3820:
3808:
1000:
182:
20:
3128:
1478:
1459:
2843:. Further, the portion of this image in the upper half-plane depicts the following situation: The double-loops are a reduced set of congruence classes for the central Steiner conics in the hyperbolic plane produced by direct collineations; and each single-loop is the locus of points
2216:
1305:
3293:
3448:
3598:
2290:
3788:
1999:
1814:
884:
570:
2456:.) Instead these curves actually correspond to the (plane sections of) equipotential sets of two infinite wires with equal constant line charge density, or alternatively, to the level sets of the sums of the
2074:
3105:
989:
751:
268:
2450:
1310:
2463:
The single-loop and double loop
Cassini curves can be represented as the orthogonal trajectories of each other when each family is coaxal but not confocal. If the single-loops are described by
2719:
1676:
1259:. These points are biflecnodes, meaning that the curve has two distinct tangents at these points and each branch of the curve has a point of inflection there. From this information and
1270:: degree = 4, class = 8, number of nodes = 2, number of cusps = 0, number of double tangents = 8, number of points of inflection = 12, genus = 1.
2341:
1302:
the curve has class eight, which implies that there should be a total of eight real foci. Six of these have been accounted for in the two triple foci and the remaining two are at
1133:
1167:
2525:
371:
2051:
show that these curves are hyperbolas. A more detailed investigation reveals that the hyperbolas are rectangular. In order to get normals, which are independent from parameter
3011:
1877:
1588:
1544:
613:
411:
3640:
2841:
2806:
2045:
2069:
2925:
3152:
2632:
2577:
464:
1909:
438:
2606:
3666:
3322:
2887:
2771:
2745:
2551:
1454:{\displaystyle {\begin{aligned}\left(\pm a{\sqrt {1-e^{4}}},0\right)&\quad (e<1),\\\left(0,\pm a{\sqrt {e^{4}-1}}\right)&\quad (e>1).\end{aligned}}}
2965:
2945:
2861:
1914:
300:
3143:, but only when the cutting plane is parallel to the axis of the torus and its distance to the axis equals the radius of the generating circle (see picture).
2773:. Each curve, up to similarity, appears twice in the image, which now resembles the field lines and potential curves for four equal point charges, located at
3330:
3459:
1685:
2221:
3682:
1295:. So the foci are, in fact, foci in the sense defined by Plücker. The circular points are points of inflection so these are triple foci. When
762:
472:
4011:
Cohen, I. Bernard (1962). "Leibniz on elliptical orbits: as seen in his correspondence with the Académie Royale des
Sciences in 1700".
3028:
899:
161:
around another body traveled on one of these ovals, with the body it orbited around at one focus of the oval. Other names include
641:
188:
2365:
1494:
of curves are curves which intersect all given curves orthogonally. For example the orthogonal trajectories of a pencil of
4183:
2211:{\displaystyle g(x,y)={\frac {x^{2}-y^{2}-1}{xy}}-\lambda ={\frac {x}{y}}-{\frac {y}{x}}-{\frac {1}{xy}}-\lambda =0\;.}
4061:
2637:
1601:
3836:
3022:
4112:
3819:
2295:
1102:
1138:
4107:
2466:
1252:
309:
3676:
Cassini's method is easy to generalize to curves and surfaces with an arbitrarily many defining points:
4163:
154:
4148:
4119:
2048:
4178:
3999:
De l'Origine et du progrès de l'astronomie et de son usage dans la géographie et dans la navigation
2970:
1256:
4102:
4078:
3807:
1829:
4173:
1553:
1509:
1495:
1486:
1081:
578:
376:
3606:
3288:{\displaystyle \left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4R^{2}\!\left(x^{2}+y^{2}\right)}
2811:
2776:
2015:
147:
2054:
4168:
2892:
1819:
1547:
139:
138:
of the distances is constant, rather than the product. Cassini ovals are the special case of
123:
4053:
4042:
2611:
2556:
443:
4087:
4003:
3997:
3895:
2457:
1882:
1491:
416:
2582:
8:
3645:
3301:
2866:
2750:
2724:
2530:
2351:
115:
111:
4072:
4049:
4028:
3974:
2950:
2930:
2846:
285:
1260:
4126:
4057:
3978:
3966:
3946:
1099:, the curve is a single, connected loop enclosing both foci. It is peanut-shaped for
1073:, the curve consists of two disconnected loops, each of which contains a focus. When
890:
3443:{\displaystyle \left(x^{2}+z^{2}+R^{2}\right)^{2}=4R^{2}\!\left(x^{2}+r^{2}\right).}
4020:
3958:
3798:
3593:{\displaystyle \left(x^{2}+z^{2}\right)^{2}-2R^{2}(x^{2}-z^{2})=4R^{2}r^{2}-R^{4},}
2359:
127:
119:
4129:
4143:
3794:
2453:
4024:
3962:
2285:{\displaystyle \operatorname {grad} f(x,y)\cdot \operatorname {grad} g(x,y)=0}
4157:
3970:
3132:
3112:
2355:
1085:
157:
who studied them in the late 17th century. Cassini believed that a planet
3783:{\displaystyle |PP_{1}|\times |PP_{2}|\times \cdots \times |PP_{n}|=b^{n}}
1994:{\displaystyle x^{2}-y^{2}-\lambda xy-1=0,\ \ \ \lambda \in \mathbb {R} .}
3131:
Cassini ovals as planar sections of a torus (the torus on the right is a
1223:
4032:
2350:
The image depicting the
Cassini ovals and the hyperbolas looks like the
143:
23:
Three
Cassini ovals, differing by the range within which the parameter
4085:
Lawden, D. F., "Families of ovals and their orthogonal trajectories",
4134:
2343:. Hence the Cassini ovals and the hyperbolas intersect orthogonally.
