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2217:." It also describes Klein disc model distances there. This is inconsistent with the metric described in the "Metric and curvature" section. I suspect that the "Distance" section was accidentally copied from another article and never updated for the model described in this article.
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I think that this article needs some clarification. It is too abstract and short for a real understanding. If only the specialists understand it, then it is not very useful. As a physicist, I would prefer that this work be done by a mathematician rather than by me ...
218:
Sometimes the
Poincaré model is called the Poincaré ball model or the conformal ball model. Perhaps the Poincaré ball model is a better name than Poincaré disk model? The name "ball model" clearly shows that the article also treats the higher-dimensional cases.
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about the circle that represents the geodesic. Rotations and translations can be represented as a combination of two reflections about different geodesics. In the case of rotations, the two geodesics intersect, while in the case of translations, they do not.
2177:{\displaystyle {\frac {(1+2\mathbf {v} \cdot \mathbf {x} +\left|\mathbf {x} \right|^{2})\mathbf {v} +(1-\left|\mathbf {v} \right|^{2})\mathbf {x} }{1+2\mathbf {v} \cdot \mathbf {x} +\left|\mathbf {v} \right|^{2}\left|\mathbf {x} \right|^{2}}}.}
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I went ahead an removed the merge tags. It still seems like there's a significant amount of overlap between the two, however, and I'm sure that each could benefit from incorporating some of that material from the other. For example, the
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article is by no means more general; in a lot of ways less general. It focuses on the hyperbolic plane, and on complex analysis. Merging the two would be difficult, and in general a bad idea, I think.
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article seems to be more general, and includes the disk model in its discussion. This article has a lot more details on that specific model, but nothing that wouldn't fit into the larger setting. --
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page simply says that the geodesics are "circular arcs whose endpoints are orthogonal to the boundary of the disk." This article goes into lots more detail, with specific equations. --
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1158:{\displaystyle g_{22}={\frac {4\left(y_{1}^{4}+2y_{1}^{2}\left(-1+y_{2}^{2}\right)+\left(1+y_{2}^{2}\right){}^{2}\right)}{\left(1-y_{1}^{2}-y_{2}^{2}\right){}^{4}}}}
802:{\displaystyle g_{11}={\frac {4\left(y_{1}^{4}+\left(-1+y_{2}^{2}\right){}^{2}+2y_{1}^{2}\left(1+y_{2}^{2}\right)\right)}{\left(1-y_{1}^{2}-y_{2}^{2}\right){}^{4}}}}
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1880:. If we put the above coordinate transformation in it, we get the usual Poincaré metric. In fact, if one changes x3 into i*x3 one gets the coordinates
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I found the problem: the metric above is the metric of the regular surface of a hyperboloid, of which the general equation in three dimensions is
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The analog of a reflection about a line in hyperbolic space is a reflection about a geodesic, which can be represented in the model as a
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Setting a = b = c = 1 and x = x1, y = x2 we get the above metric with the coordinates of the hyperboloid (graph of the regular surface)
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578:{\displaystyle g_{ij}=\langle {\frac {\partial {\vec {w}}}{\partial y_{i}}},{\frac {\partial {\vec {w}}}{\partial y_{j}}}\rangle }
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One result of this is that if hyperbolic space is translated such that the origin in the unit
Poincaré disk is translated to
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Given Klein's geometry program, a discussion of the isometries being Mobius transformations would be nice.
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The
Poincaré metric on the other hand comes from the hyperbolic (or Minkowski) line element (or metric)
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Should the
Poincaré metric article also discuss the higher-dimensional case or what is the intention?
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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935:{\displaystyle g_{12}=g_{21}={\frac {16y_{1}y_{2}}{\left(1-y_{1}^{2}-y_{2}^{2}\right){}^{4}}}}
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1462:{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=-1}
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What is right with it? What do your variables mean? Where did you get the equation for
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model, there can be a section on higher-dimensional cases. but keep it as disk model.
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1272:{\displaystyle g_{11}=g_{22}={\frac {4}{\left(1-y_{1}^{2}-y_{2}^{2}\right){}^{2}}}}
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374:{\displaystyle {\vec {w}}=\left({\begin{array}{c}x_{1}\\x_{2}\end{array}}\right)}
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1777:{\displaystyle t=x_{3}={\frac {1+y_{1}^{2}+y_{2}^{2}}{1-y_{1}^{2}-y_{2}^{2}}}}
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In the quotation that follows, he actually describes the 3D version of it.
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I don't understand something here, may be someone can answer this:
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This also applies for higher dimensions.(end of removed text)
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464:{\displaystyle x_{i}={\frac {2y_{i}}{1-y_{1}^{2}-y_{2}^{2}}}}
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which is not the metric mentionned in the article, that is:
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There he describes a world, now known as the
Poincaré disk
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The "Distance" section says "Distances in this model are
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1985:{\displaystyle \mathbf {x} }
1963:{\displaystyle \mathbf {v} }
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2209:Distance Section Incorrect?
