2941:
2123:
2936:{\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}}}
20:
1502:
2048:
4359:
1226:
3571:
1759:
43:
3255:
3217:
1497:{\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.}
2043:{\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 }
1655:
891:
2987:
4366:
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
3566:{\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}}}
663:
357:
1750:
1509:
699:
1183:
1001:
474:
2128:
3212:{\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}}}
517:
211:
4327:
4232:
1659:
3838:
2118:
3974:
3260:
2992:
3897:
1065:
1218:
3712:
900:
385:
4077:
4014:
3752:
1070:
3650:
1650:{\displaystyle {\frac {\partial \mathbf {r} }{\partial t}}=\left\langle {\frac {dy}{dt}}\cos(\theta ),{\frac {dy}{dt}}\sin(\theta ),{\frac {dx}{dt}}\right\rangle ,}
4137:
4117:
4097:
4057:
4037:
3994:
3732:
886:{\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}
111:
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
4422:
2943:
where the same trigonometric identity was used again. The derivation for a surface obtained by revolving around the y-axis is similar.
4237:
4142:
3757:
2053:
3902:
658:{\displaystyle A_{x}=2\pi \int _{a}^{b}y(t)\,{\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt.}
352:{\displaystyle A_{y}=2\pi \int _{a}^{b}x(t)\,{\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt,}
4546:
4513:
115:, and if the circle is rotated around an axis that does not intersect the interior of a circle, then it generates a
1745:{\displaystyle {\frac {\partial \mathbf {r} }{\partial \theta }}=\langle -y\sin(\theta ),y\cos(\theta ),0\rangle }
4701:
3843:
1067:
then its corresponding surface of revolution when revolved around the x-axis has
Cartesian coordinates given by
3594:
1020:
94:
the generatrix, except at its endpoints). The volume bounded by the surface created by this revolution is the
3576:
1188:
135:. Any meridional section can be considered to be the generatrix in the plane determined by it and the axis.
379:
3655:
138:
The sections of the surface of revolution made by planes that are perpendicular to the axis are circles.
4696:
3580:
4139:, then the surface of revolution obtained by revolving the curve around the x-axis is described by
150:
4234:, and the surface of revolution obtained by revolving the curve around the y-axis is described by
4503:
91:
4675:
4569:
1178:{\displaystyle \mathbf {r} (t,\theta )=\langle y(t)\cos(\theta ),y(t)\sin(\theta ),x(t)\rangle }
4341:
3598:
3587:
4538:
4531:
4062:
3999:
3737:
149:
are surfaces of revolution. These may be identified as those quadratic surfaces all of whose
59:
3620:
4395:
4337:
3590:
is finding the curve between two points that produces this minimal surface of revolution.
996:{\displaystyle A_{y}=2\pi \int _{a}^{b}x{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx}
8:
4660:
477:
146:
104:
96:
4479:
Thomas, George B. "6.7: Area of a
Surface of Revolution; 6.11: The Theorems of Pappus".
4122:
4102:
4082:
4042:
4022:
3979:
3717:
469:{\displaystyle {\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}}
1017:
This can also be derived from multivariable integration. If a plane curve is given by
4657:
4629:
4591:
4566:
4542:
4509:
4484:
4446:
4406:
4390:
496:
is the path of (the centroid of) this small segment, as required by Pappus' theorem.
131:
The sections of the surface of revolution made by planes through the axis are called
81:
19:
3840:. If instead we revolve the curve around the y-axis, then the curve is described by
4417:
4400:
3602:
1221:
162:
86:
4384:
4340:
are always geodesics on a surface of revolution. Other geodesics are governed by
108:
63:
4594:
4690:
1753:
4632:
3583:
196:
142:
112:
4560:
4558:
3579:
is the surface of revolution of the curve between two given points which
4555:
481:
120:
76:
4564:
4665:
4637:
4599:
4574:
50:
as a square revolved around an axis parallel to one of its diagonals.
103:
Examples of surfaces of revolution generated by a straight line are
4412:
4358:
3606:
206:
67:
480:
and represents a small segment of the arc of the curve, as in the
4488:
4450:
4368:
4353:
3617:
A surface of revolution given by rotating a curve described by
2947:
191:
ranging over some interval , and the axis of revolution is the
42:
4655:
4372:
116:
71:
47:
4537:(Alternate ed.). Prindle, Weber & Schmidt. p.
4441:
Middlemiss; Marks; Smart. "15-4. Surfaces of
Revolution".
4322:{\displaystyle (x(t)\cos(\theta ),y(t),x(t)\sin(\theta ))}
4227:{\displaystyle (x(t),y(t)\cos(\theta ),y(t)\sin(\theta ))}
3833:{\displaystyle (x,f(x)\cos(\theta ),f(x)\sin(\theta ))}
2113:{\displaystyle \sin ^{2}(\theta )+\cos ^{2}(\theta )=1}
4240:
4145:
4125:
4105:
4085:
4065:
4045:
4025:
4002:
3982:
3905:
3846:
3760:
3740:
3720:
3658:
3623:
3258:
2990:
2126:
2056:
1762:
1662:
1512:
1229:
1191:
1073:
1023:
903:
702:
667:
If the continuous curve is described by the function
520:
388:
214:
3969:{\displaystyle (x\cos(\theta ),f(x),x\sin(\theta ))}
4468:(Revised ed.), D.C. Heath and Co., p. 227
4409:, another generalization of a surface of revolution
4680:Encyclopédie des Formes Mathématiques Remarquables
4530:
4321:
4226:
4131:
4111:
4091:
4071:
4051:
4031:
4008:
3988:
3968:
3891:
3832:
3746:
3726:
3706:
3652:around the x-axis may be most simply described by
3644:
3565:
3211:
2935:
2112:
2042:
1744:
1649:
1496:
1212:
1177:
1059:
995:
885:
657:
468:
351:
4589:
16:Surface created by rotating a curve about an axis
4688:
4440:
3221:For the case of the spherical curve with radius
4627:
4618:, 2nd ed., Springer, London, 2012, pp. 227–230.
