4769:
made with straight lines fit well together and save space. And we can move easily — physically or mentally — around things made with round lines. But we are in a straitjacket, having to accept one or the other, when often some intermediate form would be better. To draw something freehand — such as the patchwork traffic circle they tried in
Stockholm — will not do. It isn't fixed, isn't definite like a circle or square. You don't know what it is. It isn't esthetically satisfying. The super-ellipse solved the problem. It is neither round nor rectangular, but in between. Yet it is fixed, it is definite — it has a unity.
3969:
1189:
993:
3155:
4663:
813:
3761:
33:
2139:
4708:
1851:
1953:
4492:
4768:
Man is the animal that draws lines which he himself then stumbles over. In the whole pattern of civilization there have been two tendencies, one toward straight lines and rectangular patterns and one toward circular lines. There are reasons, mechanical and psychological, for both tendencies. Things
3670:
4646:
scaling involves scaling the shape differently along different axes, providing additional control over the geometry. This approach can be applied to superellipses, superellipsoids, and their higher-dimensional analogues to produce a wider variety of forms and better fit specific requirements in
2537:
1701:
554:
3349:
4144:
4647:
applications such as computer graphics, structural design, and data visualization. For instance, anisotropic scaling allows the creation of shapes that can model real-world objects more accurately by adjusting the proportions along each axis independently.
3185:
The generalization of these shapes can involve several approaches.The generalizations of the superellipse in higher dimensions retain the fundamental mathematical structure of the superellipse while adapting it to different contexts and applications.
2258:
2644:
4629:
are a family of shapes that include superellipsoids as a special case. They are used in computer graphics and geometric modeling to create complex, smooth shapes with easily adjustable parameters. While not a direct generalization of superellipses,
3141:
4178:, a super-ellipsoid is obtained. If any two or all three powers differ from each other, a solid is obtained that may possess more flexibility in representing real structural data than the super ellipsoid. A three-dimensional super-ellipsoid with
3869:
3516:
4320:
223:
2134:{\displaystyle {\begin{aligned}x\left(t\right)&={\left|\cos t\right|}^{\frac {2}{n}}\cdot a\operatorname {sgn}(\cos t)\\y\left(t\right)&={\left|\sin t\right|}^{\frac {2}{n}}\cdot b\operatorname {sgn}(\sin t)\end{aligned}}}
3006:
2411:
420:
4774:
Sergels Torg was completed in 1967. Meanwhile, Piet Hein went on to use the superellipse in other artifacts, such as beds, dishes, tables, etc. By rotating a superellipse around the longest axis, he created the
2391:
3243:
4634:
also share the concept of extending geometric shapes into higher dimensions. These related shapes demonstrate the versatility and broad applicability of the fundamental principles underlying superellipses.
2148:
1958:
1709:
1846:{\displaystyle \left.{\begin{aligned}x\left(t\right)&=\pm a\cos ^{\frac {2}{n}}t\\y\left(t\right)&=\pm b\sin ^{\frac {2}{n}}t\end{aligned}}\right\}\qquad 0\leq t\leq {\frac {\pi }{2}}}
4041:
3202:
extends the superellipse into three dimensions, creating shapes that vary between ellipsoids and rectangular solids with rounded edges. The superellipsoid is defined as the set of all points
3015:
4616:
3511:
3194:
The generalizations of the superellipse in higher dimensions retain the fundamental mathematical structure of the superellipse while adapting it to different contexts and applications.
4551:
3729:
2549:
1949:
2685:
is also an option that balances parsimony and accuracy. Consider a superellipse centered on the origin of a 2D plane. Now, imagine that the superellipse (with shape parameter
1358:
146:
2930:
75:
970:
695:
415:
375:
1543:
1499:
1469:
3774:
4208:
3963:
3238:
2845:
1439:
1410:
1299:
1182:
1153:
932:
903:
4487:{\displaystyle \left|{\frac {x_{1}}{a_{1}}}\right|^{N_{1}}\!\!+\left|{\frac {x_{2}}{a_{2}}}\right|^{N_{2}}\!+\ldots +\left|{\frac {x_{N}}{a_{N}}}\right|^{N_{N}}\!=1}
2755:
2729:
1269:
1239:
1116:
343:
4292:
4266:
4234:
4176:
2819:
2787:
1650:
807:
737:
141:
1871:
5253:
3937:
3375:
2306:
1618:
1586:
1325:
1084:
1048:
868:
773:
317:
4315:
4036:
4015:
3994:
3910:
3889:
3749:
3442:
3415:
3395:
2913:
2885:
2865:
2703:
2278:
1911:
1891:
1696:
1378:
658:
638:
618:
291:
263:
243:
3665:{\displaystyle \left|{\frac {x_{1}}{a_{1}}}\right|^{n}\!\!+\left|{\frac {x_{2}}{a_{2}}}\right|^{n}\!+\ldots +\left|{\frac {x_{d}}{a_{d}}}\right|^{n}\!=1,}
4990:
Shi, Pei-Jian; Huang, Jian-Guo; Hui, Cang; Grissino-Mayer, Henri D.; Tardif, Jacques C.; Zhai, Li-Hong; Wang, Fu-Sheng; Li, Bai-Lian (15 October 2015).
5671:
1665:
5517:
5127:
2532:{\displaystyle \mathrm {Area} =4ab{\frac {\left(\Gamma \left(1+{\tfrac {1}{n}}\right)\right)^{2}}{\Gamma \left(1+{\tfrac {2}{n}}\right)}},}
2311:
4814:
2280:
is not the angle between the positive horizontal axis and the ray from the origin to the point, since the tangent of this angle equals
1668:. If the numerator is not even, then the curve is pieced together from portions of the same algebraic curve in different orientations.
4842:
consists of a tilted superellipse matching the proportions of
Sergels Torg. Three connected superellipses are used in the logo of the
549:{\displaystyle r=\left(\left|{\frac {\cos(\theta )}{a}}\right|^{n}\!\!+\left|{\frac {\sin(\theta )}{b}}\right|^{n}\!\right)^{-1/n}\!.}
4828:
5541:
5220:
270:
5195:
640:
are the semi-diameters or semi-axes of the curve. The overall shape of the curve is determined by the value of the exponent
3968:
3965:, the curve possesses more flexibility of behavior, and is better possible fit to describe some experimental information.
5629:
3344:{\displaystyle \left|{\frac {x}{a}}\right|^{n}\!\!+\left|{\frac {y}{b}}\right|^{n}\!+\left|{\frac {z}{c}}\right|^{n}\!=1,}
4139:{\displaystyle \left|{\frac {x}{a}}\right|^{m}\!\!+\left|{\frac {y}{b}}\right|^{n}\!+\left|{\frac {z}{c}}\right|^{p}\!=1}
104:, and symmetry about them, but defined by an equation that allows for various shapes between a rectangle and an ellipse.
5624:
108:
2847:. Then, the arc length of the superellipse within that single quadrant is approximated as the following function of
4556:
3451:
5676:
4497:
3675:
4666:
The outer outlines of the letters 'o' and 'O' in Zapf's Melior typeface are described by superellipses with
2253:{\displaystyle \operatorname {sgn}(w)={\begin{cases}-1,&w<0\\0,&w=0\\+1,&w>0.\end{cases}}}
382:
2639:{\displaystyle \mathrm {Area} ={\frac {4ab}{n}}\mathrm {B} \!\left({\frac {1}{n}},{\frac {1}{n}}+1\right).}
5571:
2663:
2891:
h + (((((n-0.88487077) * h + 0.2588574 / h) ^ exp(n / -0.90069205)) + h) + 0.09919785) ^ (-1.4812293 / n)
5713:
5554:
Tobler, Waldo (1973), "The hyperelliptical and other new pseudocylindrical equal area map projections",
3768:
Using different exponents for each term in the equation, allowing more flexibility in shape formation.
3136:{\displaystyle (a\cos \theta )^{\tfrac {n}{n-1}}+(b\sin \theta )^{\tfrac {n}{n-1}}=r^{\tfrac {n}{n-1}}.}
2675:
1916:
5339:
4992:"Capturing spiral radial growth of conifers using the superellipse to model tree-ring geometric shape"
4795:
1334:
5563:
5305:
4832:
2671:
5576:
5164:
2175:
5662:
4952:
4843:
4732:
2895:
This single-quadrant arc length approximation is accurate to within ±0.2% for across all values of
2682:
2667:
1675:
985:
39:
5666:
5470:
5282:
5133:
5105:
5078:
3864:{\displaystyle \left|{\frac {x}{a}}\right|^{m}\!\!+\left|{\frac {y}{b}}\right|^{n}\!=1;m,n>0,}
1380:
1328:
943:
668:
388:
348:
5619:
4735:(1905–1996) though he did not discover it as it is sometimes claimed. In 1959, city planners in
4696:
would build the ability to choose between true ellipses and superellipses (both approximated by
1504:
1188:
1474:
1444:
992:
5643:
5513:
5509:
5496:
5462:
5398:
5359:
5123:
5070:
5031:
5013:
4924:
4685:
4181:
3942:
3205:
2824:
1415:
1386:
1278:
1158:
1129:
908:
879:
5581:
5454:
5390:
5351:
5313:
5115:
5062:
5021:
5003:
3009:
2734:
2708:
1248:
1218:
1095:
322:
4779:, a solid egg-like shape that could stand upright on a flat surface, and was marketed as a
4271:
4239:
4213:
4149:
2792:
2760:
1623:
778:
708:
114:
4820:
4701:
1856:
1589:
1558:
101:
97:
5693:
5502:
Mathematical
Carnival. A New Round-Up of Tantalizers and Puzzles from Scientific American
4656:
3916:
3354:
2283:
1595:
1563:
1304:
1063:
1027:
847:
752:
296:
89:
5635:
5567:
5309:
218:{\displaystyle \left|{\frac {x}{a}}\right|^{n}\!\!+\left|{\frac {y}{b}}\right|^{n}\!=1,}
5491:
5276:
5026:
4991:
4942:
4854:
4824:
4803:
4799:
4300:
4021:
4000:
3979:
3895:
3874:
3734:
3427:
3421:
3400:
3380:
3199:
2898:
2870:
2850:
2688:
2405:
2263:
1896:
1876:
1681:
1363:
643:
623:
603:
276:
248:
228:
3154:
3001:{\displaystyle \left|{\frac {x}{a}}\right|^{n}\!+\left|{\frac {y}{b}}\right|^{n}\!=1,}
5707:
5505:
5137:
4938:
4894:
4626:
2540:
2142:
5227:
5698:
5416:
5082:
4932:
4780:
4748:
4697:
4693:
4678:
4631:
1657:
564:
266:
5646:
4662:
4294:
represents the structure of the
National Centre for the Performing Arts in China.
