5610:
28:
5202:
3177:
4563:
3989:
5100:
2523:
4326:
3757:
2968:
4332:
1745:
5605:{\displaystyle {\begin{aligned}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}&{\text{(zero part)}}\\z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}&{\text{(oval)}}\\z_{1}^{2}+z_{2}^{2}-z_{3}^{2}-z_{4}^{2}&{\text{(ring)}}\\-z_{1}^{2}-z_{2}^{2}-z_{3}^{2}+z_{4}^{2}\\-z_{1}^{2}-z_{2}^{2}-z_{3}^{2}-z_{4}^{2}\end{aligned}}}
2962:
4883:
363:
For the distance between two points in the interval, the Cayley–Klein metric uses the logarithm of the ratio of the points. As a ratio is preserved when numerator and denominator are equally re-proportioned, so the logarithm of such ratios is preserved. This flexibility of ratios enables the movement
1909:
The question recently arose in conversation whether a dissertation of 2 lines could deserve and get a
Fellowship. ... Cayley's projective definition of length is a clear case if we may interpret "2 lines" with reasonable latitude. ... With Cayley the importance of the idea is obvious at first sight.
4864:
In his 1873 paper he pointed out the relation between the Cayley metric and transformation groups. In particular, quadratic equations with real coefficients, corresponding to surfaces of second degree, can be transformed into a sum of squares, of which the difference between the number of positive
135:
of the geometry. Klein (1871, 1873) removed the last remnants of metric concepts from von Staudt's work and combined it with Cayley's theory, in order to base Cayley's new metric on logarithm and the cross-ratio as a number generated by the geometric arrangement of four points. This procedure is
2326:
2320:
4006:
3412:
6111:
for hyperbolic space. Klein's lectures on non-Euclidean geometry were posthumously republished as one volume and significantly edited by
Walther Rosemann in 1928. An historical analysis of Klein's work on non-Euclidean geometry was given by A'Campo and Papadopoulos (2014).
1560:
4739:, which relates to hyperbolic geometry when real, and to elliptic geometry when imaginary. The transformations leaving invariant this form represent motions in the respective non–Euclidean space. Alternatively, he used the equation of the circle in the form
193:
There are three absolutes in the real projective line, seven in the real projective plane, and 18 in real projective space. All classical non-Euclidean projective spaces as hyperbolic, elliptic, Galilean and
Minkowskian and their duals can be defined this
3172:{\displaystyle {\begin{array}{c}\cos ^{-1}{\dfrac {\sum a_{\alpha \beta }x_{\alpha }y_{\beta }}{{\sqrt {\sum a_{\alpha \beta }x_{\alpha }x_{\beta }}}{\sqrt {\sum a_{\alpha \beta }y_{\alpha }y_{\beta }}}}},\ \\\left(\alpha ,\beta =1,2,3\right)\end{array}}}
4558:{\displaystyle {\begin{matrix}\Omega _{xx}=\sum a_{\alpha \beta }x_{\alpha }x_{\beta }=0\\\Omega _{yy}=\sum a_{\alpha \beta }y_{\alpha }y_{\beta }=0\\\Omega _{xy}=\sum a_{\alpha \beta }x_{\alpha }y_{\beta }\\\left(\alpha ,\beta =1,2\right)\end{matrix}}}
1565:
3984:{\displaystyle c\log {\frac {\Omega _{xy}+{\sqrt {\Omega _{xy}^{2}-\Omega _{xx}\Omega _{yy}}}}{\Omega _{xy}-{\sqrt {\Omega _{xy}^{2}-\Omega _{xx}\Omega _{yy}}}}}=2ic\cdot \arccos {\frac {\Omega _{xy}}{\sqrt {\Omega _{xx}\cdot \Omega _{yy}}}}}
3685:
5197:
2718:
5095:{\displaystyle \sum _{\alpha ,\beta =1}^{3}a_{\alpha \beta }x_{\alpha }x_{\beta }=0\rightarrow {\begin{matrix}x^{2}+y^{2}+4k^{2}t^{2}=0&{\text{(elliptic)}}\\x^{2}+y^{2}-4k^{2}t^{2}=0&{\text{(hyperbolic)}}\end{matrix}}}
2092:
2518:{\displaystyle {\begin{array}{c}\cos ^{-1}{\dfrac {\sum a_{\alpha \beta }x_{\alpha }y_{\beta }}{{\sqrt {\sum a_{\alpha \beta }x_{\alpha }x_{\beta }}}{\sqrt {\sum a_{\alpha \beta }y_{\alpha }y_{\beta }}}}}\\\left\end{array}}}
136:
necessary to avoid a circular definition of distance if cross-ratio is merely a double ratio of previously defined distances. In particular, he showed that non-Euclidean geometries can be based on the Cayley–Klein metric.
4321:{\displaystyle {\begin{matrix}\Omega _{xx}=ax_{1}^{2}+2bx_{1}x_{2}+cx_{2}^{2}\\\Omega _{yy}=ay_{1}^{2}+2by_{1}y_{2}+cy_{2}^{2}\\\Omega _{xy}=ax_{1}y_{1}+b\left(x_{1}y_{2}+x_{2}y_{1}\right)+cx_{2}y_{2}\end{matrix}}}
2741:
3248:
339:
441:
5958:
3594:
6050:
5796:
5704:
1291:
4880:
In the first volume of his lectures on non-Euclidean geometry in the winter semester 1889/90 (published 1892/1893), he discussed the non-Euclidean plane, using these expressions for the absolute:
5207:
4737:
4815:
2115:
1435:
6906:
4857:
is negative. In space, he discussed fundamental surfaces of second degree, according to which imaginary ones refer to elliptic geometry, real and rectilinear ones correspond to a one-sheet
1113:
1199:
2631:
1422:
1029:
788:
704:
630:
5861:
31:
The metric distance between two points inside the absolute is the logarithm of the cross ratio formed by these two points and the two intersections of their line with the absolute
6109:
3473:
3241:
2007:
1740:{\textstyle {\bigl (}{\frac {dx}{dt}}{\bigr )}{\vphantom {)}}^{2}+{\bigl (}{\frac {dy}{dt}}{\bigr )}{\vphantom {)}}^{2}+{\bigl (}{\frac {dz}{dt}}{\bigr )}{\vphantom {)}}^{2}=1}
1350:
5116:
4628:
4598:
3750:
3720:
222:. Ordinarily projective geometry is not associated with metric geometry, but a device with homography and natural logarithm makes the connection. Start with two points
7182:
71:
in papers in 1871 and 1873, and subsequent books and papers. The Cayley–Klein metrics are a unifying idea in geometry since the method is used to provide metrics in
4660:
5113:
In the second volume containing the lectures of the summer semester 1890 (also published 1892/1893), Klein discussed non-Euclidean space with the Cayley metric
4855:
4835:
3600:
127:
of a cross-ratio. Eventually, Cayley (1859) formulated relations to express distance in terms of a projective metric, and related them to general quadrics or
3486:(1871) reformulated Cayley's expressions as follows: He wrote the absolute (which he called fundamental conic section) in terms of homogeneous coordinates:
2637:
162:. Indeed, cross-ratio is invariant under any collineation, and the stable absolute enables the metric comparison, which will be equality. For example, the
4869:). If the sign of all squares is the same, the surface is imaginary with positive curvature. If one sign differs from the others, the surface becomes an
2013:
5199:
and went on to show that variants of this quaternary quadratic form can be brought into one of the following five forms by real linear transformations
6789:
5863:. He eventually discussed their invariance with respect to collineations and Möbius transformations representing motions in Non-Euclidean spaces.
6692:
6666:
2957:{\displaystyle \cos ^{-1}{\frac {(a,\dots )(x,y,z)\left(x',y',z'\right)}{{\sqrt {(a,\dots )(x,y,z)^{2}}}{\sqrt {(a,\dots )(x',y',z')^{2}}}}}}
360:)/2 goes to . The natural logarithm takes the image of the interval to the real line, with the log of the image of the midpoint being 0.
1926:(1859) defined the "absolute" upon which he based his projective metric as a general equation of a surface of second degree in terms of
6965:
240:
378:
5876:
1776:
of special relativity were pointed out by Klein in 1910, as well as in the 1928 edition of his lectures on non-Euclidean geometry.
3502:
5963:
5709:
5617:
7154:
7119:
6948:
6823:
3407:{\displaystyle \cos ^{-1}{\frac {xx'+yy'+zz'}{{\sqrt {x^{2}+y^{2}+z^{2}}}{\sqrt {x^{\prime 2}+y^{\prime 2}+z^{\prime 2}}}}}}
1204:
6847:
2315:{\displaystyle \cos ^{-1}{\frac {(a,b,c)(x,y)\left(x',y'\right)}{{\sqrt {(a,b,c)(x,y)^{2}}}{\sqrt {(a,b,c)(x',y')^{2}}}}}}
1555:{\textstyle {\bigl (}{\frac {dx}{dt}}{\bigr )}{\vphantom {)}}^{2}+{\bigl (}{\frac {dy}{dt}}{\bigr )}{\vphantom {)}}^{2}=1}
471:. Frequently cross ratio is introduced as a function of four values. Here three define a homography and the fourth is the
7092:
A'Campo, N.; Papadopoulos, A. (2014). "On Klein's So-called Non-Euclidean geometry". In Ji, L.; Papadopoulos, A. (eds.).
4665:
17:
1769:
is confined to the interior of a unit sphere, and the surface of the sphere forms the Cayley absolute for the geometry.
4742:
950:
1432:. Similarly, the equations of the unit circle or unit sphere in hyperbolic geometry correspond to physical velocities
901:
in this disk, there is a unique generalized circle that meets the unit circle at right angles, say intersecting it at
7137:
Nielsen, Frank; Muzellec, Boris; Nock, Richard (2016), "Classification with mixtures of curved mahalanobis metrics",
4861:
with no relation to one of the three main geometries, while real and non-rectilinear ones refer to hyperbolic space.
1034:
6969:
1890:
for the pseudo-Euclidean geometry. These generalized complex numbers associate with their geometries as ordinary
1130:
2546:
6888:
159:
4866:
1785:
108:
722:
635:
552:
7205:
5801:
1355:
962:
6724:
Vorlesungen über die
Theorie der automorphen Functionen – Erster Band: Die gruppentheoretischen Grundlagen
475:
of the homography. The distance of this fourth point from 0 is the logarithm of the evaluated homography.
183:
6798:
6055:
3419:
3187:
6774:
6753:, Book VI Chapter 1: Theory of Distance, pp. 347–70, especially Section 199 Cayley's Theory of Distance.
1946:
6940:
5706:
was used by Klein as the Cayley absolute of elliptic geometry, while to hyperbolic geometry he related
6617:
6584:
955:
In his lectures on the history of mathematics from 1919/20, published posthumously 1926, Klein wrote:
6758:
1296:
7210:
6741:
171:
96:
7180:
Drösler, Jan (1979), "Foundations of multidimensional metric scaling in Cayley–Klein geometries",
5107:
4603:
4573:
3725:
3695:
886:
1927:
1833:
1116:
799:
147:
44:
6746:
1772:
Additional details about the relation between the Cayley–Klein metric for hyperbolic space and
1429:
934:
472:
219:
84:
938:
167:
6557:
3752:
for two elements, he defined the metrical distance between them in terms of the cross ratio:
807:
7019:
Martini, Horst; Spirova, Margarita (2008). "Circle geometry in affine Cayley–Klein planes".