4039:, then we obtain a parametrization in terms of
4019:If x and y are defined in terms of a parameter
4508:(6 ed.). Tata McGraw-Hill. p. 6.90.
3714:. This yields the parametrization in terms of
378:. This formula is the calculus equivalent of
4387:, a generalisation of a surface of revolution
3109:
3086:
3070:
3047:
863:
838:
4463:
1739:
1688:
1172:
1097:
1054:
1024:
4614:Pressley, Andrew. “Chapter 9 - Geodesics.”
4483:(3rd ed.). pp. 206–209, 217–219.
4423:Translation surface (differential geometry)
2950:with unit radius is generated by the curve
499:Likewise, when the axis of rotation is the
3892:{\displaystyle y=f({\sqrt {x^{2}+z^{2}}})}
2120:was used. With this cross product, we get
4528:
4403:, surface of revolution of a circular arc
3612:
3558:
3525:
3481:
3446:
3418:
3374:
3324:
3177:
3122:
1506:Computing the partial derivatives yields
1060:{\displaystyle \langle x(t),y(t)\rangle }
986:
876:
785:
645:
567:
339:
261:
4357:
1220:. Then the surface area is given by the
514:is never negative, the area is given by
370:is never negative between the endpoints
153:perpendicular to the axis are circular.
41:
18:
1213:{\displaystyle 0\leq \theta \leq 2\pi }
4689:
3601:which are also minimal surfaces): the
1014:). These come from the above formula.
4656:
4628:
4590:
4565:
4501:
2984:ranges over . Its area is therefore
3707:{\displaystyle y^{2}+z^{2}=f(x)^{2}}
4464:Wilson, W.A.; Tracey, J.I. (1925),
119:which does not intersect itself (a
13:
4478:
2224:
2214:
2199:
2189:
1801:
1791:
1776:
1766:
1676:
1666:
1526:
1516:
1459:
1449:
1434:
1424:
1348:
1338:
1323:
1313:
145:(of either one or two sheets) and
14:
4713:
4649:
2050:where the trigonometric identity
161:If the curve is described by the
4616:Elementary Differential Geometry
4362:A toroid generated from a square
2218:
2193:
1795:
1770:
1670:
1520:
1453:
1428:
1342:
1317:
1075:
4570:"Minimal Surface of Revolution"
4533:Calculus with analytic geometry
156:
4621:
4608:
4583:
4522:
4495:
4472:
4457:
4434:
4371:, then the object is called a
4316:
4313:
4307:
4298:
4292:
4283:
4277:
4268:
4262:
4253:
4247:
4241:
4221:
4218:
4212:
4203:
4197:
4188:
4182:
4173:
4167:
4158:
4152:
4146:
3963:
3960:
3954:
3939:
3933:
3924:
3918:
3906:
3886:
3856:
3827:
3824:
3818:
3809:
3803:
3794:
3788:
3779:
3773:
3761:
3695:
3688:
3639:
3633:
3595:minimal surfaces of revolution
3174:
3168:
3103:
3097:
3064:
3058:
3040:
3034:
2572:
2566:
2512:
2506:
2423:
2368:
2362:
2327:
2321:
2300:
2234:
2182:
2101:
2095:
2076:
2070:
1989:
1983:
1951:
1945:
1874:
1868:
1833:
1827:
1730:
1724:
1709:
1703:
1613:
1607:
1575:
1569:
1469:
1417:
1358:
1306:
1300:
1285:
1279:
1267:
1169:
1163:
1154:
1148:
1139:
1133:
1124:
1118:
1109:
1103:
1091:
1079:
1051:
1045:
1036:
1030:
857:
851:
825:
819:
564:
558:
258:
252:
1:
4445:(3rd ed.). p. 378.
4428:
3577:minimal surface of revolution
126:
4332:
696:, then the integral becomes
7:
4529:Swokowski, Earl W. (1983).
4378:
3976:in terms of the parameters
10:
4718:
4351:
4347:
3899:, yielding the expression
1003:for revolution around the
893:for revolution around the
3586:. A basic problem in the
380:Pappus's centroid theorem
503:-axis and provided that
4676:"Surface de révolution"
4661:"Surface of Revolution"
4505:Engineering Mathematics
4072:{\displaystyle \theta }
4009:{\displaystyle \theta }
3747:{\displaystyle \theta }
23:A portion of the curve
4702:Surfaces of revolution
4363:
4323:
4228:
4133:
4113:
4093:
4073:
4053:
4033:
4010:
3990:
3970:
3893:
3834:
3748:
3728:
3708:
3646:
3645:{\displaystyle y=f(x)}
3613:Coordinate expressions
3599:surfaces of revolution
3588:calculus of variations
3567:
3213:
2937:
2114:
2044:
1746:
1651:
1498:
1214:
1179:
1061:
997:
887:
659:
484:formula. The quantity
470:
353:
141:Some special cases of
51:
39:
4361:
4324:
4229:
4134:
4114:
4094:
4074:
4054:
4034:
4011:
3991:
3971:
3894:
3835:
3749:
3729:
3709:
3647:
3568:
3214:
2938:
2115:
2045:
1747:
1652:
1499:
1215:
1180:
1062:
998:
888:
660:
471:
354:
56:surface of revolution
45:
22:
4502:Singh, R.R. (1993).
