6766:
31:
277:, so that the linear dilatation of any body is proportional to its absolute temperature. Finally, I shall assume that a body transported from one point to another of different temperature is instantaneously in thermal equilibrium with its new environment. ... If they construct a geometry, it will not be like ours, which is the study of the movements of our invariable solids; it will be the study of the changes of position which they will have thus distinguished, and will be 'non-Euclidean displacements,' and
39:
1025:
6765:
5332:(the set of all points in a plane that are at a given distance from a given point, its center) is a circle completely inside the disk not touching or intersecting its boundary. The hyperbolic center of the circle in the model does not in general correspond to the Euclidean center of the circle, but they are on the same radius of the Poincare disk. (The Euclidean center is always closer to the center of the disk than the hyperbolic center.)
303:
6870:
1403:
3955:
5363:
733:
1644:
7261:
2317:
3541:
1020:{\displaystyle {\begin{aligned}d(u,v)&=\operatorname {arcosh} (1+\delta (u,v))\\&=2\operatorname {arsinh} {\sqrt {\frac {\delta (u,v)}{2}}}\\\,&=2\ln {\frac {\lVert u-v\rVert +{\sqrt {\lVert u\rVert ^{2}\lVert v\rVert ^{2}-2u\cdot v+1}}}{\sqrt {(1-\lVert u\rVert ^{2})(1-\lVert v\rVert ^{2})}}}.\end{aligned}}}
5400:), is a circle inside the disk that is tangent to the boundary circle of the disk. The point where it touches the boundary circle is not part of the horocycle. It is an ideal point and is the hyperbolic center of the horocycle. It is also the point to which all the perpendicular geodesics converge.
1030:
Such a distance function is defined for any two vectors of norm less than one, and makes the set of such vectors into a metric space which is a model of hyperbolic space of constant curvature −1. The model has the conformal property that the angle between two intersecting curves in hyperbolic space
5403:
In the
Poincaré disk model, the Euclidean points representing opposite "ends" of a horocycle converge to its center on the boundary circle, but in the hyperbolic plane every point of a horocycle is infinitely far from its center, and opposite ends of the horocycle are not connected. (Euclidean
1428:
1361:
2120:
4203:
7024:
Milnor, John W. "Hyperbolic geometry: the first 150 years." Bulletin of the
American Mathematical Society 6, no. 1 (1982): 9-24. B. Riemann, "Ueber die Hypothesen welche der Geometrie zu Grunde liegen", Abh. K. G. Wiss. Göttingen 13 (from his Inaugural Address of 1854).
2131:
3950:{\displaystyle {\begin{aligned}x^{2}+y^{2}&{}+{\frac {u_{2}(v_{1}^{2}+v_{2}^{2}+1)-v_{2}(u_{1}^{2}+u_{2}^{2}+1)}{u_{1}v_{2}-u_{2}v_{1}}}x\\&{}+{\frac {v_{1}(u_{1}^{2}+u_{2}^{2}+1)-u_{1}(v_{1}^{2}+v_{2}^{2}+1)}{u_{1}v_{2}-u_{2}v_{1}}}y+1=0\,.\end{aligned}}}
696:
6076:
1769:
5765:
5419:
533:
6512:
6343:
189:"Suppose, for example, a world enclosed in a large sphere and subject to the following laws: The temperature is not uniform; it is greatest at their centre, and gradually decreases as we move towards the circumference of the sphere, where it is
6745:
6843:
5619:
1870:
5346:(the set of all points in a plane that are on one side and at a given distance from a given line, its axis) is a Euclidean circle arc or chord of the boundary circle that intersects the boundary circle at a positive but non-
1639:{\displaystyle ds^{2}=4{\frac {\sum _{i}dx_{i}^{2}}{\left(1-\sum _{i}x_{i}^{2}\right)^{2}}}={\frac {4\,\lVert d\mathbf {x} \rVert {\vphantom {l}}^{2}}{{\bigl (}1-\lVert \mathbf {x} \rVert {\vphantom {l}}^{2}{\bigr )}^{2}}}}
5124:
1108:
1227:
6849:
6848:
6845:
6844:
6850:
4984:
4885:
4786:
4610:
1981:
3974:
3546:
285:
Poincaré's disk was an important piece of evidence for the hypothesis that the choice of spatial geometry is conventional rather than factual, especially in the influential philosophical discussions of
4684:
4418:
2312:{\displaystyle \Omega =d\omega +\omega \wedge \omega =d\omega +0={\frac {-4\,dx\wedge dy}{{\bigl (}1-|\mathbf {x} |{\vphantom {l}}^{2}{\bigr )}^{2}}}{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}.}
5963:
179:. Henri Poincaré employed it in his 1882 treatment of hyperbolic, parabolic and elliptic functions, but it became widely known following Poincaré's presentation in his 1905 philosophical treatise,
6847:
738:
6122:
is a complex number of norm less than one representing a point of the
Poincaré disk model, then the corresponding point of the half-plane model is given by the inverse of the Cayley transform:
2399:
1953:
417:
5936:
6619:
6615:
4305:
1184:
3506:
1128:
570:
6175:
5240:
5182:
1667:
725:
5635:
4472:
6533:. If we have a point on the upper sheet of the hyperboloid of the hyperboloid model, thereby defining a point in the hyperboloid model, we may project it onto the hyperplane
185:. There he describes a world, now known as the Poincaré disk, in which space was Euclidean, but which appeared to its inhabitants to satisfy the axioms of hyperbolic geometry:
7027:
Eugenio
Beltrami. "Teoria fondamentale degli spazii di curvatura costante", Annali di mat. ser. II 2, 232-255 (Op. Mat. 1, 406-429; Ann. École Norm. Sup. 6 (1869), 345-375).
5814:
4511:
1222:
6359:
6190:
5858:
271:
1392:
3248:
3178:
2803:
2733:
2660:
2610:
4348:
3104:
2536:
2485:
1973:
1908:
1041:
5266:: circles, horocycles, hypercycles, and geodesics (or "hyperbolic lines"). In the Poincaré disk model, all of these are represented by straight lines or circles.
3077:
3029:
6120:
5958:
5881:
3416:
3393:
3373:
3353:
3333:
3310:
3290:
3270:
3220:
3200:
3150:
3128:
3051:
3002:
2971:
2948:
2928:
2908:
2888:
2865:
2845:
2825:
2775:
2755:
2702:
2682:
2632:
2578:
2556:
2509:
2452:
2432:
231:
211:
3431:
is to find a line through two given points. In the
Poincaré disk model, lines in the plane are defined by portions of circles having equations of the form
1780:
5502:
4237:, by means of a formula. Since the ideal points are the same in the Klein model and the Poincaré disk model, the formulas are identical for each model.
5883:
is a vector of norm less than one representing a point of the
Poincaré disk model, then the corresponding point of the Klein disk model is given by:
537:
The vertical bars indicate
Euclidean length of the line segment connecting the points between them in the model (not along the circle arc); ln is the
6846:
5003:
2455:
7145:
7118:
7281:
6972:
5960:
of norm less than one representing a point of the
Beltrami–Klein model, the corresponding point of the Poincaré disk model is given by:
4893:
4794:
4695:
4519:
2115:{\displaystyle \omega ={\frac {2(y\,dx-x\,dy)}{1-|\mathbf {x} |{\vphantom {l}}^{2}}}{\begin{pmatrix}0&1\\-1&0\end{pmatrix}},}
7009:
4198:{\displaystyle x^{2}+y^{2}+{\frac {2(u_{2}-v_{2})}{u_{1}v_{2}-u_{2}v_{1}}}x+{\frac {2(v_{1}-u_{1})}{u_{1}v_{2}-u_{2}v_{1}}}y+1=0\,.}
1356:{\displaystyle \ln \left({\frac {1+r}{1-r}}\cdot {\frac {1-r'}{1+r'}}\right)=2(\operatorname {artanh} r-\operatorname {artanh} r').}
5442:(also known as the Beltrami–Klein model) and the Poincaré disk model are both models that project the whole hyperbolic plane in a
7265:
6911:
5886:
5342:
6565:
6537: = 0 by intersecting it with a line drawn through . The result is the corresponding point of the Poincaré disk model.
4621:
140:, because his rediscovery of this representation fourteen years later became better known than the original work of Beltrami.
4356:
3535:
in the disk which do not lie on a diameter, we can solve for the circle of this form passing through both points, and obtain
43:
6125:
1411:
7286:
6947:
6092:
2328:
121:
1917:
1131:
7291:
6897:
explored the concept of representing infinity on a two-dimensional plane. Discussions with
Canadian mathematician
4990:
93:
The group of orientation preserving isometries of the disk model is given by the projective special unitary group
274:
6942:
1034:
Specializing to the case where one of the points is the origin and the
Euclidean distance between the points is
691:{\displaystyle \delta (u,v)=2{\frac {\lVert u-v\rVert ^{2}}{(1-\lVert u\rVert ^{2})(1-\lVert v\rVert ^{2})}}\,,}
6800:
6750:
6071:{\displaystyle u={\frac {s}{1+{\sqrt {1-s\cdot s}}}}={\frac {\left(1-{\sqrt {1-s\cdot s}}\right)s}{s\cdot s}}.}
5459:
4262:
281:. So that beings like ourselves, educated in such a world, will not have the same geometry as ours." (pp.65-68)
3437:
1764:{\displaystyle e_{i}={\frac {1}{2}}{\Bigl (}1-|\mathbf {x} |^{2}{\Bigr )}{\frac {\partial }{\partial x^{i}}},}
330:
30:
6928:, a roguelike game, uses the hyperbolic plane for its world geometry, and also uses the Poincaré disk model.
5760:{\textstyle \left({\frac {x}{1+{\sqrt {1-x^{2}-y^{2}}}}}\ ,\ \ {\frac {y}{1+{\sqrt {1-x^{2}-y^{2}}}}}\right)}
1113:
7216:
5455:
3511:
which is the general form of a circle orthogonal to the unit circle, or else by diameters. Given two points
5190:
5132:
6530:
5404:
intuition can be misleading because the scale of the model increases to infinity at the boundary circle.)
1887:
704:
181:
3968:
are points on the boundary of the disk not lying at the endpoints of a diameter, the above simplifies to
6856:
Animation of a partial {7,3} hyperbolic tiling of the hyperboloid rotated into the Poincare perspective.
6541:
528:{\displaystyle d(p,q)=\ln {\frac {\left|aq\right|\,\left|pb\right|}{\left|ap\right|\,\left|qb\right|}}.}
38:
6836:. The red geodesic in the Poincaré disk model projects to the brown geodesic on the green hyperboloid.
4429:
233:
the distance of the point considered from the centre, the absolute temperature will be proportional to
6507:{\textstyle \left({\frac {2x}{x^{2}+(1+y)^{2}}}\ ,\ {\frac {x^{2}+y^{2}-1}{x^{2}+(1+y)^{2}}}\right)\,}
6338:{\textstyle \left({\frac {2x}{x^{2}+(1-y)^{2}}}\ ,\ {\frac {1-x^{2}-y^{2}}{x^{2}+(1-y)^{2}}}\right)\,}
7107:
Carus, A. W.; Friedman, Michael; Kienzler, Wolfgang; Richardson, Alan; Schlotter, Sven (2019-06-25).
