1283:
1323:
740:
1307:
1295:
3912:
1268:
759:
133:
1164:
787:
348:
1339:
491:
36:
795:
502:
4065:
992:
711:), such that, if the paraboloid is a mirror, light (or other waves) from a point source at the focus is reflected into a parallel beam, parallel to the axis of the paraboloid. This also works the other way around: a parallel beam of light that is parallel to the axis of the paraboloid is concentrated at the focal point. For a proof, see
2620:
2093:
2896:
2378:
1857:
2373:
1849:
3468:. This is sometimes called the "linear diameter", and equals the diameter of a flat, circular sheet of material, usually metal, which is the right size to be cut and bent to make the dish. Two intermediate results are useful in the calculation:
2735:
2217:
1695:
3899:
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which are both always positive, have their maximum at the origin, become smaller as a point on the surface moves further away from the origin, and tend asymptotically to zero as the said point moves infinitely away from the origin.
739:
3777:
is the aperture area of the dish, the area enclosed by the rim, which is proportional to the amount of sunlight a reflector dish can intercept. The surface area of a parabolic dish can be found using the area formula for a
1211:
3345:
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588:
442:
318:
2701:
1306:
1071:
2222:
1504:
1426:
1703:
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3155:
1282:
965:
3028:
2102:
1580:
3688:
2957:
2615:{\displaystyle H(u,v)={\frac {-a^{2}+b^{2}-{\frac {4u^{2}}{a^{2}}}+{\frac {4v^{2}}{b^{2}}}}{a^{2}b^{2}{\sqrt {\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{3}}}}}.}
2088:{\displaystyle H(u,v)={\frac {a^{2}+b^{2}+{\frac {4u^{2}}{a^{2}}}+{\frac {4v^{2}}{b^{2}}}}{a^{2}b^{2}{\sqrt {\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{3}}}}}}
1563:
830:
This property makes it simple to manufacture a hyperbolic paraboloid from a variety of materials and for a variety of purposes, from concrete roofs to snack foods. In particular,
3785:
3206:
1267:
1127:
3437:
1537:
771:
901:
3057:
2891:{\displaystyle z=\left({\frac {x^{2}+y^{2}}{2}}\right)\left({\frac {1}{a^{2}}}-{\frac {1}{b^{2}}}\right)+xy\left({\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}\right)}
3218:
474:
1294:
3545:
3643:
The volume of the dish, the amount of liquid it could hold if the rim were horizontal and the vertex at the bottom (e.g. the capacity of a paraboloidal
516:
370:
246:
2632:
1002:
1431:
1353:
3083:
4044:
4017:
3977:
822:
a hyperbolic paraboloid is a surface that may be generated by a moving line that is parallel to a fixed plane and crosses two fixed
119:
2975:
1273:
3650:
820:
These properties characterize hyperbolic paraboloids and are used in one of the oldest definitions of hyperbolic paraboloids:
2911:
57:
1199:
100:
1251:
813:. The lines in each family are parallel to a common plane, but not to each other. Hence the hyperbolic paraboloid is a
758:
72:
510:
53:
1205:
210:
into two different linear factors. The paraboloid is hyperbolic if the factors are real; elliptic if the factors are
906:
3160:
1183:
are often hyperbolic paraboloids as they are easily constructed from straight sections of material. Some examples:
195:, or a single point (in the case of a section by a tangent plane). A paraboloid is either elliptic or hyperbolic.
79:
46:
4085:
3635:
2368:{\displaystyle K(u,v)={\frac {-4}{a^{2}b^{2}\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{2}}}}
3400:
1844:{\displaystyle K(u,v)={\frac {4}{a^{2}b^{2}\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{2}}}}
1509:
770:
3451:
is the depth of the dish (measured along the axis of symmetry from the vertex to the plane of the rim), and
726:
86:
3033:
1542:
4069:
3937:
3779:
3348:
980:
857:
611:
3911:
68:
3064:
1347:
1313:
1229:
806:
360:
3690:
where the symbols are defined as above. This can be compared with the formulae for the volumes of a
2212:{\displaystyle {\vec {\sigma }}(u,v)=\left(u,v,{\frac {u^{2}}{a^{2}}}-{\frac {v^{2}}{b^{2}}}\right)}
1690:{\displaystyle {\vec {\sigma }}(u,v)=\left(u,v,{\frac {u^{2}}{a^{2}}}+{\frac {v^{2}}{b^{2}}}\right)}
1257:
3943:
3691:
1241:
853:
846:
692:
629:
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4013:
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3623:
3213:
1168:
865:
696:
641:
367:. In a suitable coordinate system, a hyperbolic paraboloid can be represented by the equation
222:
211:
3542:
are defined as above. The diameter of the dish, measured along the surface, is then given by
3461:. If two of these three lengths are known, this equation can be used to calculate the third.
