1272:
1312:
729:
1296:
1284:
3901:
1257:
748:
122:
1153:
776:
337:
1328:
480:
25:
784:
491:
4054:
981:
700:), 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
2609:
2082:
2885:
2367:
1846:
2362:
1838:
3457:. 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:
2724:
2206:
1684:
3888:
2084:
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.
728:
3766:
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
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577:
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307:
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1060:
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1415:
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954:
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2091:
1569:
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2946:
2604:{\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}}}}}.}
2077:{\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}}}}}}
1552:
819:
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,
3774:
3195:
1256:
1116:
3426:
1526:
760:
890:
3046:
2880:{\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)}
3207:
463:
1283:
3534:
3632:
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
505:
359:
235:
2621:
991:
1420:
1342:
3072:
4033:
4006:
3966:
811:
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
108:
2964:
1262:
3639:
809:
These properties characterize hyperbolic paraboloids and are used in one of the oldest definitions of hyperbolic paraboloids:
2900:
46:
1188:
89:
1240:
802:. The lines in each family are parallel to a common plane, but not to each other. Hence the hyperbolic paraboloid is a
747:
61:
499:
42:
1194:
199:
into two different linear factors. The paraboloid is hyperbolic if the factors are real; elliptic if the factors are
895:
3149:
1172:
are often hyperbolic paraboloids as they are easily constructed from straight sections of material. Some examples:
184:, or a single point (in the case of a section by a tangent plane). A paraboloid is either elliptic or hyperbolic.
68:
35:
4074:
3624:
2357:{\displaystyle K(u,v)={\frac {-4}{a^{2}b^{2}\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{2}}}}
3389:
1833:{\displaystyle K(u,v)={\frac {4}{a^{2}b^{2}\left(1+{\frac {4u^{2}}{a^{4}}}+{\frac {4v^{2}}{b^{4}}}\right)^{2}}}}
1498:
759:
3440:
is the depth of the dish (measured along the axis of symmetry from the vertex to the plane of the rim), and
715:
75:
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1531:
4058:
3926:
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969:
846:
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3900:
57:
3053:
1336:
1302:
1218:
795:
349:
3679:
where the symbols are defined as above. This can be compared with the formulae for the volumes of a
2201:{\displaystyle {\vec {\sigma }}(u,v)=\left(u,v,{\frac {u^{2}}{a^{2}}}-{\frac {v^{2}}{b^{2}}}\right)}
1679:{\displaystyle {\vec {\sigma }}(u,v)=\left(u,v,{\frac {u^{2}}{a^{2}}}+{\frac {v^{2}}{b^{2}}}\right)}
1246:
3932:
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1230:
842:
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3202:
1157:
854:
685:
630:
356:. In a suitable coordinate system, a hyperbolic paraboloid can be represented by the equation
211:
200:
3531:
are defined as above. The diameter of the dish, measured along the surface, is then given by
3450:. If two of these three lengths are known, this equation can be used to calculate the third.
1249:, Sokol district, Moscow, Russia (1960). Architect V.Volodin, engineer N.Drozdov. Demolished.
1176:
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Parallel rays coming into a circular paraboloidal mirror are reflected to the focal point,
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831:
138:
121:
3985:
3883:{\displaystyle A={\frac {\pi R\left({\sqrt {(R^{2}+4D^{2})^{3}}}-R^{3}\right)}{6D^{2}}}.}
176:, or two crossing lines (in the case of a section by a tangent plane). The paraboloid is
82:
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196:
142:
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775:
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161:
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3984:
Weisstein, Eric W. "Hyperbolic
Paraboloid." From MathWorld--A Wolfram Web Resource.
336:
1212:
610:
A circular paraboloid contains circles. This is also true in the general case (see
483:
1327:
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is negative at every point. Therefore, although it is a ruled surface, it is not
3920:
1169:
714:
The surface of a rotating liquid is also a circular paraboloid. This is used in
479:
345:
24:
3896:
3386:
The dimensions of a symmetrical paraboloidal dish are related by the equation
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812:
799:
3048:
which can be thought of as the geometric representation (a three-dimensional
466:, as it can be generated by a moving parabola directed by a second parabola.
333:
planes respectively. In this position, the elliptic paraboloid opens upward.
187:
Equivalently, a paraboloid may be defined as a quadric surface that is not a
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3914:
1182:
783:
708:
622:
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3997:
Thomas, George B.; Maurice D. Weir; Joel Hass; Frank R. Giordiano (2005).