5119:
2915:, and can be used to efficiently estimate the total perimeter of a superellipse.
5340:"Generalization of the super ellipsoid concept and its application in mechanics"
4946:
4817:
in New York City consisted of a large, superellipse-shaped overhanging balcony.
4807:
4791:
2924:
812:
5394:
5097:
5050:
4655:
The general
Cartesian notation of the form comes from the French mathematician
3760:
5355:
4744:
4643:
975:
5682:
5466:
5402:
5378:
5363:
5074:
5066:
5017:
3731:
are positive numbers referred to as the semi-axes of the hyperellipsoid, and
3448:(and by extension, a superellipsoid). It is defined as the set of all points
3397:
are positive numbers referred to as the semi-axes of the superellipsoid, and
5651:
5585:
5338:
Ni, B. Y.; Elishakoff, I.; Jiang, C.; Fu, C. M.; Han, X. (1 November 2016).
5008:
4839:
4736:
3445:
2655:
1121:
981:
568:
5035:
5531:
5102:
2022 IEEE/CVF Conference on
Computer Vision and Pattern Recognition (CVPR)
4707:
700:
32:
17:
4970:
4906:
4776:
4682:
1192:
740:
4711:
The central fountain of
Sergels Torg is outlined by a superellipse with
2678:
is another option to obtain perimeter estimates at arbitrary precision.
5688:
5474:
5297:
4866:
4728:
2659:
1053:
873:
746:
93:
5377:
Cheng, Xinyu; Li, Chengbo; Peng, Yixue; Zhao, Chuang (17 April 2021).
5317:
5600:
5442:
5098:"Robust and Accurate Superquadric Recovery: A Probabilistic Approach"
4740:
1057:
5458:
5096:
Liu, Weixiao; Wu, Yuwei; Ruan, Sipu; Chirikjian, Gregory S. (2022).
2927:
is relatively straightforward to compute. Specifically, the pedal of
2386:{\textstyle {\frac {y}{x}}={\frac {b}{a}}(\tan t)^{2/n}\neq \tan t.}
5110:
4787:
4706:
3967:
3759:
2670:. Exact solutions for the perimeter of a superellipse exist using
1187:
991:
811:
4794:
could not agree on the shape of the negotiating table, Balinski,
4659:(1795–1870), who generalized the equation for the ellipse.
4751:. Piet Hein's winning proposal was based on a superellipse with
3976:
For the three-dimensional case, three different positive powers
2401:
1588:(in lowest terms), then each quadrant of the superellipse is a
4850:
4798:
and Holt suggested a superelliptical table in a letter to the
3149:
974:
The curve looks like a rhombus with the same corners but with
345:, the shape is more rectangular with rounded corners, and for
4553:
may differ from each other. It is the superellipsoid only if
1664:. In that case it is non-singular, but in general it will be
4688:, published in 1952, uses superellipses for letters such as
2674:; these could be truncated to obtain approximate solutions.
2246:
1706:
5441:
Land, Richard; Foley, James D.; Dam, Andries Van (1984).
2404:
inside the superellipse can be expressed in terms of the
739:, in particular, each of the four arcs is a segment of a
5304:. SPIE Proceedings. Vol. 0848. SPIE. p. 358.
4853:
uses a superellipse curve for app icons, replacing the
4802:. The superellipse was used for the shape of the 1968
3166:
5379:"Discrete element simulation of super-ellipse systems"
5296:
Boult, Terrance E.; Gross, Ari D. (19 February 1988).
3110:
3082:
3039:
2507:
2464:
2314:
1893:
gives four points on the curve. Equivalently, letting
385:, the superellipse equation is (the set of all points
4949:, the three-dimensional "relatives" of superellipses.
4559:
4500:
4323:
4303:
4274:
4242:
4216:
4184:
4152:
4044:
4024:
4003:
3982:
3945:
3919:
3898:
3877:
3777:
3764:
Variations of a superellipse with different exponents
3737:
3678:
3519:
3454:
3430:
3403:
3383:
3357:
3246:
3208:
3018:
2933:
2901:
2873:
2853:
2827:
2795:
2763:
2737:
2711:
2691:
2552:
2414:
2286:
2266:
2151:
1956:
1919:
1899:
1879:
1859:
1704:
1684:
1626:
1598:
1566:
1507:
1477:
1447:
1418:
1389:
1366:
1337:
1307:
1281:
1251:
1221:
1161:
1132:
1098:
1066:
1030:
946:
911:
882:
850:
781:
755:
711:
671:
646:
626:
606:
423:
391:
351:
325:
299:
293:
is a positive parameter that defines the shape. When
279:
251:
231:
149:
117:
111:, a superellipse is defined as the set of all points
42:
5051:"Superquadrics and Angle-Preserving Transformations"
2705:) is stretched such that the first quadrant (e.g.,
699:The superellipse looks like a four-armed star with
5495:
4610:
4545:
4486:
4309:
4286:
4260:
4228:
4202:
4170:
4138:
4030:
4009:
3988:
3957:
3931:
3904:
3883:
3863:
3743:
3723:
3664:
3505:
3436:
3409:
3389:
3369:
3343:
3232:
3135:
3000:
2907:
2879:
2859:
2839:
2813:
2781:
2749:
2723:
2697:
2638:
2531:
2385:
2300:
2272:
2252:
2133:
1943:
1905:
1885:
1865:
1845:
1690:
1644:
1612:
1580:
1537:
1493:
1463:
1433:
1404:
1372:
1352:
1319:
1293:
1263:
1233:
1176:
1147:
1110:
1078:
1042:
964:
926:
897:
862:
801:
767:
731:
689:
652:
632:
612:
548:
409:
369:
337:
311:
285:
257:
237:
217:
135:
69:
5300:. In Casasent, David P.; Hall, Ernest L. (eds.).
4477:
4421:
4371:
4370:
4129:
4100:
4071:
4070:
3833:
3804:
3803:
3652:
3603:
3560:
3559:
3331:
3302:
3273:
3272:
2988:
2959:
2593:
542:
519:
476:
475:
205:
176:
175:
5689:Superellipse Calculator & Template Generator
5298:"Recovery of Superquadrics from 3-D Information"
5443:"Fundamentals of Interactive Computer Graphics"
4765:
3751:is a positive parameter that defines the shape.
3417:is a positive parameter that defines the shape.
1873:can be chosen separately so that each value of
319:, the superellipse is an ordinary ellipse. For
5538:The Guide to Life, The Universe and Everything
1698:having no elementary geometric interpretation)
1383:. The extreme points of the superellipse are (
8:
4611:{\displaystyle n_{1}=n_{2}=\ldots =n_{N}=n}
3506:{\displaystyle (x_{1},x_{2},\ldots ,x_{d})}
5486:
5484:
5575:
5302:Intelligent Robots and Computer Vision VI
5109:
5025:
5007:
4596:
4577:
4564:
4558:
4546:{\displaystyle n_{1},n_{2},\ldots ,n_{N}}
4537:
4518:
4505:
4499:
4469:
4464:
4452:
4442:
4436:
4413:
4408:
4396:
4386:
4380:
4362:
4357:
4345:
4335:
4329:
4322:
4302:
4273:
4241:
4215:
4183:
4151:
4123:
4109:
4094:
4080:
4064:
4050:
4043:
4023:
4002:
3981:
3944:
3918:
3897:
3876:
3827:
3813:
3797:
3783:
3776:
3771:For two-dimentional case the equation is
3736:
3724:{\displaystyle a_{1},a_{2},\ldots ,a_{d}}
3715:
3696:
3683:
3677:
3646:
3634:
3624:
3618:
3597:
3585:
3575:
3569:
3553:
3541:
3531:
3525:
3518:
3494:
3475:
3462:
3453:
3429:
3402:
3382:
3356:
3325:
3311:
3296:
3282:
3266:
3252:
3245:
3207:
3109:
3081:
3038:
3017:
2982:
2968:
2953:
2939:
2932:
2900:
2872:
2852:
2826:
2794:
2762:
2736:
2710:
2690:
2681:A closed-form approximation obtained via
2612:
2599:
2588:
2570:
2553:
2551:
2506:
2485:
2463:
2441:
2415:
2413:
2358:
2354:
2328:
2315:
2313:
2290:
2285:
2265:
2170:
2150:
2089:
2068:
2004:
1983:
1957:
1955:
1918:
1898:
1878:
1858:
1833:
1795:
1743:
1708:
1703:
1683:
1625:
1602:
1597:
1570:
1565:
1525:
1518:
1506:
1485:
1476:
1455:
1446:
1417:
1388:
1365:
1344:
1340:
1339:
1336:
1306:
1280:
1250:
1220:
1160:
1131:
1097:
1065:
1029:
945:
910:
881:
849:
791:
780:
754:
721:
710:
670:
645:
625:
605:
532:
525:
513:
485:
469:
441:
422:
390:
350:
324:
298:
278:
250:
230:
199:
185:
169:
155:
148:
116:
41:
4921:, looks like "The Four-Cornered Wheel."