4993:
and the fact that these are unit vectors we may rewrite the above expressions purely in terms of the
1141:
4256:, then we are merely finding the angle between two unit vectors, and the formula for the angle θ is
6937:
6740:{\displaystyle (t,x_{i})={\frac {\left(1+\sum {y_{i}^{2}},\,2y_{i}\right)}{1-\sum {y_{i}^{2}}}}\,.}
5773:
5447:
5427:
5413:
4480:
1189:
1135:
63:
7069:
6967:
5819:
5251:
1407:
323:
236:
160:
133:
6952:
1881:
7001:
7141:
7114:
7061:
7005:
6962:
6902:
6874:
6527:
6096:
5393:
5385:
3428:
538:
137:
5476:
When projecting the same lines in both models on one disk both lines go through the same two
4333:
1958:
1893:
341:
inside the disk, the unique hyperbolic line connecting them intersects the boundary at two
7051:
6993:
6917:
6889:
6086:
5466:
5451:
5439:
5307:
5259:
3107:
2488:
291:
176:
172:
125:
67:
7177:"Comparing metric tensors of the Poincare and the Klein disk models of hyperbolic geometry"
5443:
2977:
If P and Q are on a diameter of the boundary circle that diameter is the hyperbolic line.
1366:
560:
with the usual Euclidean norm, both of which have norm less than 1, then we may define an
129:
3225:
3155:
2780:
2710:
2637:
2587:
7091:
5465:
An advantage of the Klein disk model is that lines in this model are Euclidean straight
3084:
3059:
3011:
2516:
2465:
7181:
6898:
6869:
6105:
5943:
5866:
5485:
5481:
3401:
3378:
3358:
3338:
3318:
3295:
3275:
3255:
3205:
3185:
3135:
3113:
3036:
2987:
2956:
2933:
2913:
2893:
2873:
2850:
2830:
2810:
2760:
2740:
2687:
2667:
2617:
2563:
2541:
2494:
2437:
2417:
1911:
302:
216:
196:
6994:
1402:
7275:
7073:
5614:{\textstyle \left({\frac {2x}{1+x^{2}+y^{2}}}\ ,\ {\frac {2y}{1+x^{2}+y^{2}}}\right)}
5470:
5389:
4327:
3053:
2704:
1865:{\displaystyle \theta ^{i}={\frac {2}{1-|\mathbf {x} |{\vphantom {l}}^{2}}}\,dx^{i}.}
1419:
307:
287:
190:
75:
6957:
6894:
6885:
6878:
5484:
of the chord in the Klein disk model is the center of the circle that contains the
4214:
79:
7135:
7108:
6523:
5477:
5397:
5371:
5362:
5351:
5347:
5262:. In the hyperbolic plane, there are 4 distinct types of generalized circles or
4994:
1890:, the connection forms are given by the unique skew-symmetric matrix of 1-forms
342:
155:-dimensional hyperbolic geometry in which the points of the geometry are in the
117:
5370:
and some red normals. The normals converge asymptotically to the upper central
5119:{\displaystyle P=(u-v)\cdot (s-t)+(u\cdot t)(v\cdot s)-(u\cdot s)(v\cdot t)\,.}
6925:
6905:, which are regular tilings of the hyperbolic plane. Escher's wood engravings
5418:
5288:
83:
7110:
Rudolf Carnap: Early Writings: The Collected Works of Rudolf Carnap, Volume 1
7065:
1103:{\displaystyle \ln \left({\frac {1+r}{1-r}}\right)=2\operatorname {artanh} r}
17:
6826:
5380:
322:
consist of all arcs of Euclidean circles contained within the disk that are
319:
71:
7196:
7176:
7260:
3005:
2581:
561:
87:
51:
1661:
An orthonormal frame with respect to this Riemannian metric is given by
7056:
273:. Further, I shall suppose that in this world all bodies have the same
98:
7039:
5328:
5255:
175:
in an 1854 lecture (published 1868), which inspired an 1868 paper by
6909:
demonstrate this concept between 1958 and 1960, the final one being
7197:"Mapping the Poincare disk model to the Poincare half plane model"
6868:
5417:
5361:
4979:{\displaystyle R=(s-t)\cdot (s-t)-(s\wedge t)\cdot (s\wedge t)\,.}
4880:{\displaystyle Q=(u-v)\cdot (u-v)-(u\wedge v)\cdot (u\wedge v)\,,}
4781:{\displaystyle P=(u-v)\cdot (s-t)-(u\wedge v)\cdot (s\wedge t)\,,}
4605:{\displaystyle R=(s-t)\cdot (s-t)-(s\wedge t)\cdot (s\wedge t)\,.}
1401:
301:
37:
29:
6996:
The Road To Reality: A Complete Guide to the Laws of the Universe
727:
denotes the usual Euclidean norm. Then the distance function is
4615:
If both chords are not diameters, the general formula obtains
1658:
are the Cartesian coordinates of the ambient Euclidean space.
5254:(curves of constant curvature) are lines and circles. On the
326:
to the boundary of the disk, plus all diameters of the disk.
5458:
to the hemisphere model while the Poincaré disk model is a
5350:. Its axis is the hyperbolic line that shares the same two
128:
who used these models to show that hyperbolic geometry was
7165:. Translated by Cole, M.; Levy, S. Springer. p. 339.
6799:. It can be used to construct a Poincaré disk model as a
6771:
The hyperboloid model can be represented as the equation
5280:
that is inside the disk and tangent to the boundary is a
4679:{\displaystyle \cos ^{2}(\theta )={\frac {P^{2}}{QR}}\,,}
6915:
in 1960. According to Bruno Ernst, the best of them is
5414:
Hyperbolic geometry § Connection between the models
1886:
In two dimensions, with respect to these frames and the
4413:{\displaystyle \cos ^{2}(\theta )={\frac {P^{2}}{QR}},}
7240:
Teoria fondamentale degli spazii di curvatura costante
6362:
6193:
5638:
5505:
5480:. (the ideal points remain on the same spot) also the
5314:
that goes through the center is a hyperbolic line; and
2272:
2075:
6622:
6568:
6128:
6108:
5966:
5946:
5889:
5869:
5822:
5776:
5469:. A disadvantage is that the Klein disk model is not
5193:
5135:
5006:
4896:
4797:
4698:
4624:
4522:
4483:
4432:
4359:
4336:
4265:
3977:
3544:
3440:
3404:
3381:
3361:
3341:
3321:
3298:
3278:
3258:
3228:
3208:
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3158:
3138:
3116:
3087:
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3014:
2990:
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2813:
2783:
2763:
2743:
2713:
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2670:
2640:
2620:
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2544:
2519:
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2440:
2420:
2331:
2134:
1984:
1961:
1920:
1896:
1783:
1670:
1431:
1369:
1230:
1192:
1144:
1138:. If the two points lie on the same radius and point
1116:
1044:
736:
707:
573:
420:
239:
219:
199:
5317:
that does not go through the center is a hypercycle.
2322:Therefore, the curvature of the hyperbolic disk is
7096:. Robarts - University of Toronto. London W. Scott.
5298:that intersects the boundary non-orthogonally is a
7161:Berger, Marcel (1987) . "9.6 The Poincaré Model".
6739:
6609:
6506:
6337:
6169:
6114:
6070:
5952:
5930:
5875:
5852:
5808:
5759:
5613:
5234:
5176:
5118:
4978:
4879:
4780:
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4604:
4505:
4466:
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4342:
4299:
4197:
3949:
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3410:
3387:
3367:
3347:
3327:
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3172:
3144:
3122:
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3071:
3045:
3023:
2996:
2965:
2942:
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2882:
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2797:
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2727:
2696:
2676:
2654:
2626:
2604:
2572:
2550:
2530:
2503:
2479:
2446:
2426:
2394:{\displaystyle K=\Omega _{2}^{1}(e_{1},e_{2})=-1.}
2393:
2311:
2114:
1967:
1947:
1902:
1864:
1763:
1638:
1386:
1355:
1216:
1178:
1122:
1102:
1019:
719:
690:
527:
265:
225:
205:
6825:, projecting the upper half hyperboloid onto the
2538:be the inversion in the boundary circle of point
1731:
1696:
2454:not on a diameter of the boundary circle can be
1948:{\displaystyle 0=d\theta +\omega \wedge \theta }
1914:-free, i.e., that satisfies the matrix equation
353:. Label them so that the points are, in order,
193:. The law of this temperature is as follows: If
5408:Relation to other models of hyperbolic geometry
3418:that is inside the disk is the hyperbolic line.
2973:that is inside the disk is the hyperbolic line.
5354:. This is also known as an equidistant curve.
2414:The unique hyperbolic line through two points
6562:) on the plane, the conversion formulas are:
4240:If both models' lines are diameters, so that
2253:
2206:
1622:
1579:
1363:This reduces to the previous special case if
8:
7242:, Annali. di Mat., ser II 2 (1868), 232–255.
5396:, all converging asymptotically to the same
1598:
1590:
1554:
1543:
995:
988:
967:
960:
917:
910:
901:
894:
886:
874:
714:
708:
669:
662:
641:
634:
614:
601:
97:, the quotient of the special unitary group
34:Poincaré disk with hyperbolic parallel lines
5931:{\displaystyle s={\frac {2u}{1+u\cdot u}}.}
6901:around 1956 inspired Escher's interest in
6610:{\displaystyle y_{i}={\frac {x_{i}}{1+t}}}
6087:Cayley transform § Complex homography
7055:
6733:
6723:
6718:
6713:
6692:
6684:
6674:
6669:
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6648:
6636:
6621:
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6573:
6567:
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6489:
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6426:
6408:
6383:
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6334:
6320:
6295:
6283:
6270:
6257:
6239:
6214:
6199:
6192:
6135:
6127:
6107:
6081:Relation to the Poincaré half-plane model
6024:
6010:
5985:
5973:
5965:
5945:
5896:
5888:
5868:
5821:
5794:
5781:
5775:
5741:
5728:
5716:
5704:
5681:
5668:
5656:
5644:
5637:
5597:
5584:
5563:
5545:
5532:
5511:
5504:
5228:
5222:
5192:
5170:
5164:
5134:
5112:
5005:
4972:
4895:
4873:
4796:
4774:
4697:
4672:
4656:
4650:
4629:
4623:
4598:
4521:
4499:
4482:
4460:
4431:
4391:
4385:
4364:
4358:
4335:
4300:{\displaystyle \cos(\theta )=u\cdot s\,.}
4293:
4264:
4191:
4167:
4157:
4144:
4134:
4119:
4106:
4093:
4078:
4068:
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4030:
4017:
4004:
3995:
3982:
3976:
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3086:
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2217:
2205:
2204:
2186:
2177:
2133:
2070:
2061:
2051:
2050:
2044:
2039:
2034:
2016:
2003:
1991:
1983:
1960:
1919:
1895:
1853:
1845:
1836:
1826:
1825:
1819:
1814:
1809:
1797:
1788:
1782:
1749:
1736:
1730:
1729:
1723:
1718:
1712:
1707:
1695:
1694:
1684:
1675:
1669:
1627:
1621:
1620:
1613:
1603:
1602:
1593:
1578:
1577:
1569:
1559:
1558:
1549:
1542:
1536:
1525:
1514:
1509:
1499:
1476:
1471:
1458:
1451:
1439:
1430:
1368:
1271:
1242:
1229:
1191:
1143:
1115:
1055:
1043:
998:
970:
920:
904:
892:
871:
854:
821:
737:
735:
706:
684:
672:
644:
617:
598:
572:
502:
467:
448:
419:
257:
244:
238:
218:
198:
7000:. Great Britain: Jonathan Cape. p.