1260:, Sokol district, Moscow, Russia (1960). Architect V.Volodin, engineer N.Drozdov. Demolished.
1187:
972:
730:
622:
203:
157:
874:
745:
Parallel rays coming into a circular paraboloidal mirror are reflected to the focal point,
3743:
2729:
1235:
842:
149:
132:
3996:
3894:{\displaystyle A={\frac {\pi R\left({\sqrt {(R^{2}+4D^{2})^{3}}}-R^{3}\right)}{6D^{2}}}.}
187:, or two crossing lines (in the case of a section by a tangent plane). The paraboloid is
93:
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702:
207:
153:
1163:
786:
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172:
165:
3995:
Weisstein, Eric W. "Hyperbolic
Paraboloid." From MathWorld--A Wolfram Web Resource.
347:
1223:
621:
A circular paraboloid contains circles. This is also true in the general case (see
494:
17:
1338:
845:
is negative at every point. Therefore, although it is a ruled surface, it is not
3931:
1180:
725:
The surface of a rotating liquid is also a circular paraboloid. This is used in
490:
356:
35:
3907:
3397:
The dimensions of a symmetrical paraboloidal dish are related by the equation
1217:
823:
810:
3059:
which can be thought of as the geometric representation (a three-dimensional
477:, as it can be generated by a moving parabola directed by a second parabola.
344:
planes respectively. In this position, the elliptic paraboloid opens upward.
198:
Equivalently, a paraboloid may be defined as a quadric surface that is not a
3965:
3925:
1193:
794:
719:
633:
184:
4064:
4008:
Thomas, George B.; Maurice D. Weir; Joel Hass; Frank R. Giordiano (2005).
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615:
199:
161:
141:
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991:
218:
192:
3464:
A more complex calculation is needed to find the diameter of the dish
3340:{\displaystyle f(z)={\frac {z^{2}}{2}}=f(x+yi)=z_{1}(x,y)+iz_{2}(x,y)}
3705:
814:
444:
In this position, the hyperbolic paraboloid opens downward along the
364:
3608:{\displaystyle {\frac {RQ}{P}}+P\ln \left({\frac {R+Q}{P}}\right),}
27:
Quadric surface with one axis of symmetry and no center of symmetry
1161:
990:
968:
793:
785:
701:
On the axis of a circular paraboloid, there is a point called the
500:
489:
346:
131:
718:
Therefore, the shape of a circular paraboloid is widely used in
583:{\displaystyle z={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}.}
437:{\displaystyle z={\frac {y^{2}}{b^{2}}}-{\frac {x^{2}}{a^{2}}}.}
313:{\displaystyle z={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}.}
3940: – Parabolic-shaped speaker producing coherent plane waves
2696:{\displaystyle z={\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}}
1334:
Cylinder between pencils of elliptic and hyperbolic paraboloids
1328:
Markham Moor
Service Station roof, Nottinghamshire (2009 photo)
1066:{\displaystyle z={\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}}
3644:
1342:
elliptic paraboloid, parabolic cylinder, hyperbolic paraboloid
29:
217:
An elliptic paraboloid is shaped like an oval cup and has a
999:
A plane section of a hyperbolic paraboloid with equation
221:
or minimum point when its axis is vertical. In a suitable
3928: – Quadric surface that looks like a deformed sphere
834:
fried snacks resemble a truncated hyperbolic paraboloid.
801:
fried snacks are in the shape of a hyperbolic paraboloid.
