG theorem
2700:
2699:
2697:
2695:
2691:
2685:
2682:
2680:
2677:
2675:
2672:
2670:
2667:
2665:
2662:
2661:
2659:
2655:
2645:
2642:
2640:
2637:
2636:
2634:
2630:
2624:
2621:
2619:
2616:
2614:
2611:
2609:
2606:
2604:
2601:
2599:
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2594:
2591:
2589:
2586:
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2581:
2579:
2576:
2574:
2571:
2570:
2568:
2564:
2558:
2555:
2553:
2550:
2548:
2545:
2543:
2540:
2539:
2537:
2533:
2530:
2528:
2524:
2518:
2517:Twisted cubic
2515:
2513:
2510:
2508:
2505:
2503:
2500:
2498:
2495:
2494:
2492:
2490:
2486:
2482:
2474:
2469:
2467:
2462:
2460:
2455:
2454:
2451:
2445:
2442:
2441:
2437:
2431:
2425:
2420:
2419:
2413:
2412:Porteous, Ian
2409:
2405:
2399:
2395:
2391:
2387:
2386:Giblin, Peter
2382:
2381:
2377:
2373:
2370:
2368:
2365:
2364:
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2358:
2356:
2352:
2348:
2346:
2342:
2338:
2333:
2331:
2327:
2323:
2319:
2315:
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2302:
2295:
2293:
2287:
2278:
2269:
2260:
2251:
2246:
2238:
2219:
2215:
2211:
2206:
2201:
2197:
2174:
2170:
2166:
2161:
2156:
2152:
2139:
2130:
2125:
2117:
2108:
2099:
2090:
2081:
2072:
2063:
2054:
2049:
2041:
2022:
2018:
2014:
2009:
2004:
2000:
1993:
1989:
1983:
1979:
1975:
1970:
1962:
1958:
1954:
1951:
1948:
1937:
1921:
1917:
1911:
1907:
1903:
1898:
1890:
1886:
1882:
1879:
1876:
1870:
1867:
1864:
1861:
1856:
1852:
1844:to show that
1843:
1815:
1812:
1809:
1806:
1801:
1797:
1779:
1769:
1763:
1754:
1745:
1740:
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1687:
1683:
1678:
1674:
1670:
1667:
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1651:
1647:
1644:
1636:
1632:
1624:
1607:
1604:
1601:
1596:
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1583:
1579:
1558:
1555:
1550:
1546:
1542:
1537:
1533:
1529:
1524:
1520:
1511:
1507:
1503:
1496:
1492:
1488:
1481:
1477:
1473:
1469:
1467:
1456:is linear in
1453:
1449:
1445:
1438:
1434:
1430:
1421:
1420:Taylor series
1417:
1416:
1402:
1395:
1380:
1376:
1372:
1363:
1346:
1343:
1340:
1335:
1331:
1327:
1322:
1318:
1309:
1308:ordinary cusp
1305:
1304:
1287:
1283:
1279:
1274:
1270:
1262:
1257:
1250:
1248:
1244:
1223:
1220:
1216:
1204:
1186:
1181:
1178:
1174:
1151:
1146:
1143:
1140:
1137:
1133:
1129:
1124:
1120:
1097:
1094:
1091:
1087:
1083:
1078:
1074:
1048:
1045:
1042:
1039:
1027:
1024:
1021:
992:
987:
984:
980:
953:
948:
945:
941:
914:
909:
905:
895:for the type
894:
876:
873:
870:
866:
862:
857:
853:
828:
823:
820:
817:
813:
809:
804:
800:
775:
770:
766:
733:
728:
723:
719:
709:
704:
702:
698:
694:
690:
686:
682:
678:
674:
670:
666:
662:
654:
640:
636:
632:
627:
623:
620:
612:
610:
608:
604:
600:
596:
592:
587:
583:
577:
572:
568:
563:
559:. The number
557:
551:
547:
543:
537:
533:
528:
525:
520:
515:
510:
486:
480:
474:
471:
469:
464:
455:
451:
447:
444:
442:
437:
426:
425:
424:
422:
418:
414:
410:
405:
400:
396:
391:
386:
382:
363:
360:
357:
351:
348:
345:
339:
332:
331:
330:
329:
324:
321:
316:
294:
287:
284:
279:
272:
265:
262:
247:
243:
238:
232:
227:
204:
198:
192:
189:
187:
182:
172:
166:
163:
161:
156:
145:
144:
143:
142:
138:
134:
129:
127:
123:
119:
115:
111:
102:
92:
89:
81:
71:
67:
61:
60:
54:
50:
46:
41:
32:
31:
19:
3024:
3004:
2981:Prym variety
2955:Stable curve
2945:Hodge bundle
2935:ELSV formula
2737:Fermat curve
2694:Plane curves
2657:Higher genus
2632:Applications
2557:Modular form
2417:
2389:
2349:
2334:
2318:smooth curve
2311:
2296:Applications
2285:
2276:
2267:
2258:
2249:
2236:
2137:
2128:
2115:
2106:
2097:
2088:
2079:
2070:
2061:
2052:
2039:
1789:is given by
1777:
1770:
1761:
1752:
1743:
1738:
1727:
1714:
1711:
1622:
1505:
1494:
1490:
1486:
1479:
1475:
1471:
1465:
1451:
1447:
1443:
1436:
1432:
1428:
1400:
1393:
1378:
1374:
1370:
1361:
1310:is given by
1307:
1242:
1205:
893:normal forms
705:
701:group action
679:in both the
667:upon by the
653:real numbers
638:
634:
630:
616:
607:neighborhood
588:
581:
575:
571:multiplicity
570:
566:
561:
555:
549:
542:power series
535:
531:
518:
508:
505:
421:neighborhood
417:parametrized
403:
389:
378:
325:
319:
314:
236:
230:
223:
130:
117:
113:
107:
84:
75:
64:Please help
56:
3010:singularity
2856:Polar curve
2355:wave fronts
1712:For a type
1635:wave fronts
617:Consider a
514:real number
226:derivatives
133:plane curve
110:mathematics
70:introducing
18:Double cusp
3082:Categories
2851:Dual curve
2479:Topics in
2378:References
2314:projecting
1508:(from
1396:(0, 0) = 0
1203:notation.
708:denoted by
677:coordinate
413:polynomial
78:April 2021
2964:Morphisms
2712:Bitangent
2328:(and for
1997:→
1976:−
1952:±
1904:−
1880:±
1862:±
1840:. We can
1807:±
1781:and that
1589:−
1543:−
1530:−
1328:−
1187:±
1130:−
1046:−
1034:→
993:−
915:±
863:±
810:±
776:±
729:±
626:variables
591:René Thom
379:which is
2414:(1994).
2388:(1984).
2372:Cardioid
2361:See also
2351:Caustics
2135:divides
2086:divides
2077:is that
1734:), that
1631:caustics
1441:, where
1251:Examples
691:up into
683:and the
661:function
288:′
266:′
137:analytic
3035:Tacnode
3020:Crunode
2306:caustic
2245:quintic
2124:tacnode
2048:quartic
1238:
1208:
1201:
1166:
1064:
1011:
1007:
972:
968:
933:
929:
897:
790:
758:
746:
711:
699:of the
695:, i.e.
624:of two
569:or the
527:integer
419:, in a
246:tangent
118:spinode
66:improve
3015:Acnode
2928:Moduli
2426:
2400:
2234:where
2126:). If
2037:where
1828:where
1413:(0, 0)
748:where
697:orbits
685:target
681:source
643:where
628:, say
619:smooth
529:, and
506:where
407:is an
381:smooth
131:For a
2265:. If
2095:. If
1510:Greek
669:group
665:acted
659:is a
655:. So
605:of a
567:order
546:order
540:is a
512:is a
234:and
122:curve
51:, or
3025:Cusp
2424:ISBN
2398:ISBN
2353:and
2339:and
2256:and
2059:and
1836:and
1739:does
1633:and
1460:and
1387:and
651:are
647:and
593:and
524:even
114:cusp
112:, a
2243:is
2046:is
1768:).
1466:and
1422:of
1411:at
1306:An
1245:≥ 1
1112:to
703:.
