3128:
3350:
3108:
3205:
3421:
3441:
3178:
3293:
3316:
3266:
3398:
3247:
3159:
3374:
588:
898:
3228:
2161:
890:
2169:
52:
31:
40:
1763:
4087:
1554:
2627:
1057:
885:
754:
4093:
4091:
4088:
2861:
4092:
2440:
1758:{\displaystyle g_{\alpha }^{\pm }:\mathbf {x} (t)={\begin{pmatrix}a\cos \alpha \\b\sin \alpha \\0\end{pmatrix}}+t\cdot {\begin{pmatrix}-a\sin \alpha \\b\cos \alpha \\\pm c\end{pmatrix}}\ ,\quad t\in \mathbb {R} ,\ 0\leq \alpha \leq 2\pi \ }
4243:
Minkowski used the term "four-dimensional hyperboloid" only once, in a posthumously-published typescript and this was non-standard usage, as
Minkowski's hyperboloid is a three-dimensional submanifold of a four-dimensional Minkowski space
1261:
4090:
357:
462:
253:
1547:
928:
3127:
2687:
of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive. In spite of its positive curvature, the hyperboloid of two sheets with another suitably chosen metric can also be used as a
770:
4007:
3806:
642:
2711:
775:
647:
3868:
2260:
3204:
2924:
3315:
1971:
4089:
3177:
1190:
3349:
507:
4040:
3909:
1421:. The one-sheet hyperboloid has two positive eigenvalues and one negative eigenvalue. The two-sheet hyperboloid has one positive eigenvalue and two negative eigenvalues.
1856:
1824:
1419:
1382:
1345:
4272:
1115:
1086:
258:
2622:{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1,\quad {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=-1}
2400:
2366:
2336:
2295:
2138:
2104:
2070:
2036:
2002:
366:
154:
2672:
1792:
1144:
1171:
923:
548:
at the point consists of two branches of curve that have distinct tangents at the point. In the case of the one-sheet hyperboloid, these branches of curves are
3107:
2944:... the velocity vectors always lie on a surface which Minkowski calls a four-dimensional hyperboloid since, expressed in terms of purely real coordinates
359:
The coordinate axes are axes of symmetry of the hyperboloid and the origin is the center of symmetry of the hyperboloid. In any case, the hyperboloid is
1150:
One can obtain a parametric representation of a hyperboloid with a different coordinate axis as the axis of symmetry by shuffling the position of the
3265:
3397:
3447:
1973:
are considered. Because a hyperboloid in general position is an affine image of the unit hyperboloid, the result applies to the general case, too.
1451:
3642:. The study of these two hyperboloids is, therefore, in this way connected very simply, through biquaternions, with the study of the sphere; ...
3440:
3373:
3420:
905:
The following parametric representation includes hyperboloids of one sheet, two sheets, and their common boundary cone, each with the
3922:
3086:; thus, it can be built with straight steel beams, producing a strong structure at a lower cost than other methods. Examples include
3721:
1052:{\displaystyle \mathbf {x} (s,t)=\left({\begin{array}{lll}a{\sqrt {s^{2}+d}}\cos t\\b{\sqrt {s^{2}+d}}\sin t\\cs\end{array}}\right)}
3292:
2410:
Obviously, any two-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case (see
2187:
2148:
Obviously, any one-sheet hyperboloid of revolution contains circles. This is also true, but less obvious, in the general case (see
2876:
880:{\displaystyle {\begin{aligned}x&=a\sinh v\cos \theta \\y&=b\sinh v\sin \theta \\z&=\pm c\cosh v\end{aligned}}}
1870:
3246:
749:{\displaystyle {\begin{aligned}x&=a\cosh v\cos \theta \\y&=b\cosh v\sin \theta \\z&=c\sinh v\end{aligned}}}
3813:
3227:
2856:{\displaystyle q(x)=\left(x_{1}^{2}+\cdots +x_{k}^{2}\right)-\left(x_{k+1}^{2}+\cdots +x_{n}^{2}\right),\quad k<n.}
4297:
4213:
624:
3210:
1901:
3158:
3095:
572:
3336:
2270:
A plane with slope less than 1 (1 is the slope of the asymptotes of the generating hyperbola) intersects
148:
3670:
117:
of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a
4323:
4290:
4192:
4180:
3183:
1873:(see picture; rotating the hyperbola around its other axis gives a two-sheet hyperbola of revolution).
4439:
3476:. The following passage from page 673 shows how Hamilton uses biquaternion algebra and vectors from
3356:
2700:
Imaginary hyperboloids are frequently found in mathematics of higher dimensions. For example, in a
3252:
469:
4012:
3881:
1794:
the hyperboloid is a surface of revolution and can be generated by rotating one of the two lines
893:
hyperboloid of one sheet: generation by a rotating hyperbola (top) and line (bottom: red or blue)
90:
is the surface obtained from a hyperboloid of revolution by deforming it by means of directional
3682:
3465:
2870:
2701:
1829:
1797:
1387:
1350:
1313:
1307:
4226:
579:
in the sense that the tangent plane at every point intersects the surface only in this point.