332:
are constants that dictate the level of curvature in the
179:
to the axis of symmetry is a parabola. The paraboloid is
1499:{\displaystyle z=x^{2}-{\frac {y^{2}}{b^{2}}},\ b>0,}
1421:{\displaystyle z=x^{2}+{\frac {y^{2}}{b^{2}}},\ b>0,}
3457:
is the radius of the rim. They must all be in the same
3997:
http://mathworld.wolfram.com/HyperbolicParaboloid.html
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995:
A hyperbolic paraboloid with hyperbolas and parabolas
909:
877:
652:
The plane sections of an elliptic paraboloid can be:
519:
373:
249:
3946: – Reflector that has the shape of a paraboloid
3150:{\displaystyle z_{1}(x,y)={\frac {x^{2}-y^{2}}{2}}}
355:A hyperbolic paraboloid (not to be confused with a
191:if every other nonempty plane section is either an
60:. Unsourced material may be challenged and removed.
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790:A hyperbolic paraboloid with lines contained in it
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312:
206:whose part of degree two may be factored over the
1276:, an example of a hyperbolic paraboloid structure
722:for parabolic reflectors and parabolic antennas.
4039:, Jones & Bartlett Publishers, p. 649,
2625:Geometric representation of multiplication table
2099:The hyperbolic paraboloid, when parametrized as
1577:The elliptic paraboloid, parametrized simply as
713:Parabola § Proof of the reflective property
1300:Restaurante Los Manantiales, Xochimilco, Mexico
960:{\displaystyle z={\tfrac {a}{2}}(x^{2}-y^{2})}
473:Any paraboloid (elliptic or hyperbolic) is a
8:
1288:Surface illustrating a hyperbolic paraboloid
4033:Zill, Dennis G.; Wright, Warren S. (2011),
3991:
3989:
3023:{\displaystyle z={\frac {x^{2}-y^{2}}{2}}.}
729:and in making solid telescope mirrors (see
463:opens upward and the parabola in the plane
1312:Hyperbolic paraboloid thin-shell roofs at
1254:, A1(southbound), Nottinghamshire, England
513:, an elliptic paraboloid has the equation
456:-axis (that is, the parabola in the plane
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1433:
1428:and the pencil of hyperbolic paraboloids
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1212:Cathedral of Saint Mary of the Assumption
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183:if every other plane section is either a
168:that has a similar property of symmetry.
120:Learn how and when to remove this message
3683:{\displaystyle {\frac {\pi }{2}}R^{2}D,}
2972:, we see that the hyperbolic paraboloid
1337:
1242:Waterworld Leisure & Activity Centre
243:, it can be represented by the equation
160:. The term "paraboloid" is derived from
4012:. Pearson Education, Inc. p. 896.
3972:. Pearson Education, Inc. p. 892.
3956:
2952:{\displaystyle z={\frac {2xy}{a^{2}}}.}
1263:
1085:-axis, and has an equation of the form
809:: it contains two families of mutually
735:
660:, if the plane is parallel to the axis,
1214:, San Francisco, California, US (1971)
1140:-axis, and the section is not a line,
7:
3964:Thomas, George B.; Maurice D. Weir;
1558:{\displaystyle b\rightarrow \infty }
871:A hyperbolic paraboloid of equation
58:adding citations to reliable sources
1552:
1220:in Calgary, Alberta, Canada (1983)
1136:, if the plane is parallel to the
1081:, if the plane is parallel to the
25:
3934: – Unbounded quadric surface
3393:Dimensions of a paraboloidal dish
1252:Markham Moor Service Station roof
977:rectangular hyperbolic paraboloid
4063:
3910:
1321:
1305:
1293:
1281:
1266:
1208:, in Ham, London, England (1966)
769:
757:
738:
34:
4036:Calculus: Early Transcendentals
1274:Warszawa Ochota railway station
805:The hyperbolic paraboloid is a
632:, an elliptic paraboloid is an
45:needs additional citations for
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1590:
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1316:, Valencia, Spain (taken 2019)
954:
928:
602:, an elliptic paraboloid is a
1:
3968:; Frank R. Giordiano (2005).