4661:
1327:, the superellipse is the boundary of a
662:
31:
5672:MacTutor History of Mathematics Archive
5055:IEEE Computer Graphics and Applications
4982:
4935:, a generalization of the superellipse.
3972:National Centre for the Performing Arts
5694:Superellipse fitting toolbox in MATLAB
4849:In computing, mobile operating system
1126:The curvature is zero at the points (
988:as one approaches its extreme points.
143:on the curve that satisfy the equation
96:, retaining the geometric features of
5333:
5331:
5329:
5327:
5281:. London: MacMillan and Co. pp.
3939:, it is the Lamé's superellipses. If
1120:The curve looks superficially like a
7:
5190:
5188:
4891:, is a hypocycloid with four cusps.
2308:while in the parametric expressions
1545:(sometimes called the "superness").
265:are positive numbers referred to as
27:Family of closed mathematical curves
4743:announced a design challenge for a
4317:–dimensional case, the equation is
2658:of a superellipse, like that of an
660:, as shown in the following table:
417:on the curve satisfy the equation):
92:, is a closed curve resembling the
4831:, published in 1973, in which the
4727:The superellipse was named by the
2589:
2563:
2560:
2557:
2554:
2492:
2449:
2425:
2422:
2419:
2416:
25:
5196:"Ellipsoids in Higher Dimensions"
4829:Tobler hyperelliptical projection
4813:The second floor of the original
3891:either equals to or differs from
1944:{\displaystyle 0\leq t<2\pi ,}
1656:is an even integer, then it is a
5699:C code for fitting superellipses
3153:
1353:{\displaystyle \mathbb {R} ^{2}}
5556:Journal of Geophysical Research
1820:
1441:), and its four "corners" are (
5417:"SuperQuadrics - Applications"
5344:Applied Mathematical Modelling
3500:
3455:
3227:
3209:
3078:
3062:
3035:
3019:
2808:
2796:
2776:
2764:
2351:
2338:
2164:
2158:
2124:
2112:
2039:
2027:
1241:, the figure is also called a
500:
494:
456:
450:
404:
392:
130:
118:
36:Examples of superellipses for
1:
4927:, "The Three-Cornered Wheel."
4786:In 1968, when negotiators in
5120:10.1109/CVPR52688.2022.00270
4857:style used up to version 6.
4038:can be used in the equation
3444:-dimensional analogue of an
70:{\displaystyle a=1,\ b=0.75}
5625:Encyclopedia of Mathematics
5601:"The iOS Design Guidelines"
5221:"Superellipse (Lame curve)"
4835:are arcs of superellipses.
4763:= 6/5. As he explained it:
965:{\displaystyle 1<n<2}
698:
690:{\displaystyle 0<n<1}
410:{\displaystyle (r,\theta )}
370:{\displaystyle 0<n<2}
109:Cartesian coordinate system
5730:
5497:"Piet Hein's Superellipse"
5395:10.1007/s10035-021-01107-4
4996:Frontiers in Plant Science
4838:The logo for news company
3513:that satisfy the equation:
3240:that satisfy the equation:
1674:The curve is given by the
1538:{\displaystyle s=2^{-1/n}}
5356:10.1016/j.apm.2016.06.011
1494:{\displaystyle \pm s_{b}}
1464:{\displaystyle \pm s_{a}}
1052:The curve is an ordinary
273:of the superellipse, and
5677:University of St Andrews
5067:10.1109/MCG.1981.1673799
4909:, the superellipse with
4869:, the superellipse with
1195:, the superellipse with
978:(outwards-curved) sides.
5618:Sokolov, D.D. (2001) ,
5586:10.1029/JB078i011p01753
5009:10.3389/fpls.2015.00856
4955:, equation of the form
4692:. Thirty years later
4236:and the semi-diameters
4203:{\displaystyle m=n=2.2}
3958:{\displaystyle m\neq n}
3233:{\displaystyle (x,y,z)}
2840:{\displaystyle h\geq 1}
1549:Mathematical properties
1434:{\displaystyle 0,\pm b}
1405:{\displaystyle \pm a,0}
1294:{\displaystyle n\geq 1}
1177:{\displaystyle 0,\pm b}
1148:{\displaystyle \pm a,0}
927:{\displaystyle 0,\pm b}
898:{\displaystyle \pm a,0}
703:(inwards-curved) sides.
563:This formula defines a
383:polar coordinate system
377:, it is more pointed.
5104:. pp. 2666–2675.
4804:Azteca Olympic Stadium
4772:
4724:
4675:
4612:
4547:
4488:
4311:
4288:
4262:
4230:
4204:
4172:
4140:
4032:
4011:
3990:
3973:
3959:
3933:
3906:
3885:
3865:
3765:
3745:
3725:
3666:
3507:
3438:
3411:
3391:
3371:
3345:
3234:
3137:
3002:
2909:
2881:
2861:
2841:
2815:
2783:
2751:
2750:{\displaystyle y>0}
2725:
2724:{\displaystyle x>0}
2699:
2640:
2533:
2387:
2302:
2274:
2254:
2135:
1945:
1907:
1887:
1867:
1847:
1692:
1646:
1620:. In particular, when
1614:
1582:
1539:
1495:
1465:
1435:
1406:
1374:
1354:
1321:
1295:
1265:
1264:{\displaystyle n>2}
1235:
1234:{\displaystyle n<2}
1208:
1178:
1149:
1112:
1111:{\displaystyle n>2}
1080:
1044:
1019:
996:The superellipse with
966:
928:
899:
864:
839:
816:The superellipse with
803:
769:
733:
691:
654:
634:
614:
550:
411:
371:
339:
338:{\displaystyle n>2}
313:
287:
259:
239:
219:
137:
77:
71:
5278:Differential Calculus
4897:, the hypocycloid of
4747:in their city square
4710:
4665:
4613:
4548:
4489:
4312:
4289:
4287:{\displaystyle c=0.5}
4263:
4261:{\displaystyle a=b=1}
4231:
4229:{\displaystyle p=2.4}
4205:
4173:
4171:{\displaystyle m=n=p}
4141:
4033:
4012:
3991:
3971:
3960:
3934:
3907:
3886:
3866:
3763:
3746:
3726:
3667:
3508:
3439:
3412:
3392:
3372:
3346:
3235:
3138:
3003:
2910:
2882:
2862:
2842:
2816:
2814:{\displaystyle (0,h)}
2784:
2782:{\displaystyle (1,0)}
2752:
2726:
2700:
2676:Numerical integration
2641:
2534:
2388:
2303:
2275:
2255:
2136:
1946:
1908:
1888:
1868:
1848:
1693:
1647:
1645:{\displaystyle a=b=1}
1615:
1590:plane algebraic curve
1583:
1540:
1496:
1466:
1436:
1407:
1375:
1355:
1322:
1296:
1266:
1236:
1191:
1179:
1150:
1124:with rounded corners.
1113:
1081:
1045:
995:
967:
929:
900:
865:
815:
804:
802:{\displaystyle n=2/3}
770:
734:
732:{\displaystyle n=1/2}
692:
655:
635:
615:
551:
412:
372:
340:
314:
288:
260:
240:
220:
138:
136:{\displaystyle (x,y)}
72:
35:
5663:Robertson, Edmund F.
4672:log(1/2) / log (7/9)
4557:
4498:
4494:, where In general,
4321:
4301:
4272:
4240:
4214:
4182:
4150:
4042:
4022:
4001:
3980:
3943:
3917:
3896:
3875:
3775:
3735:
3676:
3517:
3452:
3428:
3401:
3381:
3355:
3244:
3206:
3016:
2931:
2899:
2871:
2851:
2825:
2793:
2761:
2735:
2709:
2689:
2668:elementary functions
2664:closed-form solution
2550:
2412:
2312:
2284:
2264:
2149:
1954:
1917:
1897:
1877:
1866:{\displaystyle \pm }
1857:
1702:
1682:
1676:parametric equations
1624:
1596:
1564:
1505:
1475:
1445:
1416:
1387:
1364:
1335:
1305:
1279:
1249:
1219:
1159:
1130:
1096:
1064:
1028:
944:
909:
880:
848:
779:
753:
749:is the special case
709:
669:
644:
624:
604:
421:
389:
349:
323:
297:
277:
249:
229:
147:
115:
40:
5661:O'Connor, John J.;
5568:1973JGR....78.1753T
5310:1988SPIE..848..358B
5275:J. Edwards (1892).
