2952:
2134:
2947:{\displaystyle {\begin{aligned}A_{x}&=\int _{a}^{b}\int _{0}^{2\pi }\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt\\&=\int _{a}^{b}\int _{0}^{2\pi }\left\|y\left\langle y\cos(\theta ){\frac {dx}{dt}},y\sin(\theta ){\frac {dx}{dt}},y{\frac {dy}{dt}}\right\rangle \right\|\ d\theta \ dt\\&=\int _{a}^{b}\int _{0}^{2\pi }y{\sqrt {\cos ^{2}(\theta )\left({\frac {dx}{dt}}\right)^{2}+\sin ^{2}(\theta )\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ d\theta \ dt\\&=\int _{a}^{b}\int _{0}^{2\pi }y{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ d\theta \ dt\\&=\int _{a}^{b}2\pi y{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ dt\end{aligned}}}
31:
1513:
2059:
4370:
1237:
3582:
1770:
54:
3266:
3228:
1508:{\displaystyle A_{x}=\iint _{S}dS=\iint _{\times }\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt=\int _{a}^{b}\int _{0}^{2\pi }\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt.}
2054:{\displaystyle {\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}=\left\langle y\cos(\theta ){\frac {dx}{dt}},y\sin(\theta ){\frac {dx}{dt}},y{\frac {dy}{dt}}\right\rangle =y\left\langle \cos(\theta ){\frac {dx}{dt}},\sin(\theta ){\frac {dx}{dt}},{\frac {dy}{dt}}\right\rangle }
1666:
902:
2998:
4377:
A surface of revolution with a hole in, where the axis of revolution does not intersect the surface, is called a toroid. For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow square-section ring is produced. If the revolved figure is a
3577:{\displaystyle {\begin{aligned}A&=2\pi \int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,{\sqrt {1+{\frac {x^{2}}{r^{2}-x^{2}}}}}\,dx\\&=2\pi r\int _{-r}^{r}\,{\sqrt {r^{2}-x^{2}}}\,{\sqrt {\frac {1}{r^{2}-x^{2}}}}\,dx\\&=2\pi r\int _{-r}^{r}\,dx\\&=4\pi r^{2}\,\end{aligned}}}
674:
368:
1761:
1520:
710:
1194:
1012:
485:
2139:
3223:{\displaystyle {\begin{aligned}A&{}=2\pi \int _{0}^{\pi }\sin(t){\sqrt {{\big (}\cos(t){\big )}^{2}+{\big (}\sin(t){\big )}^{2}}}\,dt\\&{}=2\pi \int _{0}^{\pi }\sin(t)\,dt\\&{}=4\pi .\end{aligned}}}
528:
222:
4338:
4243:
1670:
3849:
2129:
3985:
3271:
3003:
3908:
1076:
1229:
3723:
911:
396:
4088:
4025:
3763:
1081:
3661:
1661:{\displaystyle {\frac {\partial \mathbf {r} }{\partial t}}=\left\langle {\frac {dy}{dt}}\cos(\theta ),{\frac {dy}{dt}}\sin(\theta ),{\frac {dx}{dt}}\right\rangle ,}
4148:
4128:
4108:
4068:
4048:
4005:
3743:
897:{\displaystyle A_{x}=2\pi \int _{a}^{b}y{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx=2\pi \int _{a}^{b}f(x){\sqrt {1+{\big (}f'(x){\big )}^{2}}}\,dx}
122:
depending on whether or not the line is parallel to the axis. A circle that is rotated around any diameter generates a sphere of which it is then a
4433:
2954:
where the same trigonometric identity was used again. The derivation for a surface obtained by revolving around the y-axis is similar.
4248:
4153:
17:
3768:
2064:
3913:
669:{\displaystyle A_{x}=2\pi \int _{a}^{b}y(t)\,{\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt.}
363:{\displaystyle A_{y}=2\pi \int _{a}^{b}x(t)\,{\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt,}
4557:
4524:
126:, and if the circle is rotated around an axis that does not intersect the interior of a circle, then it generates a
1756:{\displaystyle {\frac {\partial \mathbf {r} }{\partial \theta }}=\langle -y\sin(\theta ),y\cos(\theta ),0\rangle }
4712:
3854:
1078:
then its corresponding surface of revolution when revolved around the x-axis has
Cartesian coordinates given by
3605:
1031:
105:
the generatrix, except at its endpoints). The volume bounded by the surface created by this revolution is the
3587:
1199:
146:. Any meridional section can be considered to be the generatrix in the plane determined by it and the axis.
390:
3666:
149:
The sections of the surface of revolution made by planes that are perpendicular to the axis are circles.
4707:
3591:
4150:, then the surface of revolution obtained by revolving the curve around the x-axis is described by
161:
4245:, and the surface of revolution obtained by revolving the curve around the y-axis is described by
4514:
102:
4686:
4580:
1189:{\displaystyle \mathbf {r} (t,\theta )=\langle y(t)\cos(\theta ),y(t)\sin(\theta ),x(t)\rangle }
4352:
3609:
3598:
4549:
4542:
4073:
4010:
3748:
160:
are surfaces of revolution. These may be identified as those quadratic surfaces all of whose
70:
3631:
4406:
4348:
3601:
is finding the curve between two points that produces this minimal surface of revolution.
1007:{\displaystyle A_{y}=2\pi \int _{a}^{b}x{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx}
8:
4671:
488:
157:
115:
107:
4490:
Thomas, George B. "6.7: Area of a
Surface of Revolution; 6.11: The Theorems of Pappus".
4133:
4113:
4093:
4053:
4033:
3990:
3728:
480:{\displaystyle {\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}}
1028:
This can also be derived from multivariable integration. If a plane curve is given by
4668:
4640:
4602:
4577:
4553:
4520:
4495:
4457:
4417:
4401:
507:
is the path of (the centroid of) this small segment, as required by Pappus' theorem.
142:
The sections of the surface of revolution made by planes through the axis are called
92:
30:
3851:. If instead we revolve the curve around the y-axis, then the curve is described by
4428:
4411:
3613:
1232:
173:
97:
4395:
4351:
are always geodesics on a surface of revolution. Other geodesics are governed by
119:
74:
4605:
4701:
1764:
4643:
3594:
207:
153:
123:
4571:
4569:
3590:
is the surface of revolution of the curve between two given points which
4566:
492:
131:
87:
4575:
4676:
4648:
4610:
4585:
61:
as a square revolved around an axis parallel to one of its diagonals.
114:
Examples of surfaces of revolution generated by a straight line are
4423:
4369:
3617:
217:
78:
491:
and represents a small segment of the arc of the curve, as in the
4499:
4461:
4379:
4364:
3628:
A surface of revolution given by rotating a curve described by
2958:
202:
ranging over some interval , and the axis of revolution is the
53:
4666:
4383:
127:
82:
58:
4548:(Alternate ed.). Prindle, Weber & Schmidt. p.
