95:
4240:
420:
5497:
3633:
2690:
1034:
1884:
3028:
2051:
62:
which fixes the points on the circle and exchanges the points in the interior and exterior, the center of the circle exchanged with the point at infinity. In inversive geometry a straight line is considered to be a generalized circle containing the point at infinity; inversion of the plane with
4215:
One should not expect that the axioms above define the classical real Möbius plane. There are many axiomatic Möbius planes which are different from the classical one (see below). Similar to the minimal model of an affine plane is the "minimal model" of a Möbius plane. It consists of
102:
Affine planes are systems of points and lines that satisfy, amongst others, the property that two points determine exactly one line. This concept can be generalized to systems of points and circles, with each circle being determined by three non-collinear points. However, three
2347:
4656:
4457:
The connection between the classical Möbius plane and the real affine plane is similar to that between the minimal model of a Möbius plane and the minimal model of an affine plane. This strong connection is typical for Möbius planes and affine planes (see below).
411:. All these miquelian Möbius planes can be described by space models. The classical real Möbius plane can be considered as the geometry of circles on the unit sphere. The essential advantage of the space model is that any cycle is just a circle (on the sphere).
4388:
921:
1740:
910:
1731:
341:. But the classical Möbius plane is not the only geometrical structure that satisfies the properties of an axiomatic Möbius plane. A simple further example of a Möbius plane can be achieved if one replaces the real numbers by
2807:
2430:
1896:
786:
6374:
and bears the same geometric properties as a sphere in a projective 3-space: 1) a line intersects an ovoid in none, one or two points and 2) at any point of the ovoid the set of the tangent lines form a plane, the
2224:
2137:
5123:
4957:
4515:
3694:
6379:. A simple ovoid in real 3-space can be constructed by glueing together two suitable halves of different ellipsoids, such that the result is not a quadric. Even in the finite case there exist ovoids (see
2644:
4775:
2791:
2745:
2588:
2534:
1526:
1141:
1088:
839:
2262:
4550:
4453:
460:
5312:
2468:
4888:
5888:
4694:
4269:
1608:
6303:
5746:
5688:
5646:
5544:
5491:
5247:
6251:
792:
The geometry of lines and circles of the euclidean plane can be homogenized (similarly to the projective completion of an affine plane) by embedding it into the incidence structure
6156:
524:
6020:
3337:
3227:
3180:
1650:
6526:
The only known finite values for the order of a Möbius plane are prime or prime powers. The only known finite Möbius planes are constructed within finite projective geometries.
5174:
4987:
4844:
4813:
4545:
4026:
1409:
5786:
3265:
2680:
2252:
2165:
5946:
5060:
3402:
557:
6498:
5600:
which can be assigned to the vertices of a cube such that the points in 5 faces correspond to concyclical quadruples than the sixth quadruple of points is concyclical, too.
5598:
6187:
6095:
6057:
3983:
3748:
3721:
2082:
1366:
1029:{\displaystyle {\mathcal {Z}}:=\{g\cup \{\infty \}\mid g{\text{ line of }}{\mathfrak {A}}(\mathbb {R} )\}\cup \{k\mid k{\text{ circle of }}{\mathfrak {A}}(\mathbb {R} )\}}
393:
5388:
4210:
3540:
2494:
5448:
3921:
3487:
1879:{\displaystyle {\mathcal {Z}}:=\{\{z\in \mathbb {C} \mid az+{\overline {az}}+b=0\ {\text{(line)}}\ \}\cup \{\infty \}\mid \ 0\neq a\in \mathbb {C} ,b\in \mathbb {R} \}}
1304:
5029:
3619:
5818:
4714:
4261:
4142:
4071:
3946:
3578:
3440:
3088:
1454:
1329:
265:
3895:
1561:
1278:
600:
5343:
3843:
3791:
3050:
2370:
1229:
1177:
3404:. That means the image of a circle is a plane section of the sphere and hence a circle (on the sphere) again. The corresponding planes do not contain the center,
626:
291:
3115:
850:
6518:
6441:
6421:
6333:
5408:
5366:
5194:
5143:
4234:
4166:
4094:
4046:
3869:
3811:
1670:
1477:
1429:
1252:
1197:
331:
311:
240:
220:
200:
180:
160:
4725:
This theorem allows to use the many results on affine planes for investigations on Möbius planes and gives rise to an equivalent definition of a Möbius plane:
1678:
3023:{\displaystyle \Phi :\ (x,y)\rightarrow \left({\frac {x}{1+x^{2}+y^{2}}},{\frac {y}{1+x^{2}+y^{2}}},{\frac {x^{2}+y^{2}}{1+x^{2}+y^{2}}}\right)=(u,v,w)\ .}
2046:{\displaystyle \cup \{\{z\in \mathbb {C} \mid (z-z_{0}){\overline {(z-z_{0})}}=d\ {\text{(circle)}}\mid z_{0}\in \mathbb {C} ,d\in \mathbb {R} ,d>0\}.}
2380:
638:
138:. Two completed lines touch if they have only the point at infinity in common, so they are parallel. The touching relation has the property
2175:
2091:
5072:
4906:
4464:
3643:
6611:
6597:
122:
In an affine plane the parallel relation between lines is essential. In the geometry of cycles, this relation is generalized to the
6575:
3629:
The incidence behavior of the classical real Möbius plane gives rise to the following definition of an axiomatic Möbius plane.
6523:
These finite block designs satisfy the axioms defining a Möbius plane, when a circle is interpreted as a block of the design.
2609:
2342:{\displaystyle z\rightarrow \displaystyle {\frac {1}{z}},\ z\neq 0,\ \ 0\rightarrow \infty ,\ \ \infty \rightarrow 0,\quad }
5450:
for describing the circles does not work in general. For details one should look into the lecture note below. So, only for
4651:{\displaystyle {\mathfrak {A}}_{P}:=({\mathcal {P}}\setminus \{P\},\{z\setminus \{P\}\mid P\in z\in {\mathcal {Z}}\},\in )}
6312:
5322:
Looking for further examples of Möbius planes it seems promising to generalize the classical construction starting with a
4734:
2750:
2704:
2547:
2499:
1485:
1100:
1047:
798:
4393:
6666:
6661:
6628:
6671:
430:
32:
5252:
4383:{\displaystyle {\mathcal {P}}:=\{A,B,C,D,\infty \},\quad {\mathcal {Z}}:=\{z\mid z\subset {\mathcal {P}},|z|=3\}.}
2439:
4853:
6263:
5823:
4668:
2798:
1566:
64:
59:
6540:
6269:
5712:
5654:
5612:
5510:
5457:
5199:
2747:
which omits the formal difference between cycles defined by lines and cycles defined by circles: The geometry
2541:
6192:
349:(instead of the real numbers) does not lead to a Möbius plane, because in the complex affine plane the curve
6100:
468:
5951:
5493:. They are (as the classical model) characterized by huge homogeneity and the following theorem of Miquel.
3272:
3185:
3120:
1613:
5148:
4962:
4818:
4788:
4520:
5751:
3232:
2649:
4716:
is the underlying real affine plane. The essential meaning of the residue shows the following theorem.
2229:
2142:
5896:
5038:
3342:
533:
6584:
6446:
5551:
6164:
6072:
6034:
3988:
3729:
3702:
2063:
1371:
6345:
5346:
408:
396:
352:
116:
71:
6643:
6558:
5371:
4171:
3492:
2477:
6535:
5413:
2698:
2603:
52:
48:
2060:
The advantage of this description is, that one checks easily that the following permutations of
3951:
3900:
3448:
1334:
1283:
6607:
6593:
4992:
3583:
108:
40:
5791:
4699:
4246:
4109:
3545:
3407:
3055:
3874:
2599:
1531:
1257:
570:
5328:
3816:
3764:
3035:
2352:
1202:
1150:
905:{\displaystyle {\mathcal {P}}:=\mathbb {R} ^{2}\cup \{\infty \},\infty \notin \mathbb {R} }
6349:
2471:
527:
342:
94:
83:
36:
605:
270:
6362:(see weblink below). The class which is most similar to miquelian Möbius planes are the
4051:
3926:
3097:
1434:
1309:
245:
6503:
6426:
6406:
6384:
6318:
5393:
5351:
5323:
5179:
5128:
4219:
4151:
4079:
4031:
3854:
3796:
2797:
to the geometry of circles on a sphere. The isomorphism can be performed by a suitable
1726:{\displaystyle {\mathcal {P}}:=\mathbb {C} \cup \{\infty \},\infty \notin \mathbb {C} }
1655:
1462:
1414:
1237:
1182:
463:
400:
346:
316:
296:
225:
205:
185:
165:
145:
79:
6560:
Planar Circle
Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.