4128:
3187:
3079:
3073:
1977:
A plane with a slope less than 1 (1 is the slope of the lines on the hyperboloid) intersects
1883:
596:
106:
95:
75:
20:
4394:
4375:
4247:
1094:
1065:
4337:
4143:
3134:
3083:
2378:
2344:
2314:
2273:
2116:
2082:
2048:
2014:
1980:
1441:
1270:
544:
at every point. This implies near every point the intersection of the hyperboloid and its
8:
4413:
4148:
3214:
3114:
3060:
is also used in this context since the sphere and hyperboloid have some commonality (See
2651:
1878:
1771:
1256:{\displaystyle (\mathbf {x} -\mathbf {v} )^{\mathrm {T} }A(\mathbf {x} -\mathbf {v} )=1,}
1123:
141:
137:
4206:
Emergence of the Theory of Lie Groups: an essay in the history of mathematics, 1869—1926
1153:
4133:
3659:
3329:
2684:
2308:
A plane with slope equal to 1 containing the origin (midpoint of the hyperboloid) does
908:
541:
91:
4449:
4444:
4410:
4391:
4372:
4353:
4293:
4209:
3275:
3233:
2689:
3271:
1865:. The more common generation of a one-sheet hyperboloid of revolution is rotating a
4329:
4308:
4138:
4119:
4097:
3524:. The equation of the sphere then breaks up into the system of the two following,
3380:
3298:
2411:
2149:
1860:
1286:
960:
130:
126:
3665:
A modern view of the unification of the sphere and hyperboloid uses the idea of a
4356:
4282:
4193:
CDKG: Computerunterstützte
Darstellende und Konstruktive Geometrie (TU Darmstadt)
4181:
CDKG: Computerunterstützte
Darstellende und Konstruktive Geometrie (TU Darmstadt)
4109:
3674:
3332:
3164:
3141:
549:
118:
102:
44:
587:
4046:
3712:
3237:
2705:
352:{\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=-1.}
3078:
One-sheeted hyperboloids are used in construction, with the structures called
457:{\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=0.}
248:{\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=1,}
83:
4433:
4208:, §9.3 "The Mathematization of Physics at Göttingen", see page 340, Springer
3666:
3325:
3137:
3091:
3087:
1858:, which are skew to the rotation axis (see picture). This property is called
1542:{\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=1}
545:
537:
134:
4052:
3678:
3473:
3427:
2648:
to the z-axis and symmetric to any plane containing the z-axis, in case of
553:
3279:
3149:
2180:
contain lines. The discussion of plane sections can be performed for the
1293:
151:
such that the hyperboloid is defined by one of the following equations:
4124:
3871:
3477:
3360:
1303:
360:
114:
595:
Cartesian coordinates for the hyperboloids can be defined, similar to
4418:
4399:
4380:
4361:
4114:
3384:
3191:
2263:
1866:
897:
79:
3403:
3322:
3195:
2341:
A plane with slope equal to 1 not containing the origin intersects
122:
110:
63:
2045:
A plane with a slope equal 1 not containing the origin intersects
1440:
A hyperboloid of one sheet contains two pencils of lines. It is a
3411:
3388:
3364:
3340:
3302:
3218:
3145:
3118:
2160:
2011:
A plane with a slope equal to 1 containing the origin intersects
1181:
More generally, an arbitrarily oriented hyperboloid, centered at
889:
600:
575:
and a positive
Gaussian curvature at every point. The surface is
4389:
4370:
2168:
4002:{\displaystyle Q=\lbrace p\ :\ w^{2}+z^{2}=x^{2}+y^{2}\rbrace }
3481:
3451:
3407:
3306:
3283:
3256:
51:
30:
3801:{\displaystyle P=\left\{p\;:\;w^{2}=x^{2}+y^{2}+z^{2}\right\}}
2113:
A non-tangential plane with a slope greater than 1 intersects
4408:
4101:
3431:
3168:
2164:
hyperboloid of two sheets: generation by rotating a hyperbola
510:
39:
3652:
is the operator giving the scalar part of a quaternion, and
4318:
David A. Brannan, M. F. Esplen, & Jeremy J Gray (1999)
1302:
define the principal directions of the hyperboloid and the
2266:
around one of its axes (the one that cuts the hyperbola)
1173:
term to the appropriate component in the equation above.
4315:, Kapital V: "Quadriken", Wolfenbutteler Verlagsanstalt.
524:
There are two kinds of hyperboloids. In the first case (
1658:
1598:
4250:
4231:
The
Symbolic Universe: Geometry and Physics 1890-1930
4015:
3925:
3884:
3816:
3724:
3628:, then the locus of the extremity of the real vector
2879:
2714:
2654:
2443:
2381:
2347:
2317:
2276:
2190:
2119:
2085:
2051:
2017:
1983:
1904:
1832:
1800:
1774:
1557:
1454:
1390:
1353:
1316:
1193:
1156:
1126:
1097:
1068:
931:
911:
773:
645:
472:
369:
261:
157:
4351:
4227:"The non-Euclidean style of Minkowskian relativity"
466:One has a hyperboloid of revolution if and only if
4266:
4171:Vandenhoeck & Ruprecht, Göttingen 1967, p. 218
4034:
4001:
3903:
3862:
3800:
2918:
2855:
2666:
2621:
2394:
2360:
2330:
2289:
2254:
2132:
2098:
2064:
2030:
1996:
1965:
1850:
1818:
1786:
1757:
1541:
1413:
1376:
1339:
1255:
1165:
1138:
1109:
1080:
1051:
917:
879:
748:
501:
456:
351:
247:
133:into hyperbolas. A hyperboloid has three pairwise
4431:
4224:
3863:{\displaystyle H_{r}=\lbrace p\ :\ w=r\rbrace ,}
2695:
3563:as two real and rectangular vectors, such that
3480:to produce hyperboloids from the equation of a
2940:As an example, consider the following passage:
3917:. On the other hand, the conical hypersurface
2375:A plane with slope greater than 1 intersects
2255:{\displaystyle H_{2}:\ x^{2}+y^{2}-z^{2}=-1.}
582:
3996:
3932:
3854:
3830:
3618:; and that if, on the other hand, we assume
3585:Hence it is easy to infer that if we assume
2919:{\displaystyle \lbrace x\ :\ q(x)=c\rbrace }
2913:
2880:
2155:
3102:Gallery of one sheet hyperboloid structures
3061:
1966:{\displaystyle \ H_{1}:x^{2}+y^{2}-z^{2}=1}
1429:
563:in the right-hand side of the equation): a
528:in the right-hand side of the equation): a
4287:Clifford Algebras and the Classical Groups
4233:, Oxford University Press, pp. 91–127
3743:
3739:
3715:. First consider the conical hypersurface
3640:equilateral but single-sheeted hyperboloid
3616:double-sheeted and equilateral hyperboloid
509:Otherwise, the axes are uniquely defined (
3459:
3067:
2172:hyperboloid of two sheets: plane sections
1894:For simplicity the plane sections of the
1724:
4169:Vorlesungen der Darstellenden Geometrie.
4085:
2167:
2159:
1176:
901:hyperboloid of one sheet: plane sections
896:
888:
591:Animation of a hyperboloid of revolution
586:
552:and thus the one-sheet hyperboloid is a
3406:, (BMW World), museum and event venue,
1434:
1088:one obtains a hyperboloid of one sheet,
4432:
2873:, then the part of the space given by
147:Given a hyperboloid, one can choose a
4409:
4390:
4371:
4352:
3601:is a vector in a given position, the
2930:. The degenerate case corresponds to
2262:which can be generated by a rotating
1448:If the hyperboloid has the equation
4056:is the subset of a quadratic space
3614:will terminate on the surface of a
2427:
2176:The hyperboloid of two sheets does
140:, and three pairwise perpendicular
13:
2640:symmetric to the coordinate planes
1216:
625:hyperbolic trigonometric functions
14:
4461:
4345:
2437:The hyperboloids with equations
1889:
1310:of the squares of the semi-axes:
4072:such that the quadratic norm of
3439:
3419:
3396:
3372:
3348:
3314:
3291:
3264:
3245:
3226:
3203:
3176:
3157:
3126:
3106:
1577:
1237:
1229:
1206:
1198:
1117:a hyperboloid of two sheets, and
933:
50:
38:
29:
4144:Split-quaternion § Profile
3472:which included presentation of
3211:Newcastle International Airport
3013:, analogous to the hyperboloid
2840:
2531:
2420:A hyperboloid of two sheets is
1716:
925:-axis as the axis of symmetry:
4276:
4237:
4218:
4198:
4186:
4174:
4161:
2904:
2898:
2724:
2718:
2182:unit hyperboloid of two sheets
2079:A tangential plane intersects
1876:A hyperboloid of one sheet is
1765:are contained in the surface.
1587:
1581:
1241:
1225:
1211:
1194:
949:
937:
363:to the cone of the equations:
25:
1:
4154:
3551:and suggests our considering
3062:§ Relation to the sphere
2696:In more than three dimensions
2432:
1424:
1187:, is defined by the equation
4100:hyperboloid tower (1898) in
3658:is the "tensor", now called
2678:
2674:(hyperboloid of revolution).