7077:
7048:
Struve, Horst; Struve, Rolf (2010), "Non-euclidean geometries: the Cayley–Klein approach",
7004:
6958:
5870:
4633:
1887:
104:
40:
7094:
Sophus Lie and Felix Klein: The
Erlangen Program and Its Impact in Mathematics and Physics
6895:
6876:
3680:{\displaystyle \Omega =\sum _{\alpha ,\beta =1,2}a_{\alpha \beta }x_{\alpha }x_{\beta }=0}
8:
6865:
6854:
6711:
6529:
175:
72:
5869:
and Klein summarized all of this in the introduction to the first volume of lectures on
7160:
7125:
7097:
7081:
7036:
7008:
6781:
6722:
6686:
6660:
6637:
6604:
4840:
4820:
1120:
882:
80:
123:(1853), who showed that the Euclidean angle between two lines can be expressed as the
7150:
7115:
7085:
7065:
7012:
6992:
6944:
6819:
6676:
6652:
Nicht-Euklidische
Geometrie I, Vorlesung gehalten während des Wintersemesters 1889–90
6650:
6641:
6608:
5192:{\displaystyle \sum _{\alpha ,\beta =1}^{4}a_{\alpha \beta }x_{\alpha }x_{\beta }=0,}
2713:{\displaystyle \sum _{\alpha ,\beta =1}^{3}a_{\alpha \beta }x_{\alpha }x_{\beta }=0,}
523:
189:
The extent of Cayley–Klein geometry was summarized by Horst and Rolf Struve in 2004:
76:
7040:
6925:
6540:
1852:= (0,1,0), then a parabola with diameter parallel to y-axis is an isotropic circle.
456:
7164:
7142:
7129:
7107:
7057:
7028:
6984:
6975:
Struve, Horst; Struve, Rolf (2004), "Projective spaces with Cayley–Klein metrics",
6921:
6837:
6811:
6770:
6732:
6678:
Nicht-Euklidische
Geometrie II, Vorlesung gehalten während des Sommersemesters 1890
6629:
6596:
6569:
930:
143:
116:
52:
6750:
2087:{\displaystyle \sum _{\alpha ,\beta =1,2}a_{\alpha \beta }x_{\alpha }x_{\beta }=0}
7073:
7000:
6954:
6934:
6841:
6815:
1789:
1773:
199:
120:
100:
6797:
6121:
1891:
1748:
791:
7146:
7061:
7032:
6988:
6342:
7199:
7069:
6996:
5866:
1923:
890:
714:
503:
128:
60:
6574:
5103:
506:
as it permutes the points of the space. Such a homography induces one on P(
155:
115:
as fundamental to the measure on a line. Another important insight was the
4570:
In the plane, the same relations for metrical distances hold, except that
1428:, and its transformation leaving the absolute invariant can be related to
889:, the source of the motions of this disk that leave the unit circle as an
4874:
4858:
3483:
1883:
1293:
in hyperbolic geometry (as discussed above), correspond to the intervals
710:
163:
112:
68:
56:
1879:
on the line at infinity. These curves are the pseudo-Euclidean circles.
6633:
6600:
6499:
214:
Cayley–Klein metric is first illustrated on the real projective line P(
203:
150:. It depends upon the selection of a quadric or conic that becomes the
7111:
6315:
6313:
885:
in the disk of the complex plane. This class of curves is permuted by
4870:
1784:
In 2008 Horst
Martini and Margarita Spirova generalized the first of
1425:
522:, the cross ratio remains invariant. The higher homographies provide
179:
124:
858:
is the logarithm of the value of the homography, generated above by
63:'s essay "On the theory of distance" where he calls the quadric the
6310:
875:
542:
Suppose a unit circle is selected for the absolute. It may be in P(
531:
7102:
6878:
Vorlesungen über die
Entwicklung der Mathematik im 19. Jahrhundert
6439:
234:). In the canonical embedding they are and . The homographic map
4662:
each. As fundamental conic section he discussed the special case
1796:
If the absolute contains a line, then one obtains a subfamily of
334:{\displaystyle {\begin{pmatrix}-1&1\\p&-q\end{pmatrix}}=}
48:
4837:
is positive (Beltrami–Klein model) or to elliptic geometry when
27:
6368:
6139:
6137:
87:
rests largely on the footing provided by Cayley–Klein metrics.
436:{\displaystyle {\begin{pmatrix}1&0\\0&w\end{pmatrix}}}
7139:
2016 IEEE International
Conference on Image Processing (ICIP)
6487:
6475:
5953:{\displaystyle e\left(z_{1}^{2}+z_{2}^{2}\right)-z_{3}^{2}=0}
6134:
3589:{\displaystyle \Omega =ax_{1}^{2}+2bx_{1}x_{2}+cx_{2}^{2}=0}
6799:"Über die geometrischen Grundlagen der Lorentzgruppe"
6324:
447:
The composition of the first and second homographies takes
7183:
British Journal of Mathematical and Statistical Psychology
6380:
842:). Then they lie on a line which intersects the circle at
818:) introduced in the previous section is available since P(
455:
in the interval. The composed homographies are called the
6562:
Philosophical Transactions of the Royal Society of London
6286:
6045:{\displaystyle z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0}
5791:{\displaystyle z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0}
5699:{\displaystyle z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}=0}
103:(1847) is an approach to geometry that is independent of
6274:
7091:
6319:
6875:
Klein, F. (1926). Courant, R.; Neugebauer, O. (eds.).
6759:"Analytische Beiträge zur nichteuklidischen Geometrie"
6193:
4957:
4337:
4011:
2973:
2331:
1568:
1438:
1358:
1299:
1286:{\textstyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0}
1207:
1133:
965:
402:
264:
67:. The construction was developed in further detail by
6298:
6058:
5966:
5879:
5804:
5712:
5620:
5205:
5119:
4886:
4865:
and negative signs remains equal (this is now called
4843:
4823:
4745:
4668:
4636:
4606:
4576:
4335:
4009:
3760:
3728:
3698:
3603:
3505:
3422:
3251:
3190:
2993:
2971:
2744:
2640:
2549:
2351:
2329:
2118:
2016:
1949:
1037:
725:
638:
555:
381:
243:
6804:
Jahresbericht der Deutschen Mathematiker-Vereinigung
6463:
6451:
6415:
6403:
6391:
6347:
6262:
6250:
6238:
1119:) has recently won special significance through the
929:, and finally uses logarithm. The two models of the
7136:
6427:
6229:
6227:
4732:{\displaystyle \Omega _{xx}=z_{1}z_{2}-z_{3}^{2}=0}
6618:"Ueber die sogenannte Nicht-Euklidische Geometrie"
6585:"Ueber die sogenannte Nicht-Euklidische Geometrie"
6103:
6044:
5952:
5855:
5798:and alternatively the equation of the unit sphere
5790:
5698:
5604:
5191:
5094:
4849:
4829:
4809:
4731:
4654:
4622:
4592:
4557:
4320:
3983:
3744:
3714:
3679:
3588:
3467:
3406:
3235:
3171:
2956:
2712:
2625:
2517:
2314:
2086:
2001:
1739:
1554:
1416:
1344:
1285:
1193:
1107:
1023:
782:
698:
624:
435:
333:
6848:On the geometric foundations of the Lorentz group
6790:Proceedings of the Edinburgh Mathematical Society
6182:
6171:
4810:{\displaystyle \Omega _{xx}=x^{2}+y^{2}-4c^{2}=0}
2099:The distance between two points is then given by
7197:
6224:
6213:
6160:
6149:
5102:and discussed their invariance with respect to
7018:
6885:Development of Mathematics in the 19th Century
6386:
5110:representing motions in non-Euclidean spaces.
364:of the zero point for distance: To move it to
6914:Bulletin of the American Mathematical Society
6870:. Berlin & Leipzig: Berlin W. de Gruyter.
1867:is as above. A rectangular hyperbola in the (
1708:
1681:
1653:
1626:
1598:
1571:
1523:
1496:
1468:
1441:
1108:{\displaystyle dx^{2}+dy^{2}+dz^{2}-dt^{2}=0}
7047:
6974:
6897:Vorlesungen über nicht-Euklidische Geometrie
6330:
4817:, which relates to hyperbolic geometry when
1194:{\textstyle x_{1}^{2}+x_{2}^{2}-x_{3}^{2}=0}
777:
726:
619:
556:
498:. A homography on the larger space may have
368:, apply the above homography, say obtaining
154:of the space. This group is obtained as the
6852:
6720:
6505:
6199:
209:
6932:
6691:: CS1 maint: location missing publisher (
6665:: CS1 maint: location missing publisher (
6527:
2626:{\displaystyle (a,b,c,f,g,h)(x,y,z)^{2}=0}
1915:
1882:The treatment by Martini and Spirova uses
530:, with the motion preserving distance, an
7101:
6846:English translation by David Delphenich:
6756:
6675:Klein, F. (1893b). Schilling, Fr. (ed.).
6649:Klein, F. (1893a). Schilling, Fr. (ed.).
6573:
881:On the other hand, geodesics are arcs of
6904:
6863:
6538:
6280:
1747:in relativity, which are bounded by the
917:one first constructs the homography for
352:to infinity. Furthermore, the midpoint (
26:
7179:
6737:An Essay on the Foundations of Geometry
6731:
6709:
6304:
5960:as the absolute in plane geometry, and
3184:of which he discussed the special case
1115:(to remain in three dimensions and use
14:
7198:
6907:"Non-euclidean geometry, a retrospect"
6894:Klein, F. (1928). Rosemann, W. (ed.).
6674:
6648:
6555:
6493:
6481:
6469:
6457:
6445:
6188:
6177:
6143:
1871:) plane is considered to pass through
1417:{\textstyle x^{2}+y^{2}+z^{2}-t^{2}=0}
1024:{\textstyle x^{2}+y^{2}+z^{2}-t^{2}=0}
783:{\displaystyle \{z:|z|^{2}=zz^{*}=1\}}
699:{\displaystyle (x/z)^{2}+(y/z)^{2}=1.}
625:{\displaystyle \{:x^{2}+y^{2}=z^{2}\}}
107:. The idea was to use the relation of
6893:
6874:
6834:Gesammelte mathematische Abhandlungen
6831:
6795:
6615:
6582:
6433:
6421:
6409:
6397:
6374:
6353:
6292:
6268:
6256:
6244:
6233:
6219:
6166:
6155:
5856:{\displaystyle x^{2}+y^{2}+z^{2}-1=0}
4630:are now related to three coordinates
1800:. If the absolute consists of a line
1792:associated with the Cayley absolute:
1779:
944:
6702:
1755:, so that for any physical velocity
537:
451:to 1, thus normalizing an arbitrary
6784:(1910/11) "Cayley–Klein metrics in
6697:(second print, first print in 1892)
6671:(second print, first print in 1892)
6104:{\displaystyle X^{2}+Y^{2}+Z^{2}=1}
3468:{\displaystyle x^{2}+y^{2}+z^{2}=1}
3236:{\displaystyle x^{2}+y^{2}+z^{2}=0}
1788:and other Euclidean geometry using
478:In a projective space containing P(
146:that leave the Cayley–Klein metric
59:. The construction originated with
24:
7172:
6545:Nouvelles annales de mathématiques
4747:
4670:
4608:
4578:
4465:
4403:
4341:
4191:
4103:
4015:
3966:
3950:
3936:
3895:
3882:
3861:
3843:
3826:
3813:
3792:
3774:
3730:
3700:
3604:
3506:
3391:
3375:
3359:
2002:{\displaystyle (a,b,c)(x,y)^{2}=0}
951:History of Lorentz transformations
25:
7222:
6836:. Vol. 1. pp. 533–552.