4452:
Middlemiss; Marks; Smart. "15-4. Surfaces of
Revolution".
4333:{\displaystyle (x(t)\cos(\theta ),y(t),x(t)\sin(\theta ))}
4238:{\displaystyle (x(t),y(t)\cos(\theta ),y(t)\sin(\theta ))}
3844:{\displaystyle (x,f(x)\cos(\theta ),f(x)\sin(\theta ))}
2124:{\displaystyle \sin ^{2}(\theta )+\cos ^{2}(\theta )=1}
4251:
4156:
4136:
4116:
4096:
4076:
4056:
4036:
4013:
3993:
3916:
3857:
3771:
3751:
3731:
3669:
3634:
3269:
3001:
2137:
2067:
1773:
1673:
1523:
1240:
1202:
1084:
1034:
914:
713:
678:
If the continuous curve is described by the function
531:
399:
225:
3980:{\displaystyle (x\cos(\theta ),f(x),x\sin(\theta ))}
4479:(Revised ed.), D.C. Heath and Co., p. 227
4420:, another generalization of a surface of revolution
4691:Encyclopédie des Formes Mathématiques Remarquables
4541:
4332:
4237:
4142:
4122:
4102:
4082:
4062:
4042:
4019:
3999:
3979:
3902:
3843:
3757:
3737:
3717:
3663:around the x-axis may be most simply described by
3655:
3576:
3222:
2946:
2123:
2053:
1755:
1660:
1507:
1223:
1188:
1070:
1006:
896:
668:
479:
362:
4600:
27:Surface created by rotating a curve about an axis
4699:
4451:
3232:For the case of the spherical curve with radius
4638:
4629:, 2nd ed., Springer, London, 2012, pp. 227–230.
4050:, then we obtain a parametrization in terms of
4030:If x and y are defined in terms of a parameter
4519:(6 ed.). Tata McGraw-Hill. p. 6.90.
3725:. This yields the parametrization in terms of
389:. This formula is the calculus equivalent of
4398:, a generalisation of a surface of revolution
3120:
3097:
3081:
3058:
874:
849:
4474:
1750:
1699:
1183:
1108:
1065:
1035:
4625:Pressley, Andrew. “Chapter 9 - Geodesics.”
4494:(3rd ed.). pp. 206–209, 217–219.
4434:Translation surface (differential geometry)
2961:with unit radius is generated by the curve
510:Likewise, when the axis of rotation is the
3903:{\displaystyle y=f({\sqrt {x^{2}+z^{2}}})}
2131:was used. With this cross product, we get
4539:
4414:, surface of revolution of a circular arc
3623:
3569:
3536:
3492:
3457:
3429:
3385:
3335:
3188:
3133:
1517:Computing the partial derivatives yields
1071:{\displaystyle \langle x(t),y(t)\rangle }
997:
887:
796:
656:
578:
350:
272:
4368:
1231:. Then the surface area is given by the
525:is never negative, the area is given by
381:is never negative between the endpoints
164:perpendicular to the axis are circular.
52:
29:
1224:{\displaystyle 0\leq \theta \leq 2\pi }
14:
4700:
3612:which are also minimal surfaces): the
1025:). These come from the above formula.
4667:
4639:
4601:
4576:
4512:
2995:ranges over . Its area is therefore
3718:{\displaystyle y^{2}+z^{2}=f(x)^{2}}
4475:Wilson, W.A.; Tracey, J.I. (1925),
130:which does not intersect itself (a
24:
4489:
2235:
2225:
2210:
2200:
1812:
1802:
1787:
1777:
1687:
1677:
1537:
1527:
1470:
1460:
1445:
1435:
1359:
1349:
1334:
1324:
156:(of either one or two sheets) and
25:
4724:
4660:
2061:where the trigonometric identity
172:If the curve is described by the
4627:Elementary Differential Geometry
4373:A toroid generated from a square
2229:
2204:
1806:
1781:
1681:
1531:
1464:
1439:
1353:
1328:
1086:
4581:"Minimal Surface of Revolution"
4544:Calculus with analytic geometry
167:
4632:
4619:
4594:
4533:
4506:
4483:
4468:
4445:
4382:, then the object is called a
4327:
4324:
4318:
4309:
4303:
4294:
4288:
4279:
4273:
4264:
4258:
4252:
4232:
4229:
4223:
4214:
4208:
4199:
4193:
4184:
4178:
4169:
4163:
4157:
3974:
3971:
3965:
3950:
3944:
3935:
3929:
3917:
3897:
3867:
3838:
3835:
3829:
3820:
3814:
3805:
3799:
3790:
3784:
3772:
3706:
3699:
3650:
3644:
3606:minimal surfaces of revolution
3185:
3179:
3114:
3108:
3075:
3069:
3051:
3045:
2583:
2577:
2523:
2517:
2434:
2379:
2373:
2338:
2332:
2311:
2245:
2193:
2112:
2106:
2087:
2081:
2000:
1994:
1962:
1956:
1885:
1879:
1844:
1838:
1741:
1735:
1720:
1714:
1624:
1618:
1586:
1580:
1480:
1428:
1369:
1317:
1311:
1296:
1290:
1278:
1180:
1174:
1165:
1159:
1150:
1144:
1135:
1129:
1120:
1114:
1102:
1090:
1062:
1056:
1047:
1041:
868:
862:
836:
830:
575:
569:
269:
263:
13:
1:
4456:(3rd ed.). p. 378.
4439:
3588:minimal surface of revolution
137:
4343:
707:, then the integral becomes
7:
4540:Swokowski, Earl W. (1983).
4389:
3987:in terms of the parameters
10:
4729:
4362:
4358:
3910:, yielding the expression
1014:for revolution around the
904:for revolution around the
3597:. A basic problem in the
391:Pappus's centroid theorem
514:-axis and provided that
4687:"Surface de révolution"
4672:"Surface of Revolution"
4516:Engineering Mathematics
4083:{\displaystyle \theta }
4020:{\displaystyle \theta }
3758:{\displaystyle \theta }
34:A portion of the curve
4713:Surfaces of revolution
4374:
4334:
4239:
4144:
4124:
4104:
4084:
4064:
4044:
4021:
4001:
3981:
3904:
3845:
3759:
3739:
3719:
3657:
3656:{\displaystyle y=f(x)}
3624:Coordinate expressions
3610:surfaces of revolution
3599:calculus of variations
3578:
3224:
2948:
2125:
2055:
1757:
1662:
1509:
1225:
1190:
1072:
1008:
898:
670:
495:formula. The quantity
481:
364:
152:Some special cases of
62:
50:
18:Surfaces of revolution
4372:
4335:
4240:
4145:
4125:
4105:
4085:
4065:
4045:
4022:
4002:
3982:
3905:
3846:
3760:
3740:
3720:
3658:
3579:
3225:
2949:
2126:
2056:
1758:
1663:
1510:
1226:
1191:
1073:
1009:
899:
671:
482:
365:
67:surface of revolution
56:
33:
4513:Singh, R.R. (1993).