502:{\displaystyle a^{2}=b^{2}.}
27:
7:
4081:
4035:{\displaystyle Q\cap H_{r}}
3904:{\displaystyle P\cap H_{r}}
3490:equation of the unit sphere
3328:for the now decommissioned
3052:of three-dimensional space.
617:, but changing inclination
149:Cartesian coordinate system
94:, or more generally, of an
10:
4466:
4324:Cambridge University Press
4291:Cambridge University Press
4289:, pages 22, 24 & 106,
4195:(PDF; 3,4 MB), S. 122
4183:(PDF; 3,4 MB), S. 116
3911:is the sphere with radius
3184:Saint Louis Science Center
3071:
2156:Hyperboloid of two sheets
2108:pair of intersecting lines
583:Parametric representations
56:Hyperboloid of two sheets
18:
4395:"Two-sheeted hyperboloid"
4376:"One-sheeted hyperboloid"
4225:Walter, Scott A. (1999),
3671:slice of a quadratic form
3357:Corporation Street Bridge
3234:Ještěd Transmission Tower
2692:for hyperbolic geometry.
1851:{\displaystyle g_{0}^{-}}
1819:{\displaystyle g_{0}^{+}}
1414:{\displaystyle {1/c^{2}}}
1377:{\displaystyle {1/b^{2}}}
1340:{\displaystyle {1/a^{2}}}
758:Two-surface hyperboloid:
630:One-surface hyperboloid:
68:hyperboloid of revolution
35:Hyperboloid of one sheet
16:Unbounded quadric surface
4334:Introduction to Geometry
3498:, and change the vector
3133:The first 1916 patented
2424:equivalent to a sphere.
1430:Hyperboloid of one sheet
129:, and intersecting many
78:generated by rotating a
23:, a saddle-like surface.
19:Not to be confused with
4322:, pp. 39–41
3677:, one requires conical
3470:Lectures on Quaternions
3272:Hyperboloid water tower
540:, which has a negative
4414:"Elliptic Hyperboloid"
4268:
4267:{\displaystyle M^{4}.}
4204:Thomas Hawkins (2000)
4105:
4036:
4003:
3905:
3864:
3802:
3683:four-dimensional space
3644:
3466:William Rowan Hamilton
3460:Relation to the sphere
3080:hyperboloid structures
3068:Hyperboloid structures
3054:
2920:
2857:
2702:pseudo-Euclidean space
2668:
2623:
2396:
2362:
2332:
2291:
2256:
2173:
2165:
2134:
2100:
2066:
2040:pair of parallel lines
2032:
1998:
1967:
1852:
1820:
1788:
1759:
1543:
1415:
1378:
1341:
1257:
1167:
1140:
1111:
1110:{\displaystyle d<0}
1082:
1081:{\displaystyle d>0}
1053:
919:
902:
894:
881:
750:
592:
571:. The surface has two
534:hyperbolic hyperboloid
503:
458:
353:
249:
4338:John Wiley & Sons
4313:Analytische Geometrie
4269:
4129:Hyperbolic paraboloid
4096:
4037:
4004:
3906:
3865:
3803:
3486:
3253:Cathedral of Brasília
3188:McDonnell Planetarium
3096:many other structures
3082:. A hyperboloid is a
3074:Hyperboloid structure
2942:
2921:
2858:
2704:one has the use of a
2669:
2624:
2397:
2395:{\displaystyle H_{2}}
2363:
2361:{\displaystyle H_{2}}
2333:
2331:{\displaystyle H_{2}}
2292:
2290:{\displaystyle H_{2}}
2257:
2171:
2163:
2135:
2133:{\displaystyle H_{1}}
2101:
2099:{\displaystyle H_{1}}
2067:
2065:{\displaystyle H_{1}}
2033:
2031:{\displaystyle H_{1}}
1999:
1997:{\displaystyle H_{1}}
1968:
1884:hyperbolic paraboloid
1853:
1821:
1789:
1760:
1544:
1416:
1379:
1342:
1258:
1177:Generalised equations
1168:
1141:
1112:
1083:
1054:
920:
900:
892:
882:
751:
597:spherical coordinates
590:
565:two-sheet hyperboloid
530:one-sheet hyperboloid
504:
459:
354:
250:
96:affine transformation
70:, sometimes called a
21:hyperbolic paraboloid
4248:
4229:, in J. Gray (ed.),
4013:
3923:
3882:
3814:
3722:
3084:doubly ruled surface
2877:
2712:
2652:
2646:rotational symmetric
2441:
2379:
2345:
2315:
2274:
2188:
2117:
2083:
2049:
2015:
1981:
1902:
1830:
1798:
1772:
1555:
1452:
1442:doubly ruled surface
1435:Lines on the surface
1388:
1351:
1314:
1191:
1154:
1124:
1095:
1066:
929:
909:
771:
643:
573:connected components
569:elliptic hyperboloid
559:In the second case (
513:the exchange of the
470:
367:
259:
155:
72:circular hyperboloid
4149:Translation of axes
3662:, of a quaternion.
3383:observation tower,
3215:Newcastle upon Tyne
3115:Adziogol Lighthouse
2831:
2807:
2773:
2749:
2667:{\displaystyle a=b}
1847:
1815:
1787:{\displaystyle a=b}
1572:
1139:{\displaystyle d=0}
637:(−∞, ∞)
101:A hyperboloid is a
4411:Weisstein, Eric W.
4392:Weisstein, Eric W.
4373:Weisstein, Eric W.
4354:Weisstein, Eric W.
4264:
4106:
4062:consisting of the
4032:
3999:
3901:
3860:
3798:
3056:However, the term
2962:, its equation is
2916:
2853:
2817:
2787:
2759:
2735:
2685:Gaussian curvature
2664:
2619:
2392:
2358:
2328:
2287:
2252:
2174:
2166:
2130:
2096:
2062:
2028:
1994:
1963:
1848:
1833:
1816:
1801:
1784:
1755:
1704:
1638:
1558:
1539:
1411:
1374:
1337:
1253:
1166:{\displaystyle cs}
1163:
1136:
1107:
1078:
1049:
1043:
915:
903:
895:
877:
875:
746:
744:
593:
542:Gaussian curvature
499:
454:
349:
245:
142:planes of symmetry
127:center of symmetry
82:around one of its
4094:
4045:In the theory of
4042:is a hyperboloid.
