3383:
1702:
416:
2844:
4421:
2385:
24:
408:
4432:
3782:
2374:
1258:
4130:
4183:
4746:
82:
If a quadric contains a circle, then every intersection of the quadric with a plane parallel to this circle is also a circle, provided it contains at least two points. Except for spheres, the circles contained in a quadric, if any, are all parallel to one of two fixed planes (which are equal in the
2112:
931:
5232:
2833:
3550:
2155:
1038:
3372:
62:
contains circles as sections with planes that are orthogonal to its axis; it does not contain any other circles, if it is not a sphere. More hidden are circles on other quadrics, such as tri-axial ellipsoids, elliptic cylinders, etc. Nevertheless, it is true that:
3539:
2646:
1846:
3942:
5421:
5012:
3931:
3091:
487:
4416:{\displaystyle \left(-{\frac {a^{2}}{b^{2}c}}+{\frac {1}{c}}\right)\;y^{2}+\left({\frac {a^{2}}{c^{3}}}+{\frac {1}{c}}\right)\;z^{2}=0\ \quad \leftrightarrow \quad z=\pm {\frac {c}{b}}{\sqrt {\frac {a^{2}-b^{2}}{a^{2}+c^{2}}}}\;y\ .}
4553:
594:
4568:
304:). A real plane contains these two points if and only if it is perpendicular to the axis of revolution. Thus the circular sections are the plane sections by a plane perpendicular to the axis, that have at least two real points.
1929:
748:
2493:
5023:
2689:
3777:{\displaystyle -{\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}+{\frac {(z+c)^{2}}{c^{2}}}=1\quad \leftrightarrow \quad \ -{\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}+{\frac {2z}{c}}=0\ .}
2942:
325:
is thus a circle, if these two points are one of these two pairs of complex conjugate points on the ombilic. Each of these pairs defines a real line (passing through the points), which is the intersection of
2369:{\displaystyle \left({\frac {a^{2}}{b^{2}}}-1\right)\;y^{2}-\left({\frac {a^{2}}{c^{2}}}+1\right)\;z^{2}=0\ \quad \leftrightarrow \quad z=\pm {\frac {c}{b}}{\sqrt {\frac {a^{2}-b^{2}}{a^{2}+c^{2}}}}\;y\ ,\ }
1253:{\displaystyle \left({\frac {b^{2}}{a^{2}}}-1\right)\;x^{2}+\left({\frac {b^{2}}{c^{2}}}-1\right)\;z^{2}=0\ \quad \leftrightarrow \quad z=\pm {\tfrac {c}{a}}{\sqrt {\tfrac {a^{2}-b^{2}}{b^{2}-c^{2}}}}\;x\ ,}
3206:
1419:
3257:
1678:
5291:
3400:
2504:
1921:
4824:
710:
4125:{\displaystyle \left(-{\frac {r}{a^{2}}}+{\frac {1}{c}}\right)\;x^{2}+\left(-{\frac {r}{b^{2}}}+{\frac {1}{c}}\right)\;y^{2}+\left({\frac {r}{c^{2}}}+{\frac {1}{c}}\right)z^{2}=0\ .}
4175:
1719:
225:
3249:
1547:
715:
The sphere's radius has to be chosen such that the intersection with the ellipsoid is contained in two planes through the origin. Multiplication of the ellipsoid's equation by
635:
5299:
1500:
375:
If the intersection of a plane and a quadric is a circle, then any parallel plane, that contains at least two points of the quadric, intersects the quadric in a circle, too.
4862:
1293:
3793:
2953:
2681:
2147:
1580:
1030:
998:
966:
422:
4854:
740:
1448:
1325:
4741:{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-(z-1)^{2}=0\quad \leftrightarrow \quad {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-z^{2}+2z=1.}
4449:
498:
2107:{\displaystyle \left({\frac {r^{2}}{a^{2}}}-1\right)\;x^{2}+\left({\frac {r^{2}}{b^{2}}}-1\right)\;y^{2}-\left({\frac {r^{2}}{c^{2}}}+1\right)\;z^{2}=0\ .}
926:{\displaystyle \left({\frac {r^{2}}{a^{2}}}-1\right)\;x^{2}+\left({\frac {r^{2}}{b^{2}}}-1\right)\;y^{2}+\left({\frac {r^{2}}{c^{2}}}-1\right)\;z^{2}=0\ .}
2402:
1298:
The diagram gives an impression of more common intersections between a sphere and an ellipsoid and highlights the exceptional circular case (blue).
165:
1334:
Turning the ellipsoid around the y-axis such that one of the two circles (blue) lies in the x-y-plane results in a new equation of the ellipsoid:
5227:{\displaystyle {\frac {a^{2}-b^{2}}{b^{2}}}\;y^{2}-(1+a^{2})\left(z-{\frac {a^{2}}{1+a^{2}}}\right)^{2}=a^{2}-{\frac {a^{4}}{1+a^{2}}}-r^{2}\ .}
2828:{\displaystyle \left({\frac {a^{2}}{b^{2}}}-1\right)\;y^{2}-z^{2}=0\ \quad \leftrightarrow \quad z=\pm {\frac {\sqrt {a^{2}-b^{2}}}{b}}\;y\ .\ }
5543:
5522:
5501:
5478:
5456:
5441:
411:
tri-axial ellipsoid with circular sections (blue and green) and the auxiliary sphere (red), which intersects the quadric in the blue circles
2861:
5559:
3367:{\displaystyle \left({\frac {b}{a}}-1\right)\;y^{2}-z^{2}=0\quad \leftrightarrow \quad z=\pm {\sqrt {\frac {b-a}{a}}}\;y\ .\ }
3102:
1340:
5558:
H. Wiener, P. Treutlein: Models of a tri-axial ellipsoid and an elliptic paraboloid using circular sections (see p. 15)
1588:
5240:
319:
is a curve of degree two, its intersection with the plane at infinity consists of two points, possibly equal. The curve
3534:{\displaystyle -{\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1\ ,\quad a>b\ ,\ c>0\ ,}
2641:{\displaystyle \left({\frac {r^{2}}{a^{2}}}-1\right)\;x^{2}+\left({\frac {r^{2}}{b^{2}}}-1\right)\;y^{2}-z^{2}=0\ .}
358:
In order to find the planes, which contain circular sections of a given quadric, one uses the following statements:
5584:
5237:
In order to get the equation of a pair of planes, the right part of the equation has to be zero, which is true for
1854:
4757:
643:
55:
section of the quadric, as this circle is the intersection with the quadric of the plane containing the circle.
1841:{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1\ ,\quad a>b\ ,c>0}
1301:
If the values of the semi-axes are approaching, the two pencils of planes (and circles) approach either. For
102:
of the quadric, as it is done in the following sections. They may also be characterised and studied by using
5579:
4138:
3787:
Analogously to the paraboloid case one chooses a sphere containing the origin with center on the z-axis:
171:
5416:{\displaystyle z={\frac {a^{2}}{1+a^{2}}}\pm {\frac {1}{b}}{\sqrt {\frac {a^{2}-b^{2}}{1+a^{2}}}}\;y\ .}
3214:
1689:
1505:
293:
243:
59:
602:
1453:
5007:{\displaystyle \left({\frac {a^{2}}{b^{2}}}-1\right)\;y^{2}-(1+a^{2})\;z^{2}+2a^{2}z=a^{2}-r^{2}\ .}
106:
283:, its intersection with the plane at infinity is the ombilic, and all plane sections are circles.
3926:{\displaystyle x^{2}+y^{2}+(z-r)^{2}=r^{2}\quad \leftrightarrow \quad x^{2}+y^{2}+z^{2}-2zr=0\ .}
3086:{\displaystyle x^{2}+y^{2}+(z-r)^{2}=r^{2}\quad \leftrightarrow \quad x^{2}+y^{2}+z^{2}-2rz=0\ .}
1266:
247:
103:
2947:
one chooses a sphere containing the vertex (origin) and with center on the axis (z-axis) :
482:{\displaystyle c<{\color {seagreen}r_{1}}<{\color {blue}b}<{\color {purple}r_{2}}<a}
5574:
5539:
5518:
5497:
5474:
5452:
5437:
297:
161:
99:
2654:
2120:
1552:
1003:
971:
939:
936:
This equation describes a pair of planes, if one of the 3 coefficients is zero. In case of
147:
52:
4832:
718:
58:
Any plane section of a sphere is a circular section, if it contains at least 2 points. Any
143:
87:
17:
1427:
1304:
4548:{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-z^{2}=0\ ,\quad a>b\ ,}
369:
are contained in a pair of planes, then the intersection curve consists of two circles.
