31:
3835:
539:
531:
447:
1161:
805:
2815:
1156:{\displaystyle {\begin{aligned}\operatorname {dist} (p_{1},p_{2})&=2\operatorname {arsinh} {\frac {\|p_{2}-p_{1}\|}{2{\sqrt {y_{1}y_{2}}}}}\\&=2\operatorname {artanh} {\frac {\|p_{2}-p_{1}\|}{\|p_{2}-{\tilde {p}}_{1}\|}}\\&=2\ln {\frac {\|p_{2}-p_{1}\|+\|p_{2}-{\tilde {p}}_{1}\|}{2{\sqrt {y_{1}y_{2}}}}},\end{aligned}}}
5535:
4890:
2332:
4424:
2647:
2498:
2119:
6053:
1669:
2155:
5320:
5339:
4762:
3738:-axis which passes through the given central point. Draw a line tangent to the circle which passes through the given non-central point. Draw a horizontal line through that point of tangency and find its intersection with the vertical line.
3725:
The midpoint between the intersection of the tangent with the vertical line and the given non-central point is the center of the model circle. Draw the model circle around that new center and passing through the given non-central point.
2361:
4176:
1926:
3549:
1520:
1426:
1698:
with one point at the origin, analogous to finding arclength on the sphere by taking a stereographic projection centered on one point and measuring the
Euclidean distance in the plane from the origin to the other point.
2926:
1291:
2810:{\displaystyle \operatorname {dist} (\langle x_{1},r\rangle ,\langle x_{1}\pm r\sin \phi ,r\cos \phi \rangle )={2\operatorname {artanh} }{\bigl (}{\tan {\tfrac {1}{2}}\phi }{\bigr )}=\operatorname {gd} ^{-1}\phi ,}
124:
5743:
5059:
810:
5927:
3445:
405:
763:
3741:
The midpoint between that intersection and the given non-central point is the center of the model circle. Draw the model circle around that new center and passing through the given non-central point.
6232:
Flavors of
Geometry, MSRI Publications, Volume 31, 1997, Hyperbolic Geometry, J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry, page 87, Figure 19. Constructing the hyperbolic center of a circle
5920:
5848:
306:
5622:
4735:
4532:
3722:-axis which passes through the given central point. Draw a horizontal line through the non-central point. Construct the tangent to the circle at its intersection with that horizontal line.
666:
607:
4022:
5186:
The geodesics for this metric tensor are circular arcs perpendicular to the real axis (half-circles whose origin is on the real axis) and straight vertical lines ending on the real axis.
5171:
5114:
4972:
4479:
4146:
3331:
1865:
1810:
3706:(half-circle) between the two given points as in the previous case. Construct a tangent to that line at the non-central point. Drop a perpendicular from the given center point to the
450:
The distance between two points in the half-plane model can be computed in terms of
Euclidean distances in an isosceles trapezoid formed by the points and their reflection across the
2618:
3710:-axis. Find the intersection of these two lines to get the center of the model circle. Draw the model circle around that new center and passing through the given non-central point.
2847:
1692:
1547:
2539:
4635:
4921:
4682:
5530:{\displaystyle \gamma (t)={\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}{\begin{pmatrix}e^{t/2}&0\\0&e^{-t/2}\\\end{pmatrix}}\cdot i={\frac {aie^{t}+b}{cie^{t}+d}}.}
4885:{\displaystyle {\rm {SO}}(2)=\left.\left\{{\begin{pmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{pmatrix}}\right|\theta \in \mathbb {R} \right\}.}
3964:) is contained as an index-two normal subgroup, the other coset being the set of 2×2 matrices with real entries whose determinant equals −1, modulo plus or minus the identity.
1560:
5199:
3133:
4575:
4168:
3638:
3594:
3274:
3168:
2984:
3685:
Or in the special case where the two given points lie on a vertical line, draw that vertical line through the two points and erase the part which is on or below the
3358:
3059:
4604:
6090:
2641:
2327:{\displaystyle \operatorname {dist} (\langle x,y_{1}\rangle ,\langle x,y_{2}\rangle )=\left|\ln {\frac {y_{2}}{y_{1}}}\right|=\left|\ln(y_{2})-\ln(y_{1})\right|.}
1352:
692:
229:
4419:{\displaystyle {\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}\cdot z={\frac {az+b}{cz+d}}={\frac {(ac|z|^{2}+bd+(ad+bc)\Re (z))+i(ad-bc)\Im (z)}{|cz+d|^{2}}}.}
1919:
1892:
1754:
1727:
1322:
790:
3079:
2354:
2148:
5629:
3896:
which consists of the set of 2×2 matrices with real entries whose determinant equals +1. Note that many texts (including
Knowledge (XXG)) often say SL(2,
3456:
2493:{\displaystyle \operatorname {dist} \left(\langle x_{1},y\rangle ,\langle x_{2},y\rangle \right)=2\operatorname {arsinh} {\frac {|x_{2}-x_{1}|}{2y}}.}
1435:
1356:
534:
Distance between two points can alternately be computed using ratios of
Euclidean distances to the ideal points at the ends of the hyperbolic line.
3666:. For example, how to construct the half-circle in the Euclidean half-plane which models a line on the hyperbolic plane through two given points.
2856:
6254:
3988:). This includes both the orientation preserving and the orientation-reversing isometries. The orientation-reversing map (the mirror map) is
3659:
3653:
1168:
6154:
2114:{\displaystyle \operatorname {dist} (p_{1},p_{2})=\left|\ln {\frac {\|p_{2}-p_{0}\|\|p_{1}-p_{3}\|}{\|p_{1}-p_{0}\|\|p_{2}-p_{3}\|}}\right|.}
56:
4985:
5329:) acts transitively by isometries of the upper half-plane, this geodesic is mapped into the other geodesics through the action of PSL(2,
3674:
Draw the line segment between the two points. Construct the perpendicular bisector of the line segment. Find its intersection with the
3373:
6343:
6325:
6307:
6289:
30:
314:
4099:
3919:
697:
5759:
for this metric tensor, i.e. curves which minimize the distance) are represented in this model by circular arcs normal to the
3678:-axis. Draw the circle around the intersection which passes through the given points. Erase the part which is on or below the
6048:{\displaystyle \operatorname {dist} (p_{1},p_{2})=2\operatorname {arsinh} {\frac {\|p_{2}-p_{1}\|}{2{\sqrt {z_{1}z_{2}}}}}.}
5853:
5781:
258:
6365:
3557:
when the circle is completely inside the halfplane and touches the boundary a horocycle centered around the ideal point
6360:
5571:
4687:
4484:
2929:
1523:
1429:
191:
which means that the angles measured at a point are the same in the model as they are in the actual hyperbolic plane.
6370:
4649:
4095:
3991:
612:
553:
6273:
is freely available. On page 52 one can see an example of the semicircle diagrams so characteristic of the model.
5129:
5072:
4930:
4436:
4104:
3839:
3290:
422:
for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs
1815:
4754:
1763:
188:
3855:
3911:) consisting of the set of 2×2 matrices with real entries whose determinant equals +1 or −1. Note that SL(2,
3881:
that act on the upper half-plane by fractional linear transformations and preserve the hyperbolic distance.
3847:
2548:
3224:(a curve whose normals all converge asymptotically in the same direction, its center) is modeled by either:
2822:
1695:
1674:
1664:{\textstyle \operatorname {chord} (p_{1},p_{2})=2\sinh {\tfrac {1}{2}}\operatorname {dist} (p_{1},p_{2}),}
1529:
173:
39:
2505:
6159:
4609:
3178:
2850:
543:
5315:{\displaystyle \gamma (t)={\begin{pmatrix}e^{t/2}&0\\0&e^{-t/2}\\\end{pmatrix}}\cdot i=ie^{t}.}
4898:
4659:
538:
530:
6114:
3886:
127:
3834:
446:
6144:
5066:
3730:
If the two given points lie on a vertical line and the given center is below the other given point:
3714:
If the two given points lie on a vertical line and the given center is above the other given point:
3663:
3091:
139:
4537:
4151:
4067:
1294:
181:
135:
131:
3604:
3560:
3253:
3138:
2963:
6339:
6321:
6303:
6285:
6260:
6134:
4430:
3871:
3867:
3814:
Creating the one or two points in the intersection of a line and a circle (if they intersect):
3207:-axis at the same point as the vertical line which models its axis, but at an acute or obtuse
1554:
161:
3008:, geodesics (the shortest path between the points contained within it) are modeled by either:
6265:
6246:
6093:
5120:
1550:
232:
195:
165:
47:
3336:
3032:
5549:
5548:) on the upper half-plane. Starting with this model, one can obtain the flow on arbitrary
4580:
4059:
3960:) is again a projective group, and again, modulo plus or minus the identity matrix. PSL(2,
2542:
177:
6069:
208:
17:
4058:). This group is important in two ways. First, it is a symmetry group of the square 2x2
3818:
Find the intersection of the given semicircle (or vertical line) with the given circle.
2623:
1327:
671:
6331:
6295:
6186:"Distance formula for points in the Poincare half plane model on a "vertical geodesic""
6139:
6129:
6124:
4044:
3064:
1897:
1870:
1732:
1705:
1300:
768:
150:
6346:. An elementary introduction to the Poincaré half-plane model of the hyperbolic plane.
4086:) can be used to tessellate the hyperbolic plane into cells of equal (Poincaré) area.
3822:
Creating the one or two points in the intersection of two circles (if they intersect):
3806:
Creating the point which is the intersection of two existing lines, if they intersect:
6354:
5565:
5541:
5124:
4051:
423:
252:
146:
6270:
6149:
5062:
4063:
2339:
2133:
3544:{\displaystyle {\frac {1}{2}}\ln \left({\frac {y_{e}+r_{e}}{y_{e}-r_{e}}}\right).}
3364:
when the circle is completely inside the halfplane a hyperbolic circle with center
6119:
5553:
5545:
5181:
3247:
3236:
3194:
2987:
2942:
2124:
1515:{\textstyle \operatorname {artanh} x={\frac {1}{2}}\ln \left((1+x)/(1-x)\right)}
169:
1421:{\textstyle \operatorname {arsinh} x=\ln {\bigl (}x+{\sqrt {x^{2}+1}}{\bigr )}}
6269:
v.1, p. 1. First article in a series exploiting the half-plane model. An
4979:
3599:
2991:
542:
Distance from the apex of a semicircle to another point on it is the inverse
6277:
5756:
3878:
3220:
2921:{\textstyle \operatorname {artanh} x={\tfrac {1}{2}}\ln {\dfrac {1+x}{1-x}}}
419:
6185:
6208:
4082:), and thus has a hyperbolic behavior embedded in it. In particular, SL(2,
3182:(a curve equidistant from a straight line, its axis) is modeled by either:
1671:
analogous to finding arclength on a sphere in terms of chord length. This
1286:{\textstyle \|p_{2}-p_{1}\|={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}}
3973:
199:
3810:
Find the intersection of the two given semicircles (or vertical lines).