6522:The Poincaré disk model, as well as the
5273:that is completely inside the disk is a
3501:{\displaystyle x^{2}+y^{2}+ax+by+1=0\,,}
6984:
6761:
4326:, the formula becomes, in terms of the
1422:of the Poincaré disk model is given by
1410:' model view of the hyperbolic regular
1123:{\displaystyle \operatorname {artanh} }
1031:is the same as the angle in the model.
6170:{\displaystyle s={\frac {u+i}{iu+1}}.}
5426:), and their relations with the other
171:The disk model was first described by
5499:) in the Poincaré disk model maps to
5235:{\displaystyle R=(1-s\cdot t)^{2}\,.}
5177:{\displaystyle Q=(1-u\cdot v)^{2}\,,}
4213:We may compute the angle between the
7:
7085:
7083:
5473:(circles and angles are distorted).
5450:through a projection on or from the
720:{\displaystyle \lVert \cdot \rVert }
6973:Uniform tilings in hyperbolic plane
279:this will be non-Euclidean geometry
82:contained within the disk that are
4229:, and the arc whose endpoints are
2339:
2135:
1742:
1738:
1186:lies between the origin and point
25:
7235:, second edition, Springer, 2005.
7217:Escher's Circle Limit Exploration
6518:Relation to the hyperboloid model
6356:) in the halfplane model maps to
4467:{\displaystyle P=u\cdot (s-t)\,,}
213:be the radius of the sphere, and
7259:
7137:The Philosophy of Space and Time
7134:Reichenbach, Hans (2012-03-13).
6912:Circle Limit IV: Heaven and Hell
6841:
6764:
6091:The Poincaré disk model and the
5434:Relation to the Klein disk model
3110:in the boundary circle of point
2491:in the boundary circle of point
2223:
2040:
1815:
1713:
1594:
1550:
404:The hyperbolic distance between
7040:"Théorie des groupes fuchsiens"
5422:the Poincaré disk model (line
1224:, their hyperbolic distance is
1179:{\displaystyle x'=(r',\theta )}
1038:, the hyperbolic distance is:
6753:between a sphere and a plane.
6642:
6623:
6486:
6473:
6405:
6392:
6317:
6304:
6236:
6223:
5219:
5200:
5161:
5142:
5109:
5097:
5094:
5082:
5076:
5064:
5061:
5049:
5043:
5031:
5025:
5013:
4969:
4957:
4951:
4939:
4933:
4921:
4915:
4903:
4870:
4858:
4852:
4840:
4834:
4822:
4816:
4804:
4771:
4759:
4753:
4741:
4735:
4723:
4717:
4705:
4644:
4638:
4595:
4583:
4577:
4565:
4559:
4547:
4541:
4529:
4457:
4445:
4379:
4373:
4278:
4272:
4125:
4099:
4036:
4010:
3873:
3831:
3815:
3773:
3694:
3652:
3636:
3594:
2379:
2353:
2228:
2218:
2125:where the curvature matrix is
2045:
2035:
2023:
1997:
1820:
1810:
1719:
1708:
1347:
1318:
1211:
1199:
1173:
1156:
1004:
979:
976:
951:
840:
828:
799:
796:
784:
772:
756:
744:
678:
653:
650:
625:
589:
577:
436:
424:
62:, is a model of 2-dimensional
44:truncated triheptagonal tiling
1:
5809:{\displaystyle x^{2}+y^{2}=1}
5632:) in the Klein model maps to
5454:. The Klein disk model is an
5287:that intersects the boundary
4506:{\displaystyle Q=u\cdot u\,,}
1774:with dual coframe of 1-forms
1217:{\displaystyle x=(r,\theta )}
6187:) in the disk model maps to
5767:in the Poincaré disk model.
5488:in the Poincaré disk model.
4221:) are given by unit vectors
2235:
2052:
1955:. Solving this equation for
1827:
1604:
1560:
333:. Given two distinct points
329:Distances in this model are
27:Model of hyperbolic geometry
7249:, Jones and Bartlett, 1993.
7113:. Oxford University Press.
7038:Poincaré, H. (1882-12-01).
5853:{\displaystyle x=x\ ,\ y=y}
5250:In the Euclidean plane the
3152:be the midpoint of segment
2634:be the midpoint of segment
2410:By compass and straightedge
1132:inverse hyperbolic function
310:(hyperbolic) straight lines
266:{\displaystyle R^{2}-r^{2}}
42:Poincaré disk model of the
7308:
7282:Multi-dimensional geometry
6883:
6555:) on the hyperboloid and (
6084:
5940:Conversely, from a vector
5448:The two models are related
5411:
1879:
556:-dimensional vector space
275:co-efficient of dilatation
6948:Poincaré half-plane model
6749:Compare the formulas for
6093:Poincaré half-plane model
5860:so the points are fixed.
3222:perpendicular to segment
2777:perpendicular to segment
147:is the similar model for
122:Poincaré half-space model
7090:Poincaré, Henri (1905).
6903:hyperbolic tessellations
6751:stereographic projection
6345:in the halfplane model.
5816:and the formulas become
5460:stereographic projection
5366:A blue horocycle in the
5310:of the boundary circle:
3427:A basic construction of
552:are two vectors in real
7247:The Poincaré Half-Plane
7140:. Courier Corporation.
6992:Penrose, Roger (2004).
6873:The (6,4,2) triangular
5456:orthographic projection
5260:great and small circles
4343:{\displaystyle \wedge }
1968:{\displaystyle \omega }
1903:{\displaystyle \omega }
7093:Science and hypothesis
6881:
6741:
6611:
6508:
6339:
6171:
6116:
6072:
5954:
5932:
5877:
5854:
5810:
5761:
5615:
5430:
5375:
5236:
5178:
5120:
4980:
4881:
4782:
4680:
4606:
4507:
4468:
4414:
4344:
4301:
4199:
3951:
3502:
3412:
3389:
3369:
3349:
3329:
3306:
3286:
3266:
3244:
3216:
3196:
3174:
3146:
3124:
3100:
3073:
3047:
3025:
2998:
2967:
2944:
2924:
2904:
2884:
2861:
2841:
2821:
2799:
2771:
2751:
2729:
2698:
2678:
2656:
2628:
2606:
2574:
2552:
2532:
2505:
2481:
2448:
2428:
2395:
2313:
2116:
1969:
1949:
1904:
1888:Levi-Civita connection
1866:
1765:
1640:
1415:
1388:
1357:
1218:
1180:
1124:
1104:
1021:
721:
692:
529:
311:
283:
267:
227:
207:
182:Science and Hypothesis
86:to the unit circle or
47:
35:
6872:
6865:Artistic realizations
6742:
6612:
6542:Cartesian coordinates
6526:, are related to the
6509:
6340:
6172:
6117:
6095:are both named after
6073:
5955:
5933:
5878:
5855:
5811:
5762:
5616:
5421:
5365:
5237:
5179:
5121:
4991:Binet–Cauchy identity
4981:
4882:
4783:
4681:
4607:
4508:
4469:
4415:
4345:
4302:
4200:
3952:
3503:
3413:
3390:
3370:
3350:
3330:
3307:
3287:
3267:
3245:
3217:
3197:
3175:
3147:
3125:
3101:
3074:
3048:
3026:
2999:
2968:
2945:
2925:
2905:
2885:
2862:
2842:
2822:
2800:
2772:
2752:
2730:
2699:
2679:
2657:
2629:
2607:
2575:
2553:
2533:
2506:
2482:
2449:
2429:
2405:Construction of lines
2396:
2314:
2117:
1970:
1950:
1905:
1867:
1766:
1641:
1412:icosahedral honeycomb
1405:
1389:
1358:
1219:
1181:
1125:
1105:
1022:
722:
693:
530:
306:Poincaré disk with 3
305:
268:
228:
208:
187:
124:, it was proposed by
41:
33:
7268:at Wikimedia Commons
7266:Poincaré disk models
6943:Beltrami–Klein model
6620:
6566:
6360:
6191:
6126:
6106:
5964:
5944:
5887:
5867:
5820:
5774:
5636:
5621:in the Klein model.
5503:
5269:A Euclidean circle:
5191:
5133:
5004:
4894:
4795:
4696:
4622:
4520:
4481:
4430:
4357:
4334:
4263:
3975:
3542:
3438:
3423:By analytic geometry
3402:
3379:
3359:
3339:
3319:
3296:
3276:
3256:
3226:
3206:
3186:
3156:
3136:
3114:
3085:
3060:
3037:
3033:Draw line m through
3012:
2988:
2957:
2934:
2914:
2894:
2874:
2851:
2831:
2811:
2781:
2761:
2741:
2711:
2688:
2668:
2638:
2618:
2588:
2564:
2542:
2517:
2495:
2466:
2438:
2418:
2329:
2240:
2132:
2057:
1982:
1959:
1918:
1894:
1832:
1781:
1668:
1609:
1565:
1429:
1398:Metric and curvature
1387:{\displaystyle r'=0}
1367:
1228:
1190:
1142:
1114:
1042:
734:
705:
571:
418:
331:Cayley–Klein metrics
237:
217:
197:
136:. It is named after
90:of the unit circle.
60:conformal disk model
7287:Hyperbolic geometry
7233:Hyperbolic Geometry
7231:James W. Anderson,
6938:Hyperbolic geometry
6728:
6679:
6514:in the disk model.