671:of
584:= 2
544:of
387:of
256:lim
228:of
108:In
3084::
2396:.
2392:.
2074:≥4
1922:4.
1756:≥4
1747:≥3
1731:≥2
1559:0.
1504:A
1493:,
1478:,
1464:,
1450:,
1435:,
1409:f
1377:,
637:,
586:.
516:,
139:,
128:.
55:,
47:,
3007:k
3005:A
2472:e
2465:t
2458:v
2432:.
2406:.
2289:4
2286:A
2280:2
2277:P
2271:2
2268:x
2262:2
2259:y
2253:2
2250:x
2240:2
2237:P
2220:2
2216:P
2212:+
2207:2
2202:2
2198:x
2175:1
2171:P
2167:+
2162:2
2157:1
2153:x
2141:1
2138:P
2132:1
2129:x
2119:3
2116:A
2110:1
2107:P
2101:1
2098:x
2092:1
2089:P
2083:1
2080:x
2071:A
2065:1
2062:y
2056:1
2053:x
2043:1
2040:P
2023:1
2019:P
2015:+
2010:2
2005:1
2001:x
1994:4
1990:/
1984:4
1980:Q
1971:2
1967:)
1963:2
1959:/
1955:Q
1949:L
1946:(
1918:/
1912:4
1908:Q
1899:2
1895:)
1891:2
1887:/
1883:Q
1877:L
1874:(
1871:=
1868:Q
1865:L
1857:2
1853:L
1838:y
1834:x
1830:Q
1816:,
1813:Q
1810:L
1802:2
1798:L
1787:f
1783:L
1778:L
1773:f
1765:4
1762:A
1753:A
1744:A
1736:L
1728:A
1723:f
1718:4
1715:A
1697:.
1692:5
1688:x
1684:+
1679:4
1675:x
1671:a
1668:=
1665:y
1661:,
1656:2
1652:t
1648:=
1645:x
1626:4
1623:A
1608:,
1605:0
1602:=
1597:5
1593:y
1584:2
1580:x
1556:=
1551:5
1547:y
1538:4
1534:x
1525:2
1521:x
1499:.
1497:)
1495:y
1491:x
1489:(
1487:f
1482:)
1480:y
1476:x
1474:(
1472:L
1462:y
1458:x
1454:)
1452:y
1448:x
1446:(
1444:L
1439:)
1437:y
1433:x
1431:(
1429:L
1424:f
1404:2
1401:A
1394:f
1389:y
1385:x
1381:)
1379:y
1375:x
1373:(
1371:f
1365:2
1362:A
1347:,
1344:0
1341:=
1336:3
1332:y
1323:2
1319:x
1288:3
1284:x
1280:=
1275:2
1271:y
1243:n
1224:n
1221:2
1217:A
1182:n
1179:2
1175:A
1152:.
1147:1
1144:+
1141:n
1138:2
1134:y
1125:2
1121:x
1098:1
1095:+
1092:k
1088:y
1084:+
1079:2
1075:x
1052:)
1049:y
1043:,
1040:x
1037:(
1031:)
1028:y
1025:,
1022:x
1019:(
988:n
985:2
981:A
954:+
949:n
946:2
942:A
910:k
906:A
877:1
874:+
871:k
867:y
858:2
854:x
843:f
829:,
824:1
821:+
818:k
814:y
805:2
801:x
771:k
767:A
754:f
750:k
734:,
724:k
720:A
657:f
649:y
645:x
641:)
639:y
635:x
633:(
631:f
582:m
576:F
562:m
556:m
550:k
538:)
536:t
534:(
532:S
519:m
509:a
487:,
484:)
481:t
478:(
475:S
472:=
465:y
456:m
452:t
448:a
445:=
438:x
404:F
390:F
364:,
361:0
358:=
355:)
352:y
349:,
346:x
343:(
340:F
320:t
301:)
298:)
295:t
292:(
285:f
280:/
276:)
273:t
270:(
263:g
259:(
237:g
231:f
205:,
202:)
199:t
196:(
193:g
190:=
183:y
176:)
173:t
170:(
167:f
164:=
157:x
91:)
85:(
80:)
76:(
62:.
20:)
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