4062:
of points. Thus, functions that are periodic on a square grid, such as
3922:
2946:(points at infinity) in the Poincaré half-plane model are of two kinds:
145:
Equivalently the
Poincaré half-plane model is sometimes described as a
119:{\displaystyle =\{\langle x,y\rangle \mid y>0;x,y\in \mathbb {R} \}}
5738:{\displaystyle (ds)^{2}={\frac {(dx)^{2}+(dy)^{2}+(dz)^{2}}{z^{2}}}\,}
5054:{\displaystyle \mathbb {H} ={\rm {PSL}}(2,\mathbb {R} )/{\rm {SO}}(2)}
3968:
The relationship of these groups to the
Poincaré model is as follows:
1756:
are on a hyperbolic line (Euclidean half-circle) which intersects the
3889:
3662:
in the model to achieve the effect of the basic constructions in the
3643:
when the circle intersects the boundary non- orthogonal a hypercycle.
5065:
of unit-length tangent vectors on the upper half-plane, called the
3734:
Draw a circle around the intersection of the vertical line and the
3718:
Draw a circle around the intersection of the vertical line and the
3197:
as the half-circle which models its axis but at an acute or obtuse
1694:
formula can be thought of as coming from
Euclidean distance in the
3833:
3208:
3198:
537:
529:
445:
29:
34:
Parallel rays in
Poincare half-plane model of hyperbolic geometry
3440:{\displaystyle \left(x_{e},{\sqrt {y_{e}^{2}-r_{e}^{2}}}\right)}
5189:
The unit-speed geodesic going up vertically, through the point
3786:
going through the point where the tangent and the circle meet.
3693:
Creating the circle through one point with center another point
2130:
Some special cases can be simplified. Two points with the same
800:
between the two points under the hyperbolic-plane metric is:
4789:
400:{\displaystyle (ds)^{2}={\frac {(dx)^{2}+(dy)^{2}}{y^{2}}}}
27:
Upper-half plane model of hyperbolic non-Euclidean geometry
758:{\textstyle {\tilde {p}}_{1}=\langle x_{1},-y_{1}\rangle }
202:
between the half-plane model and the
Poincaré disk model.
3866:), the transforms with real coefficients, and these act
414:
measures the length along a (possibly curved) line. The
3870:
and isometrically on the upper half-plane, making it a
434:-axis) and straight vertical rays perpendicular to the
6251:
Teoria fondamentale degli spazi di curvatura constante
5856:
5784:
5574:
5399:
5363:
5223:
4801:
4185:
3029:(curves equidistant from a central point) with center
2873:
2859:
2825:
2760:
2626:
2551:
2508:
2342:
2136:
1900:
1873:
1818:
1766:
1735:
1708:
1677:
1612:
1563:
1532:
1438:
1359:
1330:
1303:
1171:
771:
700:
674:
615:
556:
6072:
5930:
5632:
5342:
5333:). Thus, the general unit-speed geodesic is given by
5202:
5132:
5075:
4988:
4933:
4901:
4765:
4690:
4662:
4612:
4583:
4540:
4487:
4439:
4179:
4154:
4107:
3994:
3607:
3563:
3459:
3376:
3339:
3293:
3256:
3141:
3094:
3067:
3035:
2966:
2891:
2650:
2364:
2158:
1929:
1553:
length in the Minkowski metric between points in the
808:
317:
261:
211:
59:
5915:{\textstyle p_{2}=\langle x_{2},y_{2},z_{2}\rangle }
5843:{\textstyle p_{1}=\langle x_{1},y_{1},z_{1}\rangle }
301:{\displaystyle \{\langle x,y\rangle \mid y>0\},}
6084:
6047:
5922:measured in this metric along such a geodesic is:
5914:
5842:
5751:measures length along a possibly curved line. The
5737:
5616:
5529:
5314:
5165:
5108:
5053:
4966:
4915:
4884:
4729:
4676:
4629:
4598:
4569:
4526:
4473:
4418:
4162:
4140:
4043:Important subgroups of the isometry group are the
4027:The group of orientation-preserving isometries of
4016:
3632:
3588:
3543:
3439:
3352:
3325:
3268:
3162:
3127:
3073:
3053:
2978:
2920:
2841:
2809:
2635:
2612:
2533:
2492:
2348:
2326:
2142:
2113:
1913:
1886:
1859:
1804:
1748:
1721:
1686:
1663:
1541:
1514:
1420:
1346:
1316:
1285:
1155:
784:
757:
686:
660:
601:
399:
300:
223:
118:
5767:-plane) and straight vertical rays normal to the
5617:{\textstyle \{\langle x,y,z\rangle \mid z>0\}}
4730:{\displaystyle g\in {\rm {PSL}}(2,\mathbb {R} )}
4527:{\displaystyle g\in {\rm {PSL}}(2,\mathbb {R} )}
3826:Find the intersection of the two given circles.
3771:Draw a line tangent to the circle going through
4974:, this means that the isotropy subgroup of any
3858:. The subgroup that maps the upper half-plane,
3753:from the Euclidean center of the circle to the
3018:straight vertical rays orthogonal to the x-axis
661:{\textstyle p_{2}=\langle x_{2},y_{2}\rangle }
602:{\textstyle p_{1}=\langle x_{1},y_{1}\rangle }
4017:{\displaystyle z\rightarrow -{\overline {z}}}
3698:If the two points are not on a vertical line:
3670:Creating the line through two existing points
2777:
2747:
1549:formula can be thought of as coming from the
1413:
1380:
430:-axis (half-circles whose centers are on the
160:The Poincaré half-plane model is named after
8:
6007:
5981:
5909:
5870:
5837:
5798:
5763:-plane (half-circles whose origin is on the
5611:
5596:
5578:
5575:
3320:
3294:
2728:
2685:
2679:
2660:
2528:
2509:
2420:
2401:
2395:
2376:
2212:
2193:
2187:
2168:
2097:
2071:
2068:
2042:
2037:
2011:
2008:
1982:
1851:
1832:
1799:
1780:
1198:
1172:
1111:
1076:
1070:
1044:
1016:
981:
976:
950:
893:
867:
752:
723:
655:
629:
596:
570:
292:
277:
265:
262:
113:
78:
66:
63:
6188:. mathematics stackexchange. August 6, 2015
6066:This model can be generalized to model an
5544:on the unit-length tangent bundle (complex
5166:{\displaystyle {\rm {SL}}(2,\mathbb {Z} ).}
5109:{\displaystyle {\rm {PSL}}(2,\mathbb {R} )}
4967:{\displaystyle {\rm {PSL}}(2,\mathbb {R} )}
4474:{\displaystyle z_{1},z_{2}\in \mathbb {H} }
4141:{\displaystyle {\rm {PSL}}(2,\mathbb {R} )}
3941:) modulo plus or minus the identity matrix.
3789:The (hyperbolic) center is the point where
3745:Given a circle find its (hyperbolic) center
3326:{\displaystyle \langle x_{e},y_{e}\rangle }
1860:{\textstyle p_{3}=\langle x_{3},0\rangle ,}
205:This model can be generalized to model an
3015:half-circles whose origin is on the x-axis
1805:{\textstyle p_{0}=\langle x_{0},0\rangle }
6209:"Tools to work with the Half-Plane model"
6071:
6031:
6021:
6015:
6001:
5988:
5978:
5957:
5944:
5929:
5903:
5890:
5877:
5861:
5855:
5831:
5818:
5805:
5789:
5783:
5734:
5726:
5715:
5693:
5671:
5655:
5646:
5631:
5573:
5540:This provides a basic description of the
5509:
5485:
5472:
5445:
5438:
5410:
5406:
5394:
5358:
5341:
5303:
5269:
5262:
5234:
5230:
5218:
5201:
5153:
5152:
5134:
5133:
5131:
5119:The upper half-plane is tessellated into
5099:
5098:
5077:
5076:
5074:
5033:
5032:
5027:
5020:
5019:
4998:
4997:
4990:
4989:
4987:
4957:
4956:
4935:
4934:
4932:
4909:
4908:
4900:
4870:
4869:
4796:
4767:
4766:
4764:
4720:
4719:
4698:
4697:
4689:
4670:
4669:
4661:
4620:
4619:
4611:
4582:
4561:
4548:
4539:
4517:
4516:
4495:
4494:
4486:
4467:
4466:
4457:
4444:
4438:
4404:
4399:
4381:
4286:
4281:
4272:
4260:
4225:
4180:
4178:
4156:
4155:
4153:
4131:
4130:
4109:
4108:
4106:
4004:
3993:
3764:be the intersection of this line and the
3615:
3606:
3571:
3562:
3525:
3512:
3500:
3487:
3480:
3460:
3458:
3424:
3419:
3406:
3401:
3395:
3386:
3375:
3344:
3338:
3314:
3301:
3292:
3255:
3140:
3093:
3066:
3034:
2965:
2890:
2872:
2858:
2830:
2824:
2789:
2776:
2775:
2759:
2752:
2746:
2745:
2737:
2692:
2667:
2649:
2625:
2601:
2588:
2575:
2565:
2550:
2516:
2507:
2471:
2465:
2452:
2443:
2440:
2408:
2383:
2363:
2341:
2307:
2282:
2248:
2238:
2232:
2206:
2181:
2157:
2135:
2091:
2078:
2062:
2049:
2031:
2018:
2002:
1989:
1979:
1956:
1943:
1928:
1905:
1899:
1878:
1872:
1839:
1823:
1817:
1787:
1771:
1765:
1740:
1734:
1713:
1707:
1676:
1649:
1636:
1611:
1590:
1577:
1562:
1531:
1487:
1451:
1437:
1412:
1411:
1397:
1391:
1379:
1378:
1358:
1335:
1329:
1308:
1302:
1275:
1265:
1252:
1236:
1226:
1213:
1204:
1192:
1179:
1170:
1135:
1125:
1119:
1105:
1094:
1093:
1083:
1064:
1051:
1041:
1010:
999:
998:
988:
970:
957:
947:
917:
907:
901:
887:
874:
864:
839:
826:
809:
807:
776:
770:
746:
730:
714:
703:
702:
699:
673:
649:
636:
620:
614:
590:
577:
561:
555:
389:
378:
356:
340:
331:
316:
260:
210:
157:coordinate mentioned above) is positive.