3946:
3940:
3844:
3838:
2894:
2888:
2608:
2581:
2554:
2520:
2493:
2466:
2206:
1907:
1754:
1733:
1712:
1531:
1504:
1477:
1020:
984:
918:{\displaystyle z}
567:, also called an
538:connected surface
446:
419:
392:
338:
311:
284:
234:
207:
180:
60:
59:
4457:
4440:Geometric shapes
4424:
4423:
4405:
4404:
4386:
4385:
4367:
4366:
4330:H. S. M. Coxeter
4309:Wilhelm Blaschke
4300:
4280:
4274:
4273:
4271:
4270:
4265:
4260:
4259:
4241:
4235:
4234:
4222:
4216:
4202:
4196:
4190:
4184:
4178:
4172:
4165:
4139:Rotation of axes
4120:List of surfaces
4095:
4077:
4071:
4061:
4041:
4039:
4038:
4033:
4031:
4030:
4008:
4006:
4005:
4000:
3995:
3994:
3982:
3981:
3969:
3968:
3956:
3955:
3944:
3938:
3916:
3910:
3908:
3907:
3902:
3900:
3899:
3869:
3867:
3866:
3861:
3842:
3836:
3826:
3825:
3807:
3805:
3804:
3799:
3797:
3793:
3792:
3791:
3779:
3778:
3766:
3765:
3753:
3752:
3710:
3657:
3651:
3646:In this passage
3637:
3627:
3613:
3600:
3594:
3581:
3562:
3556:
3547:
3536:
3523:
3522:
3521:
3503:
3497:
3443:
3423:
3400:
3376:
3352:
3318:
3299:Roy Thomson Hall
3295:
3268:
3249:
3230:
3207:
3180:
3161:
3130:
3110:
3090:, especially of
3051:
3049:
3048:
3037:
3036:
3025:
3024:
3012:
3010:
3009:
2998:
2997:
2986:
2985:
2974:
2973:
2961:
2936:
2925:
2923:
2922:
2917:
2892:
2886:
2868:
2862:
2860:
2859:
2854:
2836:
2832:
2830:
2825:
2806:
2801:
2778:
2774:
2772:
2767:
2748:
2743:
2673:
2671:
2670:
2665:
2628:
2626:
2625:
2620:
2609:
2607:
2606:
2597:
2596:
2587:
2582:
2580:
2579:
2570:
2569:
2560:
2555:
2553:
2552:
2543:
2542:
2533:
2521:
2519:
2518:
2509:
2508:
2499:
2494:
2492:
2491:
2482:
2481:
2472:
2467:
2465:
2464:
2455:
2454:
2445:
2428:Other properties
2412:circular section
2401:
2399:
2398:
2393:
2391:
2390:
2367:
2365:
2364:
2359:
2357:
2356:
2337:
2335:
2334:
2329:
2327:
2326:
2296:
2294:
2293:
2288:
2286:
2285:
2261:
2259:
2258:
2253:
2242:
2241:
2229:
2228:
2216:
2215:
2204:
2200:
2199:
2150:circular section
2139:
2137:
2136:
2131:
2129:
2128:
2105:
2103:
2102:
2097:
2095:
2094:
2071:
2069:
2068:
2063:
2061:
2060:
2037:
2035:
2034:
2029:
2027:
2026:
2003:
2001:
2000:
1995:
1993:
1992:
1972:
1970:
1969:
1964:
1956:
1955:
1943:
1942:
1930:
1929:
1917:
1916:
1905:
1896:unit hyperboloid
1882:equivalent to a
1857:
1855:
1854:
1849:
1846:
1841:
1825:
1823:
1822:
1817:
1814:
1809:
1793:
1791:
1790:
1785:
1764:
1762:
1761:
1756:
1752:
1731:
1727:
1710:
1709:
1708:
1643:
1642:
1580:
1571:
1566:
1548:
1546:
1545:
1540:
1532:
1530:
1529:
1520:
1519:
1510:
1505:
1503:
1502:
1493:
1492:
1483:
1478:
1476:
1475:
1466:
1465:
1456:
1420:
1418:
1417:
1412:
1410:
1409:
1408:
1399:
1383:
1381:
1380:
1375:
1373:
1372:
1371:
1362:
1346:
1344:
1343:
1338:
1336:
1335:
1334:
1325:
1301:
1284:
1278:
1268:
1262:
1260:
1259:
1254:
1240:
1232:
1221:
1220:
1219:
1209:
1201:
1186:
1172:
1170:
1169:
1164:
1145:
1143:
1142:
1137:
1116:
1114:
1113:
1108:
1087:
1085:
1084:
1079:
1058:
1056:
1055:
1050:
1048:
1044:
1021:
1013:
1012:
1003:
985:
977:
976:
967:
936:
924:
922:
921:
916:
886:
884:
883:
878:
876:
767:
766:
765:[0, ∞)
755:
753:
752:
747:
745:
639:
638:
622:
616:
615:
562:
532:, also called a
527:
508:
506:
505:
500:
495:
494:
482:
481:
463:
461:
460:
455:
447:
445:
444:
435:
434:
425:
420:
418:
417:
408:
407:
398:
393:
391:
390:
381:
380:
371:
358:
356:
355:
350:
339:
337:
336:
327:
326:
317:
312:
310:
309:
300:
299:
290:
285:
283:
282:
273:
272:
263:
254:
252:
251:
246:
235:
233:
232:
223:
222:
213:
208:
206:
205:
196:
195:
186:
181:
179:
178:
169:
168:
159:
138:axes of symmetry
54:
42:
33:
26:
4465:
4464:
4460:
4459:
4458:
4456:
4455:
4454:
4430:
4429:
4348:
4343:
4336:, p. 130,
4304:
4303:
4283:Ian R. Porteous
4281:
4277:
4255:
4251:
4249:
4246:
4245:
4242:
4238:
4223:
4219:
4203:
4199:
4191:
4187:
4179:
4175:
4167:K. Strubecker:
4166:
4162:
4157:
4110:De Sitter space
4086:
4084:
4073:
4063:
4057:
4047:quadratic forms
4043:
4026:
4022:
4014:
4011:
4010:
3990:
3986:
3977:
3973:
3964:
3960:
3951:
3947:
3924:
3921:
3920:
3912:
3895:
3891:
3883:
3880:
3879:
3821:
3817:
3815:
3812:
3811:
3787:
3783:
3774:
3770:
3761:
3757:
3748:
3744:
3735:
3731:
3723:
3720:
3719:
3713:quadratic forms
3686:
3675:conical surface
3673:. Instead of a
3653:
3647:
3629:
3619:
3605:
3603:new real vector
3596:
3586:
3583:
3566:
3558:
3552:
3549:
3538:
3527:
3519:
3517:
3509:
3499:
3492:
3462:
3455:
3444:
3435:
3424:
3415:
3401:
3392:
3377:
3368:
3353:
3344:
3333:nuclear reactor
3319:
3310:
3296:
3287:
3269:
3260:
3250:
3241:
3231:
3222:
3213:control tower,
3208:
3199:
3181:
3172:
3165:Kobe Port Tower
3162:
3153:
3150:The Netherlands
3131:
3122:
3111:
3076:
3070:
3047:
3044:
3043:
3042:
3035:
3032:
3031:
3030:
3023:
3020:
3019:
3018:
3014:
3008:
3005:
3004:
3003:
2996:
2993:
2992:
2991:
2984:
2981:
2980:
2979:
2972:
2969:
2968:
2967:
2963:
2959:
2952:
2945:
2931:
2878:
2875:
2874:
2864:
2826:
2821:
2802:
2791:
2786:
2782:
2768:
2763:
2744:
2739:
2734:
2730:
2713:
2710:
2709:
2698:
2681:
2653:
2650:
2649:
2602:
2598:
2592:
2588:
2586:
2575:
2571:
2565:
2561:
2559:
2548:
2544:
2538:
2534:
2532:
2514:
2510:
2504:
2500:
2498:
2487:
2483:
2477:
2473:
2471:
2460:
2456:
2450:
2446:
2444:
2442:
2439:
2438:
2435:
2430:
2386:
2382:
2380:
2377:
2376:
2352:
2348:
2346:
2343:
2342:
2322:
2318:
2316:
2313:
2312:
2281:
2277:
2275:
2272:
2271:
2237:
2233:
2224:
2220:
2211:
2207:
2195:
2191:
2189:
2186:
2185:
2158:
2124:
2120:
2118:
2115:
2114:
2090:
2086:
2084:
2081:
2080:
2056:
2052:
2050:
2047:
2046:
2022:
2018:
2016:
2013:
2012:
1988:
1984:
1982:
1979:
1978:
1951:
1947:
1938:
1934:
1925:
1921:
1912:
1908:
1903:
1900:
1899:
1892:
1871:semi-minor axis
1842:
1837:
1831:
1828:
1827:
1810:
1805:
1799:
1796:
1795:
1773:
1770:
1769:
1723:
1703:
1702:
1693:
1692:
1677:
1676:
1654:
1653:
1637:
1636:
1630:
1629:
1614:
1613:
1594:
1593:
1576:
1567:
1562:
1556:
1553:
1552:
1549:then the lines
1525:
1521:
1515:
1511:
1509:
1498:
1494:
1488:
1484:
1482:
1471:
1467:
1461:
1457:
1455:
1453:
1450:
1449:
1437:
1432:
1427:
1404:
1400:
1395:
1391:
1389:
1386:
1385:
1367:
1363:
1358:
1354:
1352:
1349:
1348:
1330:
1326:
1321:
1317:
1315:
1312:
1311:
1297:
1280:
1274:
1264:
1236:
1228:
1215:
1214:
1210:
1205:
1197:
1192:
1189:
1188:
1182:
1179:
1155:
1152:
1151:
1125:
1122:
1121:
1096:
1093:
1092:
1067:
1064:
1063:
1042:
1041:
1032:
1031:
1008:
1004:
1002:
996:
995:
972:
968:
966:
959:
955:
932:
930:
927:
926:
910:
907:
906:
874:
873:
851:
845:
844:
816:
810:
809:
781:
774:
772:
769:
768:
764:
759:
743:
742:
723:
717:
716:
688:
682:
681:
653:
646:
644:
641:
640:
636:
631:
618:
609:
604:
585:
560:
525:
490:
486:
477:
473:
471:
468:
467:
440:
436:
430:
426:
424:
413:
409:
403:
399:
397:
386:
382:
376:
372:
370:
368:
365:
364:
332:
328:
322:
318:
316:
305:
301:
295:
291:
289:
278:
274:
268:
264:
262:
260:
257:
256:
228:
224:
218:
214:
212:
201:
197:
191:
187:
185:
174:
170:
164:
160:
158:
156:
153:
152:
109:defined as the
103:quadric surface
55:
45:conical surface
43:
34:
24:
17:
12:
11:
5:
4463:
4453:
4452:
4447:
4442:
4428:
4427:
4426:
4425:
4406:
4387:
4347:
4346:External links
4344:
4342:
4341:
4327:
4316:
4305:
4302:
4301:
4275:
4263:
4258:
4254:
4236:
4217:
4197:
4185:
4173:
4159:
4158:
4156:
4153:
4152:
4151:
4146:
4141:
4136:
4131:
4122:
4117:
4112:
4083:
4080:
4029:
4025:
4021:
4018:
4009:provides that
3998:
3993:
3989:
3985:
3980:
3976:
3972:
3967:
3963:
3959:
3954:
3950:
3943:
3937:
3934:
3931:
3928:
3919:
3898:
3894:
3890:
3887:
3876:
3875:
3859:
3856:
3853:
3850:
3847:
3841:
3835:
3832:
3829:
3824:
3820:
3809:
3796:
3790:
3786:
3782:
3777:
3773:
3769:
3764:
3760:
3756:
3751:
3747:
3742:
3738:
3734:
3730:
3727:
3711:determined by
3565:
3526:
3468:published his
3461:
3458:
3457:
3456:
3448:Essarts-le-Roi
3445:
3438:
3436:
3425:
3418:
3416:
3402:
3395:
3393:
3378:
3371:
3369:
3354:
3347:
3345:
3320:
3313:
3311:
3297:
3290:
3288:
3270:
3263:
3261:
3251:
3244:
3242:
3238:Czech Republic
3232:
3225:
3223:
3209:
3202:
3200:
3182:
3175:
3173:
3163:
3156:
3154:
3132:
3125:
3123:
3112:
3105:
3103:
3092:power stations
3088:cooling towers
3072:Main article:
3069:
3066:
3045:
3033:
3021:
3006:
2994:
2982:
2970:
2957:
2950:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2891:
2885:
2882:
2852:
2849:
2846:
2843:
2839:
2835:
2829:
2824:
2820:
2816:
2813:
2810:
2805:
2800:
2797:
2794:
2790:
2785:
2781:
2777:
2771:
2766:
2762:
2758:
2755:
2752:
2747:
2742:
2738:
2733:
2729:
2726:
2723:
2720:
2717:
2706:quadratic form
2697:
2694:
2680:
2677:
2676:
2675:
2663:
2660:
2657:
2643:
2637:
2636:to the origin,
2634:pointsymmetric
2618:
2615:
2612:
2605:
2601:
2595:
2591:
2585:
2578:
2574:
2568:
2564:
2558:
2551:
2547:
2541:
2537:
2530:
2527:
2524:
2517:
2513:
2507:
2503:
2497:
2490:
2486:
2480:
2476:
2470:
2463:
2459:
2453:
2449:
2434:
2431:
2429:
2426:
2408:
2407:
2389:
2385:
2373:
2355:
2351:
2339:
2325:
2321:
2306:
2305:or not at all,
2284:
2280:
2251:
2248:
2245:
2240:
2236:
2232:
2227:
2223:
2219:
2214:
2210:
2203:
2198:
2194:
2184:with equation
2157:
2154:
2146:
2145:
2127:
2123:
2111:
2093:
2089:
2077:
2059:
2055:
2043:
2025:
2021:
2009:
1991:
1987:
1962:
1959:
1954:
1950:
1946:
1941:
1937:
1933:
1928:
1924:
1920:
1915:
1911:
1898:with equation
1891:
1890:Plane sections
1888:
1845:
1840:
1836:
1813:
1808:
1804:
1783:
1780:
1777:
1751:
1748:
1745:
1742:
1739:
1736:
1730:
1726:
1722:
1719:
1715:
1707:
1701:
1698:
1695:
1694:
1691:
1688:
1685:
1682:
1679:
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1659:
1657:
1652:
1649:
1646:
1641:
1635:
1632:
1631:
1628:
1625:
1622:
1619:
1616:
1615:
1612:
1609:
1606:
1603:
1600:
1599:
1597:
1592:
1589:
1586:
1583:
1579:
1575:
1570:
1565:
1561:
1538:
1535:
1528:
1524:
1518:
1514:
1508:
1501:
1497:
1491:
1487:
1481:
1474:
1470:
1464:
1460:
1446:
1445:
1436:
1433:
1431:
1428:
1426:
1423:
1407:
1403:
1398:
1394:
1370:
1366:
1361:
1357:
1333:
1329:
1324:
1320:
1252:
1249:
1246:
1243:
1239:
1235:
1231:
1227:
1224:
1218:
1213:
1208:
1204:
1200:
1196:
1178:
1175:
1162:
1159:
1148:
1147:
1146:a double cone.