6713:Die nicht-euklidischen Raumformen
6320:A'Campo & Papadopoulos (2014)
1863:= (0,1,0) be on the absolute, so
1031:in the four-dimensional world or
933:obtained in this fashion are the
6853:Veblen, O.; Young, J.W. (1918).
6541:"Note sur la théorie des foyers"
1345:{\textstyle x^{2}+y^{2}-t^{2}=0}
838:are interior to the circle in P(
806:), something different from the
202:are affine diagrams with linear
7021:Periodica Mathematica Hungarica
6970:Georgia Institute of Technology
6926:10.1090/S0002-9904-1930-04885-5
6359:
6336:
1886:for the isotropic geometry and
909:. Again, for the distance from
878:in the disk are line segments.
6763:Leipziger Math.-Phys. Berichte
6721:Fricke, R.; Klein, F. (1897).
6558:"A sixth memoir upon quantics"
6204:
4953:
2940:
2906:
2903:
2891:
2878:
2859:
2856:
2844:
2797:
2779:
2776:
2764:
2608:
2589:
2586:
2550:
2298:
2275:
2272:
2254:
2241:
2228:
2225:
2207:
2171:
2159:
2156:
2138:
1984:
1971:
1968:
1950:
1798:affine Cayley–Klein geometries
1718:
1663:
1608:
1533:
1478:
745:
736:
681:
666:
654:
639:
577:
559:
394:
382:
328:
304:
256:
244:
109:projective harmonic conjugates
90:
13:
1:
6534:. Nürnberg: Nürnberg F. Korn.
6520:
6515:
6508:, pp. 1–60, Introduction
3692:and by forming the absolutes
814:). The distance notion for P(
372:. Then form this homography:
6842:10.1007/978-3-642-51960-4_31
6816:10.1007/978-3-642-51960-4_31
6387:Martini & Spirova (2008)
6127:
5873:in 1897, in which they used
4623:{\displaystyle \Omega _{yy}}
4593:{\displaystyle \Omega _{xx}}
3745:{\displaystyle \Omega _{yy}}
3715:{\displaystyle \Omega _{xx}}
3416:He also alluded to the case
1894:do with Euclidean geometry.
1716:
1661:
1606:
1531:
1476:
7:
6933:Littlewood, J. E. (1986) ,
6448:, pp. 64, 94, 109, 138
6115:
10:
7227:
6941:Cambridge University Press
6867:Nichteuklidische Geometrie
6331:Struve & Struve (2004)
4867:Sylvester's law of inertia
1897:
1786:Clifford's circle theorems
948:
158:for which the absolute is
7147:10.1109/ICIP.2016.7532355
7062:10.1007/s00022-010-0053-z
7033:10.1007/s10998-008-8197-5
6989:10.1007/s00022-004-1679-5
6506:Fricke & Klein (1897)
6200:Fricke & Klein (1897)
4877:with negative curvature.
1902:
526:of the region bounded by
184:Poincaré half-plane model
55:which is defined using a
6742:Dover Publications, Inc.
3478:
822:) is included in both P(
210:Cross ratio and distance
6936:Littlewood's miscellany
6883:; English translation:
6775:2027/hvd.32044092889328
6528:von Staudt, K. (1847).
6146:, p. 82, §§209–229
1928:homogeneous coordinates
1834:homogeneous coordinates
1430:Lorentz transformations
1127:That is, the absolutes
1117:homogeneous coordinates
925:, then evaluates it at
874:. In this instance the
800:complex projective line
709:On the other hand, the
182:is the absolute of the
166:is the absolute of the
6757:Hausdorff, F. (1899).
6747:Alfred North Whitehead
6575:10.1098/rstl.1859.0004
6496:, pp. 82ff, 142ff
6484:, pp. 76ff, 108ff
6105:
6046:
5954:
5857:
5792:
5700:
5606:
5193:
5146:
5108:Möbius transformations
5096:
4913:
4851:
4831:
4811:
4733:
4656:
4624:
4594:
4559:
4322:
3985:
3746:
3716:
3681:
3590:
3469:
3408:
3237:
3173:
2958:
2714:
2667:
2627:
2519:
2316:
2088:
2003:
1912:
1741:
1556:
1418:
1346:
1287:
1195:
1109:
1025:
887:Möbius transformations
784:
700:
626:
457:cross ratio homography
437:
335:
220:projective coordinates
85:non-Euclidean geometry
32:
6964:Harvey Lipkin (1985)
6905:Pierpont, J. (1930).
6864:Liebmann, H. (1923).
6832:Klein, Felix (1921).
6796:Klein, Felix (1921).
6788:-dimensional space",
6622:Mathematische Annalen
6589:Mathematische Annalen
6539:Laguerre, E. (1853).
6106:
6047:
5955:
5871:automorphic functions
5858:
5793:
5701:
5607:
5194:
5120:
5097:
4887:
4852:
4832:
4812:
4734:
4657:
4655:{\displaystyle x,y,z}
4625:
4595:
4560:
4323:
3986:
3747:
3717:
3682:
3591:
3470:
3409:
3238:
3174:
2959:
2715:
2641:
2628:
2520:
2317:
2089:
2004:
1907:
1888:split-complex numbers
1742:
1557:
1419:
1347:
1288:
1196:
1110:
1026:
808:real projective plane
785:
701:
632:which corresponds to
627:
438:
336:
140:Cayley–Klein geometry
30:
7141:, pp. 241–245,
6710:Killing, W. (1885).
6056:
5964:
5877:
5802:
5710:
5618:
5203:
5117:
4884:
4841:
4821:
4743:
4666:
4634:
4604:
4574:
4333:
4007:
3758:
3726:
3696:
3601:
3503:
3420:
3249:
3188:
2969:
2742:
2638:
2547:
2327:
2116:
2014:
1947:
1820:is a conic touching
1721:
1666:
1611:
1566:
1536:
1481:
1436:
1356:
1297:
1205:
1131:
1035:
963:
850:. The distance from
798:and is found in the
723:
636:
553:
379:
241:
172:Beltrami–Klein model
142:is the study of the
7206:Projective geometry
7096:. pp. 91–136.
7050:Journal of Geometry
6977:Journal of Geometry
6900:. Berlin: Springer.
6881:. Berlin: Springer.
6856:Projective geometry
6727:. Leipzig: Teubner.
6716:. Leipzig: Teubner.
6556:Cayley, A. (1859).
6295:, pp. 163, 304
6035:
6017:
5999:
5981:
5943:
5920:
5902:
5781:
5763:
5745:
5727:
5689:
5671:
5653:
5635:
5597:
5579:
5561:
5543:
5521:
5503:
5485:
5467:
5438:
5420:
5402:
5384:
5358:
5340:
5322:
5304:
5278:
5260:
5242:
5224:
4722:
4185:
4135:
4097:
4047:
3877:
3808:
3579:
3529:
1812:, then we have the
1722:
1717:
1667:
1662:
1612:
1607:
1537:
1532:
1482:
1477:
1276:
1258:
1240:
1222:
1184:
1166:
1148:
939:Poincaré disk model
883:generalized circles
482:), suppose a conic
176:hyperbolic geometry
168:Poincaré disk model
73:hyperbolic geometry
37:Cayley–Klein metric
18:Absolute (geometry)
6782:Duncan Sommerville
6740:re-issued 1956 by
6634:10.1007/BF01443189
6616:Klein, F. (1873).
6601:10.1007/BF02100583
6583:Klein, F. (1871).
6531:Geometrie der Lage
6101:
6042:
6021:
6003:
5985:
5967:
5950:
5929:
5906:
5888:
5853:
5788:
5767:
5749:
5731:
5713:
5696:
5675:
5657:
5639:
5621:
5602:
5600:
5583:
5565:
5547:
5529:
5507:
5489:
5471:
5453:
5424:
5406:
5388:
5370:
5344:
5326:
5308:
5290:
5264:
5246:
5228:
5210:
5189:
5092:
5090:
4847:
4827:
4807:
4729:
4708:
4652:
4620:
4590:
4555:
4553:
4318:
4316:
4171:
4121:
4083:
4033:
3981:
3860:
3791:
3742:
3712:
3677:
3637:
3586:
3565:
3515:
3465:
3404:
3243:with the distance
3233:
3169:
3167:
3116:
2954:
2725:with the distance
2710:
2623:
2530:In two dimensions
2515:
2513:
2474:
2312:
2084:
2044:
1999:
1814:isotropic geometry
1780:Affine CK-geometry
1737:
1552:
1414:
1342:
1283:
1262:
1244:
1226:
1208:
1191:
1170:
1152:
1134:
1105:
1021:
945:Special relativity
935:Cayley–Klein model
870:, when applied to
780:
696:
622:
433:
427:
331:
295:
81:Euclidean geometry
35:In mathematics, a
33:
7156:978-1-4673-9961-6
7121:978-3-03719-148-4
6966:Metrical Geometry
6950:978-0-521-33058-9
6825:978-3-642-51898-0
6751:Universal Algebra
6733:Russell, Bertrand
6703:Secondary sources
5444:
5364:
5284:
5086:
5020:
4850:{\displaystyle c}
4830:{\displaystyle c}
4568:
4567:
3979:
3978:
3910:
3907:
3838:
3690:
3689:
3610:
3402:
3399:
3350:
3182:
3181:
3123:
3115:
3112:
3072:
2952:
2949:
2887:
2723:
2722:
2528:
2527:
2473:
2470:
2430:
2310:
2307:
2250:
2097:
2096:
2017:
1918:, pp. 39–40)
1704:
1649:
1594:
1519:
1464:
1121:relativity theory
538:Disk applications
443:which takes to .