4239:
74:
with the same incidence relationships as the classical Möbius plane. It is one of the
6655:
6646:
Planar Circle
Geometries', an Introduction to Möbius-, Laguerre- and Minkowski Planes
6380:
6371:
6367:
1097:
Within the new structure the completed lines play no special role anymore. Obviously
395:
is not a circle-like curve, but a hyperbola-like one. Fortunately there are a lot of
6400:
6396:
2595:
104:
2425:{\displaystyle z\rightarrow {\overline {z}},\ \ \infty \rightarrow \infty .\quad }
419:
781:{\displaystyle \rho (x-x_{0},y-y_{0})=(x-x_{0})^{2}+(y-y_{0})^{2}=r^{2},\ r>0}
5496:
107:
points determine a line, not a circle. This drawback can be removed by adding a
6623:
2794:
75:
17:
6634:
3632:
2689:
6589:
2537:
3542:. So, the image of a line is a circle (on the sphere) through the point
6344:
A proof of Miquel's theorem for the classical (real) case can be found
4777:
is a Möbius plane if and only if the following property is fulfilled:
135:
131:
130:
each other if they have just one point in common. This is true for two
6366:. An ovoidal Möbius plane is the geometry of the plane sections of an
4102:
Any cycle contains at least three points. There is at least one cycle.
2219:{\displaystyle z\rightarrow z+s,\ \ \infty \rightarrow \infty ,\quad }
5704:
of a Möbius plane is miquelian. It is isomorphic to the Möbius plane
2132:{\displaystyle z\rightarrow rz,\ \ \infty \rightarrow \infty ,\quad }
5118:{\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}},\in )}
4952:{\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}},\in )}
4510:{\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}},\in )}
3689:{\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}},\in )}
4238:
58:
An inversion of the Möbius plane with respect to any circle is an
403:
that lead to Möbius planes (see below). Such examples are called
4894:
Any two cycles of a Möbius plane have the same number of points.
111:
to every line. If we call both circles and such completed lines
5368:
for defining circles. But, just to replace the real numbers
5263:
5210:
5101:
5091:
4974:
4935:
4925:
4864:
4800:
4753:
4743:
4631:
4576:
4532:
4493:
4483:
4404:
4347:
4322:
4275:
3735:
3708:
3672:
3662:
2769:
2759:
2723:
2713:
2566:
2556:
2496:
one recognizes that the mappings (1)-(3) generate the group
2069:
1746:
1684:
1504:
1494:
1119:
1109:
1066:
1056:
927:
856:
817:
807:
6399:
with the parameters of the one-point extension of a finite
119:
in which every three points determine exactly one cycle.
47:
because it is closed under inversion with respect to any
4243:
Möbius plane: minimal model (only the cycles containing
2639:{\displaystyle z\rightarrow {\tfrac {1}{\overline {z}}}}
2602:. Hence from (4) we get: For any cycle there exists an
3243:
3205:
2620:
6506:
6449:
6429:
6409:
6321:
6272:
6195:
6167:
6103:
6075:
6037:
5954:
5899:
5826:
5794:
5754:
5715:
5657:
5615:
5554:
5513:
5460:
5454:
of fields and quadratic forms one gets Möbius planes
5416:
5396:
5374:
5354:
5331:
5255:
5202:
5182:
5151:
5131:
5075:
5041:
4995:
4965:
4909:
4856:
4821:
4791:
4737:
4702:
4671:
4553:
4523:
4467:
4396:
4272:
4249:
4222:
4174:
4154:
4112:
4082:
4054:
4034:
3991:
3954:
3929:
3903:
3877:
3857:
3819:
3799:
3767:
3732:
3705:
3646:
3586:
3548:
3495:
3451:
3410:
3345:
3275:
3235:
3188:
3123:
3100:
3058:
3038:
2810:
2753:
2707:
2652:
2612:
2550:
2502:
2480:
2442:
2383:
2355:
2272:
2265:
2232:
2178:
2145:
2094:
2066:
1899:
1743:
1681:
1658:
1616:
1569:
1534:
1488:
1465:
1437:
1417:
1374:
1337:
1312:
1286:
1260:
1240:
1205:
1185:
1153:
1103:
1050:
924:
853:
801:
641:
608:
573:
536:
471:
433:
355:
319:
299:
273:
248:
228:
208:
188:
168:
148:
4770:{\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}
2786:{\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}
2740:{\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}
2583:{\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}
2529:{\displaystyle \operatorname {PGL} (2,\mathbb {C} )}
1521:{\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}
1136:{\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}
1083:{\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}
834:{\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}
6512:
6492:
6435:
6415:
6383:). Ovoidal Möbius planes are characterized by the
6348:. It is elementary and based on the theorem of an
6327:
6297:
6245:
6181:
6150:
6089:
6051:
6014:
5940:
5882:
5812:
5780:
5740:
5682:
5640:
5592:
5538:
5485:
5442:
5402:
5382:
5360:
5337:
5306:
5241:
5188:
5168:
5137:
5117:
5054:
5023:
4981:
4951:
4882:
4838:
4807:
4769:
4722:Any residue of a Möbius plane is an affine plane.
4708:
4688:
4650:
4539:
4509:
4448:{\displaystyle |{\mathcal {Z}}|={5 \choose 3}=10.}
4447:
4382:
4255:
4228:
4204:
4160:
4136:
4088:
4065:
4040:
4020:
3977:
3940:
3915:
3889:
3863:
3837:
3805:
3785:
3742:
3715:
3688:
3613:
3572:
3534:
3481:
3434:
3396:
3331:
3259:
3221:
3174:
3109:
3082:
3044:
3022:
2785:
2739:
2682:. This property gives rise to the alternate name
2674:
2638:
2582:
2528:
2488:
2462:
2424:
2364:
2341:
2246:
2218:
2159:
2131:
2076:
2045:
1878:
1725:
1664:
1644:
1602:
1555:
1520:
1471:
1448:
1423:
1403:
1360:
1323:
1298:
1272:
1246:
1223:
1191:
1171:
1135:
1082:
1028:
904:
833:
780:
620:
594:
551:
518:
454:
387:
325:
305:
285:
259:
234:
214:
194:
174:
154:
4433:
4420:
2432:(reflection or inversion through the real axis)
632:is a set of points that fulfills an equation
6315:of index 1) in projective 3-space over field
2646:is the inversion which fixes the unit circle
1528:can be described using the complex numbers.
455:{\displaystyle {\mathfrak {A}}(\mathbb {R} )}
8:
6009:
5955:
5807:
5795:
5307:{\displaystyle |{\mathcal {Z}}|=n(n^{2}+1).}
4636:
4611:
4605:
4596:
4590:
4584:
4374:
4330:
4313:
4283:
4015:
4009:
2701:there exists a space model for the geometry
2463:{\displaystyle \mathbb {C} \cup \{\infty \}}
2457:
2451:
2037:
1906:
1903:
1873:
1833:
1827:
1821:
1757:
1754:
1706:
1700:
1398:
1392:
1023:
988:
982:
950:
944:
935:
885:
879:
6305:is isomorphic to the geometry of the plane
5651:Because of the last Theorem a Möbius plane
4883:{\displaystyle |{\mathcal {P}}|<\infty }
4263:are drawn. Any set of 3 points is a cycle.)