1823:
1499:
3947:"Conics in the hyperbolic plane intrinsic to the collineation group"
100:
3453:
After partially resolving the first bracket one gets the equation
1809:{\displaystyle f(x,y)=(x^{2}+y^{2})^{2}-2(x^{2}-y^{2})+1-b^{4}=0.}
999:
879:{\displaystyle (x^{2}+y^{2})^{2}-2a^{2}(x^{2}-y^{2})+a^{4}=b^{4}.}
565:{\displaystyle \{P:|PP_{1}|\!\!\;\times \!\!\;|PP_{2}|=b^{2}\}\ .}
1089:
181:
131:
19:
3127:
1184:
1477:
3140:
3100:{\displaystyle L_{2}=\{c:\operatorname {abs} (c^{2}+c)=ER\}.}
984:{\displaystyle r^{4}-2a^{2}r^{2}\cos 2\theta =b^{4}-a^{4}.\,}
158:
1506:
The orthogonal trajectories of the
Cassini curves with foci
1481:
Cassini ovals and their orthogonal trajectories (hyperbolas)
1472:
2362:. But for the potential of two equal point charges one has
106:
2460:
for the
Laplacian in two dimensions centered at the foci.
2071:
the following implicit representation is more convenient
1263:
it is possible to deduce the Plücker numbers for the case
1183:), in which case the foci coincide with each other, is a
746:{\displaystyle ((x-a)^{2}+y^{2})((x+a)^{2}+y^{2})=b^{4}.}
263:{\displaystyle |PP_{1}|\!\!\;\times \!\!\;|PP_{2}|=b^{2}}
3603:
which is the equation of a
Cassini oval with parameters
1590:
with the same center as the
Cassini ovals (see picture).
2445:{\displaystyle 1/|PP_{1}|+1/|PP_{2}|={\text{constant}}}
4124:
4013:
Journal of the
History of Medicine and Allied Sciences
3685:
3648:
3609:
3462:
3333:
3304:
3155:
3031:
2973:
2953:
2933:
2895:
2869:
2849:
2814:
2779:
2753:
2727:
2640:
2614:
2585:
2559:
2533:
2469:
2368:
2298:
2224:
2077:
2057:
2018:
1917:
1885:
1832:
1688:
1604:
1556:
1512:
1308:
1141:
1105:
902:
765:
644:
581:
475:
446:
419:
379:
312:
288:
191:
1084:having the shape of a sideways figure eight with a
4074:An Elementary Treatise on Cubic and Quartic Curves
4041:
4039:
3782:
3660:
3634:
3592:
3442:
3316:
3287:
3099:
3005:
2959:
2939:
2919:
2881:
2855:
2835:
2800:
2765:
2739:
2713:
2626:
2600:
2571:
2545:
2519:
2444:
2335:
2284:
2210:
2063:
2039:
1993:
1903:
1871:
1808:
1670:
1582:
1538:
1453:
1161:
1127:
983:
878:
745:
607:
564:
458:
432:
405:
365:
294:
262:
3403:
3251:
3119:is equal to zero, which are the values 0 and −1.
1273:The tangents at the circular points are given by
515:
514:
509:
508:
222:
221:
216:
215:
4155:
3858:
4149:"MacTutor History of Mathematics" Famous Curves
1502:with the same foci. For Cassini ovals one has:
2714:{\displaystyle (x^{2}+y^{2})+1=b(x^{2}-y^{2})}
1465:-axis when the curve has two loops and on the
130:) is constant. This may be contrasted with an
3995:
3115:that have orbits where every second value of
2004:These conic sections have no points with the
635:, 0), then the equation of the curve is
153:Cassini ovals are named after the astronomer
4070:
3896:"Cassini oval - Encyclopedia of Mathematics"
3146:The intersection of the torus with equation
3091:
3045:
553:
476:
282:is a set of points, such that for any point
3139:Cassini ovals appear as planar sections of
3025:is a Cassini oval defined by the equation
2204:
1671:{\displaystyle P_{1}=(1,0),\,P_{2}=(-1,0)}
1287:which have real points of intersection at
516:
510:
223:
217:
4077:. London: Deighton Bell and Co. pp.
2947:is the foot of the perpendicular through
2358:together with the lines of the generated
2317:
1984:
1636:
1473:Cassini ovals and orthogonal trajectories
1155:
980:
339:
3126:
3122:
1476:
1469:-axis when the curve has a single loop.
1052:The curve depends, up to similarity, on
998:
180:
18:
2634:; if the double-loops are described by
2527:then the foci are variable on the axis
4156:
126:of the distances to two fixed points (
4125:
4010:
3944:
3864:
2336:{\displaystyle (x,y),\,x\neq 0\neq y}
1128:{\displaystyle 1<e<{\sqrt {2}}}
575:As with an ellipse, the fixed points
1682:The Cassini ovals have the equation
1162:{\displaystyle e\geq {\sqrt {2}}\,.}
176:
2520:{\displaystyle (x^{2}+y^{2})-1=axy}
1251:The curve has double points at the
366:{\displaystyle |PP_{1}|,\,|PP_{2}|}
13:
3671:
2008:-axis in common and intersect the
1461:So the additional foci are on the
413:is a constant, usually written as
14:
4195:
4095:
4044:A catalog of special plane curves
2721:then the axes are, respectively,
1911:can be described by the equation
4002:. L’Imprimerie Royale. pp.
3818:
3806:
3793:describes in the planar case an
3023:lemniscate of the Mandelbrot set
2218:A simple calculation shows that
3989:
3938:
3825:surface with 6 defining points
1428:
1359:
3929:
3920:
3911:
3902:
3888:
3879:
3870:
3849:
3837:Two-center bipolar coordinates
3763:
3745:
3731:
3713:
3705:
3687:
3545:
3519:
3079:
3060:
2914:
2902:
2830:
2815:
2795:
2780:
2708:
2682:
2667:
2641:
2496:
2470:
2430:
2412:
2396:
2378:
2311:
2299:
2273:
2261:
2246:
2234:
2093:
2081:
2034:
2019:
1898:
1886:
1866:
1851:
1845:
1833:
1778:
1752:
1737:
1710:
1704:
1692:
1665:
1650:
1630:
1618:
1441:
1429:
1372:
1360:
1255:, in other words the curve is
844:
818:
793:
766:
724:
702:
689:
686:
683:
661:
648:
645:
536:
518:
504:
486:
359:
341:
332:
314:
243:
225:
211:
193:
1:
3842:
3006:{\displaystyle x^{2}+y^{2}=1}
1248:-intercepts being imaginary.
16:Class of quartic plane curves
3813:curve with 3 defining points
1872:{\displaystyle (1,0),(-1,0)}
1826:are rectangular) containing
622:
7:
4108:Encyclopedia of Mathematics
4091:83, November 1999, 410–420.