109:
108:
58:
6338:, pp. 100–104, Springer-Verlag, NY
6263:(1882) "Théorie des Groupes Fuchsiens",
3598:when the circle intersects the boundary
2620:and another point at a central angle of
6177:
4074:) symmetry from the grid. Second, SL(2,
2613:{\textstyle (x-x_{1})^{2}+y^{2}=r^{2},}
176:, to show that hyperbolic geometry was
6255:Annali di Matematica Pura ed Applicata
6213:Tools to work with the Half-Plane mode
3937:}, consisting of the matrices in SL(2,
3660:compass and straightedge constructions
3654:Compass and straightedge constructions
3648:Compass and straightedge constructions
3246:-axis, in this case the center is the
5552:, as described in the article on the
3239:of intersection, which is its center)
3203:a straight line which intersects the
2842:{\textstyle \operatorname {gd} ^{-1}}
1687:{\textstyle 2\operatorname {artanh} }
1542:{\textstyle 2\operatorname {arsinh} }
796:-axis into the lower half plane, the
7:
6104:dimensional Euclidean vector space.
3854:) acts on the Riemann sphere by the
3189:a circular arc which intersects the
2534:{\textstyle \langle x_{1},r\rangle }
243:dimensional Euclidean vector space.
6302:(2002), Springer-Verlag, New York.
6284:(1980), Springer-Verlag, New York.
5138:
5135:
5084:
5081:
5078:
5037:
5034:
5005:
5002:
4999:
4942:
4939:
4936:
4771:
4768:
4705:
4702:
4699:
4630:{\displaystyle z\in \mathbb {H} ,}
4577:. It is also faithful, in that if
4502:
4499:
4496:
4367:
4325:
4116:
4113:
4110:
4078:) is of course a subgroup of SL(2,
3263:
2973:
25:
4916:{\displaystyle z\in \mathbb {H} }
4677:{\displaystyle z\in \mathbb {H} }
668:are two points in the half-plane
5568:of the model on the half- space
255:of the model on the half-plane,
4100:projective special linear group
3920:projective special linear group
3877:There are four closely related
3287:A Euclidean circle with center
6145:Models of the hyperbolic plane
5963:
5937:
5712:
5702:
5690:
5680:
5668:
5658:
5643:
5633:
5352:
5346:
5212:
5206:
5157:
5143:
5103:
5089:
5048:
5042:
5024:
5010:
4961:
4947:
4782:
4776:
4724:
4710:
4521:
4507:
4400:
4382:
4376:
4370:
4364:
4346:
4337:
4334:
4328:
4322:
4304:
4282:
4273:
4263:
4135:
4121:
3998:
3915:) is a subgroup of this group.
3900:) when they really mean PSL(2,
3627:
3608:
3583:
3564:
3157:
3151:
3122:
3119:
3113:
3095:
3048:
3036:
2731:
2657:
2572:
2552:
2472:
2444:
2313:
2300:
2288:
2275:
2215:
2165:
1962:
1936:
1655:
1629:
1596:
1570:
1504:
1492:
1484:
1472:
1272:
1245:
1233:
1206:
1099:
1004:
845:
819:
708:
375:
365:
353:
343:
328:
318:
1:
6096:by replacing the real number
5560:The model in three dimensions
4050:One also frequently sees the
3128:{\displaystyle (x,y\cosh(r))}
235:by replacing the real number
6320:, Jones and Bartlett, 1993,
6155:Schwarz–Ahlfors–Pick theorem
4570:{\displaystyle gz_{1}=z_{2}}
4163:{\displaystyle \mathbb {H} }
4070:, will thus inherit an SL(2,
4031:, sometimes denoted as Isom(
4009:
3980:, sometimes denoted as Isom(
168:who used it, along with the
6387:
6257:, ser II 2 (1868), 232–255
5179:
4035:), is isomorphic to PSL(2,
3984:), is isomorphic to PSL(2,
3651:
2930:inverse hyperbolic tangent
1760:-axis at the ideal points
1524:inverse hyperbolic tangent
5755:in the hyperbolic space (
3633:{\displaystyle (y_{e}=0)}
3589:{\displaystyle (x_{e},0)}
3269:{\displaystyle y=\infty }
3235:-axis (but excluding the
3163:{\displaystyle y\sinh(r)}
2979:{\displaystyle y=\infty }
2936:Special points and curves
2336:Two points with the same
418:in the hyperbolic plane (
164:, but it originated with
44:Poincaré half-plane model
18:Poincaré half-space model
6300:Compact Riemann Surfaces
6207:Bochaca, Judit Abardia.
4429:Note that the action is
3658:Here is how one can use
3231:a circle tangent to the
6318:The Poincaré Half-Plane
3862:, onto itself is PSL(2,
3848:projective linear group
3242:a line parallel to the
2960:one imaginary point at
1430:inverse hyperbolic sine
454:-axis: a "side length"
6276:Hershel M. Farkas and
6086:
6049:
5916:
5844:
5739:
5618:
5531:
5316:
5167:
5110:
5055:
4968:
4917:
4886:
4731:
4678:
4631:
4600:
4571:
4528:
4475:
4420:
4164:
4142:
4018:
3856:Möbius transformations
3843:
3634:
3590:
3545:
3441:
3354:
3327:
3270:
3193:-axis at the same two
3164:
3129:
3075:
3055:
2980:
2922:
2843:
2811:
2637:
2614:
2535:
2494:
2350:
2328:
2144:
2115:
1915:
1888:
1861:
1806:
1750:
1723:
1688:
1665:
1543:
1516:
1422:
1348:
1318:
1287:
1157:
786:
759:
688:
662:
603:
547:
535:
527:
480:. It is the logarithm
401:
302:
225:
120:
40:non-Euclidean geometry
35:
6160:Ultraparallel theorem
6087:
6050:
5917:
5845:
5740:
5619:
5532:
5317:
5168:
5111:
5061:. Alternatively, the
5056:
4969:
4918:
4887:
4732:
4679:
4632:
4601:
4572:
4529:
4476:
4421:
4165:
4143:
4019:
3837:
3778:Draw the half circle
3749:Drop a perpendicular
3635:
3591:
3546:
3442:
3355:
3353:{\displaystyle r_{e}}
3328:
3271:
3165:
3130:
3088:a circle with center
3076:
3056:
3054:{\displaystyle (x,y)}
2981:
2923:
2851:Gudermannian function
2844:
2812:
2638:
2615:
2536:
2495:
2351:
2329:
2145:
2116:
1916:
1889:
1862:
1807:
1751:
1724:
1689:
1666:
1544:
1517:
1423:
1349:
1319:
1288:
1158:
787:
765:is the reflection of
760:
689:
663:
604:
546:of the central angle.