5200:analyticphysics.com
4953:Superelliptic curve
4913: = 4 and
4844:Pittsburgh Steelers
4731:poet and scientist
4639:Anisotropic Scaling
3932:{\displaystyle m=n}
3756:Different Exponents
3370:{\displaystyle a,b}
2683:symbolic regression
2672:infinite summations
2539:or in terms of the
2301:{\displaystyle y/x}
1613:{\displaystyle p/q}
1581:{\displaystyle p/q}
1320:{\displaystyle a=b}
1079:{\displaystyle a=b}
1043:{\displaystyle n=2}
863:{\displaystyle n=1}
768:{\displaystyle a=b}
312:{\displaystyle n=2}
107:In two dimensional
5644:Weisstein, Eric W.
4815:World Trade Center
4725:
4676:
4608:
4543:
4484:
4307:
4284:
4258:
4226:
4200:
4168:
4136:
4028:
4007:
3986:
3974:
3955:
3929:
3902:
3881:
3861:
3766:
3741:
3721:
3662:
3503:
3434:
3407:
3387:
3367:
3341:
3230:
3165:. You can help by
3133:
3127:
3099:
3056:
2998:
2905:
2877:
2857:
2837:
2811:
2779:
2747:
2721:
2695:
2636:
2529:
2516:
2473:
2383:
2298:
2270:
2250:
2245:
2131:
2129:
1941:
1903:
1883:
1863:
1843:
1814:
1688:
1642:
1610:
1578:
1535:
1491:
1461:
1431:
1402:
1370:
1350:
1317:
1291:
1261:
1231:
1209:
1174:
1145:
1108:
1076:
1056:(in particular, a
1040:
1020:
984:increases without
962:
924:
895:
860:
840:
799:
765:
729:
687:
650:
630:
610:
600:. The parameters
546:
407:
367:
335:
309:
283:
255:
235:
215:
133:
84:, also known as a
78:
67:
5562:(11): 1753–1759,
5519:978-0-394-72349-5
5350:(21): 9427–9444.
5318:10.1117/12.942759
5129:978-1-6654-6946-3
4925:Reuleaux triangle
4458:
4402:
4351:
4310:{\displaystyle N}
4117:
4088:
4058:
4031:{\displaystyle p}
4010:{\displaystyle n}
3989:{\displaystyle m}
3905:{\displaystyle n}
3884:{\displaystyle m}
3821:
3791:
3744:{\displaystyle n}
3640:
3591:
3547:
3437:{\displaystyle d}
3410:{\displaystyle n}
3390:{\displaystyle c}
3319:
3290:
3260:
3190:Higher Dimensions
3183:
3182:
3126:
3098:
3055:
3010:polar coordinates
2976:
2947:
2908:{\displaystyle n}
2880:{\displaystyle n}
2860:{\displaystyle h}
2757:) is an arc from
2698:{\displaystyle n}
2662:, does not admit
2620:
2607:
2586:
2524:
2515:
2472:
2336:
2323:
2273:{\displaystyle t}
2097:
2012:
1906:{\displaystyle t}
1886:{\displaystyle t}
1841:
1803:
1751:
1691:{\displaystyle t}
1373:{\displaystyle n}
1213:
1212:
653:{\displaystyle n}
633:{\displaystyle b}
613:{\displaystyle a}
567:contained in the
507:
463:
286:{\displaystyle n}
258:{\displaystyle b}
238:{\displaystyle a}
193:
163:
57:
16:(Redirected from
5721:
5679:
5657:
5656:
5632:
5605:
5604:
5599:Mynttinen, Ivo.
5596:
5590:
5589:
5579:
5551:
5545:
5533:The Superellipse
5529:
5523:
5522:
5499:
5488:
5479:
5478:
5438:
5432:
5431:
5429:
5427:
5421:www.cs.mcgill.ca
5413:
5407:
5406:
5374:
5368:
5367:
5335:
5322:
5321:
5293:
5287:
5286:
5272:
5266:
5265:
5263:
5261:
5249:
5243:
5242:
5240:
5238:
5233:on 31 March 2022
5232:
5226:. Archived from
5225:
5217:
5211:
5210:
5208:
5206:
5192:
5183:
5182:
5180:
5178:
5169:
5161:
5155:
5152:The METAFONTbook
5148:
5142:
5141:
5113:
5093:
5087:
5086:
5046:
5040:
5039:
5029:
5011:
4987:
4882:
4881:
4877:
4796:Kieron Underwood
4673:
4617:
4615:
4614:
4609:
4601:
4600:
4582:
4581:
4569:
4568:
4552:
4550:
4549:
4544:
4542:
4541:
4523:
4522:
4510:
4509:
4493:
4491:
4490:
4485:
4476:
4475:
4474:
4473:
4463:
4459:
4457:
4456:
4447:
4446:
4437:
4420:
4419:
4418:
4417:
4407:
4403:
4401:
4400:
4391:
4390:
4381:
4369:
4368:
4367:
4366:
4356:
4352:
4350:
4349:
4340:
4339:
4330:
4316:
4314:
4313:
4308:
4293:
4291:
4290:
4285:
4267:
4265:
4264:
4259:
4235:
4233:
4232:
4227:
4209:
4207:
4206:
4201:
4177:
4175:
4174:
4169:
4145:
4143:
4142:
4137:
4128:
4127:
4122:
4118:
4110:
4099:
4098:
4093:
4089:
4081:
4069:
4068:
4063:
4059:
4051:
4037:
4035:
4034:
4029:
4016:
4014:
4013:
4008:
3995:
3993:
3992:
3987:
3964:
3962:
3961:
3956:
3938:
3936:
3935:
3930:
3911:
3909:
3908:
3903:
3890:
3888:
3887:
3882:
3870:
3868:
3867:
3862:
3832:
3831:
3826:
3822:
3814:
3802:
3801:
3796:
3792:
3784:
3750:
3748:
3747:
3742:
3730:
3728:
3727:
3722:
3720:
3719:
3701:
3700:
3688:
3687:
3671:
3669:
3668:
3663:
3651:
3650:
3645:
3641:
3639:
3638:
3629:
3628:
3619:
3602:
3601:
3596:
3592:
3590:
3589:
3580:
3579:
3570:
3558:
3557:
3552:
3548:
3546:
3545:
3536:
3535:
3526:
3512:
3510:
3509:
3504:
3499:
3498:
3480:
3479:
3467:
3466:
3443:
3441:
3440:
3435:
3416:
3414:
3413:
3408:
3396:
3394:
3393:
3388:
3376:
3374:
3373:
3368:
3350:
3348:
3347:
3342:
3330:
3329:
3324:
3320:
3312:
3301:
3300:
3295:
3291:
3283:
3271:
3270:
3265:
3261:
3253:
3239:
3237:
3236:
3231:
3178:
3175:
3157:
3150:
3142:
3140:
3139:
3134:
3129:
3128:
3125:
3111:
3101:
3100:
3097:
3083:
3058:
3057:
3054:
3040:
3007:
3005:
3004:
2999:
2987:
2986:
2981:
2977:
2969:
2958:
2957:
2952:
2948:
2940:
2914:
2912:
2911:
2906:
2892:
2886:
2884:
2883:
2878:
2866:
2864:
2863:
2858:
2846:
2844:
2843:
2838:
2820:
2818:
2817:
2812:
2788:
2786:
2785:
2780:
2756:
2754:
2753:
2748:
2730:
2728:
2727:
2722:
2704:
2702:
2701:
2696:
2645:
2643:
2642:
2637:
2632:
2628:
2621:
2613:
2608:
2600:
2592:
2587:
2582:
2571:
2566:
2538:
2536:
2535:
2530:
2525:
2523:
2522:
2518:
2517:
2508:
2490:
2489:
2484:
2480:
2479:
2475:
2474:
2465:
2442:
2428:
2392:
2390:
2389:
2384:
2367:
2366:
2362:
2337:
2329:
2324:
2316:
2307:
2305:
2304:
2299:
2294:
2279:
2277:
2276:
2271:
2259:
2257:
2256:
2251:
2249:
2248:
2140:
2138:
2137:
2132:
2130:
2099:
2098:
2090:
2088:
2087:
2083:
2059:
2014:
2013:
2005:
2003:
2002:
1998:
1974:
1950:
1948:
1947:
1942:
1912:
1910:
1909:
1904:
1892:
1890:
1889:
1884:
1872:
1870:
1869:
1864:
1852:
1850:
1849:
1844:
1842:
1834:
1819:
1815:
1805:
1804:
1796:
1777:
1753:
1752:
1744:
1725:
1697:
1695:
1694:
1689:
1678:(with parameter
1651:
1649:
1648:
1643:
1619:
1617:
1616:
1611:
1606:
1587:
1585:
1584:
1579:
1574:
1544:
1542:
1541:
1536:
1534:
1533:
1529:
1500:
1498:
1497:
1492:
1490:
1489:
1470:
1468:
1467:
1462:
1460:
1459:
1440:
1438:
1437:
1432:
1411:
1409:
1408:
1403:
1379:
1377:
1376:
1371:
1359:
1357:
1356:
1351:
1349:
1348:
1343:
1326:
1324:
1323:
1318:
1300:
1298:
1297:
1292:
1270:
1268:
1267:
1262:
1240:
1238:
1237:
1232:
1199: = 4,
1183:
1181:
1180:
1175:
1154:
1152:
1151:
1146:
1117:
1115:
1114:
1109:
1085:
1083:
1082:
1077:
1049:
1047:
1046:
1041:
1009:
1008:
1004:
971:
969:
968:
963:
933:
931:
930:
925:
904:
902:
901:
896:
869:
867:
866:
861:
829:
828:
824:
808:
806:
805:
800:
795:
774:
772:
771:
766:
738:
736:
735:
730:
725:
696:
694:
693:
688:
663:
659:
657:
656:
651:
639:
637:
636:
631:
619:
617:
616:
611:
599:
584:
555:
553:
552:
547:
541:
540:
536:
524:
520:
518:
517:
512:
508:
503:
486:
474:
473:
468:
464:
459:
442:
416:
414:
413:
408:
376:
374:
373:
368:
344:
342:
341:
336:
318:
316:
315:
310:
292:
290:
289:
284:
264:
262:
261:
256:
244:
242:
241:
236:
224:
222:
221:
216:
204:
203:
198:
194:
186:
174:
173:
168:
164:
156:
142:
140:
139:
134:
76:
74:
73:
68:
55:
21:
5729:
5728:
5724:
5723:
5722:
5720:
5719:
5718:
5704:
5703:
5685:on 2dcurves.com
5683:"Super Ellipse"
5660:
5642:
5641:
5617:
5614:
5609:
5608:
5598:
5597:
5593:
5577:10.1.1.495.6424
5553:
5552:
5548:
5530:
5526:
5520:
5492:Gardner, Martin
5490:
5489:
5482:
5459:10.2307/1574879
5440:
5439:
5435:
5425:
5423:
5415:
5414:
5410:
5383:Granular Matter
5376:
5375:
5371:
5337:
5336:
5325:
5295:
5294:
5290:
5274:
5273:
5269:
5259:
5257:
5252:Sharpe, Peter.