3632:, i.e. its logarithm to base
3201:{\displaystyle z_{2}(x,y)=xy}
2732:), the result is the surface
2629:If the hyperbolic paraboloid
856:, a hyperbolic paraboloid is
837:A hyperbolic paraboloid is a
3030:is congruent to the surface
2728:direction (according to the
1226:in Gothenburg, Sweden (1971)
1122:{\displaystyle bx\pm ay+b=0}
1506:approach the same surface
1200:St. Mary's Cathedral, Tokyo
1171:hyperbolic paraboloid model
511:Cartesian coordinate system
481:Properties and applications
450:-axis and upward along the
175:of a paraboloid by a plane
4117:
3466:measured along its surface
3432:{\displaystyle 4FD=R^{2},}
2703:is rotated by an angle of
852:From the point of view of
690:
628:From the point of view of
1232:in Valencia, Spain (2003)
1190:Expo '58, Brussels (1958)
776:Rotating water in a glass
4010:Thomas' Calculus 11th ed
3970:Thomas' Calculus 11th ed
3212:, and together form the
2908:then this simplifies to
1350:of elliptic paraboloids
1206:St Richard's Church, Ham
1176:Examples in architecture
727:liquid-mirror telescopes
614:obtained by revolving a
608:paraboloid of revolution
497:of a circular paraboloid
136:Paraboloid of revolution
2219:has Gaussian curvature
1532:{\displaystyle z=x^{2}}
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981:rectangular hyperbolas
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498:
438:
352:
314:
137:
3938:Parabolic loudspeaker
3896:
3780:surface of revolution
3685:
3610:
3445:is the focal length,
3434:
3349:analytic continuation
3342:
3203:
3152:
3070:The two paraboloidal
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896:{\displaystyle z=axy}
858:one-sheet hyperboloid
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782:Hyperbolic paraboloid
612:surface of revolution
585:
504:
493:
439:
351:Hyperbolic paraboloid
350:
315:
152:that has exactly one
135:
4072:at Wikimedia Commons
3786:
3651:
3546:
3478:(or the equivalent:
3401:
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3161:
3084:
3065:multiplication table
3052:{\displaystyle z=xy}
3034:
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1147:, if the plane is a
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907:
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807:doubly ruled surface
667:, if the plane is a
517:
371:
361:doubly ruled surface
247:
164:, which refers to a
54:improve this article
3944:Parabolic reflector
3361:parabolic function
3210:harmonic conjugates
3063:, as it were) of a
2375:and mean curvature
854:projective geometry
764:Parabolic reflector
693:Parabolic reflector
687:Parabolic reflector
630:projective geometry
604:circular paraboloid
505:Circular paraboloid
486:Elliptic paraboloid
475:translation surface
18:Circular paraboloid
3918:Mathematics portal
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1699:Gaussian curvature
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1567:parabolic cylinder
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1496:
1418:
1344:
1173:
1145:intersecting lines
1119:
1063:
997:
979:, by analogy with
975:) may be called a
967:(this is the same
957:
926:
893:
803:
792:
580:
507:
499:
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353:
310:
158:center of symmetry
138:
4068:Media related to
3886:
3850:
3662:
3624:natural logarithm
3596:
3562:
3252:
3214:analytic function
3145:
3015:
2959:Finally, letting
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2881:
2861:
2825:
2805:
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2607:
2604:
2591:
2559:
2494:
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2115:
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2035:
1970:
1938:
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1593:
1483:
1476:
1405:
1398:
1196:- Dogra Hall Roof
1061:
1034:
925:
866:plane at infinity
697:parabolic antenna
642:plane at infinity
618:around its axis.
575:
548:
470:opens downward).