178:. Similarly, the
97:algebra of throws
77:elliptic geometry
16:(Redirected from
7218:
7191:
7167:
7133:
7105:
7088:
7044:
7015:
6961:
6929:
6911:
6901:
6887:by M. Ackerman,
6882:
6871:
6860:
6845:
6829:
6801:
6778:
6739:
6728:
6717:
6696:
6690:
6682:
6670:
6664:
6656:
6645:
6612:
6579:
6577:
6552:
6535:
6509:
6503:
6497:
6491:
6485:
6479:
6473:
6467:
6461:
6455:
6449:
6443:
6437:
6431:
6425:
6419:
6413:
6407:
6401:
6395:
6389:
6384:
6378:
6377:, chapter XI, §5
6372:
6366:
6363:
6357:
6351:
6345:
6340:
6334:
6328:
6322:
6317:
6308:
6302:
6296:
6290:
6284:
6278:
6272:
6266:
6260:
6254:
6248:
6242:
6236:
6231:
6222:
6217:
6211:
6208:
6202:
6197:
6191:
6186:
6180:
6175:
6169:
6164:
6158:
6153:
6147:
6141:
6110:
6108:
6107:
6102:
6094:
6093:
6081:
6080:
6068:
6067:
6051:
6049:
6048:
6043:
6034:
6029:
6016:
6011:
5998:
5993:
5980:
5975:
5959:
5957:
5956:
5951:
5942:
5937:
5925:
5921:
5919:
5914:
5901:
5896:
5862:
5860:
5859:
5854:
5840:
5839:
5827:
5826:
5814:
5813:
5797:
5795:
5794:
5789:
5780:
5775:
5762:
5757:
5744:
5739:
5726:
5721:
5705:
5703:
5702:
5697:
5688:
5683:
5670:
5665:
5652:
5647:
5634:
5629:
5611:
5609:
5608:
5603:
5601:
5596:
5591:
5578:
5573:
5560:
5555:
5542:
5537:
5520:
5515:
5502:
5497:
5484:
5479:
5466:
5461:
5445:
5442:
5437:
5432:
5419:
5414:
5401:
5396:
5383:
5378:
5365:
5362:
5357:
5352:
5339:
5334:
5321:
5316:
5303:
5298:
5285:
5282:
5277:
5272:
5259:
5254:
5241:
5236:
5223:
5218:
5198:
5196:
5195:
5190:
5179:
5178:
5169:
5168:
5159:
5158:
5145:
5140:
5101:
5099:
5098:
5093:
5091:
5087:
5084:
5074:
5073:
5064:
5063:
5048:
5047:
5035:
5034:
5021:
5018:
5008:
5007:
4998:
4997:
4982:
4981:
4969:
4968:
4946:
4945:
4936:
4935:
4926:
4925:
4912:
4907:
4856:
4854:
4853:
4848:
4836:
4834:
4833:
4828:
4816:
4814:
4813:
4808:
4800:
4799:
4784:
4783:
4771:
4770:
4758:
4757:
4738:
4736:
4735:
4730:
4721:
4716:
4704:
4703:
4694:
4693:
4681:
4680:
4661:
4659:
4658:
4653:
4629:
4627:
4626:
4621:
4619:
4618:
4599:
4597:
4596:
4591:
4589:
4588:
4564:
4562:
4561:
4556:
4554:
4550:
4546:
4515:
4514:
4505:
4504:
4495:
4494:
4476:
4475:
4453:
4452:
4443:
4442:
4433:
4432:
4414:
4413:
4391:
4390:
4381:
4380:
4371:
4370:
4352:
4351:
4327:
4325:
4324:
4319:
4317:
4313:
4312:
4303:
4302:
4287:
4283:
4282:
4281:
4272:
4271:
4259:
4258:
4249:
4248:
4228:
4227:
4218:
4217:
4202:
4201:
4184:
4179:
4164:
4163:
4154:
4153:
4134:
4129:
4114:
4113:
4096:
4091:
4076:
4075:
4066:
4065:
4046:
4041:
4026:
4025:
3993:
3992:
3990:
3988:
3987:
3982:
3980:
3977:
3976:
3961:
3960:
3948:
3947:
3946:
3934:
3911:
3909:
3908:
3906:
3905:
3893:
3892:
3876:
3871:
3859:
3854:
3853:
3840:
3839:
3837:
3836:
3824:
3823:
3807:
3802:
3790:
3785:
3784:
3771:
3751:
3749:
3748:
3743:
3741:
3740:
3721:
3719:
3718:
3713:
3711:
3710:
3686:
3684:
3683:
3678:
3670:
3669:
3660:
3659:
3650:
3649:
3636:
3595:
3593:
3592:
3587:
3578:
3573:
3558:
3557:
3548:
3547:
3528:
3523:
3489:
3488:
3474:
3472:
3471:
3466:
3458:
3457:
3445:
3444:
3432:
3431:
3413:
3411:
3410:
3405:
3403:
3401:
3400:
3398:
3397:
3382:
3381:
3366:
3365:
3353:
3351:
3349:
3348:
3336:
3335:
3323:
3322:
3313:
3310:
3309:
3295:
3281:
3269:
3264:
3263:
3242:
3240:
3239:
3234:
3226:
3225:
3213:
3212:
3200:
3199:
3178:
3176:
3175:
3170:
3168:
3164:
3160:
3121:
3117:
3114:
3113:
3111:
3110:
3101:
3100:
3091:
3090:
3075:
3073:
3071:
3070:
3061:
3060:
3051:
3050:
3035:
3032:
3031:
3030:
3021:
3020:
3011:
3010:
2994:
2988:
2987:
2963:
2961:
2960:
2955:
2953:
2951:
2950:
2948:
2947:
2938:
2927:
2916:
2890:
2888:
2886:
2885:
2843:
2840:
2839:
2835:
2834:
2823:
2812:
2762:
2757:
2756:
2728:
2727:
2719:
2717:
2716:
2711:
2700:
2699:
2690:
2689:
2680:
2679:
2666:
2661:
2632:
2630:
2629:
2624:
2616:
2615:
2533:
2532:
2524:
2522:
2521:
2516:
2514:
2510:
2506:
2475:
2472:
2471:
2469:
2468:
2459:
2458:
2449:
2448:
2433:
2431:
2429:
2428:
2419:
2418:
2409:
2408:
2393:
2390:
2389:
2388:
2379:
2378:
2369:
2368:
2352:
2346:
2345:
2321:
2319:
2318:
2313:
2311:
2309:
2308:
2306:
2305:
2296:
2285:
2253:
2251:
2249:
2248:
2206:
2203:
2202:
2198:
2197:
2186:
2136:
2131:
2130:
2102:
2101:
2093:
2091:
2090:
2085:
2077:
2076:
2067:
2066:
2057:
2056:
2043:
2008:
2006:
2005:
2000:
1992:
1991:
1933:
1932:
1919:
1916:Littlewood (1986
1818:isotropic circle
1768:
1758:
1754:
1746:
1744:
1743:
1738:
1730:
1729:
1724:
1723:
1712:
1711:
1705:
1703:
1695:
1687:
1685:
1684:
1675:
1674:
1669:
1668:
1657:
1656:
1650:
1648:
1640:
1632:
1630:
1629:
1620:
1619:
1614:
1613:
1602:
1601:
1595:
1593:
1585:
1577:
1575:
1574:
1561:
1559:
1558:
1553:
1545:
1544:
1539:
1538:
1527:
1526:
1520:
1518:
1510:
1502:
1500:
1499:
1490:
1489:
1484:
1483:
1472:
1471:
1465:
1463:
1455:
1447:
1445:
1444:
1423:
1421:
1420:
1415:
1407:
1406:
1394:
1393:
1381:
1380:
1368:
1367:
1351:
1349:
1348:
1343:
1335:
1334:
1322:
1321:
1309:
1308:
1292:
1290:
1289:
1284:
1275:
1270:
1257:
1252:
1239:
1234:
1221:
1216:
1200:
1198:
1197:
1192:
1183:
1178:
1165:
1160:
1147:
1142:
1114:
1112:
1111:
1106:
1098:
1097:
1082:
1081:
1066:
1065:
1050:
1049:
1030:
1028:
1027:
1022:
1014:
1013:
1001:
1000:
988:
987:
975:
974:
931:hyperbolic plane
789:
787:
786:
781:
770:
769:
754:
753:
748:
739:
713:in the ordinary
705:
703:
702:
697:
689:
688:
676:
662:
661:
649:
631:
629:
628:
623:
618:
617:
605:
604:
592:
591:
442:
440:
439:
434:
432:
431:
340:
338:
337:
332:
300:
299:
200:Voronoi diagrams
144:group of motions
117:Laguerre formula
53:projective space
21:
7226:
7225:
7221:
7220:
7219:
7217:
7216:
7215:
7211:Metric geometry
7196:
7195:
7194:
7175:
7173:Further reading
7170:
7157:
7122:
7112:10.4171/148-1/5
6951:
6909:
6859:. Boston: Ginn.
6826:
6705:
6700:
6684:
6683:
6658:
6657:
6523:
6518:
6513:
6512:
6504:
6500:
6492:
6488:
6480:
6476:
6468:
6464:
6456:
6452:
6444:
6440:
6432:
6428:
6420:
6416:
6408:
6404:
6396:
6392:
6385:
6381:
6373:
6369:
6364:
6360:
6352:
6348:
6341:
6337:
6329:
6325:
6318:
6311:
6303:
6299:
6291:
6287:
6281:Pierpont (1930)
6279:
6275:
6267:
6263:
6255:
6251:
6243:
6239:
6232:
6225:
6218:
6214:
6209:
6205:
6198:
6194:
6187:
6183:
6176:
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6165:
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6150:
6142:
6135:
6130:
6118:
6089:
6085:
6076:
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6063:
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6025:
6012:
6007:
5994:
5989:
5976:
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5938:
5933:
5915:
5910:
5897:
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5878:
5875:
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5835:
5831:
5822:
5818:
5809:
5805:
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5800:
5799:
5776:
5771:
5758:
5753:
5740:
5735:
5722:
5717:
5711:
5708:
5707:
5684:
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5630:
5625:
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5538:
5533:
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5522:
5516:
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5439:
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5410:
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5374:
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5348:
5335:
5330:
5317:
5312:
5299:
5294:
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5281:
5279:
5273:
5268:
5255:
5250:
5237:
5232:
5219:
5214:
5206:
5204:
5201:
5200:
5174:
5170:
5164:
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5147:
5141:
5124:
5118:
5115:
5114:
5089:
5088:
5083:
5081:
5069:
5065:
5059:
5055:
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5039:
5030:
5026:
5023:
5022:
5017:
5015:
5003:
4999:
4993:
4989:
4977:
4973:
4964:
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4941:
4937:
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4918:
4914:
4908:
4891:
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4882:
4881:
4842:
4839:
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4795:
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4779:
4775:
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4717:
4712:
4699:
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4669:
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4635:
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4607:
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4506:
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4460:
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4386:
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4359:
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4340:
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4330:
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4267:
4263:
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4244:
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4219:
4213:
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4008:
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3969:
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3638:
3614:
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3598:
3574:
3569:
3553:
3549:
3543:
3539:
3524:
3519:
3504:
3501:
3500:
3481:
3475:(unit sphere).