151:
589:{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1}
5568:
260:
are real (intersection of a real plane with real lines). Thus the plane sections of
1263:
because only in this case the remaining coefficients have different signs (due to:
1000:
the equation is only fulfilled by either the x-axis or the z-axis. Only in case of
301:
71:
Equivalently, all quadric surfaces contain circles except parabolic and hyperbolic
3382:
2488:{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1\ ,\quad a>b\ ,}
313:
with the ombilic consists of two different pairs of complex conjugate points. As
3391:
1710:
1701:
415:
48:
398:
that are parallel to the detected ones deliver the remaining circular sections.
2852:
76:
1688:
or a point or the empty set. Center and radius of the circle can be found be
2843:
44:
236:
with the plane at infinity consists of one or two real lines, that is when
2384:
350:
is parallel to one of two directions defined by these lines at infinity).
23:
2393:
332:
with the plane at infinity. Thus, one has a circular section if and only
251:
72:
407:
267:
40:
4431:
3544:
is shifted at first such that one vertex is the origin (s. diagram):
280:
36:
2937:{\displaystyle ax^{2}+by^{2}-z=0\ ,\quad a{\color {red}{<}}b\ ,}
1502:, which has to be the equation of a circle. This is only true, if
414:
67:
Any quadric surface which contains ellipses contains circles, too.
22:
4440:
1549:. The intersection of the ellipsoid by a plane with equation
1327:
all the planes are orthogonal to the z-axis (rotation axis).
3936:
After elimination of the linear parts one gets the equation
3096:
After elimination of the linear parts one gets the equation
230:
The first case to be considered is when the intersection of
98:
The circular sections of a quadric may be computed from the
1851:
analogously one gets for the intersection with the sphere
296:, its intersection with the ombilic consists of a pair of
3201:{\displaystyle (2ra-1)\;x^{2}+(2rb-1)\;y^{2}-z^{2}=0\ .}
1414:{\displaystyle Ax^{2}+By^{2}+Cz^{2}+D{\color {red}xz}=E}
5251:
4149:
3225:
1184:
1171:
5302:
5243:
5026:
4865:
4835:
4760:
4571:
4452:
4186:
4141:
3945:
3796:
3553:
3403:
3260:
3217:
3105:
2956:
2864:
2692:
2657:
2507:
2405:
2158:
2123:
1932:
1857:
1722:
1591:
1555:
1508:
1456:
1430:
1343:
1307:
1269:
1041:
1006:
974:
942:
751:
721:
646:
605:
501:
425:
174:
160:
is a circle if and only if its intersection with the
5494:
Physikalisches Wörterbuch: Zwei Teile in Einem Band.
4751:
Now a sphere with center at the origin is suitable:
344:
contains one of these lines at infinity (that is if
1673:{\displaystyle A(x^{2}+y^{2})+Dz_{0}x=E-Cz_{0}^{2}}
391:
that intersects the quadric in a pair of planes and
5415:
5286:{\displaystyle r={\tfrac {a}{\sqrt {1+a^{2}}}}\ .}
5285:
5226:
5006:
4848:
4818:
4740:
4547:
4415:
4169:
4124:
3925:
3776:
3533:
3366:
3243:
3200:
3085:
2936:
2827:
2675:
2640:
2487:
2368:
2141:
2106:
1915:
1840:
1672:
1574:
1541:
1494:
1442:
1413:
1319:
1287:
1252:
1024:
992:
960:
925:
734:
704:
629:
588:
481:
219:
5471:Elemente der Linearen Algebra und der Analysis.
1582:, (parallel to the x-y-plane) has the equation
354:Determination of circular sections of a quadric
742:and subtracting the sphere's equation yields:
8:
637:one uses an auxiliary sphere with equation
383:for the detection of circular sections is:
5515:Die Festigkeitserscheinungen der Kristalle.
1916:{\displaystyle \ x^{2}+y^{2}+z^{2}=r^{2}\ }
27:tri-axial ellipsoid with a circular section
5403:
5066:
5017:In this case completing the square gives:
4942:
4906:
4819:{\displaystyle x^{2}+y^{2}+z^{2}=r^{2}\ .}
4403:
4303:
4242:
4046:
3989:
3351:
3287:
3162:
3127:
2812:
2733:
2602:
2548:
2353:
2253:
2199:
2081:
2027:
1973:
1240:
1136:
1082:
1032:one gets a pair of planes with equation
900:
846:
792:
705:{\displaystyle x^{2}+y^{2}+z^{2}=r^{2}\ .}
5393:
5375:
5362:
5354:
5344:
5332:
5315:
5309:
5301:
5267:
5250:
5242:
5212:
5196:
5179:
5173:
5164:
5151:
5137:
5120:
5114:
5093:
5071:
5058:
5047:
5034:
5027:
5025:
4992:
4979:
4963:
4947:
4933:
4911:
4887:
4877:
4871:
4864:
4840:
4834:
4804:
4791:
4778:
4765:
4759:
4717:
4702:
4692:
4686:
4675:
4665:
4659:
4642:
4615:
4605:
4599:
4588:
4578:
4572:
4570:
4511:
4496:
4486:
4480:
4469:
4459:
4453:
4451:
4393:
4380:
4368:
4355:
4347:
4337:
4308:
4288:
4277:
4267:
4261:
4247:
4227:
4212:
4201:
4195:
4185:
4155:
4148:
4140:
4104:
4085:
4074:
4065:
4051:
4031:
4020:
4011:
3994:
3974:
3963:
3954:
3944:
3893:
3880:
3867:
3852:
3839:
3814:
3801:
3795:
3747:
3736:
3726:
3720:
3709:
3699:
3693:
3682:
3672:
3666:
3641:
3630:
3611:
3600:
3590:
3584:
3573:
3563:
3557:
3552:
3477:
3467:
3461:
3450:
3440:
3434:
3423:
3413:
3407:
3402:
3331:
3305:
3292:
3266:
3259:
3224:
3216:
3180:
3167:
3132:
3104:
3053:
3040:
3027:
3012:
2999:
2974:
2961:
2955:
2918:
2916:
2888:
2872:
2863:
2800:
2787:
2780:
2751:
2738:
2714:
2704:
2698:
2691:
2656:
2620:
2607:
2583:
2573:
2567:
2553:
2529:
2519:
2513:
2506:
2449:
2439:
2433:
2422:
2412:
2406:
2404:
2343:
2330:
2318:
2305:
2297:
2287:
2258:
2234:
2224:
2218:
2204:
2180:
2170:
2164:
2157:
2122:
2086:
2062:
2052:
2046:
2032:
2008:
1998:
1992:
1978:
1954:
1944:
1938:
1931:
1904:
1891:
1878:
1865:
1856:
1793:
1783:
1777:
1766:
1756:
1750:
1739:
1729:
1723:
1721:
1664:
1659:
1634:
1615:
1602:
1590:
1566:
1554:
1507:
1480:
1464:
1455:
1429:
1395:
1383:
1367:
1351:
1342:
1306:
1268:
1229:
1216:
1204:
1191:
1182:
1170:
1141:
1117:
1107:
1101:
1087:
1063:
1053:
1047:
1040:
1005:
973:
941:
905:
881:
871:
865:
851:
827:
817:
811:
797:
773:
763:
757:
750:
726:
720:
690:
677:
664:
651:
645:
604:
572:
562:
556:
545:
535:
529:
518:
508:
502:
500:
465:
459:
449:
438:
432:
424:
365:If the common points of a quadric with a
254:. In this case the points at infinity of
205:
192:
179:
173:
118:be the intersection of a quadric surface
4430:
3381:
2842:
2383:
1700:
406:
307:In the other cases, the intersection of
5485:
5449:Analytical Geometry of Three Dimensions
433:
142:are surfaces in the three-dimensional
5536:Lehrbuch der Allgemeinen Metallkunde.
4170:{\displaystyle r={\tfrac {a^{2}}{c}}}
7:
5451:, Cambridge University Press, 1959,
5436:, Cambridge University Press, 2010,
3386:elliptical hyperboloid of two sheets
3378:Elliptical hyperboloid of two sheets
460:
154:. Under these hypotheses, the curve
4558:is shifted such that the vertex is
1697:Elliptical hyperboloid of one sheet
220:{\displaystyle x^{2}+y^{2}+z^{2}=0}
168:(the curve at infinity of equation
5466:Göschen-Verlag, 1962, p. 143.