544:Gudermannian function
541:
533:
449:
402:
303:
226:
121:
33:
6336:Numbers and Geometry
6115:Angle of parallelism
6070:
5928:
5854:
5782:
5630:
5572:
5340:
5200:
5130:
5073:
4986:
4931:
4899:
4763:
4749:. The stabilizer of
4688:
4660:
4610:
4599:{\displaystyle gz=z}
4581:
4538:
4485:
4437:
4177:
4152:
4105:
3992:
3887:special linear group
3605:
3561:
3457:
3374:
3337:
3291:
3254:
3139:
3092:
3065:
3033:
2964:
2857:
2823:
2648:
2624:
2549:
2506:
2362:
2340:
2156:
2134:
1927:
1898:
1871:
1816:
1764:
1733:
1706:
1675:
1561:
1530:
1436:
1357:
1328:
1301:
1169:
806:
769:
698:
672:
613:
554:
462:, and two "heights"
442:Distance calculation
315:
259:
209:
57:
6366:Hyperbolic geometry
6085:{\displaystyle n+1}
5778:between two points
5069:, is isomorphic to
5067:unit tangent bundle
4927:by some element of
3801:Other constructions
3429:
3411:
2990:to which all lines
2953:the points on the
2636:{\textstyle \phi .}
1696:Poincaré disk model
1347:{\textstyle p_{2},}
687:{\textstyle y>0}
224:{\displaystyle n+1}
174:Poincaré disk model
140:hyperbolic geometry
138:of two-dimensional
50:, denoted below as
6361:Conformal geometry
6100:by a vector in an
6082:
6045:
5912:
5840:
5735:
5614:
5527:
5457:
5388:
5312:
5281:
5163:
5106:
5051:
4964:
4913:
4895:Since any element
4882:
4853:
4727:
4674:
4627:
4596:
4567:
4524:
4471:
4416:
4210:
4160:
4138:
4090:Isometric symmetry
4068:elliptic functions
4014:
3844:
3838:Stellated regular
3630:
3586:
3541:
3437:
3415:
3397:
3350:
3323:
3283:Euclidean synopsis
3266:
3160:
3125:
3071:
3051:
2976:
2918:
2916:
2882:
2839:
2807:
2769:
2633:
2610:
2545:of the semicircle
2531:
2490:
2346:
2324:
2140:
2111:
1914:{\textstyle p_{2}}
1911:
1887:{\textstyle p_{1}}
1884:
1867:the distance from
1857:
1802:
1749:{\textstyle p_{2}}
1746:
1722:{\textstyle p_{1}}
1719:
1702:If the two points
1684:
1661:
1621:
1539:
1512:
1418:
1344:
1317:{\textstyle p_{1}}
1314:
1295:Euclidean distance
1283:
1153:
1151:
785:{\textstyle p_{1}}
782:
755:
684:
658:
599:
548:
536:
528:
397:
298:
239:by a vector in an
221:
182:Euclidean geometry
134:, that makes it a
126:, together with a
116:
36:
6311:(See Section 2.3)
6135:Hyperbolic motion
6040:
6037:
5732:
5522:
5121:free regular sets
4654:isotropy subgroup
4481:, there exists a
4411:
4255:
4012:
3972:The group of all
3872:homogeneous space
3840:heptagonal tiling
3640:a hyperbolic line
3532:
3468:
3430:
3074:{\displaystyle r}
2915:
2881:
2768:
2485:
2254:
2101:
1620:
1555:hyperboloid model
1459:
1409:
1281:
1144:
1141:
1102:
1020:
1007:
926:
923:
711:
395:
16:(Redirected from
6378:
6282:Riemann Surfaces
6266:Acta Mathematica
6247:Eugenio Beltrami
6233:
6230:
6224:
6223:
6221:
6219:
6204:
6198:
6197:
6195:
6193:
6182:
6094:hyperbolic space
6091:
6089:
6088:
6083:
6054:
6052:
6051:
6046:
6041:
6039:
6038:
6036:
6035:
6026:
6025:
6016:
6010:
6006:
6005:
5993:
5992:
5979:
5962:
5961:
5949:
5948:
5921:
5919:
5918:
5913:
5908:
5907:
5895:
5894:
5882:
5881:
5866:
5865:
5849:
5847:
5846:
5841:
5836:
5835:
5823:
5822:
5810:
5809:
5794:
5793:
5744:
5742:
5741:
5736:
5733:
5731:
5730:
5721:
5720:
5719:
5698:
5697:
5676:
5675:
5656:
5651:
5650:
5623:
5621:
5620:
5615:
5550:Riemann surfaces
5536:
5534:
5533:
5528:
5523:
5521:
5514:
5513:
5497:
5490:
5489:
5473:
5462:
5461:
5454:
5453:
5449:
5419:
5418:
5414:
5393:
5392:
5321:
5319:
5318:
5313:
5308:
5307:
5286:
5285:
5278:
5277:
5273:
5243:
5242:
5238:
5172:
5170:
5169:
5164:
5156:
5142:
5141:
5115:
5113:
5112:
5107:
5102:
5088:
5087:
5060:
5058:
5057:
5052:
5041:
5040:
5031:
5023:
5009:
5008:
4993:
4982:to SO(2). Thus,
4973:
4971:
4970:
4965:
4960:
4946:
4945:
4922:
4920:
4919:
4914:
4912:
4891:
4889:
4888:
4883:
4878:
4874:
4873:
4862:
4858:
4857:
4775:
4774:
4736:
4734:
4733:
4728:
4723:
4709:
4708:
4683:
4681:
4680:
4675:
4673:
4636:
4634:
4633:
4628:
4623:
4605:
4603:
4602:
4597:
4576:
4574:
4573:
4568:
4566:
4565:
4553:
4552:
4533:
4531:
4530:
4525:
4520:
4506:
4505:
4480:
4478:
4477:
4472:
4470:
4462:
4461:
4449:
4448:
4425:
4423:
4422:
4417:
4412:
4410:
4409:
4408:
4403:
4385:
4379:
4291:
4290:
4285:
4276:
4261:
4256:
4254:
4240:
4226:
4215:
4214:
4169:
4167:
4166:
4161:
4159:
4147:
4145:
4144:
4139:
4134:
4120:
4119:
4023:
4021:
4020:
4015:
4013:
4005:
3944:The group PSL(2,
3907:The group S*L(2,
3702:Draw the radial
3664:hyperbolic plane
3639:
3637:
3636:
3631:
3620:
3619:
3595:
3593:
3592:
3587:
3576:
3575:
3550:
3548:
3547:
3542:
3537:
3533:
3531:
3530:
3529:
3517:
3516:
3506:
3505:
3504:
3492:
3491:
3481:
3469:
3461:
3446:
3444:
3443:
3438:
3436:
3432:
3431:
3428:
3423:
3410:
3405:
3396:
3391:
3390:
3359:
3357:
3356:
3351:
3349:
3348:
3332:
3330:
3329:
3324:
3319:
3318:
3306:
3305:
3275:
3273:
3272:
3267:
3169:
3167:
3166:
3161:
3134:
3132:
3131:
3126:
3080:
3078:
3077:
3072:
3060:
3058:
3057:
3052:
2985:
2983:
2982:
2977:
2927:
2925:
2924:
2919:
2917:
2914:
2903:
2892:
2883:
2874:
2848:
2846:
2845:
2840:
2838:
2837:
2816:
2814:
2813:
2808:
2797:
2796:
2781:
2780:
2774:
2770:
2761:
2751:
2750:
2744:
2697:
2696:
2672:
2671:
2642:
2640:
2639:
2634:
2619:
2617:
2616:
2611:
2606:
2605:
2593:
2592:
2580:
2579:
2570:
2569:
2540:
2538:
2537:
2532:
2521:
2520:
2499:
2497:
2496:
2491:
2486:
2484:
2476:
2475:
2470:
2469:
2457:
2456:
2447:
2441:
2427:
2423:
2413:
2412:
2388:
2387:
2355:
2353:
2352:
2347:
2333:
2331:
2330:
2325:
2320:
2316:
2312:
2311:
2287:
2286:
2260:
2256:
2255:
2253:
2252:
2243:
2242:
2233:
2211:
2210:
2186:
2185:
2149:
2147:
2146:
2141:
2120:
2118:
2117:
2112:
2107:
2103:
2102:
2100:
2096:
2095:
2083:
2082:
2067:
2066:
2054:
2053:
2040:
2036:
2035:
2023:
2022:
2007:
2006:
1994:
1993:
1980:
1961:
1960:
1948:
1947:
1920:
1918:
1917:
1912:
1910:
1909:
1893:
1891:
1890:
1885:
1883:
1882:
1866:
1864:
1863:
1858:
1844:
1843:
1828:
1827:
1811:
1809:
1808:
1803:
1792:
1791:
1776:
1775:
1755:
1753:
1752:
1747:
1745:
1744:
1728:
1726:
1725:
1720:
1718:
1717:
1693:
1691:
1690:
1685:
1670:
1668:
1667:
1662:
1654:
1653:
1641:
1640:
1622:
1613:
1595:
1594:
1582:
1581:
1548:
1546:
1545:
1540:
1521:
1519:
1518:
1513:
1511:
1507:
1491:
1460:
1452:
1427:
1425:
1424:
1419:
1417:
1416:
1410:
1402:
1401:
1392:
1384:
1383:
1353:
1351:
1350:
1345:
1340:
1339:
1323:
1321:
1320:
1315:
1313:
1312:
1292:
1290:
1289:
1284:
1282:
1280:
1279:
1270:
1269:
1257:
1256:
1241:
1240:
1231:
1230:
1218:
1217:
1205:
1197:
1196:
1184:
1183:
1162:
1160:
1159:
1154:
1152:
1145:
1143:
1142:
1140:
1139:
1130:
1129:
1120:
1114:
1110:
1109:
1104:
1103:
1095:
1088:
1087:
1069:
1068:
1056:
1055:
1042:
1025:
1021:
1019:
1015:
1014:
1009:
1008:
1000:
993:
992:
979:
975:
974:
962:
961:
948:
931:
927:
925:
924:
922:
921:
912:
911:
902:
896:
892:
891:
879:
878:
865:
844:
843:
831:
830:
791:
789:
788:
783:
781:
780:
764:
762:
761:
756:
751:
750:
735:
734:
719:
718:
713:
712:
704:
693:
691:
690:
685:
667:
665:
664:
659:
654:
653:
641:
640:
625:
624:
608:
606:
605:
600:
595:
594:
582:
581:
566:
565:
526:
525:
522:
499:
479:
470:
461:
457:
453:
406:
404:
403:
398:
396:
394:
393:
384:
383:
382:
361:
360:
341:
336:
335:
307:
305:
304:
299:
233:hyperbolic space
230:
228:
227:
222:
196:Cayley transform
166:Eugenio Beltrami
125:
123:
122:
117:
112:
48:upper half-plane
21:
6386:
6385:
6381:
6380:
6379:
6377:
6376:
6375:
6351:
6350:
6349:
6237:
6236:
6231:
6227:
6217:
6215:
6206:
6205:
6201:
6191:
6189:
6184:
6183:
6179:
6169:
6164:
6110:
6068:
6067:
6064:
6027:
6017:
6011:
5997:
5984:
5980:
5953:
5940:
5926:
5925:
5899:
5886:
5873:
5857:
5852:
5851:
5827:
5814:
5801:
5785:
5780:
5779:
5722:
5711:
5689:
5667:
5657:
5642:
5628:
5627:
5570:
5569:
5562:
5505:
5498:
5481:
5474:
5456:
5455:
5434:
5432:
5426:
5425:
5420:
5402:
5395:
5387:
5386:
5381:
5375:
5374:
5369:
5359:
5338:
5337:
5299:
5280:
5279:
5258:
5256:
5250:
5249:
5244:
5226:
5219:
5198:
5197:
5184:
5178:
5128:
5127:
5071:
5070:
4984:
4983:
4929:
4928:
4897:
4896:
4852:
4851:
4840:
4825:
4824:
4813:
4797:
4792:
4791:
4788:
4761:
4760:
4686:
4685:
4658:
4657:
4608:
4607:
4579:
4578:
4557:
4544:
4536:
4535:
4483:
4482:
4453:
4440:
4435:
4434:
4398:
4380:
4280:
4262:
4241:
4227:
4209:
4208:
4203:
4197:
4196:
4191:
4181:
4175:
4174:
4150:
4149:
4103:
4102:
4092:
4045:Fuchsian groups
3990:
3989:
3832:
3830:Symmetry groups
3803:
3747:
3695:
3672:
3656:
3650:
3611:
3603:
3602:
3567:
3559:
3558:
3521:
3508:
3507:
3496:
3483:
3482:
3476:
3455:
3454:
3382:
3381:
3377:
3372:
3371:
3340:
3335:
3334:
3310:
3297:
3289:
3288:
3285:
3252:
3251:
3137:
3136:
3090:
3089:
3063:
3062:
3031:
3030:
2998:-axis converge.