6505:
6457:
6448:
6428:
6408:
6320:
6274:
6273:
6271:
6237:
6221:
6194:
6175:
6174:
6166:
6142:
6129:
6102:
6083:
6082:
6074:
6045:
6044:
6036:
5953:
5926:
5904:
5898:
5883:{\displaystyle \rho (x,y)=x^{2}+xy+y^{2}}
5874:
5852:
5825:
5793:
5761:
5753:
5717:
5716:
5714:
5659:
5658:
5656:
5617:
5616:
5614:
5584:
5559:
5553:
5515:
5514:
5512:
5462:
5461:
5459:
5434:
5421:
5415:
5410:and to keep the classical quadratic form
5395:
5376:
5375:
5373:
5353:
5330:
5286:
5268:
5262:
5261:
5256:
5254:
5227:
5215:
5209:
5208:
5203:
5201:
5181:
5160:
5154:
5153:
5150:
5130:
5100:
5099:
5090:
5089:
5077:
5076:
5074:
5043:
5042:
5040:
5010:
5002:
4994:
4973:
4972:
4964:
4934:
4933:
4924:
4923:
4911:
4910:
4908:
4869:
4863:
4862:
4857:
4855:
4830:
4824:
4823:
4820:
4799:
4798:
4790:
4752:
4751:
4742:
4741:
4736:
4701:
4689:{\displaystyle {\mathfrak {A}}_{\infty }}
4680:
4674:
4673:
4670:
4630:
4629:
4575:
4574:
4562:
4556:
4555:
4552:
4531:
4530:
4522:
4492:
4491:
4482:
4481:
4469:
4468:
4466:
4432:
4419:
4417:
4409:
4403:
4402:
4397:
4395:
4363:
4355:
4346:
4345:
4321:
4320:
4274:
4273:
4271:
4248:
4221:
4173:
4153:
4111:
4081:
4053:
4033:
3990:
3953:
3928:
3902:
3876:
3856:
3818:
3798:
3766:
3734:
3733:
3731:
3707:
3706:
3704:
3671:
3670:
3661:
3660:
3648:
3647:
3645:
3585:
3547:
3494:
3450:
3409:
3344:
3293:
3280:
3274:
3242:
3234:
3204:
3187:
3154:
3141:
3128:
3122:
3099:
3057:
3037:
2976:
2963:
2945:
2932:
2925:
2913:
2900:
2884:
2872:
2859:
2843:
2809:
2768:
2767:
2758:
2757:
2752:
2722:
2721:
2712:
2711:
2706:
2656:
2651:
2619:
2611:
2565:
2564:
2555:
2554:
2549:
2519:
2518:
2501:
2482:
2481:
2479:
2444:
2443:
2441:
2390:
2382:
2354:
2273:
2264:
2240:
2239:
2231:
2177:
2153:
2152:
2144:
2093:
2068:
2067:
2065:
2021:
2020:
2007:
2006:
1997:
1985:
1961:
1945:
1936:
1916:
1915:
1898:
1869:
1868:
1855:
1854:
1813:
1783:
1767:
1766:
1745:
1744:
1742:
1719:
1718:
1693:
1692:
1683:
1682:
1680:
1657:
1617:
1615:
1603:{\displaystyle (x,y)\in \mathbb {R} ^{2}}
1594:
1590:
1589:
1568:
1533:
1503:
1502:
1493:
1492:
1487:
1464:
1436:
1416:
1373:
1336:
1311:
1285:
1259:
1239:
1204:
1184:
1152:
1118:
1117:
1108:
1107:
1102:
1065:
1064:
1055:
1054:
1049:
1016:
1015:
1006:
1005:
1000:
975:
974:
965:
964:
959:
926:
925:
923:
898:
897:
870:
866:
865:
855:
854:
852:
816:
815:
806:
805:
800:
757:
744:
734:
712:
702:
677:
658:
640:
607:
572:
543:
539:
538:
535:
510:
497:
470:
445:
444:
435:
434:
432:
373:
360:
354:
318:
298:
272:
247:
227:
207:
187:
167:
147:
6298:{\displaystyle {\mathfrak {M}}(K,\rho )}
5741:{\displaystyle {\mathfrak {M}}(K,\rho )}
5683:{\displaystyle {\mathfrak {M}}(K,\rho )}
5641:{\displaystyle {\mathfrak {M}}(K,\rho )}
5539:{\displaystyle {\mathfrak {M}}(K,\rho )}
5495:
5486:{\displaystyle {\mathfrak {M}}(K,\rho )}
5242:{\displaystyle |{\mathcal {P}}|=n^{2}+1}
4898:This justifies the following definition:
3631:
2688:
418:
93:
51:, and thus a natural setting for planar
6572:Vorlesungen über Geometrie der Algebren
6551:
6358:There are many Möbius planes which are
6246:{\displaystyle \rho (x,y)=x^{2}-2y^{2}}
6059:the field of complex numbers, there is
4602:
4581:
337:These properties essentially define an
6391:Finite Möbius planes and block designs
6151:{\displaystyle \rho (x,y)=x^{2}+y^{2}}
519:{\displaystyle \rho (x,y)=x^{2}+y^{2}}
6015:{\displaystyle \{(0,1),(1,0),(1,1)\}}
3332:{\displaystyle x^{2}+y^{2}-ax-by-c=0}
3222:{\displaystyle (0,0,{\tfrac {1}{2}})}
3175:{\displaystyle u^{2}+v^{2}+w^{2}-w=0}
3117:-plane onto the sphere with equation
70:More generally, a Möbius plane is an
7:
6500:-design, is a Möbius plane of order
6311:sections on a sphere (nondegenerate
6189:(the field of rational numbers) and
6097:(the field of rational numbers) and
4665:For the classical model the residue
1645:{\displaystyle {\overline {z}}=x-iy}
427:We start from the real affine plane
6275:
5718:
5660:
5618:
5516:
5463:
5169:{\displaystyle {\mathfrak {A}}_{P}}
5155:
5078:
5044:
4982:{\displaystyle z\in {\mathcal {Z}}}
4912:
4890:, we have (as with affine planes):
4839:{\displaystyle {\mathfrak {A}}_{P}}
4825:
4808:{\displaystyle P\in {\mathcal {P}}}
4675:
4557:
4540:{\displaystyle P\in {\mathcal {P}}}
4470:
3649:
1007:
966:
436:
423:classical Moebius plane:2d/3d-model
5781:{\displaystyle K=\mathrm {GF} (2)}
5765:
5762:
4877:
4703:
4681:
4424:
4310:
4250:
3260:{\displaystyle r={\tfrac {1}{2}};}
3039:
2811:
2697:Similarly to the space model of a
2675:{\displaystyle z{\overline {z}}=1}
2454:
2415:
2409:
2325:
2313:
2209:
2203:
2122:
2116:
1830:
1712:
1703:
947:
891:
882:
25:
5648:satisfies the Theorem of Miquel.
2247:{\displaystyle s\in \mathbb {C} }
2160:{\displaystyle r\in \mathbb {C} }
399:(numbers) together with suitable
63:respect to a line is a Euclidean
5941:{\displaystyle x^{2}+xy+y^{2}=1}
5055:{\displaystyle {\mathfrak {M}}.}
3397:{\displaystyle au+bv-(1+c)w+c=0}
552:{\displaystyle \mathbb {R} ^{2}}
6563:(PDF; 891 kB), S. 60.
6493:{\displaystyle (n^{2}+1,n+1,1)}
5893:(For example, the unit circle
5593:{\displaystyle P_{1},...,P_{8}}
4850:For finite Möbius planes, i.e.
4319:
3923:there exists exactly one cycle
2421:
2337:
2215:
2128:
1306:there exists exactly one cycle
98:Möbius-plane: touching relation
6487:
6450:
6292:
6280:
6211:
6199:
6182:{\displaystyle K=\mathbb {Q} }
6119:
6107:
6090:{\displaystyle K=\mathbb {Q} }
6052:{\displaystyle K=\mathbb {C} }
6006:
5994:
5988:
5976:
5970:
5958:
5842:
5830:
5775:
5769:
5735:
5723:
5677:
5665:
5635:
5623:
5533:
5521:
5480:
5468:
5298:
5279:
5269:
5257:
5216:
5204:
5112:
5086:
5011:
5003:
4946:
4920:
4870:
4858:
4764:
4738:
4645:
4571:
4504:
4478:
4410:
4398:
4364:
4356:
4021:{\displaystyle z\cap z'=\{P\}}
3754:if the following axioms hold:
3743:{\displaystyle {\mathcal {Z}}}
3716:{\displaystyle {\mathcal {P}}}
3683:
3657:
3636:Möbius plane: axioms (A1),(A2)
3605:
3587:
3567:
3549:
3429:
3411:
3376:
3364:
3216:
3189:
3077:
3059:
3011:
2993:
2835:
2832:
2820:
2780:
2754:
2734:
2708:
2616:
2577:
2551:
2523:
2509:
2412:
2387:
2328:
2310:
2269:
2206:
2182:
2119:
2098:
2077:{\displaystyle {\mathcal {P}}}
1967:
1948:
1942:
1923:
1582:
1570:
1515:
1489:
1404:{\displaystyle z\cap z'=\{P\}}
1143:has the following properties.
1130:
1104:
1077:
1051:
1020:
1012:
979:
971:
828:
802:
741:
721:
709:
689:
683:
645:
487:
475:
449:
441:
27:In mathematics, the classical
1:
2699:desarguesian projective plane
388:{\displaystyle x^{2}+y^{2}=1}
5383:{\displaystyle \mathbb {R} }
5176:is an affine plane of order
4205:{\displaystyle A,B,C,D\in z}
3535:{\displaystyle au+bv-cw+c=0}
3052:is a projection with center
2661:
2629:
2590:is a homogeneous structure,
2489:{\displaystyle \mathbb {C} }
2395:
1971:
1793:
1652:is the complex conjugate of
1622:
1147:For any set of three points
6639:Encyclopedia of Mathematics
6629:Encyclopedia of Mathematics
5603:The converse is true, too.
5443:{\displaystyle x^{2}+y^{2}}
5125:be a Möbius plane of order
5065:From combinatorics we get:
3793:there is exactly one cycle
1179:there is exactly one cycle
1092:classical real Möbius plane
567:are described by equations
415:Classical real Möbius plane
242:there is exactly one cycle
6688:
5345:on an affine plane over a
4903:For a finite Möbius plane
6606:, Springer-Verlag (1968)
3978:{\displaystyle P,Q\in z'}
3916:{\displaystyle Q\notin z}
3482:{\displaystyle ax+by+c=0}
3269:the circle with equation
1361:{\displaystyle P,Q\in z'}
1299:{\displaystyle Q\notin z}
90:Relation to affine planes
39:supplemented by a single
6264:stereographic projection
5024:{\displaystyle n:=|z|-1}
3625:Axioms of a Möbius plane
3614:{\displaystyle (0,0,1).}
2799:stereographic projection
2693:stereographic projection
2084:map cycles onto cycles.