4040:J. Dennis Lawrence (1972).
3830:
3107:Its foci are at the points
3016:
1598:For simplicity one chooses
1583:{\displaystyle P_{1},P_{2}}
1539:{\displaystyle P_{1},P_{2}}
1253:circular points at infinity
756:When expanded this becomes
608:{\displaystyle P_{1},P_{2}}
406:{\displaystyle P_{1},P_{2}}
10:
4200:
3945:Sarli, John (April 2012).
1498:ellipses are the confocal
4184:Giovanni Domenico Cassini
3963:10.1007/s00022-012-0115-5
3635:{\displaystyle b^{2}=2Rr}
2967:on the line described by
2836:{\displaystyle (0,\pm 1)}
2801:{\displaystyle (\pm 1,0)}
2040:{\displaystyle (\pm 1,0)}
1222:there are two additional
155:Giovanni Domenico Cassini
4144:2Dcurves.com description
2064:{\displaystyle \lambda }
994:
4025:10.1093/jhmas/xvii.1.72
2920:{\displaystyle O=(0,1)}
1487:Orthogonal trajectories
1240:-intercepts, all other
1082:lemniscate of Bernoulli
3996:J.-D. Cassini (1693).
3784:
3662:
3636:
3594:
3444:
3318:
3289:
3136:
3101:
3007:
2961:
2941:
2921:
2883:
2857:
2837:
2802:
2767:
2741:
2715:
2628:
2627:{\displaystyle a<0}
2602:
2573:
2572:{\displaystyle a>0}
2547:
2521:
2446:
2337:
2286:
2212:
2065:
2041:
1995:
1905:
1873:
1820:equilateral hyperbolas
1810:
1672:
1584:
1548:equilateral hyperbolas
1540:
1482:
1455:
1163:
1129:
1092:) at the origin. When
1049:
985:
880:
747:
609:
566:
460:
459:{\displaystyle b>0}
434:
407:
367:
296:
275:
264:
140:polynomial lemniscates
96:
4071:A. B. Basset (1901).
3785:
3663:
3637:
3595:
3445:
3319:
3290:
3130:
3123:Cassini ovals on tori
3102:
3008:
2962:
2942:
2922:
2884:
2858:
2838:
2803:
2768:
2742:
2716:
2629:
2603:
2574:
2548:
2522:
2447:
2338:
2287:
2213:
2066:
2042:
1996:
1906:
1904:{\displaystyle (0,0)}
1874:
1811:
1673:
1585:
1541:
1480:
1456:
1229:-intercepts and when
1190:The curve always has
1169:The limiting case of
1164:
1130:
1003:Some Cassini ovals. (
1002:
986:
881:
748:
619:of the Cassini oval.
610:
567:
461:
435:
433:{\displaystyle b^{2}}
408:
368:
297:
265:
184:
22:
4120:MacTutor description
4088:Mathematical Gazette
3683:
3646:
3607:
3460:
3331:
3302:
3153:
3029:
2971:
2951:
2931:
2893:
2867:
2863:such that the angle
2847:
2812:
2777:
2751:
2725:
2638:
2612:
2601:{\displaystyle y=-x}
2583:
2557:
2531:
2467:
2366:
2354:curves of two equal
2296:
2222:
2075:
2055:
2016:
1915:
1883:
1830:
1686:
1602:
1554:
1510:
1306:
1139:
1103:
900:
763:
642:
579:
473:
444:
417:
377:
373:to two fixed points
310:
286:
270:for any location of
189:
3951:Journal of Geometry
3661:{\displaystyle a=R}
3317:{\displaystyle y=r}
2889:is constant, where
2882:{\displaystyle OPQ}
2766:{\displaystyle x=0}
2740:{\displaystyle y=0}
2546:{\displaystyle y=x}
1236:there are two real
1080:, the curve is the
112:quartic plane curve
4127:Weisstein, Eric W.
4050:Dover Publications
3797:and in 3-space an
3780:
3658:
3632:
3590:
3440:
3314:
3285:
3137:
3097:
3003:
2957:
2937:
2917:
2879:
2853:
2833:
2798:
2763:
2737:
2711:
2624:
2598:
2569:
2543:
2517:
2442:
2333:
2282:
2208:
2061:
2037:
1991:
1901:
1869:
1806:
1668:
1580:
1536:
1483:
1451:
1449:
1261:Plücker's formulas
1159:
1125:
1050:
981:
876:
743:
605:
562:
456:
430:
403:
363:
292:
276:
260:
97:
2960:{\displaystyle P}
2940:{\displaystyle Q}
2856:{\displaystyle P}
2458:Green’s functions
2440:
2190:
2172:
2159:
2140:
1976:
1973:
1970:
1417:
1342:
1153:
1123:
1088:(specifically, a
627:If the foci are (
558:
306:of the distances
295:{\displaystyle P}
177:Formal definition
118:of points in the
4191:
4164:Eponymous curves
4140:
4139:
4116:
4082:
4067:
4047:
4036:
4007:
3983:
3982:
3942:
3936:
3933:
3927:
3926:See Basset p. 47
3924:
3918:
3915:
3909:
3906:
3900:
3899:
3892:
3886:
3883:
3877:
3874:
3868:
3862:
3856:
3853:
3822:
3810:
3799:implicit surface
3789:
3787:
3786:
3781:
3779:
3778:
3766:
3761:
3760:
3748:
3734:
3729:
3728:
3716:
3708:
3703:
3702:
3690:
3667:
3665:
3664:
3659:
3641:
3639:
3638:
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3619:
3618:
3599:
3597:
3596:
3591:
3586:
3585:
3573:
3572:
3563:
3562:
3544:
3543:
3531:
3530:
3518:
3517:
3502:
3501:
3496:
3492:
3491:
3490:
3478:
3477:
3449:
3447:
3446:
3441:
3436:
3432:
3431:
3430:
3418:
3417:
3402:
3401:
3386:
3385:
3380:
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3374:
3362:
3361:
3349:
3348:
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3321:
3320:
3315:
3294:
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3291:
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3284:
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3279:
3278:
3266:
3265:
3250:
3249:
3234:
3233:
3228:
3224:
3223:
3222:
3210:
3209:
3197:
3196:
3184:
3183:
3171:
3170:
3106:
3104:
3103:
3098:
3072:
3071:
3041:
3040:
3012:
3010:
3009:
3004:
2996:
2995:
2983:
2982:
2966:
2964:
2963:
2958:
2946:
2944:
2943:
2938:
2926:
2924:
2923:
2918:
2888:
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2880:
2862:
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2859:
2854:
2842:
2840:
2839:
2834:
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2804:
2799:
2772:
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2769:
2764:
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2744:
2743:
2738:
2720:
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2717:
2712:
2707:
2706:
2694:
2693:
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2625:
2607:
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2604:
2599:
2578:
2576:
2575:
2570:
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2550:
2549:
2544:
2526:
2524:
2523:
2518:
2495:
2494:
2482:
2481:
2451:
2449:
2448:
2443:
2441:
2438:
2433:
2428:
2427:
2415:
2410:
2399:
2394:
2393:
2381:
2376:
2360:electrical field
2342:
2340:
2339:
2334:
2291:
2289:
2288:
2283:
2217:
2215:
2214:
2209:
2191:
2189:
2178:
2173:
2165:
2160:
2152:
2141:
2139:
2131:
2124:
2123:
2111:
2110:
2100:
2070:
2068:
2067:
2062:
2046:
2044:
2043:
2038:
2000:
1998:
1997:
1992:
1987:
1974:
1971:
1968:
1940:
1939:
1927:
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1910:
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1902:
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1813:
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1807:
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1798:
1777:
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1763:
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1744:
1735:
1734:
1722:
1721:
1677:
1675:
1674:
1669:
1646:
1645:
1614:
1613:
1589:
1587:
1586:
1581:
1579:
1578:
1566:
1565:
1545:
1543:
1542:
1537:
1535:
1534:
1522:
1521:
1460:
1458:
1457:
1452:
1450:
1423:
1419:
1418:
1410:
1409:
1400:
1354:
1350:
1343:
1341:
1340:
1325:
1301:
1294:
1286:
1269:
1235:
1221:
1214:
1200:
1182:
1175:
1168:
1166:
1165:
1160:
1154:
1149:
1134:
1132:
1131:
1126:
1124:
1119:
1098:
1079:
1072:
1065:
1047:
1040:
1033:
1026:
1020:
1013:
990:
988:
987:
982:
976:
975:
963:
962:
938:
937:
928:
927:
912:
911:
885:
883:
882:
877:
872:
871:
859:
858:
843:
842:
830:
829:
817:
816:
801:
800:
791:
790:
778:
777:
752:
750:
749:
744:
739:
738:
723:
722:
710:
709:
682:
681:
669:
668:
631:, 0) and (−
614:
612:
611:
606:
604:
603:
591:
590:
571:
569:
568:
563:
556:
552:
551:
539:
534:
533:
521:
507:
502:
501:
489:
465:
463:
462:
457:
439:
437:
436:
431:
429:
428:
412:
410:
409:
404:
402:
401:
389:
388:
372:
370:
369:
364:
362:
357:
356:
344:
335:
330:
329:
317:
302:of the set, the
301:
299:
298:
293:
269:
267:
266:
261:
259:
258:
246:
241:
240:
228:
214:
209:
208:
196:
171:ovals of Cassini
167:Cassinian curves
134:, for which the
94:
93:
92:
80:
79:
78:
66:
61:
54:
49:
41:
36:
26:
4199:
4198:
4194:
4193:
4192:
4190:
4189:
4188:
4179:Spiric sections
4154:
4153:
4130:"Cassini Ovals"
4101:
4098:
4064:
3992:
3987:
3986:
3943:
3939:
3934:
3930:
3925:
3921:
3916:
3912:
3907:
3903:
3894:
3893:
3889:
3884:
3880:
3875:
3871:
3863:
3859:
3854:
3850:
3845:
3833:
3826:
3823:
3814:
3811:
3774:
3770:
3762:
3756:
3752:
3744:
3730:
3724:
3720:
3712:
3704:
3698:
3694:
3686:
3684:
3681:
3680:
3674:
3672:Generalizations
3647:
3644:
3643:
3614:
3610:
3608:
3605:
3604:
3581:
3577:
3568:
3564:
3558:
3554:
3539:
3535:
3526:
3522:
3513:
3509:
3497:
3486:
3482:
3473:
3469:
3468:
3464:
3463:
3461:
3458:
3457:
3426:
3422:
3413:
3409:
3408:
3404:
3397:
3393:
3381:
3370:
3366:
3357:
3353:
3344:
3340:
3339:
3335:
3334:
3332:
3329:
3328:
3303:
3300:
3299:
3274:
3270:
3261:
3257:
3256:
3252:
3245:
3241:
3229:
3218:
3214:
3205:
3201:
3192:
3188:
3179:
3175:
3166:
3162:
3161:
3157:
3156:
3154:
3151:
3150:
3125:
3067:
3063:
3036:
3032:
3030:
3027:
3026:
3019:
2991:
2987:
2978:
2974:
2972:
2969:
2968:
2952:
2949:
2948:
2932:
2929:
2928:
2894:
2891:
2890:
2868:
2865:
2864:
2848:
2845:
2844:
2813:
2810:
2809:
2778:
2775:
2774:
2752:
2749:
2748:
2726:
2723:
2722:
2702:
2698:
2689:
2685:
2661:
2657:
2648:
2644:
2639:
2636:
2635:
2613:
2610:
2609:
2584:
2581:
2580:
2558:
2555:
2554:
2532:
2529:
2528:
2490:
2486:
2477:
2473:
2468:
2465:
2464:
2437:
2429:
2423:
2419:
2411:
2406:
2395:
2389:
2385:
2377:
2372:
2367:
2364:
2363:
2349:
2297:
2294:
2293:
2223:
2220:
2219:
2182:
2177:
2164:
2151:
2132:
2119:
2115:
2106:
2102:
2101:
2099:
2076:
2073:
2072:
2056:
2053:
2052:
2017:
2014:
2013:
1983:
1935:
1931:
1922:
1918:
1916:
1913:
1912:
1884:
1881:
1880:
1831:
1828:
1827:
1794:
1790:
1772:
1768:
1759:
1755:
1740:
1736:
1730:
1726:
1717:
1713:
1687:
1684:
1683:
1641:
1637:
1609:
1605:
1603:
1600:
1599:
1597:
1574:
1570:
1561:
1557:
1555:
1552:
1551:
1530:
1526:
1517:
1513:
1511:
1508:
1507:
1475:
1448:
1447:
1424:
1405:
1401:
1399:
1386:
1382:
1379:
1378:
1355:
1336:
1332:
1324:
1317:
1313:
1309:
1307:
1304:
1303:
1296:
1288:
1274:
1264:
1230:
1216:
1202:
1195:
1194:-intercepts at
1177:
1170:
1148:
1140:
1137:
1136:
1135:and convex for
1118:
1104:
1101:
1100:
1093:
1074:
1067:
1053:
1042:
1035:
1028:
1022:
1015:
1008:
997:
971:
967:
958:
954:
933:
929:
923:
919:
907:
903:
901:
898:
897:
889:The equivalent
867:
863:
854:
850:
838:
834:
825:
821:
812:
808:
796:
792:
786:
782:
773:
769:
764:
761:
760:
734:
730:
718:
714:
705:
701:
677:
673:
664:
660:
643:
640:
639:
625:
615:are called the
599:
595:
586:
582:
580:
577:
576:
547:
543:
535:
529:
525:
517:
503:
497:
493:
485:
474:
471:
470:
445:
442:
441:
424:
420:
418:
415:
414:
397:
393:
384:
380:
378:
375:
374:
358:
352:
348:
340:
331:
325:
321:
313:
311:
308:
307:
287:
284:
283:
254:
250:
242:
236:
232:
224:
210:
204:
200:
192:
190:
187:
186:
179:
163:Cassinian ovals
114:defined as the
90:
88:
83:
81:
76:
74:
68:
64:
62:
56:
52:
50:
43:
39:
28:
24:
17:
12:
11:
5:
4197:
4187:
4186:
4181:
4176:
4174:Quartic curves
4171:
4166:
4152:
4151:
4146:
4141:
4122:
4117:
4103:"Cassini oval"
4097:
4096:External links
4094:
4093:
4092:
4083:
4068:
4062:
4037:
4008:
3991:
3988:
3985:
3984:
3957:(1): 131–148.
3937:
3928:
3919:
3910:
3901:
3887:
3878:
3869:
3857:
3847:
3846:
3844:
3841:
3840:
3839:
3832:
3829:
3828:
3827:
3824:
3817:
3815:
3812:
3805:
3795:implicit curve
3791:
3790:
3777:
3773:
3769:
3765:
3759:
3755:
3751:
3747:
3743:
3740:
3737:
3733:
3727:
3723:
3719:
3715:
3711:
3707:
3701:
3697:
3693:
3689:
3673:
3670:
3657:
3654:
3651:
3631:
3628:
3625:
3622:
3617:
3613:
3601:
3600:
3589:
3584:
3580:
3576:
3571:
3567:
3561:
3557:
3553:
3550:
3547:
3542:
3538:
3534:
3529:
3525:
3521:
3516:
3512:
3508:
3505:
3500:
3495:
3489:
3485:
3481:
3476:
3472:
3467:
3451:
3450:
3439:
3435:
3429:
3425:
3421:
3416:
3412:
3407:
3400:
3396:
3392:
3389:
3384:
3379:
3373:
3369:
3365:
3360:
3356:
3352:
3347:
3343:
3338:
3313:
3310:
3307:
3298:and the plane
3296:
3295:
3283:
3277:
3273:
3269:
3264:
3260:
3255:
3248:
3244:
3240:
3237:
3232:
3227:
3221:
3217:
3213:
3208:
3204:
3200:
3195:
3191:
3187:
3182:
3178:
3174:
3169:
3165:
3160:
3124:
3121:
3096:
3093:
3090:
3087:
3084:
3081:
3078:
3075:
3070:
3066:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3039:
3035:
3018:
3015:
3002:
2999:
2994:
2990:
2986:
2981:
2977:
2956:
2936:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2878:
2875:
2872:
2852:
2832:
2829:
2826:
2823:
2820:
2817:
2797:
2794:
2791:
2788:
2785:
2782:
2762:
2759:
2756:
2736:
2733:
2730:
2710:
2705:
2701:
2697:
2692:
2688:
2684:
2681:
2678:
2675:
2672:
2669:
2664:
2660:
2656:
2651:
2647:
2643:
2623:
2620:
2617:
2597:
2594:
2591:
2588:
2568:
2565:
2562:
2542:
2539:
2536:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2493:
2489:
2485:
2480:
2476:
2472:
2454:Implicit curve
2436:
2432:
2426:
2422:
2418:
2414:
2409:
2405:
2402:
2398:
2392:
2388:
2384:
2380:
2375:
2371:
2332:
2329:
2326:
2323:
2320:
2316:
2313:
2310:
2307:
2304:
2301:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2230:
2227:
2207:
2203:
2200:
2197:
2194:
2188:
2185:
2181:
2176:
2171:
2168:
2163:
2158:
2155:
2150:
2147:
2144:
2138:
2135:
2130:
2127:
2122:
2118:
2114:
2109:
2105:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2060:
2036:
2033:
2030:
2027:
2024:
2021:
2002:
2001:
1990:
1986:
1982:
1979:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1938:
1934:
1930:
1925:
1921:
1900:
1897:
1894:
1891:
1888:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1816:
1805:
1802:
1797:
1793:
1789:
1786:
1783:
1780:
1775:
1771:
1767:
1762:
1758:
1754:
1751:
1748:
1743:
1739:
1733:
1729:
1725:
1720:
1716:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1644:
1640:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1612:
1608:
1592:
1591:
1577:
1573:
1569:
1564:
1560:
1533:
1529:
1525:
1520:
1516:
1474:
1471:
1446:
1443:
1440:
1437:
1434:
1431:
1427:
1425:
1422:
1416:
1413:
1408:
1404:
1398:
1395:
1392:
1389:
1385:
1381:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1358:
1356:
1353:
1349:
1346:
1339:
1335:
1331:
1328:
1323:
1320:
1316:
1312:
1311:
1158:
1152:
1147:
1144:
1122:
1117:
1114:
1111:
1108:
996:
993:
992:
991:
979:
974:
970:
966:
961:
957:
953:
950:
947:
944:
941:
936:
932:
926:
922:
918:
915:
910:
906:
887:
886:
875:
870:
866:
862:
857:
853:
849:
846:
841:
837:
833:
828:
824:
820:
815:
811:
807:
804:
799:
795:
789:
785:
781:
776:
772:
768:
754:
753:
742:
737:
733:
729:
726:
721:
717:
713:
708:
704:
700:
697:
694:
691:
688:
685:
680:
676:
672:
667:
663:
659:
656:
653:
650:
647:
624:
621:
602:
598:
594:
589:
585:
573:
572:
561:
555:
550:
546:
542:
538:
532:
528:
524:
520:
513:
506:
500:
496:
492:
488:
484:
481:
478:
455:
452:
449:
427:
423:
400:
396:
392:
387:
383:
361:
355:
351:
347:
343:
338:
334:
328:
324:
320:
316:
291:
257:
253:
249:
245:
239:
235:
231:
227:
220:
213:
207:
203:
199:
195:
185:Cassini oval:
178:
175:
122:such that the
63:
51:
38:
15:
9:
6:
4:
3:
2:
4196:
4185:
4182:
4180:
4177:
4175:
4172:
4170:
4167:
4165:
4162:
4161:
4159:
4150:
4147:
4145:
4142:
4137:
4136:
4131:
4128:
4123:
4121:
4118:
4114:
4110:
4109:
4104:
4100:
4099:
4090:
4089:
4084:
4080:
4076:
4075:
4069:
4065:
4063:0-486-60288-5
4059:
4055:
4051:
4046:
4045:
4038:
4034:
4030:
4026:
4022:
4018:
4014:
4009:
4005:
4001:
4000:
3994:
3993:
3980:
3976:
3972:
3968:
3964:
3960:
3956:
3952:
3948:
3941:
3935:Basset p. 164
3932:
3923:
3917:Basset p. 163
3914:
3908:Basset p. 163
3905:
3897:
3891:
3882:
3876:Basset p. 163
3873:
3866:
3861:
3852:
3848:
3838:
3835:
3834:
3821:
3816:
3809:
3804:
3803:
3802:
3800:
3796:
3775:
3771:
3767:
3757:
3753:
3749:
3741:
3738:
3735:
3725:
3721:
3717:
3709:
3699:
3695:
3691:
3679:
3678:
3677:
3669:
3655:
3652:
3649:
3629:
3626:
3623:
3620:
3615:
3611:
3587:
3582:
3578:
3574:
3569:
3565:
3559:
3555:
3551:
3548:
3540:
3536:
3532:
3527:
3523:
3514:
3510:
3506:
3503:
3498:
3493:
3487:
3483:
3479:
3474:
3470:
3465:
3456:
3455:
3454:
3437:
3433:
3427:
3423:
3419:
3414:
3410:
3405:
3398:
3394:
3390:
3387:
3382:
3377:
3371:
3367:
3363:
3358:
3354:
3350:
3345:
3341:
3336:
3327:
3326:
3325:
3311:
3308:
3305:
3281:
3275:
3271:
3267:
3262:
3258:
3253:
3246:
3242:
3238:
3235:
3230:
3225:
3219:
3215:
3211:
3206:
3202:
3198:
3193:
3189:
3185:
3180:
3176:
3172:
3167:
3163:
3158:
3149:
3148:
3147:
3144:
3142:
3134:
3133:spindle torus
3129:
3120:
3118:
3114:
3113:complex plane
3110:
3094:
3088:
3085:
3082:
3076:
3073:
3068:
3064:
3057:
3054:
3051:
3048:
3042:
3037:
3033:
3024:
3014:
3000:
2997:
2992:
2988:
2984:
2979:
2975:
2954:
2934:
2911:
2908:
2905:
2899:
2896:
2876:
2873:
2870:
2850:
2827:
2824:
2821:
2818:
2792:
2789:
2786:
2783:
2760:
2757:
2754:
2734:
2731:
2728:
2703:
2699:
2695:
2690:
2686:
2679:
2676:
2673:
2670:
2662:
2658:
2654:
2649:
2645:
2621:
2618:
2615:
2595:
2592:
2589:
2586:
2566:
2563:
2560:
2540:
2537:
2534:
2514:
2511:
2508:
2505:
2502:
2499:
2491:
2487:
2483:
2478:
2474:
2461:
2459:
2455:
2434:
2424:
2420:
2416:
2407:
2403:
2400:
2390:
2386:
2382:
2373:
2369:
2361:
2357:
2356:point charges
2353:
2352:equipotential
2348:
2344:
2330:
2327:
2324:
2321:
2318:
2314:
2308:
2305:
2302:
2279:
2276:
2270:
2267:
2264:
2258:
2255:
2252:
2249:
2243:
2240:
2237:
2231:
2228:
2225:
2205:
2201:
2198:
2195:
2192:
2186:
2183:
2179:
2174:
2169:
2166:
2161:
2156:
2153:
2148:
2145:
2142:
2136:
2133:
2128:
2125:
2120:
2116:
2112:
2107:
2103:
2096:
2090:
2087:
2084:
2078:
2058:
2050:
2049:discriminants
2031:
2028:
2025:
2022:
2011:
2007:
1988:
1980:
1977:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1936:
1932:
1928:
1923:
1919:
1895:
1892:
1889:
1863:
1860:
1857:
1854:
1848:
1842:
1839:
1836:
1825:
1821:
1817:
1803:
1800:
1795:
1791:
1787:
1784:
1781:
1773:
1769:
1765:
1760:
1756:
1749:
1746:
1741:
1731:
1727:
1723:
1718:
1714:
1707:
1701:
1698:
1695:
1689:
1681:
1680:
1679:
1662:
1659:
1656:
1653:
1647:
1642:
1638:
1633:
1627:
1624:
1621:
1615:
1610:
1606:
1596:
1575:
1571:
1567:
1562:
1558:
1549:
1531:
1527:
1523:
1518:
1514:
1505:
1504:
1503:
1501:
1497:
1493:
1489:
1488:
1479:
1470:
1468:
1464:
1444:
1438:
1435:
1432:
1426:
1420:
1414:
1411:
1406:
1402:
1396:
1393:
1390:
1387:
1383:
1375:
1369:
1366:
1363:
1357:
1351:
1347:
1344:
1337:
1333:
1329:
1326:
1321:
1318:
1314:
1299:
1292:
1285:
1281:
1277:
1271:
1267:
1262:
1258:
1254:
1249:
1247:
1243:
1239:
1233:
1228:
1225:
1219:
1213:
1209:
1205:
1199:
1193:
1188:
1186:
1180:
1173:
1156:
1150:
1145:
1142:
1120:
1115:
1112:
1109:
1106:
1096:
1091:
1087:
1083:
1077:
1070:
1064:
1060:
1056:
1046:
1039:
1032:
1025:
1019:
1012:
1007: =
1006:
1001:
977:
972:
968:
964:
959:
955:
951:
948:
945:
942:
939:
934:
930:
924:
920:
916:
913:
908:
904:
896:
895:
894:
892:
873:
868:
864:
860:
855:
851:
847:
839:
835:
831:
826:
822:
813:
809:
805:
802:
797:
787:
783:
779:
774:
770:
759:
758:
757:
740:
735:
731:
727:
719:
715:
711:
706:
698:
695:
692:
678:
674:
670:
665:
657:
654:
651:
638:
637:
636:
634:
630:
620:
618:
600:
596:
592:
587:
583:
559:
548:
544:
540:
530:
526:
522:
511:
498:
494:
490:
482:
479:
469:
468:
467:
453:
450:
447:
425:
421:
398:
394:
390:
385:
381:
353:
349:
345:
336:
326:
322:
318:
305:
289:
281:
273:
255:
251:
247:
237:
233:
229:
218:
205:
201:
197:
183:
174:
172:
168:
164:
160:
156:
151:
149:
145:
141:
137:
133:
129:
125:
121:
117:
113:
109:
108:
102:
86:
72:
59:
47:
35:
31:
21:
4169:Plane curves
4133:
4106:
4086:
4073:
4043:
4019:(1): 72–82.