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130:
1234:
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A more complex calculation is needed to find the diameter of the dish
3329:{\displaystyle f(z)={\frac {z^{2}}{2}}=f(x+yi)=z_{1}(x,y)+iz_{2}(x,y)}
3694:
803:
433:
In this position, the hyperbolic paraboloid opens downward along the
353:
3597:{\displaystyle {\frac {RQ}{P}}+P\ln \left({\frac {R+Q}{P}}\right),}
16:
Quadric surface with one axis of symmetry and no center of symmetry
1150:
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782:
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690:
On the axis of a circular paraboloid, there is a point called the
489:
478:
335:
120:
707:
Therefore, the shape of a circular paraboloid is widely used in
572:{\displaystyle z={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}.}
426:{\displaystyle z={\frac {y^{2}}{b^{2}}}-{\frac {x^{2}}{a^{2}}}.}
302:{\displaystyle z={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}.}
3929: – Parabolic-shaped speaker producing coherent plane waves
2685:{\displaystyle z={\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}}
1323:
Cylinder between pencils of elliptic and hyperbolic paraboloids
1317:
Markham Moor
Service Station roof, Nottinghamshire (2009 photo)
1055:{\displaystyle z={\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}}
3633:
1331:
elliptic paraboloid, parabolic cylinder, hyperbolic paraboloid
18:
206:
An elliptic paraboloid is shaped like an oval cup and has a
988:
A plane section of a hyperbolic paraboloid with equation
210:
or minimum point when its axis is vertical. In a suitable
3917: – Quadric surface that looks like a deformed sphere
823:
fried snacks resemble a truncated hyperbolic paraboloid.
790:
fried snacks are in the shape of a hyperbolic paraboloid.
321:
are constants that dictate the level of curvature in the
168:
to the axis of symmetry is a parabola. The paraboloid is
1488:{\displaystyle z=x^{2}-{\frac {y^{2}}{b^{2}}},\ b>0,}
1410:{\displaystyle z=x^{2}+{\frac {y^{2}}{b^{2}}},\ b>0,}
3446:
is the radius of the rim. They must all be in the same
3986:
http://mathworld.wolfram.com/HyperbolicParaboloid.html
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1080:
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A hyperbolic paraboloid with hyperbolas and parabolas
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866:
641:
The plane sections of an elliptic paraboloid can be:
508:
362:
238:
3935: – Reflector that has the shape of a paraboloid
3139:{\displaystyle z_{1}(x,y)={\frac {x^{2}-y^{2}}{2}}}
344:A hyperbolic paraboloid (not to be confused with a
180:if every other nonempty plane section is either an
49:. Unsourced material may be challenged and removed.
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779:A hyperbolic paraboloid with lines contained in it
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195:whose part of degree two may be factored over the
1265:, an example of a hyperbolic paraboloid structure
711:for parabolic reflectors and parabolic antennas.
4028:, Jones & Bartlett Publishers, p. 649,
2614:Geometric representation of multiplication table
2088:The hyperbolic paraboloid, when parametrized as
1566:The elliptic paraboloid, parametrized simply as
702:Parabola § Proof of the reflective property
1289:Restaurante Los Manantiales, Xochimilco, Mexico
949:{\displaystyle z={\tfrac {a}{2}}(x^{2}-y^{2})}
462:Any paraboloid (elliptic or hyperbolic) is a
8:
1277:Surface illustrating a hyperbolic paraboloid
4022:Zill, Dennis G.; Wright, Warren S. (2011),
3980:
3978:
3012:{\displaystyle z={\frac {x^{2}-y^{2}}{2}}.}
718:and in making solid telescope mirrors (see
452:opens upward and the parabola in the plane
1301:Hyperbolic paraboloid thin-shell roofs at
1243:, A1(southbound), Nottinghamshire, England
502:, an elliptic paraboloid has the equation
445:-axis (that is, the parabola in the plane
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1417:and the pencil of hyperbolic paraboloids
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1201:Cathedral of Saint Mary of the Assumption
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172:if every other plane section is either a
157:that has a similar property of symmetry.
109:Learn how and when to remove this message
3672:{\displaystyle {\frac {\pi }{2}}R^{2}D,}
2961:, we see that the hyperbolic paraboloid
1326:
1231:Waterworld Leisure & Activity Centre
232:, it can be represented by the equation
149:. The term "paraboloid" is derived from
4001:. Pearson Education, Inc. p. 896.