3453:
3449:
3440:
3436:
3427:
3423:
3421:
3418:
3417:
3390:
3386:
3374:
3370:
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3003:
2999:
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2841:
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2800:
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2749:
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2739:
2695:
2691:
2685:
2681:
2672:
2668:
2662:
2645:
2639:
2636:
2635:
2611:
2607:
2548:
2545:
2544:
2512:
2511:
2484:
2480:
2477:
2476:
2464:
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2454:
2450:
2441:
2437:
2432:
2424:
2420:
2414:
2410:
2401:
2397:
2392:
2391:
2384:
2380:
2374:
2370:
2361:
2357:
2353:
2350:
2338:
2334:
2330:
2328:
2325:
2324:
2301:
2297:
2289:
2278:
2252:
2244:
2240:
2205:
2204:
2190:
2179:
2178:
2174:
2137:
2135:
2123:
2119:
2117:
2114:
2113:
2072:
2068:
2062:
2058:
2049:
2045:
2021:
2015:
2012:
2011:
1987:
1983:
1948:
1945:
1944:
1921:
1914:
1905:
1900:
1892:complex numbers
1844:at infinity is
1790:affine geometry
1782:
1774:Minkowski space
1760:
1756:
1752:
1725:
1715:
1714:
1713:
1707:
1706:
1696:
1688:
1686:
1680:
1679:
1670:
1660:
1659:
1658:
1652:
1651:
1641:
1633:
1631:
1625:
1624:
1615:
1605:
1604:
1603:
1597:
1596:
1586:
1578:
1576:
1570:
1569:
1567:
1564:
1563:
1540:
1530:
1529:
1528:
1522:
1521:
1511:
1503:
1501:
1495:
1494:
1485:
1475:
1474:
1473:
1467:
1466:
1456:
1448:
1446:
1440:
1439:
1437:
1434:
1433:
1402:
1398:
1389:
1385:
1376:
1372:
1363:
1359:
1357:
1354:
1353:
1330:
1326:
1317:
1313:
1304:
1300:
1298:
1295:
1294:
1271:
1266:
1253:
1248:
1235:
1230:
1217:
1212:
1206:
1203:
1202:
1179:
1174:
1161:
1156:
1143:
1138:
1132:
1129:
1128:
1093:
1089:
1077:
1073:
1061:
1057:
1045:
1041:
1036:
1033:
1032:
1009:
1005:
996:
992:
983:
979:
970:
966:
964:
961:
960:
953:
947:
765:
761:
749:
744:
743:
735:
724:
721:
720:
684:
680:
672:
657:
653:
645:
637:
634:
633:
613:
609:
600:
596:
587:
583:
554:
551:
550:
540:
486:is given, with
426:
425:
420:
414:
413:
408:
398:
397:
380:
377:
376:
294:
293:
285:
279:
278:
273:
260:
259:
242:
239:
238:
212:
131:serving as the
121:Edmond Laguerre
101:Karl von Staudt
93:
83:. The field of
23:
22:
15:
12:
11:
5:
7224:
7214:
7213:
7208:
7193:
7192:
7176:
7174:
7171:
7169:
7168:
7155:
7134:
7120:
7089:
7056:(1): 151–170,
7045:
7027:(2): 197–206.
7016:
6983:(1): 155–167,
6972:
6962:
6949:
6930:
6902:
6891:
6889:Math Sci Press
6872:
6861:
6850:
6824:
6793:
6779:
6754:
6744:
6729:
6718:
6706:
6704:
6701:
6699:
6698:
6672:
6646:
6628:(2): 112–145.
6613:
6595:(4): 573–625.
6580:
6553:
6536:
6524:
6522:
6519:
6517:
6514:
6511:
6510:
6498:
6486:
6474:
6462:
6450:
6438:
6426:
6414:
6402:
6390:
6379:
6367:
6358:
6346:
6343:Nielsen (2016)
6335:
6323:
6309:
6305:Russell (1898)
6297:
6285:
6283:, p. 67ff
6273:
6261:
6249:
6237:
6223:
6212:
6203:
6192:
6181:
6170:
6159:
6148:
6132:
6131:
6129:
6126:
6125:
6124:
6122:Hilbert metric
6117:
6114:
6100:
6097:
6092:
6088:
6084:
6079:
6075:
6071:
6066:
6062:
6041:
6038:
6033:
6028:
6024:
6020:
6015:
6010:
6006:
6002:
5997:
5992:
5988:
5984:
5979:
5974:
5970:
5949:
5946:
5941:
5936:
5932:
5928:
5924:
5918:
5913:
5909:
5905:
5900:
5895:
5891:
5886:
5882:
5852:
5849:
5846:
5843:
5838:
5834:
5830:
5825:
5821:
5817:
5812:
5808:
5787:
5784:
5779:
5774:
5770:
5766:
5761:
5756:
5752:
5748:
5743:
5738:
5734:
5730:
5725:
5720:
5716:
5695:
5692:
5687:
5682:
5678:
5674:
5669:
5664:
5660:
5656:
5651:
5646:
5642:
5638:
5633:
5628:
5624:
5595:
5590:
5586:
5582:
5577:
5572:
5568:
5564:
5559:
5554:
5550:
5546:
5541:
5536:
5532:
5528:
5525:
5524:
5519:
5514:
5510:
5506:
5501:
5496:
5492:
5488:
5483:
5478:
5474:
5470:
5465:
5460:
5456:
5452:
5449:
5448:
5440:
5436:
5431:
5427:
5423:
5418:
5413:
5409:
5405:
5400:
5395:
5391:
5387:
5382:
5377:
5373:
5369:
5368:
5360:
5356:
5351:
5347:
5343:
5338:
5333:
5329:
5325:
5320:
5315:
5311:
5307:
5302:
5297:
5293:
5289:
5288:
5280:
5276:
5271:
5267:
5263:
5258:
5253:
5249:
5245:
5240:
5235:
5231:
5227:
5222:
5217:
5213:
5209:
5208:
5188:
5185:
5182:
5177:
5173:
5167:
5163:
5157:
5154:
5150:
5144:
5139:
5136:
5133:
5130:
5127:
5123:
5082:
5080:
5077:
5072:
5068:
5062:
5058:
5054:
5051:
5046:
5042:
5038:
5033:
5029:
5025:
5024:
5016:
5014:
5011:
5006:
5002:
4996:
4992:
4988:
4985:
4980:
4976:
4972:
4967:
4963:
4959:
4958:
4955:
4952:
4949:
4944:
4940:
4934:
4930:
4924:
4921:
4917:
4911:
4906:
4903:
4900:
4897:
4894:
4890:
4846:
4826:
4806:
4803:
4798:
4794:
4790:
4787:
4782:
4778:
4774:
4769:
4765:
4761:
4756:
4753:
4749:
4728:
4725:
4720:
4715:
4711:
4707:
4702:
4698:
4692:
4688:
4684:
4679:
4676:
4672:
4651:
4648:
4645:
4642:
4639:
4617:
4614:
4610:
4587:
4584:
4580:
4566:
4565:
4549:
4545:
4542:
4539:
4536:
4533:
4530:
4527:
4523:
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4518:
4513:
4509:
4503:
4499:
4493:
4490:
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4479:
4474:
4471:
4467:
4463:
4462:
4459:
4456:
4451:
4447:
4441:
4437:
4431:
4428:
4424:
4420:
4417:
4412:
4409:
4405:
4401:
4400:
4397:
4394:
4389:
4385:
4379:
4375:
4369:
4366:
4362:
4358:
4355:
4350:
4347:
4343:
4339:
4338:
4328:
4311:
4307:
4301:
4297:
4293:
4290:
4286:
4280:
4276:
4270:
4266:
4262:
4257:
4253:
4247:
4243:
4238:
4234:
4231:
4226:
4222:
4216:
4212:
4208:
4205:
4200:
4197:
4193:
4189:
4188:
4183:
4178:
4174:
4170:
4167:
4162:
4158:
4152:
4148:
4144:
4141:
4138:
4133:
4128:
4124:
4120:
4117:
4112:
4109:
4105:
4101:
4100:
4095:
4090:
4086:
4082:
4079:
4074:
4070:
4064:
4060:
4056:
4053:
4050:
4045:
4040:
4036:
4032:
4029:
4024:
4021:
4017:
4013:
4012:
4001:
4000:
3997:
3975:
3972:
3968:
3964:
3959:
3956:
3952:
3945:
3942:
3938:
3932:
3929:
3926:
3923:
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3914:
3904:
3901:
3897:
3891:
3888:
3884:
3880:
3875:
3870:
3867:
3863:
3857:
3852:
3849:
3845:
3835:
3832:
3828:
3822:
3819:
3815:
3811:
3806:
3801:
3798:
3794:
3788:
3783:
3780:
3776:
3769:
3766:
3763:
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3736:
3732:
3709:
3706:
3702:
3688:
3687:
3676:
3673:
3668:
3664:
3658:
3654:
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3645:
3641:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3613:
3609:
3606:
3596:
3585:
3582:
3577:
3572:
3568:
3564:
3561:
3556:
3552:
3546:
3542:
3538:
3535:
3532:
3527:
3522:
3518:
3514:
3511:
3508:
3497:
3496:
3493:
3480:
3477:
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3461:
3456:
3452:
3448:
3443:
3439:
3435:
3430:
3426:
3396:
3393:
3389:
3385:
3380:
3377:
3373:
3369:
3364:
3361:
3357:
3347:
3343:
3339:
3334:
3330:
3326:
3321:
3317:
3308:
3305:
3301:
3298:
3294:
3291:
3287:
3284:
3280:
3277:
3273:
3267:
3262:
3259:
3255:
3232:
3229:
3224:
3220:
3216:
3211:
3207:
3203:
3198:
3194:
3180:
3179:
3163:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3131:
3127:
3126:
3120:
3109:
3105:
3099:
3095:
3089:
3086:
3082:
3078:
3069:
3065:
3059:
3055:
3049:
3046:
3042:
3038:
3029:
3025:
3019:
3015:
3009:
3006:
3002:
2998:
2991:
2986:
2983:
2979:
2975:
2974:
2964:
2946:
2942:
2937:
2934:
2930:
2926:
2923:
2919:
2915:
2912:
2908:
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2899:
2896:
2893:
2884:
2880:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2849:
2846:
2838:
2833:
2830:
2826:
2822:
2819:
2815:
2811:
2808:
2803:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2760:
2755:
2752:
2748:
2736:
2735:
2732:
2721:
2720:
2709:
2706:
2703:
2698:
2694:
2688:
2684:
2678:
2675:
2671:
2665:
2660:
2657:
2654:
2651:
2648:
2644:
2633:
2622:
2619:
2614:
2610:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2541:
2540:
2537:
2526:
2525:
2509:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2483:
2479:
2478:
2467:
2463:
2457:
2453:
2447:
2444:
2440:
2436:
2427:
2423:
2417:
2413:
2407:
2404:
2400:
2396:
2387:
2383:
2377:
2373:
2367:
2364:
2360:
2356:
2349:
2344:
2341:
2337:
2333:
2332:
2322:
2304:
2300:
2295:
2292:
2288:
2284:
2281:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2247:
2243:
2239:
2236:
2233:
2230:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2201:
2196:
2193:
2189:
2185:
2182:
2177:
2173:
2170:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2134:
2129:
2126:
2122:
2110:
2109:
2106:
2095:
2094:
2083:
2080:
2075:
2071:
2065:
2061:
2055:
2052:
2048:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2020:
2009:
1998:
1995:
1990:
1986:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1941:
1940:
1937:
1906:
1904:
1901:
1899:
1896:
1859:= (1,0,0) and
1830:
1829:
1781:
1778:
1749:speed of light
1736:
1733:
1728:
1720:
1710:
1702:
1699:
1694:
1691:
1683:
1678:
1673:
1665:
1655:
1647:
1644:
1639:
1636:
1628:
1623:
1618:
1610:
1600:
1592:
1589:
1584:
1581:
1573:
1551:
1548:
1543:
1535:
1525:
1517:
1514:
1509:
1506:
1498:
1493:
1488:
1480:
1470:
1462:
1459:
1454:
1451:
1443:
1413:
1410:
1405:
1401:
1397:
1392:
1388:
1384:
1379:
1375:
1371:
1366:
1362:
1341:
1338:
1333:
1329:
1325:
1320:
1316:
1312:
1307:
1303:
1282:
1279:
1274:
1269:
1265:
1261:
1256:
1251:
1247:
1243:
1238:
1233:
1229:
1225:
1220:
1215:
1211:
1190:
1187:
1182:
1177:
1173:
1169:
1164:
1159:
1155:
1151:
1146:
1141:
1137:
1125:
1124:
1104:
1101:
1096:
1092:
1088:
1085:
1080:
1076:
1072:
1069:
1064:
1060:
1056:
1053:
1048:
1044:
1040:
1020:
1017:
1012:
1008:
1004:
999:
995:
991:
986:
982:
978:
973:
969:
949:Main article:
946:
943:
796:
795:
792:complex number
779:
776:
773:
768:
764:
760:
757:
752:
747:
742:
738:
734:
731:
728:
707:
706:
695:
692:
687:
683:
679:
675:
671:
668:
665:
660:
656:
652:
648:
644:
641:
621:
616:
612:
608:
603:
599:
595:
590:
586:
582:
579:
576:
573:
570:
567:
564:
561:
558:
539:
536:
445:
444:
430:
424:
421:
419:
416:
415:
412:
409:
407:
404:
403:
401:
396:
393:
390:
387:
384:
342:
341:
330:
327:
324:
321:
318:
315:
312:
309:
306:
303:
298:
292:
289:
286:
284:
281:
280:
277:
274:
272:
269:
266:
265:
263:
258:
255:
252:
249:
246:
211:
208:
196:
195:
92:
89:
9:
6:
4:
3:
2:
7223:
7212:
7209:
7207:
7204:
7203:
7201:
7189:
7185:
7184:
7178:
7177:
7166:
7162:
7158:
7152:
7148:
7144:
7140:
7135:
7131:
7127:
7123:
7117:
7113:
7109:
7104:
7099:
7095:
7090:
7087:
7083:
7079:
7075:
7071:
7067:
7063:
7059:
7055:
7051:
7046:
7042:
7038:
7034:
7030:
7026:
7022:
7017:
7014:
7010:
7006:
7002:
6998:
6994:
6990:
6986:
6982:
6978:
6973:
6971:
6967:
6963:
6960:
6956:
6952:
6946:
6942:
6938:
6937:
6931:
6927:
6923:
6919:
6915:
6908:
6903:
6899:
6898:
6892:
6890:
6886:
6880:
6879:
6873:
6869:
6868:
6862:
6858:
6857:
6851:
6849:
6843:
6839:
6835:
6830:Reprinted in
6827:
6821:
6817:
6813:
6809:
6805:
6800:
6794:
6791:
6787:
6783:
6780:
6776:
6772:
6768:
6764:
6760:
6755:
6752:
6748:
6745:
6743:
6738:
6734:
6730:
6726:
6725:
6719:
6715:
6714:
6708:
6707:
6694:
6688:
6680:
6679:
6673:
6668:
6662:
6654:
6653:
6647:
6643:
6639:
6635:
6631:
6627:
6623:
6619:
6614:
6610:
6606:
6602:
6598:
6594:
6590:
6586:
6581:
6576:
6571:
6567:
6563:
6559:
6554:
6550:
6546:
6542:
6537:
6533:
6532:
6526:
6525:
6507:
6502:
6495:
6494:Klein (1893b)
6490:
6483:
6482:Klein (1893b)
6478:
6471:
6470:Klein (1893b)
6466:
6459:
6458:Klein (1893b)
6454:
6447:
6446:Klein (1893a)
6442:
6435:
6430:
6424:, p. 618
6423:
6418:
6412:, p. 601
6411:
6406:
6400:, p. 587
6399:
6394:
6388:
6383:
6376:
6371:
6362:
6356:, p. 138
6355:
6350:
6344:
6339:
6333:, p. 157
6332:
6327:
6321:
6316:
6314:
6306:
6301:
6294:
6289:
6282:
6277:
6271:, p. 303
6270:
6265:
6259:, p. 138
6258:
6253:
6247:, p. 163
6246:
6241:
6235:
6230:
6228:
6221:
6216:
6207:
6201:
6196:
6190:
6189:Klein (1893b)
6185:
6179:
6178:Klein (1893a)
6174:
6168:
6163:
6157:
6152:
6145:
6144:Cayley (1859)
6140:
6138:
6133:
6123:
6120:
6119:
6113:
6098:
6095:
6090:
6086:
6082:
6077:
6073:
6069:
6064:
6060:
6039:
6036:
6031:
6026:
6022:
6018:
6013:
6008:
6004:
6000:
5995:
5990:
5986:
5982:
5977:
5972:
5968:
5947:
5944:
5939:
5934:
5930:
5926:
5922:
5916:
5911:
5907:
5903:
5898:
5893:
5889:
5884:
5880:
5872:
5868:
5867:Robert Fricke
5864:
5850:
5847:
5844:
5841:
5836:
5832:
5828:
5823:
5819:
5815:
5810:
5806:
5785:
5782:
5777:
5772:
5768:
5764:
5759:
5754:
5750:
5746:
5741:
5736:
5732:
5728:
5723:
5718:
5714:
5693:
5690:
5685:
5680:
5676:
5672:
5667:
5662:
5658:
5654:
5649:
5644:
5640:
5636:
5631:
5626:
5622:
5612:
5593:
5588:
5584:
5580:
5575:
5570:
5566:
5562:
5557:
5552:
5548:
5544:
5539:
5534:
5530:
5526:
5517:
5512:
5508:
5504:
5499:
5494:
5490:
5486:
5481:
5476:
5472:
5468:
5463:
5458:
5454:
5450:
5434:
5429:
5425:
5421:
5416:
5411:
5407:
5403:
5398:
5393:
5389:
5385:
5380:
5375:
5371:
5354:
5349:
5345:
5341:
5336:
5331:
5327:
5323:
5318:
5313:
5309:
5305:
5300:
5295:
5291:
5274:
5269:
5265:
5261:
5256:
5251:
5247:
5243:
5238:
5233:
5229:
5225:
5220:
5215:
5211:
5186:
5183:
5180:
5175:
5171:
5165:
5161:
5155:
5152:
5148:
5142:
5137:
5134:
5131:
5128:
5125:
5121:
5111:
5109:
5105:
5104:collineations
5078:
5075:
5070:
5066:
5060:
5056:
5052:
5049:
5044:
5040:
5036:
5031:
5027:
5012:
5009:
5004:
5000:
4994:
4990:
4986:
4983:
4978:
4974:
4970:
4965:
4961:
4950:
4947:
4942:
4938:
4932:
4928:
4922:
4919:
4915:
4909:
4904:
4901:
4898:
4895:
4892:
4888:
4878:
4876:
4873:or two-sheet
4872:
4868:
4862:
4860:
4844:
4824:
4804:
4801:
4796:
4792:
4788:
4785:
4780:
4776:
4772:
4767:
4763:
4759:
4754:
4751:
4726:
4723:
4718:
4713:
4709:
4705:
4700:
4696:
4690:
4686:
4682:
4677:
4674:
4649:
4646:
4643:
4640:
4637:
4615:
4612:
4585:
4582:
4547:
4543:
4540:
4537:
4534:
4531:
4528:
4525:
4521:
4511:
4507:
4501:
4497:
4491:
4488:
4484:
4480:
4477:
4472:
4469:
4457:
4454:
4449:
4445:
4439:
4435:
4429:
4426:
4422:
4418:
4415:
4410:
4407:
4395:
4392:
4387:
4383:
4377:
4373:
4367:
4364:
4360:
4356:
4353:
4348:
4345:
4329:
4309:
4305:
4299:
4295:
4291:
4288:
4284:
4278:
4274:
4268:
4264:
4260:
4255:
4251:
4245:
4241:
4236:
4232:
4229:
4224:
4220:
4214:
4210:
4206:
4203:
4198:
4195:
4181:
4176:
4172:
4168:
4165:
4160:
4156:
4150:
4146:
4142:
4139:
4136:
4131:
4126:
4122:
4118:
4115:
4110:
4107:
4093:
4088:
4084:
4080:
4077:
4072:
4068:
4062:
4058:
4054:
4051:
4048:
4043:
4038:
4034:
4030:
4027:
4022:
4019:
4003:
4002:
3998:
3995:
3994:
3991:
3973:
3970:
3962:
3957:
3954:
3943:
3940:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3902:
3899:
3889:
3886:
3878:
3873:
3868:
3865:
3855:
3850:
3847:
3833:
3830:
3820:
3817:
3809:
3804:
3799:
3796:
3786:
3781:
3778:
3767:
3764:
3761:
3753:
3737:
3734:
3707:
3704:
3674:
3671:
3666:
3662:
3656:
3652:
3646:
3643:
3639:
3633:
3630:
3627:
3624:
3621:
3618:
3615:
3611:
3607:
3597:
3583:
3580:
3575:
3570:
3566:
3562:
3559:
3554:
3550:
3544:
3540:
3536:
3533:
3530:
3525:
3520:
3516:
3512:
3509:
3499:
3498:
3494:
3491:
3490:
3487:
3485:
3476:
3462:
3459:
3454:
3450:
3446:
3441:
3437:
3433:
3428:
3424:
3414:
3394:
3387:
3383:
3378:
3371:
3367:
3362:
3355:
3345:
3341:
3337:
3332:
3328:
3324:
3319:
3315:
3306:
3303:
3299:
3296:
3292:
3289:
3285:
3282:
3278:
3275:
3271:
3265:
3260:
3257:
3253:
3244:
3230:
3227:
3222:
3218:
3214:
3209:
3205:
3201:
3196:
3192:
3161:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3129:
3118:
3107:
3103:
3097:
3093:
3087:
3084:
3080:
3076:
3067:
3063:
3057:
3053:
3047:
3044:
3040:
3036:
3027:
3023:
3017:
3013:
3007:
3004:
3000:
2996:
2989:
2984:
2981:
2977:
2965:
2944:
2935:
2932:
2928:
2924:
2921:
2917:
2913:
2910:
2900:
2897:
2894:
2882:
2874:
2871:
2868:
2865:
2862:
2853:
2850:
2847:
2836:
2831:
2828:
2824:
2820:
2817:
2813:
2809:
2806:
2801:
2794:
2791:
2788:
2785:
2782:
2773:
2770:
2767:
2758:
2753:
2750:
2746:
2738:
2737:
2733:
2730:
2729:
2726:
2707:
2704:
2701:
2696:
2692:
2686:
2682:
2676:
2673:
2669:
2663:
2658:
2655:
2652:
2649:
2646:
2642:
2634:
2620:
2617:
2612:
2604:
2601:
2598:
2595:
2592:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2543:
2542:
2538:
2535:
2534:
2531:
2507:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2481:
2465:
2461:
2455:
2451:
2445:
2442:
2438:
2434:
2425:
2421:
2415:
2411:
2405:
2402:
2398:
2394:
2385:
2381:
2375:
2371:
2365:
2362:
2358:
2354:
2347:
2342:
2339:
2335:
2323:
2302:
2293:
2290:
2286:
2282:
2279:
2269:
2266:
2263:
2260:
2257:
2245:
2237:
2234:
2231:
2222:
2219:
2216:
2213:
2210:
2199:
2194:
2191:
2187:
2183:
2180:
2175:
2168:
2165:
2162:
2153:
2150:
2147:
2144:
2141:
2132:
2127:
2124:
2120:
2112:
2111:
2107:
2104:
2103:
2100:
2081:
2078:
2073:
2069:
2063:
2059:
2053:
2050:
2046:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2018:
2010:
1996:
1993:
1988:
1980:
1977:
1974:
1965:
1962:
1959:
1956:
1953:
1943:
1942:
1938:
1935:
1934:
1931:
1929:
1925:
1924:Arthur Cayley
1920:
1917:
1911:
1895:
1893:
1889:
1885:
1880:
1878:
1874:
1870:
1866:
1862:
1858:
1853:
1851:
1847:
1843:
1839:
1835:
1827:
1823:
1819:
1815:
1811:
1807:
1803:
1799:
1795:
1794:
1793:
1791:
1787:
1777:
1775:
1770:
1767:
1763:
1750:
1734:
1731:
1726:
1700:
1697:
1692:
1689:
1676:
1671:
1645:
1642:
1637:
1634:
1621:
1616:
1590:
1587:
1582:
1579:
1549:
1546:
1541:
1515:
1512:
1507:
1504:
1491:
1486:
1460:
1457:
1452:
1449:
1431:
1427:
1411:
1408:
1403:
1399:
1395:
1390:
1386:
1382:
1377:
1373:
1369:
1364:
1360:
1339:
1336:
1331:
1327:
1323:
1318:
1314:
1310:
1305:
1301:
1280:
1277:
1272:
1267:
1263:
1259:
1254:
1249:
1245:
1241:
1236:
1231:
1227:
1223:
1218:
1213:
1209:
1188:
1185:
1180:
1175:
1171:
1167:
1162:
1157:
1153:
1149:
1144:
1139:
1135:
1122:
1118:
1102:
1099:
1094:
1090:
1086:
1083:
1078:
1074:
1070:
1067:
1062:
1058:
1054:
1051:
1046:
1042:
1038:
1018:
1015:
1010:
1006:
1002:
997:
993:
989:
984:
980:
976:
971:
967:
958:
957:
956:
952:
942:
940:
936:
932:
928:
924:
920:
916:
912:
908:
904:
900:
896:
892:
891:invariant set
888:
884:
879:
877:
873:
869:
865:
861:
857:
853:
849:
845:
841:
837:
833:
829:
825:
821:
817:
813:
809:
805:
801:
793:
774:
771:
766:
762:
758:
755:
750:
740:
732:
729:
719:
718:
717:
716:
715:complex plane
712:
693:
690:
685:
677:
673:
669:
663:
658:
650:
646:
642:
614:
610:
606:
601:
597:
593:
588:
584:
580:
574:
571:
568:
565:
562:
549:
548:
547:
545:
535:
533:
529:
525:
521:
517:
513:
510:), and since
509:
505:
504:invariant set
501:
497:
493:
489:
485:
481:
476:
474:
470:
466:
462:
458:
454:
450:
428:
422:
417:
410:
405:
399:
391:
388:
385:
375:
374:
373:
371:
367:
361:
359:
355:
351:
347:
325:
322:
319:
316:
313:
310:
307:
301:
296:
290:
287:
282:
275:
270:
267:
261:
253:
250:
247:
237:
236:
235:
233:
229:
225:
221:
217:
207:
205:
201:
198:Cayley-Klein
192:
191:
190:
187:
185:
181:
177:
173:
169:
165:
161:
157:
156:collineations
153:
149:
145:
141:
137:
134:
130:
126:
122:
118:
114:
110:
106:
102:
98:
88:
86:
82:
78:
74:
70:
66:
62:
61:Arthur Cayley
58:
54:
50:
46:
42:
38:
29:
19:
7190:(2): 185–211
7187:
7181:
7138:
7093:
7053:
7049:
7024:
7020:
6980:
6976:
6935:
6920:(2): 66–76.
6917:
6913:
6896:
6884:
6877:
6866:
6855:
6833:
6807:
6803:
6785:
6766:
6762:
6736:
6723:
6712:
6681:. Göttingen.
6677:
6655:. Göttingen.
6651:
6625:
6621:
6592:
6588:
6565:
6561:
6548:
6544:
6530:
6501:
6489:
6477:
6472:, p. 64
6465:
6460:, p. 61
6453:
6441:
6434:Klein (1873)
6429:
6422:Klein (1871)
6417:
6410:Klein (1871)
6405:
6398:Klein (1871)
6393:
6382:
6375:Klein (1928)
6370:
6365:Klein (1910)
6361:
6354:Klein (1926)
6349:
6338:
6326:
6307:, p. 32
6300:
6293:Klein (1928)
6288:
6276:
6269:Klein (1928)
6264:
6257:Klein (1928)
6252:
6245:Klein (1928)
6240:
6234:Klein (1928)
6220:Klein (1926)
6215:
6210:Klein (1910)
6206:
6195:
6184:
6173:
6167:Klein (1873)
6162:
6156:Klein (1871)
6151:
5865:
5613:
5112:
5085:(hyperbolic)
4879:
4863:
4569:
3754:
3691:
3482:
3415:
3245:
3183:
2724:
2529:
2098:
1922:
1913:
1908:
1884:dual numbers
1881:
1876:
1872:
1868:
1864:
1860:
1856:
1854:
1849:
1845:
1841:
1837:
1831:
1825:
1821:
1817:
1813:
1809:
1805:
1804:and a point
1801:
1797:
1783:
1771:
1765:
1761:
1759:, the ratio
1126:
954:
926:
922:
918:
914:
910:
906:
902:
898:
894:
880:
871:
867:
863:
859:
855:
851:
847:
843:
839:
835:
831:
827:
823:
819:
815:
811:
803:
797:
708:
543:
541:
527:
519:
515:
511:
507:
499:
495:
491:
487:
483:
479:
477:
468:
464:
460:
452:
448:
446:
369:
365:
362:
357:
353:
349:
348:to zero and
345:
343:
231:
227:
223:
215:
213:
197:
188:
151:
139:
138:
132:
113:cross-ratios
94:
64:
36:
34:
6810:: 533–552.
6769:: 161–214.
6052:as well as
5283:(zero part)
4875:hyperboloid
4859:hyperboloid
3484:Felix Klein
1123:of physics.
711:unit circle
206:bisectors.
164:unit circle
91:Foundations
69:Felix Klein
57:cross-ratio
47:of a fixed
7200:Categories
6521:Historical
6516:References
5019:(elliptic)
794:arithmetic
204:hyperplane
45:complement
7103:1406.7309
7086:123015988
7070:0047-2468
7013:121783102
6997:0047-2468
6792:28:25–41.
6687:cite book
6661:cite book
6642:123810749
6609:119465069
6568:: 61–90.
6128:Citations
6019:−
5927:−
5842:−
5765:−
5614:The form
5581:−
5563:−
5545:−
5527:−
5487:−
5469:−
5451:−
5422:−
5404:−
5342:−
5176:β
5166:α
5156:β
5153:α
5132:β
5126:α
5122:∑
5050:−
4954:→
4943:β
4933:α
4923:β
4920:α
4899:β
4893:α
4889:∑
4871:ellipsoid
4786:−
4748:Ω
4706:−
4671:Ω
4609:Ω
4579:Ω
4532:β
4526:α
4512:β
4502:α
4492:β
4489:α
4481:∑
4466:Ω
4450:β
4440:α
4430:β
4427:α
4419:∑
4404:Ω
4388:β
4378:α
4368:β
4365:α
4357:∑
4342:Ω
4192:Ω
4104:Ω
4016:Ω
3996:original
3967:Ω
3963:⋅
3951:Ω
3937:Ω
3931:
3925:⋅
3896:Ω
3883:Ω
3879:−
3862:Ω
3856:−
3844:Ω
3827:Ω
3814:Ω
3810:−
3793:Ω
3775:Ω
3768:
3731:Ω
3701:Ω
3667:β
3657:α
3647:β
3644:α
3622:β
3616:α
3612:∑
3605:Ω
3507:Ω
3492:original
3392:′
3376:′
3360:′
3266:
3258:−
3140:β
3134:α
3108:β
3098:α
3088:β
3085:α
3077:∑
3068:β
3058:α
3048:β
3045:α
3037:∑
3028:β
3018:α
3008:β
3005:α
2997:∑
2990:
2982:−
2901:…
2854:…
2774:…
2759:
2751:−
2731:original
2697:β
2687:α
2677:β
2674:α
2653:β
2647:α
2643:∑
2536:original
2492:β
2486:α
2466:β
2456:α
2446:β
2443:α
2435:∑
2426:β
2416:α
2406:β
2403:α
2395:∑
2386:β
2376:α
2366:β
2363:α
2355:∑
2348:
2340:−
2133:
2125:−
2105:original
2074:β
2064:α
2054:β
2051:α
2029:β
2023:α
2019:∑
1936:original
1426:spacetime
1396:−
1324:−
1260:−
1168:−
1084:−
1003:−
959:The case
876:geodesics
767:∗
323:−
311:−
288:−
268:−
180:real line
148:invariant
125:logarithm
7041:31045705
6735:(1898),
6551:: 57–66.
6116:See also
3307:′
3293:′
3279:′
2936:′
2925:′
2914:′
2832:′
2821:′
2810:′
2294:′
2283:′
2195:′
2184:′
1848:= 0. If
1840:). Line
937:and the
893:. Given
826:) and P(
532:isometry
518:stay on
473:argument
170:and the
152:absolute
133:absolute
65:absolute
7165:7481968
7130:6389531
7078:2739193
7005:2134074
6959:0872858
6749:(1898)
3999:modern
3495:modern
2734:modern
2539:modern
2108:modern
1939:modern
1898:History
830:). Say
524:motions
49:quadric
43:on the
7163:
7153:
7128:
7118:
7084:
7076:
7068:
7039:
7011:
7003:
6995:
6957:
6947:
6822:
6640:
6607:
5443:(ring)
5363:(oval)
3928:arccos
3122:
1903:Cayley
1751:
921:, and
866:, and
546:) as
502:as an
344:takes
218:) and
160:stable
129:conics
105:metric
79:, and
41:metric
7161:S2CID
7126:S2CID
7098:arXiv
7082:S2CID
7037:S2CID
7009:S2CID
6968:from
6910:(PDF)
6638:S2CID
6605:S2CID
3479:Klein
1838:x,y,z
1816:. An
790:uses
230:on P(
51:in a
39:is a
7151:ISBN
7116:ISBN
7066:ISSN
6993:ISSN
6945:ISBN
6820:ISBN
6693:link
6667:link
6436:, §7
5106:and
4600:and
3722:and
1875:and
1855:Let
1832:Use
919:p, q
905:and
897:and
846:and
834:and
514:and
490:and
467:and
226:and
194:way.
111:and
95:The
7143:doi
7108:doi
7058:doi
7029:doi
6985:doi
6922:doi
6838:doi
6812:doi
6771:hdl
6630:doi
6597:doi
6570:doi
6566:149
3765:log
3254:cos
2978:cos
2747:cos
2336:cos
2121:cos
1869:x,y
1824:at
1808:on
1562:or
1424:in
1352:or
1201:or
913:to
854:to
494:on
459:of
174:in
119:by
99:by
7202::
7188:32
7186:,
7159:,
7149:,
7124:.
7114:.
7106:.
7080:,
7074:MR
7072:,
7064:,
7054:89
7052:,
7035:.
7025:57
7023:.
7007:,
7001:MR
6999:,
6991:,
6981:81
6979:,
6955:MR
6953:,
6943:,
6939:,
6918:36
6916:.
6912:.
6818:.