4407:Generalized helicoid
4249:
4154:
4134:
4114:
4094:
4074:
4054:
4034:
4011:
3991:
3914:
3855:
3769:
3749:
3729:
3667:
3632:
3267:
2999:
2135:
2065:
1771:
1671:
1521:
1238:
1200:
1082:
1032:
912:
711:
529:
397:
223:
158:elliptic paraboloids
4353:Clairaut's relation
3604:There are only two
3535:
3428:
3307:
3172:
3038:
2844:
2721:
2703:
2498:
2480:
2309:
2291:
2191:
2173:
1426:
1408:
948:
826:
747:
565:
489:Pythagorean theorem
259:
144:meridional sections
108:solid of revolution
45:rotated around the
4669:Weisstein, Eric W.
4641:Weisstein, Eric W.
4603:Weisstein, Eric W.
4578:Weisstein, Eric W.
4375:
4330:
4235:
4140:
4120:
4100:
4080:
4060:
4040:
4017:
3997:
3977:
3900:
3841:
3755:
3735:
3715:
3653:
3574:
3572:
3518:
3411:
3290:
3259:rotated about the
3220:
3218:
3158:
3024:
2944:
2942:
2830:
2704:
2689:
2481:
2466:
2292:
2277:
2174:
2159:
2121:
2051:
1763:and computing the
1753:
1658:
1505:
1409:
1394:
1221:
1186:
1068:
1004:
934:
894:
812:
733:
666:
551:
477:
360:
245:
63:
51:
4708:Integral calculus
4477:Analytic Geometry
4454:Analytic Geometry
4418:Liouville surface
4143:{\displaystyle t}
4130:are functions of
4123:{\displaystyle y}
4103:{\displaystyle x}
4063:{\displaystyle t}
4043:{\displaystyle t}
4000:{\displaystyle x}
3895:
3738:{\displaystyle x}
3490:
3489:
3455:
3383:
3381:
3333:
3131:
2959:spherical surface
2957:For example, the
2933:
2929:
2917:
2879:
2813:
2804:
2800:
2788:
2750:
2672:
2663:
2659:
2647:
2609:
2549:
2449:
2440:
2426:
2400:
2359:
2260:
2251:
2242:
2217:
2044:
2021:
1983:
1932:
1906:
1865:
1819:
1794:
1694:
1648:
1610:
1572:
1544:
1495:
1486:
1477:
1452:
1384:
1375:
1366:
1341:
995:
983:
885:
794:
782:
654:
642:
604:
475:
463:
425:
348:
336:
298:
16:(Redirected from
4720:
4694:
4682:
4681:
4655:
4654:
4653:
4636:
4630:
4623:
4617:
4616:
4615:
4598:
4592:
4591:
4590:
4573:
4564:
4563:
4547:
4537:
4531:
4530:
4510:
4504:
4503:
4487:
4481:
4480:
4472:
4466:
4465:
4449:
4429:Surface integral
4412:Lemon (geometry)
4339:
4337:
4336:
4331:
4244:
4242:
4241:
4236:
4149:
4147:
4146:
4141:
4129:
4127:
4126:
4121:
4109:
4107:
4106:
4101:
4089:
4087:
4086:
4081:
4069:
4067:
4066:
4061:
4049:
4047:
4046:
4041:
4026:
4024:
4023:
4018:
4006:
4004:
4003:
3998:
3986:
3984:
3983:
3978:
3909:
3907:
3906:
3901:
3896:
3894:
3893:
3881:
3880:
3871:
3850:
3848:
3847:
3842:
3764:
3762:
3761:
3756:
3744:
3742:
3741:
3736:
3724:
3722:
3721:
3716:
3714:
3713:
3692:
3691:
3679:
3678:
3662:
3660:
3659:
3654:
3583:
3581:
3580:
3575:
3573:
3568:
3567:
3546:
3534:
3529:
3502:
3491:
3488:
3487:
3486:
3474:
3473:
3460:
3459:
3456:
3454:
3453:
3441:
3440:
3431:
3427:
3422:
3395:
3384:
3382:
3380:
3379:
3378:
3366:
3365:
3355:
3354:
3345:
3337:
3334:
3332:
3331:
3319:
3318:
3309:
3306:
3301:
3262:
3258:
3257:
3256:
3235:
3229:
3227:
3226:
3221:
3219:
3203:
3198:
3171:
3166:
3148:
3143:
3132:
3130:
3129:
3124:
3123:
3101:
3100:
3091:
3090:
3085:
3084:
3062:
3061:
3055:
3037:
3032:
3014:
2994:
2990:
2975:
2953:
2951:
2950:
2945:
2943:
2931:
2930:
2928:
2927:
2922:
2918:
2916:
2908:
2900:
2890:
2889:
2884:
2880:
2878:
2870:
2862:
2855:
2843:
2838:
2823:
2811:
2802:
2801:
2799:
2798:
2793:
2789:
2787:
2779:
2771:
2761:
2760:
2755:
2751:
2749:
2741:
2733:
2726:
2720:
2712:
2702:
2697:
2682:
2670:
2661:
2660:
2658:
2657:
2652:
2648:
2646:
2638:
2630:
2620:
2619:
2614:
2610:
2608:
2600:
2592:
2573:
2572:
2560:
2559:
2554:
2550:
2548:
2540:
2532:
2513:
2512:
2503:
2497:
2489:
2479:
2474:
2459:
2447:
2438:
2437:
2433:
2432:
2428:
2427:
2425:
2417:
2409:
2401:
2399:
2391:
2383:
2360:
2358:
2350:
2342:
2308:
2300:
2290:
2285:
2270:
2258:
2249:
2248:
2244:
2243:
2241:
2233:
2232:
2223:
2218:
2216:
2208:
2207:
2198:
2190:
2182:
2172:
2167:
2151:
2150:
2130:
2128:
2127:
2122:
2102:
2101:
2077:
2076:
2060:
2058:
2057:
2052:
2050:
2046:
2045:
2043:
2035:
2027:
2022:
2020:
2012:
2004:
1984:
1982:
1974:
1966:
1938:
1934:
1933:
1931:
1923:
1915:
1907:
1905:
1897:
1889:
1866:
1864:
1856:
1848:
1820:
1818:
1810:
1809:
1800:
1795:
1793:
1785:
1784:
1775:
1762:
1760:
1759:
1754:
1695:
1693:
1685:
1684:
1675:
1667:
1665:
1664:
1659:
1654:
1650:
1649:
1647:
1639:
1631:
1611:
1609:
1601:
1593:
1573:
1571:
1563:
1555:
1545:
1543:
1535:
1534:
1525:
1514:
1512:
1511:
1506:
1493:
1484:
1483:
1479:
1478:
1476:
1468:
1467:
1458:
1453:
1451:
1443:
1442:
1433:
1425:
1417:
1407:
1402:
1382:
1373:
1372:
1368:
1367:
1365:
1357:
1356:
1347:
1342:
1340:
1332:
1331:
1322:
1315:
1314:
1263:
1262:
1250:
1249:
1233:surface integral
1230:
1228:
1227:
1222:
1195:
1193:
1192:
1187:
1089:
1077:
1075:
1074:
1069:
1024:
1018:-axis (provided
1013:
1011:
1010:
1005:
996:
994:
993:
988:
984:
982:
974:
966:
953:
947:
942:
924:
923:
907:
903:
901:
900:
895:
886:
884:
883:
878:
877:
861:
853:
852:
840:
825:
820:
795:
793:
792:
787:
783:
781:
773:
765:
752:
746:
741:
723:
722:
706:
692:
675:
673:
672:
667:
655:
653:
652:
647:
643:
641:
633:
625:
615:
614:
609:
605:
603:
595:
587:
580:
564:
559:
541:
540:
524:
513:
506:
486:
484:
483:
478:
476:
474:
473:
468:
464:
462:
454:
446:
436:
435:
430:
426:
424:
416:
408:
401:
388:
384:
380:
369:
367:
366:
361:
349:
347:
346:
341:
337:
335:
327:
319:
309:
308:
303:
299:
297:
289:
281:
274:
258:
253:
235:
234:
216:is given by the
215:
206:-axis, then the
205:
201:
197:
186:
120:conical surfaces
98:axis of rotation
48:
44:
21:
4728:
4727:
4723:
4722:
4721:
4719:
4718:
4717:
4698:
4697:
4685:
4663:
4658:
4637:
4633:
4624:
4620:
4599:
4595:
4574:
4567:
4560:
4538:
4534:
4527:
4511:
4507:
4488:
4484:
4473:
4469:
4450:
4446:
4442:
4396:Channel surface
4392:
4367:
4361:
4346:
4250:
4247:
4246:
4155:
4152:
4151:
4135:
4132:
4131:
4115:
4112:
4111:
4095:
4092:
4091:
4075:
4072:
4071:
4055:
4052:
4051:
4035:
4032:
4031:
4012:
4009:
4008:
3992:
3989:
3988:
3915:
3912:
3911:
3889:
3885:
3876:
3872:
3870:
3856:
3853:
3852:
3770:
3767:
3766:
3750:
3747:
3746:
3730:
3727:
3726:
3709:
3705:
3687:
3683:
3674:
3670:
3668:
3665:
3664:
3633:
3630:
3629:
3626:
3571:
3570:
3563:
3559:
3544:
3543:
3530:
3522:
3500:
3499:
3482:
3478:
3469:
3465:
3464:
3458:
3449:
3445:
3436:
3432:
3430:
3423:
3415:
3393:
3392:
3374:
3370:
3361:
3357:
3356:
3350:
3346:
3344:
3336:
3327:
3323:
3314:
3310:
3308:
3302:
3294:
3277:
3270:
3268:
3265:
3264:
3260:
3248:
3246:
3237:
3233:
3217:
3216:
3202:
3196:
3195:
3167:
3162:
3147:
3141:
3140:
3125:
3119:
3118:
3117:
3096:
3095:
3086:
3080:
3079:
3078:
3057:
3056:
3054:
3033:
3028:
3013:
3009:
3002:
3000:
2997:
2996:
2992:
2977:
2962:
2941:
2940:
2923:
2909:
2901:
2899:
2895:
2894:
2885:
2871:
2863:
2861:
2857:
2856:
2854:
2839:
2834:
2821:
2820:
2794:
2780:
2772:
2770:
2766:
2765:
2756:
2742:
2734:
2732:
2728:
2727:
2725:
2713:
2708:
2698:
2693:
2680:
2679:
2653:
2639:
2631:
2629:
2625:
2624:
2615:
2601:
2593:
2591:
2587:
2586:
2568:
2564:
2555:
2541:
2533:
2531:
2527:
2526:
2508:
2504:
2502:
2490:
2485:
2475:
2470:
2457:
2456:
2418:
2410:
2408:
2392:
2384:
2382:
2351:
2343:
2341:
2322:
2318:
2314:
2310:
2301:
2296:
2286:
2281:
2268:
2267:
2234:
2228:
2224:
2222:
2209:
2203:
2199:
2197:
2196:
2192:
2183:
2178:
2168:
2163:
2152:
2146:
2142:
2138:
2136:
2133:
2132:
2097:
2093:
2072:
2068:
2066:
2063:
2062:
2036:
2028:
2026:
2013:
2005:
2003:
1975:
1967:
1965:
1949:
1945:
1924:
1916:
1914:
1898:
1890:
1888:
1857:
1849:
1847:
1828:
1824:
1811:
1805:
1801:
1799:
1786:
1780:
1776:
1774:
1772:
1769:
1768:
1686:
1680:
1676:
1674:
1672:
1669:
1668:
1640:
1632:
1630:
1602:
1594:
1592:
1564:
1556:
1554:
1553:
1549:
1536:
1530:
1526:
1524:
1522:
1519:
1518:
1469:
1463:
1459:
1457:
1444:
1438:
1434:
1432:
1431:
1427:
1418:
1413:
1403:
1398:
1358:
1352:
1348:
1346:
1333:
1327:
1323:
1321:
1320:
1316:
1277:
1273:
1258:
1254:
1245:
1241:
1239:
1236:
1235:
1201:
1198:
1197:
1085:
1083:
1080:
1079:
1033:
1030:
1029:
1019:
989:
975:
967:
965:
961:
960:
952:
943:
938:
919:
915:
913:
910:
909:
905:
879:
873:
872:
871:
854:
848:
847:
839:
821:
816:
788:
774:
766:
764:
760:
759:
751:
742:
737:
718:
714:
712:
709:
708:
694:
679:
648:
634:
626:
624:
620:
619:
610:
596:
588:
586:
582:
581:
579:
560:
555:
536:
532:
530:
527:
526:
515:
511:
496:
487:comes from the
469:
455:
447:
445:
441:
440:
431:
417:
409:
407:
403:
402:
400:
398:
395:
394:
393:. The quantity
386:
382:
371:
342:
328:
320:
318:
314:
313:
304:
290:
282:
280:
276:
275:
273:
254:
249:
230:
226:
224:
221:
220:
214:
210:
203:
199:
188:
177:
170:
140:
75:Euclidean space
46:
35:
28:
23:
22:
15:
12:
11:
5:
4726:
4716:
4715:
4710:
4696:
4695:
4683:
4662:
4661:External links
4659:
4657:
4656:
4631:
4618:
4593:
4565:
4558:
4532:
4525:
4505:
4482:
4467:
4443:
4441:
4438:
4437:
4436:
4431:
4426:
4421:
4415:
4409:
4404:
4402:Gabriel's Horn
4399:
4391:
4388:
4363:Main article:
4360:
4357:
4345:
4342:
4329:
4326:
4323:
4320:
4317:
4314:
4311:
4308:
4305:
4302:
4299:
4296:
4293:
4290:
4287:
4284:
4281:
4278:
4275:
4272:
4269:
4266:
4263:
4260:
4257:
4254:
4234:
4231:
4228:
4225:
4222:
4219:
4216:
4213:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4180:
4177:
4174:
4171:
4168:
4165:
4162:
4159:
4139:
4119:
4099:
4079:
4059:
4039:
4016:
3996:
3976:
3973:
3970:
3967:
3964:
3961:
3958:
3955:
3952:
3949:
3946:
3943:
3940:
3937:
3934:
3931:
3928:
3925:
3922:
3919:
3899:
3892:
3888:
3884:
3879:
3875:
3869:
3866:
3863:
3860:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3754:
3734:
3712:
3708:
3704:
3701:
3698:
3695:
3690:
3686:
3682:
3677:
3673:
3652:
3649:
3646:
3643:
3640:
3637:
3625:
3622:
3566:
3562:
3558:
3555:
3552:
3549:
3547:
3545:
3542:
3539:
3533:
3528:
3525:
3521:
3517:
3514:
3511:
3508:
3505:
3503:
3501:
3498:
3495:
3485:
3481:
3477:
3472:
3468:
3463:
3452:
3448:
3444:
3439:
3435:
3426:
3421:
3418:
3414:
3410:
3407:
3404:
3401:
3398:
3396:
3394:
3391:
3388:
3377:
3373:
3369:
3364:
3360:
3353:
3349:
3343:
3340:
3330:
3326:
3322:
3317:
3313:
3305:
3300:
3297:
3293:
3289:
3286:
3283:
3280:
3278:
3276:
3273:
3272:
3215:
3212:
3209:
3206:
3201:
3199:
3197:
3194:
3191:
3187:
3184:
3181:
3178:
3175:
3170:
3165:
3161:
3157:
3154:
3151:
3146:
3144:
3142:
3139:
3136:
3128:
3122:
3116:
3113:
3110:
3107:
3104:
3099:
3094:
3089:
3083:
3077:
3074:
3071:
3068:
3065:
3060:
3053:
3050:
3047:
3044:
3041:
3036:
3031:
3027:
3023:
3020:
3017:
3012:
3010:
3008:
3005:
3004:
2939:
2936:
2926:
2921:
2915:
2912:
2907:
2904:
2898:
2893:
2888:
2883:
2877:
2874:
2869:
2866:
2860:
2853:
2850:
2847:
2842:
2837:
2833:
2829:
2826:
2824:
2822:
2819:
2816:
2810:
2807:
2797:
2792:
2786:
2783:
2778:
2775:
2769:
2764:
2759:
2754:
2748:
2745:
2740:
2737:
2731:
2724:
2719:
2716:
2711:
2707:
2701:
2696:
2692:
2688:
2685:
2683:
2681:
2678:
2675:
2669:
2666:
2656:
2651:
2645:
2642:
2637:
2634:
2628:
2623:
2618:
2613:
2607:
2604:
2599:
2596:
2590:
2585:
2582:
2579:
2576:
2571:
2567:
2563:
2558:
2553:
2547:
2544:
2539:
2536:
2530:
2525:
2522:
2519:
2516:
2511:
2507:
2501:
2496:
2493:
2488:
2484:
2478:
2473:
2469:
2465:
2462:
2460:
2458:
2455:
2452:
2446:
2443:
2436:
2431:
2424:
2421:
2416:
2413:
2407:
2404:
2398:
2395:
2390:
2387:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2357:
2354:
2349:
2346:
2340:
2337:
2334:
2331:
2328:
2325:
2321:
2317:
2313:
2307:
2304:
2299:
2295:
2289:
2284:
2280:
2276:
2273:
2271:
2269:
2266:
2263:
2257:
2254:
2247:
2240:
2237:
2231:
2227:
2221:
2215:
2212:
2206:
2202:
2195:
2189:
2186:
2181:
2177:
2171:
2166:
2162:
2158:
2155:
2153:
2149:
2145:
2141:
2140:
2120:
2117:
2114:
2111:
2108:
2105:
2100:
2096:
2092:
2089:
2086:
2083:
2080:
2075:
2071:
2049:
2042:
2039:
2034:
2031:
2025:
2019:
2016:
2011:
2008:
2002:
1999:
1996:
1993:
1990:
1987:
1981:
1978:
1973:
1970:
1964:
1961:
1958:
1955:
1952:
1948:
1944:
1941:
1937:
1930:
1927:
1922:
1919:
1913:
1910:
1904:
1901:
1896:
1893:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1863:
1860:
1855:
1852:
1846:
1843:
1840:
1837:
1834:
1831:
1827:
1823:
1817:
1814:
1808:
1804:
1798:
1792:
1789:
1783:
1779:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1731:
1728:
1725:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1692:
1689:
1683:
1679:
1657:
1653:
1646:
1643:
1638:
1635:
1629:
1626:
1623:
1620:
1617:
1614:
1608:
1605:
1600:
1597:
1591:
1588:
1585:
1582:
1579:
1576:
1570:
1567:
1562:
1559:
1552:
1548:
1542:
1539:
1533:
1529:
1504:
1501:
1498:
1492:
1489:
1482:
1475:
1472:
1466:
1462:
1456:
1450:
1447:
1441:
1437:
1430:
1424:
1421:
1416:
1412:
1406:
1401:
1397:
1393:
1390:
1387:
1381:
1378:
1371:
1364:
1361:
1355:
1351:
1345:
1339:
1336:
1330:
1326:
1319:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1276:
1272:
1269:
1266:
1261:
1257:
1253:
1248:
1244:
1220:
1217:
1214:
1211:
1208:
1205:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1088:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1003:
1000:
992:
987:
981:
978:
973:
970:
964:
959:
956:
951:
946:
941:
937:
933:
930:
927:
922:
918:
893:
890:
882:
876:
870:
867:
864:
860:
857:
851:
846:
843:
838:
835:
832:
829:
824:
819:
815:
811:
808:
805:
802:
799:
791:
786:
780:
777:
772:
769:
763:
758:
755:
750:
745:
740:
736:
732:
729:
726:
721:
717:
665:
662:
659:
651:
646:
640:
637:
632:
629:
623:
618:
613:
608:
602:
599:
594:
591:
585:
577:
574:
571:
568:
563:
558:
554:
550:
547:
544:
539:
535:
472:
467:
461:
458:
453:
450:
444:
439:
434:
429:
423:
420:
415:
412:
406:
370:provided that
359:
356:
353:
345:
340:
334:
331:
326:
323:
317:
312:
307:
302:
296:
293:
288:
285:
279:
271:
268:
265:
262:
257:
252:
248:
244:
241:
238:
233:
229:
212:
169:
166:
162:cross sections
139:
136:
101:(normally not
26:
9:
6:
4:
3:
2:
4725:
4714:
4711:
4709:
4706:
4705:
4703:
4692:
4688:
4684:
4679:
4678:
4673:
4670:
4665:
4664:
4651:
4650:
4645:
4642:
4635:
4628:
4622:
4613:
4612:
4607:
4604:
4597:
4588:
4587:
4582:
4579:
4572:
4570:
4561:
4559:0-87150-341-7
4555:
4551:
4546:
4545:
4536:
4528:
4526:0-07-014615-2
4522:
4518:
4517:
4509:
4501:
4497:
4493:
4486:
4478:
4471:
4463:
4459:
4455:
4448:
4444:
4435:
4432:
4430:
4427:
4425:
4422:
4419:
4416:
4413:
4410:
4408:
4405:
4403:
4400:
4397:
4394:
4393:
4387:
4385:
4381:
4371:
4366:
4356:
4354:
4350:
4341:
4321:
4315:
4312:
4306:
4300:
4297:
4291:
4285:
4282:
4276:
4270:
4267:
4261:
4255:
4226:
4220:
4217:
4211:
4205:
4202:
4196:
4190:
4187:
4181:
4175:
4172:
4166:
4160:
4137:
4117:
4097:
4077:
4057:
4037:
4028:
4014:
3994:
3968:
3962:
3959:
3956:
3953:
3947:
3941:
3938:
3932:
3926:
3923:
3920:
3890:
3886:
3882:
3877:
3873:
3864:
3861:
3858:
3832:
3826:
3823:
3817:
3811:
3808:
3802:
3796:
3793:
3787:
3781:
3778:
3775:
3752:
3732:
3710:
3702:
3696:
3693:
3688:
3684:
3680:
3675:
3671:
3647:
3641:
3638:
3635:
3621:
3619:
3615:
3611:
3607:
3602:
3600:
3596:
3593:
3589:
3584:
3564:
3560:
3556:
3553:
3550:
3548:
3540:
3537:
3531:
3526:
3523:
3519:
3515:
3512:
3509:
3506:
3504:
3496:
3493:
3483:
3479:
3475:
3470:
3466:
3461:
3450:
3446:
3442:
3437:
3433:
3424:
3419:
3416:
3412:
3408:
3405:
3402:
3399:
3397:
3389:
3386:
3375:
3371:
3367:
3362:
3358:
3351:
3347:
3341:
3338:
3328:
3324:
3320:
3315:
3311:
3303:
3298:
3295:
3291:
3287:
3284:
3281:
3279:
3274:
3255:
3251:
3244:
3240:
3230:
3213:
3210:
3207:
3204:
3200:
3192:
3189:
3182:
3176:
3173:
3168:
3163:
3159:
3155:
3152:
3149:
3145:
3137:
3134:
3126:
3111:
3105:
3102:
3092:
3087:
3072:
3066:
3063:
3048:
3042:
3039:
3034:
3029:
3025:
3021:
3018:
3015:
3011:
3006:
2988:
2984:
2980:
2973:
2969:
2965:
2960:
2955:
2937:
2934:
2924:
2919:
2913:
2910:
2905:
2902:
2896:
2891:
2886:
2881:
2875:
2872:
2867:
2864:
2858:
2851:
2848:
2845:
2840:
2835:
2831:
2827:
2825:
2817:
2814:
2808:
2805:
2795:
2790:
2784:
2781:
2776:
2773:
2767:
2762:
2757:
2752:
2746:
2743:
2738:
2735:
2729:
2722:
2717:
2714:
2709:
2705:
2699:
2694:
2690:
2686:
2684:
2676:
2673:
2667:
2664:
2654:
2649:
2643:
2640:
2635:
2632:
2626:
2621:
2616:
2611:
2605:
2602:
2597:
2594:
2588:
2580:
2574:
2569:
2565:
2561:
2556:
2551:
2545:
2542:
2537:
2534:
2528:
2520:
2514:
2509:
2505:
2499:
2494:
2491:
2486:
2482:
2476:
2471:
2467:
2463:
2461:
2453:
2450:
2444:
2441:
2429:
2422:
2419:
2414:
2411:
2405:
2402:
2396:
2393:
2388:
2385:
2376:
2370:
2367:
2364:
2361:
2355:
2352:
2347:
2344:
2335:
2329:
2326:
2323:
2319:
2315:
2305:
2302:
2297:
2293:
2287:
2282:
2278:
2274:
2272:
2264:
2261:
2255:
2252:
2238:
2219:
2213:
2187:
2184:
2179:
2175:
2169:
2164:
2160:
2156:
2154:
2147:
2143:
2118:
2115:
2109:
2103:
2098:
2094:
2090:
2084:
2078:
2073:
2069:
2047:
2040:
2037:
2032:
2029:
2023:
2017:
2014:
2009:
2006:
1997:
1991:
1988:
1985:
1979:
1976:
1971:
1968:
1959:
1953:
1950:
1946:
1942:
1939:
1935:
1928:
1925:
1920:
1917:
1911:
1908:
1902:
1899:
1894:
1891:
1882:
1876:
1873:
1870:
1867:
1861:
1858:
1853:
1850:
1841:
1835:
1832:
1829:
1825:
1821:
1815:
1796:
1790:
1766:
1765:cross product
1747:
1744:
1738:
1732:
1729:
1726:
1723:
1717:
1711:
1708:
1705:
1702:
1696:
1690:
1655:
1651:
1644:
1641:
1636:
1633:
1627:
1621:
1615:
1612:
1606:
1603:
1598:
1595:
1589:
1583:
1577:
1574:
1568:
1565:
1560:
1557:
1550:
1546:
1540:
1515:
1502:
1499:
1496:
1490:
1487:
1473:
1454:
1448:
1422:
1419:
1414:
1410:
1404:
1399:
1395:
1391:
1388:
1385:
1379:
1376:
1362:
1343:
1337:
1308:
1305:
1302:
1299:
1293:
1287:
1284:
1281:
1274:
1270:
1267:
1264:
1259:
1255:
1251:
1246:
1242:
1234:
1218:
1215:
1212:
1209:
1206:
1203:
1177:
1171:
1168:
1162:
1156:
1153:
1147:
1141:
1138:
1132:
1126:
1123:
1117:
1111:
1105:
1099:
1096:
1093:
1059:
1053:
1050:
1044:
1038:
1026:
1022:
1017:
1001:
998:
990:
985:
979:
976:
971:
968:
962:
957:
954:
949:
944:
939:
935:
931:
928:
925:
920:
916:
891:
888:
880:
865:
858:
855:
844:
841:
833:
827:
822:
817:
813:
809:
806:
803:
800:
797:
789:
784:
778:
775:
770:
767:
761:
756:
753:
748:
743:
738:
734:
730:
727:
724:
719:
715:
705:
701:
697:
690:
686:
682:
676:
663:
660:
657:
649:
644:
638:
635:
630:
627:
621:
616:
611:
606:
600:
597:
592:
589:
583:
572:
566:
561:
556:
552:
548:
545:
542:
537:
533:
522:
518:
508:
504:
500:
494:
490:
470:
465:
459:
456:
451:
448:
442:
437:
432:
427:
421:
418:
413:
410:
404:
392:
378:
374:
357:
354:
351:
343:
338:
332:
329:
324:
321:
315:
310:
305:
300:
294:
291:
286:
283:
277:
266:
260:
255:
250:
246:
242:
239:
236:
231:
227:
219:
209:
195:
191:
184:
180:
175:
165:
163:
159:
155:
150:
147:
145:
135:
133:
129:
125:
121:
117:
112:
110:
109:
104:
100:
99:
94:
90:
89:
84:
80:
76:
72:
68:
60:
55:
42:
38:
32:
19:
4693:(in French).