1135:
1132:
1129:
1118:
1106:
1103:
1100:
1089:
1077:
1074:
1071:
1047:
1040:
1037:
1034:
1033:
1030:
1027:
1024:
1019:
1016:
1011:
1007:
1001:
998:
997:
994:
991:
988:
983:
980:
975:
971:
965:
962:
961:
958:
954:
951:
948:
945:
942:
939:
935:
914:
872:
869:
866:
863:
860:
857:
854:
852:
850:
847:
846:
843:
840:
837:
834:
831:
828:
825:
822:
819:
817:
815:
812:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
782:
780:
777:
776:
741:
738:
735:
732:
729:
726:
724:
722:
719:
718:
715:
712:
709:
706:
703:
700:
697:
694:
691:
689:
687:
684:
683:
680:
677:
674:
671:
668:
665:
662:
659:
656:
654:
652:
649:
648:
599:, keeping the
584:
581:
517:-axis and the
498:
493:
489:
485:
480:
476:
453:
450:
443:
439:
433:
429:
423:
416:
412:
406:
402:
396:
389:
385:
379:
375:
348:
345:
342:
335:
331:
325:
321:
315:
308:
304:
298:
294:
288:
281:
277:
271:
267:
244:
241:
238:
231:
227:
221:
217:
211:
204:
200:
194:
190:
184:
177:
173:
167:
163:
84:principal axes
58:
57:
48:
36:
15:
9:
6:
4:
3:
2:
4462:
4451:
4448:
4446:
4443:
4441:
4438:
4437:
4435:
4421:
4420:
4415:
4412:
4407:
4402:
4401:
4396:
4393:
4388:
4383:
4382:
4377:
4374:
4369:
4368:
4364:
4363:
4358:
4357:"Hyperboloid"
4355:
4350:
4349:
4339:
4335:
4331:
4328:
4325:
4321:
4317:
4314:
4310:
4307:
4306:
4299:
4298:0-521-55177-3
4295:
4292:
4288:
4284:
4279:
4261:
4256:
4252:
4240:
4232:
4228:
4221:
4215:
4214:0-387-98963-3
4211:
4207:
4201:
4194:
4189:
4182:
4177:
4170:
4164:
4160:
4150:
4147:
4145:
4142:
4140:
4137:
4135:
4132:
4130:
4126:
4123:
4121:
4118:
4116:
4113:
4111:
4108:
4107:
4103:
4099:
4079:
4076:
4070:
4066:
4060:
4055:
4054:
4048:
4027:
4023:
4019:
4016:
3991:
3987:
3983:
3978:
3974:
3970:
3965:
3961:
3957:
3952:
3948:
3941:
3935:
3929:
3926:
3918:
3915:
3896:
3892:
3888:
3885:
3873:
3857:
3851:
3848:
3845:
3839:
3833:
3827:
3822:
3818:
3810:
3794:
3788:
3784:
3780:
3775:
3771:
3767:
3762:
3758:
3754:
3749:
3745:
3740:
3736:
3732:
3728:
3725:
3718:
3717:
3716:
3714:
3709:
3705:
3701:
3697:
3693:
3689:
3684:
3680:
3679:hypersurfaces
3676:
3672:
3668:
3667:conic section
3663:
3661:
3656:
3650:
3643:
3641:
3636:
3632:
3626:
3622:
3617:
3612:
3608:
3604:
3599:
3593:
3589:
3579:
3576:
3572:
3569:
3564:
3561:
3555:
3545:
3541:
3534:
3530:
3525:
3516:
3512:
3507:
3506:bivector form
3502:
3495:
3491:
3485:
3483:
3479:
3475:
3474:biquaternions
3471:
3467:
3453:
3450:water tower,
3449:
3442:
3437:
3433:
3429:
3422:
3417:
3413:
3409:
3405:
3399:
3394:
3390:
3386:
3382:
3375:
3370:
3366:
3362:
3358:
3351:
3346:
3342:
3338:
3334:
3331:
3327:
3326:cooling tower
3324:
3317:
3312:
3308:
3304:
3300:
3294:
3289:
3285:
3281:
3277:
3273:
3267:
3262:
3258:
3254:
3248:
3243:
3239:
3235:
3229:
3224:
3220:
3216:
3212:
3206:
3201:
3197:
3193:
3189:
3185:
3179:
3174:
3170:
3166:
3160:
3155:
3151:
3147:
3143:
3139:
3138:cooling tower
3136:
3129:
3124:
3120:
3116:
3109:
3104:
3101:
3100:
3099:
3097:
3093:
3089:
3085:
3081:
3075:
3065:
3063:
3059:
3053:
3041:
3029:
3017:
3002:
2990:
2978:
2966:
2956:
2949:
2941:
2938:
2934:
2929:
2910:
2907:
2901:
2895:
2889:
2883:
2872:
2867:
2850:
2847:
2844:
2841:
2837:
2833:
2827:
2822:
2818:
2814:
2811:
2808:
2803:
2798:
2795:
2792:
2788:
2783:
2779:
2775:
2769:
2764:
2760:
2756:
2753:
2750:
2745:
2740:
2736:
2731:
2727:
2721:
2715:
2707:
2703:
2693:
2691:
2686:
2661:
2658:
2655:
2647:
2644:
2641:
2638:
2635:
2632:
2631:
2630:
2616:
2613:
2610:
2603:
2599:
2593:
2589:
2583:
2576:
2572:
2566:
2562:
2556:
2549:
2545:
2539:
2535:
2528:
2525:
2522:
2515:
2511:
2505:
2501:
2495:
2488:
2484:
2478:
2474:
2468:
2461:
2457:
2451:
2447:
2425:
2423:
2419:
2415:
2413:
2405:
2387:
2383:
2374:
2371:
2353:
2349:
2340:
2323:
2319:
2311:
2310:not intersect
2307:
2304:
2300:
2297:either in an
2282:
2278:
2269:
2268:
2267:
2265:
2249:
2246:
2243:
2238:
2234:
2230:
2225:
2221:
2217:
2212:
2208:
2201:
2196:
2192:
2183:
2179:
2170:
2162:
2153:
2151:
2143:
2125:
2121:
2112:
2109:
2091:
2087:
2078:
2075:
2057:
2053:
2044:
2041:
2023:
2019:
2010:
2007:
1989:
1985:
1976:
1975:
1974:
1960:
1957:
1952:
1948:
1944:
1939:
1935:
1931:
1926:
1922:
1918:
1913:
1909:
1897:
1887:
1885:
1881:
1880:
1874:
1872:
1868:
1864:
1862:
1843:
1838:
1834:
1811:
1806:
1802:
1781:
1778:
1775:
1766:
1749:
1746:
1743:
1740:
1737:
1734:
1728:
1720:
1717:
1713:
1705:
1699:
1696:
1689:
1686:
1683:
1680:
1673:
1670:
1667:
1664:
1661:
1655:
1650:
1647:
1644:
1639:
1633:
1626:
1623:
1620:
1617:
1610:
1607:
1604:
1601:
1595:
1590:
1584:
1573:
1568:
1563:
1559:
1550:
1536:
1533:
1526:
1522:
1516:
1512:
1506:
1499:
1495:
1489:
1485:
1479:
1472:
1468:
1462:
1458:
1443:
1439:
1438:
1422:
1405:
1401:
1396:
1392:
1368:
1364:
1359:
1355:
1331:
1327:
1322:
1318:
1309:
1306:of A are the
1305:
1300:
1295:
1290:
1288:
1283:
1277:
1272:
1267:
1250:
1247:
1244:
1233:
1222:
1202:
1185:
1174:
1160:
1157:
1133:
1130:
1127:
1119:
1104:
1101:
1098:
1090:
1075:
1072:
1069:
1061:
1060:
1059:
1045:
1038:
1035:
1028:
1025:
1022:
1017:
1014:
1009:
1005:
999:
992:
989:
986:
981:
978:
973:
969:
963:
956:
952:
946:
943:
940:
912:
899:
891:
887:
870:
867:
864:
861:
858:
855:
853:
848:
841:
838:
835:
832:
829:
826:
823:
820:
818:
813:
806:
803:
800:
797:
794:
791:
788:
785:
783:
778:
762:
756:
739:
736:
733:
730:
727:
725:
720:
713:
710:
707:
704:
701:
698:
695:
692:
690:
685:
678:
675:
672:
669:
666:
663:
660:
657:
655:
650:
634:
628:
626:
621:
613:
607:
602:
598:
589:
580:
578:
574:
570:
566:
557:
555:
551:
547:
546:tangent plane
543:
539:
535:
531:
522:
520:
516:
512:
496:
491:
487:
483:
478:
474:
464:
451:
448:
441:
437:
431:
427:
421:
414:
410:
404:
400:
394:
387:
383:
377:
373:
362:
346:
343:
340:
333:
329:
323:
319:
313:
306:
302:
296:
292:
286:
279:
275:
269:
265:
242:
239:
236:
229:
225:
219:
215:
209:
202:
198:
192:
188:
182:
175:
171:
165:
161:
150:
145:
143:
139:
136:
135:perpendicular
132:
128:
124:
120:
116:
112:
108:
105:, that is, a
104:
99:
97:
93:
89:
85:
81:
77:
73:
69:
65:
53:
49:
46:
41:
37:
32:
28:
22:
4417:
4398:
4379:
4360:
4333:
4319:
4312:
4286:
4278:
4239:
4230:
4220:
4205:
4200:
4188:
4176:
4168:
4163:
4074:
4068:
4064:
4058:
4053:quasi-sphere
4050:
4044:
3913:
3877:
3707:
3703:
3699:
3695:
3691:
3687:
3685:with points
3664:
3654:
3648:
3645:
3639:
3634:
3630:
3624:
3620:
3615:
3610:
3606:
3602:
3597:
3591:
3587:
3584:
3577:
3574:
3570:
3567:
3559:
3553:
3550:
3543:
3539:
3532:
3528:
3514:
3510:
3505:
3500:
3493:
3489:
3487:
3469:
3463:
3428:Canton Tower
3186:'s James S.
3077:
3058:quasi-sphere
3057:
3055:
3039:
3027:
3015:
3000:
2988:
2976:
2964:
2954:
2947:
2943:
2939:
2932:
2927:
2926:is called a
2865:
2699:
2683:Whereas the
2682:
2645:
2639:
2633:
2436:
2422:projectively
2421:
2417:
2416:
2409:
2403:
2369:
2309:
2302:
2298:
2181:
2177:
2175:
2147:
2141:
2107:
2073:
2039:
2005:
1895:
1893:
1879:projectively
1877:
1875:
1859:
1767:
1551:
1447:
1298:
1294:eigenvectors
1291:
1281:
1275:
1265:
1183:
1180:
1149:
904:
760:
757:
632:
629:
619:
611:
605:
594:
576:
568:
564:
558:
554:doubly ruled
533:
529:
523:
518:
514:
465:
146:
100:
87:
71:
67:
61:
3870:which is a
3638:will be an
3580:− 1 )
3478:quaternions
3135:Van Iterson
2928:hyperboloid
1869:around its
1308:reciprocals
1304:eigenvalues
125:, having a
88:hyperboloid
47:in between
4434:Categories
4155:References
4125:Paraboloid
3872:hyperplane
3706:) ∈
3508:, such as
3381:Killesberg
3361:Manchester
3339:-Uentrop,
2433:Symmetries
1863:'s theorem
1425:Properties
536:. It is a
361:asymptotic
115:polynomial
4419:MathWorld
4400:MathWorld
4381:MathWorld
4362:MathWorld
4115:Ellipsoid
4020:∩
3889:∩
3385:Stuttgart
3280:Ciechanów
3192:St. Louis
2812:⋯
2780:−
2754:⋯
2679:Curvature
2614:−
2584:−
2496:−
2404:hyperbola
2264:hyperbola
2247:−
2231:−
2142:hyperbola
1945:−
1867:hyperbola
1844:−
1750:π
1744:≤
1741:α
1738:≤
1721:∈
1697:±
1690:α
1687:
1674:α
1671:
1662:−
1651:⋅
1627:α
1624:
1611:α
1608:
1569:±
1564:α
1507:−
1234:−
1203:−
1026:
990:
868:
859:±
842:θ
839:
830:
807:θ
804:
795:
737:
714:θ
711:
702:
679:θ
676:
667:
610:[0, 2
556:surface.
422:−
344:−
314:−
210:−
80:hyperbola
74:, is the
4450:Quadrics
4445:Surfaces
4320:Geometry
4104:, Russia
4082:See also
4078:is one.
4067:∈
3595:, where
3531:−
3488:... the
3464:In 1853
3404:BMW Welt
3343:, 1983.
3323:THTR-300
3276:toroidal
3196:Missouri
3142:DSM Emma
3064:below).
2871:constant
2370:parabola
2301:or in a
2074:parabola
1768:In case
521:-axis).
123:cylinder
111:zero set
92:scalings
64:geometry
4332:(1961)
4311:(1948)
4285:(1995)
4134:Regulus
4098:Shukhov
3535:+ 1 = 0
3518:√
3496:+ 1 = 0
3434:, 2010.
3414:, 2007.
3412:Germany
3391:, 2001.
3389:Germany
3367:, 1999.
3365:England
3341:Germany
3330:thorium
3309:, 1982.
3303:Toronto
3286:, 1972.
3259:, 1970.
3240:, 1968.
3221:, 1967.
3219:England
3198:, 1963.
3171:, 1963.
3146:Heerlen
3121:, 1911.