5368:Poincaré disk model
5252:generalized circles
3866:
3848:
3808:
3790:
3687:
3669:
3629:
3611:
3398:The part of circle
3243:{\displaystyle PP'}
3173:{\displaystyle PP'}
2953:The part of circle
2798:{\displaystyle QQ'}
2728:{\displaystyle PP'}
2655:{\displaystyle QQ'}
2605:{\displaystyle PP'}
2352:
2241:
2236:
2058:
2053:
1833:
1828:
1610:
1605:
1566:
1561:
1519:
1481:
562:isometric invariant
369:, that is, so that
145:Poincaré ball model
64:hyperbolic geometry
56:Poincaré disk model
7238:Eugenio Beltrami,
7057:10.1007/BF02592124
6968:Inversive geometry
6882:
6737:
6714:
6665:
6607:
6504:
6335:
6167:
6112:
6068:
5950:
5928:
5873:
5850:
5806:
5757:
5611:
5431:
5394:limiting parallels
5376:
5232:
5174:
5116:
4976:
4877:
4778:
4676:
4602:
4503:
4464:
4410:
4340:
4297:
4195:
3947:
3945:
3852:
3834:
3794:
3776:
3673:
3655:
3615:
3597:
3498:
3408:
3385:
3365:
3355:and going through
3345:
3325:
3302:
3282:
3262:
3240:
3212:
3192:
3170:
3142:
3120:
3099:{\displaystyle P'}
3096:
3072:{\displaystyle PQ}
3069:
3043:
3024:{\displaystyle PQ}
3021:
2994:
2963:
2940:
2920:
2910:and going through
2900:
2880:
2857:
2837:
2817:
2795:
2767:
2747:
2725:
2694:
2674:
2652:
2624:
2602:
2570:
2548:
2531:{\displaystyle Q'}
2528:
2501:
2480:{\displaystyle P'}
2477:
2444:
2424:
2391:
2338:
2309:
2300:
2112:
2103:
1965:
1945:
1900:
1862:
1761:
1636:
1505:
1504:
1467:
1463:
1416:
1384:
1353:
1214:
1176:
1136:hyperbolic tangent
1120:
1100:
1017:
1015:
717:
688:
525:
393:| > |
377:| > |
312:
298:Lines and distance
263:
223:
203:
134:Euclidean geometry
58:, also called the
48:
36:
7264:Media related to
7147:978-0-486-13803-9
7120:978-0-19-106526-2
6963:Hyperboloid model
6907:Circle Limit I–IV
6875:hyperbolic tiling
6851:
6731:
6605:
6528:hyperboloid model
6496:
6425:
6419:
6415:
6327:
6256:
6250:
6246:
6162:
6115:{\displaystyle u}
6063:
6041:
6005:
6002:
5953:{\displaystyle s}
5923:
5876:{\displaystyle u}
5840:
5834:
5770:For ideal points
5750:
5747:
5703:
5700:
5694:
5690:
5687:
5604:
5562:
5556:
5552:
5275:hyperbolic circle
4670:
4405:
4217:whose endpoints (
4174:
4085:
3922:
3743:
3429:analytic geometry
3411:{\displaystyle c}
3388:{\displaystyle Q}
3368:{\displaystyle P}
3348:{\displaystyle C}
3328:{\displaystyle c}
3305:{\displaystyle n}
3285:{\displaystyle m}
3265:{\displaystyle C}
3215:{\displaystyle N}
3195:{\displaystyle n}
3145:{\displaystyle N}
3123:{\displaystyle P}
3046:{\displaystyle M}
2997:{\displaystyle M}
2966:{\displaystyle c}
2943:{\displaystyle Q}
2923:{\displaystyle P}
2903:{\displaystyle C}
2883:{\displaystyle c}
2860:{\displaystyle n}
2840:{\displaystyle m}
2820:{\displaystyle C}
2770:{\displaystyle N}
2750:{\displaystyle n}
2697:{\displaystyle M}
2677:{\displaystyle m}
2627:{\displaystyle N}
2573:{\displaystyle M}
2551:{\displaystyle Q}
2504:{\displaystyle P}
2447:{\displaystyle Q}
2427:{\displaystyle P}
2265:
2068:
1876:In two dimensions
1843:
1756:
1692:
1634:
1531:
1495:
1454:
1305:
1266:
1079:
1008:
1007:
947:
848:
847:
682:
544:Equivalently, if
539:natural logarithm
520:
226:{\displaystyle r}
206:{\displaystyle R}
16:(Redirected from
7299:
7263:
7219:
7214:
7208:
7207:
7205:
7203:
7193:
7187:
7186:
7173:
7167:
7166:
7158:
7152:
7151:
7131:
7125:
7124:
7104:
7098:
7097:
7087:
7078:
7077:
7059:
7044:Acta Mathematica
7035:
7029:
7022:
7016:
7015:
6999:
6989:
6918:Circle Limit III
6890:Circle Limit III
6853:
6852:
6835:
6824:
6798:
6791:
6768:
6746:
6744:
6743:
6738:
6732:
6730:
6729:
6727:
6722:
6702:
6698:
6697:
6696:
6680:
6678:
6673:
6649:
6641:
6640:
6616:
6614:
6613:
6608:
6606:
6604:
6593:
6592:
6583:
6578:
6577:
6513:
6511:
6510:
6505:
6502:
6498:
6497:
6495:
6494:
6493:
6469:
6468:
6458:
6451:
6450:
6438:
6437:
6427:
6423:
6417:
6416:
6414:
6413:
6412:
6388:
6387:
6377:
6369:
6344:
6342:
6341:
6336:
6333:
6329:
6328:
6326:
6325:
6324:
6300:
6299:
6289:
6288:
6287:
6275:
6274:
6258:
6254:
6248:
6247:
6245:
6244:
6243:
6219:
6218:
6208:
6200:
6176:
6174:
6173:
6168:
6163:
6161:
6147:
6136:
6121:
6119:
6118:
6113:
6077:
6075:
6074:
6069:
6064:
6062:
6051:
6047:
6043:
6042:
6025:
6011:
6006:
6004:
6003:
5986:
5974:
5959:
5957:
5956:
5951:
5937:
5935:
5934:
5929:
5924:
5922:
5905:
5897:
5882:
5880:
5879:
5874:
5859:
5857:
5856:
5851:
5838:
5832:
5815:
5813:
5812:
5807:
5799:
5798:
5786:
5785:
5766:
5764:
5763:
5758:
5756:
5752:
5751:
5749:
5748:
5746:
5745:
5733:
5732:
5717:
5705:
5701:
5698:
5692:
5691:
5689:
5688:
5686:
5685:
5673:
5672:
5657:
5645:
5620:
5618:
5617:
5612:
5610:
5606:
5605:
5603:
5602:
5601:
5589:
5588:
5572:
5564:
5560:
5554:
5553:
5551:
5550:
5549:
5537:
5536:
5520:
5512:
5452:hemisphere model
5440:Klein disk model
5241:
5239:
5238:
5233:
5227:
5226:
5183:
5181:
5180:
5175:
5169:
5168:
5125:
5123:
5122:
5117:
4985:
4983:
4982:
4977:
4886:
4884:
4883:
4878:
4787:
4785:
4784:
4779:
4685:
4683:
4682:
4677:
4671:
4669:
4661:
4660:
4651:
4634:
4633:
4611:
4609:
4608:
4603:
4512:
4510:
4509:
4504:
4473:
4471:
4470:
4465:
4419:
4417:
4416:
4411:
4406:
4404:
4396:
4395:
4386:
4369:
4368:
4349:
4347:
4346:
4341:
4306:
4304:
4303:
4298:
4204:
4202:
4201:
4196:
4175:
4173:
4172:
4171:
4162:
4161:
4149:
4148:
4139:
4138:
4128:
4124:
4123:
4111:
4110:
4094:
4086:
4084:
4083:
4082:
4073:
4072:
4060:
4059:
4050:
4049:
4039:
4035:
4034:
4022:
4021:
4005:
4000:
3999:
3987:
3986:
3956:
3954:
3953:
3948:
3946:
3923:
3921:
3920:
3919:
3910:
3909:
3897:
3896:
3887:
3886:
3876:
3865:
3860:
3847:
3842:
3830:
3829:
3807:
3802:
3789:
3784:
3772:
3771:
3761:
3756:
3751:
3744:
3742:
3741:
3740:
3731:
3730:
3718:
3717:
3708:
3707:
3697:
3686:
3681:
3668:
3663:
3651:
3650:
3628:
3623:
3610:
3605:
3593:
3592:
3582:
3577:
3571:
3570:
3558:
3557:
3507:
3505:
3504:
3499:
3463:
3462:
3450:
3449:
3417:
3415:
3414:
3409:
3394:
3392:
3391:
3386:
3374:
3372:
3371:
3366:
3354:
3352:
3351:
3346:
3334:
3332:
3331:
3326:
3311:
3309:
3308:
3303:
3291:
3289:
3288:
3283:
3271:
3269:
3268:
3263:
3249:
3247:
3246:
3241:
3239:
3221:
3219:
3218:
3213:
3201:
3199:
3198:
3193:
3179:
3177:
3176:
3171:
3169:
3151:
3149:
3148:
3143:
3129:
3127:
3126:
3121:
3105:
3103:
3102:
3097:
3095:
3078:
3076:
3075:
3070:
3052:
3050:
3049:
3044:
3030:
3028:
3027:
3022:
3003:
3001:
3000:
2995:
2980:Another way is:
2972:
2970:
2969:
2964:
2949:
2947:
2946:
2941:
2929:
2927:
2926:
2921:
2909:
2907:
2906:
2901:
2889:
2887:
2886:
2881:
2866:
2864:
2863:
2858:
2846:
2844:
2843:
2838:
2826:
2824:
2823:
2818:
2804:
2802:
2801:
2796:
2794:
2776:
2774:
2773:
2768:
2756:
2754:
2753:
2748:
2734:
2732:
2731:
2726:
2724:
2703:
2701:
2700:
2695:
2683:
2681:
2680:
2675:
2661:
2659:
2658:
2653:
2651:
2633:
2631:
2630:
2625:
2611:
2609:
2608:
2603:
2601:
2579:
2577:
2576:
2571:
2557:
2555:
2554:
2549:
2537:
2535:
2534:
2529:
2527:
2510:
2508:
2507:
2502:
2486:
2484:
2483:
2478:
2476:
2453:
2451:
2450:
2445:
2433:
2431:
2430:
2425:
2400:
2398:
2397:
2392:
2378:
2377:
2365:
2364:
2351:
2346:
2318:
2316:
2315:
2310:
2305:
2304:
2266:
2264:
2263:
2262:
2257:
2256:
2249:
2248:
2243:
2242:
2231:
2226:
2221:
2210:
2209:
2202:
2178:
2121:
2119:
2118:
2113:
2108:
2107:
2069:
2067:
2066:
2065:
2060:
2059:
2048:
2043:
2038:
2026:
1992:
1974:
1972:
1971:
1966:
1954:
1952:
1951:
1946:
1909:
1907:
1906:
1901:
1871:
1869:
1868:
1863:
1858:
1857:
1844:
1842:
1841:
1840:
1835:
1834:
1823:
1818:
1813:
1798:
1793:
1792:
1770:
1768:
1767:
1762:
1757:
1755:
1754:
1753:
1737:
1735:
1734:
1728:
1727:
1722:
1716:
1711:
1700:
1699:
1693:
1685:
1680:
1679:
1645:
1643:
1642:
1637:
1635:
1633:
1632:
1631:
1626:
1625:
1618:
1617:
1612:
1611:
1597:
1583:
1582:
1575:
1574:
1573:
1568:
1567:
1553:
1537:
1532:
1530:
1529:
1524:
1520:
1518:
1513:
1503:
1482:
1480:
1475:
1462:
1452:
1444:
1443:
1393:
1391:
1390:
1385:
1377:
1362:
1360:
1359:
1354:
1346:
1311:
1307:
1306:
1304:
1303:
1288:
1287:
1272:
1267:
1265:
1254:
1243:
1223:
1221:
1220:
1215:
1185:
1183:
1182:
1177:
1166:
1152:
1129:
1127:
1126:
1121:
1109:
1107:
1106:
1101:
1084:
1080:
1078:
1067:
1056:
1026:
1024:
1023:
1018:
1016:
1009:
1003:
1002:
975:
974:
950:
949:
948:
925:
924:
909:
908:
893:
872:
849:
843:
823:
822:
805:
726:
724:
723:
718:
697:
695:
694:
689:
683:
681:
677:
676:
649:
648:
623:
622:
621:
599:
534:
532:
531:
526:
521:
519:
518:
514:
501:
497:
484:
483:
479:
466:
462:
449:
400:
398:
392:
384:
382:
376:
292:Hans Reichenbach
272:
270:
269:
264:
262:
261:
249:
248:
232:
230:
229:
224:
212:
210:
209:
204:
177:Eugenio Beltrami
173:Bernhard Riemann
126:Eugenio Beltrami
112:
96:
21:
7307:
7306:
7302:
7301:
7300:
7298:
7297:
7296:
7272:
7271:
7256:
7228:
7226:Further reading
7223:
7222:
7215:
7211:
7201:
7199:
7195:
7194:
7190:
7185:. May 23, 2015.