4690:
4675:
4647:
4634:
4626:
4621:
4609:
4596:
4584:
4543:
4535:
4515:
4508:
4491:
4485:
4476:
4470:
4453:
4447:
4376:
4347:
4029:
3627:
3603:
3595:surface area
3585:
3253:
3249:
3242:
3238:
3231:
2986:
2982:
2978:
2971:
2967:
2963:
2956:
1516:
1027:
1020:
1015:
703:
699:
695:
688:
684:
680:
677:
520:
516:
509:
502:
498:
376:
372:
208:surface area
193:
189:
182:
178:
171:
168:Area formula
154:hyperboloids
151:
148:
143:
141:
124:great circle
113:
106:
103:intersecting
96:
86:
66:
64:
40:
36:
908:-axis, and
116:cylindrical
91:) one full
77:created by
4702:Categories
4606:"Catenoid"
4440:References
493:arc length
176:functions
174:parametric
138:Properties
132:ring torus
95:around an
93:revolution
88:generatrix
39:= 2 + cos(
4677:MathWorld
4649:MathWorld
4611:MathWorld
4586:MathWorld
4349:Meridians
4344:Geodesics
4322:θ
4316:
4277:θ
4271:
4227:θ
4221:
4197:θ
4191:
4078:θ
4015:θ
3969:θ
3963:
3933:θ
3927:
3833:θ
3827:
3803:θ
3797:
3753:θ
3592:minimizes
3557:π
3524:−
3520:∫
3513:π
3476:−
3443:−
3417:−
3413:∫
3406:π
3368:−
3321:−
3296:−
3292:∫
3288:π
3211:π
3177:
3169:π
3160:∫
3156:π
3106:
3067:
3043:
3035:π
3026:∫
3022:π
2849:π
2832:∫
2809:θ
2718:π
2706:∫
2691:∫
2668:θ
2581:θ
2575:
2521:θ
2515:
2495:π
2483:∫
2468:∫
2445:θ
2377:θ
2371:
2336:θ
2330:
2306:π
2294:∫
2279:∫
2256:θ
2239:θ
2236:∂
2226:∂
2220:×
2211:∂
2201:∂
2188:π
2176:∫
2161:∫
2110:θ
2104:
2085:θ
2079:
1998:θ
1992:
1960:θ
1954:
1883:θ
1877:
1842:θ
1836:
1816:θ
1813:∂
1803:∂
1797:×
1788:∂
1778:∂
1751:⟩
1739:θ
1733:
1718:θ
1712:
1703:−
1700:⟨
1691:θ
1688:∂
1678:∂
1622:θ
1616:
1584:θ
1578:
1538:∂
1528:∂
1491:θ
1474:θ
1471:∂
1461:∂
1455:×
1446:∂
1436:∂
1423:π
1411:∫
1396:∫
1380:θ
1363:θ
1360:∂
1350:∂
1344:×
1335:∂
1325:∂
1309:π
1294:×
1275:∬
1256:∬
1219:π
1213:≤
1210:θ
1207:≤
1184:⟩
1163:θ
1157:
1133:θ
1127:
1109:⟨
1100:θ
1066:⟩
1036:⟨
936:∫
932:π
814:∫
810:π
735:∫
731:π
553:∫
549:π
247:∫
243:π
4644:"Toroid"
4500:69016407
4492:Calculus
4462:68015472
4424:Spheroid
4390:See also
3618:catenoid
3616:and the
2985:) = cos(
2970:) = sin(
2435:‖
2430:⟩
2320:⟨
2312:‖
2246:‖
2194:‖
2048:⟩
1947:⟨
1936:⟩
1826:⟨
1767:yields
1652:⟩
1551:⟨
1481:‖
1429:‖
1370:‖
1318:‖
859:′
218:integral
79:rotating
4359:Toroids
3247:√
2991:, when
198:, with
71:surface
4556:
4523:
4498:
4460:
4380:circle
4365:Toroid
3263:-axis
2932:
2812:
2803:
2671:
2662:
2448:
2439:
2259:
2250:
1494:
1485:
1383:
1374:
4384:torus
4090:. If
3614:plane
1196:with
128:torus
85:(the
83:curve
69:is a
59:torus
49:-axis
4554:ISBN
4521:ISBN
4496:LCCN
4458:LCCN
4110:and
4070:and
4007:and
3745:and
3245:) =
385:and
118:and
4550:617
4313:sin
4268:cos
4218:sin
4188:cos
4027:.
3960:sin
3924:cos
3824:sin
3794:cos
3765:as
3174:sin
3103:sin
3064:cos
3040:sin
2566:sin
2506:cos
2368:sin
2327:cos
2095:cos
2070:sin
1989:sin
1951:cos
1874:sin
1833:cos
1730:cos
1709:sin
1613:sin
1575:cos
1154:sin
1124:cos
1023:≥ 0
134:).
73:in
4704::
4689:.
4674:.
4646:.
4608:.
4583:.
4568:^
4552:.
4386:.
4355:.
4340:.
3620:.
3586:A
3252:−
3236:,
2976:,
702:≤
698:≤
693:,
683:=
497:2π
187:,
111:.
81:a
65:A
57:A
4680:.
4652:.
4614:.
4589:.
4562:.
4529:.
4502:.
4464:.