3119:Ukraine
2953:, ...,
2869:is any
2418:Remark:
2299:ellipse
2006:ellipse
1287:vectors
601:azimuth
107:surface
76:surface
4296:
4212:
3945:
3939:
3843:
3837:
3482:sphere
3452:France
3408:Munich
3307:Canada
3284:Poland
3278:tank,
3257:Brazil
3152:, 1918
3094:, and
2893:
2887:
2205:
2004:in an
1906:
1753:
1732:
1711:
1271:matrix
1263:where
603:angle
577:convex
131:planes
4102:Vyksa
4051:unit
3878:Then
3669:as a
3504:to a
3432:China
3274:with
3169:Japan
2863:When
2690:model
2402:in a
2368:in a
2303:point
2140:in a
2106:in a
2072:in a
2038:in a
1269:is a
623:into
550:lines
511:up to
121:or a
113:of a
4294:ISBN
4210:ISBN
4049:, a
3660:norm
3557:and
3446:The
3426:The
3379:The
3355:The
3337:Hamm
3321:The
3113:The
3050:= −1
3011:= −1
2845:<
2629:are
1861:Wren
1384:and
1292:The
1285:are
1273:and
1120:For
1102:<
1091:For
1073:>
1062:For
865:cosh
827:sinh
792:sinh
734:sinh
699:cosh
664:cosh
119:cone
86:. A
66:, a
3808:and
3690:= (
3681:in
3623:||
3590:||
3573:= (
3546:= 0
3335:in
3144:in
3140:of
2935:= 0
2642:and
2414:).
2178:not
2152:).
1826:or
1684:cos
1668:sin
1621:sin
1605:cos
1296:of
1023:sin
987:cos
836:sin
801:cos
708:sin
673:cos
255:or
62:In
4436::
4416:.
4397:.
4378:.
4359:.
4127:/
3702:,
3698:,
3694:,
3633:+
3609:+
3544:στ
3537:,
3520:−1
3513:+
3484::
3430:,
3410:,
3387:,
3363:,
3359:,
3305:,
3301:,
3282:,
3255:,
3236:,
3217:,
3194:,
3190:,
3167:,
3148:,
3117:,
3098:.
3038:−
3026:+
2999:−
2987:+
2975:+
2937:.
2708::
2250:1.
1886:.
1347:,
1289:.
1279:,
763:∈
635:∈
627::
608:∈
561:−1
526:+1
452:0.
347:1.
144:.
98:.
4422:.
4403:.
4384:.
4365:.
4340:.
4326:.
4262:.
4257:4
4253:M
4075:x
4069:X
4065:x
4059:X
4028:r
4024:H
4017:Q
3997:}
3992:2
3988:y
3984:+
3979:2
3975:x
3971:=
3966:2
3962:z
3958:+
3953:2
3949:w
3942::
3936:p
3933:{
3930:=
3927:Q
3914:r
3897:r
3893:H
3886:P
3874:.
3858:,
3855:}
3852:r
3849:=
3846:w
3840::
3834:p
3831:{
3828:=
3823:r
3819:H
3795:}
3789:2
3785:z
3781:+
3776:2
3772:y
3768:+
3763:2
3759:x
3755:=
3750:2
3746:w
3741::
3737:p
3733:{
3729:=
3726:P
3708:R
3704:z
3700:y
3696:x
3692:w
3688:p
3655:T
3649:S
3635:τ
3631:σ
3625:λ
3621:τ
3611:τ
3607:σ
3598:λ
3592:λ
3588:σ
3582:.
3578:σ
3575:T
3571:τ
3568:T
3560:τ
3554:σ
3548:;
3542:.
3540:S
3533:τ
3529:σ
3515:τ
3511:σ
3501:ρ
3494:ρ
3454:.
3046:3
3040:y
3034:2
3028:y
3022:1
3016:y
3007:4
3001:y
2995:3
2989:y
2983:2
2977:y
2971:1
2965:y
2960:)
2958:4
2955:y
2951:1
2948:y
2946:(
2933:c
2914:}
2911:c
2908:=
2905:)
2902:x
2899:(
2896:q
2890::
2884:x
2881:{
2866:c
2851:.
2848:n
2842:k
2838:,
2834:)
2828:2
2823:n
2819:x
2815:+
2809:+
2804:2
2799:1
2796:+
2793:k
2789:x
2784:(
2776:)
2770:2
2765:k
2761:x
2757:+
2751:+
2746:2
2741:1
2737:x
2732:(
2728:=
2725:)
2722:x
2719:(
2716:q
2662:b
2659:=
2656:a
2617:1
2611:=
2604:2
2600:c
2594:2
2590:z
2577:2
2573:b
2567:2
2563:y
2557:+
2550:2
2546:a
2540:2
2536:x
2529:,
2526:1
2523:=
2516:2
2512:c
2506:2
2502:z
2489:2
2485:b
2479:2
2475:y
2469:+
2462:2
2458:a
2452:2
2448:x
2406:.
2388:2
2384:H
2372:,
2354:2
2350:H
2338:,
2324:2
2320:H
2283:2
2279:H
2244:=
2239:2
2235:z
2226:2
2222:y
2218:+
2213:2
2209:x
2202::
2197:2
2193:H
2144:.
2126:1
2122:H
2110:,
2092:1
2088:H
2076:,
2058:1
2054:H
2042:,
2024:1
2020:H
2008:,
1990:1
1986:H
1961:1
1958:=
1953:2
1949:z
1940:2
1936:y
1932:+
1927:2
1923:x
1919::
1914:1
1910:H
1839:0
1835:g
1812:+
1807:0
1803:g
1782:b
1779:=
1776:a
1747:2
1735:0
1729:,
1725:R
1718:t
1714:,
1706:)
1700:c
1681:b
1665:a
1656:(
1648:t
1645:+
1640:)
1634:0
1618:b
1602:a
1596:(
1591:=
1588:)
1585:t
1582:(
1578:x
1574::
1560:g
1537:1
1534:=
1527:2
1523:c
1517:2
1513:z
1500:2
1496:b
1490:2
1486:y
1480:+
1473:2
1469:a
1463:2
1459:x
1444:.
1406:2
1402:c
1397:/
1393:1
1369:2
1365:b
1360:/
1356:1
1332:2
1328:a
1323:/
1319:1
1299:A
1282:v
1276:x
1266:A
1251:,
1248:1
1245:=
1242:)
1238:v
1230:x
1226:(
1223:A
1217:T
1212:)
1207:v
1199:x
1195:(
1184:v
1161:s
1158:c
1134:0
1131:=
1128:d
1105:0
1099:d
1076:0
1070:d
1046:)
1039:s
1036:c
1029:t
1018:d
1015:+
1010:2
1006:s
1000:b
993:t
982:d
979:+
974:2
970:s
964:a
957:(
953:=
950:)
947:t
944:,
941:s
938:(
934:x
913:z
871:v
862:c
856:=
849:z
833:v
824:b
821:=
814:y
798:v
789:a
786:=
779:x
761:v
740:v
731:c
728:=
721:z
705:v
696:b
693:=
686:y
670:v
661:a
658:=
651:x
633:v
620:v
614:)
612:π
606:θ
519:y
515:x
497:.
492:2
488:b
484:=
479:2
475:a
449:=
442:2
438:c
432:2
428:z
415:2
411:b
405:2
401:y
395:+
388:2
384:a
378:2
374:x
341:=
334:2
330:c
324:2
320:z
307:2
303:b
297:2
293:y
287:+
280:2
276:a
270:2
266:x
243:,
240:1
237:=
230:2
226:c
220:2
216:z
203:2
199:b
193:2
189:y
183:+
176:2
172:a
166:2
162:x
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