4396:Generalized helicoid
4238:
4143:
4123:
4103:
4083:
4063:
4043:
4023:
4000:
3980:
3903:
3844:
3758:
3738:
3718:
3656:
3621:
3256:
2988:
2124:
2054:
1760:
1660:
1510:
1227:
1189:
1071:
1021:
901:
700:
518:
386:
212:
147:elliptic paraboloids
4342:Clairaut's relation
3593:There are only two
3524:
3417:
3296:
3161:
3027:
2833:
2710:
2692:
2487:
2469:
2298:
2280:
2180:
2162:
1415:
1397:
937:
815:
736:
554:
478:Pythagorean theorem
248:
133:meridional sections
97:solid of revolution
34:rotated around the
4658:Weisstein, Eric W.
4630:Weisstein, Eric W.
4592:Weisstein, Eric W.
4567:Weisstein, Eric W.
4364:
4319:
4224:
4129:
4109:
4089:
4069:
4049:
4029:
4006:
3986:
3966:
3889:
3830:
3744:
3724:
3704:
3642:
3563:
3561:
3507:
3400:
3279:
3248:rotated about the
3209:
3207:
3147:
3013:
2933:
2931:
2819:
2693:
2678:
2470:
2455:
2281:
2266:
2163:
2148:
2110:
2040:
1752:and computing the
1742:
1647:
1494:
1398:
1383:
1210:
1175:
1057:
993:
923:
883:
801:
722:
655:
540:
466:
349:
234:
52:
40:
4697:Integral calculus
4466:Analytic Geometry
4443:Analytic Geometry
4407:Liouville surface
4132:{\displaystyle t}
4119:are functions of
4112:{\displaystyle y}
4092:{\displaystyle x}
4052:{\displaystyle t}
4032:{\displaystyle t}
3989:{\displaystyle x}
3884:
3727:{\displaystyle x}
3479:
3478:
3444:
3372:
3370:
3322:
3120:
2948:spherical surface
2946:For example, the
2922:
2918:
2906:
2868:
2802:
2793:
2789:
2777:
2739:
2661:
2652:
2648:
2636:
2598:
2538:
2438:
2429:
2415:
2389:
2348:
2249:
2240:
2231:
2206:
2033:
2010:
1972:
1921:
1895:
1854:
1808:
1783:
1683:
1637:
1599:
1561:
1533:
1484:
1475:
1466:
1441:
1373:
1364:
1355:
1330:
984:
972:
874:
783:
771:
643:
631:
593:
464:
452:
414:
337:
325:
287:
4709:
4683:
4671:
4670:
4644:
4643:
4642:
4625:
4619:
4612:
4606:
4605:
4604:
4587:
4581:
4580:
4579:
4562:
4553:
4552:
4536:
4526:
4520:
4519:
4499:
4493:
4492:
4476:
4470:
4469:
4461:
4455:
4454:
4438:
4418:Surface integral
4401:Lemon (geometry)
4328:
4326:
4325:
4320:
4233:
4231:
4230:
4225:
4138:
4136:
4135:
4130:
4118:
4116:
4115:
4110:
4098:
4096:
4095:
4090:
4078:
4076:
4075:
4070:
4058:
4056:
4055:
4050:
4038:
4036:
4035:
4030:
4015:
4013:
4012:
4007:
3995:
3993:
3992:
3987:
3975:
3973:
3972:
3967:
3898:
3896:
3895:
3890:
3885:
3883:
3882:
3870:
3869:
3860:
3839:
3837:
3836:
3831:
3753:
3751:
3750:
3745:
3733:
3731:
3730:
3725:
3713:
3711:
3710:
3705:
3703:
3702:
3681:
3680:
3668:
3667:
3651:
3649:
3648:
3643:
3572:
3570:
3569:
3564:
3562:
3557:
3556:
3535:
3523:
3518:
3491:
3480:
3477:
3476:
3475:
3463:
3462:
3449:
3448:
3445:
3443:
3442:
3430:
3429:
3420:
3416:
3411:
3384:
3373:
3371:
3369:
3368:
3367:
3355:
3354:
3344:
3343:
3334:
3326:
3323:
3321:
3320:
3308:
3307:
3298:
3295:
3290:
3251:
3247:
3246:
3245:
3224:
3218:
3216:
3215:
3210:
3208:
3192:
3187:
3160:
3155:
3137:
3132:
3121:
3119:
3118:
3113:
3112:
3090:
3089:
3080:
3079:
3074:
3073:
3051:
3050:
3044:
3026:
3021:
3003:
2983:
2979:
2964:
2942:
2940:
2939:
2934:
2932:
2920:
2919:
2917:
2916:
2911:
2907:
2905:
2897:
2889:
2879:
2878:
2873:
2869:
2867:
2859:
2851:
2844:
2832:
2827:
2812:
2800:
2791:
2790:
2788:
2787:
2782:
2778:
2776:
2768:
2760:
2750:
2749:
2744:
2740:
2738:
2730:
2722:
2715:
2709:
2701:
2691:
2686:
2671:
2659:
2650:
2649:
2647:
2646:
2641:
2637:
2635:
2627:
2619:
2609:
2608:
2603:
2599:
2597:
2589:
2581:
2562:
2561:
2549:
2548:
2543:
2539:
2537:
2529:
2521:
2502:
2501:
2492:
2486:
2478:
2468:
2463:
2448:
2436:
2427:
2426:
2422:
2421:
2417:
2416:
2414:
2406:
2398:
2390:
2388:
2380:
2372:
2349:
2347:
2339:
2331:
2297:
2289:
2279:
2274:
2259:
2247:
2238:
2237:
2233:
2232:
2230:
2222:
2221:
2212:
2207:
2205:
2197:
2196:
2187:
2179:
2171:
2161:
2156:
2140:
2139:
2119:
2117:
2116:
2111:
2091:
2090:
2066:
2065:
2049:
2047:
2046:
2041:
2039:
2035:
2034:
2032:
2024:
2016:
2011:
2009:
2001:
1993:
1973:
1971:
1963:
1955:
1927:
1923:
1922:
1920:
1912:
1904:
1896:
1894:
1886:
1878:
1855:
1853:
1845:
1837:
1809:
1807:
1799:
1798:
1789:
1784:
1782:
1774:
1773:
1764:
1751:
1749:
1748:
1743:
1684:
1682:
1674:
1673:
1664:
1656:
1654:
1653:
1648:
1643:
1639:
1638:
1636:
1628:
1620:
1600:
1598:
1590:
1582:
1562:
1560:
1552:
1544:
1534:
1532:
1524:
1523:
1514:
1503:
1501:
1500:
1495:
1482:
1473:
1472:
1468:
1467:
1465:
1457:
1456:
1447:
1442:
1440:
1432:
1431:
1422:
1414:
1406:
1396:
1391:
1371:
1362:
1361:
1357:
1356:
1354:
1346:
1345:
1336:
1331:
1329:
1321:
1320:
1311:
1304:
1303:
1252:
1251:
1239:
1238:
1222:surface integral
1219:
1217:
1216:
1211:
1184:
1182:
1181:
1176:
1078:
1066:
1064:
1063:
1058:
1013:
1007:-axis (provided
1002:
1000:
999:
994:
985:
983:
982:
977:
973:
971:
963:
955:
942:
936:
931:
913:
912:
896:
892:
890:
889:
884:
875:
873:
872:
867:
866:
850:
842:
841:
829:
814:
809:
784:
782:
781:
776:
772:
770:
762:
754:
741:
735:
730:
712:
711:
695:
681:
664:
662:
661:
656:
644:
642:
641:
636:
632:
630:
622:
614:
604:
603:
598:
594:
592:
584:
576:
569:
553:
548:
530:
529:
513:
502:
495:
475:
473:
472:
467:
465:
463:
462:
457:
453:
451:
443:
435:
425:
424:
419:
415:
413:
405:
397:
390:
377:
373:
369:
358:
356:
355:
350:
338:
336:
335:
330:
326:
324:
316:
308:
298:
297:
292:
288:
286:
278:
270:
263:
247:
242:
224:
223:
205:is given by the
204:
195:-axis, then the
194:
190:
186:
175:
109:conical surfaces
87:axis of rotation
37:
33:
4717:
4716:
4712:
4711:
4710:
4708:
4707:
4706:
4687:
4686:
4674:
4652:
4647:
4626:
4622:
4613:
4609:
4588:
4584:
4563:
4556:
4549:
4527:
4523:
4516:
4500:
4496:
4477:
4473:
4462:
4458:
4439:
4435:
4431:
4385:Channel surface
4381:
4356:
4350:
4335:
4239:
4236:
4235:
4144:
4141:
4140:
4124:
4121:
4120:
4104:
4101:
4100:
4084:
4081:
4080:
4064:
4061:
4060:
4044:
4041:
4040:
4024:
4021:
4020:
4001:
3998:
3997:
3981:
3978:
3977:
3904:
3901:
3900:
3878:
3874:
3865:
3861:
3859:
3845:
3842:
3841:
3759:
3756:
3755:
3739:
3736:
3735:
3719:
3716:
3715:
3698:
3694:
3676:
3672:
3663:
3659:
3657:
3654:
3653:
3622:
3619:
3618:
3615:
3560:
3559:
3552:
3548:
3533:
3532:
3519:
3511:
3489:
3488:
3471:
3467:
3458:
3454:
3453:
3447:
3438:
3434:
3425:
3421:
3419:
3412:
3404:
3382:
3381:
3363:
3359:
3350:
3346:
3345:
3339:
3335:
3333:
3325:
3316:
3312:
3303:
3299:
3297:
3291:
3283:
3266:
3259:
3257:
3254:
3253:
3249:
3237:
3235:
3226:
3222:
3206:
3205:
3191:
3185:
3184:
3156:
3151:
3136:
3130:
3129:
3114:
3108:
3107:
3106:
3085:
3084:
3075:
3069:
3068:
3067:
3046:
3045:
3043:
3022:
3017:
3002:
2998:
2991:
2989:
2986:
2985:
2981:
2966:
2951:
2930:
2929:
2912:
2898:
2890:
2888:
2884:
2883:
2874:
2860:
2852:
2850:
2846:
2845:
2843:
2828:
2823:
2810:
2809:
2783:
2769:
2761:
2759:
2755:
2754:
2745:
2731:
2723:
2721:
2717:
2716:
2714:
2702:
2697:
2687:
2682:
2669:
2668:
2642:
2628:
2620:
2618:
2614:
2613:
2604:
2590:
2582:
2580:
2576:
2575:
2557:
2553:
2544:
2530:
2522:
2520:
2516:
2515:
2497:
2493:
2491:
2479:
2474:
2464:
2459:
2446:
2445:
2407:
2399:
2397:
2381:
2373:
2371:
2340:
2332:
2330:
2311:
2307:
2303:
2299:
2290:
2285:
2275:
2270:
2257:
2256:
2223:
2217:
2213:
2211:
2198:
2192:
2188:
2186:
2185:
2181:
2172:
2167:
2157:
2152:
2141:
2135:
2131:
2127:
2125:
2122:
2121:
2086:
2082:
2061:
2057:
2055:
2052:
2051:
2025:
2017:
2015:
2002:
1994:
1992:
1964:
1956:
1954:
1938:
1934:
1913:
1905:
1903:
1887:
1879:
1877:
1846:
1838:
1836:
1817:
1813:
1800:
1794:
1790:
1788:
1775:
1769:
1765:
1763:
1761:
1758:
1757:
1675:
1669:
1665:
1663:
1661:
1658:
1657:
1629:
1621:
1619:
1591:
1583:
1581:
1553:
1545:
1543:
1542:
1538:
1525:
1519:
1515:
1513:
1511:
1508:
1507:
1458:
1452:
1448:
1446:
1433:
1427:
1423:
1421:
1420:
1416:
1407:
1402:
1392:
1387:
1347:
1341:
1337:
1335:
1322:
1316:
1312:
1310:
1309:
1305:
1266:
1262:
1247:
1243:
1234:
1230:
1228:
1225:
1224:
1190:
1187:
1186:
1074:
1072:
1069:
1068:
1022:
1019:
1018:
1008:
978:
964:
956:
954:
950:
949:
941:
932:
927:
908:
904:
902:
899:
898:
894:
868:
862:
861:
860:
843:
837:
836:
828:
810:
805:
777:
763:
755:
753:
749:
748:
740:
731:
726:
707:
703:
701:
698:
697:
683:
668:
637:
623:
615:
613:
609:
608:
599:
585:
577:
575:
571:
570:
568:
549:
544:
525:
521:
519:
516:
515:
504:
500:
485:
476:comes from the
458:
444:
436:
434:
430:
429:
420:
406:
398:
396:
392:
391:
389:
387:
384:
383:
382:. The quantity
375:
371:
360:
331:
317:
309:
307:
303:
302:
293:
279:
271:
269:
265:
264:
262:
243:
238:
219:
215:
213:
210:
209:
203:
199:
192:
188:
177:
166:
159:
129:
64:Euclidean space
35:
24:
17:
12:
11:
5:
4715:
4705:
4704:
4699:
4685:
4684:
4672:
4651:
4650:External links
4648:
4646:
4645:
4620:
4607:
4582:
4554:
4547:
4521:
4514:
4494:
4471:
4456:
4432:
4430:
4427:
4426:
4425:
4420:
4415:
4410:
4404:
4398:
4393:
4391:Gabriel's Horn
4388:
4380:
4377:
4352:Main article:
4349:
4346:
4334:
4331:
4318:
4315:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4291:
4288:
4285:
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4279:
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4196:
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4163:
4160:
4157:
4154:
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4148:
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4068:
4048:
4028:
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3808:
3805:
3802:
3799:
3796:
3793:
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3787:
3784:
3781:
3778:
3775:
3772:
3769:
3766:
3763:
3743:
3723:
3701:
3697:
3693:
3690:
3687:
3684:
3679:
3675:
3671:
3666:
3662:
3641:
3638:
3635:
3632:
3629:
3626:
3614:
3611:
3555:
3551:
3547:
3544:
3541:
3538:
3536:
3534:
3531:
3528:
3522:
3517:
3514:
3510:
3506:
3503:
3500:
3497:
3494:
3492:
3490:
3487:
3484:
3474:
3470:
3466:
3461:
3457:
3452:
3441:
3437:
3433:
3428:
3424:
3415:
3410:
3407:
3403:
3399:
3396:
3393:
3390:
3387:
3385:
3383:
3380:
3377:
3366:
3362:
3358:
3353:
3349:
3342:
3338:
3332:
3329:
3319:
3315:
3311:
3306:
3302:
3294:
3289:
3286:
3282:
3278:
3275:
3272:
3269:
3267:
3265:
3262:
3261:
3204:
3201:
3198:
3195:
3190:
3188:
3186:
3183:
3180:
3176:
3173:
3170:
3167:
3164:
3159:
3154:
3150:
3146:
3143:
3140:
3135:
3133:
3131:
3128:
3125:
3117:
3111:
3105:
3102:
3099:
3096:
3093:
3088:
3083:
3078:
3072:
3066:
3063:
3060:
3057:
3054:
3049:
3042:
3039:
3036:
3033:
3030:
3025:
3020:
3016:
3012:
3009:
3006:
3001:
2999:
2997:
2994:
2993:
2928:
2925:
2915:
2910:
2904:
2901:
2896:
2893:
2887:
2882:
2877:
2872:
2866:
2863:
2858:
2855:
2849:
2842:
2839:
2836:
2831:
2826:
2822:
2818:
2815:
2813:
2811:
2808:
2805:
2799:
2796:
2786:
2781:
2775:
2772:
2767:
2764:
2758:
2753:
2748:
2743:
2737:
2734:
2729:
2726:
2720:
2713:
2708:
2705:
2700:
2696:
2690:
2685:
2681:
2677:
2674:
2672:
2670:
2667:
2664:
2658:
2655:
2645:
2640:
2634:
2631:
2626:
2623:
2617:
2612:
2607:
2602:
2596:
2593:
2588:
2585:
2579:
2574:
2571:
2568:
2565:
2560:
2556:
2552:
2547:
2542:
2536:
2533:
2528:
2525:
2519:
2514:
2511:
2508:
2505:
2500:
2496:
2490:
2485:
2482:
2477:
2473:
2467:
2462:
2458:
2454:
2451:
2449:
2447:
2444:
2441:
2435:
2432:
2425:
2420:
2413:
2410:
2405:
2402:
2396:
2393:
2387:
2384:
2379:
2376:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2346:
2343:
2338:
2335:
2329:
2326:
2323:
2320:
2317:
2314:
2310:
2306:
2302:
2296:
2293:
2288:
2284:
2278:
2273:
2269:
2265:
2262:
2260:
2258:
2255:
2252:
2246:
2243:
2236:
2229:
2226:
2220:
2216:
2210:
2204:
2201:
2195:
2191:
2184:
2178:
2175:
2170:
2166:
2160:
2155:
2151:
2147:
2144:
2142:
2138:
2134:
2130:
2129:
2109:
2106:
2103:
2100:
2097:
2094:
2089:
2085:
2081:
2078:
2075:
2072:
2069:
2064:
2060:
2038:
2031:
2028:
2023:
2020:
2014:
2008:
2005:
2000:
1997:
1991:
1988:
1985:
1982:
1979:
1976:
1970:
1967:
1962:
1959:
1953:
1950:
1947:
1944:
1941:
1937:
1933:
1930:
1926:
1919:
1916:
1911:
1908:
1902:
1899:
1893:
1890:
1885:
1882:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1852:
1849:
1844:
1841:
1835:
1832:
1829:
1826:
1823:
1820:
1816:
1812:
1806:
1803:
1797:
1793:
1787:
1781:
1778:
1772:
1768:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1681:
1678:
1672:
1668:
1646:
1642:
1635:
1632:
1627:
1624:
1618:
1615:
1612:
1609:
1606:
1603:
1597:
1594:
1589:
1586:
1580:
1577:
1574:
1571:
1568:
1565:
1559:
1556:
1551:
1548:
1541:
1537:
1531:
1528:
1522:
1518:
1493:
1490:
1487:
1481:
1478:
1471:
1464:
1461:
1455:
1451:
1445:
1439:
1436:
1430:
1426:
1419:
1413:
1410:
1405:
1401:
1395:
1390:
1386:
1382:
1379:
1376:
1370:
1367:
1360:
1353:
1350:
1344:
1340:
1334:
1328:
1325:
1319:
1315:
1308:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1265:
1261:
1258:
1255:
1250:
1246:
1242:
1237:
1233:
1209:
1206:
1203:
1200:
1197:
1194:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1077:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
992:
989:
981:
976:
970:
967:
962:
959:
953:
948:
945:
940:
935:
930:
926:
922:
919:
916:
911:
907:
882:
879:
871:
865:
859:
856:
853:
849:
846:
840:
835:
832:
827:
824:
821:
818:
813:
808:
804:
800:
797:
794:
791:
788:
780:
775:
769:
766:
761:
758:
752:
747:
744:
739:
734:
729:
725:
721:
718:
715:
710:
706:
654:
651:
648:
640:
635:
629:
626:
621:
618:
612:
607:
602:
597:
591:
588:
583:
580:
574:
566:
563:
560:
557:
552:
547:
543:
539:
536:
533:
528:
524:
461:
456:
450:
447:
442:
439:
433:
428:
423:
418:
412:
409:
404:
401:
395:
359:provided that
348:
345:
342:
334:
329:
323:
320:
315:
312:
306:
301:
296:
291:
285:
282:
277:
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268:
260:
257:
254:
251:
246:
241:
237:
233:
230:
227:
222:
218:
201:
158:
155:
151:cross sections
128:
125:
90:(normally not
15:
9:
6:
4:
3:
2:
4714:
4703:
4700:
4698:
4695:
4694:
4692:
4681:
4677:
4673:
4668:
4667:
4662:
4659:
4654:
4653:
4640:
4639:
4634:
4631:
4624:
4617:
4611:
4602:
4601:
4596:
4593:
4586:
4577:
4576:
4571:
4568:
4561:
4559:
4550:
4548:0-87150-341-7
4544:
4540:
4535:
4534:
4525:
4517:
4515:0-07-014615-2
4511:
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2136:
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2079:
2073:
2067:
2062:
2058:
2036:
2029:
2026:
2021:
2018:
2012:
2006:
2003:
1998:
1995:
1986:
1980:
1977:
1974:
1968:
1965:
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1948:
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1785:
1779:
1755:
1754:cross product
1736:
1733:
1727:
1721:
1718:
1715:
1712:
1706:
1700:
1697:
1694:
1691:
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1399:
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1368:
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1248:
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1207:
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1198:
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1166:
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1157:
1151:
1145:
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1136:
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1127:
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1115:
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1094:
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1042:
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1033:
1027:
1015:
1011:
1006:
990:
987:
979:
974:
968:
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951:
946:
943:
938:
933:
928:
924:
920:
917:
914:
909:
905:
880:
877:
869:
854:
847:
844:
833:
830:
822:
816:
811:
806:
802:
798:
795:
792:
789:
786:
778:
773:
767:
764:
759:
756:
750:
745:
742:
737:
732:
727:
723:
719:
716:
713:
708:
704:
694:
690:
686:
679:
675:
671:
665:
652:
649:
646:
638:
633:
627:
624:
619:
616:
610:
605:
600:
595:
589:
586:
581:
578:
572:
561:
555:
550:
545:
541:
537:
534:
531:
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522:
511:
507:
497:
493:
489:
483:
479:
459:
454:
448:
445:
440:
437:
431:
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421:
416:
410:
407:
402:
399:
393:
381:
367:
363:
346:
343:
340:
332:
327:
321:
318:
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304:
299:
294:
289:
283:
280:
275:
272:
266:
255:
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244:
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231:
228:
225:
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216:
208:
198:
184:
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173:
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139:
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122:
118:
114:
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88:
83:
79:
78:
73:
69:
65:
61:
57:
49:
44:
31:
27:
21:
4682:(in French).