6808:19
6806:.
6802:.
6767:51
6765:.
6761:.
6689:}}
6685:{{
6663:}}
6659:{{
6636:.
6624:.
6620:.
6603:.
6591:.
6587:.
6564:.
6560:.
6549:12
6547:.
6543:.
6312:^
6226:^
6136:^
1930::
941:.
862:,
810:P(
802:P(
694:1.
534:.
463:,
186:.
75:,
7145::
7132:.
7110::
7100::
7060::
7043:.
7031::
6987::
6928:.
6924::
6844:.
6840::
6828:.
6814::
6786:n
6777:.
6773::
6695:)
6669:)
6644:.
6632::
6626:6
6611:.
6599::
6593:4
6578:.
6572::
6099:1
6096:=
6091:2
6087:Z
6083:+
6078:2
6074:Y
6070:+
6065:2
6061:X
6040:0
6037:=
6032:2
6027:4
6023:z
6014:2
6009:3
6005:z
6001:+
5996:2
5991:2
5987:z
5983:+
5978:2
5973:1
5969:z
5948:0
5945:=
5940:2
5935:3
5931:z
5923:)
5917:2
5912:2
5908:z
5904:+
5899:2
5894:1
5890:z
5885:(
5881:e
5851:0
5848:=
5845:1
5837:2
5833:z
5829:+
5824:2
5820:y
5816:+
5811:2
5807:x
5786:0
5783:=
5778:2
5773:4
5769:z
5760:2
5755:3
5751:z
5747:+
5742:2
5737:2
5733:z
5729:+
5724:2
5719:1
5715:z
5694:0
5691:=
5686:2
5681:4
5677:z
5673:+
5668:2
5663:3
5659:z
5655:+
5650:2
5645:2
5641:z
5637:+
5632:2
5627:1
5623:z
5594:2
5589:4
5585:z
5576:2
5571:3
5567:z
5558:2
5553:2
5549:z
5540:2
5535:1
5531:z
5518:2
5513:4
5509:z
5505:+
5500:2
5495:3
5491:z
5482:2
5477:2
5473:z
5464:2
5459:1
5455:z
5435:2
5430:4
5426:z
5417:2
5412:3
5408:z
5399:2
5394:2
5390:z
5386:+
5381:2
5376:1
5372:z
5355:2
5350:4
5346:z
5337:2
5332:3
5328:z
5324:+
5319:2
5314:2
5310:z
5306:+
5301:2
5296:1
5292:z
5275:2
5270:4
5266:z
5262:+
5257:2
5252:3
5248:z
5244:+
5239:2
5234:2
5230:z
5226:+
5221:2
5216:1
5212:z
5187:,
5184:0
5181:=
5172:x
5162:x
5149:a
5143:4
5138:1
5135:=
5129:,
5079:0
5076:=
5071:2
5067:t
5061:2
5057:k
5053:4
5045:2
5041:y
5037:+
5032:2
5028:x
5013:0
5010:=
5005:2
5001:t
4995:2
4991:k
4987:4
4984:+
4979:2
4975:y
4971:+
4966:2
4962:x
4951:0
4948:=
4939:x
4929:x
4916:a
4910:3
4905:1
4902:=
4896:,
4845:c
4825:c
4805:0
4802:=
4797:2
4793:c
4789:4
4781:2
4777:y
4773:+
4768:2
4764:x
4760:=
4755:x
4752:x
4727:0
4724:=
4719:2
4714:3
4710:z
4701:2
4697:z
4691:1
4687:z
4683:=
4678:x
4675:x
4650:z
4647:,
4644:y
4641:,
4638:x
4616:y
4613:y
4586:x
4583:x
4548:)
4544:2
4541:,
4538:1
4535:=
4529:,
4522:(
4508:y
4498:x
4485:a
4478:=
4473:y
4470:x
4458:0
4455:=
4446:y
4436:y
4423:a
4416:=
4411:y
4408:y
4396:0
4393:=
4384:x
4374:x
4361:a
4354:=
4349:x
4346:x
4310:2
4306:y
4300:2
4296:x
4292:c
4289:+
4285:)
4279:1
4275:y
4269:2
4265:x
4261:+
4256:2
4252:y
4246:1
4242:x
4237:(
4233:b
4230:+
4225:1
4221:y
4215:1
4211:x
4207:a
4204:=
4199:y
4196:x
4182:2
4177:2
4173:y
4169:c
4166:+
4161:2
4157:y
4151:1
4147:y
4143:b
4140:2
4137:+
4132:2
4127:1
4123:y
4119:a
4116:=
4111:y
4108:y
4094:2
4089:2
4085:x
4081:c
4078:+
4073:2
4069:x
4063:1
4059:x
4055:b
4052:2
4049:+
4044:2
4039:1
4035:x
4031:a
4028:=
4023:x
4020:x
3974:y
3971:y
3958:x
3955:x
3944:y
3941:x
3922:c
3919:i
3916:2
3913:=
3903:y
3900:y
3890:x
3887:x
3874:2
3869:y
3866:x
3851:y
3848:x
3834:y
3831:y
3821:x
3818:x
3805:2
3800:y
3797:x
3787:+
3782:y
3779:x
3762:c
3738:y
3735:y
3708:x
3705:x
3675:0
3672:=
3663:x
3653:x
3640:a
3634:2
3631:,
3628:1
3625:=
3619:,
3608:=
3584:0
3581:=
3576:2
3571:2
3567:x
3563:c
3560:+
3555:2
3551:x
3545:1
3541:x
3537:b
3534:2
3531:+
3526:2
3521:1
3517:x
3513:a
3510:=
3463:1
3460:=
3455:2
3451:z
3447:+
3442:2
3438:y
3434:+
3429:2
3425:x
3395:2
3388:z
3384:+
3379:2
3372:y
3368:+
3363:2
3356:x
3346:2
3342:z
3338:+
3333:2
3329:y
3325:+
3320:2
3316:x
3304:z
3300:z
3297:+
3290:y
3286:y
3283:+
3276:x
3272:x
3261:1
3231:0
3228:=
3223:2
3219:z
3215:+
3210:2
3206:y
3202:+
3197:2
3193:x
3162:)
3158:3
3155:,
3152:2
3149:,
3146:1
3143:=
3137:,
3130:(
3119:,
3104:y
3094:y
3081:a
3064:x
3054:x
3041:a
3024:y
3014:x
3001:a
2985:1
2945:2
2941:)
2933:z
2929:,
2922:y
2918:,
2911:x
2907:(
2904:)
2898:,
2895:a
2892:(
2883:2
2879:)
2875:z
2872:,
2869:y
2866:,
2863:x
2860:(
2857:)
2851:,
2848:a
2845:(
2837:)
2829:z
2825:,
2818:y
2814:,
2807:x
2802:(
2798:)
2795:z
2792:,
2789:y
2786:,
2783:x
2780:(
2777:)
2771:,
2768:a
2765:(
2754:1
2708:,
2705:0
2702:=
2693:x
2683:x
2670:a
2664:3
2659:1
2656:=
2650:,
2621:0
2618:=
2613:2
2609:)
2605:z
2602:,
2599:y
2596:,
2593:x
2590:(
2587:)
2584:h
2581:,
2578:g
2575:,
2572:f
2569:,
2566:c
2563:,
2560:b
2557:,
2554:a
2551:(
2508:]
2504:2
2501:,
2498:1
2495:=
2489:,
2482:[
2462:y
2452:y
2439:a
2422:x
2412:x
2399:a
2382:y
2372:x
2359:a
2343:1
2303:2
2299:)
2291:y
2287:,
2280:x
2276:(
2273:)
2270:c
2267:,
2264:b
2261:,
2258:a
2255:(
2246:2
2242:)
2238:y
2235:,
2232:x
2229:(
2226:)
2223:c
2220:,
2217:b
2214:,
2211:a
2208:(
2200:)
2192:y
2188:,
2181:x
2176:(
2172:)
2169:y
2166:,
2163:x
2160:(
2157:)
2154:c
2151:,
2148:b
2145:,
2142:a
2139:(
2128:1
2082:0
2079:=
2070:x
2060:x
2047:a
2041:2
2038:,
2035:1
2032:=
2026:,
1997:0
1994:=
1989:2
1985:)
1981:y
1978:,
1975:x
1972:(
1969:)
1966:c
1963:,
1960:b
1957:,
1954:a
1951:(
1877:Q
1873:P
1865:f
1861:Q
1857:P
1850:F
1846:z
1842:f
1836:(
1828:.
1826:F
1822:f
1810:f
1806:F
1802:f
1766:c
1764:/
1762:v
1757:v
1753:c
1735:1
1732:=
1727:2
1719:)
1709:)
1701:t
1698:d
1693:z
1690:d
1682:(
1677:+
1672:2
1664:)
1654:)
1646:t
1643:d
1638:y
1635:d
1627:(
1622:+
1617:2
1609:)
1599:)
1591:t
1588:d
1583:x
1580:d
1572:(
1550:1
1547:=
1542:2
1534:)
1524:)
1516:t
1513:d
1508:y
1505:d
1497:(
1492:+
1487:2
1479:)
1469:)
1461:t
1458:d
1453:x
1450:d
1442:(
1412:0
1409:=
1404:2
1400:t
1391:2
1387:z
1383:+
1378:2
1374:y
1370:+
1365:2
1361:x
1340:0
1337:=
1332:2
1328:t
1319:2
1315:y
1311:+
1306:2
1302:x
1281:0
1278:=
1273:2
1268:4
1264:x
1255:2
1250:3
1246:x
1242:+
1237:2
1232:2
1228:x
1224:+
1219:2
1214:1
1210:x
1189:0
1186:=
1181:2
1176:3
1172:x
1163:2
1158:2
1154:x
1150:+
1145:2
1140:1
1136:x
1103:0
1100:=
1095:2
1091:t
1087:d
1079:2
1075:z
1071:d
1068:+
1063:2
1059:y
1055:d
1052:+
1047:2
1043:x
1039:d
1019:0
1016:=
1011:2
1007:t
998:2
994:z
990:+
985:2
981:y
977:+
972:2
968:x
927:b
923:a
915:b
911:a
907:q
903:p
899:b
895:a
872:b
868:a
864:q
860:p
856:b
852:a
848:q
844:p
840:R
836:b
832:a
828:C
824:R
820:R
816:R
812:R
804:C
778:}
775:1
772:=
763:z
759:z
756:=
751:2
746:|
741:z
737:|
733::
730:z
727:{
691:=
686:2
682:)
678:z
674:/
670:y
667:(
664:+
659:2
655:)
651:z
647:/
643:x
640:(
620:}
615:2
611:z
607:=
602:2
598:y
594:+
589:2
585:x
581::
578:]
575:z
572::
569:y
566::
563:x
560:[
557:{
544:R
528:K
520:K
516:q
512:p
508:R
500:K
496:K
492:q
488:p
484:K
480:R
469:a
465:q
461:p
453:a
449:a
429:)
423:w
418:0
411:0
406:1
400:(
395:]
392:1
389::
386:z
383:[
370:w
366:a
358:q
356:+
354:p
350:q
346:p
329:]
326:q
320:z
317::
314:z
308:p
305:[
302:=
297:)
291:q
283:p
276:1
271:1
262:(
257:]
254:1
251::
248:z
245:[
232:R
228:q
224:p
216:R
20:)
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