5251:
5250:
5246:
5236:
5234:
5230:
5223:
5219:
5218:
5214:
5204:
5202:
5194:
5193:
5186:
5176:
5174:
5167:
5163:
5162:
5158:
5149:
5145:
5130:
5095:
5094:
5090:
5048:
5047:
5043:
4989:
4988:
4984:
4979:
4943:superellipsoids
4879:
4875:
4874:
4863:
4855:rounded corners
4821:Waldo R. Tobler
4702:Computer Modern
4671:
4653:
4641:
4624:
4592:
4573:
4560:
4555:
4554:
4533:
4514:
4501:
4496:
4495:
4465:
4448:
4438:
4432:
4431:
4409:
4392:
4382:
4376:
4375:
4358:
4341:
4331:
4325:
4324:
4319:
4318:
4299:
4298:
4297:In the general
4270:
4269:
4238:
4237:
4212:
4211:
4180:
4179:
4148:
4147:
4105:
4104:
4076:
4075:
4046:
4045:
4040:
4039:
4020:
4019:
3999:
3998:
3978:
3977:
3941:
3940:
3915:
3914:
3894:
3893:
3873:
3872:
3809:
3808:
3779:
3778:
3773:
3772:
3758:
3733:
3732:
3711:
3692:
3679:
3674:
3673:
3630:
3620:
3614:
3613:
3581:
3571:
3565:
3564:
3537:
3527:
3521:
3520:
3515:
3514:
3490:
3471:
3458:
3450:
3449:
3426:
3425:
3399:
3398:
3379:
3378:
3353:
3352:
3307:
3306:
3278:
3277:
3248:
3247:
3242:
3241:
3204:
3203:
3192:
3179:
3173:
3170:
3163:needs expansion
3148:
3146:Generalizations
3115:
3105:
3087:
3077:
3044:
3034:
3014:
3013:
2964:
2963:
2935:
2934:
2929:
2928:
2921:
2897:
2896:
2890:
2869:
2868:
2849:
2848:
2823:
2822:
2791:
2790:
2759:
2758:
2733:
2732:
2707:
2706:
2687:
2686:
2652:
2598:
2594:
2572:
2548:
2547:
2499:
2495:
2491:
2456:
2452:
2448:
2444:
2443:
2410:
2409:
2398:
2350:
2310:
2309:
2282:
2281:
2262:
2261:
2244:
2243:
2232:
2220:
2219:
2208:
2199:
2198:
2187:
2171:
2147:
2146:
2128:
2127:
2073:
2069:
2067:
2060:
2049:
2043:
2042:
1988:
1984:
1982:
1975:
1964:
1952:
1951:
1915:
1914:
1895:
1894:
1875:
1874:
1855:
1854:
1813:
1812:
1791:
1778:
1767:
1761:
1760:
1739:
1726:
1715:
1705:
1700:
1699:
1680:
1679:
1671:
1622:
1621:
1594:
1593:
1562:
1561:
1559:rational number
1551:
1514:
1503:
1502:
1481:
1473:
1472:
1451:
1443:
1442:
1414:
1413:
1385:
1384:
1362:
1361:
1338:
1333:
1332:
1303:
1302:
1277:
1276:
1247:
1246:
1217:
1216:
1157:
1156:
1128:
1127:
1125:
1094:
1093:
1062:
1061:
1026:
1025:
1006:
1002:
1001:
979:
942:
941:
907:
906:
878:
877:
876:with corners (
872:The curve is a
846:
845:
826:
822:
821:
777:
776:
751:
750:
744:
707:
706:
704:
667:
666:
642:
641:
622:
621:
602:
601:
586:
571:
561:
487:
481:
480:
443:
437:
436:
435:
431:
430:
419:
418:
387:
386:
380:
347:
346:
321:
320:
295:
294:
275:
274:
247:
246:
227:
226:
181:
180:
151:
150:
145:
144:
113:
112:
102:semi-minor axis
98:semi-major axis
38:
37:
28:
23:
22:
15:
12:
11:
5:
5727:
5725:
5717:
5716:
5706:
5705:
5702:
5701:
5696:
5691:
5686:
5680:
5658:
5647:"Superellipse"
5639:
5633:
5613:
5612:External links
5610:
5607:
5606:
5591:
5546:
5544:(27 June 2003)
5524:
5518:
5480:
5433:
5408:
5369:
5323:
5288:
5267:
5244:
5212:
5184:
5156:
5150:Donald Knuth:
5143:
5128:
5088:
5041:
4981:
4980:
4978:
4975:
4974:
4973:
4968:
4950:
4936:
4930:
4929:
4928:
4904:
4903:
4902:
4862:
4859:
4825:map projection
4800:New York Times
4652:
4649:
4640:
4637:
4623:
4622:Related shapes
4620:
4607:
4604:
4599:
4595:
4591:
4588:
4585:
4580:
4576:
4572:
4567:
4563:
4540:
4536:
4532:
4529:
4526:
4521:
4517:
4513:
4508:
4504:
4483:
4480:
4472:
4468:
4462:
4455:
4451:
4445:
4441:
4435:
4430:
4427:
4424:
4416:
4412:
4406:
4399:
4395:
4389:
4385:
4379:
4374:
4365:
4361:
4355:
4348:
4344:
4338:
4334:
4328:
4306:
4283:
4280:
4277:
4257:
4254:
4251:
4248:
4245:
4225:
4222:
4219:
4199:
4196:
4193:
4190:
4187:
4167:
4164:
4161:
4158:
4155:
4135:
4132:
4126:
4121:
4116:
4113:
4108:
4103:
4097:
4092:
4087:
4084:
4079:
4074:
4067:
4062:
4057:
4054:
4049:
4027:
4006:
3985:
3954:
3951:
3948:
3928:
3925:
3922:
3901:
3880:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3830:
3825:
3820:
3817:
3812:
3807:
3800:
3795:
3790:
3787:
3782:
3757:
3754:
3753:
3752:
3740:
3718:
3714:
3710:
3707:
3704:
3699:
3695:
3691:
3686:
3682:
3661:
3658:
3655:
3649:
3644:
3637:
3633:
3627:
3623:
3617:
3612:
3609:
3606:
3600:
3595:
3588:
3584:
3578:
3574:
3568:
3563:
3556:
3551:
3544:
3540:
3534:
3530:
3524:
3502:
3497:
3493:
3489:
3486:
3483:
3478:
3474:
3470:
3465:
3461:
3457:
3433:
3422:hyperellipsoid
3418:
3406:
3386:
3366:
3363:
3360:
3340:
3337:
3334:
3328:
3323:
3318:
3315:
3310:
3305:
3299:
3294:
3289:
3286:
3281:
3276:
3269:
3264:
3259:
3256:
3251:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3200:superellipsoid
3191:
3188:
3181:
3180:
3160:
3158:
3147:
3144:
3132:
3124:
3121:
3118:
3114:
3108:
3104:
3096:
3093:
3090:
3086:
3080:
3076:
3073:
3070:
3067:
3064:
3061:
3053:
3050:
3047:
3043:
3037:
3033:
3030:
3027:
3024:
3021:
2997:
2994:
2991:
2985:
2980:
2975:
2972:
2967:
2962:
2956:
2951:
2946:
2943:
2938:
2920:
2917:
2904:
2876:
2856:
2836:
2833:
2830:
2810:
2807:
2804:
2801:
2798:
2778:
2775:
2772:
2769:
2766:
2746:
2743:
2740:
2720:
2717:
2714:
2694:
2651:
2648:
2647:
2646:
2635:
2631:
2627:
2624:
2619:
2616:
2611:
2606:
2603:
2597:
2591:
2585:
2581:
2578:
2575:
2569:
2565:
2562:
2559:
2556:
2528:
2521:
2514:
2511:
2505:
2502:
2498:
2494:
2488:
2483:
2478:
2471:
2468:
2462:
2459:
2455:
2451:
2447:
2440:
2437:
2434:
2431:
2427:
2424:
2421:
2418:
2406:gamma function
2397:
2394:
2382:
2379:
2376:
2373:
2370:
2365:
2361:
2357:
2353:
2349:
2346:
2343:
2340:
2335:
2332:
2327:
2322:
2319:
2297:
2293:
2289:
2269:
2247:
2242:
2239:
2236:
2233:
2231:
2228:
2225:
2222:
2221:
2218:
2215:
2212:
2209:
2207:
2204:
2201:
2200:
2197:
2194:
2191:
2188:
2186:
2183:
2180:
2177:
2176:
2174:
2169:
2166:
2163:
2160:
2157:
2154:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2096:
2093:
2086:
2082:
2079:
2076:
2072:
2066:
2063:
2061:
2058:
2055:
2052:
2048:
2045:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2011:
2008:
2001:
1997:
1994:
1991:
1987:
1981:
1978:
1976:
1973:
1970:
1967:
1963:
1960:
1959:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1902:
1882:
1862:
1840:
1837:
1832:
1829:
1826:
1823:
1818:
1811:
1808:
1802:
1799:
1794:
1790:
1787:
1784:
1781:
1779:
1776:
1773:
1770:
1766:
1763:
1762:
1759:
1756:
1750:
1747:
1742:
1738:
1735:
1732:
1729:
1727:
1724:
1721:
1718:
1714:
1711:
1710:
1707:
1687:
1641:
1638:
1635:
1632:
1629:
1609:
1605:
1601:
1577:
1573:
1569:
1557:is a positive
1550:
1547:
1532:
1528:
1524:
1521:
1517:
1513:
1510:
1488:
1484:
1480:
1458:
1454:
1450:
1430:
1427:
1424:
1421:
1401:
1398:
1395:
1392:
1369:
1347:
1342:
1316:
1313:
1310:
1290:
1287:
1284:
1260:
1257:
1254:
1230:
1227:
1224:
1211:
1210:
1207: = 1
1185:
1173:
1170:
1167:
1164:
1144:
1141:
1138:
1135:
1118:
1107:
1104:
1101:
1090:
1089:
1087:
1075:
1072:
1069:
1050:
1039:
1036:
1033:
1022:
1021:
1018: = 1
989:
972:
961:
958:
955:
952:
949:
938:
937:
935:
923:
920:
917:
914:
894:
891:
888:
885:
870:
859:
856:
853:
842:
841:
838: = 1
809:
798:
794:
790:
787:
784:
764:
761:
758:
728:
724:
720:
717:
714:
697:
686:
683:
680:
677:
674:
649:
629:
609:
560:
559:Specific cases
557:
545:
539:
535:
531:
528:
523:
516:
511:
506:
502:
499:
496:
493:
490:
484:
479:
472:
467:
462:
458:
455:
452:
449:
446:
440:
434:
429:
426:
406:
403:
400:
397:
394:
366:
363:
360:
357:
354:
334:
331:
328:
308:
305:
302:
282:
267:semi-diameters
254:
234:
214:
211:
208:
202:
197:
192:
189:
184:
179:
172:
167:
162:
159:
154:
132:
129:
126:
123:
120:
66:
63:
60:
54:
51:
48:
45:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5726:
5715:
5712:
5711:
5709:
5700:
5697:
5695:
5692:
5690:
5687:
5684:
5681:
5678:
5674:
5673:
5668:
5667:"Lame Curves"
5664:
5659:
5654:
5653:
5648:
5645:
5640:
5638:at MathCurve.
5637:
5634:
5631:
5627:
5626:
5621:
5616:
5615:
5611:
5602:
5595:
5592:
5587:
5583:
5578:
5573:
5569:
5565:
5561:
5557:
5550:
5547:
5543:
5539:
5535:
5534:
5528:
5525:
5521:
5515:
5511:
5507:
5506:Vintage Press
5503:
5498:
5493:
5487:
5485:
5481:
5476:
5472:
5468:
5464:
5460:
5456:
5452:
5448:
5444:
5437:
5434:
5422:
5418:
5412:
5409:
5404:
5400:
5396:
5392:
5388:
5384:
5380:
5373:
5370:
5365:
5361:
5357:
5353:
5349:
5345:
5341:
5334:
5332:
5330:
5328:
5324:
5319:
5315:
5311:
5307:
5303:
5299:
5292:
5289:
5284:
5280:
5279:
5271:
5268:
5255:
5254:"AeroSandbox"
5248:
5245:
5229:
5222:
5216:
5213:
5201:
5197:
5191:
5189:
5185:
5173:
5166:
5160:
5157:
5153:
5147:
5144:
5139:
5135:
5131:
5125:
5121:
5117:
5112:
5107:
5103:
5099:
5092:
5089:
5084:
5080:
5076:
5072:
5068:
5064:
5060:
5056:
5052:
5049:Barr (1981).
5045:
5042:
5037:
5033:
5028:
5023:
5019:
5015:
5010:
5005:
5001:
4997:
4993:
4986:
4983:
4976:
4972:
4969:
4966:
4962:
4958:
4954:
4951:
4948:
4944:
4940:
4939:Superquadrics
4937:
4934:
4931:
4926:
4923:
4922:
4920:
4917: =
4916:
4912:
4908:
4905:
4900:
4896:
4895:Deltoid curve
4893:
4892:
4890:
4887: =
4886:
4873: =
4872:
4868:
4865:
4864:
4860:
4858:
4856:
4852:
4847:
4845:
4841:
4836:
4834:
4830:
4826:
4822:
4818:
4816:
4811:
4809:
4805:
4801:
4797:
4793:
4789:
4784:
4782:
4778:
4771:
4770:
4764:
4762:
4758:
4754:
4750:
4746:
4742:
4738:
4734:
4730:
4722:
4718:
4714:
4709:
4705:
4704:type family.
4703:
4699:
4698:cubic splines
4695:
4691:
4687:
4684:
4680:
4669:
4664:
4660:
4658:
4650:
4648:
4645:
4638:
4636:
4633:
4628:
4627:Superquadrics
4621:
4619:
4605:
4602:
4597:
4593:
4589:
4586:
4583:
4578:
4574:
4570:
4565:
4561:
4538:
4534:
4530:
4527:
4524:
4519:
4515:
4511:
4506:
4502:
4481:
4478:
4470:
4466:
4460:
4453:
4449:
4443:
4439:
4433:
4428:
4425:
4422:
4414:
4410:
4404:
4397:
4393:
4387:
4383:
4377:
4372:
4363:
4359:
4353:
4346:
4342:
4336:
4332:
4326:
4304:
4295:
4281:
4278:
4275:
4255:
4252:
4249:
4246:
4243:
4223:
4220:
4217:
4197:
4194:
4191:
4188:
4185:
4165:
4162:
4159:
4156:
4153:
4133:
4130:
4124:
4119:
4114:
4111:
4106:
4101:
4095:
4090:
4085:
4082:
4077:
4072:
4065:
4060:
4055:
4052:
4047:
4025:
4017:
4004:
3983:
3970:
3966:
3952:
3949:
3946:
3926:
3923:
3920:
3912:
3899:
3878:
3858:
3855:
3852:
3849:
3846:
3843:
3840:
3837:
3834:
3828:
3823:
3818:
3815:
3810:
3805:
3798:
3793:
3788:
3785:
3780:
3769:
3762:
3755:
3738:
3716:
3712:
3708:
3705:
3702:
3697:
3693:
3689:
3684:
3680:
3659:
3656:
3653:
3647:
3642:
3635:
3631:
3625:
3621:
3615:
3610:
3607:
3604:
3598:
3593:
3586:
3582:
3576:
3572:
3566:
3561:
3554:
3549:
3542:
3538:
3532:
3528:
3522:
3495:
3491:
3487:
3484:
3481:
3476:
3472:
3468:
3463:
3459:
3447:
3431:
3423:
3419:
3404:
3384:
3364:
3361:
3358:
3338:
3335:
3332:
3326:
3321:
3316:
3313:
3308:
3303:
3297:
3292:
3287:
3284:
3279:
3274:
3267:
3262:
3257:
3254:
3249:
3224:
3221:
3218:
3215:
3212:
3201:
3197:
3196:
3195:
3189:
3187:
3177:
3168:
3164:
3161:This section
3159:
3156:
3152:
3151:
3145:
3143:
3130:
3122:
3119:
3116:
3112:
3106:
3102:
3094:
3091:
3088:
3084:
3074:
3071:
3068:
3065:
3059:
3051:
3048:
3045:
3041:
3031:
3028:
3025:
3022:
3011:
2995:
2992:
2989:
2983:
2978:
2973:
2970:
2965:
2960:
2954:
2949:
2944:
2941:
2936:
2926:
2918:
2916:
2902:
2893:
2888:
2874:
2854:
2834:
2831:
2828:
2805:
2802:
2799:
2773:
2770:
2767:
2744:
2741:
2738:
2718:
2715:
2712:
2692:
2684:
2679:
2677:
2673:
2669:
2666:purely using