4016:
4012:
3998:
3990:Bibliography
3954:
3950:
3940:
3931:
3922:
3913:
3904:
3890:
3881:
3872:
3860:
3851:
3792:
3675:
3602:
3452:
3297:
3145:
3138:
3116:
3108:
3020:
2462:
2346:
2345:
2009:
2005:
2003:
1879:with center
1594:
1593:
1485:
1484:
1466:
1462:
1297:
1290:
1283:
1279:
1275:
1272:
1265:
1250:
1245:
1241:
1237:
1231:
1226:
1217:
1211:
1207:
1203:
1197:
1191:
1189:
1178:
1171:
1094:
1086:double point
1075:
1068:
1062:
1058:
1054:
1051:
1044:
1037:
1030:
1023:
1017:
1010:
1004:
893:equation is
888:
755:
632:
628:
626:
616:
574:
303:
280:Cassini oval
279:
277:
274:on the curve
271:
170:
166:
162:
152:
135:
104:
98:
84:
70:
57:
45:
33:
29:
4052:. pp.
3021:The second
1550:containing
1490:of a given
82:Not shown:
4158:Categories
4054:5, 153–155
3865:Cohen 1962
3843:References
1824:asymptotes
1500:hyperbolas
1282:= ±
1257:bicircular
144:polynomial
27:(equal to
4135:MathWorld
4113:EMS Press
3979:253597755
3971:0047-2468
3742:×
3739:⋯
3736:×
3710:×
3575:−
3533:−
3504:−
3212:−
3058:
2825:±
2784:±
2696:−
2593:−
2500:−
2328:≠
2322:≠
2256:
2250:⋅
2229:
2196:λ
2193:−
2175:−
2162:−
2146:λ
2143:−
2126:−
2113:−
2059:λ
2023:±
2012:-axis at
1981:∈
1978:λ
1954:−
1945:λ
1942:−
1929:−
1855:−
1788:−
1766:−
1747:−
1654:−
1412:−
1394:±
1330:−
1319:±
1289:(±
1181:→ ∞
1146:≥
965:−
949:θ
943:
914:−
832:−
803:−
655:−
623:Equations
512:×
219:×
146:used has
142:when the
95:(convex).
37:) falls:
4115:. 2001 .
4033:24620858
3831:See also
3017:Examples
2439:constant
2292:for all
2047:. Their
1546:are the
1496:confocal
1196:±
159:orbiting
105:Cassini
101:geometry
3855:Cassini
3324:yields
3111:on the
2452:. (See
2347:Remark:
1822:(their
1215:. When
1176:(hence
1090:crunode
1066:. When
304:product
132:ellipse
124:product
89:√
75:√
69:1 <
44:0 <
4060:
4031:
3977:
3969:
3885:Lawden
1975:
1972:
1969:
1595:Proof:
1492:pencil
1244:- and
1234:> 1
1220:< 1
1201:where
1185:circle
1097:> 1
1071:< 1
557:
440:where
148:degree
67:
65:
55:
53:
48:< 1
42:
40:
4029:JSTOR
3975:S2CID
995:Shape
891:polar
120:plane
116:locus
110:is a
73:<
4058:ISBN
3967:ISSN
3642:and
3141:tori
2927:and
2808:and
2747:and
2619:<
2564:>
2253:grad
2226:grad
1818:The
1436:>
1367:<
1293:, 0)
1224:real
1116:<
1110:<
617:foci
451:>
169:and
128:foci
107:oval
103:, a
4081:ff.
4079:162
4021:doi
3959:doi
3955:103
3055:abs
2608:if
2553:if
1678:.
1300:≠ 1
1268:≠ 1
1174:→ 0
1078:= 1
1043:1.6
1036:1.4
1029:1.2
1016:0.8
1009:0.6
940:cos
466::
150:2.
136:sum
99:In
60:= 1
4160::
4132:.
4111:.
4105:.
4056:.
4048:.
4027:.
4017:17
4015:.
4004:36
3973:.
3965:.
3953:.
3949:.
3801:.
3668:.
3013:.
2579:,
1804:0.
1280:iy
1278:±
1210:+
1206:=
1187:.
1057:=
1041:,
1034:,
1027:,
1021:,
1014:,
278:A
173:.
165:,
87:≥
4138:.
4066:.
4035:.
4023::
4006:.
3981:.
3961::
3898:.
3867:.