4328:)
4325:)
4319:(
4310:)
4307:t
4304:(
4301:x
4298:,
4295:)
4292:t
4289:(
4286:y
4283:,
4280:)
4274:(
4265:)
4262:t
4259:(
4256:x
4253:(
4233:)
4230:)
4224:(
4215:)
4212:t
4209:(
4206:y
4203:,
4200:)
4194:(
4185:)
4182:t
4179:(
4176:y
4173:,
4170:)
4167:t
4164:(
4161:x
4158:(
4138:t
4118:y
4098:x
4058:t
4038:t
3995:x
3975:)
3972:)
3966:(
3957:x
3954:,
3951:)
3948:x
3945:(
3942:f
3939:,
3936:)
3930:(
3921:x
3918:(
3898:)
3891:2
3887:z
3883:+
3878:2
3874:x
3868:(
3865:f
3862:=
3859:y
3839:)
3836:)
3830:(
3821:)
3818:x
3815:(
3812:f
3809:,
3806:)
3800:(
3791:)
3788:x
3785:(
3782:f
3779:,
3776:x
3773:(
3733:x
3711:2
3707:)
3703:x
3700:(
3697:f
3694:=
3689:2
3685:z
3681:+
3676:2
3672:y
3651:)
3648:x
3645:(
3642:f
3639:=
3636:y
3608:(
3565:2
3561:r
3554:4
3551:=
3541:x
3538:d
3532:r
3527:r
3516:r
3510:2
3507:=
3497:x
3494:d
3484:2
3480:x
3471:2
3467:r
3462:1
3451:2
3447:x
3438:2
3434:r
3425:r
3420:r
3409:r
3403:2
3400:=
3390:x
3387:d
3376:2
3372:x
3363:2
3359:r
3352:2
3348:x
3342:+
3339:1
3329:2
3325:x
3316:2
3312:r
3304:r
3299:r
3285:2
3282:=
3275:A
3261:x
3254:x
3250:r
3243:x
3241:(
3239:y
3234:r
3214:.
3208:4
3205:=
3193:t
3190:d
3186:)
3183:t
3180:(
3164:0
3153:2
3150:=
3138:t
3135:d
3127:2
3121:)
3115:)
3112:t
3109:(
3098:(
3093:+
3088:2
3082:)
3076:)
3073:t
3070:(
3059:(
3052:)
3049:t
3046:(
3030:0
3019:2
3016:=
3007:A
2993:t
2989:)
2987:t
2983:t
2981:(
2979:x
2974:)
2972:t
2968:t
2966:(
2964:y
2938:t
2935:d
2925:2
2920:)
2914:t
2911:d
2906:y
2903:d
2897:(
2892:+
2887:2
2882:)
2876:t
2873:d
2868:x
2865:d
2859:(
2852:y
2846:2
2841:b
2836:a
2828:=
2818:t
2815:d
2806:d
2796:2
2791:)
2785:t
2782:d
2777:y
2774:d
2768:(
2763:+
2758:2
2753:)
2747:t
2744:d
2739:x
2736:d
2730:(
2723:y
2715:2
2710:0
2700:b
2695:a
2687:=
2677:t
2674:d
2665:d
2655:2
2650:)
2644:t
2641:d
2636:y
2633:d
2627:(
2622:+
2617:2
2612:)
2606:t
2603:d
2598:x
2595:d
2589:(
2584:)
2578:(
2570:2
2562:+
2557:2
2552:)
2546:t
2543:d
2538:x
2535:d
2529:(
2524:)
2518:(
2510:2
2500:y
2492:2
2487:0
2477:b
2472:a
2464:=
2454:t
2451:d
2442:d
2423:t
2420:d
2415:y
2412:d
2406:y
2403:,
2397:t
2394:d
2389:x
2386:d
2380:)
2374:(
2365:y
2362:,
2356:t
2353:d
2348:x
2345:d
2339:)
2333:(
2324:y
2316:y
2303:2
2298:0
2288:b
2283:a
2275:=
2265:t
2262:d
2253:d
2230:r
2214:t
2205:r
2185:2
2180:0
2170:b
2165:a
2157:=
2148:x
2144:A
2119:1
2116:=
2113:)
2107:(
2099:2
2091:+
2088:)
2082:(
2074:2
2041:t
2038:d
2033:y
2030:d
2024:,
2018:t
2015:d
2010:x
2007:d
2001:)
1995:(
1986:,
1980:t
1977:d
1972:x
1969:d
1963:)
1957:(
1943:y
1940:=
1929:t
1926:d
1921:y
1918:d
1912:y
1909:,
1903:t
1900:d
1895:x
1892:d
1886:)
1880:(
1871:y
1868:,
1862:t
1859:d
1854:x
1851:d
1845:)
1839:(
1830:y
1822:=
1807:r
1791:t
1782:r
1748:0
1745:,
1742:)
1736:(
1727:y
1724:,
1721:)
1715:(
1706:y
1697:=
1682:r
1656:,
1645:t
1642:d
1637:x
1634:d
1628:,
1625:)
1619:(
1607:t
1604:d
1599:y
1596:d
1590:,
1587:)
1581:(
1569:t
1566:d
1561:y
1558:d
1547:=
1541:t
1532:r
1503:.
1500:t
1497:d
1488:d
1465:r
1449:t
1440:r
1420:2
1415:0
1405:b
1400:a
1392:=
1389:t
1386:d
1377:d
1354:r
1338:t
1329:r
1312:]
1306:2
1303:,
1300:0
1297:[
1291:]
1288:b
1285:,
1282:a
1279:[
1271:=
1268:S
1265:d
1260:S
1252:=
1247:x
1243:A
1216:2
1204:0
1181:)
1178:t
1175:(
1172:x
1169:,
1166:)
1160:(
1151:)
1148:t
1145:(
1142:y
1139:,
1136:)
1130:(
1121:)
1118:t
1115:(
1112:y
1106:=
1103:)
1097:,
1094:t
1091:(
1087:r
1063:)
1060:t
1057:(
1054:y
1051:,
1048:)
1045:t
1042:(
1039:x
1021:a
1016:y
1002:x
999:d
991:2
986:)
980:x
977:d
972:y
969:d
963:(
958:+
955:1
950:x
945:b
940:a
929:2
926:=
921:y
917:A
906:x
892:x
889:d
881:2
875:)
869:)
866:x
863:(
856:f
850:(
845:+
842:1
837:)
834:x
831:(
828:f
823:b
818:a
807:2
804:=
801:x
798:d
790:2
785:)
779:x
776:d
771:y
768:d
762:(
757:+
754:1
749:y
744:b
739:a
728:2
725:=
720:x
716:A
704:b
700:x
696:a
691:)
689:x
687:(
685:f
681:y
664:.
661:t
658:d
650:2
645:)
639:t
636:d
631:y
628:d
622:(
617:+
612:2
607:)
601:t
598:d
593:x
590:d
584:(
576:)
573:t
570:(
567:y
562:b
557:a
546:2
543:=
538:x
534:A
523:)
521:t
519:(
517:y
512:x
505:)
503:t
501:(
499:x
471:2
466:)
460:t
457:d
452:y
449:d
443:(
438:+
433:2
428:)
422:t
419:d
414:x
411:d
405:(
387:b
383:a
379:)
377:t
375:(
373:x
358:,
355:t
352:d
344:2
339:)
333:t
330:d
325:y
322:d
316:(
311:+
306:2
301:)
295:t
292:d
287:x
284:d
278:(
270:)
267:t
264:(
261:x
256:b
251:a
240:2
237:=
232:y
228:A
213:y
211:A
204:y
200:t
196:)
194:t
192:(
190:y
185:)
183:t
181:(
179:x
47:z
43:)
41:z
37:x
20:)
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