3961:. Pearson Education, Inc. p. 892.
3945:
2941:{\displaystyle z={\frac {2xy}{a^{2}}}.}
1252:
1074:-axis, and has an equation of the form
798:: it contains two families of mutually
724:
649:, if the plane is parallel to the axis,
1203:, San Francisco, California, US (1971)
1129:-axis, and the section is not a line,
7:
3953:Thomas, George B.; Maurice D. Weir;
1547:{\displaystyle b\rightarrow \infty }
860:A hyperbolic paraboloid of equation
47:adding citations to reliable sources
1541:
1209:in Calgary, Alberta, Canada (1983)
1125:, if the plane is parallel to the
1070:, if the plane is parallel to the
14:
3923: – Unbounded quadric surface
3382:Dimensions of a paraboloidal dish
1241:Markham Moor Service Station roof
966:rectangular hyperbolic paraboloid
4052:
3899:
1310:
1294:
1282:
1270:
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1197:, in Ham, London, England (1966)
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23:
4025:Calculus: Early Transcendentals
1263:Warszawa Ochota railway station
794:The hyperbolic paraboloid is a
621:, an elliptic paraboloid is an
34:needs additional citations for
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1305:, Valencia, Spain (taken 2019)
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917:
591:, an elliptic paraboloid is a
1:
3957:; Frank R. Giordiano (2005).
3621:, i.e. its logarithm to base
3190:{\displaystyle z_{2}(x,y)=xy}
2721:), the result is the surface
2618:If the hyperbolic paraboloid
845:, a hyperbolic paraboloid is
826:A hyperbolic paraboloid is a
3019:is congruent to the surface
2717:direction (according to the
1215:in Gothenburg, Sweden (1971)
1111:{\displaystyle bx\pm ay+b=0}
1495:approach the same surface
1189:St. Mary's Cathedral, Tokyo
1160:hyperbolic paraboloid model
500:Cartesian coordinate system
470:Properties and applications
439:-axis and upward along the
164:of a paraboloid by a plane
4106:
3455:measured along its surface
3421:{\displaystyle 4FD=R^{2},}
2692:is rotated by an angle of
841:From the point of view of
679:
617:From the point of view of
1221:in Valencia, Spain (2003)
1179:Expo '58, Brussels (1958)
765:Rotating water in a glass
3999:Thomas' Calculus 11th ed
3959:Thomas' Calculus 11th ed
3201:, and together form the
2897:then this simplifies to
1339:of elliptic paraboloids
1195:St Richard's Church, Ham
1165:Examples in architecture
716:liquid-mirror telescopes
603:obtained by revolving a
597:paraboloid of revolution
486:of a circular paraboloid
125:Paraboloid of revolution
2208:has Gaussian curvature
1521:{\displaystyle z=x^{2}}
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126:
3927:Parabolic loudspeaker
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3769:surface of revolution
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3434:is the focal length,
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3338:analytic continuation
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3059:The two paraboloidal
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885:{\displaystyle z=axy}
847:one-sheet hyperboloid
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771:Hyperbolic paraboloid
601:surface of revolution
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428:
340:Hyperbolic paraboloid
339:
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141:that has exactly one
124:
4061:at Wikimedia Commons
3775:
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3535:
3467:(or the equivalent:
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3054:multiplication table
3041:{\displaystyle z=xy}
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1136:, if the plane is a
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796:doubly ruled surface
656:, if the plane is a
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360:
350:doubly ruled surface
236:
153:, which refers to a
43:improve this article
3933:Parabolic reflector
3350:parabolic function
3199:harmonic conjugates
3052:, as it were) of a
2364:and mean curvature
843:projective geometry
753:Parabolic reflector
682:Parabolic reflector
676:Parabolic reflector
619:projective geometry
593:circular paraboloid
494:Circular paraboloid
475:Elliptic paraboloid
464:translation surface
3907:Mathematics portal
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1134:intersecting lines
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968:, by analogy with
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147:center of symmetry
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4057:Media related to
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3613:natural logarithm
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3203:analytic function
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3004:
2948:Finally, letting
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1185:- Dogra Hall Roof
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855:plane at infinity
686:parabolic antenna
631:plane at infinity
607:around its axis.
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537:
459:opens downward).