429:
402:
305:
278:
223:coordinate system
212:complex conjugate
204:implicit equation
130:
129:
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104:
16:(Redirected from
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4086:Geometric shapes
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2950:
2945:
2943:
2942:
2933:
2922:
2907:
2897:
2895:
2894:
2889:
2887:
2883:
2882:
2880:
2879:
2867:
2862:
2860:
2859:
2847:
2831:
2827:
2826:
2824:
2823:
2811:
2806:
2804:
2803:
2791:
2784:
2780:
2775:
2774:
2773:
2761:
2760:
2750:
2727:
2720:
2719:
2717:
2716:
2713:
2710:
2702:
2700:
2699:
2694:
2692:
2690:
2689:
2680:
2679:
2670:
2665:
2663:
2662:
2653:
2652:
2643:
2621:
2619:
2618:
2613:
2608:
2606:
2605:
2603:
2602:
2597:
2593:
2592:
2590:
2589:
2580:
2579:
2578:
2565:
2560:
2558:
2557:
2548:
2547:
2546:
2533:
2519:
2517:
2516:
2507:
2506:
2496:
2495:
2493:
2492:
2483:
2482:
2481:
2468:
2463:
2461:
2460:
2451:
2450:
2449:
2436:
2431:
2430:
2418:
2417:
2404:
2374:
2372:
2371:
2366:
2364:
2362:
2361:
2360:
2355:
2351:
2350:
2348:
2347:
2338:
2337:
2336:
2323:
2318:
2316:
2315:
2306:
2305:
2304:
2291:
2277:
2276:
2267:
2266:
2256:
2248:
2218:
2216:
2215:
2210:
2208:
2204:
2203:
2201:
2200:
2191:
2190:
2181:
2176:
2174:
2173:
2164:
2163:
2154:
2117:
2116:
2108:
2094:
2092:
2091:
2086:
2084:
2082:
2081:
2079:
2078:
2073:
2069:
2068:
2066:
2065:
2056:
2055:
2054:
2041:
2036:
2034:
2033:
2024:
2023:
2022:
2009:
1995:
1993:
1992:
1983:
1982:
1972:
1971:
1969:
1968:
1959:
1958:
1957:
1944:
1939:
1937:
1936:
1927:
1926:
1925:
1912:
1907:
1906:
1894:
1893:
1883:
1850:
1848:
1847:
1842:
1840:
1838:
1837:
1836:
1831:
1827:
1826:
1824:
1823:
1814:
1813:
1812:
1799:
1794:
1792:
1791:
1782:
1781:
1780:
1767:
1753:
1752:
1743:
1742:
1729:
1696:
1694:
1693:
1688:
1686:
1682:
1681:
1679:
1678:
1669:
1668:
1659:
1654:
1652:
1651:
1642:
1641:
1632:
1595:
1594:
1586:
1564:
1562:
1561:
1556:
1538:
1536:
1535:
1530:
1528:
1527:
1505:
1503:
1502:
1497:
1481:
1477:
1475:
1474:
1465:
1464:
1455:
1450:
1449:
1427:
1425:
1424:
1419:
1403:
1399:
1397:
1396:
1387:
1386:
1377:
1372:
1371:
1325:
1309:
1297:
1285:
1270:
1238:, England (2011)
1188:Philips Pavilion
1166:
1139:
1128:
1126:
1125:
1120:
1084:
1072:
1070:
1069:
1064:
1062:
1060:
1059:
1050:
1049:
1040:
1035:
1033:
1032:
1023:
1022:
1013:
973:rotation of axes
966:
964:
963:
958:
953:
952:
940:
939:
927:
918:
902:
900:
899:
894:
773:
761:
748:
742:
731:rotating furnace
623:Circular section
601:
589:
587:
586:
581:
576:
574:
573:
564:
563:
554:
549:
547:
546:
537:
536:
527:
469:
462:
455:
449:
443:
441:
440:
435:
430:
428:
427:
418:
417:
408:
403:
401:
400:
391:
390:
381:
343:
337:
331:
325:
319:
317:
316:
311:
306:
304:
303:
294:
293:
284:
279:
277:
276:
267:
266:
257:
242:
236:
230:
225:with three axes
154:axis of symmetry
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
4116:
4115:
4111:
4110:
4109:
4107:
4106:
4105:
4076:
4075:
4060:
4055:
4054:
4047:
4032:
4031:
4027:
4020:
4007:
4006:
4002:
3994:
3987:
3980:
3963:
3962:
3958:
3953:
3916:
3909:
3906:
3875:
3871:
3855:
3840:
3830:
3814:
3808:
3804:
3797:
3784:
3783:
3771:
3757:
3754:
3751:
3750:
3748:
3747:
3733:
3719:
3716:
3713:
3712:
3710:
3709:
3695:
3664:
3649:
3648:
3647:), is given by
3633:
3627:
3616:
3582:
3576:
3551:
3544:
3543:
3537:
3531:
3525:
3513:
3511:
3506:
3495:
3492:
3487:
3486:
3484:
3479:
3469:
3452:
3446:
3440:
3416:
3399:
3398:
3395:
3382:
3379:
3374:
3373:
3371:
3362:
3352:
3312:
3281:
3239:
3217:
3216:
3164:
3159:
3158:
3131:
3118:
3117:
3087:
3082:
3081:
3071:
3032:
3031:
3001:
2988:
2987:
2974:
2973:
2967:
2965:
2960:
2934:
2923:
2910:
2909:
2899:
2871:
2851:
2845:
2841:
2815:
2795:
2789:
2785:
2765:
2752:
2751:
2745:
2734:
2733:
2730:right hand rule
2722:
2714:
2711:
2708:
2707:
2705:
2704:
2681:
2671:
2654:
2644:
2631:
2630:
2627:
2581:
2570:
2566:
2549:
2538:
2534:
2525:
2521:
2520:
2508:
2498:
2497:
2484:
2473:
2469:
2452:
2441:
2437:
2422:
2409:
2405:
2377:
2376:
2339:
2328:
2324:
2307:
2296:
2292:
2283:
2279:
2278:
2268:
2258:
2257:
2249:
2221:
2220:
2192:
2182:
2165:
2155:
2140:
2136:
2101:
2100:
2057:
2046:
2042:
2025:
2014:
2010:
2001:
1997:
1996:
1984:
1974:
1973:
1960:
1949:
1945:
1928:
1917:
1913:
1898:
1885:
1884:
1856:
1855:
1815:
1804:
1800:
1783:
1772:
1768:
1759:
1755:
1754:
1744:
1734:
1733:
1702:
1701:
1670:
1660:
1643:
1633:
1618:
1614:
1579:
1578:
1575:
1541:
1540:
1519:
1508:
1507:
1466:
1456:
1441:
1430:
1429:
1388:
1378:
1363:
1352:
1351:
1336:
1329:
1326:
1317:
1310:
1301:
1298:
1289:
1286:
1277:
1271:
1236:London Velopark
1178:
1162:
1137:
1087:
1086:
1082:
1051:
1041:
1024:
1014:
1001:
1000:
944:
931:
905:
904:
873:
872:
843:Gauss curvature
784:
777:
774:
765:
762:
753:
746:
743:
699:
691:Main articles:
689:
593:
565:
555:
538:
528:
515:
514:
488:
483:
464:
457:
451:
445:
419:
409:
392:
382:
369:
368:
339:
333:
327:
321:
295:
285:
268:
258:
245:
244:
238:
232:
226:
208:complex numbers
150:quadric surface
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
4114:
4112:
4104:
4103:
4098:
4093:
4088:
4078:
4077:
4074:
4073:
4059:
4058:External links
4056:
4053:
4052:
4045:
4025:
4018:
4000:
3985:
3978:
3955:
3954:
3952:
3949:
3948:
3947:
3941:
3935:
3929:
3922:
3921:
3905:
3902:
3890:
3882:
3878:
3874:
3868:
3862:
3858:
3854:
3847:
3843:
3837:
3833:
3829:
3826:
3821:
3817:
3813:
3807:
3803:
3800:
3794:
3791:
3679:
3676:
3671:
3667:
3661:
3658:
3604:
3600:
3595:
3591:
3588:
3585:
3579:
3575:
3572:
3569:
3566:
3561:
3557:
3554:
3459:unit of length
3428:
3423:
3419:
3415:
3412:
3409:
3406:
3394:
3391:
3336:
3333:
3330:
3327:
3324:
3319:
3315:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3288:
3284:
3280:
3277:
3274:
3271:
3268:
3265:
3262:
3259:
3256:
3251:
3246:
3242:
3236:
3233:
3230:
3227:
3224:
3197:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3171:
3167:
3144:
3138:
3134:
3130:
3125:
3121:
3114:
3111:
3108:
3105:
3102:
3099:
3094:
3090:
3048:
3045:
3042:
3039:
3019:
3014:
3008:
3004:
3000:
2995:
2991:
2984:
2981:
2948:
2941:
2937:
2932:
2929:
2926:
2920:
2917:
2886:
2878:
2874:
2870:
2865:
2858:
2854:
2850:
2844:
2840:
2837:
2834:
2830:
2822:
2818:
2814:
2809:
2802:
2798:
2794:
2788:
2783:
2778:
2772:
2768:
2764:
2759:
2755:
2748:
2744:
2741:
2688:
2684:
2678:
2674:
2668:
2661:
2657:
2651:
2647:
2641:
2638:
2626:
2623:
2611:
2601:
2596:
2588:
2584:
2577:
2573:
2569:
2563:
2556:
2552:
2545:
2541:
2537:
2531:
2528:
2524:
2515:
2511:
2505:
2501:
2491:
2487:
2480:
2476:
2472:
2466:
2459:
2455:
2448:
2444:
2440:
2434:
2429:
2425:
2421:
2416:
2412:
2408:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2359:
2354:
2346:
2342:
2335:
2331:
2327:
2321:
2314:
2310:
2303:
2299:
2295:
2289:
2286:
2282:
2275:
2271:
2265:
2261:
2255:
2252:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2207:
2199:
2195:
2189:
2185:
2179:
2172:
2168:
2162:
2158:
2152:
2149:
2146:
2143:
2139:
2135:
2132:
2129:
2126:
2123:
2120:
2114:
2111:
2077:
2072:
2064:
2060:
2053:
2049:
2045:
2039:
2032:
2028:
2021:
2017:
2013:
2007:
2004:
2000:
1991:
1987:
1981:
1977:
1967:
1963:
1956:
1952:
1948:
1942:
1935:
1931:
1924:
1920:
1916:
1910:
1905:
1901:
1897:
1892:
1888:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1853:mean curvature
1835:
1830:
1822:
1818:
1811:
1807:
1803:
1797:
1790:
1786:
1779:
1775:
1771:
1765:
1762:
1758:
1751:
1747:
1741:
1737:
1732:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1685:
1677:
1673:
1667:
1663:
1657:
1650:
1646:
1640:
1636:
1630:
1627:
1624:
1621:
1617:
1613:
1610:
1607:
1604:
1601:
1598:
1592:
1589:
1574:
1571:
1554:
1551:
1548:
1526:
1522:
1518:
1515:
1495:
1492:
1489:
1486:
1480:
1473:
1469:
1463:
1459:
1453:
1448:
1444:
1440:
1437:
1417:
1414:
1411:
1408:
1402:
1395:
1391:
1385:
1381:
1375:
1370:
1366:
1362:
1359:
1335:
1332:
1331:
1330:
1327:
1320:
1318:
1314:L'OceanogrĂ fic
1311:
1304:
1302:
1299:
1292:
1290:
1287:
1280:
1278:
1272:
1265:
1262:
1261:
1255:
1249:
1248:, Wales (1970)
1239:
1233:
1230:L'OceanogrĂ fic
1227:
1221:
1215:
1209:
1203:
1202:, Japan (1964)
1197:
1191:
1177:
1174:
1160:
1159:
1152:
1141:
1130:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1058:
1054:
1048:
1044:
1038:
1031:
1027:
1021:
1017:
1011:
1008:
989:
988:
987:Plane sections
956:
951:
947:
943:
938:
934:
930:
924:
921:
915:
912:
892:
889:
886:
883:
880:
839:saddle surface
783:
780:
779:
778:
775:
768:
766:
763:
756:
754:
744:
737:
688:
685:
684:
683:
672:
661:
650:
649:
648:Plane sections
579:
572:
568:
562:
558:
552:
545:
541:
535:
531:
525:
522:
509:In a suitable
487:
484:
482:
479:
433:
426:
422:
416:
412:
406:
399:
395:
389:
385:
379:
376:
363:shaped like a
309:
302:
298:
292:
288:
282:
275:
271:
265:
261:
255:
252:
128:
127:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4113:
4102:
4099:
4097:
4094:
4092:
4089:
4087:
4084:
4083:
4081:
4071:
4066:
4062:
4061:
4057:
4048:
4046:9781449644482
4042:
4038:
4037:
4029:
4026:
4021:
4019:0-321-18558-7
4015:
4011:
4004:
4001:
3998:
3992:
3990:
3986:
3981:
3979:0-321-18558-7
3975:
3971:
3967:
3960:
3957:
3950:
3945:
3942:
3939:
3936:
3933:
3930:
3927:
3924:
3923:
3919:
3913:
3908:
3903:
3901:
3888:
3880:
3876:
3872:
3866:
3860:
3856:
3852:
3845:
3835:
3831:
3827:
3824:
3819:
3815:
3805:
3801:
3798:
3792:
3789:
3781:
3775:
3768:
3765:
3745:
3740:
3736:
3730:
3727:
3707:
3702:
3699:
3693:
3677:
3674:
3669:
3665:
3659:
3656:
3646:
3641:
3638:
3637:
3630:
3625:
3620:
3602:
3598:
3593:
3589:
3586:
3583:
3577:
3573:
3570:
3567:
3564:
3559:
3555:
3552:
3540:
3534:
3528:
3520:
3516:
3509:
3499:
3490:
3482:
3476:
3472:
3467:
3462:
3460:
3455:
3449:
3443:
3426:
3421:
3417:
3413:
3410:
3407:
3404:
3392:
3390:
3377:
3369:
3365:
3359:
3355:
3350:
3347:which is the
3331:
3328:
3325:
3317:
3313:
3309:
3306:
3300:
3297:
3294:
3286:
3282:
3278:
3272:
3269:
3266:
3263:
3257:
3254:
3249:
3244:
3240:
3234:
3228:
3222:
3215:
3211:
3195:
3192:
3189:
3183:
3180:
3177:
3169:
3165:
3142:
3136:
3132:
3128:
3123:
3119:
3112:
3106:
3103:
3100:
3092:
3088:
3078:
3074:
3068:
3066:
3062:
3046:
3043:
3040:
3037:
3017:
3012:
3006:
3002:
2998:
2993:
2989:
2982:
2979:
2963:
2946:
2939:
2935:
2930:
2927:
2924:
2918:
2915:
2906:
2902:
2884:
2876:
2872:
2868:
2863:
2856:
2852:
2848:
2842:
2838:
2835:
2832:
2828:
2820:
2816:
2812:
2807:
2800:
2796:
2792:
2786:
2781:
2776:
2770:
2766:
2762:
2757:
2753:
2746:
2742:
2739:
2731:
2726:
2686:
2682:
2676:
2672:
2666:
2659:
2655:
2649:
2645:
2639:
2636:
2624:
2622:
2609:
2599:
2594:
2586:
2582:
2575:
2571:
2567:
2561:
2554:
2550:
2543:
2539:
2535:
2529:
2526:
2522:
2513:
2509:
2503:
2499:
2489:
2485:
2478:
2474:
2470:
2464:
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65:Find sources:
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43:This article
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69:"Paraboloid"
64:
52:Please help
47:verification
44:
3932:Hyperboloid
847:developable
709:focal point
357:hyperboloid
4080:Categories
4070:Paraboloid
3951:References
3706:hemisphere
3622:means the
3080:functions
1218:Saddledome
1143:a pair of
824:skew lines
811:skew lines
751:vice versa
610:. It is a
181:hyperbolic
146:paraboloid
80:newspapers
4101:Parabolas
3966:Joel Hass
3926:Ellipsoid
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3657:π
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3129:−
3061:nomograph
2999:−
2808:−
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2251:−
2178:−
2113:→
2110:σ
1591:→
1588:σ
1573:Curvature
1553:∞
1550:→
1452:−
1194:IIT Delhi
1156:hyperbola
1099:±
1037:−
942:−
841:, as its
720:astronomy
634:ellipsoid
405:−
185:hyperbola
110:June 2020
4096:Quadrics
4091:Surfaces
3904:See also
3732:, where
3692:cylinder
3524:, where
1134:parabola
860:that is
832:Pringles
799:Pringles
658:parabola
636:that is
616:parabola
200:cylinder
189:elliptic
177:parallel
162:parabola
142:geometry
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864:to the
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676:ellipse
640:to the
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219:maximum
193:ellipse
156:and no
94:scholar
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1348:pencil
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365:saddle
320:where
237:, and
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969:up to
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101:JSTOR
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