3244:{\displaystyle r={\tfrac {1}{2a}}}
1542:{\displaystyle A=B\neq 0,\ E>0}
450:
419:Ellipsoid intersected by spheres:
83:case of a quadric of revolution).
14:
3251:one gets a pair of planes :
2917:
1396:
492:For the ellipsoid with equation
338:has at least two real points and
5434:Principles of Geometry, Volume 3
630:{\displaystyle a>b>c>0}
5538:Springer-Verlag, Berlin, 1950,
4658:
4654:
4529:
4327:
4323:
3862:
3858:
3659:
3655:
3497:
3321:
3317:
3022:
3018:
2912:
2770:
2766:
2469:
2277:
2273:
1813:
1495:{\displaystyle Ax^{2}+By^{2}=E}
1160:
1156:
5099:
5080:
4939:
4920:
4655:
4639:
4626:
4324:
3859:
3836:
3823:
3656:
3627:
3614:
3318:
3159:
3141:
3124:
3106:
3019:
2996:
2983:
2767:
2274:
1621:
1595:
1157:
86:Circular sections are used in
1:
5517:Springer-Verlag, Wien, 1949,
3394:of two sheets with equation
5473:Spektrum, Heidelberg, 2009,
1713:of one sheet with equation
146:, which are extended to the
4177:one gets a pair of planes:
2683:one gets a pair of planes:
2149:one gets a pair of planes:
1288:{\displaystyle a>b>c}
266:cannot be circles (neither
5601:
5293:The solution for z gives:
4562:the origin (see diagram):
1684:This equation describes a
15:
94:Using projective geometry
1705:hyperboloid of one sheet
16:Not to be confused with
5496:Springer-Verlag, 1952,
5469:H. Scheid, W. Schwarz:
2676:{\displaystyle \ r=a\ }
2142:{\displaystyle \ r=a\ }
1575:{\displaystyle z=z_{0}}
1025:{\displaystyle \ r=b\ }
993:{\displaystyle \ r=c\ }
961:{\displaystyle \ r=a\ }
5464:Analytische Geometrie.
5447:D. M. Y. Sommerville:
5417:
5287:
5228:
5008:
4850:
4820:
4742:
4549:
4436:
4417:
4171:
4126:
3927:
3778:
3535:
3387:
3368:
3245:
3202:
3087:
2938:
2848:
2829:
2677:
2642:
2498:one gets the equation
2489:
2389:
2370:
2143:
2108:
1917:
1842:
1706:
1674:
1576:
1543:
1496:
1444:
1415:
1331:Proof of property (P)
1321:
1289:
1254:
1026:
994:
962:
927:
736:
706:
631:
590:
489:
483:
412:
221:
28:
5418:
5288:
5229:
5009:
4851:
4849:{\displaystyle x^{2}}
4821:
4743:
4550:
4434:
4418:
4172:
4127:
3928:
3779:
3536:
3385:
3369:
3246:
3203:
3088:
2939:
2847:elliptical paraboloid
2846:
2839:Elliptical paraboloid
2830:
2678:
2643:
2490:
2387:
2371:
2144:
2109:
1918:
1843:
1704:
1690:completing the square
1675:
1577:
1544:
1497:
1445:
1416:
1322:
1290:
1255:
1027:
995:
963:
928:
737:
735:{\displaystyle r^{2}}
707:
632:
591:
484:
418:
410:
294:surface of revolution
244:hyperbolic paraboloid
222:
60:quadric of revolution
26:
5300:
5241:
5024:
4863:
4833:
4758:
4569:
4450:
4184:
4139:
3943:
3794:
3551:
3401:
3258:
3215:
3103:
2954:
2862:
2690:
2655:
2505:
2403:
2156:
2121:
1930:
1855:
1720:
1589:
1553:
1506:
1454:
1428:
1341:
1305:
1267:
1039:
1004:
972:
940:
749:
719:
644:
603:
499:
423:
172:
43:surface (such as an
2851:For the elliptical
2392:For the elliptical
2388:elliptical cylinder
2380:Elliptical cylinder
1669:
1443:{\displaystyle z=0}
1320:{\displaystyle a=b}
599:and the semi-axes
403:Tri-axial ellipsoid
252:hyperbolic cylinder
164:is included in the
130:. In this section,
107:projective geometry
51:). It is a special
5544:978-3-642-52-993-1
5462:K. P. Grotemeyer:
5413:
5283:
5275:
5224:
5004:
4846:
4816:
4738:
4545:
4437:
4413:
4167:
4165:
4122:
3923:
3774:
3531:
3388:
3364:
3241:
3239:
3198:
3083:
2934:
2923:
2849:
2825:
2673:
2638:
2485:
2390:
2366:
2139:
2104:
1913:
1838:
1707:
1670:
1655:
1572:
1539:
1492:
1440:
1411:
1403:
1332:
1317:
1285:
1250:
1237:
1180:
1022:
990:
958:
923:
732:
702:
627:
586:
490:
479:
471:
454:
444:
413:
300:points (which are
248:parabolic cylinder
217:
29:
5585:Analytic geometry
5523:978-3-211-80120-8
5502:978-3-662-12707-0
5479:978-3-8274-1971-2
5457:978-1-316-60190-7
5442:978-1-108-01779-4
5409:
5401:
5400:
5352:
5339:
5279:
5274:
5273:
5220:
5203:
5144:
5064:
5000:
4893:
4812:
4708:
4681:
4621:
4594:
4541:
4525:
4502:
4475:
4409:
4401:
4400:
4345:
4322:
4296:
4283:
4235:
4222:
4164:
4118:
4093:
4080:
4039:
4026:
3982:
3969:
3919:
3770:
3760:
3742:
3715:
3688:
3662:
3647:
3606:
3579:
3527:
3515:
3509:
3493:
3483:
3456:
3429:
3363:
3357:
3349:
3348:
3274:
3238:
3194:
3079:
2930:
2908:
2824:
2818:
2810:
2806:
2765:
2720:
2672:
2660:
2634:
2589:
2535:
2481:
2465:
2455:
2428:
2365:
2359:
2351:
2350:
2295:
2272:
2240:
2186:
2138:
2126:
2100:
2068:
2014:
1960:
1912:
1860:
1825:
1809:
1799:
1772:
1745:
1529:
1330:
1246:
1238:
1236:
1179:
1155:
1123:
1069:
1021:
1009:
989:
977:
957:
945:
919:
887:
833:
779:
698:
578:
551:
524:
298:complex conjugate
162:plane at infinity
100:implicit equation
5592:
5547:
5532:
5526:
5511:
5505:
5492:W. H. Westphal:
5490:
5422:
5420:
5419:
5414:
5407:
5402:
5399:
5398:
5397:
5381:
5380:
5379:
5367:
5366:
5356:
5355:
5353:
5345:
5340:
5338:
5337:
5336:
5320:
5319:
5310:
5292:
5290:
5289:
5284:
5277:
5276:
5272:
5271:
5256:
5252:
5233:
5231:
5230:
5225:
5218:
5217:
5216:
5204:
5202:
5201:
5200:
5184:
5183:
5174:
5169:
5168:
5156:
5155:
5150:
5146:
5145:
5143:
5142:
5141:
5125:
5124:
5115:
5098:
5097:
5076:
5075:
5065:
5063:
5062:
5053:
5052:
5051:
5039:
5038:
5028:
5013:
5011:
5010:
5005:
4998:
4997:
4996:
4984:
4983:
4968:
4967:
4952:
4951:
4938:
4937:
4916:
4915:
4905:
4901:
4894:
4892:
4891:
4882:
4881:
4872:
4855:
4853:
4852:
4847:
4845:
4844:
4825:
4823:
4822:
4817:
4810:
4809:
4808:
4796:
4795:
4783:
4782:
4770:
4769:
4747:
4745:
4744:
4739:
4722:
4721:
4709:
4707:
4706:
4697:
4696:
4687:
4682:
4680:
4679:
4670:
4669:
4660:
4647:
4646:
4622:
4620:
4619:
4610:
4609:
4600:
4595:
4593:
4592:
4583:
4582:
4573:
4554:
4552:
4551:
4546:
4539:
4523:
4516:
4515:
4503:
4501:
4500:
4491:
4490:
4481:
4476:
4474:
4473:
4464:
4463:
4454:
4422:
4420:
4419:
4414:
4407:
4402:
4399:
4398:
4397:
4385:
4384:
4374:
4373:
4372:
4360:
4359:
4349:
4348:
4346:
4338:
4320:
4313:
4312:
4302:
4298:
4297:
4289:
4284:
4282:
4281:
4272:
4271:
4262:
4252:
4251:
4241:
4237:
4236:
4228:
4223:
4221:
4217:
4216:
4206:
4205:
4196:
4176:
4174:
4173:
4168:
4166:
4160:
4159:
4150:
4131:
4129:
4128:
4123:
4116:
4109:
4108:
4099:
4095:
4094:
4086:
4081:
4079:
4078:
4066:
4056:
4055:
4045:
4041:
4040:
4032:
4027:
4025:
4024:
4012:
3999:
3998:
3988:
3984:
3983:
3975:
3970:
3968:
3967:
3955:
3932:
3930:
3929:
3924:
3917:
3898:
3897:
3885:
3884:
3872:
3871:
3857:
3856:
3844:
3843:
3819:
3818:
3806:
3805:
3783:
3781:
3780:
3775:
3768:
3761:
3756:
3748:
3743:
3741:
3740:
3731:
3730:
3721:
3716:
3714:
3713:
3704:
3703:
3694:
3689:
3687:
3686:
3677:
3676:
3667:
3660:
3648:
3646:
3645:
3636:
3635:
3634:
3612:
3607:
3605:
3604:
3595:
3594:
3585:
3580:
3578:
3577:
3568:
3567:
3558:
3540:
3538:
3537:
3532:
3525:
3513:
3507:
3491:
3484:
3482:
3481:
3472:
3471:
3462:
3457:
3455:
3454:
3445:
3444:
3435:
3430:
3428:
3427:
3418:
3417:
3408:
3373:
3371:
3370:
3365:
3361:
3355:
3350:
3344:
3333:
3332:
3310:
3309:
3297:
3296:
3286:
3282:
3275:
3267:
3250:
3248:
3247:
3242:
3240:
3237:
3226:
3207:
3205:
3204:
3199:
3192:
3185:
3184:
3172:
3171:
3137:
3136:
3092:
3090:
3089:
3084:
3077:
3058:
3057:
3045:
3044:
3032:
3031:
3017:
3016:
3004:
3003:
2979:
2978:
2966:
2965:
2943:
2941:
2940:
2935:
2928:
2924:
2922:
2906:
2893:
2892:
2877:
2876:
2834:
2832:
2831:
2826:
2822:
2816:
2811:
2805:
2804:
2792:
2791:
2782:
2781:
2763:
2756:
2755:
2743:
2742:
2732:
2728:
2721:
2719:
2718:
2709:
2708:
2699:
2682:
2680:
2679:
2674:
2670:
2658:
2647:
2645:
2644:
2639:
2632:
2625:
2624:
2612:
2611:
2601:
2597:
2590:
2588:
2587:
2578:
2577:
2568:
2558:
2557:
2547:
2543:
2536:
2534:
2533:
2524:
2523:
2514:
2494:
2492:
2491:
2486:
2479:
2463:
2456:
2454:
2453:
2444:
2443:
2434:
2429:
2427:
2426:
2417:
2416:
2407:
2375:
2373:
2372:
2367:
2363:
2357:
2352:
2349:
2348:
2347:
2335:
2334:
2324:
2323:
2322:
2310:
2309:
2299:
2298:
2296:
2288:
2270:
2263:
2262:
2252:
2248:
2241:
2239:
2238:
2229:
2228:
2219:
2209:
2208:
2198:
2194:
2187:
2185:
2184:
2175:
2174:
2165:
2148:
2146:
2145:
2140:
2136:
2124:
2113:
2111:
2110:
2105:
2098:
2091:
2090:
2080:
2076:
2069:
2067:
2066:
2057:
2056:
2047:
2037:
2036:
2026:
2022:
2015:
2013:
2012:
2003:
2002:
1993:
1983:
1982:
1972:
1968:
1961:
1959:
1958:
1949:
1948:
1939:
1922:
1920:
1919:
1914:
1910:
1909:
1908:
1896:
1895:
1883:
1882:
1870:
1869:
1858:
1847:
1845:
1844:
1839:
1823:
1807:
1800:
1798:
1797:
1788:
1787:
1778:
1773:
1771:
1770:
1761:
1760:
1751:
1746:
1744:
1743:
1734:
1733:
1724:
1679:
1677:
1676:
1671:
1668:
1663:
1639:
1638:
1620:
1619:
1607:
1606:
1581:
1579:
1578:
1573:
1571:
1570:
1548:
1546:
1545:
1540:
1527:
1501:
1499:
1498:
1493:
1485:
1484:
1469:
1468:
1449:
1447:
1446:
1441:
1420:
1418:
1417:
1412:
1404:
1388:
1387:
1372:
1371:
1356:
1355:
1326:
1324:
1323:
1318:
1294:
1292:
1291:
1286:
1259:
1257:
1256:
1251:
1244:
1239:
1235:
1234:
1233:
1221:
1220:
1210:
1209:
1208:
1196:
1195:
1185:
1183:
1181:
1172:
1153:
1146:
1145:
1135:
1131:
1124:
1122:
1121:
1112:
1111:
1102:
1092:
1091:
1081:
1077:
1070:
1068:
1067:
1058:
1057:
1048:
1031:
1029:
1028:
1023:
1019:
1007:
999:
997:
996:
991:
987:
975:
967:
965:
964:
959:
955:
943:
932:
930:
929:
924:
917:
910:
909:
899:
895:
888:
886:
885:
876:
875:
866:
856:
855:
845:
841:
834:
832:
831:
822:
821:
812:
802:
801:
791:
787:
780:
778:
777:
768:
767:
758:
741:
739:
738:
733:
731:
730:
711:
709:
708:
703:
696:
695:
694:
682:
681:
669:
668:
656:
655:
636:
634:
633:
628:
595:
593:
592:
587:
579:
577:
576:
567:
566:
557:
552:
550:
549:
540:
539:
530:
525:
523:
522:
513:
512:
503:
488:
486:
485:
480:
472:
470:
469:
455:
445:
443:
442:
349:
343:
337:
331:
324:
318:
312:
291:
278:
265:
259:
241:
235:
226:
224:
223:
218:
210:
209:
197:
196:
184:
183:
159:
148:projective space
141:
135:
129:
123:
117:
33:circular section
5600:
5599:
5595:
5594:
5593:
5591:
5590:
5589:
5565:
5564:
5555:
5550:
5533:
5529:
5512:
5508:
5491:
5487:
5429:
5389:
5382:
5371:
5358:
5357:
5328:
5321:
5311:
5298:
5297:
5263:
5239:
5238:
5208:
5192:
5185:
5175:
5160:
5133:
5126:
5116:
5107:
5103:
5102:
5089:
5067:
5054:
5043:
5030:
5029:
5022:
5021:
4988:
4975:
4959:
4943:
4929:
4907:
4883:
4873:
4870:
4866:
4861:
4860:
4836:
4831:
4830:
4829:Elimination of
4800:
4787:
4774:
4761:
4756:
4755:
4713:
4698:
4688:
4671:
4661:
4638:
4611:
4601:
4584:
4574:
4567:
4566:
4507:
4492:
4482:
4465:
4455:
4448:
4447:
4443:with equation
4439:The elliptical
4435:elliptical cone
4429:
4427:Elliptical cone
4389:
4376:
4375:
4364:
4351:
4350:
4304:
4273:
4263:
4260:
4256:
4243:
4208:
4207:
4197:
4191:
4187:
4182:
4181:
4151:
4137:
4136:
4100:
4070:
4064:
4060:
4047:
4016:
4007:
4003:
3990:
3959:
3950:
3946:
3941:
3940:
3889:
3876:
3863:
3848:
3835:
3810:
3797:
3792:
3791:
3749:
3732:
3722:
3705:
3695:
3678:
3668:
3637:
3626:
3613:
3596:
3586:
3569:
3559:
3549:
3548:
3473:
3463:
3446:
3436:
3419:
3409:
3399:
3398:
3380:
3334:
3301:
3288:
3265:
3261:
3256:
3255:
3230:
3213:
3212:
3176:
3163:
3128:
3101:
3100:
3049:
3036:
3023:
3008:
2995:
2970:
2957:
2952:
2951:
2884:
2868:
2860:
2859:
2841:
2796:
2783:
2747:
2734:
2710:
2700:
2697:
2693:
2688:
2687:
2653:
2652:
2616:
2603:
2579:
2569:
2566:
2562:
2549:
2525:
2515:
2512:
2508:
2503:
2502:
2445:
2435:
2418:
2408:
2401:
2400:
2382:
2339:
2326:
2325:
2314:
2301:
2300:
2254:
2230:
2220:
2217:
2213:
2200:
2176:
2166:
2163:
2159:
2154:
2153:
2119:
2118:
2082:
2058:
2048:
2045:
2041:
2028:
2004:
1994:
1991:
1987:
1974:
1950:
1940:
1937:
1933:
1928:
1927:
1900:
1887:
1874:
1861:
1853:
1852:
1789:
1779:
1762:
1752:
1735:
1725:
1718:
1717:
1699:
1694:
1630:
1611:
1598:
1587:
1586:
1562:
1551:
1550:
1504:
1503:
1476:
1460:
1452:
1451:
1426:
1425:
1379:
1363:
1347:
1339:
1338:
1303:
1302:
1265:
1264:
1225:
1212:
1211:
1200:
1187:
1186:
1137:
1113:
1103:
1100:
1096:
1083:
1059:
1049:
1046:
1042:
1037:
1036:
1002:
1001:
970:
969:
938:
937:
901:
877:
867:
864:
860:
847:
823:
813:
810:
806:
793:
769:
759:
756:
752:
747:
746:
722:
717:
716:
686:
673:
660:
647:
642:
641:
601:
600:
568:
558:
541:
531:
514:
504:
497:
496:
461:
434:
421:
420:
405:
356:
345:
339:
333:
327:
320:
314:
308:
287:
274:
261:
255:
237:
231:
201:
188:
175:
170:
169:
155:
152:complex numbers
144:Euclidean space
137:
131:
125:
119:
113:
96:
88:crystallography
75:and hyperbolic
31:In geometry, a
21:
18:circular sector
12:
11:
5:
5598:
5596:
5588:
5587:
5582:
5580:Conic sections
5577:
5567:
5566:
5563:
5562:
5554:
5553:External links
5551:
5549:
5548:
5546:, p. 355.