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212:coordinate system
201:complex conjugate
193:implicit equation
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2922:
2911:
2896:
2886:
2884:
2883:
2878:
2876:
2872:
2871:
2869:
2868:
2856:
2851:
2849:
2848:
2836:
2820:
2816:
2815:
2813:
2812:
2800:
2795:
2793:
2792:
2780:
2773:
2769:
2764:
2763:
2762:
2750:
2749:
2739:
2716:
2709:
2708:
2706:
2705:
2702:
2699:
2691:
2689:
2688:
2683:
2681:
2679:
2678:
2669:
2668:
2659:
2654:
2652:
2651:
2642:
2641:
2632:
2610:
2608:
2607:
2602:
2597:
2595:
2594:
2592:
2591:
2586:
2582:
2581:
2579:
2578:
2569:
2568:
2567:
2554:
2549:
2547:
2546:
2537:
2536:
2535:
2522:
2508:
2506:
2505:
2496:
2495:
2485:
2484:
2482:
2481:
2472:
2471:
2470:
2457:
2452:
2450:
2449:
2440:
2439:
2438:
2425:
2420:
2419:
2407:
2406:
2393:
2363:
2361:
2360:
2355:
2353:
2351:
2350:
2349:
2344:
2340:
2339:
2337:
2336:
2327:
2326:
2325:
2312:
2307:
2305:
2304:
2295:
2294:
2293:
2280:
2266:
2265:
2256:
2255:
2245:
2237:
2207:
2205:
2204:
2199:
2197:
2193:
2192:
2190:
2189:
2180:
2179:
2170:
2165:
2163:
2162:
2153:
2152:
2143:
2106:
2105:
2097:
2083:
2081:
2080:
2075:
2073:
2071:
2070:
2068:
2067:
2062:
2058:
2057:
2055:
2054:
2045:
2044:
2043:
2030:
2025:
2023:
2022:
2013:
2012:
2011:
1998:
1984:
1982:
1981:
1972:
1971:
1961:
1960:
1958:
1957:
1948:
1947:
1946:
1933:
1928:
1926:
1925:
1916:
1915:
1914:
1901:
1896:
1895:
1883:
1882:
1872:
1839:
1837:
1836:
1831:
1829:
1827:
1826:
1825:
1820:
1816:
1815:
1813:
1812:
1803:
1802:
1801:
1788:
1783:
1781:
1780:
1771:
1770:
1769:
1756:
1742:
1741:
1732:
1731:
1718:
1685:
1683:
1682:
1677:
1675:
1671:
1670:
1668:
1667:
1658:
1657:
1648:
1643:
1641:
1640:
1631:
1630:
1621:
1584:
1583:
1575:
1553:
1551:
1550:
1545:
1527:
1525:
1524:
1519:
1517:
1516:
1494:
1492:
1491:
1486:
1470:
1466:
1464:
1463:
1454:
1453:
1444:
1439:
1438:
1416:
1414:
1413:
1408:
1392:
1388:
1386:
1385:
1376:
1375:
1366:
1361:
1360:
1314:
1298:
1286:
1274:
1259:
1227:, England (2011)
1177:Philips Pavilion
1155:
1128:
1117:
1115:
1114:
1109:
1073:
1061:
1059:
1058:
1053:
1051:
1049:
1048:
1039:
1038:
1029:
1024:
1022:
1021:
1012:
1011:
1002:
962:rotation of axes
955:
953:
952:
947:
942:
941:
929:
928:
916:
907:
891:
889:
888:
883:
762:
750:
737:
731:
720:rotating furnace
612:Circular section
590:
578:
576:
575:
570:
565:
563:
562:
553:
552:
543:
538:
536:
535:
526:
525:
516:
458:
451:
444:
438:
432:
430:
429:
424:
419:
417:
416:
407:
406:
397:
392:
390:
389:
380:
379:
370:
332:
326:
320:
314:
308:
306:
305:
300:
295:
293:
292:
283:
282:
273:
268:
266:
265:
256:
255:
246:
231:
225:
219:
214:with three axes
143:axis of symmetry
114:
107:
103:
100:
94:
92:
51:
27:
19:
4105:
4104:
4100:
4099:
4098:
4096:
4095:
4094:
4065:
4064:
4049:
4044:
4043:
4036:
4021:
4020:
4016:
4009:
3996:
3995:
3991:
3983:
3976:
3969:
3952:
3951:
3947:
3942:
3905:
3898:
3895:
3864:
3860:
3844:
3829:
3819:
3803:
3797:
3793:
3786:
3773:
3772:
3760:
3746:
3743:
3740:
3739:
3737:
3736:
3722:
3708:
3705:
3702:
3701:
3699:
3698:
3684:
3653:
3638:
3637:
3636:), is given by
3622:
3616:
3605:
3571:
3565:
3540:
3533:
3532:
3526:
3520:
3514:
3502:
3500:
3495:
3484:
3481:
3476:
3475:
3473:
3468:
3458:
3441:
3435:
3429:
3405:
3388:
3387:
3384:
3371:
3368:
3363:
3362:
3360:
3351:
3341:
3301:
3270:
3228:
3206:
3205:
3153:
3148:
3147:
3120:
3107:
3106:
3076:
3071:
3070:
3060:
3021:
3020:
2990:
2977:
2976:
2963:
2962:
2956:
2954:
2949:
2923:
2912:
2899:
2898:
2888:
2860:
2840:
2834:
2830:
2804:
2784:
2778:
2774:
2754:
2741:
2740:
2734:
2723:
2722:
2719:right hand rule
2711:
2703:
2700:
2697:
2696:
2694:
2693:
2670:
2660:
2643:
2633:
2620:
2619:
2616:
2570:
2559:
2555:
2538:
2527:
2523:
2514:
2510:
2509:
2497:
2487:
2486:
2473:
2462:
2458:
2441:
2430:
2426:
2411:
2398:
2394:
2366:
2365:
2328:
2317:
2313:
2296:
2285:
2281:
2272:
2268:
2267:
2257:
2247:
2246:
2238:
2210:
2209:
2181:
2171:
2154:
2144:
2129:
2125:
2090:
2089:
2046:
2035:
2031:
2014:
2003:
1999:
1990:
1986:
1985:
1973:
1963:
1962:
1949:
1938:
1934:
1917:
1906:
1902:
1887:
1874:
1873:
1845:
1844:
1804:
1793:
1789:
1772:
1761:
1757:
1748:
1744:
1743:
1733:
1723:
1722:
1691:
1690:
1659:
1649:
1632:
1622:
1607:
1603:
1568:
1567:
1564:
1530:
1529:
1508:
1497:
1496:
1455:
1445:
1430:
1419:
1418:
1377:
1367:
1352:
1341:
1340:
1325:
1318:
1315:
1306:
1299:
1290:
1287:
1278:
1275:
1266:
1260:
1225:London Velopark
1167:
1151:
1126:
1076:
1075:
1071:
1040:
1030:
1013:
1003:
990:
989:
933:
920:
894:
893:
862:
861:
832:Gauss curvature
773:
766:
763:
754:
751:
742:
735:
732:
688:
680:Main articles:
678:
582:
554:
544:
527:
517:
504:
503:
477:
472:
453:
446:
440:
434:
408:
398:
381:
371:
358:
357:
328:
322:
316:
310:
284:
274:
257:
247:
234:
233:
227:
221:
215:
197:complex numbers
139:quadric surface
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
4103:
4101:
4093:
4092:
4087:
4082:
4077:
4067:
4066:
4063:
4062:
4048:
4047:External links
4045:
4042:
4041:
4034:
4014:
4007:
3989:
3974:
3967:
3944:
3943:
3941:
3938:
3937:
3936:
3930:
3924:
3918:
3911:
3910:
3894:
3891:
3879:
3871:
3867:
3863:
3857:
3851:
3847:
3843:
3836:
3832:
3826:
3822:
3818:
3815:
3810:
3806:
3802:
3796:
3792:
3789:
3783:
3780:
3668:
3665:
3660:
3656:
3650:
3647:
3593:
3589:
3584:
3580:
3577:
3574:
3568:
3564:
3561:
3558:
3555:
3550:
3546:
3543:
3448:unit of length
3417:
3412:
3408:
3404:
3401:
3398:
3395:
3383:
3380:
3325:
3322:
3319:
3316:
3313:
3308:
3304:
3300:
3297:
3294:
3291:
3288:
3285:
3282:
3277:
3273:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3240:
3235:
3231:
3225:
3222:
3219:
3216:
3213:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3160:
3156:
3133:
3127:
3123:
3119:
3114:
3110:
3103:
3100:
3097:
3094:
3091:
3088:
3083:
3079:
3037:
3034:
3031:
3028:
3008:
3003:
2997:
2993:
2989:
2984:
2980:
2973:
2970:
2937:
2930:
2926:
2921:
2918:
2915:
2909:
2906:
2875:
2867:
2863:
2859:
2854:
2847:
2843:
2839:
2833:
2829:
2826:
2823:
2819:
2811:
2807:
2803:
2798:
2791:
2787:
2783:
2777:
2772:
2767:
2761:
2757:
2753:
2748:
2744:
2737:
2733:
2730:
2677:
2673:
2667:
2663:
2657:
2650:
2646:
2640:
2636:
2630:
2627:
2615:
2612:
2600:
2590:
2585:
2577:
2573:
2566:
2562:
2558:
2552:
2545:
2541:
2534:
2530:
2526:
2520:
2517:
2513:
2504:
2500:
2494:
2490:
2480:
2476:
2469:
2465:
2461:
2455:
2448:
2444:
2437:
2433:
2429:
2423:
2418:
2414:
2410:
2405:
2401:
2397:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2348:
2343:
2335:
2331:
2324:
2320:
2316:
2310:
2303:
2299:
2292:
2288:
2284:
2278:
2275:
2271:
2264:
2260:
2254:
2250:
2244:
2241:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2196:
2188:
2184:
2178:
2174:
2168:
2161:
2157:
2151:
2147:
2141:
2138:
2135:
2132:
2128:
2124:
2121:
2118:
2115:
2112:
2109:
2103:
2100:
2066:
2061:
2053:
2049:
2042:
2038:
2034:
2028:
2021:
2017:
2010:
2006:
2002:
1996:
1993:
1989:
1980:
1976:
1970:
1966:
1956:
1952:
1945:
1941:
1937:
1931:
1924:
1920:
1913:
1909:
1905:
1899:
1894:
1890:
1886:
1881:
1877:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1842:mean curvature
1824:
1819:
1811:
1807:
1800:
1796:
1792:
1786:
1779:
1775:
1768:
1764:
1760:
1754:
1751:
1747:
1740:
1736:
1730:
1726:
1721:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1674:
1666:
1662:
1656:
1652:
1646:
1639:
1635:
1629:
1625:
1619:
1616:
1613:
1610:
1606:
1602:
1599:
1596:
1593:
1590:
1587:
1581:
1578:
1563:
1560:
1543:
1540:
1537:
1515:
1511:
1507:
1504:
1484:
1481:
1478:
1475:
1469:
1462:
1458:
1452:
1448:
1442:
1437:
1433:
1429:
1426:
1406:
1403:
1400:
1397:
1391:
1384:
1380:
1374:
1370:
1364:
1359:
1355:
1351:
1348:
1324:
1321:
1320:
1319:
1316:
1309:
1307:
1303:L'OceanogrĂ fic
1300:
1293:
1291:
1288:
1281:
1279:
1276:
1269:
1267:
1261:
1254:
1251:
1250:
1244:
1238:
1237:, Wales (1970)
1228:
1222:
1219:L'OceanogrĂ fic
1216:
1210:
1204:
1198:
1192:
1191:, Japan (1964)
1186:
1180:
1166:
1163:
1149:
1148:
1141:
1130:
1119:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1047:
1043:
1037:
1033:
1027:
1020:
1016:
1010:
1006:
1000:
997:
978:
977:
976:Plane sections
945:
940:
936:
932:
927:
923:
919:
913:
910:
904:
901:
881:
878:
875:
872:
869:
828:saddle surface
772:
769:
768:
767:
764:
757:
755:
752:
745:
743:
733:
726:
677:
674:
673:
672:
661:
650:
639:
638:
637:Plane sections
568:
561:
557:
551:
547:
541:
534:
530:
524:
520:
514:
511:
498:In a suitable
476:
473:
471:
468:
422:
415:
411:
405:
401:
395:
388:
384:
378:
374:
368:
365:
352:shaped like a
298:
291:
287:
281:
277:
271:
264:
260:
254:
250:
244:
241:
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
4102:
4091:
4088:
4086:
4083:
4081:
4078:
4076:
4073:
4072:
4070:
4060:
4055:
4051:
4050:
4046:
4037:
4035:9781449644482
4031:
4027:
4026:
4018:
4015:
4010:
4008:0-321-18558-7
4004:
4000:
3993:
3990:
3987:
3981:
3979:
3975:
3970:
3968:0-321-18558-7
3964:
3960:
3956:
3949:
3946:
3939:
3934:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3912:
3908:
3902:
3897:
3892:
3890:
3877:
3869:
3865:
3861:
3855:
3849:
3845:
3841:
3834:
3824:
3820:
3816:
3813:
3808:
3804:
3794:
3790:
3787:
3781:
3778:
3770:
3764:
3757:
3754:
3734:
3729:
3725:
3719:
3716:
3696:
3691:
3688:
3682:
3666:
3663:
3658:
3654:
3648:
3645:
3635:
3630:
3627:
3626:
3619:
3614:
3609:
3591:
3587:
3582:
3578:
3575:
3572:
3566:
3562:
3559:
3556:
3553:
3548:
3544:
3541:
3529:
3523:
3517:
3509:
3505:
3498:
3488:
3479:
3471:
3465:
3461:
3456:
3451:
3449:
3444:
3438:
3432:
3415:
3410:
3406:
3402:
3399:
3396:
3393:
3381:
3379:
3366:
3358:
3354:
3348:
3344:
3339:
3336:which is the
3320:
3317:
3314:
3306:
3302:
3298:
3295:
3289:
3286:
3283:
3275:
3271:
3267:
3261:
3258:
3255:
3252:
3246:
3243:
3238:
3233:
3229:
3223:
3217:
3211:
3204:
3200:
3184:
3181:
3178:
3172:
3169:
3166:
3158:
3154:
3131:
3125:
3121:
3117:
3112:
3108:
3101:
3095:
3092:
3089:
3081:
3077:
3067:
3063:
3057:
3055:
3051:
3035:
3032:
3029:
3026:
3006:
3001:
2995:
2991:
2987:
2982:
2978:
2971:
2968:
2952:
2935:
2928:
2924:
2919:
2916:
2913:
2907:
2904:
2895:
2891:
2873:
2865:
2861:
2857:
2852:
2845:
2841:
2837:
2831:
2827:
2824:
2821:
2817:
2809:
2805:
2801:
2796:
2789:
2785:
2781:
2775:
2770:
2765:
2759:
2755:
2751:
2746:
2742:
2735:
2731:
2728:
2720:
2715:
2675:
2671:
2665:
2661:
2655:
2648:
2644:
2638:
2634:
2628:
2625:
2613:
2611:
2598:
2588:
2583:
2575:
2571:
2564:
2560:
2556:
2550:
2543:
2539:
2532:
2528:
2524:
2518:
2515:
2511:
2502:
2498:
2492:
2488:
2478:
2474:
2467:
2463:
2459:
2453:
2446:
2442:
2435:
2431:
2427:
2421:
2416:
2412:
2408:
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54:Find sources:
48:
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32:This article
30:
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1170:Saddle roofs
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58:"Paraboloid"
53:
41:Please help
36:verification
33:
3921:Hyperboloid
836:developable
698:focal point
346:hyperboloid
4069:Categories
4059:Paraboloid
3940:References
3695:hemisphere
3611:means the
3069:functions
1207:Saddledome
1132:a pair of
813:skew lines
800:skew lines
740:vice versa
599:. It is a
170:hyperbolic
135:paraboloid
69:newspapers
4090:Parabolas
3955:Joel Hass
3915:Ellipsoid
3842:−
3788:π
3731:), and a
3646:π
3563:
3118:−
3050:nomograph
2988:−
2797:−
2656:−
2422:−
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2240:−
2167:−
2102:→
2099:σ
1580:→
1577:σ
1562:Curvature
1542:∞
1539:→
1441:−
1183:IIT Delhi
1145:hyperbola
1088:±
1026:−
931:−
830:, as its
709:astronomy
623:ellipsoid
394:−
174:hyperbola
99:June 2020
4085:Quadrics
4080:Surfaces
3893:See also
3721:, where
3681:cylinder
3513:, where
1123:parabola
849:that is
821:Pringles
788:Pringles
647:parabola
625:that is
605:parabola
189:cylinder
178:elliptic
166:parallel
151:parabola
131:geometry
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208:maximum
182:ellipse
145:and no
83:scholar
4032:
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3604:where
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3428:where
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1337:pencil
804:conoid
354:saddle
309:where
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160:Every
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958:up to
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693:focus
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137:is a
90:JSTOR
76:books
4030:ISBN
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3733:cone
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3197:are
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