4679:
4664:
4636:
4623:
4615:
4610:
4598:
4585:
4573:
4532:
4524:
4504:
4497:
4480:
4474:
4465:
4459:
4442:
4436:
4365:
4336:
4018:
3616:
3592:
3584:surface area
3574:
3242:
3238:
3231:
3227:
3220:
2975:
2971:
2967:
2960:
2956:
2952:
2945:
1505:
1016:
1009:
1004:
692:
688:
684:
677:
673:
669:
666:
509:
505:
498:
491:
487:
365:
361:
197:surface area
182:
178:
171:
167:
160:
157:Area formula
143:hyperboloids
140:
137:
132:
130:
113:great circle
102:
95:
92:intersecting
85:
75:
55:
53:
29:
25:
897:-axis, and
105:cylindrical
80:) one full
66:created by
4691:Categories
4595:"Catenoid"
4429:References
482:arc length
165:functions
163:parametric
127:Properties
121:ring torus
84:around an
82:revolution
77:generatrix
28:= 2 + cos(
4666:MathWorld
4638:MathWorld
4600:MathWorld
4575:MathWorld
4338:Meridians
4333:Geodesics
4311:θ
4305:
4266:θ
4260:
4216:θ
4210:
4186:θ
4180:
4067:θ
4004:θ
3958:θ
3952:
3922:θ
3916:
3822:θ
3816:
3792:θ
3786:
3742:θ
3581:minimizes
3546:π
3513:−
3509:∫
3502:π
3465:−
3432:−
3406:−
3402:∫
3395:π
3357:−
3310:−
3285:−
3281:∫
3277:π
3200:π
3166:
3158:π
3149:∫
3145:π
3095:
3056:
3032:
3024:π
3015:∫
3011:π
2838:π
2821:∫
2798:θ
2707:π
2695:∫
2680:∫
2657:θ
2570:θ
2564:
2510:θ
2504:
2484:π
2472:∫
2457:∫
2434:θ
2366:θ
2360:
2325:θ
2319:
2295:π
2283:∫
2268:∫
2245:θ
2228:θ
2225:∂
2215:∂
2209:×
2200:∂
2190:∂
2177:π
2165:∫
2150:∫
2099:θ
2093:
2074:θ
2068:
1987:θ
1981:
1949:θ
1943:
1872:θ
1866:
1831:θ
1825:
1805:θ
1802:∂
1792:∂
1786:×
1777:∂
1767:∂
1740:⟩
1728:θ
1722:
1707:θ
1701:
1692:−
1689:⟨
1680:θ
1677:∂
1667:∂
1611:θ
1605:
1573:θ
1567:
1527:∂
1517:∂
1480:θ
1463:θ
1460:∂
1450:∂
1444:×
1435:∂
1425:∂
1412:π
1400:∫
1385:∫
1369:θ
1352:θ
1349:∂
1339:∂
1333:×
1324:∂
1314:∂
1298:π
1283:×
1264:∬
1245:∬
1208:π
1202:≤
1199:θ
1196:≤
1173:⟩
1152:θ
1146:
1122:θ
1116:
1098:⟨
1089:θ
1055:⟩
1025:⟨
925:∫
921:π
803:∫
799:π
724:∫
720:π
542:∫
538:π
236:∫
232:π
4633:"Toroid"
4489:69016407
4481:Calculus
4451:68015472
4413:Spheroid
4379:See also
3607:catenoid
3605:and the
2974:) = cos(
2959:) = sin(
2424:‖
2419:⟩
2309:⟨
2301:‖
2235:‖
2183:‖
2037:⟩
1936:⟨
1925:⟩
1815:⟨
1756:yields
1641:⟩
1540:⟨
1470:‖
1418:‖
1359:‖
1307:‖
848:′
207:integral
68:rotating
4348:Toroids
3236:√
2980:, when
187:, with
60:surface
4545:
4512:
4487:
4449:
4369:circle
4354:Toroid
3252:-axis
2921:
2801:
2792:
2660:
2651:
2437:
2428:
2248:
2239:
1483:
1474:
1372:
1363:
4373:torus
4079:. If
3603:plane
1185:with
117:torus
74:(the
72:curve
58:is a
48:torus
38:-axis
4543:ISBN
4510:ISBN
4485:LCCN
4447:LCCN
4099:and
4059:and
3996:and
3734:and
3234:) =
374:and
107:and
4539:617
4302:sin
4257:cos
4207:sin
4177:cos
4016:.
3949:sin
3913:cos
3813:sin
3783:cos
3754:as
3163:sin
3092:sin
3053:cos
3029:sin
2555:sin
2495:cos
2357:sin
2316:cos
2084:cos
2059:sin
1978:sin
1940:cos
1863:sin
1822:cos
1719:cos
1698:sin
1602:sin
1564:cos
1143:sin
1113:cos
1012:≥ 0
123:).
62:in
4693::
4678:.
4663:.
4635:.
4597:.
4572:.
4557:^
4541:.
4375:.
4344:.
4329:.
3609:.
3575:A
3241:−
3225:,
2965:,
691:≤
687:≤
682:,
672:=
486:2π
176:,
100:.
70:a
54:A
46:A
4669:.
4641:.
4603:.
4578:.
4551:.
4518:.
4491:.
4453:.
4317:)
4314:)
4308:(
4299:)
4296:t
4293:(
4290:x
4287:,
4284:)
4281:t
4278:(
4275:y
4272:,
4269:)
4263:(
4254:)
4251:t
4248:(
4245:x
4242:(
4222:)
4219:)
4213:(
4204:)
4201:t
4198:(
4195:y
4192:,
4189:)
4183:(
4174:)
4171:t
4168:(
4165:y
4162:,
4159:)
4156:t
4153:(
4150:x
4147:(
4127:t
4107:y
4087:x
4047:t
4027:t
3984:x
3964:)
3961:)
3955:(
3946:x
3943:,
3940:)
3937:x
3934:(
3931:f
3928:,
3925:)
3919:(
3910:x
3907:(
3887:)
3880:2
3876:z
3872:+
3867:2
3863:x
3857:(
3854:f
3851:=
3848:y
3828:)
3825:)
3819:(
3810:)
3807:x
3804:(
3801:f
3798:,
3795:)
3789:(
3780:)
3777:x
3774:(
3771:f
3768:,
3765:x
3762:(
3722:x
3700:2
3696:)
3692:x
3689:(
3686:f
3683:=
3678:2
3674:z
3670:+
3665:2
3661:y
3640:)
3637:x
3634:(
3631:f
3628:=
3625:y
3597:(
3554:2
3550:r
3543:4
3540:=
3530:x
3527:d
3521:r
3516:r
3505:r
3499:2
3496:=
3486:x
3483:d
3473:2
3469:x
3460:2
3456:r
3451:1
3440:2
3436:x
3427:2
3423:r
3414:r
3409:r
3398:r
3392:2
3389:=
3379:x
3376:d
3365:2
3361:x
3352:2
3348:r
3341:2
3337:x
3331:+
3328:1
3318:2
3314:x
3305:2
3301:r
3293:r
3288:r
3274:2
3271:=
3264:A
3250:x
3243:x
3239:r
3232:x
3230:(
3228:y
3223:r
3203:.