5527:
5506:
5504:, p. 350.
5484:
5483:
5482:
5481:, p. 132.
5467:
5460:
5459:, p. 204.
5445:
5428:
5425:
5424:
5423:
5412:
5406:
5396:
5392:
5388:
5385:
5378:
5374:
5370:
5365:
5361:
5351:
5348:
5343:
5335:
5331:
5327:
5324:
5318:
5314:
5308:
5305:
5282:
5270:
5266:
5262:
5259:
5255:
5249:
5246:
5235:
5234:
5223:
5215:
5211:
5207:
5199:
5195:
5191:
5188:
5182:
5178:
5172:
5167:
5163:
5159:
5154:
5149:
5140:
5136:
5132:
5129:
5123:
5119:
5113:
5110:
5106:
5101:
5096:
5092:
5088:
5085:
5082:
5079:
5074:
5070:
5061:
5057:
5050:
5046:
5042:
5037:
5033:
5015:
5014:
5003:
4995:
4991:
4987:
4982:
4978:
4974:
4971:
4966:
4962:
4958:
4955:
4950:
4946:
4941:
4936:
4932:
4928:
4925:
4922:
4919:
4914:
4910:
4904:
4900:
4897:
4890:
4886:
4880:
4876:
4869:
4843:
4839:
4827:
4826:
4815:
4807:
4803:
4799:
4794:
4790:
4786:
4781:
4777:
4773:
4768:
4764:
4749:
4748:
4737:
4734:
4731:
4728:
4725:
4720:
4716:
4712:
4705:
4701:
4695:
4691:
4685:
4678:
4674:
4668:
4664:
4657:
4653:
4650:
4645:
4641:
4637:
4634:
4631:
4628:
4625:
4618:
4614:
4608:
4604:
4598:
4591:
4587:
4581:
4577:
4556:
4555:
4544:
4538:
4535:
4532:
4528:
4522:
4519:
4514:
4510:
4506:
4499:
4495:
4489:
4485:
4479:
4472:
4468:
4462:
4458:
4428:
4425:
4424:
4423:
4412:
4406:
4396:
4392:
4388:
4383:
4379:
4371:
4367:
4363:
4358:
4354:
4344:
4341:
4336:
4333:
4330:
4326:
4319:
4316:
4311:
4307:
4301:
4295:
4292:
4287:
4280:
4276:
4270:
4266:
4259:
4255:
4250:
4246:
4240:
4234:
4231:
4226:
4220:
4215:
4211:
4204:
4200:
4194:
4190:
4163:
4158:
4154:
4147:
4144:
4133:
4132:
4121:
4115:
4112:
4107:
4103:
4098:
4092:
4089:
4084:
4077:
4073:
4069:
4063:
4059:
4054:
4050:
4044:
4038:
4035:
4030:
4023:
4019:
4015:
4010:
4006:
4002:
3997:
3993:
3987:
3981:
3978:
3973:
3966:
3962:
3958:
3953:
3949:
3934:
3933:
3922:
3916:
3913:
3910:
3907:
3904:
3901:
3896:
3892:
3888:
3883:
3879:
3875:
3870:
3866:
3861:
3855:
3851:
3847:
3842:
3838:
3834:
3831:
3828:
3825:
3822:
3817:
3813:
3809:
3804:
3800:
3785:
3784:
3773:
3767:
3764:
3759:
3755:
3752:
3746:
3739:
3735:
3729:
3725:
3719:
3712:
3708:
3702:
3698:
3692:
3685:
3681:
3675:
3671:
3665:
3658:
3654:
3651:
3644:
3640:
3633:
3629:
3625:
3622:
3619:
3616:
3610:
3603:
3599:
3593:
3589:
3583:
3576:
3572:
3566:
3562:
3556:
3542:
3541:
3530:
3524:
3521:
3518:
3512:
3506:
3503:
3500:
3496:
3490:
3487:
3480:
3476:
3470:
3466:
3460:
3453:
3449:
3443:
3439:
3433:
3426:
3422:
3416:
3412:
3406:
3379:
3376:
3375:
3374:
3360:
3354:
3347:
3343:
3340:
3337:
3330:
3327:
3324:
3320:
3316:
3313:
3308:
3304:
3300:
3295:
3291:
3285:
3281:
3278:
3273:
3270:
3264:
3236:
3233:
3229:
3223:
3220:
3209:
3208:
3197:
3191:
3188:
3183:
3179:
3175:
3170:
3166:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3140:
3135:
3131:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3094:
3093:
3082:
3076:
3073:
3070:
3067:
3064:
3061:
3056:
3052:
3048:
3043:
3039:
3035:
3030:
3026:
3021:
3015:
3011:
3007:
3002:
2998:
2994:
2991:
2988:
2985:
2982:
2977:
2973:
2969:
2964:
2960:
2945:
2944:
2933:
2927:
2921:
2915:
2911:
2905:
2902:
2899:
2896:
2891:
2887:
2883:
2880:
2875:
2871:
2867:
2855:with equation
2840:
2837:
2836:
2835:
2821:
2815:
2809:
2803:
2799:
2795:
2790:
2786:
2779:
2776:
2773:
2769:
2762:
2759:
2754:
2750:
2746:
2741:
2737:
2731:
2727:
2724:
2717:
2713:
2707:
2703:
2696:
2669:
2666:
2663:
2649:
2648:
2637:
2631:
2628:
2623:
2619:
2615:
2610:
2606:
2600:
2596:
2593:
2586:
2582:
2576:
2572:
2565:
2561:
2556:
2552:
2546:
2542:
2539:
2532:
2528:
2522:
2518:
2511:
2496:
2495:
2484:
2478:
2475:
2472:
2468:
2462:
2459:
2452:
2448:
2442:
2438:
2432:
2425:
2421:
2415:
2411:
2396:with equation
2381:
2378:
2377:
2376:
2362:
2356:
2346:
2342:
2338:
2333:
2329:
2321:
2317:
2313:
2308:
2304:
2294:
2291:
2286:
2283:
2280:
2276:
2269:
2266:
2261:
2257:
2251:
2247:
2244:
2237:
2233:
2227:
2223:
2216:
2212:
2207:
2203:
2197:
2193:
2190:
2183:
2179:
2173:
2169:
2162:
2135:
2132:
2129:
2115:
2114:
2103:
2097:
2094:
2089:
2085:
2079:
2075:
2072:
2065:
2061:
2055:
2051:
2044:
2040:
2035:
2031:
2025:
2021:
2018:
2011:
2007:
2001:
1997:
1990:
1986:
1981:
1977:
1971:
1967:
1964:
1957:
1953:
1947:
1943:
1936:
1907:
1903:
1899:
1894:
1890:
1886:
1881:
1877:
1873:
1868:
1864:
1849:
1848:
1837:
1834:
1831:
1828:
1822:
1819:
1816:
1812:
1806:
1803:
1796:
1792:
1786:
1782:
1776:
1769:
1765:
1759:
1755:
1749:
1742:
1738:
1732:
1728:
1698:
1695:
1682:
1681:
1667:
1662:
1658:
1654:
1651:
1648:
1645:
1642:
1637:
1633:
1629:
1626:
1623:
1618:
1614:
1610:
1605:
1601:
1597:
1594:
1569:
1565:
1561:
1558:
1538:
1535:
1532:
1526:
1523:
1520:
1517:
1514:
1511:
1491:
1488:
1483:
1479:
1475:
1472:
1467:
1463:
1459:
1439:
1436:
1433:
1422:
1421:
1410:
1407:
1402:
1399:
1394:
1391:
1386:
1382:
1378:
1375:
1370:
1366:
1362:
1359:
1354:
1350:
1346:
1329:
1316:
1313:
1310:
1284:
1281:
1278:
1275:
1272:
1261:
1260:
1249:
1243:
1232:
1228:
1224:
1219:
1215:
1207:
1203:
1199:
1194:
1190:
1178:
1175:
1169:
1166:
1163:
1159:
1152:
1149:
1144:
1140:
1134:
1130:
1127:
1120:
1116:
1110:
1106:
1099:
1095:
1090:
1086:
1080:
1076:
1073:
1066:
1062:
1056:
1052:
1045:
1018:
1015:
1012:
986:
983:
980:
954:
951:
948:
934:
933:
922:
916:
913:
908:
904:
898:
894:
891:
884:
880:
874:
870:
863:
859:
854:
850:
844:
840:
837:
830:
826:
820:
816:
809:
805:
800:
796:
790:
786:
783:
776:
772:
766:
762:
755:
729:
725:
713:
712:
701:
693:
689:
685:
680:
676:
672:
667:
663:
659:
654:
650:
626:
623:
620:
617:
614:
611:
608:
597:
596:
585:
582:
575:
571:
565:
561:
555:
548:
544:
538:
534:
528:
521:
517:
511:
507:
478:
475:
468:
464:
458:
453:
448:
441:
437:
431:
428:
404:
401:
400:
399:
392:
377:
376:
370:
355:
352:
216:
213:
208:
204:
200:
195:
191:
187:
182:
178:
95:
92:
69:
68:
13:
10:
9:
6:
4:
3:
2:
5597:
5586:
5583:
5581:
5578:
5576:
5573:
5572:
5570:
5560:
5557:
5556:
5552:
5545:
5541:
5537:
5531:
5528:
5525:, p. 87.
5524:
5520:
5516:
5510:
5507:
5503:
5499:
5495:
5489:
5486:
5480:
5476:
5472:
5468:
5465:
5461:
5458:
5454:
5450:
5446:
5443:
5439:
5435:
5432:H. F. Baker:
5431:
5430:
5426:
5410:
5404:
5394:
5390:
5386:
5383:
5376:
5372:
5368:
5363:
5359:
5349:
5346:
5341:
5333:
5329:
5325:
5322:
5316:
5312:
5306:
5303:
5296:
5295:
5294:
5280:
5268:
5264:
5260:
5257:
5253:
5247:
5244:
5221:
5213:
5209:
5205:
5197:
5193:
5189:
5186:
5180:
5176:
5170:
5165:
5161:
5157:
5152:
5147:
5138:
5134:
5130:
5127:
5121:
5117:
5111:
5108:
5104:
5094:
5090:
5086:
5083:
5077:
5072:
5068:
5059:
5055:
5048:
5044:
5040:
5035:
5031:
5020:
5019:
5018:
5001:
4993:
4989:
4985:
4980:
4976:
4972:
4969:
4964:
4960:
4956:
4953:
4948:
4944:
4934:
4930:
4926:
4923:
4917:
4912:
4908:
4902:
4898:
4895:
4888:
4884:
4878:
4874:
4867:
4859:
4858:
4857:
4841:
4837:
4813:
4805:
4801:
4797:
4792:
4788:
4784:
4779:
4775:
4771:
4766:
4762:
4754:
4753:
4752:
4735:
4732:
4729:
4726:
4723:
4718:
4714:
4710:
4703:
4699:
4693:
4689:
4683:
4676:
4672:
4666:
4662:
4651:
4648:
4643:
4635:
4632:
4629:
4623:
4616:
4612:
4606:
4602:
4596:
4589:
4585:
4579:
4575:
4565:
4564:
4563:
4561:
4542:
4536:
4533:
4530:
4526:
4520:
4517:
4512:
4508:
4504:
4497:
4493:
4487:
4483:
4477:
4470:
4466:
4460:
4456:
4446:
4445:
4444:
4442:
4433:
4426:
4410:
4404:
4394:
4390:
4386:
4381:
4377:
4369:
4365:
4361:
4356:
4352:
4342:
4339:
4334:
4331:
4328:
4317:
4314:
4309:
4305:
4299:
4293:
4290:
4285:
4278:
4274:
4268:
4264:
4257:
4253:
4248:
4244:
4238:
4232:
4229:
4224:
4218:
4213:
4209:
4202:
4198:
4192:
4188:
4180:
4179:
4178:
4161:
4156:
4152:
4145:
4142:
4119:
4113:
4110:
4105:
4101:
4096:
4090:
4087:
4082:
4075:
4071:
4067:
4061:
4057:
4052:
4048:
4042:
4036:
4033:
4028:
4021:
4017:
4013:
4008:
4004:
4000:
3995:
3991:
3985:
3979:
3976:
3971:
3964:
3960:
3956:
3951:
3947:
3939:
3938:
3937:
3920:
3914:
3911:
3908:
3905:
3902:
3899:
3894:
3890:
3886:
3881:
3877:
3873:
3868:
3864:
3853:
3849:
3845:
3840:
3832:
3829:
3826:
3820:
3815:
3811:
3807:
3802:
3798:
3790:
3789:
3788:
3771:
3765:
3762:
3757:
3753:
3750:
3744:
3737:
3733:
3727:
3723:
3717:
3710:
3706:
3700:
3696:
3690:
3683:
3679:
3673:
3669:
3663:
3652:
3649:
3642:
3638:
3631:
3623:
3620:
3617:
3608:
3601:
3597:
3591:
3587:
3581:
3574:
3570:
3564:
3560:
3554:
3547:
3546:
3545:
3528:
3522:
3519:
3516:
3510:
3504:
3501:
3498:
3494:
3488:
3485:
3478:
3474:
3468:
3464:
3458:
3451:
3447:
3441:
3437:
3431:
3424:
3420:
3414:
3410:
3404:
3397:
3396:
3395:
3393:
3384:
3377:
3358:
3352:
3345:
3341:
3338:
3335:
3328:
3325:
3322:
3314:
3311:
3306:
3302:
3298:
3293:
3289:
3283:
3279:
3276:
3271:
3268:
3262:
3254:
3253:
3252:
3234:
3231:
3227:
3221:
3218:
3195:
3189:
3186:
3181:
3177:
3173:
3168:
3164:
3156:
3153:
3150:
3147:
3144:
3138:
3133:
3129:
3121:
3118:
3115:
3112:
3109:
3099:
3098:
3097:
3080:
3074:
3071:
3068:
3065:
3062:
3059:
3054:
3050:
3046:
3041:
3037:
3033:
3028:
3024:
3013:
3009:
3005:
3000:
2992:
2989:
2986:
2980:
2975:
2971:
2967:
2962:
2958:
2950:
2949:
2948:
2931:
2925:
2919:
2913:
2909:
2903:
2900:
2897:
2894:
2889:
2885:
2881:
2878:
2873:
2869:
2865:
2858:
2857:
2856:
2854:
2845:
2838:
2819:
2813:
2807:
2801:
2797:
2793:
2788:
2784:
2777:
2774:
2771:
2760:
2757:
2752:
2748:
2744:
2739:
2735:
2729:
2725:
2722:
2715:
2711:
2705:
2701:
2694:
2686:
2685:
2684:
2667:
2664:
2661:
2635:
2629:
2626:
2621:
2617:
2613:
2608:
2604:
2598:
2594:
2591:
2584:
2580:
2574:
2570:
2563:
2559:
2554:
2550:
2544:
2540:
2537:
2530:
2526:
2520:
2516:
2509:
2501:
2500:
2499:
2482:
2476:
2473:
2470:
2466:
2460:
2457:
2450:
2446:
2440:
2436:
2430:
2423:
2419:
2413:
2409:
2399:
2398:
2397:
2395:
2386:
2379:
2360:
2354:
2344:
2340:
2336:
2331:
2327:
2319:
2315:
2311:
2306:
2302:
2292:
2289:
2284:
2281:
2278:
2267:
2264:
2259:
2255:
2249:
2245:
2242:
2235:
2231:
2225:
2221:
2214:
2210:
2205:
2201:
2195:
2191:
2188:
2181:
2177:
2171:
2167:
2160:
2152:
2151:
2150:
2133:
2130:
2127:
2101:
2095:
2092:
2087:
2083:
2077:
2073:
2070:
2063:
2059:
2053:
2049:
2042:
2038:
2033:
2029:
2023:
2019:
2016:
2009:
2005:
1999:
1995:
1988:
1984:
1979:
1975:
1969:
1965:
1962:
1955:
1951:
1945:
1941:
1934:
1926:
1925:
1924:
1923:the equation
1905:
1901:
1897:
1892:
1888:
1884:
1879:
1875:
1871:
1866:
1862:
1835:
1832:
1829:
1826:
1820:
1817:
1814:
1810:
1804:
1801:
1794:
1790:
1784:
1780:
1774:
1767:
1763:
1757:
1753:
1747:
1740:
1736:
1730:
1726:
1716:
1715:
1714:
1712:
1703:
1696:
1693:
1691:
1687:
1665:
1660:
1656:
1652:
1649:
1646:
1643:
1640:
1635:
1631:
1627:
1624:
1616:
1612:
1608:
1603:
1599:
1592:
1585:
1584:
1583:
1567:
1563:
1559:
1556:
1536:
1533:
1530:
1524:
1521:
1518:
1515:
1512:
1509:
1489:
1486:
1481:
1477:
1473:
1470:
1465:
1461:
1457:
1437:
1434:
1431:
1408:
1405:
1400:
1397:
1392:
1389:
1384:
1380:
1376:
1373:
1368:
1364:
1360:
1357:
1352:
1348:
1344:
1337:
1336:
1335:
1328:
1314:
1311:
1308:
1299:
1296:
1282:
1279:
1276:
1273:
1270:
1247:
1241:
1230:
1226:
1222:
1217:
1213:
1205:
1201:
1197:
1192:
1188:
1176:
1173:
1167:
1164:
1161:
1150:
1147:
1142:
1138:
1132:
1128:
1125:
1118:
1114:
1108:
1104:
1097:
1093:
1088:
1084:
1078:
1074:
1071:
1064:
1060:
1054:
1050:
1043:
1035:
1034:
1033:
1016:
1013:
1010:
984:
981:
978:
952:
949:
946:
920:
914:
911:
906:
902:
896:
892:
889:
882:
878:
872:
868:
861:
857:
852:
848:
842:
838:
835:
828:
824:
818:
814:
807:
803:
798:
794:
788:
784:
781:
774:
770:
764:
760:
753:
745:
744:
743:
727:
723:
699:
691:
687:
683:
678:
674:
670:
665:
661:
657:
652:
648:
640:
639:
638:
624:
621:
618:
615:
612:
609:
606:
583:
580:
573:
569:
563:
559:
553:
546:
542:
536:
532:
526:
519:
515:
509:
505:
495:
494:
493:
476:
473:
466:
462:
456:
451:
446:
439:
435:
429:
426:
417:
409:
402:
397:
393:
390:
386:
385:
384:
382:
374:
371:
368:
364:
361:
360:
359:
353:
351:
348:
342:
336:
330:
323:
317:
311:
305:
303:
302:double points
299:
295:
290:
284:
282:
277:
271:
269:
264:
258:
253:
249:
245:
240:
234:
228:
214:
211:
206:
202:
198:
193:
189:
185:
180:
176:
167:
163:
158:
153:
149:
145:
140:
134:
128:
122:
116:
110:
108:
105:
101:
93:
91:
89:
84:
80:
78:
74:
66:
65:
64:
61:
56:
54:
50:
46:
42:
38:
34:
25:
19:
5535:
5530:
5514:
5513:H. Tertsch:
5509:
5493:
5488:
5470:
5463:
5448:
5433:
5236:
5016:
4828:
4750:
4559:
4557:
4438:
4134:
3935:
3786:
3543:
3389:
3210:
3095:
2946:
2850:
2650:
2497:
2391:
2116:
1850:
1708:
1685:
1683:
1423:
1333:
1300:
1297:
1262:
935:
714:
598:
491:
395:
388:
380:
378:
372:
366:
362:
357:
346:
340:
334:
328:
321:
315:
309:
306:
288:
285:
275:
272:
262:
256:
242:is either a
238:
232:
229:
156:
138:
132:
126:
124:and a plane
120:
114:
111:
97:
85:
81:
70:
57:
32:
30:
5534:G. Masing:
3392:hyperboloid
1711:hyperboloid
77:paraboloids
49:hyperboloid
5569:Categories
5427:References
4135:Only for
2853:paraboloid
387:1) Find a
379:Hence the
5369:−
5342:±
5206:−
5171:−
5112:−
5078:−
5041:−
4986:−
4918:−
4896:−
4711:−
4656:↔
4633:−
4624:−
4505:−
4362:−
4335:±
4325:↔
4193:−
4009:−
3952:−
3900:−
3860:↔
3830:−
3691:−
3664:−
3657:↔
3582:−
3555:−
3432:−
3405:−
3339:−
3329:±
3319:↔
3299:−
3277:−
3211:Only for
3174:−
3154:−
3119:−
3060:−
3020:↔
2990:−
2895:−
2794:−
2778:±
2768:↔
2745:−
2723:−
2651:Only for
2614:−
2592:−
2538:−
2312:−
2285:±
2275:↔
2211:−
2189:−
2117:Only for
2039:−
2017:−
1963:−
1775:−
1709:For the
1650:−
1519:≠
1450:one gets
1223:−
1198:−
1168:±
1158:↔
1126:−
1072:−
890:−
836:−
782:−
150:over the
104:synthetic
73:cylinders
45:ellipsoid
5575:Quadrics
4856:yields:
2394:cylinder
381:strategy
268:ellipses
394:2) The
166:ombilic
41:quadric
5561:(PDF).
5542:
5521:
5500:
5477:
5455:
5440:
5408:
5278:
5219:
4999:
4811:
4540:
4524:
4408:
4321:
4117:
3918:
3769:
3661:
3526:
3514:
3508:
3492:
3362:
3356:
3193:
3078:
2929:
2907:
2823:
2817:
2764:
2671:
2659:
2633:
2480:
2464:
2364:
2358:
2271:
2137:
2125:
2099:
1911:
1859:
1824:
1808:
1686:circle
1528:
1245:
1154:
1020:
1008:
988:
976:
956:
944:
918:
697:
396:planes
389:sphere
367:sphere
281:sphere
37:circle
292:is a
279:is a
250:or a
53:plane
39:on a
35:is a
5540:ISBN
5519:ISBN
5498:ISBN
5475:ISBN
5453:ISBN
5438:ISBN
4534:>
4441:cone
3520:>
3502:>
3390:The
2920:<
2474:>
1833:>
1818:>
1534:>
1424:For
1280:>
1274:>
622:>
616:>
610:>
474:<
457:<
447:<
430:<
373:(P:)
363:(S:)
246:, a
136:and
112:Let
4560:not
1295:).
968:or
286:If
273:If
270:).
227:).
47:or
5571::
4736:1.
1692:.
109:.
90:.
79:.
5444:.
5411:.
5405:y
5395:2
5391:a
5387:+
5384:1
5377:2
5373:b
5364:2
5360:a
5350:b
5347:1
5334:2
5330:a
5326:+
5323:1
5317:2
5313:a
5307:=
5304:z
5281:.
5269:2
5265:a
5261:+
5258:1
5254:a
5248:=
5245:r
5222:.
5214:2
5210:r
5198:2
5194:a
5190:+
5187:1
5181:4
5177:a
5166:2
5162:a
5158:=
5153:2
5148:)
5139:2
5135:a
5131:+
5128:1
5122:2
5118:a
5109:z
5105:(
5100:)
5095:2
5091:a
5087:+
5084:1
5081:(
5073:2
5069:y
5060:2
5056:b
5049:2
5045:b
5036:2
5032:a
5002:.
4994:2
4990:r
4981:2
4977:a
4973:=
4970:z
4965:2
4961:a
4957:2
4954:+
4949:2
4945:z
4940:)
4935:2
4931:a
4927:+
4924:1
4921:(
4913:2
4909:y
4903:)
4899:1
4889:2
4885:b
4879:2
4875:a
4868:(
4842:2
4838:x
4814:.