2962:
2961:
2938:
2904:
2893:
2855:
2854:
2849:is the inverse
2826:
2821:
2820:
2785:
2688:
2663:
2646:
2645:
2622:
2621:
2597:
2584:
2571:
2561:
2547:
2546:
2512:
2504:
2503:
2477:
2461:
2448:
2442:
2404:
2379:
2375:
2371:
2360:
2359:
2338:
2337:
2303:
2278:
2268:
2264:
2244:
2234:
2225:
2221:
2202:
2177:
2154:
2153:
2132:
2131:
2087:
2074:
2058:
2045:
2041:
2027:
2014:
1998:
1985:
1981:
1972:
1968:
1952:
1939:
1925:
1924:
1901:
1896:
1895:
1874:
1869:
1868:
1835:
1819:
1814:
1813:
1783:
1767:
1762:
1761:
1736:
1731:
1730:
1709:
1704:
1703:
1673:
1672:
1645:
1632:
1586:
1573:
1559:
1558:
1528:
1527:
1471:
1467:
1434:
1433:
1393:
1355:
1354:
1331:
1326:
1325:
1304:
1299:
1298:
1297:between points
1271:
1261:
1248:
1232:
1222:
1209:
1188:
1175:
1167:
1166:
1150:
1149:
1131:
1121:
1115:
1092:
1079:
1060:
1047:
1043:
1023:
1022:
997:
984:
980:
966:
953:
949:
929:
928:
913:
903:
897:
883:
870:
866:
848:
835:
822:
804:
803:
772:
767:
766:
742:
726:
701:
696:
695:
670:
669:
645:
632:
616:
611:
610:
586:
573:
557:
552:
551:
523:
521:
515:
500:
497:
495:
488:
481:
478:
472:
469:
463:
459:
458:, a "diagonal"
455:
451:
444:
385:
374:
352:
342:
327:
313:
312:
257:
256:
249:
207:
206:
132:Poincaré metric
55:
54:
28:
23:
22:
15:
12:
11:
5:
6384:
6382:
6374:
6373:
6371:Henri Poincaré
6368:
6363:
6353:
6352:
6348:
6347:
6332:John Stillwell
6329:
6314:
6293:
6274:
6261:Henri Poincaré
6258:
6243:
6242:
6241:
6235:
6234:
6225:
6199:
6176:
6175:
6174:
6173:
6168:
6165:
6163:
6162:
6157:
6152:
6147:
6142:
6140:Kleinian model
6137:
6132:
6130:Fuchsian model
6127:
6125:Fuchsian group
6122:
6117:
6111:
6109:
6106:
6081:
6078:
6075:
6063:
6056:
6044:
6034:
6030:
6024:
6020:
6014:
6009:
6004:
6000:
5996:
5991:
5987:
5983:
5977:
5974:
5971:
5968:
5965:
5960:
5956:
5952:
5947:
5943:
5939:
5936:
5933:
5911:
5906:
5902:
5898:
5893:
5889:
5885:
5880:
5876:
5872:
5869:
5864:
5860:
5839:
5834:
5830:
5826:
5821:
5817:
5813:
5808:
5804:
5800:
5797:
5792:
5788:
5753:straight lines
5729:
5725:
5718:
5714:
5710:
5707:
5704:
5701:
5696:
5692:
5688:
5685:
5682:
5679:
5674:
5670:
5666:
5663:
5660:
5654:
5649:
5645:
5641:
5638:
5635:
5613:
5610:
5607:
5604:
5601:
5598:
5595:
5592:
5589:
5586:
5583:
5580:
5577:
5561:
5558:
5538:
5537:
5526:
5520:
5517:
5512:
5508:
5504:
5501:
5496:
5493:
5488:
5484:
5480:
5477:
5471:
5468:
5465:
5460:
5452:
5448:
5444:
5441:
5437:
5433:
5431:
5428:
5427:
5424:
5421:
5417:
5413:
5409:
5405:
5401:
5400:
5398:
5391:
5385:
5382:
5380:
5377:
5376:
5373:
5370:
5368:
5365:
5364:
5362:
5357:
5354:
5351:
5348:
5345:
5325:Because PSL(2,
5323:
5322:
5311:
5306:
5302:
5298:
5295:
5292:
5289:
5284:
5276:
5272:
5268:
5265:
5261:
5257:
5255:
5252:
5251:
5248:
5245:
5241:
5237:
5233:
5229:
5225:
5224:
5222:
5217:
5214:
5211:
5208:
5205:
5180:Main article:
5177:
5174:
5162:
5159:
5155:
5151:
5148:
5145:
5140:
5137:
5105:
5101:
5097:
5094:
5091:
5086:
5083:
5080:
5050:
5047:
5044:
5039:
5036:
5030:
5026:
5022:
5018:
5015:
5012:
5007:
5004:
5001:
4996:
4992:
4963:
4959:
4955:
4952:
4949:
4944:
4941:
4938:
4911:
4907:
4904:
4893:
4892:
4881:
4877:
4872:
4868:
4865:
4861:
4856:
4850:
4847:
4844:
4841:
4839:
4836:
4833:
4830:
4827:
4826:
4823:
4820:
4817:
4814:
4812:
4809:
4806:
4803:
4802:
4800:
4795:
4790:
4787:
4784:
4781:
4778:
4773:
4770:
4755:rotation group
4726:
4722:
4718:
4715:
4712:
4707:
4704:
4701:
4696:
4693:
4684:is the set of
4672:
4668:
4665:
4656:of an element
4626:
4622:
4618:
4615:
4595:
4592:
4589:
4586:
4564:
4560:
4556:
4551:
4547:
4543:
4523:
4519:
4515:
4512:
4509:
4504:
4501:
4498:
4493:
4490:
4469:
4465:
4460:
4456:
4452:
4447:
4443:
4427:
4426:
4415:
4407:
4402:
4397:
4394:
4391:
4388:
4384:
4378:
4375:
4372:
4369:
4366:
4363:
4360:
4357:
4354:
4351:
4348:
4345:
4342:
4339:
4336:
4333:
4330:
4327:
4324:
4321:
4318:
4315:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4289:
4284:
4279:
4275:
4271:
4268:
4265:
4259:
4253:
4250:
4247:
4244:
4239:
4236:
4233:
4230:
4224:
4221:
4218:
4213:
4207:
4204:
4202:
4199:
4198:
4195:
4192:
4190:
4187:
4186:
4184:
4170:is defined by
4158:
4137:
4133:
4129:
4126:
4123:
4118:
4115:
4112:
4091:
4088:
4041:
4040:
4025:
4011:
4008:
4003:
4000:
3997:
3966:
3965:
3942:
3916:
3905:
3831:
3828:
3824:
3823:
3816:
3815:
3808:
3807:
3802:
3799:
3746:
3743:
3732:
3731:
3716:
3715:
3700:
3699:
3694:
3691:
3671:
3668:
3649:
3646:
3645:
3644:
3641:
3629:
3626:
3623:
3618:
3614:
3610:
3596:
3585:
3582:
3579:
3574:
3570:
3566:
3554:
3553:
3552:
3551:
3540:
3536:
3528:
3524:
3520:
3515:
3511:
3503:
3499:
3495:
3490:
3486:
3479:
3475:
3472:
3467:
3464:
3449:
3448:
3447:
3435:
3427:
3422:
3418:
3414:
3409:
3404:
3400:
3394:
3389:
3385:
3380:
3366:
3365:
3347:
3343:
3322:
3317:
3313:
3309:
3304:
3300:
3296:
3284:
3281:
3280:
3279:
3278:
3277:
3265:
3262:
3259:
3240:
3226:
3225:
3215:
3214:
3213:
3212:
3201:
3184:
3183:
3173:
3172:
3171:
3170:
3159:
3156:
3153:
3150:
3147:
3144:
3124:
3121:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3083:
3082:
3081:is modeled by:
3070:
3050:
3047:
3044:
3041:
3038:
3022:
3021:
3020:
3019:
3016:
3010:
3009:
3006:Straight lines
3002:
3001:
3000:
2999:
2975:
2972:
2969:
2958:
2948:
2947:
2937:
2934:
2913:
2910:
2907:
2902:
2899:
2896:
2889:
2886:
2880:
2877:
2871:
2868:
2865:
2862:
2836:
2833:
2829:
2806:
2803:
2800:
2795:
2792:
2788:
2784:
2779:
2773:
2767:
2764:
2758:
2755:
2749:
2743:
2740:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2695:
2691:
2687:
2684:
2681:
2678:
2675:
2670:
2666:
2662:
2659:
2656:
2653:
2632:
2629:
2609:
2604:
2600:
2596:
2591:
2587:
2583:
2578:
2574:
2568:
2564:
2560:
2557:
2554:
2530:
2527:
2524:
2519:
2515:
2511:
2489:
2483:
2480:
2474:
2468:
2464:
2460:
2455:
2451:
2446:
2439:
2436:
2433:
2430:
2426:
2422:
2419:
2416:
2411:
2407:
2403:
2400:
2397:
2394:
2391:
2386:
2382:
2378:
2374:
2370:
2367:
2349:{\textstyle y}
2345:
2323:
2319:
2315:
2310:
2306:
2302:
2299:
2296:
2293:
2290:
2285:
2281:
2277:
2274:
2271:
2267:
2263:
2259:
2251:
2247:
2241:
2237:
2231:
2228:
2224:
2220:
2217:
2214:
2209:
2205:
2201:
2198:
2195:
2192:
2189:
2184:
2180:
2176:
2173:
2170:
2167:
2164:
2161:
2143:{\textstyle x}
2139:
2110:
2106:
2099:
2094:
2090:
2086:
2081:
2077:
2073:
2070:
2065:
2061:
2057:
2052:
2048:
2044:
2039:
2034:
2030:
2026:
2021:
2017:
2013:
2010:
2005:
2001:
1997:
1992:
1988:
1984:
1978:
1975:
1971:
1967:
1964:
1959:
1955:
1951:
1946:
1942:
1938:
1935:
1932:
1908:
1904:
1881:
1877:
1856:
1853:
1850:
1847:
1842:
1838:
1834:
1831:
1826:
1822:
1801:
1798:
1795:
1790:
1786:
1782:
1779:
1774:
1770:
1743:
1739:
1716:
1712:
1683:
1680:
1660:
1657:
1652:
1648:
1644:
1639:
1635:
1631:
1628:
1625:
1619:
1616:
1610:
1607:
1604:
1601:
1598:
1593:
1589:
1585:
1580:
1576:
1572:
1569:
1566:
1538:
1535:
1510:
1506:
1503:
1500:
1497:
1494:
1490:
1486:
1483:
1480:
1477:
1474:
1470:
1466:
1463:
1458:
1455:
1450:
1447:
1444:
1441:
1415:
1408:
1405:
1400:
1396:
1390:
1387:
1382:
1377:
1374:
1371:
1368:
1365:
1362:
1343:
1338:
1334:
1311:
1307:
1278:
1274:
1268:
1264:
1260:
1255:
1251:
1247:
1244:
1239:
1235:
1229:
1225:
1221:
1216:
1212:
1208:
1203:
1200:
1195:
1191:
1187:
1182:
1178:
1174:
1148:
1138:
1134:
1128:
1124:
1118:
1113:
1108:
1101:
1098:
1091:
1086:
1082:
1078:
1075:
1072:
1067:
1063:
1059:
1054:
1050:
1046:
1040:
1037:
1034:
1031:
1028:
1026:
1024:
1018:
1013:
1006:
1003:
996:
991:
987:
983:
978:
973:
969:
965:
960:
956:
952:
946:
943:
940:
937:
934:
932:
930:
920:
916:
910:
906:
900:
895:
890:
886:
882:
877:
873:
869:
863:
860:
857:
854:
851:
849:
847:
842:
838:
834:
829:
825:
821:
818:
815:
812:
811:
779:
775:
754:
749:
745:
741:
738:
733:
729:
725:
722:
717:
710:
707:
683:
680:
677:
657:
652:
648:
644:
639:
635:
631:
628:
623:
619:
598:
593:
589:
585:
580:
576:
572:
569:
564:
560:
519:
513:
493:
486:
476:
467:
443:
440:
416:straight lines
408:
407:
392:
388:
381:
377:
373:
370:
367:
364:
359:
355:
351:
348:
345:
339:
334:
330:
326:
323:
320:
297:
294:
291:
288:
285:
282:
279:
276:
273:
270:
267:
264:
248:
245:
220:
217:
214:
187:This model is
178:equiconsistent
162:Henri Poincaré
151:imaginary part
115:
111:
107:
104:
101:
98:
95:
92:
89:
86:
83:
80:
77:
74:
71:
68:
65:
62:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6383:
6372:
6369:
6367:
6364:
6362:
6359:
6358:
6356:
6345:
6344:0-387-98289-2
6341:
6337:
6333:
6330:
6327:
6326:0-86720-298-X
6323:
6319:
6315:
6312:
6309:
6308:3-540-43299-X
6305:
6301:
6297:
6294:
6291:
6290:0-387-90465-4
6287:
6283:
6279:
6275:
6272:
6271:archived copy
6268:
6267:
6262:
6259:
6256:
6252:
6248:
6245:
6244:
6239:
6238:
6229:
6226:
6214:
6210:
6203:
6200:
6187:
6181:
6178:
6171:
6170:
6166:
6161:
6158:
6156:
6153:
6151:
6148:
6146:
6143:
6141:
6138:
6136:
6133:
6131:
6128:
6126:
6123:
6121:
6118:
6116:
6113:
6112:
6107:
6105:
6103:
6099:
6095:
6079:
6076:
6073:
6061:
6058:The model in
6057:
6055:
6042:
6032:
6028:
6022:
6018:
6012:
6002:
5998:
5994:
5989:
5985:
5975:
5972:
5969:
5966:
5958:
5954:
5950:
5945:
5941:
5934:
5931:
5923:
5904:
5900:
5896:
5891:
5887:
5883:
5878:
5874:
5867:
5862:
5858:
5832:
5828:
5824:
5819:
5815:
5811:
5806:
5802:
5795:
5790:
5786:
5777:
5772:
5770:
5766:
5762:
5758:
5754:
5750:
5745:
5727:
5723:
5716:
5708:
5705:
5699:
5694:
5686:
5683:
5677:
5672:
5664:
5661:
5652:
5647:
5639:
5636:
5625:
5608:
5605:
5602:
5599:
5593:
5590:
5587:
5584:
5581:
5567:
5559:
5557:
5555:
5551:
5547:
5543:
5542:geodesic flow
5524:
5518:
5515:
5510:
5506:
5502:
5499:
5494:
5491:
5486:
5482:
5478:
5475:
5469:
5466:
5463:
5458:
5450:
5446:
5442:
5439:
5435:
5429:
5422:
5415:
5411:
5407:
5403:
5396:
5389:
5383:
5378:
5371:
5366:
5360:
5355:
5349:
5343:
5336:
5335:
5334:
5332:
5328:
5309:
5304:
5300:
5296:
5293:
5290:
5287:
5282:
5274:
5270:
5266:
5263:
5259:
5253:
5246:
5239:
5235:
5231:
5227:
5220:
5215:
5209:
5203:
5196:
5195:
5194:
5192:
5187:
5183:
5175:
5173:
5160:
5149:
5146:
5126:
5125:modular group
5122:
5117:
5095:
5092:
5068:
5064:
5045:
5028:
5016:
5013:
4994:
4981:
4977:
4953:
4950:
4926:
4923:is mapped to
4905:
4902:
4879:
4875:
4866:
4863:
4859:
4854:
4848:
4845:
4842:
4837:
4834:
4831:
4828:
4821:
4818:
4815:
4810:
4807:
4804:
4798:
4793:
4785:
4779:
4759:
4758:
4757:
4756:
4752:
4748:
4744:
4740:
4716:
4713:
4694:
4691:
4666:
4663:
4655:
4651:
4646:
4644:
4640:
4624:
4616:
4613:
4593:
4590:
4587:
4584:
4562:
4558:
4554:
4549:
4545:
4541:
4513:
4510:
4491:
4488:
4463:
4458:
4454:
4450:
4445:
4441:
4432:
4413:
4405:
4395:
4392:
4389:
4386:
4373:
4361:
4358:
4355:
4352:
4349:
4343:
4340:
4331:
4319:
4316:
4313:
4310:
4307:
4301:
4298:
4295:
4292:
4287:
4277:
4269:
4266:
4257:
4251:
4248:
4245:
4242:
4237:
4234:
4231:
4228:
4222:
4219:
4216:
4211:
4205:
4200:
4193:
4188:
4182:
4173:
4172:
4171:
4127:
4124:
4101:
4097:
4089:
4087:
4085:
4081:
4077:
4073:
4069:
4065:
4064:modular forms
4061:
4057:
4053:
4052:modular group
4048:
4046:
4038:
4034:
4030:
4026:
4006:
4001:
3995:
3987:
3983:
3979:
3975:
3971:
3970:
3969:
3963:
3959:
3955:
3951:
3947:
3943:
3940:
3936:
3932:
3928:
3926:
3921:
3917:
3914:
3910:
3906:
3903:
3899:
3895:
3893:
3888:
3884:
3883:
3882:
3880:
3875:
3873:
3869:
3865:
3861:
3857:
3853:
3849:
3841:
3836:
3829:
3827:
3821:
3820:
3819:
3813:
3812:
3811:
3805:
3804:
3800:
3798:
3796:
3792:
3787:
3785:
3781:
3776:
3774:
3769:
3767:
3763:
3758:
3756:
3752:
3744:
3742:
3739:
3737:
3729:
3728:
3727:
3723:
3721:
3713:
3712:
3711:
3709:
3705:
3697:
3696:
3692:
3690:
3688:
3683:
3681:
3677:
3669:
3667:
3665:
3661:
3655:
3647:
3642:
3624:
3621:
3616:
3612:
3601:
3597:
3580:
3577:
3572:
3568:
3556:
3555:
3538:
3534:
3526:
3522:
3518:
3513:
3509:
3501:
3497:
3493:
3488:
3484:
3477:
3473:
3470:
3465:
3462:
3453:
3452:
3450:
3433:
3425:
3420:
3416:
3412:
3407:
3402:
3398:
3392:
3387:
3383:
3378:
3370:
3369:
3368:
3367:
3363:
3362:
3361:
3360:represents:
3345:
3341:
3315:
3311:
3307:
3302:
3298:
3282:
3260:
3257:
3249:
3245:
3241:
3238:
3234:
3230:
3229:
3228:
3227:
3223:
3222:
3217:
3216:
3210:
3206:
3202:
3200:
3196:
3192:
3188:
3187:
3186:
3185:
3181:
3180:
3175:
3174:
3154:
3148:
3145:
3142:
3116:
3110:
3107:
3104:
3101:
3098:
3087:
3086:
3085:
3084:
3068:
3045:
3042:
3039:
3028:
3024:
3023:
3017:
3014:
3013:
3012:
3011:
3007:
3004:
3003:
2997:
2993:
2989:
2986:which is the
2970:
2967:
2959:
2956:
2952:
2951:
2950:
2949:
2945:
2944:
2940:
2939:
2935:
2933:
2931:
2911:
2908:
2905:
2900:
2897:
2894:
2887:
2884:
2878:
2875:
2869:
2866:
2863:
2860:
2852:
2834:
2831:
2827:
2817:
2804:
2801:
2798:
2793:
2790:
2786:
2782:
2771:
2765:
2762:
2756:
2753:
2741:
2738:
2734:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2693:
2689:
2682:
2676:
2673:
2668:
2664:
2654:
2651:
2643:
2630:
2627:
2607:
2602:
2598:
2594:
2589:
2585:
2581:
2576:
2566:
2562:
2558:
2555:
2544:
2525:
2522:
2517:
2513:
2500:
2487:
2481:
2478:
2466:
2462:
2458:
2453:
2449:
2437:
2434:
2431:
2428:
2424:
2417:
2414:
2409:
2405:
2398:
2392:
2389:
2384:
2380:
2372:
2368:
2365:
2357:
2343:
2334:
2321:
2317:
2308:
2304:
2297:
2294:
2291:
2283:
2279:
2272:
2269:
2265:
2261:
2257:
2249:
2245:
2239:
2235:
2229:
2226:
2222:
2218:
2207:
2203:
2199:
2196:
2190:
2182:
2178:
2174:
2171:
2162:
2159:
2151:
2137:
2128:
2126:
2121:
2108:
2104:
2092:
2088:
2084:
2079:
2075:
2063:
2059:
2055:
2050:
2046:
2032:
2028:
2024:
2019:
2015:
2003:
1999:
1995:
1990:
1986:
1976:
1973:
1969:
1965:
1957:
1953:
1949:
1944:
1940:
1933:
1930:
1922:
1906:
1902:
1879:
1875:
1854:
1848:
1845:
1840:
1836:
1829:
1824:
1820:
1796:
1793:
1788:
1784:
1777:
1772:
1768:
1759:
1741:
1737:
1714:
1710:
1700:
1697:
1681:
1678:
1658:
1650:
1646:
1642:
1637:
1633:
1626:
1623:
1617:
1614:
1608:
1605:
1602:
1599:
1591:
1587:
1583:
1578:
1574:
1567:
1564:
1556:
1552:
1536:
1533:
1525:
1508:
1501:
1498:
1495:
1488:
1481:
1478:
1475:
1468:
1464:
1461:
1456:
1453:
1448:
1445:
1442:
1439:
1431:
1406:
1403:
1398:
1394:
1388:
1385:
1375:
1372:
1369:
1366:
1363:
1360:
1341:
1336:
1332:
1309:
1305:
1296:
1276:
1266:
1262:
1258:
1253:
1249:
1242:
1237:
1227:
1223:
1219:
1214:
1210:
1201:
1193:
1189:
1185:
1180:
1176:
1163:
1146:
1136:
1132:
1126:
1122:
1116:
1106:
1096:
1089:
1084:
1080:
1073:
1065:
1061:
1057:
1052:
1048:
1038:
1035:
1032:
1029:
1027:
1011:
1001:
994:
989:
985:
971:
967:
963:
958:
954:
944:
941:
938:
935:
933:
918:
914:
908:
904:
898:
888:
884:
880:
875:
871:
861:
858:
855:
852:
850:
840:
836:
832:
827:
823:
816:
813:
801:
799:
795:
777:
773:
747:
743:
739:
736:
731:
727:
720:
715:
705:
681:
678:
675:
650:
646:
642:
637:
633:
626:
621:
617:
591:
587:
583:
578:
574:
567:
562:
558:
545:
540:
532:
518:
512:
508:
504:
492:
485:
475:
466:
448:
441:
439:
437:
433:
429:
425:
424:perpendicular
421:
417:
413:
390:
386:
379:
371:
368:
362:
357:
349:
346:
337:
332:
324:
321:
311:
310:
309:
295:
289:
286:
283:
280:
274:
271:
268:
254:
246:
244:
242:
238:
234:
218:
215:
212:
203:
201:
197:
192:
190:
185:
183:
179:
175:
171:
167:
163:
158:
156:
152:
148:
147:complex plane
143:
141:
137:
133:
129:
105:
102:
99:
96:
93:
90:
87:
84:
81:
75:
72:
69:
60:
53:
49:
45:
41:
32:
19:
6335:
6317:
6316:Saul Stahl,
6310:
6299:
6281:
6264:
6250:
6228:
6216:. Retrieved
6212:
6202:
6192:19 September
6190:. Retrieved
6180:
6150:Pseudosphere
6101:
6097:
6092:dimensional
6065:
6059:
5924:
5775:
5773:
5768:
5764:
5760:
5752:
5748:
5746:
5626:
5624:is given by
5563:
5539:
5330:
5326:
5324:
5193:is given by
5190:
5188:
5185:
5118:
4975:
4924:
4894:
4750:
4746:
4742:
4738:
4737:which leave
4653:
4647:
4642:
4638:
4428:
4096:group action
4093:
4083:
4079:
4075:
4071:
4055:
4049:
4042:
4036:
4032:
4028:
3985:
3981:
3977:
3967:
3961:
3957:
3953:
3949:
3945:
3938:
3934:
3930:
3924:
3912:
3908:
3901:
3897:
3891:
3876:
3868:transitively
3863:
3859:
3851:
3845:
3842:of the model
3825:
3817:
3809:
3794:
3790:
3788:
3783:
3782:with center
3779:
3777:
3772:
3770:
3765:
3761:
3759:
3754:
3750:
3748:
3740:
3735:
3733:
3724:
3719:
3717:
3707:
3703:
3701:
3686:
3684:
3679:
3675:
3673:
3657:
3286:
3243:
3232:
3219:
3204:
3195:ideal points
3190:
3177:
3026:
3005:
2995:
2954:
2943:Ideal points
2941:
2818:
2644:
2501:
2358:
2356:coordinate:
2335:
2152:
2150:coordinate:
2129:
2122:
1923:
1757:
1701:
1164:
802:
797:
793:
549:
516:
510:
506:
502:
490:
483:
473:
464:
435:
431:
427:
415:
411:
409:
250:
240:
236:
231:dimensional
204:
198:provides an
193:
186:
159:
154:
144:
51:
43:
37:
6296:Jürgen Jost
6120:Anosov flow
5554:Anosov flow
5546:line bundle
5182:Anosov flow
4741:unchanged:
3797:intersect.
3451:and radius
3333:and radius
3248:ideal point
3237:ideal point
3135:and radius
3061:and radius
2988:ideal point
2125:Cross-ratio
792:across the
170:Klein model
6355:Categories
6167:References
6062:dimensions
4980:isomorphic
4650:stabilizer
4534:such that
4433:: for any
4431:transitive
3974:isometries
3879:Lie groups
3760:Let point
3652:See also:
3600:orthogonal
3179:hypercycle
2992:orthogonal
2957:-axis, and
2502:One point
149:where the
6278:Irwin Kra
6008:‖
5995:−
5982:‖
5976:
5935:
5910:⟩
5871:⟨
5838:⟩
5799:⟨
5757:geodesics
5600:∣
5597:⟩
5579:⟨
5464:⋅
5440:−
5344:γ
5288:⋅
5264:−
5204:γ
5176:Geodesics
4906:∈
4867:∈
4864:θ
4849:θ
4846:
4838:θ
4835:
4829:−
4822:θ
4819:
4811:θ
4808:
4695:∈
4667:∈
4617:∈
4492:∈
4464:∈
4368:ℑ
4356:−
4326:ℜ
4217:⋅
4010:¯
4002:−
3999:→
3948:) = SL(2,
3519:−
3474:
3413:−
3321:⟩
3295:⟨
3264:∞
3221:horocycle
3149:
3111:
2974:∞
2909:−
2888:
2864:
2832:−
2802:ϕ
2799:
2791:−
2772:ϕ
2757:
2729:⟩
2726:ϕ
2723:
2711:ϕ
2708:
2699:±
2686:⟨
2680:⟩
2661:⟨
2655:
2628:ϕ
2559:−
2529:⟩
2510:⟨
2459:−
2438:
2421:⟩
2402:⟨
2396:⟩
2377:⟨
2369:
2298:
2292:−
2273:
2230:
2213:⟩
2194:⟨
2188:⟩
2169:⟨
2163:
2098:‖
2085:−
2072:‖
2069:‖
2056:−
2043:‖
2038:‖
2025:−
2012:‖
2009:‖
1996:−
1983:‖
1977:
1934:
1852:⟩
1833:⟨
1800:⟩
1781:⟨
1627:
1609:
1568:
1499:−
1465:
1443:
1376:
1364:
1259:−
1220:−
1199:‖
1186:−
1173:‖
1112:‖
1100:~
1090:−
1077:‖
1071:‖
1058:−
1045:‖
1039:
1017:‖
1005:~
995:−
982:‖
977:‖
964:−
951:‖
945:
894:‖
881:−
868:‖
862:
817:
753:⟩
740:−
724:⟨
709:~
656:⟩
630:⟨
597:⟩
571:⟨
420:geodesics
281:∣
278:⟩
266:⟨
189:conformal
106:∈
82:∣
79:⟩
67:⟨
6108:See also
5776:distance
5771:-plane.
4606:for all
3956:}=PGL(2,
3768:- axis.
798:distance
200:isometry
172:and the
6334:(1998)
6240:Sources
6218:25 June
5123:by the
4753:is the
4098:of the
4060:lattice
3929:= SL(2,
3757:-axis.
3689:-axis.
3682:-axis.
2994:to the
2928:is the
2541:at the
1526:. This
1522:is the
1428:is the
1293:is the
496:) = log
438:-axis.
426:to the
46:is the
6342:
6324:
6306:
6288:
5973:arsinh
5747:where
5566:metric
5063:bundle
3923:PSL(2,
3850:PGL(2,
3027:circle
2861:artanh
2853:, and
2819:where
2742:artanh
2435:arsinh
1682:artanh
1537:arsinh
1440:artanh
1432:, and
1361:arsinh
1165:where
942:artanh
859:arsinh
410:where
253:metric
247:Metric
130:, the
128:metric
42:, the
6172:Notes
5769:z = 0
5765:z = 0
5761:z = 0
4637:then
4054:SL(2,
3890:SL(2,
3209:angle
3199:angle
1565:chord
1551:chord
482:dist(
180:with
153:(the
136:model
6340:ISBN
6322:ISBN
6304:ISBN
6286:ISBN
6220:2015
6194:2015
5932:dist
5850:and
5774:The
5606:>
5564:The
4648:The
4094:The
4066:and
3952:)/{±
3933:)/{±
3918:The
3885:The
3846:The
3793:and
3704:line
3146:sinh
3108:cosh
2652:dist
2543:apex
2366:dist
2160:dist
2123:Cf.
1931:dist
1921:is:
1812:and
1729:and
1624:dist
1606:sinh
1324:and
814:dist
694:and
679:>
609:and
471:and
308:is:
287:>
251:The
194:The
88:>
4978:is
4843:cos
4832:sin
4816:sin
4805:cos
4652:or
4148:on
3976:of
3250:at
2754:tan
2720:cos
2705:sin
1894:to
550:If
38:In
6357::
6298:,
6280:,
6253:,
6249:,
6211:.