2665:
2661:
2657:
2649:
2633:
2629:
2625:
2622:
2617:
2614:
2609:
2604:
2601:
2595:
2583:
2579:
2576:
2573:
2567:
2546:
2545:
2544:
2542:
2541:beta function
2526:
2519:
2512:
2509:
2503:
2500:
2496:
2486:
2481:
2476:
2469:
2466:
2460:
2457:
2453:
2445:
2438:
2435:
2432:
2429:
2407:
2403:
2395:
2393:
2380:
2377:
2374:
2371:
2368:
2363:
2359:
2355:
2347:
2344:
2341:
2333:
2330:
2325:
2320:
2317:
2295:
2291:
2287:
2267:
2240:
2237:
2234:
2229:
2226:
2223:
2216:
2213:
2210:
2205:
2202:
2195:
2192:
2189:
2184:
2181:
2178:
2172:
2167:
2161:
2155:
2152:
2144:
2143:sign function
2121:
2118:
2115:
2109:
2106:
2103:
2100:
2094:
2091:
2084:
2080:
2077:
2074:
2070:
2064:
2062:
2056:
2053:
2050:
2046:
2036:
2033:
2030:
2024:
2021:
2018:
2015:
2009:
2006:
1999:
1995:
1992:
1989:
1985:
1979:
1977:
1971:
1968:
1965:
1961:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1900:
1880:
1860:
1838:
1835:
1830:
1827:
1824:
1821:
1816:
1809:
1806:
1800:
1797:
1792:
1788:
1785:
1782:
1780:
1774:
1771:
1768:
1764:
1757:
1754:
1748:
1745:
1740:
1736:
1733:
1730:
1728:
1722:
1719:
1716:
1712:
1685:
1677:
1672:
1669:
1667:
1663:
1659:
1655:
1639:
1636:
1633:
1630:
1627:
1607:
1603:
1599:
1591:
1575:
1571:
1567:
1560:
1556:
1548:
1546:
1530:
1526:
1522:
1519:
1515:
1511:
1508:
1486:
1482:
1478:
1456:
1452:
1448:
1428:
1425:
1422:
1419:
1399:
1396:
1393:
1390:
1382:
1367:
1345:
1330:
1314:
1311:
1308:
1288:
1285:
1282:
1274:
1258:
1255:
1252:
1244:
1228:
1225:
1222:
1206:
1203: =
1202:
1198:
1194:
1190:
1186:
1171:
1168:
1165:
1162:
1142:
1139:
1136:
1133:
1123:
1119:
1105:
1102:
1099:
1092:
1091:
1088:
1073:
1070:
1067:
1059:
1055:
1051:
1037:
1034:
1031:
1024:
1023:
1017:
1014: =
1013:
1000: =
999:
994:
990:
987:
983:
977:
973:
959:
956:
953:
950:
947:
940:
939:
936:
921:
918:
915:
912:
892:
889:
886:
883:
875:
871:
857:
854:
851:
844:
843:
837:
834: =
833:
820: =
819:
814:
810:
796:
792:
788:
785:
782:
762:
759:
756:
748:
742:
726:
722:
718:
715:
712:
702:
684:
681:
678:
675:
672:
665:
664:
661:
647:
627:
607:
598:
594:
590:
583:
579:
575:
570:
566:
558:
556:
543:
537:
533:
529:
526:
521:
514:
509:
504:
497:
491:
488:
482:
477:
470:
465:
460:
453:
447:
444:
438:
432:
427:
424:
401:
398:
395:
384:
378:
364:
361:
358:
355:
352:
332:
329:
326:
306:
303:
300:
280:
272:
268:
252:
232:
212:
209:
206:
200:
195:
190:
187:
182:
177:
170:
165:
160:
157:
152:
127:
124:
121:
110:
105:
103:
99:
95:
91:
87:
83:
64:
61:
58:
52:
49:
46:
43:
34:
30:
19:
5714:Plane curves
5670:
5650:
5636:"Lamé Curve"
5623:
5620:"Lamé curve"
5594:
5559:
5555:
5549:
5537:
5532:
5527:
5504:, New York:
5501:
5450:
5446:
5436:
5424:. Retrieved
5420:
5411:
5386:
5382:
5372:
5347:
5343:
5301:
5291:
5277:
5270:
5258:. Retrieved
5247:
5235:. Retrieved
5228:the original
5215:
5203:. Retrieved
5199:
5175:. Retrieved
5171:
5159:
5151:
5146:
5101:
5091:
5061:(1): 11–23.
5058:
5054:
5044:
4999:
4995:
4985:
4964:
4960:
4956:
4947:supertoroids
4933:Superformula
4918:
4914:
4910:
4898:
4888:
4884:
4870:
4848:
4837:
4823:developed a
4819:
4812:
4785:
4773:
4767:
4766:
4760:
4756:
4752:
4749:Sergels Torg
4726:
4720:
4716:
4712:
4694:Donald Knuth
4689:
4679:Hermann Zapf
4677:
4667:
4657:Gabriel Lamé
4654:
4642:
4632:hyperspheres
4625:
4296:
3996:
3975:
3892:
3770:
3767:
3193:
3184:
3171:
3167:adding to it
3162:
3008:is given in
2922:
2894:
2889:
2680:
2653:
2399:
1673:
1670:
1661:
1658:Fermat curve
1653:
1554:
1552:
1273:hyperellipse
1272:
1242:
1214:
1204:
1200:
1196:
1015:
1011:
997:
835:
831:
817:
596:
592:
588:
581:
577:
573:
565:closed curve
562:
379:
106:
90:Gabriel Lamé
85:
82:superellipse
81:
79:
29:
5508:, pp.
4808:Mexico City
4792:Vietnam War
4781:novelty toy
4700:) into his
4644:Anisotropic
2925:pedal curve
2919:Pedal curve
1913:range over
1853:where each
1243:hypoellipse
5260:9 November
5237:9 November
5111:2111.14517
4977:References
4755:= 2.5 and
4745:roundabout
4715:= 2.5 and
2141:where the
1660:of degree
86:Lamé curve
18:Lamé curve
5652:MathWorld
5630:EMS Press
5572:CiteSeerX
5467:0024-094X
5453:(1): 59.
5403:1434-7636
5389:(2): 50.
5364:0307-904X
5165:"Astroid"
5138:244715106
5075:1558-1756
5018:1664-462X
4840:The Local
4833:meridians
4737:Stockholm
4733:Piet Hein
4587:…
4528:…
4426:…
3950:≠
3706:…
3608:…
3485:…
3446:ellipsoid
3174:June 2008
3120:−
3092:−
3075:θ
3072:
3049:−
3032:θ
3029:
2832:≥
2656:perimeter
2650:Perimeter
2493:Γ
2450:Γ
2375:
2369:≠
2345:
2179:−
2156:
2119:
2110:
2101:⋅
2078:
2034:
2025:
2016:⋅
1993:
1936:π
1924:≤
1861:±
1836:π
1831:≤
1825:≤
1807:
1786:±
1755:
1734:±
1592:of order
1520:−
1501:), where
1479:±
1449:±
1426:±
1391:±
1286:≥
1169:±
1134:±
1122:rectangle
982:curvature
919:±
884:±
569:rectangle
527:−
498:θ
492:
454:θ
448:
402:θ
271:semi-axes
5708:Category
5494:(1977),
5447:Leonardo
5256:. GitHub
5177:14 March
5172:Xah Code
5154:, p. 126
5036:26528316
4971:L spaces
4907:Squircle
4861:See also
4790:for the
4777:superegg
4683:typeface
1666:singular
1193:Squircle
741:parabola
5564:Bibcode
5510:240–254
5475:1574879
5426:18 June
5306:Bibcode
5205:19 June
5083:9389947
5027:4606055
5002:: 856.
4878:⁄
4867:Astroid
4674:≈ 2.758
4651:History
3424:is the
2821:, with
2660:ellipse
1412:) and (
1360:in the
1275:. When
1155:) and (
1054:ellipse
1005:⁄
905:) and (
874:rhombus
825:⁄
747:astroid
701:concave
587:−
572:−
381:In the
94:ellipse
5574:
5516:
5473:
5465:
5401:
5362:
5136:
5126:
5081:
5073:
5034:
5024:
5016:
4901:cusps.
4827:, the
4741:Sweden
4729:Danish
4723:= 6/5.