7175:
7174:
7170:
7160:
7159:
7155:
7148:
7133:
7132:
7128:
7121:
7106:
7105:
7101:
7089:
7088:
7081:
7037:
7036:
7032:
7023:
7019:
7012:
6991:
6990:
6986:
6981:
6953:Poincaré metric
6934:
6924:
6892:
6867:
6862:
6861:
6860:
6857:
6854:
6842:
6837:
6830:
6822:
6815:
6804:
6793:
6789:
6782:
6772:
6769:
6756:
6703:
6688:
6654:
6650:
6632:
6618:
6617:
6594:
6584:
6569:
6564:
6563:
6560:
6553:
6520:
6485:
6460:
6459:
6442:
6429:
6428:
6404:
6379:
6378:
6370:
6367:
6363:
6358:
6357:
6316:
6291:
6290:
6279:
6266:
6259:
6235:
6210:
6209:
6201:
6198:
6194:
6189:
6188:
6148:
6137:
6124:
6123:
6104:
6103:
6089:
6083:
6052:
6017:
6013:
6012:
5978:
5962:
5961:
5942:
5941:
5906:
5898:
5885:
5884:
5865:
5864:
5818:
5817:
5790:
5777:
5772:
5771:
5737:
5724:
5709:
5677:
5664:
5649:
5643:
5639:
5634:
5633:
5593:
5580:
5573:
5565:
5541:
5528:
5521:
5513:
5510:
5506:
5501:
5500:
5436:
5416:
5410:
5384:(a curve whose
5360:
5338:
5324:
5293:hyperbolic line
5248:
5218:
5189:
5188:
5160:
5131:
5130:
5002:
5001:
4892:
4891:
4793:
4792:
4694:
4693:
4662:
4652:
4625:
4620:
4619:
4518:
4517:
4479:
4478:
4428:
4427:
4397:
4387:
4360:
4355:
4354:
4332:
4331:
4261:
4260:
4211:
4163:
4153:
4140:
4130:
4129:
4115:
4102:
4095:
4074:
4064:
4051:
4041:
4040:
4026:
4013:
4006:
3991:
3978:
3973:
3972:
3944:
3943:
3911:
3901:
3888:
3878:
3877:
3821:
3763:
3762:
3749:
3748:
3732:
3722:
3709:
3699:
3698:
3642:
3584:
3583:
3572:
3562:
3549:
3540:
3539:
3532:
3528:
3520:
3516:
3454:
3441:
3436:
3435:
3425:
3400:
3399:
3377:
3376:
3357:
3356:
3337:
3336:
3317:
3316:
3294:
3293:
3274:
3273:
3254:
3253:
3232:
3224:
3223:
3204:
3203:
3184:
3183:
3162:
3154:
3153:
3134:
3133:
3112:
3111:
3088:
3083:
3082:
3058:
3057:
3035:
3034:
3010:
3009:
2986:
2985:
2955:
2954:
2932:
2931:
2912:
2911:
2892:
2891:
2872:
2871:
2849:
2848:
2829:
2828:
2809:
2808:
2787:
2779:
2778:
2759:
2758:
2739:
2738:
2717:
2709:
2708:
2686:
2685:
2666:
2665:
2644:
2636:
2635:
2616:
2615:
2594:
2586:
2585:
2562:
2561:
2540:
2539:
2520:
2515:
2514:
2493:
2492:
2469:
2464:
2463:
2436:
2435:
2416:
2415:
2412:
2407:
2369:
2356:
2327:
2326:
2299:
2298:
2293:
2284:
2283:
2278:
2268:
2250:
2232:
2203:
2179:
2130:
2129:
2102:
2101:
2096:
2087:
2086:
2081:
2071:
2049:
2027:
1993:
1980:
1979:
1957:
1956:
1916:
1915:
1892:
1891:
1884:
1882:Poincaré metric
1878:
1849:
1824:
1802:
1784:
1779:
1778:
1745:
1741:
1717:
1671:
1666:
1665:
1657:
1619:
1601:
1576:
1557:
1538:
1488:
1484:
1483:
1453:
1435:
1427:
1426:
1418:The associated
1400:
1370:
1365:
1364:
1339:
1296:
1289:
1280:
1273:
1255:
1244:
1241:
1237:
1226:
1225:
1188:
1187:
1159:
1145:
1140:
1139:
1112:
1111:
1068:
1057:
1051:
1040:
1039:
1014:
1013:
994:
966:
916:
900:
873:
855:
851:
850:
824:
803:
802:
759:
732:
731:
703:
702:
668:
640:
624:
613:
600:
569:
568:
507:
503:
490:
486:
485:
472:
468:
455:
451:
450:
416:
415:
394:
388:
386:
378:
372:
370:
300:
253:
240:
235:
234:
215:
214:
195:
194:
169:
116:Along with the
102:
94:
70:are inside the
28:
23:
22:
15:
12:
11:
5:
7305:
7303:
7295:
7294:
7292:Henri Poincaré
7289:
7284:
7274:
7273:
7270:
7269:
7255:
7254:External links
7252:
7251:
7250:
7243:
7236:
7227:
7224:
7221:
7220:
7209:
7188:
7182:Stack Exchange
7168:
7153:
7146:
7126:
7119:
7099:
7079:
7030:
7017:
7010:
6983:
6982:
6980:
6977:
6976:
6975:
6970:
6965:
6960:
6955:
6950:
6945:
6940:
6933:
6930:
6899:H.S.M. Coxeter
6877:that inspired
6866:
6863:
6859:
6858:
6855:
6840:
6838:
6820:
6813:
6787:
6780:
6770:
6763:
6760:
6759:
6758:
6736:
6726:
6721:
6717:
6712:
6709:
6706:
6701:
6695:
6691:
6687:
6683:
6677:
6672:
6668:
6663:
6660:
6657:
6653:
6647:
6644:
6639:
6635:
6631:
6628:
6625:
6603:
6600:
6597:
6591:
6587:
6581:
6576:
6572:
6558:
6551:
6519:
6516:
6501:
6492:
6488:
6484:
6481:
6478:
6475:
6472:
6467:
6463:
6457:
6454:
6449:
6445:
6441:
6436:
6432:
6422:
6411:
6407:
6403:
6400:
6397:
6394:
6391:
6386:
6382:
6376:
6373:
6366:
6332:
6323:
6319:
6315:
6312:
6309:
6306:
6303:
6298:
6294:
6286:
6282:
6278:
6273:
6269:
6265:
6262:
6253:
6242:
6238:
6234:
6231:
6228:
6225:
6222:
6217:
6213:
6207:
6204:
6197:
6166:
6160:
6157:
6154:
6151:
6146:
6143:
6140:
6134:
6131:
6111:
6097:Henri Poincaré
6082:
6079:
6067:
6061:
6058:
6055:
6050:
6046:
6040:
6037:
6034:
6031:
6028:
6023:
6020:
6016:
6009:
6001:
5998:
5995:
5992:
5989:
5984:
5981:
5977:
5972:
5969:
5949:
5927:
5921:
5918:
5915:
5912:
5909:
5904:
5901:
5895:
5892:
5872:
5849:
5846:
5843:
5837:
5831:
5828:
5825:
5805:
5802:
5797:
5793:
5789:
5784:
5780:
5755:
5744:
5740:
5736:
5731:
5727:
5723:
5720:
5715:
5712:
5708:
5697:
5684:
5680:
5676:
5671:
5667:
5663:
5660:
5655:
5652:
5648:
5642:
5609:
5600:
5596:
5592:
5587:
5583:
5579:
5576:
5571:
5568:
5559:
5548:
5544:
5540:
5535:
5531:
5527:
5524:
5519:
5516:
5509:
5435:
5432:
5409:
5406:
5392:geodesics are
5359:
5356:
5337:
5334:
5323:
5320:
5319:
5318:
5315:
5304:
5303:
5296:
5285:
5278:
5247:
5244:
5243:
5242:
5231:
5225:
5221:
5217:
5214:
5211:
5208:
5205:
5202:
5199:
5196:
5185:
5184:
5173:
5167:
5163:
5159:
5156:
5153:
5150:
5147:
5144:
5141:
5138:
5127:
5126:
5115:
5111:
5108:
5105:
5102:
5099:
5096:
5093:
5090:
5087:
5084:
5081:
5078:
5075:
5072:
5069:
5066:
5063:
5060:
5057:
5054:
5051:
5048:
5045:
5042:
5039:
5036:
5033:
5030:
5027:
5024:
5021:
5018:
5015:
5012:
5009:
4987:
4986:
4975:
4971:
4968:
4965:
4962:
4959:
4956:
4953:
4950:
4947:
4944:
4941:
4938:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4914:
4911:
4908:
4905:
4902:
4899:
4888:
4887:
4876:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4809:
4806:
4803:
4800:
4789:
4788:
4777:
4773:
4770:
4767:
4764:
4761:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4734:
4731:
4728:
4725:
4722:
4719:
4716:
4713:
4710:
4707:
4704:
4701:
4687:
4686:
4675:
4668:
4665:
4659:
4655:
4649:
4646:
4643:
4640:
4637:
4632:
4628:
4613:
4612:
4601:
4597:
4594:
4591:
4588:
4585:
4582:
4579:
4576:
4573:
4570:
4567:
4564:
4561:
4558:
4555:
4552:
4549:
4546:
4543:
4540:
4537:
4534:
4531:
4528:
4525:
4514:
4513:
4502:
4498:
4495:
4492:
4489:
4486:
4475:
4474:
4463:
4459:
4456:
4453:
4450:
4447:
4444:
4441:
4438:
4435:
4421:
4420:
4409:
4403:
4400:
4394:
4390:
4384:
4381:
4378:
4375:
4372:
4367:
4363:
4339:
4308:
4307:
4296:
4292:
4289:
4286:
4283:
4280:
4277:
4274:
4271:
4268:
4210:
4207:
4206:
4205:
4194:
4190:
4187:
4184:
4181:
4178:
4170:
4166:
4160:
4156:
4152:
4147:
4143:
4137:
4133:
4127:
4122:
4118:
4114:
4109:
4105:
4101:
4098:
4092:
4089:
4081:
4077:
4071:
4067:
4063:
4058:
4054:
4048:
4044:
4038:
4033:
4029:
4025:
4020:
4016:
4012:
4009:
4003:
3998:
3994:
3990:
3985:
3981:
3960:If the points
3958:
3957:
3942:
3938:
3935:
3932:
3929:
3926:
3918:
3914:
3908:
3904:
3900:
3895:
3891:
3885:
3881:
3875:
3872:
3869:
3864:
3859:
3855:
3851:
3846:
3841:
3837:
3833:
3828:
3824:
3820:
3817:
3814:
3811:
3806:
3801:
3797:
3793:
3788:
3783:
3779:
3775:
3770:
3766:
3759:
3754:
3752:
3750:
3747:
3739:
3735:
3729:
3725:
3721:
3716:
3712:
3706:
3702:
3696:
3693:
3690:
3685:
3680:
3676:
3672:
3667:
3662:
3658:
3654:
3649:
3645:
3641:
3638:
3635:
3632:
3627:
3622:
3618:
3614:
3609:
3604:
3600:
3596:
3591:
3587:
3580:
3575:
3573:
3569:
3565:
3561:
3556:
3552:
3548:
3547:
3530:
3526:
3518:
3514:
3509:
3508:
3497:
3493:
3490:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3466:
3461:
3457:
3453:
3448:
3444:
3424:
3421:
3420:
3419:
3407:
3396:
3384:
3364:
3344:
3324:
3313:
3301:
3281:
3272:be where line
3261:
3250:
3238:
3235:
3231:
3211:
3191:
3180:
3168:
3165:
3161:
3141:
3130:
3119:
3094:
3091:
3079:
3068:
3065:
3042:
3031:
3020:
3017:
2993:
2975:
2974:
2962:
2951:
2939:
2919:
2899:
2879:
2868:
2856:
2836:
2827:be where line
2816:
2805:
2793:
2790:
2786:
2766:
2746:
2735:
2723:
2720:
2716:
2693:
2673:
2662:
2650:
2647:
2643:
2623:
2612:
2600:
2597:
2593:
2569:
2558:
2547:
2526:
2523:
2511:
2500:
2475:
2472:
2443:
2423:
2411:
2408:
2406:
2403:
2402:
2401:
2390:
2387:
2384:
2381:
2376:
2372:
2368:
2363:
2359:
2355:
2350:
2345:
2341:
2337:
2334:
2320:
2319:
2308:
2303:
2297:
2294:
2292:
2289:
2286:
2285:
2282:
2279:
2277:
2274:
2273:
2271:
2261:
2255:
2247:
2239:
2230:
2225:
2220:
2216:
2213:
2208:
2201:
2198:
2195:
2192:
2189:
2185:
2182:
2176:
2173:
2170:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2123:
2122:
2111:
2106:
2100:
2097:
2095:
2092:
2089:
2088:
2085:
2082:
2080:
2077:
2076:
2074:
2064:
2056:
2047:
2042:
2037:
2033:
2030:
2025:
2022:
2019:
2015:
2012:
2009:
2006:
2002:
1999:
1996:
1990:
1987:
1964:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1899:
1880:Main article:
1877:
1874:
1873:
1872:
1861:
1856:
1852:
1848:
1839:
1831:
1822:
1817:
1812:
1808:
1805:
1801:
1796:
1791:
1787:
1772:
1771:
1760:
1752:
1748:
1744:
1740:
1733:
1726:
1721:
1715:
1710:
1706:
1703:
1698:
1691:
1688:
1683:
1678:
1674:
1653:
1647:
1646:
1630:
1624:
1616:
1608:
1600:
1596:
1592:
1589:
1586:
1581:
1572:
1564:
1556:
1552:
1548:
1545:
1541:
1535:
1528:
1523:
1517:
1512:
1508:
1502:
1498:
1494:
1491:
1487:
1479:
1474:
1470:
1466:
1461:
1457:
1450:
1447:
1442:
1438:
1434:
1399:
1396:
1383:
1380:
1376:
1373:
1352:
1349:
1345:
1342:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1310:
1302:
1299:
1295:
1292:
1286:
1283:
1279:
1276:
1270:
1264:
1261:
1258:
1253:
1250:
1247:
1240:
1236:
1233:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1175:
1172:
1169:
1165:
1162:
1158:
1155:
1151:
1148:
1119:
1099:
1096:
1093:
1090:
1087:
1083:
1077:
1074:
1071:
1066:
1063:
1060:
1054:
1050:
1047:
1028:
1027:
1012:
1006:
1001:
997:
993:
990:
987:
984:
981:
978:
973:
969:
965:
962:
959:
956:
953:
946:
943:
940:
937:
934:
931:
928:
923:
919:
915:
912:
907:
903:
899:
896:
891:
888:
885:
882:
879:
876:
870:
867:
864:
861:
858:
856:
853:
852:
846:
842:
839:
836:
833:
830:
827:
820:
817:
814:
811:
808:
806:
804:
801:
798:
795:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
762:
760:
758:
755:
752:
749:
746:
743:
740:
739:
716:
713:
710:
699:
698:
687:
680:
675:
671:
667:
664:
661:
658:
655:
652:
647:
643:
639:
636:
633:
630:
627:
620:
616:
612:
609:
606:
603:
597:
594:
591:
588:
585:
582:
579:
576:
524:
517:
513:
510:
506:
500:
496:
493:
489:
482:
478:
475:
471:
465:
461:
458:
454:
447:
444:
441:
438:
435:
432:
429:
426:
423:
316:straight lines
299:
296:
260:
256:
252:
247:
243:
222:
202:
168:
165:
138:Henri Poincaré
130:equiconsistent
101:by its center
76:straight lines
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7304:
7293:
7290:
7288:
7285:
7283:
7280:
7279:
7277:
7267:
7262:
7258:
7257:
7253:
7248:
7244:
7241:
7237:
7234:
7230:
7229:
7225:
7218:
7213:
7210:
7198:
7192:
7189:
7184:
7183:
7178:
7172:
7169:
7164:
7157:
7154:
7149:
7143:
7139:
7138:
7130:
7127:
7122:
7116:
7112:
7111:
7103:
7100:
7095:
7094:
7086:
7084:
7080:
7075:
7071:
7067:
7063:
7058:
7053:
7049:
7046:(in French).
7045:
7041:
7034:
7031:
7028:
7021:
7018:
7013:
7011:0-224-04447-8
7007:
7003:
6998:
6997:
6988:
6985:
6978:
6974:
6971:
6969:
6966:
6964:
6961:
6959:
6956:
6954:
6951:
6949:
6946:
6944:
6941:
6939:
6936:
6935:
6931:
6929:
6927:
6922:
6920:
6919:
6914:
6913:
6908:
6904:
6900:
6896:
6891:
6887:
6880:
6876:
6871:
6864:
6839:
6833:
6828:
6819:
6812:
6808:
6802:
6796:
6786:
6779:
6775:
6767:
6762:
6757:
6754:
6752:
6747:
6734:
6724:
6719:
6715:
6710:
6707:
6704:
6699:
6693:
6689:
6685:
6681:
6675:
6670:
6666:
6661:
6658:
6655:
6651:
6645:
6637:
6633:
6629:
6626:
6601:
6598:
6595:
6589:
6585:
6579:
6574:
6570:
6561:
6554:
6547:
6543:
6538:
6536:
6532:
6529:
6525:
6517:
6515:
6499:
6490:
6482:
6479:
6476:
6470:
6465:
6461:
6455:
6452:
6447:
6443:
6439:
6434:
6430:
6420:
6409:
6401:
6398:
6395:
6389:
6384:
6380:
6374:
6371:
6364:
6355:
6351:
6346:
6330:
6321:
6313:
6310:
6307:
6301:
6296:
6292:
6284:
6280:
6276:
6271:
6267:
6263:
6260:
6251:
6240:
6232:
6229:
6226:
6220:
6215:
6211:
6205:
6202:
6195:
6186:
6182:
6177:
6164:
6158:
6155:
6152:
6149:
6144:
6141:
6138:
6132:
6129:
6109:
6100:
6098:
6094:
6088:
6080:
6078:
6065:
6059:
6056:
6053:
6048:
6044:
6038:
6035:
6032:
6029:
6026:
6021:
6018:
6014:
6007:
5999:
5996:
5993:
5990:
5987:
5982:
5979:
5975:
5970:
5967:
5947:
5938:
5925:
5919:
5916:
5913:
5910:
5907:
5902:
5899:
5893:
5890:
5870:
5861:
5847:
5844:
5841:
5835:
5829:
5826:
5823:
5803:
5800:
5795:
5791:
5787:
5782:
5778:
5768:
5753:
5742:
5738:
5734:
5729:
5725:
5721:
5718:
5713:
5710:
5706:
5695:
5682:
5678:
5674:
5669:
5665:
5661:
5658:
5653:
5650:
5646:
5640:
5631:
5627:
5622:
5607:
5598:
5594:
5590:
5585:
5581:
5577:
5574:
5569:
5566:
5557:
5546:
5542:
5538:
5533:
5529:
5525:
5522:
5517:
5514:
5507:
5498:
5494:
5489:
5487:
5483:
5479:
5474:
5472:
5468:
5463:
5461:
5457:
5453:
5449:
5445:
5441:
5433:
5429:
5425:
5420:
5415:
5407:
5405:
5401:
5399:
5395:
5391:
5390:perpendicular
5387:
5383:
5382:
5373:
5369:
5364:
5357:
5355:
5353:
5349:
5345:
5344:
5335:
5333:
5331:
5330:
5321:
5316:
5313:
5312:
5311:
5309:
5301:
5297:
5294:
5290:
5286:
5283:
5279:
5276:
5272:
5271:
5270:
5267:
5265:
5261:
5257:
5253:
5245:
5229:
5223:
5215:
5212:
5209:
5206:
5203:
5197:
5194:
5187:
5186:
5171:
5165:
5157:
5154:
5151:
5148:
5145:
5139:
5136:
5129:
5128:
5113:
5106:
5103:
5100:
5091:
5088:
5085:
5079:
5073:
5070:
5067:
5058:
5055:
5052:
5046:
5040:
5037:
5034:
5028:
5022:
5019:
5016:
5010:
5007:
5000:
4999:
4998:
4996:
4992:
4973:
4966:
4963:
4960:
4954:
4948:
4945:
4942:
4936:
4930:
4927:
4924:
4918:
4912:
4909:
4906:
4900:
4897:
4890:
4889:
4874:
4867:
4864:
4861:
4855:
4849:
4846:
4843:
4837:
4831:
4828:
4825:
4819:
4813:
4810:
4807:
4801:
4798:
4791:
4790:
4775:
4768:
4765:
4762:
4756:
4750:
4747:
4744:
4738:
4732:
4729:
4726:
4720:
4714:
4711:
4708:
4702:
4699:
4692:
4691:
4690:
4673:
4666:
4663:
4657:
4653:
4647:
4641:
4635:
4630:
4626:
4618:
4617:
4616:
4599:
4592:
4589:
4586:
4580:
4574:
4571:
4568:
4562:
4556:
4553:
4550:
4544:
4538:
4535:
4532:
4526:
4523:
4516:
4515:
4500:
4496:
4493:
4490:
4487:
4484:
4477:
4476:
4461:
4454:
4451:
4448:
4442:
4439:
4436:
4433:
4426:
4425:
4424:
4407:
4401:
4398:
4392:
4388:
4382:
4376:
4370:
4365:
4361:
4353:
4352:
4351:
4337:
4329:
4328:wedge product
4325:
4321:
4317:
4313:
4294:
4290:
4287:
4284:
4281:
4275:
4269:
4266:
4259:
4258:
4257:
4255:
4251:
4247:
4243:
4238:
4236:
4232:
4228:
4224:
4220:
4216:
4208:
4192:
4188:
4185:
4182:
4179:
4176:
4168:
4164:
4158:
4154:
4150:
4145:
4141:
4135:
4131:
4120:
4116:
4112:
4107:
4103:
4096:
4090:
4087:
4079:
4075:
4069:
4065:
4061:
4056:
4052:
4046:
4042:
4031:
4027:
4023:
4018:
4014:
4007:
4001:
3996:
3992:
3988:
3983:
3979:
3971:
3970:
3969:
3967:
3963:
3940:
3936:
3933:
3930:
3927:
3924:
3916:
3912:
3906:
3902:
3898:
3893:
3889:
3883:
3879:
3870:
3867:
3862:
3857:
3853:
3849:
3844:
3839:
3835:
3826:
3822:
3818:
3812:
3809:
3804:
3799:
3795:
3791:
3786:
3781:
3777:
3768:
3764:
3757:
3753:
3745:
3737:
3733:
3727:
3723:
3719:
3714:
3710:
3704:
3700:
3691:
3688:
3683:
3678:
3674:
3670:
3665:
3660:
3656:
3647:
3643:
3639:
3633:
3630:
3625:
3620:
3616:
3612:
3607:
3602:
3598:
3589:
3585:
3578:
3574:
3567:
3563:
3559:
3554:
3550:
3538:
3537:
3536:
3534:
3522:
3495:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3459:
3455:
3451:
3446:
3442:
3434:
3433:
3432:
3430:
3422:
3405:
3397:
3382:
3362:
3342:
3322:
3314:
3299:
3279:
3259:
3251:
3236:
3233:
3229:
3209:
3189:
3181:
3166:
3163:
3159:
3139:
3131:
3117:
3109:
3092:
3089:
3080:
3066:
3063:
3055:
3054:perpendicular
3040:
3032:
3018:
3015:
3007:
2991:
2983:
2982:
2981:
2978:
2960:
2952:
2937:
2917:
2897:
2877:
2869:
2854:
2834:
2814:
2806:
2791:
2788:
2784:
2764:
2744:
2736:
2721:
2718:
2714:
2706:
2705:perpendicular
2691:
2671:
2663:
2648:
2645:
2641:
2621:
2613:
2598:
2595:
2591:
2583:
2567:
2559:
2545:
2524:
2521:
2512:
2498:
2490:
2473:
2470:
2461:
2460:
2459:
2457:
2441:
2421:
2409:
2404:
2388:
2385:
2382:
2374:
2370:
2366:
2361:
2357:
2348:
2343:
2335:
2332:
2325:
2324:
2323:
2306:
2301:
2295:
2290:
2287:
2280:
2275:
2269:
2259:
2245:
2237:
2214:
2211:
2199:
2196:
2193:
2190:
2187:
2183:
2180:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2128:
2127:
2126:
2109:
2104:
2098:
2093:
2090:
2083:
2078:
2072:
2062:
2054:
2031:
2028:
2020:
2017:
2013:
2010:
2007:
2004:
2000:
1994:
1988:
1985:
1978:
1977:
1976:
1962:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1913:
1897:
1889:
1883:
1875:
1859:
1854:
1850:
1846:
1837:
1829:
1806:
1803:
1799:
1794:
1789:
1785:
1777:
1776:
1775:
1758:
1750:
1746:
1724:
1704:
1701:
1689:
1686:
1681:
1676:
1672:
1664:
1663:
1662:
1659:
1656:
1652:
1628:
1614:
1606:
1587:
1584:
1570:
1562:
1546:
1539:
1533:
1526:
1521:
1515:
1510:
1506:
1500:
1496:
1492:
1489:
1485:
1477:
1472:
1468:
1464:
1459:
1455:
1448:
1445:
1440:
1436:
1432:
1425:
1424:
1423:
1421:
1420:metric tensor
1413:
1409:
1404:
1397:
1395:
1381:
1378:
1374:
1371:
1350:
1343:
1340:
1336:
1333:
1330:
1327:
1324:
1321:
1315:
1312:
1308:
1300:
1297:
1293:
1290:
1284:
1281:
1277:
1274:
1268:
1262:
1259:
1256:
1251:
1248:
1245:
1238:
1234:
1231:
1208:
1205:
1202:
1196:
1193:
1170:
1167:
1163:
1160:
1153:
1149:
1146:
1137:
1133:
1117:
1097:
1094:
1091:
1088:
1085:
1081:
1075:
1072:
1069:
1064:
1061:
1058:
1052:
1048:
1045:
1037:
1032:
1010:
999:
991:
985:
982:
971:
963:
957:
954:
944:
941:
938:
935:
932:
929:
926:
921:
913:
905:
897:
889:
883:
880:
877:
868:
865:
862:
859:
857:
844:
837:
834:
831:
825:
818:
815:
812:
809:
807:
793:
790:
787:
781:
778:
775:
769:
766:
763:
761:
753:
750:
747:
741:
730:
729:
728:
711:
685:
673:
665:
659:
656:
645:
637:
631:
628:
618:
610:
607:
604:
595:
592:
586:
583:
580:
574:
567:
566:
565:
563:
559:
555:
551:
547:
542:
540:
535:
522:
515:
511:
508:
504:
498:
494:
491:
487:
480:
476:
473:
469:
463:
459:
456:
452:
445:
442:
439:
433:
430:
427:
421:
413:
411:
407:
402:
397:
391:
381:
375:
368:
364:
360:
356:
352:
348:
344:
340:
336:
332:
327:
325:
321:
317:
309:
308:ultraparallel
304:
297:
295:
293:
289:
288:Rudolf Carnap
282:
280:
276:
258:
254:
250:
245:
241:
220:
200:
192:
191:absolute zero
186:
184:
183:
178:
174:
166:
164:
162:
159:-dimensional
158:
154:
150:
146:
141:
139:
135:
131:
127:
123:
119:
114:
110:
106:
100:
91:
89:
85:
81:
80:circular arcs
77:
73:
69:
66:in which all
65:
61:
57:
53:
45:
40:
32:
19:
18:Poincaré disk
7246:
7245:Saul Stahl,
7239:
7232:
7212:
7200:. Retrieved
7191:
7180:
7171:
7162:
7156:
7136:
7129:
7109:
7102:
7092:
7047:
7043:
7033:
7026:
7020:
6995:
6987:
6958:Pseudosphere
6923:
6916:
6910:
6906:
6895:M. C. Escher
6893:
6886:M. C. Escher
6879:M. C. Escher
6831:
6817:
6810:
6806:
6803:viewed from
6794:
6784:
6777:
6773:
6755:
6748:
6556:
6549:
6545:
6539:
6534:
6531:projectively
6521:
6353:
6349:
6347:
6184:
6180:
6178:
6101:
6090:
5939:
5862:
5769:
5629:
5625:
5623:
5496:
5492:
5490:
5478:ideal points
5475:
5464:
5437:
5423:
5402:
5379:
5377:
5367:
5352:ideal points
5341:
5339:
5327:
5325:
5306:A Euclidean
5305:
5299:
5292:
5289:orthogonally
5281:
5274:
5268:
5263:
5249:
4988:
4688:
4614:
4422:
4323:
4319:
4315:
4311:
4309:
4253:
4249:
4245:
4241:
4239:
4234:
4230:
4226:
4222:
4219:ideal points
4218:
4215:circular arc
4212:
3965:
3961:
3959:
3524:
3512:
3510:
3426:
3335:with center
3315:Draw circle
2979:
2976:
2890:with center
2870:Draw circle
2413:
2321:
2124:
1885:
1773:
1660:
1654:
1650:
1648:
1417:
1035:
1033:
1029:
700:
557:
553:
549:
545:
543:
536:
414:
409:
405:
403:
395:
389:
379:
373:
366:
362:
358:
354:
350:
346:
343:ideal points
338:
334:
328:
315:
313:
284:
278:
188:
180:
170:
156:
152:
148:
144:
142:
115:
108:
104:
92:
59:
55:
49:
7202:13 December
7163:Geometry II
7050:(1): 1–62.
6524:Klein model
5398:ideal point
5372:ideal point
5348:right angle
5336:Hypercycles
5258:, they are
4995:dot product
3056:to segment
3008:of segment
2707:to segment
2584:of segment
2456:constructed
314:Hyperbolic
118:Klein model
78:are either
7276:Categories
6979:References
6926:HyperRogue
6884:See also:
6801:projection
6085:See also:
5412:See also:
5358:Horocycles
5343:hypercycle
5300:hypercycle
4989:Using the
3312:intersect.
3182:Draw line
2867:intersect.
2737:Draw line
2664:Draw line
1649:where the
1406:Poincaré '
324:orthogonal
84:orthogonal
7074:120406828
7066:1871-2509
6827:unit disk
6711:∑
6708:−
6662:∑
6453:−
6348:A point (
6311:−
6277:−
6264:−
6230:−
6179:A point (
6057:⋅
6036:⋅
6030:−
6022:−
5997:⋅
5991:−
5917:⋅
5735:−
5722:−
5675:−
5662:−
5624:A point (
5491:A point (
5471:conformal
5381:horocycle
5282:horocycle
5213:⋅
5207:−
5155:⋅
5149:−
5104:⋅
5089:⋅
5080:−
5071:⋅
5056:⋅
5038:−
5029:⋅
5020:−
4964:∧
4955:⋅
4946:∧
4937:−
4928:−
4919:⋅
4910:−
4865:∧
4856:⋅
4847:∧
4838:−
4829:−
4820:⋅
4811:−
4766:∧
4757:⋅
4748:∧
4739:−
4730:−
4721:⋅
4712:−
4642:θ
4636:
4590:∧
4581:⋅
4572:∧
4563:−
4554:−
4545:⋅
4536:−
4494:⋅
4452:−
4443:⋅
4377:θ
4371:
4338:∧
4288:⋅
4276:θ
4270:
4151:−
4113:−
4062:−
4024:−
3899:−
3819:−
3720:−
3640:−
3292:and line
3108:inversion
2847:and line
2489:inversion
2386:−
2340:Ω
2288:−
2215:−
2194:∧
2181:−
2166:ω
2157:ω
2154:∧
2151:ω
2145:ω
2136:Ω
2091:−
2032:−
2011:−
1986:ω
1963:ω
1943:θ
1940:∧
1937:ω
1931:θ
1898:ω
1807:−
1786:θ
1743:∂
1739:∂
1705:−
1599:‖
1591:‖
1588:−
1555:‖
1544:‖
1497:∑
1493:−
1456:∑
1414:, {3,5,3}
1337:
1331:−
1325:
1278:−
1269:⋅
1260:−
1235:
1209:θ
1171:θ
1095:
1073:−
1049:
996:‖
989:‖
986:−
968:‖
961:‖
958:−
936:⋅
927:−
918:‖
911:‖
902:‖
895:‖
887:‖
881:−
875:‖
869:
826:δ
819:
782:δ
770:
715:‖
712:⋅
709:‖
670:‖
663:‖
660:−
642:‖
635:‖
632:−
615:‖
608:−
602:‖
575:δ
446:
320:geodesics
251:−
161:unit ball
88:diameters
72:unit disk
6932:See also
4318:but not
3237:′
3202:through
3167:′
3093:′
3006:midpoint
2792:′
2757:through
2722:′
2684:through
2649:′
2599:′
2582:midpoint
2525:′
2474:′
1910:that is
1375:′
1344:′
1301:′
1285:′
1164:′
1150:′
412:is then
120:and the
95:PSU(1,1)
52:geometry
6548:,
5322:Circles
3106:be the
3004:be the
2580:be the
2487:be the
1975:yields
1912:torsion
1134:of the
1130:is the
290:and of
167:History
99:SU(1,1)
7144:
7117:
7072:
7064:
7008:
6809:= −1,
6797:> 1
6424:
6418:
6255:
6249:
5839:
5833:
5702:
5699:
5693:
5561:
5555:
5467:chords
5428:models
5386:normal
5329:circle
5264:cycles
5256:sphere
5246:Cycles
4689:where
4423:where
4209:Angles
3525:v = (v
3513:u = (u
1334:artanh
1322:artanh
1118:artanh
1110:where
1092:artanh
816:arsinh
767:arcosh
701:where
399:|
387:|
383:|
371:|
74:, and
68:points
54:, the
7070:S2CID
6816:= 0,
5308:chord
5295:; and
5291:is a
4997:, as
3375:(and
2930:(and
132:with
7204:2015
7142:ISBN
7115:ISBN
7062:ISSN
7006:ISBN
6888:and
6823:= 0)
6540:For
5482:pole
5444:disk
5438:The
4248:and
4233:and
4225:and
3964:and
3523:and
3252:let
3132:let
3081:let
2984:let
2807:let
2614:let
2560:let
2513:let
2462:let
2458:by:
2434:and
1408:ball
548:and
408:and
385:and
349:and
337:and
143:The
7052:doi
6834:= 0
6829:at
6790:+ 1
6102:If
5863:If
5486:arc
5446:.