43:. It is also called the
6063:quadratic form at all.
5813:{\displaystyle \{0,1\}}
5318:Miquelian Möbius planes
4731:An incidence structure
4709:{\displaystyle \infty }
4256:{\displaystyle \infty }
4137:{\displaystyle A,B,C,D}
3640:An incidence structure
3580:but omitting the point
3573:{\displaystyle (0,0,1)}
3435:{\displaystyle (0,0,1)}
3083:{\displaystyle (0,0,1)}
2167:(rotation + dilatation)
407:, because they fulfill
33:August Ferdinand Möbius
6514:
6494:
6437:
6417:
6329:
6299:
6247:
6183:
6152:
6091:
6053:
6016:
5942:
5884:
5814:
5782:
5742:
5692:miquelian Möbius plane
5684:
5642:
5594:
5546:the following is true:
5540:
5501:
5487:
5444:
5404:
5384:
5362:
5339:
5308:
5243:
5190:
5170:
5145:. Then a) any residue
5139:
5119:
5056:
5025:
4983:
4953:
4884:
4840:
4809:
4771:
4710:
4690:
4652:
4541:
4511:
4449:
4384:
4264:
4257:
4230:
4206:
4162:
4138:
4090:
4067:
4042:
4022:
3979:
3942:
3917:
3891:
3890:{\displaystyle P\in z}
3865:
3839:
3807:
3787:
3744:
3717:
3690:
3637:
3615:
3574:
3536:
3483:
3436:
3398:
3333:
3261:
3223:
3176:
3111:
3084:
3046:
3024:
2787:
2741:
2694:
2676:
2640:
2584:
2530:
2490:
2464:
2426:
2366:
2343:
2248:
2220:
2161:
2133:
2078:
2047:
1880:
1727:
1666:
1646:
1604:
1557:
1556:{\displaystyle z=x+iy}
1522:
1473:
1450:
1425:
1405:
1362:
1325:
1300:
1274:
1273:{\displaystyle P\in z}
1248:
1225:
1193:
1173:
1137:
1084:
1030:
906:
835:
782:
622:
596:
595:{\displaystyle y=mx+b}
553:
520:
456:
424:
389:
339:axiomatic Möbius plane
327:
307:
287:
261:
236:
216:
196:
176:
156:
99:
6581:F. Buekenhout (ed.),
6541:Möbius transformation
6515:
6495:
6438:
6418:
6364:ovoidal Möbius planes
6330:
6300:
6248:
6184:
6153:
6092:
6054:
6017:
5943:
5885:
5815:
5783:
5743:
5685:
5643:
5595:
5541:
5507:For the Möbius plane
5499:
5488:
5445:
5405:
5385:
5363:
5340:
5338:{\displaystyle \rho }
5309:
5244:
5191:
5171:
5140:
5120:
5057:
5026:
4984:
4954:
4885:
4841:
4810:
4772:
4711:
4691:
4653:
4542:
4512:
4450:
4385:
4258:
4242:
4231:
4207:
4163:
4139:
4091:
4068:
4043:
4023:
3980:
3943:
3918:
3892:
3866:
3840:
3838:{\displaystyle A,B,C}
3808:
3788:
3786:{\displaystyle A,B,C}
3761:For any three points
3745:
3718:
3691:
3635:
3616:
3575:
3537:
3484:
3437:
3399:
3334:
3262:
3224:
3177:
3112:
3085:
3047:
3045:{\displaystyle \Phi }
3025:
2788:
2742:
2692:
2677:
2641:
2585:
2542:Möbius transformation
2531:
2491:
2465:
2427:
2367:
2365:{\displaystyle \pm 1}
2344:
2249:
2221:
2162:
2134:
2079:
2048:
1881:
1728:
1667:
1647:
1605:
1558:
1523:
1474:
1451:
1426:
1406:
1363:
1326:
1301:
1275:
1249:
1226:
1224:{\displaystyle A,B,C}
1194:
1174:
1172:{\displaystyle A,B,C}
1138:
1085:
1031:
1002: circle of
907:
836:
783:
623:
597:
554:
521:
457:
422:
390:
328:
308:
288:
262:
237:
217:
197:
177:
157:
126:relation. Two cycles
97:
6504:
6447:
6427:
6407:
6319:
6270:
6193:
6165:
6101:
6073:
6035:
5952:
5897:
5824:
5792:
5752:
5713:
5655:
5613:
5609:Only a Möbius plane
5552:
5548:If for any 8 points
5511:
5458:
5414:
5394:
5372:
5352:
5329:
5253:
5200:
5180:
5149:
5129:
5073:
5039:
4993:
4963:
4907:
4854:
4819:
4789:
4735:
4700:
4669:
4551:
4547:we define structure
4521:
4465:
4394:
4270:
4247:
4220:
4172:
4152:
4148:if there is a cycle
4110:
4080:
4076:each other at point
4052:
4032:
3989:
3952:
3927:
3901:
3875:
3855:
3817:
3797:
3765:
3730:
3703:
3644:
3584:
3546:
3493:
3449:
3408:
3343:
3273:
3233:
3186:
3121:
3098:
3056:
3036:
2808:
2751:
2705:
2650:
2610:
2548:
2500:
2478:
2440:
2381:
2353:
2263:
2230:
2176:
2143:
2092:
2064:
1897:
1741:
1679:
1656:
1614:
1567:
1532:
1486:
1463:
1459:each other at point
1435:
1415:
1372:
1335:
1310:
1284:
1258:
1238:
1203:
1183:
1151:
1101:
1048:
922:
851:
799:
639:
606:
571:
534:
469:
431:
353:
317:
297:
271:
246:
226:
206:
186:
166:
146:
4846:is an affine plane.
4461:For a Möbius plane
961: line of
621:{\displaystyle x=c}
286:{\displaystyle P,Q}
136:tangent to a circle
117:incidence structure
72:incidence structure
6667:Incidence geometry
6662:Classical geometry
6585:Incidence Geometry
6536:Conformal geometry
6510:
6490:
6433:
6413:
6325:
6295:
6243:
6179:
6148:
6087:
6049:
6012:
5938:
5880:
5810:
5778:
5738:
5680:
5638:
5590:
5536:
5502:
5483:
5440:
5400:
5380:
5358:
5335:
5304:
5239:
5186:
5166:
5135:
5115:
5052:
5021:
4979:
4949:
4880:
4836:
4805:
4767:
4706:
4686:
4660:residue at point P
4648:
4537:
4507:
4445:
4380:
4265:
4253:
4226:
4202:
4158:
4134:
4086:
4066:{\displaystyle z'}
4063:
4038:
4018:
3975:
3941:{\displaystyle z'}
3938:
3913:
3887:
3861:
3835:
3803:
3783:
3740:
3713:
3686:
3638:
3611:
3570:
3532:
3479:
3432:
3394:
3329:
3257:
3252:
3219:
3214:
3172:
3110:{\displaystyle xy}
3107:
3080:
3042:
3020:
2783:
2737:
2695:
2672:
2636:
2634:
2596:automorphism group
2580:
2526:
2486:
2460:
2422:
2362:
2339:
2338:
2244:
2216:
2157:
2129:
2074:
2043:
1876:
1723:
1662:
1642:
1600:
1553:
1518:
1469:
1449:{\displaystyle z'}
1446:
1421:
1401:
1358:
1324:{\displaystyle z'}
1321:
1296:
1270:
1244:
1221:
1189:
1169:
1133:
1080:
1026:
902:
831:
778:
618:
592:
549:
516:
452:
425:
385:
323:
303:
283:
267:containing points
260:{\displaystyle z'}
257:
232:
212:
192:
172:
152:
134:or a line that is
100:
53:inversive geometry
49:generalized circle
6672:Planes (geometry)
6604:Finite Geometries
6513:{\displaystyle n}
6436:{\displaystyle 3}
6416:{\displaystyle n}
6328:{\displaystyle K}
6253:is suitable, too.