4806:2
4802:r
4798:=
4793:2
4789:z
4785:+
4780:2
4776:y
4772:+
4767:2
4763:x
4733:=
4730:z
4727:2
4724:+
4719:2
4715:z
4704:2
4700:b
4694:2
4690:y
4684:+
4677:2
4673:a
4667:2
4663:x
4652:0
4649:=
4644:2
4640:)
4636:1
4630:z
4627:(
4617:2
4613:b
4607:2
4603:y
4597:+
4590:2
4586:a
4580:2
4576:x
4543:,
4537:b
4531:a
4527:,
4521:0
4518:=
4513:2
4509:z
4498:2
4494:b
4488:2
4484:y
4478:+
4471:2
4467:a
4461:2
4457:x
4411:.
4405:y
4395:2
4391:c
4387:+
4382:2
4378:a
4370:2
4366:b
4357:2
4353:a
4343:b
4340:c
4332:=
4329:z
4318:0
4315:=
4310:2
4306:z
4300:)
4294:c
4291:1
4286:+
4279:3
4275:c
4269:2
4265:a
4258:(
4254:+
4249:2
4245:y
4239:)
4233:c
4230:1
4225:+
4219:c
4214:2
4210:b
4203:2
4199:a
4189:(
4162:c
4157:2
4153:a
4146:=
4143:r
4120:.
4114:0
4111:=
4106:2
4102:z
4097:)
4091:c
4088:1
4083:+
4076:2
4072:c
4068:r
4062:(
4058:+
4053:2
4049:y
4043:)
4037:c
4034:1
4029:+
4022:2
4018:b
4014:r
4005:(
4001:+
3996:2
3992:x
3986:)
3980:c
3977:1
3972:+
3965:2
3961:a
3957:r
3948:(
3921:.
3915:0
3912:=
3909:r
3906:z
3903:2
3895:2
3891:z
3887:+
3882:2
3878:y
3874:+
3869:2
3865:x
3854:2
3850:r
3846:=
3841:2
3837:)
3833:r
3827:z
3824:(
3821:+
3816:2
3812:y
3808:+
3803:2
3799:x
3772:.
3766:0
3763:=
3758:c
3754:z
3751:2
3745:+
3738:2
3734:c
3728:2
3724:z
3718:+
3711:2
3707:b
3701:2
3697:y
3684:2
3680:a
3674:2
3670:x
3653:1
3650:=
3643:2
3639:c
3632:2
3628:)
3624:c
3621:+
3618:z
3615:(
3609:+
3602:2
3598:b
3592:2
3588:y
3575:2
3571:a
3565:2
3561:x
3529:,
3523:0
3517:c
3511:,
3505:b
3499:a
3495:,
3489:1
3486:=
3479:2
3475:c
3469:2
3465:z
3459:+
3452:2
3448:b
3442:2
3438:y
3425:2
3421:a
3415:2
3411:x
3359:.
3353:y
3346:a
3342:a
3336:b
3326:=
3323:z
3315:0
3312:=
3307:2
3303:z
3294:2
3290:y
3284:)
3280:1
3272:a
3269:b
3263:(
3235:a
3232:2
3228:1
3222:=
3219:r
3196:.
3190:0
3187:=
3182:2
3178:z
3169:2
3165:y
3160:)
3157:1
3151:b
3148:r
3145:2
3142:(
3139:+
3134:2
3130:x
3125:)
3122:1
3116:a
3113:r
3110:2
3107:(
3081:.
3075:0
3072:=
3069:z
3066:r
3063:2
3055:2
3051:z
3047:+
3042:2
3038:y
3034:+
3029:2
3025:x
3014:2
3010:r
3006:=
3001:2
2997:)
2993:r
2987:z
2984:(
2981:+
2976:2
2972:y
2968:+
2963:2
2959:x
2932:,
2926:b
2914:a
2910:,
2904:0
2901:=
2898:z
2890:2
2886:y
2882:b
2879:+
2874:2
2870:x
2866:a
2820:.
2814:y
2808:b
2802:2
2798:b
2789:2
2785:a
2775:=
2772:z
2761:0
2758:=
2753:2
2749:z
2740:2
2736:y
2730:)
2726:1
2716:2
2712:b
2706:2
2702:a
2695:(
2668:a
2665:=
2662:r
2636:.
2630:0
2627:=
2622:2
2618:z
2609:2
2605:y
2599:)
2595:1
2585:2
2581:b
2575:2
2571:r
2564:(
2560:+
2555:2
2551:x
2545:)
2541:1
2531:2
2527:a
2521:2
2517:r
2510:(
2483:,
2477:b
2471:a
2467:,
2461:1
2458:=
2451:2
2447:b
2441:2
2437:y
2431:+
2424:2
2420:a
2414:2
2410:x
2361:,
2355:y
2345:2
2341:c
2337:+
2332:2
2328:a
2320:2
2316:b
2307:2
2303:a
2293:b
2290:c
2282:=
2279:z
2268:0
2265:=
2260:2
2256:z
2250:)
2246:1
2243:+
2236:2
2232:c
2226:2
2222:a
2215:(
2206:2
2202:y
2196:)
2192:1
2182:2
2178:b
2172:2
2168:a
2161:(
2134:a
2131:=
2128:r
2102:.
2096:0
2093:=
2088:2
2084:z
2078:)
2074:1
2071:+
2064:2
2060:c
2054:2
2050:r
2043:(
2034:2
2030:y
2024:)
2020:1
2010:2
2006:b
2000:2
1996:r
1989:(
1985:+
1980:2
1976:x
1970:)
1966:1
1956:2
1952:a
1946:2
1942:r
1935:(
1906:2
1902:r
1898:=
1893:2
1889:z
1885:+
1880:2
1876:y
1872:+
1867:2
1863:x
1836:0
1830:c
1827:,
1821:b
1815:a
1811:,
1805:1
1802:=
1795:2
1791:c
1785:2
1781:z
1768:2
1764:b
1758:2
1754:y
1748:+
1741:2
1737:a
1731:2
1727:x
1680:.
1666:2
1661:0
1657:z
1653:C
1647:E
1644:=
1641:x
1636:0
1632:z
1628:D
1625:+
1622:)
1617:2
1613:y
1609:+
1604:2
1600:x
1596:(
1593:A
1568:0
1564:z
1560:=
1557:z
1537:0
1531:E
1525:,
1522:0
1516:B
1513:=
1510:A
1490:E
1487:=
1482:2
1478:y
1474:B
1471:+
1466:2
1462:x
1458:A
1438:0
1435:=
1432:z
1409:E
1406:=
1401:z
1398:x
1393:D
1390:+
1385:2
1381:z
1377:C
1374:+
1369:2
1365:y
1361:B
1358:+
1353:2
1349:x
1345:A
1315:b
1312:=
1309:a
1283:c
1277:b
1271:a
1248:,
1242:x
1231:2
1227:c
1218:2
1214:b
1206:2
1202:b
1193:2
1189:a
1177:a
1174:c
1165:=
1162:z
1151:0
1148:=
1143:2
1139:z
1133:)
1129:1
1119:2
1115:c
1109:2
1105:b
1098:(
1094:+
1089:2
1085:x
1079:)
1075:1
1065:2
1061:a
1055:2
1051:b
1044:(
1017:b
1014:=
1011:r
985:c
982:=
979:r
953:a
950:=
947:r
921:.
915:0
912:=
907:2
903:z
897:)
893:1
883:2
879:c
873:2
869:r
862:(
858:+
853:2
849:y
843:)
839:1
829:2
825:b
819:2
815:r
808:(
804:+
799:2
795:x
789:)
785:1
775:2
771:a
765:2
761:r
754:(
728:2
724:r
700:.
692:2
688:r
684:=
679:2
675:z
671:+
666:2
662:y
658:+
653:2
649:x
625:0
619:c
613:b
607:a
584:1
581:=
574:2
570:c
564:2
560:z
554:+
547:2
543:b
537:2
533:y
527:+
520:2
516:a
510:2
506:x
477:a
467:2
463:r
452:b
440:1
436:r
427:c
347:P
341:P
335:C
329:P
322:C
316:C
310:Q
289:Q
276:Q
263:Q
257:C
239:Q
233:Q
215:0
212:=
207:2
203:z
199:+
194:2
190:y
186:+
181:2
177:x
157:C
139:C
133:Q
127:P
121:Q
115:C
20:.
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