3197:4
3194:=
3182:t
3179:d
3175:)
3172:t
3169:(
3153:0
3142:2
3139:=
3127:t
3124:d
3116:2
3110:)
3104:)
3101:t
3098:(
3087:(
3082:+
3077:2
3071:)
3065:)
3062:t
3059:(
3048:(
3041:)
3038:t
3035:(
3019:0
3008:2
3005:=
2996:A
2982:t
2978:)
2976:t
2972:t
2970:(
2968:x
2963:)
2961:t
2957:t
2955:(
2953:y
2927:t
2924:d
2914:2
2909:)
2903:t
2900:d
2895:y
2892:d
2886:(
2881:+
2876:2
2871:)
2865:t
2862:d
2857:x
2854:d
2848:(
2841:y
2835:2
2830:b
2825:a
2817:=
2807:t
2804:d
2795:d
2785:2
2780:)
2774:t
2771:d
2766:y
2763:d
2757:(
2752:+
2747:2
2742:)
2736:t
2733:d
2728:x
2725:d
2719:(
2712:y
2704:2
2699:0
2689:b
2684:a
2676:=
2666:t
2663:d
2654:d
2644:2
2639:)
2633:t
2630:d
2625:y
2622:d
2616:(
2611:+
2606:2
2601:)
2595:t
2592:d
2587:x
2584:d
2578:(
2573:)
2567:(
2559:2
2551:+
2546:2
2541:)
2535:t
2532:d
2527:x
2524:d
2518:(
2513:)
2507:(
2499:2
2489:y
2481:2
2476:0
2466:b
2461:a
2453:=
2443:t
2440:d
2431:d
2412:t
2409:d
2404:y
2401:d
2395:y
2392:,
2386:t
2383:d
2378:x
2375:d
2369:)
2363:(
2354:y
2351:,
2345:t
2342:d
2337:x
2334:d
2328:)
2322:(
2313:y
2305:y
2292:2
2287:0
2277:b
2272:a
2264:=
2254:t
2251:d
2242:d
2219:r
2203:t
2194:r
2174:2
2169:0
2159:b
2154:a
2146:=
2137:x
2133:A
2108:1
2105:=
2102:)
2096:(
2088:2
2080:+
2077:)
2071:(
2063:2
2030:t
2027:d
2022:y
2019:d
2013:,
2007:t
2004:d
1999:x
1996:d
1990:)
1984:(
1975:,
1969:t
1966:d
1961:x
1958:d
1952:)
1946:(
1932:y
1929:=
1918:t
1915:d
1910:y
1907:d
1901:y
1898:,
1892:t
1889:d
1884:x
1881:d
1875:)
1869:(
1860:y
1857:,
1851:t
1848:d
1843:x
1840:d
1834:)
1828:(
1819:y
1811:=
1796:r
1780:t
1771:r
1737:0
1734:,
1731:)
1725:(
1716:y
1713:,
1710:)
1704:(
1695:y
1686:=
1671:r
1645:,
1634:t
1631:d
1626:x
1623:d
1617:,
1614:)
1608:(
1596:t
1593:d
1588:y
1585:d
1579:,
1576:)
1570:(
1558:t
1555:d
1550:y
1547:d
1536:=
1530:t
1521:r
1492:.
1489:t
1486:d
1477:d
1454:r
1438:t
1429:r
1409:2
1404:0
1394:b
1389:a
1381:=
1378:t
1375:d
1366:d
1343:r
1327:t
1318:r
1301:]
1295:2
1292:,
1289:0
1286:[
1280:]
1277:b
1274:,
1271:a
1268:[
1260:=
1257:S
1254:d
1249:S
1241:=
1236:x
1232:A
1205:2
1193:0
1170:)
1167:t
1164:(
1161:x
1158:,
1155:)
1149:(
1140:)
1137:t
1134:(
1131:y
1128:,
1125:)
1119:(
1110:)
1107:t
1104:(
1101:y
1095:=
1092:)
1086:,
1083:t
1080:(
1076:r
1052:)
1049:t
1046:(
1043:y
1040:,
1037:)
1034:t
1031:(
1028:x
1010:a
1005:y
991:x
988:d
980:2
975:)
969:x
966:d
961:y
958:d
952:(
947:+
944:1
939:x
934:b
929:a
918:2
915:=
910:y
906:A
895:x
881:x
878:d
870:2
864:)
858:)
855:x
852:(
845:f
839:(
834:+
831:1
826:)
823:x
820:(
817:f
812:b
807:a
796:2
793:=
790:x
787:d
779:2
774:)
768:x
765:d
760:y
757:d
751:(
746:+
743:1
738:y
733:b
728:a
717:2
714:=
709:x
705:A
693:b
689:x
685:a
680:)
678:x
676:(
674:f
670:y
653:.
650:t
647:d
639:2
634:)
628:t
625:d
620:y
617:d
611:(
606:+
601:2
596:)
590:t
587:d
582:x
579:d
573:(
565:)
562:t
559:(
556:y
551:b
546:a
535:2
532:=
527:x
523:A
512:)
510:t
508:(
506:y
501:x
494:)
492:t
490:(
488:x
460:2
455:)
449:t
446:d
441:y
438:d
432:(
427:+
422:2
417:)
411:t
408:d
403:x
400:d
394:(
376:b
372:a
368:)
366:t
364:(
362:x
347:,
344:t
341:d
333:2
328:)
322:t
319:d
314:y
311:d
305:(
300:+
295:2
290:)
284:t
281:d
276:x
273:d
267:(
259:)
256:t
253:(
250:x
245:b
240:a
229:2
226:=
221:y
217:A
202:y
200:A
193:y
189:t
185:)
183:t
181:(
179:y
174:)
172:t
170:(
168:x
36:z
32:)
30:z
26:x
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