5948:is the point set
5505:Theorem (Miquel):
5500:Theorem of Miquel
5403:{\displaystyle K}
5361:{\displaystyle K}
5189:{\displaystyle n}
5138:{\displaystyle n}
4431:
4229:{\displaystyle 5}
4161:{\displaystyle z}
4089:{\displaystyle P}
4041:{\displaystyle z}
3864:{\displaystyle z}
3806:{\displaystyle z}
3251:
3213:
3016:
2983:
2920:
2879:
2819:
2664:
2633:
2632:
2408:
2405:
2398:
2324:
2321:
2306:
2303:
2288:
2281:
2202:
2199:
2115:
2112:
1988:
1984:
1974:
1841:
1820:
1816:
1812:
1796:
1665:{\displaystyle z}
1625:
1563:represents point
1472:{\displaystyle P}
1424:{\displaystyle z}
1247:{\displaystyle z}
1192:{\displaystyle z}
1003:
962:
768:
526:and get the real
326:{\displaystyle P}
306:{\displaystyle z}
235:{\displaystyle z}
215:{\displaystyle Q}
195:{\displaystyle z}
175:{\displaystyle P}
155:{\displaystyle z}
109:point at infinity
41:point at infinity
16:(Redirected from
6679:
6564:
6556:
6519:
6517:
6516:
6511:
6499:
6497:
6496:
6491:
6462:
6461:
6442:
6440:
6439:
6434:
6422:
6420:
6419:
6414:
6370:. An ovoid is a
6334:
6332:
6331:
6326:
6304:
6302:
6301:
6296:
6279:
6278:
6252:
6250:
6249:
6244:
6242:
6241:
6226:
6225:
6188:
6186:
6185:
6180:
6178:
6157:
6155:
6154:
6149:
6147:
6146:
6134:
6133:
6096:
6094:
6093:
6088:
6086:
6058:
6056:
6055:
6050:
6048:
6021:
6019:
6018:
6013:
5947:
5945:
5944:
5939:
5931:
5930:
5909:
5908:
5889:
5887:
5886:
5881:
5879:
5878:
5857:
5856:
5819:
5817:
5816:
5811:
5787:
5785:
5784:
5779:
5768:
5747:
5745:
5744:
5739:
5722:
5721:
5689:
5687:
5686:
5681:
5664:
5663:
5647:
5645:
5644:
5639:
5622:
5621:
5599:
5597:
5596:
5591:
5589:
5588:
5564:
5563:
5545:
5543:
5542:
5537:
5520:
5519:
5492:
5490:
5489:
5484:
5467:
5466:
5449:
5447:
5446:
5441:
5439:
5438:
5426:
5425:
5409:
5407:
5406:
5401:
5389:
5387:
5386:
5381:
5379:
5367:
5365:
5364:
5359:
5344:
5342:
5341:
5336:
5313:
5311:
5310:
5305:
5291:
5290:
5272:
5267:
5266:
5260:
5248:
5246:
5245:
5240:
5232:
5231:
5219:
5214:
5213:
5207:
5195:
5193:
5192:
5187:
5175:
5173:
5172:
5167:
5165:
5164:
5159:
5158:
5144:
5142:
5141:
5136:
5124:
5122:
5121:
5116:
5105:
5104:
5095:
5094:
5082:
5081:
5061:
5059:
5058:
5053:
5048:
5047:
5030:
5028:
5027:
5022:
5014:
5006:
4988:
4986:
4985:
4980:
4978:
4977:
4958:
4956:
4955:
4950:
4939:
4938:
4929:
4928:
4916:
4915:
4889:
4887:
4886:
4881:
4873:
4868:
4867:
4861:
4845:
4843:
4842:
4837:
4835:
4834:
4829:
4828:
4814:
4812:
4811:
4806:
4804:
4803:
4776:
4774:
4773:
4768:
4757:
4756:
4747:
4746:
4715:
4713:
4712:
4707:
4695:
4693:
4692:
4687:
4685:
4684:
4679:
4678:
4658:and call it the
4657:
4655:
4654:
4649:
4635:
4634:
4580:
4579:
4567:
4566:
4561:
4560:
4546:
4544:
4543:
4538:
4536:
4535:
4516:
4514:
4513:
4508:
4497:
4496:
4487:
4486:
4474:
4473:
4454:
4452:
4451:
4446:
4438:
4437:
4436:
4423:
4413:
4408:
4407:
4401:
4389:
4387:
4386:
4381:
4367:
4359:
4351:
4350:
4326:
4325:
4279:
4278:
4262:
4260:
4259:
4254:
4235:
4233:
4232:
4227:
4211:
4209:
4208:
4203:
4167:
4165:
4164:
4159:
4143:
4141:
4140:
4135:
4095:
4093:
4092:
4087:
4072:
4070:
4069:
4064:
4062:
4047:
4045:
4044:
4039:
4027:
4025:
4024:
4019:
4005:
3984:
3982:
3981:
3976:
3974:
3947:
3945:
3944:
3939:
3937:
3922:
3920:
3919:
3914:
3896:
3894:
3893:
3888:
3870:
3868:
3867:
3862:
3844:
3842:
3841:
3836:
3812:
3810:
3809:
3804:
3792:
3790:
3789:
3784:
3749:
3747:
3746:
3741:
3739:
3738:
3722:
3720:
3719:
3714:
3712:
3711:
3695:
3693:
3692:
3687:
3676:
3675:
3666:
3665:
3653:
3652:
3620:
3618:
3617:
3612:
3579:
3577:
3576:
3571:
3541:
3539:
3538:
3533:
3489:into the plane
3488:
3486:
3485:
3480:
3441:
3439:
3438:
3433:
3403:
3401:
3400:
3395:
3338:
3336:
3335:
3330:
3298:
3297:
3285:
3284:
3266:
3264:
3263:
3258:
3253:
3244:
3228:
3226:
3225:
3220:
3215:
3206:
3181:
3179:
3178:
3173:
3159:
3158:
3146:
3145:
3133:
3132:
3116:
3114:
3113:
3108:
3089:
3087:
3086:
3081:
3051:
3049:
3048:
3043:
3029:
3027:
3026:
3021:
3014:
2989:
2985:
2984:
2982:
2981:
2980:
2968:
2967:
2951:
2950:
2949:
2937:
2936:
2926:
2921:
2919:
2918:
2917:
2905:
2904:
2885:
2880:
2878:
2877:
2876:
2864:
2863:
2844:
2817:
2792:
2790:
2789:
2784:
2773:
2772:
2763:
2762:
2746:
2744:
2743:
2738:
2727:
2726:
2717:
2716:
2681:
2679:
2678:
2673:
2665:
2657:
2645:
2643:
2642:
2637:
2635:
2625:
2621:
2589:
2587:
2586:
2581:
2570:
2569:
2560:
2559:
2544:). The geometry
2535:
2533:
2532:
2527:
2522:
2495:
2493:
2492:
2487:
2485:
2469:
2467:
2466:
2461:
2447:
2431:
2429:
2428:
2423:
2406:
2403:
2399:
2391:
2371:
2369:
2368:
2363:
2348:
2346:
2345:
2340:
2322:
2319:
2304:
2301:
2286:
2282:
2274:
2253:
2251:
2250:
2245:
2243:
2225:
2223:
2222:
2217:
2200:
2197:
2166:
2164:
2163:
2158:
2156:
2138:
2136:
2135:
2130:
2113:
2110:
2083:
2081:
2080:
2075:
2073:
2072:
2052:
2050:
2049:
2044:
2024:
2010:
2002:
2001:
1989:
1986:
1982:
1975:
1970:
1966:
1965:
1946:
1941:
1940:
1919:
1885:
1883:
1882:
1877:
1872:
1858:
1839:
1818:
1817:
1814:
1810:
1797:
1792:
1784:
1770:
1750:
1749:
1732:
1730:
1729:
1724:
1722:
1696:
1688:
1687:
1671:
1669:
1668:
1663:
1651:
1649:
1648:
1643:
1626:
1618:
1609:
1607:
1606:
1601:
1599:
1598:
1593:
1562:
1560:
1559:
1554:
1527:
1525:
1524:
1519:
1508:
1507:
1498:
1497:
1478:
1476:
1475:
1470:
1455:
1453:
1452:
1447:
1445:
1430:
1428:
1427:
1422:
1410:
1408:
1407:
1402:
1388:
1367:
1365:
1364:
1359:
1357:
1330:
1328:
1327:
1322:
1320:
1305:
1303:
1302:
1297:
1279:
1277:
1276:
1271:
1253:
1251:
1250:
1245:
1230:
1228:
1227:
1222:
1198:
1196:
1195:
1190:
1178:
1176:
1175:
1170:
1142:
1140:
1139:
1134:
1123:
1122:
1113:
1112:
1089:
1087:
1086:
1081:
1070:
1069:
1060:
1059:
1035:
1033:
1032:
1027:
1019:
1011:
1010:
1004:
1001:
978:
970:
969:
963:
960:
931:
930:
911:
909:
908:
903:
901:
875:
874:
869:
860:
859:
840:
838:
837:
832:
821:
820:
811:
810:
787:
785:
784:
779:
766:
762:
761:
749:
748:
739:
738:
717:
716:
707:
706:
682:
681:
663:
662:
627:
625:
624:
619:
601:
599:
598:
593:
558:
556:
555:
550:
548:
547:
542:
525:
523:
522:
517:
515:
514:
502:
501:
461:
459:
458:
453:
448:
440:
439:
409:Miquel's theorem
394:
392:
391:
386:
378:
377:
365:
364:
343:rational numbers
332:
330:
329:
324:
312:
310:
309:
304:
292:
290:
289:
284:
266:
264:
263:
258:
256:
241:
239:
238:
233:
221:
219:
218:
213:
201:
199:
198:
193:
181:
179:
178:
173:
161:
159:
158:
153:
78:: Möbius plane,
21:
6687:
6686:
6682:
6681:
6680:
6678:
6677:
6676:
6652:
6651:
6620:
6567:
6557:
6553:
6549:
6532:
6502:
6501:
6453:
6445:
6444:
6425:
6424:
6405:
6404:
6393:
6350:inscribed angle
6317:
6316:
6268:
6267:
6233:
6217:
6191:
6190:
6163:
6162:
6138:
6125:
6099:
6098:
6071:
6070:
6033:
6032:
5950:
5949:
5922:
5900:
5895:
5894:
5870:
5848:
5822:
5821:
5790:
5789:
5750:
5749:
5711:
5710:
5653:
5652:
5611:
5610:
5607:Theorem (Chen):
5580:
5555:
5550:
5549:
5547:
5509:
5508:
5456:
5455:
5430:
5417:
5412:
5411:
5392:
5391:
5370:
5369:
5350:
5349:
5327:
5326:
5320:
5282:
5251:
5250:
5223:
5198:
5197:
5178:
5177:
5152:
5147:
5146:
5127:
5126:
5071:
5070:
5037:
5036:
4991:
4990:
4961:
4960:
4905:
4904:
4899:
4852:
4851:
4822:
4817:
4816:
4787:
4786:
4778:
4733:
4732:
4698:
4697:
4672:
4667:
4666:
4554:
4549:
4548:
4519:
4518:
4463:
4462:
4418:
4392:
4391:
4268:
4267:
4245:
4244:
4218:
4217:
4170:
4169:
4150:
4149:
4108:
4107:
4078:
4077:
4055:
4050:
4049:
4030:
4029:
3998:
3987:
3986:
3967:
3950:
3949:
3930:
3925:
3924:
3899:
3898:
3873:
3872:
3853:
3852:
3815:
3814:
3795:
3794:
3763:
3762:
3728:
3727:
3701:
3700:
3642:
3641:
3627:
3582:
3581:
3544:
3543:
3491:
3490:
3447:
3446:
3406:
3405:
3341:
3340:
3339:into the plane
3289:
3276:
3271:
3270:
3231:
3230:
3184:
3183:
3150:
3137:
3124:
3119:
3118:
3096:
3095:
3054:
3053:
3034:
3033:
2972:
2959:
2952:
2941:
2928:
2927:
2909:
2896:
2889:
2868:
2855:
2848:
2842:
2838:
2806:
2805:
2801:. For example:
2749:
2748:
2703:
2702:
2684:inversive plane
2648:
2647:
2608:
2607:
2606:. For example:
2546:
2545:
2498:
2497:
2476:
2475:
2472:projective line
2438:
2437:
2379:
2378:
2351:
2350:
2349:(reflection at
2261:
2260:
2228:
2227:
2174:
2173:
2141:
2140:
2090:
2089:
2062:
2061:
1993:
1957:
1947:
1932:
1895:
1894:
1785:
1739:
1738:
1677:
1676:
1654:
1653:
1612:
1611:
1588:
1565:
1564:
1530:
1529:
1484:
1483:
1461:
1460:
1438:
1433:
1432:
1413:
1412:
1381:
1370:
1369:
1350:
1333:
1332:
1313:
1308:
1307:
1282:
1281:
1256:
1255:
1236:
1235:
1201:
1200:
1199:which contains
1181:
1180:
1149:
1148:
1099:
1098:
1046:
1045:
920:
919:
864:
849:
848:
797:
796:
753:
740:
730:
708:
698:
673:
654:
637:
636:
604:
603:
569:
568:
537:
532:
531:
528:Euclidean plane
506:
493:
467:
466:
429:
428:
417:
401:quadratic forms
369:
356:
351:
350:
347:complex numbers
345:. The usage of
315:
314:
295:
294:
269:
268:
249:
244:
243:
224:
223:
204:
203:
202:and any point
184:
183:
164:
163:
144:
143:
132:tangent circles
92:
84:Minkowski plane
45:inversive plane
37:Euclidean plane
23:
22:
15:
12:
11:
5:
6685:
6683:
6675:
6674:
6669:
6664:
6654:
6653:
6650:
6649:
6641:
6632:
6619:
6618:External links
6616:
6615:
6614:
6602:P. Dembowski,
6600:
6579:
6566:
6565:
6550:
6548:
6545:
6544:
6543:
6538:
6531:
6528:
6509:
6489:
6486:
6483:
6480:
6477:
6474:
6471:
6468:
6465:
6460:
6456:
6452:
6432:
6412:
6392:
6389:
6385:bundle theorem
6339:
6338:
6337:
6336:
6324:
6294:
6291:
6288:
6285:
6282:
6277:
6257:
6256:
6255:
6254:
6240:
6236:
6232:
6229:
6224:
6220:
6216:
6213:
6210:
6207:
6204:
6201:
6198:
6177:
6173:
6170:
6159:
6145:
6141:
6137:
6132:
6128:
6124:
6121:
6118:
6115:
6112:
6109:
6106:
6085:
6081:
6078:
6047:
6043:
6040:
6031:If we choose
6026:
6025:
6024:
6023:
6011:
6008:
6005:
6002:
5999:
5996:
5993:
5990:
5987:
5984:
5981:
5978:
5975:
5972:
5969:
5966:
5963:
5960:
5957:
5937:
5934:
5929:
5925:
5921:
5918:
5915:
5912:
5907:
5903:
5891:
5877:
5873:
5869:
5866:
5863:
5860:
5855:
5851:
5847:
5844:
5841:
5838:
5835:
5832:
5829:
5809:
5806:
5803:
5800:
5797:
5777:
5774:
5771:
5767:
5764:
5760:
5757:
5737:
5734:
5731:
5728:
5725:
5720:
5679:
5676:
5673:
5670:
5667:
5662:
5637:
5634:
5631:
5628:
5625:
5620:
5587:
5583:
5579:
5576:
5573:
5570:
5567:
5562:
5558:
5535:
5532:
5529:
5526:
5523:
5518:
5482:
5479:
5476:
5473:
5470:
5465:
5452:suitable pairs
5437:
5433:
5429:
5424:
5420:
5399:
5378:
5357:
5334:
5324:quadratic form
5319:
5316:
5315:
5314:
5303:
5300:
5297:
5294:
5289:
5285:
5281:
5278:
5275:
5271:
5265:
5259:
5238:
5235:
5230:
5226:
5222:
5218:
5212:
5206:
5185:
5163:
5157:
5134:
5114:
5111:
5108:
5103:
5098:
5093:
5088:
5085:
5080:
5063:
5062:
5051:
5046:
5031:is called the
5020:
5017:
5013:
5009:
5005:
5001:
4998:
4976:
4971:
4968:
4948:
4945:
4942:
4937:
4932:
4927:
4922:
4919:
4914:
4896:
4895:
4879:
4876:
4872:
4866:
4860:
4848:
4847:
4833:
4827:
4802:
4797:
4794:
4785:For any point
4766:
4763:
4760:
4755:
4750:
4745:
4740:
4705:
4683:
4677:
4647:
4644:
4641:
4638:
4633:
4628:
4625:
4622:
4619:
4616:
4613:
4610:
4607:
4604:
4601:
4598:
4595:
4592:
4589:
4586:
4583:
4578:
4573:
4570:
4565:
4559:
4534:
4529:
4526:
4506:
4503:
4500:
4495:
4490:
4485:
4480:
4477:
4472:
4444:
4441:
4435:
4430:
4427:
4422:
4416:
4412:
4406:
4400:
4379:
4376:
4373:
4370:
4366:
4362:
4358:
4354:
4349:
4344:
4341:
4338:
4335:
4332:
4329:
4324:
4318:
4315:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4291:
4288:
4285:
4282:
4277:
4252:
4225:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4180:
4177:
4157:
4133:
4130:
4127:
4124:
4121:
4118:
4115:
4104:
4103:
4097:
4085:
4061:
4058:
4037:
4017:
4014:
4011:
4008:
4004:
4001:
3997:
3994:
3973:
3970:
3966:
3963:
3960:
3957:
3936:
3933:
3912:
3909:
3906:
3886:
3883:
3880:
3860:
3851:For any cycle
3846:
3834:
3831:
3828:
3825:
3822:
3813:that contains
3802:
3782:
3779:
3776:
3773:
3770:
3737:
3710:
3685:
3682:
3679:
3674:
3669:
3664:
3659:
3656:
3651:
3626:
3623:
3622:
3621:
3610:
3607:
3604:
3601:
3598:
3595:
3592:
3589:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3507:
3504:
3501:
3498:
3478:
3475:
3472:
3469:
3466:
3463:
3460:
3457:
3454:
3443:
3431:
3428:
3425:
3422:
3419:
3416:
3413:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3328:
3325:
3322:
3319:
3316:
3313:
3310:
3307:
3304:
3301:
3296:
3292:
3288:
3283:
3279:
3267:
3256:
3250:
3247:
3241:
3238:
3218:
3212:
3209:
3203:
3200:
3197:
3194:
3191:
3171:
3168:
3165:
3162:
3157:
3153:
3149:
3144:
3140:
3136:
3131:
3127:
3106:
3103:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3041:
3031:
3030:
3019:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2988:
2979:
2975:
2971:
2966:
2962:
2958:
2955:
2948:
2944:
2940:
2935:
2931:
2924:
2916:
2912:
2908:
2903:
2899:
2895:
2892:
2888:
2883:
2875:
2871:
2867:
2862:
2858:
2854:
2851:
2847:
2841:
2837:
2834:
2831:
2828:
2825:
2822:
2816:
2813:
2782:
2779:
2776:
2771:
2766:
2761:
2756:
2736:
2733:
2730:
2725:
2720:
2715:
2710:
2671:
2668:
2663:
2660:
2655:
2631:
2628:
2624:
2618:
2615:
2579:
2576:
2573:
2568:
2563:
2558:
2553:
2525:
2521:
2517:
2514:
2511:
2508:
2505:
2484:
2459:
2456:
2453:
2450:
2446:
2434:
2433:
2420:
2417:
2414:
2411:
2402:
2397:
2394:
2389:
2386:
2374:
2373:
2361:
2358:
2336:
2333:
2330:
2327:
2318:
2315:
2312:
2309:
2300:
2297:
2294:
2291:
2285:
2280:
2277:
2271:
2268:
2256:
2255:
2242:
2238:
2235:
2214:
2211:
2208:
2205:
2196:
2193:
2190:
2187:
2184:
2181:
2169:
2168:
2155:
2151:
2148:
2127:
2124:
2121:
2118:
2109:
2106:
2103:
2100:
2097:
2071:
2058:
2057:
2056:
2055:
2054:
2053:
2042:
2039:
2036:
2033:
2030:
2027:
2023:
2019:
2016:
2013:
2009:
2005:
2000:
1996:
1992:
1981:
1978:
1973:
1969:
1964:
1960:
1956:
1953:
1950:
1944:
1939:
1935:
1931:
1928:
1925:
1922:
1918:
1914:
1911:
1908:
1905:
1902:
1887:
1886:
1875:
1871:
1867:
1864:
1861:
1857:
1853:
1850:
1847:
1844:
1838:
1835:
1832:
1829:
1826:
1823:
1809:
1806:
1803:
1800:
1795:
1791:
1788:
1782:
1779:
1776:
1773:
1769:
1765:
1762:
1759:
1756:
1753:
1748:
1735:
1734:
1721:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1695:
1691:
1686:
1661:
1641:
1638:
1635:
1632:
1629:
1624:
1621:
1597:
1592:
1587:
1584:
1581:
1578:
1575:
1572:
1552:
1549:
1546:
1543:
1540:
1537:
1517:
1514:
1511:
1506:
1501:
1496:
1491:
1481:
1480:
1468:
1444:
1441:
1420:
1400:
1397:
1394:
1391:
1387:
1384:
1380:
1377:
1356:
1353:
1349:
1346:
1343:
1340:
1319:
1316:
1295:
1292:
1289:
1269:
1266:
1263:
1243:
1234:For any cycle
1232:
1220:
1217:
1214:
1211:
1208:
1188:
1168:
1165:
1162:
1159:
1156:
1132:
1129:
1126:
1121:
1116:
1111:
1106:
1090:is called the
1079:
1076:
1073:
1068:
1063:
1058:
1053:
1042:
1041:
1025:
1022:
1018:
1014:
1009:
999:
996:
993:
990:
987:
984:
981:
977:
973:
968:
958:
955:
952:
949:
946:
943:
940:
937:
934:
929:
917:
900:
896:
893:
890:
887:
884:
881:
878:
873:
868:
863:
858:
842:
841:
830:
827:
824:
819:
814:
809:
804:
790:
789:
777:
774:
771:
765:
760:
756:
752:
747:
743:
737:
733:
729:
726:
723:
720:
715:
711:
705:
701:
697:
694:
691:
688:
685:
680:
676:
672:
669:
666:
661:
657:
653:
650:
647:
644:
617:
614:
611:
591:
588:
585:
582:
579:
576:
546:
541:
513:
509:
505:
500:
496:
492:
489:
486:
483:
480:
477:
474:
464:quadratic form
451:
447:
443:
438:
416:
413:
384:
381:
376:
372:
368:
363:
359:
335:
334:
322:
302:
282:
279:
276:
255:
252:
231:
211:
191:
171:
151:
142:for any cycle
91:
88:
80:Laguerre plane
24:
14:
13:
10:
9:
6:
4:
3:
2:
6684:
6673:
6670:
6668:
6665:
6663:
6660:
6659:
6657:
6648:
6647:
6644:Lecture Note
6642:
6640:
6636:
6633:
6631:
6630:
6625:
6622:
6621:
6617:
6613:
6612:3-540-61786-8
6609:
6605:
6601:
6599:
6598:0-444-88355-X
6595:
6591:
6587:
6586:
6580:
6577:
6573:
6569:
6568:
6562:
6561:
6555:
6552:
6546:
6542:
6539:
6537:
6534:
6533:
6529:
6527:
6524:
6521:
6507:
6484:
6481:
6478:
6475:
6472:
6469:
6466:
6463:
6458:
6454:
6430:
6410:
6402:
6398:
6390:
6388:
6386:
6382:
6381:quadratic set
6378:
6377:tangent plane
6373:
6372:quadratic set
6369:
6365:
6361:
6360:not miquelian
6357:
6353:
6351:
6347:
6343:
6322:
6314:
6310:
6309:
6308:
6307:
6306:
6289:
6286:
6283:
6265:
6261:
6238:
6234:
6230:
6227:
6222:
6218:
6214:
6208:
6205:
6202:
6196:
6171:
6168:
6160:
6143:
6139:
6135:
6130:
6126:
6122:
6116:
6113:
6110:
6104:
6079:
6076:
6068:
6067:
6066:
6065:
6064:
6062:
6041:
6038:
6030:
6003:
6000:
5997:
5991:
5985:
5982:
5979:
5973:
5967:
5964:
5961:
5935:
5932:
5927:
5923:
5919:
5916:
5913:
5910:
5905:
5901:
5892:
5875:
5871:
5867:
5864:
5861:
5858:
5853:
5849:
5845:
5839:
5836:
5833:
5827:
5804:
5801:
5798:
5772:
5758:
5755:
5732:
5729:
5726:
5709:
5708:
5707:
5706:
5705:
5703:
5702:minimal model
5699:
5695:
5693:
5674:
5671:
5668:
5649:
5632:
5629:
5626:
5608:
5604:
5601:
5585:
5581:
5577:
5574:
5571:
5568:
5565:
5560:
5556:
5530:
5527:
5524:
5506:
5498:
5494:
5477:
5474:
5471:
5453:
5435:
5431:
5427:
5422:
5418:
5397:
5390:by any field
5355:
5348:
5332:
5325:
5317:
5301:
5295:
5292:
5287:
5283:
5276:
5273:
5236:
5233:
5228:
5224:
5220:
5183:
5161:
5132:
5109:
5106:
5096:
5083:
5068:
5067:
5066:
5049:
5034:
5018:
5015:
5007:
4999:
4996:
4969:
4966:
4943:
4940:
4930:
4917:
4902:
4901:
4900:
4893:
4892:
4891:
4874:
4831:
4795:
4792:
4784:
4781:
4780:
4779:
4761:
4758:
4748:
4730:
4726:
4723:
4721:
4717:
4663:
4661:
4642:
4639:
4626:
4623:
4620:
4617:
4614:
4608:
4599:
4593:
4587:
4568:
4563:
4527:
4524:
4501:
4498:
4488:
4475:
4459:
4455:
4442:
4439:
4428:
4425:
4414:
4377:
4371:
4368:
4360:
4352:
4342:
4339:
4336:
4333:
4327:
4316:
4307:
4304:
4301:
4298:
4295:
4292:
4289:
4286:
4280:
4241:
4237:
4223:
4213:
4199:
4196:
4193:
4190:
4187:
4184:
4181:
4178:
4175:
4155:
4147:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4101:
4098:
4083:
4075:
4059:
4056:
4035:
4012:
4006:
4002:
3999:
3995:
3992:
3971:
3968:
3964:
3961:
3958:
3955:
3934:
3931:
3910:
3907:
3904:
3884:
3881:
3878:
3858:
3850:
3847:
3832:
3829:
3826:
3823:
3820:
3800:
3780:
3777:
3774:
3771:
3768:
3760:
3757:
3756:
3755:
3753:
3726:
3725:set of cycles
3699:
3680:
3677:
3667:
3654:
3634:
3630:
3624:
3608:
3602:
3599:
3596:
3593:
3590:
3564:
3561:
3558:
3555:
3552:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3502:
3499:
3496:
3476:
3473:
3470:
3467:
3464:
3461:
3458:
3455:
3452:
3444:
3426:
3423:
3420:
3417:
3414:
3391:
3388:
3385:
3382:
3379:
3373:
3370:
3367:
3361:
3358:
3355:
3352:
3349:
3346:
3326:
3323:
3320:
3317:
3314:
3311:
3308:
3305:
3302:
3299:
3294:
3290:
3286:
3281:
3277:
3268:
3254:
3248:
3245:
3239:
3236:
3210:
3207:
3201:
3198:
3195:
3192:
3169:
3166:
3163:
3160:
3155:
3151:
3147:
3142:
3138:
3134:
3129:
3125:
3104:
3101:
3093:
3092:
3091:
3074:
3071:
3068:
3065:
3062:
3017:
3008:
3005:
3002:
2999:
2996:
2990:
2986:
2977:
2973:
2969:
2964:
2960:
2956:
2953:
2946:
2942:
2938:
2933:
2929:
2922:
2914:
2910:
2906:
2901:
2897:
2893:
2890:
2886:
2881:
2873:
2869:
2865:
2860:
2856:
2852:
2849:
2845:
2839:
2829:
2826:
2823:
2814:
2804:
2803:
2802:
2800:
2796:
2777:
2774:
2764:
2731:
2728:
2718:
2700:
2691:
2687:
2685:
2669:
2666:
2658:
2653:
2626:
2622:
2613:
2605:
2601:
2597:
2593:
2574:
2571:
2561:
2543:
2539:
2515:
2512:
2506:
2503:
2473:
2448:
2418:
2400:
2392:
2384:
2376:
2375:
2359:
2356:
2334:
2331:
2316:
2307:
2298:
2295:
2292:
2289:
2283:
2278:
2275:
2266:
2258:
2257:
2254:(translation)
2236:
2233:
2212:
2194:
2191:
2188:
2185:
2179:
2171:
2170:
2149:
2146:
2125:
2107:
2104:
2101:
2095:
2087:
2086:
2085:
2040:
2034:
2031:
2028:
2025:
2017:
2014:
2011:
2003:
1998:
1994:
1990:
1979:
1976:
1962:
1958:
1954:
1951:
1937:
1933:
1929:
1926:
1920:
1912:
1909:
1900:
1893:
1892:
1891:
1890:
1889:
1888:
1865:
1862:
1859:
1851:
1848:
1845:
1842:
1836:
1824:
1807:
1804:
1801:
1798:
1789:
1786:
1780:
1777:
1774:
1771:
1763:
1760:
1751:
1737:
1736:
1715:
1709:
1697:
1689:
1675:
1674:
1673:
1659:
1639:
1636:
1633:
1630:
1627:
1619:
1595:
1585:
1579:
1576:
1573:
1550:
1547:
1544:
1541:
1538:
1535:
1512:
1509:
1499:
1466:
1458:
1442:
1439:
1418:
1395:
1389:
1385:
1382:
1378:
1375:
1354:
1351:
1347:
1344:
1341:
1338:
1317:
1314:
1293:
1290:
1287:
1267:
1264:
1261:
1241:
1233:
1218:
1215:
1212:
1209:
1206:
1186:
1166:
1163:
1160:
1157:
1154:
1146:
1145:
1144:
1127:
1124:
1114:
1095:
1093:
1074:
1071:
1061:
1039:
1038:set of cycles
997:
994:
991:
985:
956:
953:
941:
938:
932:
918:
915:
914:set of points
894:
888:
876:
871:
861:
847:
846:
845:
825:
822:
812:
795:
794:
793:
775:
772:
769:
763:
758:
754:
750:
745:
735:
731:
727:
724:
718:
713:
703:
699:
695:
692:
686:
678:
674:
670:
667:
664:
659:
655:
651:
648:
642:
635:
634:
633:
631:
615:
612:
609:
589:
586:
583:
580:
577:
574:
566:
562:
544:
529:
511:
507:
503:
498:
494:
490:
484:
481:
478:
472:
465:
421:
414:
412:
410:
406:
402:
398:
382:
379:
374:
370:
366:
361:
357:
348:
344:
340:
320:
300:
293:and touching
280:
277:
274:
253:
250:
229:
209:
189:
169:
149:
141:
140:
139:
137:
133:
129:
125:
120:
118:
114:
110:
106:
96:
89:
87:
85:
81:
77:
73:
68:
66:
61:
56:
54:
50:
46:
42:
38:
34:
31:(named after
30:
19:
18:Moebius plane
6645:
6638:
6627:
6624:Möbius plane
6603:
6583:Handbook of
6582:
6571:
6559:
6554:
6525:
6522:
6401:affine plane
6397:block design
6394:
6376:
6363:
6359:
6355:
6354:
6341:
6340:
6259:
6258:
6158:is suitable.
6060:
6028:
6027:
5701:
5697:
5696:
5691:
5690:is called a
5650:
5606:
5605:
5602:
5504:
5503:
5451:
5321:
5064:
5032:
4989:the integer
4959:and a cycle
4897:
4849:
4815:the residue
4782:
4728:
4727:
4724:
4719:
4718:
4664:
4659:
4460:
4456:
4266:
4214:
4145:
4106:Four points
4105:
4099:
4073:
3871:, any point
3848:
3758:
3752:Möbius plane
3751:
3750:is called a
3724:
3697:
3639:
3628:
3032:
2696:
2683:
2591:
2436:Considering
2435:
2059:
1482:
1456:
1254:, any point
1096:
1091:
1043:
1037:
913:
843:
791:
629:
564:
560:
426:
404:
338:
336:
127:
123:
121:
115:, we get an
112:
101:
69:
57:
44:
29:Möbius plane
28:
26:
6161:The choice
6069:The choice
6061:no suitable
3229:and radius
3182:, midpoint
3090:and maps
76:Benz planes
6656:Categories
6635:Benz plane
6547:References
2795:isomorphic
2600:transitive
313:(at point
65:reflection
60:involution
6570:W. Benz,
6423:, i.e. a
6403:of order
6290:ρ
6228:−
6197:ρ
6105:ρ
5828:ρ
5733:ρ
5675:ρ
5633:ρ
5531:ρ
5478:ρ
5333:ρ
5110:∈
5016:−
4970:∈
4944:∈
4878:∞
4796:∈
4762:∈
4704:∞
4696:at point
4682:∞
4643:∈
4627:∈
4621:∈
4615:∣
4603:∖
4582:∖
4528:∈
4502:∈
4343:⊂
4337:∣
4311:∞
4251:∞
4197:∈
4146:concyclic
3996:∩
3965:∈
3908:∉
3882:∈
3698:point set
3681:∈
3512:−
3445:the line
3362:−
3318:−
3309:−
3300:−
3161:−
3040:Φ
2836:→
2812:Φ
2778:∈
2732:∈
2662:¯
2630:¯
2617:→
2604:inversion
2575:∈
2507:
2455:∞
2449:∪
2416:∞
2413:→
2410:∞
2396:¯
2388:→
2357:±
2329:→
2326:∞
2314:∞
2311:→
2293:≠
2270:→
2237:∈
2210:∞
2207:→
2204:∞
2183:→
2150:∈
2123:∞
2120:→
2117:∞
2099:→
2018:∈
2004:∈
1991:∣
1972:¯
1955:−
1930:−
1921:∣
1913:∈
1901:∪
1866:∈
1852:∈
1846:≠
1837:∣
1831:∞
1825:∪
1794:¯
1772:∣
1764:∈
1716:∉
1713:∞
1704:∞
1698:∪
1634:−
1623:¯
1586:∈
1513:∈
1379:∩
1348:∈
1291:∉
1265:∈
1128:∈
1075:∈
995:∣
986:∪
954:∣
948:∞
942:∪
895:∉
892:∞
883:∞
877:∪
826:∈
728:−
696:−
671:−
652:−
643:ρ
563:set, the
473:ρ
462:with the
405:miquelian
105:collinear
35:) is the
6590:Elsevier
6576:Springer
6530:See also
4729:Theorem:
4720:Theorem:
4236:points:
4060:′
4003:′
3972:′
3935:′
2538:PGL(2,C)
1987:(circle)
1443:′
1386:′
1355:′
1318:′
254:′
162:, point
124:touching
6637:in the
6626:in the
6592:(1995)
6356:Remark:
6342:Remark:
6313:quadric
6266:shows:
6260:Remark:
6029:Remark:
5820:) and
5788:(field
5698:Remark:
4390:Hence:
2377:(4)
2259:(3)
2172:(2)
2088:(1)
1411:, i.e.
912:, the
559:is the
222:not on
6610:
6596:
6578:(1973)
5249:, c)
5196:, b)
3948:with:
3015:
2818:
2594:, its
2407:
2404:
2323:
2320:
2305:
2302:
2287:
2201:
2198:
2114:
2111:
1983:
1840:
1819:
1815:(line)
1811:
1733:, and
1331:with:
767:
630:circle
628:and a
397:fields
113:cycles
6368:ovoid
5748:with
5347:field
5033:order
4168:with
4074:touch
3696:with
2536:(see
2474:over
2226:with
2139:with
1457:touch
1044:Then
916:, and
844:with
565:lines
561:point
128:touch
6608:ISBN
6594:ISBN
6346:here
5700:The
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