4686:Melior
3871:where
3672:where
3351:where
1058:circle
976:convex
225:where
88:after
56:
5536:, in
5471:JSTOR
5231:(PDF)
5224:(PDF)
5168:(PDF)
5134:S2CID
5106:arXiv
5079:S2CID
4899:three
4806:, in
4788:Paris
4146:. If
3913:. If
2260:Here
1553:When
1381:-norm
1245:; if
986:limit
5514:ISBN
5463:ISSN
5428:2024
5399:ISSN
5360:ISSN
5262:2023
5239:2023
5207:2024
5179:2023
5124:ISBN
5071:ISSN
5032:PMID
5014:ISSN
4945:and
4883:and
4018:and
3853:>
3377:and
2923:The
2867:and
2742:>
2716:>
2654:The
2402:area
2400:The
2396:Area
2238:>
2193:<
1930:<
1652:and
1329:ball
1301:and
1271:, a
1256:>
1226:<
1103:>
980:The
957:<
951:<
705:For
682:<
676:<
620:and
585:and
362:<
356:<
330:>
245:and
100:and
65:0.75
5582:doi
5542:BBC
5540:by
5455:doi
5391:doi
5352:doi
5314:doi
5283:164
5116:doi
5063:doi
5022:PMC
5004:doi
4851:iOS
4681:'s
4282:0.5
4268:,
4224:2.4
4210:,
4198:2.2
3169:.
3069:sin
3026:cos
2789:to
2543:as
2372:tan
2342:tan
2153:sgn
2116:sin
2107:sgn
2075:sin
2031:cos
2022:sgn
1990:cos
1793:sin
1741:cos
1331:of
1215:If
1184:).
1086:).
1060:if
934:).
745:An
595:≤ +
580:≤ +
489:sin
445:cos
269:or
5710::
5675:,
5669:,
5665:,
5649:.
5628:,
5622:,
5580:,
5570:,
5560:78
5558:,
5512:,
5500:,
5483:^
5469:.
5461:.
5451:17
5449:.
5445:.
5419:.
5397:.
5387:23
5385:.
5381:.
5358:.
5348:40
5346:.
5342:.
5326:^
5312:.
5198:.
5187:^
5170:.
5132:.
5122:.
5114:.
5100:.
5077:.
5069:.
5057:.
5053:.
5030:.
5020:.
5012:.
4998:.
4994:.
4967:).
4959:=
4941::
4846:.
4810:.
4783:.
4739:,
4670:=
4618:.
3997:,
3420:A
3198:A
3012:by
2887::
2731:,
2408:as
2241:0.
2145:is
1010:,
830:,
775:,
591:≤
576:≤
80:A
5655:.
5603:.
5588:.
5584::
5566::
5477:.
5457::
5430:.
5405:.
5393::
5366:.
5354::
5320:.
5316::
5308::
5285:.
5264:.
5241:.
5209:.
5181:.
5140:.
5118::
5108::
5085:.
5065::
5059:1
5038:.
5006::
5000:6
4965:X
4963:(
4961:f
4957:Y
4919:b
4915:a
4911:n
4889:b
4885:a
4880:3
4876:2
4871:n
4761:b
4759:/
4757:a
4753:n
4721:b
4719:/
4717:a
4713:n
4690:o
4668:n
4606:n
4603:=
4598:N
4594:n
4590:=
4584:=
4579:2
4575:n
4571:=
4566:1
4562:n
4539:N
4535:n
4531:,
4525:,
4520:2
4516:n
4512:,
4507:1
4503:n
4482:1
4479:=
4471:N
4467:N
4461:|
4454:N
4450:a
4444:N
4440:x
4434:|
4429:+
4423:+
4415:2
4411:N
4405:|
4398:2
4394:a
4388:2
4384:x
4378:|
4373:+
4364:1
4360:N
4354:|
4347:1
4343:a
4337:1
4333:x
4327:|
4305:N
4279:=
4276:c
4256:1
4253:=
4250:b
4247:=
4244:a
4221:=
4218:p
4195:=
4192:n
4189:=
4186:m
4166:p
4163:=
4160:n
4157:=
4154:m
4134:1
4131:=
4125:p
4120:|
4115:c
4112:z
4107:|
4102:+
4096:n
4091:|
4086:b
4083:y
4078:|
4073:+
4066:m
4061:|
4056:a
4053:x
4048:|
4026:p
4005:n
3984:m
3953:n
3947:m
3927:n
3924:=
3921:m
3900:n
3879:m
3859:,
3856:0
3850:n
3847:,
3844:m
3841:;
3838:1
3835:=
3829:n
3824:|
3819:b
3816:y
3811:|
3806:+
3799:m
3794:|
3789:a
3786:x
3781:|
3739:n
3717:d
3713:a
3709:,
3703:,
3698:2
3694:a
3690:,
3685:1
3681:a
3660:,
3657:1
3654:=
3648:n
3643:|
3636:d
3632:a
3626:d
3622:x
3616:|
3611:+
3605:+
3599:n
3594:|
3587:2
3583:a
3577:2
3573:x
3567:|
3562:+
3555:n
3550:|
3543:1
3539:a
3533:1
3529:x
3523:|
3501:)
3496:d
3492:x
3488:,
3482:,
3477:2
3473:x
3469:,
3464:1
3460:x
3456:(
3432:d
3405:n
3385:c
3365:b
3362:,
3359:a
3339:,
3336:1
3333:=
3327:n
3322:|
3317:c
3314:z
3309:|
3304:+
3298:n
3293:|
3288:b
3285:y
3280:|
3275:+
3268:n
3263:|
3258:a
3255:x
3250:|
3228:)
3225:z
3222:,
3219:y
3216:,
3213:x
3210:(
3176:)
3172:(
3131:.
3123:1
3117:n
3113:n
3107:r
3103:=
3095:1
3089:n
3085:n
3079:)
3066:b
3063:(
3060:+
3052:1
3046:n
3042:n
3036:)
3023:a
3020:(
2996:,
2993:1
2990:=
2984:n
2979:|
2974:b
2971:y
2966:|
2961:+
2955:n
2950:|
2945:a
2942:x
2937:|
2903:n
2875:n
2855:h
2835:1
2829:h
2809:)
2806:h
2803:,
2800:0
2797:(
2777:)
2774:0
2771:,
2768:1
2765:(
2745:0
2739:y
2719:0
2713:x
2693:n
2634:.
2630:)
2626:1
2623:+
2618:n
2615:1
2610:,
2605:n
2602:1
2596:(
2590:B
2584:n
2580:b
2577:a
2574:4
2568:=
2564:a
2561:e
2558:r
2555:A
2527:,
2520:)
2513:n
2510:2
2504:+
2501:1
2497:(
2487:2
2482:)
2477:)
2470:n
2467:1
2461:+
2458:1
2454:(
2446:(
2439:b
2436:a
2433:4
2430:=
2426:a
2423:e
2420:r
2417:A
2381:.
2378:t
2364:n
2360:/
2356:2
2352:)
2348:t
2339:(
2334:a
2331:b
2326:=
2321:x
2318:y
2296:x
2292:/
2288:y
2268:t
2235:w
2230:,
2227:1
2224:+
2217:0
2214:=
2211:w
2206:,
2203:0
2196:0
2190:w
2185:,
2182:1
2173:{
2168:=
2165:)
2162:w
2159:(
2125:)
2122:t
2113:(
2104:b
2095:n
2092:2
2085:|
2081:t
2071:|
2065:=
2057:)
2054:t
2051:(
2047:y
2040:)
2037:t
2028:(
2019:a
2010:n
2007:2
2000:|
1996:t
1986:|
1980:=
1972:)
1969:t
1966:(
1962:x
1939:,
1933:2
1927:t
1921:0
1901:t
1881:t
1839:2
1828:t
1822:0
1817:}
1810:t
1801:n
1798:2
1789:b
1783:=
1775:)
1772:t
1769:(
1765:y
1758:t
1749:n
1746:2
1737:a
1731:=
1723:)
1720:t
1717:(
1713:x
1686:t
1662:n
1654:n
1640:1
1637:=
1634:b
1631:=
1628:a
1608:q
1604:/
1600:p
1576:q
1572:/
1568:p
1555:n
1531:n
1527:/
1523:1
1516:2
1512:=
1509:s
1487:b
1483:s
1471:,
1457:a
1453:s
1429:b
1423:,
1420:0
1400:0
1397:,
1394:a
1368:n
1346:2
1341:R
1315:b
1312:=
1309:a
1289:1
1283:n
1259:2
1253:n
1229:2
1223:n
1205:b
1201:a
1197:n
1172:b
1166:,
1163:0
1143:0
1140:,
1137:a
1106:2
1100:n
1074:b
1071:=
1068:a
1038:2
1035:=
1032:n
1016:b
1012:a
1007:2
1003:3
998:n
960:2
954:n
948:1
922:b
916:,
913:0
893:0
890:,
887:a
858:1
855:=
852:n
836:b
832:a
827:2
823:1
818:n
797:3
793:/
789:2
786:=
783:n
763:b
760:=
757:a
743:.
727:2
723:/
719:1
716:=
713:n
685:1
679:n
673:0
648:n
628:b
608:a
597:b
593:y
589:b
582:a
578:x
574:a
544:.
538:n
534:/
530:1
522:)
515:n
510:|
505:b
501:)
495:(
483:|
478:+
471:n
466:|
461:a
457:)
451:(
439:|
433:(
428:=
425:r
405:)
399:,
396:r
393:(
365:2
359:n
353:0
333:2
327:n
307:2
304:=
301:n
281:n
253:b
233:a
213:,
210:1
207:=
201:n
196:|
191:b
188:y
183:|
178:+
171:n
166:|
161:a
158:x
153:|
131:)
128:y
125:,
122:x
119:(
62:=
59:b
53:,
50:1
47:=
44:a
20:)
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