5388:or
4627:cos
4362:cos
4350:),
4322:= −
4314:= −
4310:If
4267:cos
4252:= −
4244:= −
564:by
318:or
151:or
107:, −
50:In
7278::
7179:.
7082:^
7068:.
7060:.
7042:.
7004:.
7002:45
6921:.
6792:,
6783:+
6776:=
6099:.
5462:.
5378:A
5340:A
5326:A
3529:,v
3517:,u
3395:).
2950:).
2389:1.
1394:.
1232:ln
1046:ln
866:ln
541:.
443:ln
401:.
396:qb
390:pb
380:ap
374:aq
365:,
361:,
357:,
345:,
294:.
163:.
113:.
7206:.
7150:.
7123:.
7076:.
7054::
7048:1
7014:.
6832:t
6821:2
6818:x
6814:1
6811:x
6807:t
6805:(
6795:t
6788:2
6785:x
6781:1
6778:x
6774:t
6735:.
6725:2
6720:i
6716:y
6705:1
6700:)
6694:i
6690:y
6686:2
6682:,
6676:2
6671:i
6667:y
6659:+
6656:1
6652:(
6646:=
6643:)
6638:i
6634:x
6630:,
6627:t
6624:(
6602:t
6599:+
6596:1
6590:i
6586:x
6580:=
6575:i
6571:y
6559:i
6557:y
6552:i
6550:x
6546:t
6544:(
6535:t
6500:)
6491:2
6487:)
6483:y
6480:+
6477:1
6474:(
6471:+
6466:2
6462:x
6456:1
6448:2
6444:y
6440:+
6435:2
6431:x
6421:,
6410:2
6406:)
6402:y
6399:+
6396:1
6393:(
6390:+
6385:2
6381:x
6375:x
6372:2
6365:(
6354:y
6352:,
6350:x
6331:)
6322:2
6318:)
6314:y
6308:1
6305:(
6302:+
6297:2
6293:x
6285:2
6281:y
6272:2
6268:x
6261:1
6252:,
6241:2
6237:)
6233:y
6227:1
6224:(
6221:+
6216:2
6212:x
6206:x
6203:2
6196:(
6185:y
6183:,
6181:x
6165:.
6159:1
6156:+
6153:u
6150:i
6145:i
6142:+
6139:u
6133:=
6130:s
6110:u
6066:.
6060:s
6054:s
6049:s
6045:)
6039:s
6033:s
6027:1
6019:1
6015:(
6008:=
6000:s
5994:s
5988:1
5983:+
5980:1
5976:s
5971:=
5968:u
5948:s
5926:.
5920:u
5914:u
5911:+
5908:1
5903:u
5900:2
5894:=
5891:s
5871:u
5848:y
5845:=
5842:y
5836:,
5830:x
5827:=
5824:x
5804:1
5801:=
5796:2
5792:y
5788:+
5783:2
5779:x
5754:)
5743:2
5739:y
5730:2
5726:x
5719:1
5714:+
5711:1
5707:y
5696:,
5683:2
5679:y
5670:2
5666:x
5659:1
5654:+
5651:1
5647:x
5641:(
5630:y
5628:,
5626:x
5608:)
5599:2
5595:y
5591:+
5586:2
5582:x
5578:+
5575:1
5570:y
5567:2
5558:,
5547:2
5543:y
5539:+
5534:2
5530:x
5526:+
5523:1
5518:x
5515:2
5508:(
5497:y
5495:,
5493:x
5424:P
5374:.
5302:.
5284:;
5277:;
5230:.
5224:2
5220:)
5216:t
5210:s
5204:1
5201:(
5198:=
5195:R
5172:,
5166:2
5162:)
5158:v
5152:u
5146:1
5143:(
5140:=
5137:Q
5114:.
5110:)
5107:t
5101:v
5098:(
5095:)
5092:s
5086:u
5083:(
5077:)
5074:s
5068:v
5065:(
5062:)
5059:t
5053:u
5050:(
5047:+
5044:)
5041:t
5035:s
5032:(
5026:)
5023:v
5017:u
5014:(
5011:=
5008:P
4974:.
4970:)
4967:t
4961:s
4958:(
4952:)
4949:t
4943:s
4940:(
4934:)
4931:t
4925:s
4922:(
4916:)
4913:t
4907:s
4904:(
4901:=
4898:R
4875:,
4871:)
4868:v
4862:u
4859:(
4853:)
4850:v
4844:u
4841:(
4835:)
4832:v
4826:u
4823:(
4817:)
4814:v
4808:u
4805:(
4802:=
4799:Q
4776:,
4772:)
4769:t
4763:s
4760:(
4754:)
4751:v
4745:u
4742:(
4736:)
4733:t
4727:s
4724:(
4718:)
4715:v
4709:u
4706:(
4703:=
4700:P
4674:,
4667:R
4664:Q
4658:2
4654:P
4648:=
4645:)
4639:(
4631:2
4600:.
4596:)
4593:t
4587:s
4584:(
4578:)
4575:t
4569:s
4566:(
4560:)
4557:t
4551:s
4548:(
4542:)
4539:t
4533:s
4530:(
4527:=
4524:R
4501:,
4497:u
4491:u
4488:=
4485:Q
4462:,
4458:)
4455:t
4449:s
4446:(
4440:u
4437:=
4434:P
4408:,
4402:R
4399:Q
4393:2
4389:P
4383:=
4380:)
4374:(
4366:2
4330:(
4324:s
4320:t
4316:u
4312:v
4295:.
4291:s
4285:u
4282:=
4279:)
4273:(
4254:s
4250:t
4246:u
4242:v
4235:t
4231:s
4227:v
4223:u
4193:.
4189:0
4186:=
4183:1
4180:+
4177:y
4169:1
4165:v
4159:2
4155:u
4146:2
4142:v
4136:1
4132:u
4126:)
4121:1
4117:u
4108:1
4104:v
4100:(
4097:2
4091:+
4088:x
4080:1
4076:v
4070:2
4066:u
4057:2
4053:v
4047:1
4043:u
4037:)
4032:2
4028:v
4019:2
4015:u
4011:(
4008:2
4002:+
3997:2
3993:y
3989:+
3984:2
3980:x
3966:v
3962:u
3941:.
3937:0
3934:=
3931:1
3928:+
3925:y
3917:1
3913:v
3907:2
3903:u
3894:2
3890:v
3884:1
3880:u
3874:)
3871:1
3868:+
3863:2
3858:2
3854:v
3850:+
3845:2
3840:1
3836:v
3832:(
3827:1
3823:u
3816:)
3813:1
3810:+
3805:2
3800:2
3796:u
3792:+
3787:2
3782:1
3778:u
3774:(
3769:1
3765:v
3758:+
3746:x
3738:1
3734:v
3728:2
3724:u
3715:2
3711:v
3705:1
3701:u
3695:)
3692:1
3689:+
3684:2
3679:2
3675:u
3671:+
3666:2
3661:1
3657:u
3653:(
3648:2
3644:v
3637:)
3634:1
3631:+
3626:2
3621:2
3617:v
3613:+
3608:2
3603:1
3599:v
3595:(
3590:2
3586:u
3579:+
3568:2
3564:y
3560:+
3555:2
3551:x
3533:)
3531:2
3527:1
3521:)
3519:2
3515:1
3496:,
3492:0
3489:=
3486:1
3483:+
3480:y
3477:b
3474:+
3471:x
3468:a
3465:+
3460:2
3456:y
3452:+
3447:2
3443:x
3406:c
3383:Q
3363:P
3343:C
3323:c
3300:n
3280:m
3260:C
3234:P
3230:P
3210:N
3190:n
3164:P
3160:P
3140:N
3118:P
3090:P
3067:Q
3064:P
3041:M
3019:Q
3016:P
2992:M
2961:c
2938:Q
2918:P
2898:C
2878:c
2855:n
2835:m
2815:C
2789:Q
2785:Q
2765:N
2745:n
2719:P
2715:P
2692:M
2672:m
2646:Q
2642:Q
2622:N
2596:P
2592:P
2568:M
2546:Q
2522:Q
2499:P
2471:P
2442:Q
2422:P
2383:=
2380:)
2375:2
2371:e
2367:,
2362:1
2358:e
2354:(
2349:1
2344:2
2336:=
2333:K
2307:.
2302:)
2296:0
2291:1
2281:1
2276:0
2270:(
2260:2
2254:)
2246:2
2238:l
2229:|
2224:x
2219:|
2212:1
2207:(
2200:y
2197:d
2191:x
2188:d
2184:4
2175:=
2172:0
2169:+
2163:d
2160:=
2148:+
2142:d
2139:=
2110:,
2105:)
2099:0
2094:1
2084:1
2079:0
2073:(
2063:2
2055:l
2046:|
2041:x
2036:|
2029:1
2024:)
2021:y
2018:d
2014:x
2008:x
2005:d
2001:y
1998:(
1995:2
1989:=
1934:+
1928:d
1925:=
1922:0
1860:.
1855:i
1851:x
1847:d
1838:2
1830:l
1821:|
1816:x
1811:|
1804:1
1800:2
1795:=
1790:i
1759:,
1751:i
1747:x
1732:)
1725:2
1720:|
1714:x
1709:|
1702:1
1697:(
1690:2
1687:1
1682:=
1677:i
1673:e
1655:i
1651:x
1629:2
1623:)
1615:2
1607:l
1595:x
1585:1
1580:(
1571:2
1563:l
1551:x
1547:d
1540:4
1534:=
1527:2
1522:)
1516:2
1511:i
1507:x
1501:i
1490:1
1486:(
1478:2
1473:i
1469:x
1465:d
1460:i
1449:4
1446:=
1441:2
1437:s
1433:d
1382:0
1379:=
1372:r
1351:.
1348:)
1341:r
1328:r
1319:(
1316:2
1313:=
1309:)
1298:r
1294:+
1291:1
1282:r
1275:1
1263:r
1257:1
1252:r
1249:+
1246:1
1239:(
1212:)
1206:,
1203:r
1200:(
1197:=
1194:x
1174:)
1168:,
1161:r
1157:(
1154:=
1147:x
1098:r
1089:2
1086:=
1082:)
1076:r
1070:1
1065:r
1062:+
1059:1
1053:(
1036:r
1011:.
1005:)
1000:2
992:v
983:1
980:(
977:)
972:2
964:u
955:1
952:(
945:1
942:+
939:v
933:u
930:2
922:2
914:v
906:2
898:u
890:+
884:v
878:u
863:2
860:=
845:2
841:)
838:v
835:,
832:u
829:(
813:2
810:=
800:)
797:)
794:v
791:,
788:u
785:(
779:+
776:1
773:(
764:=
757:)
754:v
751:,
748:u
745:(
742:d
686:,
679:)
674:2
666:v
657:1
654:(
651:)
646:2
638:u
629:1
626:(
619:2
611:v
605:u
596:2
593:=
590:)
587:v
584:,
581:u
578:(
558:R
554:n
550:v
546:u
523:.
516:|
512:b
509:q
505:|
499:|
495:p
492:a
488:|
481:|
477:b
474:p
470:|
464:|
460:q
457:a
453:|
440:=
437:)
434:q
431:,
428:p
425:(
422:d
410:q
406:p
367:b
363:q
359:p
355:a
351:b
347:a
339:q
335:p
259:2
255:r
246:2
242:R
221:r
201:R
157:n
153:n
149:3
111:}
109:I
105:I
103:{
46:.
20:)
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