3776:n
3772:b
3768:=
3764:|
3758:n
3754:P
3750:P
3746:|
3732:|
3726:2
3722:P
3718:P
3714:|
3706:|
3700:1
3696:P
3692:P
3688:|
3656:R
3653:=
3650:a
3630:r
3627:R
3624:2
3621:=
3616:2
3612:b
3588:,
3583:4
3579:R
3570:2
3566:r
3560:2
3556:R
3552:4
3549:=
3546:)
3541:2
3537:z
3528:2
3524:x
3520:(
3515:2
3511:R
3507:2
3499:2
3494:)
3488:2
3484:z
3480:+
3475:2
3471:x
3466:(
3438:.
3434:)
3428:2
3424:r
3420:+
3415:2
3411:x
3406:(
3399:2
3395:R
3391:4
3388:=
3383:2
3378:)
3372:2
3368:R
3364:+
3359:2
3355:z
3351:+
3346:2
3342:x
3337:(
3312:r
3309:=
3306:y
3282:)
3276:2
3272:y
3268:+
3263:2
3259:x
3254:(
3247:2
3243:R
3239:4
3236:=
3231:2
3226:)
3220:2
3216:r
3207:2
3203:R
3199:+
3194:2
3190:z
3186:+
3181:2
3177:y
3173:+
3168:2
3164:x
3159:(
3135:)
3117:z
3109:c
3095:.
3092:}
3089:R
3086:E
3083:=
3080:)
3077:c
3074:+
3069:2
3065:c
3061:(
3052::
3049:c
3046:{
3043:=
3038:2
3034:L
3001:1
2998:=
2993:2
2989:y
2985:+
2980:2
2976:x
2955:P
2935:Q
2915:)
2912:1
2909:,
2906:0
2903:(
2900:=
2897:O
2877:Q
2874:P
2871:O
2851:P
2831:)
2828:1
2822:,
2819:0
2816:(
2796:)
2793:0
2790:,
2787:1
2781:(
2761:0
2758:=
2755:x
2735:0
2732:=
2729:y
2709:)
2704:2
2700:y
2691:2
2687:x
2683:(
2680:b
2677:=
2674:1
2671:+
2668:)
2663:2
2659:y
2655:+
2650:2
2646:x
2642:(
2622:0
2616:a
2596:x
2590:=
2587:y
2567:0
2561:a
2541:x
2538:=
2535:y
2515:y
2512:x
2509:a
2506:=
2503:1
2497:)
2492:2
2488:y
2484:+
2479:2
2475:x
2471:(
2435:=
2431:|
2425:2
2421:P
2417:P
2413:|
2408:/
2404:1
2401:+
2397:|
2391:1
2387:P
2383:P
2379:|
2374:/
2370:1
2331:y
2325:0
2319:x
2315:,
2312:)
2309:y
2306:,
2303:x
2300:(
2280:0
2277:=
2274:)
2271:y
2268:,
2265:x
2262:(
2259:g
2247:)
2244:y
2241:,
2238:x
2235:(
2232:f
2206:.
2202:0
2199:=
2187:y
2184:x
2180:1
2170:x
2167:y
2157:y
2154:x
2149:=
2137:y
2134:x
2129:1
2121:2
2117:y
2108:2
2104:x
2097:=
2094:)
2091:y
2088:,
2085:x
2082:(
2079:g
2035:)
2032:0
2029:,
2026:1
2020:(
2010:x
2006:y
1989:.
1985:R
1966:,
1963:0
1960:=
1957:1
1951:y
1948:x
1937:2
1933:y
1924:2
1920:x
1899:)
1896:0
1893:,
1890:0
1887:(
1867:)
1864:0
1861:,
1858:1
1852:(
1849:,
1846:)
1843:0
1840:,
1837:1
1834:(
1801:=
1796:4
1792:b
1785:1
1782:+
1779:)
1774:2
1770:y
1761:2
1757:x
1753:(
1750:2
1742:2
1738:)
1732:2
1728:y
1724:+
1719:2
1715:x
1711:(
1708:=
1705:)
1702:y
1699:,
1696:x
1693:(
1690:f
1666:)
1663:0
1660:,
1657:1
1651:(
1648:=
1643:2
1639:P
1634:,
1631:)
1628:0
1625:,
1622:1
1619:(
1616:=
1611:1
1607:P
1576:2
1572:P
1568:,
1563:1
1559:P
1532:2
1528:P
1524:,
1519:1
1515:P
1467:y
1463:x
1445:.
1442:)
1439:1
1433:e
1430:(
1421:)
1415:1
1407:4
1403:e
1397:a
1391:,
1388:0
1384:(
1376:,
1373:)
1370:1
1364:e
1361:(
1352:)
1348:0
1345:,
1338:4
1334:e
1327:1
1322:a
1315:(
1298:e
1291:a
1284:a
1276:x
1266:e
1246:y
1242:x
1238:y
1232:e
1227:x
1218:e
1212:b
1208:a
1204:c
1198:c
1192:x
1179:e
1172:a
1157:.
1151:2
1143:e
1121:2
1113:e
1107:1
1095:e
1076:e
1069:e
1063:a
1061:/
1059:b
1055:e
1048:)
1045:a
1038:a
1031:a
1024:a
1018:a
1011:a
1005:b
978:.
973:4
969:a
960:4
956:b
952:=
946:2
935:2
931:r
925:2
921:a
917:2
909:4
905:r
874:.
869:4
865:b
861:=
856:4
852:a
848:+
845:)
840:2
836:y
827:2
823:x
819:(
814:2
810:a
806:2
798:2
794:)
788:2
784:y
780:+
775:2
771:x
767:(
741:.
736:4
732:b
728:=
725:)
720:2
716:y
712:+
707:2
703:)
699:a
696:+
693:x
690:(
687:(
684:)
679:2
675:y
671:+
666:2
662:)
658:a
652:x
649:(
646:(
633:a
629:a
601:2
597:P
593:,
588:1
584:P
560:.
554:}
549:2
545:b
541:=
537:|
531:2
527:P
523:P
519:|
505:|
499:1
495:P
491:P
487:|
483::
480:P
477:{
454:0
448:b
426:2
422:b
399:2
395:P
391:,
386:1
382:P
360:|
354:2
350:P
346:P
342:|
337:,
333:|
327:1
323:P
319:P
315:|
290:P
272:P
256:2
252:b
248:=
244:|
238:2
234:P
230:P
226:|
212:|
206:1
202:P
198:P
194:|
91:2
85:e
77:2
71:e
58:e
46:e
34:a
32:/
30:b
25:e
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