5556:.
5116:.
4745:=
4743:gz
4645:.
4641:=
4047:.
4039:).
3904:).
3874:.
3775:.
3471:ln
3218:A
3176:A
3025:A
2932:.
2885:ln
2828:gd
2787:gd
2295:ln
2270:ln
2227:ln
2127:.
1974:ln
1557:,
1462:ln
1373:ln
1036:ln
509:)/
505:+
489:,
184:.
142:.
6328:.
6313:.
6292:.
6222:.
6196:.
6102:n
6098:x
6080:1
6077:+
6074:n
6060:n
6043:.
6033:2
6029:z
6023:1
6019:z
6013:2
6003:1
5999:p
5990:2
5986:p
5970:2
5967:=
5964:)
5959:2
5955:p
5951:,
5946:1
5942:p
5938:(
5905:2
5901:z
5897:,
5892:2
5888:y
5884:,
5879:2
5875:x
5868:=
5863:2
5859:p
5833:1
5829:z
5825:,
5820:1
5816:y
5812:,
5807:1
5803:x
5796:=
5791:1
5787:p
5749:s
5728:2
5724:z
5717:2
5713:)
5709:z
5706:d
5703:(
5700:+
5695:2
5691:)
5687:y
5684:d
5681:(
5678:+
5673:2
5669:)
5665:x
5662:d
5659:(
5653:=
5648:2
5644:)
5640:s
5637:d
5634:(
5612:}
5609:0
5603:z
5594:z
5591:,
5588:y
5585:,
5582:x
5576:{
5525:.
5519:d
5516:+
5511:t
5507:e
5503:i
5500:c
5495:b
5492:+
5487:t
5483:e
5479:i
5476:a
5470:=
5467:i
5459:)
5451:2
5447:/
5443:t
5436:e
5430:0
5423:0
5416:2
5412:/
5408:t
5404:e
5397:(
5390:)
5384:d
5379:c
5372:b
5367:a
5361:(
5356:=
5353:)
5350:t
5347:(
5331:R
5327:R
5310:.
5305:t
5301:e
5297:i
5294:=
5291:i
5283:)
5275:2
5271:/
5267:t
5260:e
5254:0
5247:0
5240:2
5236:/
5232:t
5228:e
5221:(
5216:=
5213:)
5210:t
5207:(
5191:i
5161:.
5158:)
5154:Z
5150:,
5147:2
5144:(
5139:L
5136:S
5104:)
5100:R
5096:,
5093:2
5090:(
5085:L
5082:S
5079:P
5049:)
5046:2
5043:(
5038:O
5035:S
5029:/
5025:)
5021:R
5017:,
5014:2
5011:(
5006:L
5003:S
5000:P
4995:=
4991:H
4976:z
4962:)
4958:R
4954:,
4951:2
4948:(
4943:L
4940:S
4937:P
4925:i
4910:H
4903:z
4880:.
4876:}
4871:R
4860:|
4855:)
4799:(
4794:{
4786:=
4783:)
4780:2
4777:(
4772:O
4769:S
4751:i
4747:z
4739:z
4725:)
4721:R
4717:,
4714:2
4711:(
4706:L
4703:S
4700:P
4692:g
4671:H
4664:z
4643:e
4639:g
4625:,
4621:H
4614:z
4594:z
4591:=
4588:z
4585:g
4563:2
4559:z
4555:=
4550:1
4546:z
4542:g
4522:)
4518:R
4514:,
4511:2
4508:(
4503:L
4500:S
4497:P
4489:g
4468:H
4459:2
4455:z
4451:,
4446:1
4442:z
4414:.
4406:2
4401:|
4396:d
4393:+
4390:z
4387:c
4383:|
4377:)
4374:z
4371:(
4365:)
4362:c
4359:b
4353:d
4350:a
4347:(
4344:i
4341:+
4338:)
4335:)
4332:z
4329:(
4323:)
4320:c
4317:b
4314:+
4311:d
4308:a
4305:(
4302:+
4299:d
4296:b
4293:+
4288:2
4283:|
4278:z
4274:|
4270:c
4267:a
4264:(
4258:=
4252:d
4249:+
4246:z
4243:c
4238:b
4235:+
4232:z
4229:a
4223:=
4220:z
4212:)
4206:d
4201:c
4194:b
4189:a
4183:(
4157:H
4136:)
4132:R
4128:,
4125:2
4122:(
4117:L
4114:S
4111:P
4084:Z
4080:R
4076:Z
4072:Z
4056:Z
4037:R
4033:H
4029:H
4024:.
4007:z
3996:z
3986:R
3982:H
3978:H
3962:R
3958:R
3954:I
3950:R
3946:R
3939:R
3935:I
3931:R
3927:)
3925:R
3913:R
3909:R
3902:R
3898:R
3894:)
3892:R
3864:R
3860:H
3852:C
3795:p
3791:h
3784:q
3780:h
3773:q
3766:x
3762:q
3755:x
3751:p
3736:x
3720:x
3708:x
3687:x
3680:x
3676:x
3628:)
3625:0
3622:=
3617:e
3613:y
3609:(
3584:)
3581:0
3578:,
3573:e
3569:x
3565:(
3539:.
3535:)
3527:e
3523:r
3514:e
3510:y
3502:e
3498:r
3494:+
3489:e
3485:y
3478:(
3466:2
3463:1
3434:)
3426:2
3421:e
3417:r
3408:2
3403:e
3399:y
3393:,
3388:e
3384:x
3379:(
3346:e
3342:r
3316:e
3312:y
3308:,
3303:e
3299:x
3276:.
3261:=
3258:y
3244:x
3233:x
3211:.
3205:x
3191:x
3158:)
3155:r
3152:(
3143:y
3123:)
3120:)
3117:r
3114:(
3105:y
3102:,
3099:x
3096:(
3069:r
3049:)
3046:y
3043:,
3040:x
3037:(
2996:x
2971:=
2968:y
2955:x
2912:x
2906:1
2901:x
2898:+
2895:1
2879:2
2876:1
2870:=
2867:x
2835:1
2805:,
2794:1
2783:=
2778:)
2766:2
2763:1
2748:(
2739:2
2735:=
2732:)
2717:r
2714:,
2702:r
2694:1
2690:x
2683:,
2677:r
2674:,
2669:1
2665:x
2658:(
2631:.
2608:,
2603:2
2599:r
2595:=
2590:2
2586:y
2582:+
2577:2
2573:)
2567:1
2563:x
2556:x
2553:(
2526:r
2523:,
2518:1
2514:x
2488:.
2482:y
2479:2
2473:|
2467:1
2463:x
2454:2
2450:x
2445:|
2432:2
2429:=
2425:)
2418:y
2415:,
2410:2
2406:x
2399:,
2393:y
2390:,
2385:1
2381:x
2373:(
2344:y
2322:.
2318:|
2314:)
2309:1
2305:y
2301:(
2289:)
2284:2
2280:y
2276:(
2266:|
2262:=
2258:|
2250:1
2246:y
2240:2
2236:y
2223:|
2219:=
2216:)
2208:2
2204:y
2200:,
2197:x
2191:,
2183:1
2179:y
2175:,
2172:x
2166:(
2138:x
2109:.
2105:|
2093:3
2089:p
2080:2
2076:p
2064:0
2060:p
2051:1
2047:p
2033:3
2029:p
2020:1
2016:p
2004:0
2000:p
1991:2
1987:p
1970:|
1966:=
1963:)
1958:2
1954:p
1950:,
1945:1
1941:p
1937:(
1907:2
1903:p
1880:1
1876:p
1855:,
1849:0
1846:,
1841:3
1837:x
1830:=
1825:3
1821:p
1797:0
1794:,
1789:0
1785:x
1778:=
1773:0
1769:p
1758:x
1742:2
1738:p
1715:1
1711:p
1679:2
1659:,
1656:)
1651:2
1647:p
1643:,
1638:1
1634:p
1630:(
1618:2
1615:1
1603:2
1600:=
1597:)
1592:2
1588:p
1584:,
1579:1
1575:p
1571:(
1534:2
1509:)
1505:)
1502:x
1496:1
1493:(
1489:/
1485:)
1482:x
1479:+
1476:1
1473:(
1469:(
1457:2
1454:1
1449:=
1446:x
1414:)
1407:1
1404:+
1399:2
1395:x
1389:+
1386:x
1381:(
1370:=
1367:x
1342:,
1337:2
1333:p
1310:1
1306:p
1277:2
1273:)
1267:1
1263:y
1254:2
1250:y
1246:(
1243:+
1238:2
1234:)
1228:1
1224:x
1215:2
1211:x
1207:(
1202:=
1194:1
1190:p
1181:2
1177:p
1147:,
1137:2
1133:y
1127:1
1123:y
1117:2
1107:1
1097:p
1085:2
1081:p
1074:+
1066:1
1062:p
1053:2
1049:p
1033:2
1030:=
1012:1
1002:p
990:2
986:p
972:1
968:p
959:2
955:p
939:2
936:=
919:2
915:y
909:1
905:y
899:2
889:1
885:p
876:2
872:p
856:2
853:=
846:)
841:2
837:p
833:,
828:1
824:p
820:(
794:x
778:1
774:p
748:1
744:y
737:,
732:1
728:x
721:=
716:1
706:p
682:0
676:y
651:2
647:y
643:,
638:2
634:x
627:=
622:2
618:p
592:1
588:y
584:,
579:1
575:x
568:=
563:1
559:p
524:)
520:2
517:h
514:1
511:h
507:d
503:s
501:(
498:(
494:2
491:p
487:1
484:p
477:2
474:h
468:1
465:h
460:d
456:s
452:x
436:x
432:x
428:x
412:s
391:2
387:y
380:2
376:)
372:y
369:d
366:(
363:+
358:2
354:)
350:x
347:d
344:(
338:=
333:2
329:)
325:s
322:d
319:(
296:,
293:}
290:0
284:y
275:y
272:,
269:x
263:{
241:n
237:x
219:1
216:+
213:n
155:y
114:}
110:R
103:y
100:,
97:x
94:;
91:0
85:y
76:y
73:,
70:x
64:{
61:=
52:H
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