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1902:{\displaystyle {\vec {w}}}
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471:and calculates the metric
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1936:Isometric Transformations
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1328:So what is wrong here?
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184:06:04, 1 May 2006 (UTC)
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1915:Ulrich Utiger
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2219:— Preceding
2212:
2189:
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1947:
1939:
1931:
1911:
1786:
1571:
1474:
1471:
1365:
1362:
1345:
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1340:in terms of
1337:
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137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
258:—Preceding
112:Mathematics
103:mathematics
59:Mathematics
2286:Categories
238:WillemienH
221:Pierreback
206:Pierreback
2245:this edit
1350:JRSpriggs
585:one gets
286:today. —
2221:unsigned
260:unsigned
196:Dantheox
168:Dantheox
2196:Rgdboer
2192:SU(1,1)
288:Rgdboer
284:SU(1,1)
230:against
139:on the
30:C-class
302:Metric
36:scale.
381:with
2274:talk
2255:talk
2229:talk
2200:talk
1919:talk
1354:talk
292:talk
278:The
268:talk
242:talk
234:disk
175:The
162:The
1469:.
131:Mid
2288::
2276:)
2257:)
2231:)
2202:)
2118:⋅
2071:−
2023:⋅
1970:,
1921:)
1894:→
1850:−
1754:−
1736:−
1632:−
1614:−
1454:−
1424:−
1356:)
1348:?
1306:21
1293:12
1235:−
1217:−
1195:22
1182:11
1121:−
1103:−
1016:−
956:22
898:−
880:−
847:16
836:21
823:12
765:−
747:−
642:−
600:11
573:⟩
557:∂
549:→
540:∂
518:∂
510:→
501:∂
495:⟨
441:−
423:−
323:→
294:)
270:)
244:)
2272:(
2253:(
2227:(
2198:(
2172:.
2164:2
2159:|
2155:x
2151:|
2144:2
2139:|
2135:v
2131:|
2126:+
2122:x
2114:v
2110:2
2107:+
2104:1
2098:x
2094:)
2089:2
2084:|
2080:v
2076:|
2068:1
2065:(
2062:+
2058:v
2054:)
2049:2
2044:|
2040:x
2036:|
2031:+
2027:x
2019:v
2015:2
2012:+
2009:1
2006:(
1979:x
1957:v
1917:(
1891:w
1866:2
1861:3
1857:x
1853:d
1845:2
1840:2
1836:x
1832:d
1829:+
1824:2
1819:1
1815:x
1811:d
1808:=
1803:2
1799:s
1795:d
1767:2
1762:2
1758:y
1749:2
1744:1
1740:y
1733:1
1726:2
1721:2
1717:y
1713:+
1708:2
1703:1
1699:y
1695:+
1692:1
1686:=
1681:3
1677:x
1673:=
1670:t
1645:2
1640:2
1636:y
1627:2
1622:1
1618:y
1611:1
1604:i
1600:y
1596:2
1590:=
1585:i
1581:x
1557:)
1549:2
1544:2
1540:x
1536:+
1531:2
1526:1
1522:x
1518:+
1515:1
1510:,
1505:2
1501:x
1497:,
1492:1
1488:x
1483:(
1457:1
1451:=
1444:2
1440:c
1434:2
1430:z
1417:2
1413:b
1407:2
1403:y
1397:+
1390:2
1386:a
1380:2
1376:x
1352:(
1346:t
1342:y
1338:x
1314:0
1311:=
1302:g
1298:=
1289:g
1262:2
1254:)
1248:2
1243:2
1239:y
1230:2
1225:1
1221:y
1214:1
1210:(
1205:4
1200:=
1191:g
1187:=
1178:g
1148:4
1140:)
1134:2
1129:2
1125:y
1116:2
1111:1
1107:y
1100:1
1096:(
1089:)
1083:2
1075:)
1069:2
1064:2
1060:y
1056:+
1053:1
1049:(
1045:+
1041:)
1035:2
1030:2
1026:y
1022:+
1019:1
1012:(
1006:2
1001:1
997:y
993:2
990:+
985:4
980:1
976:y
971:(
967:4
961:=
952:g
925:4
917:)
911:2
906:2
902:y
893:2
888:1
884:y
877:1
873:(
865:2
861:y
855:1
851:y
841:=
832:g
828:=
819:g
792:4
784:)
778:2
773:2
769:y
760:2
755:1
751:y
744:1
740:(
733:)
728:)
722:2
717:2
713:y
709:+
706:1
702:(
696:2
691:1
687:y
683:2
680:+
675:2
667:)
661:2
656:2
652:y
648:+
645:1
638:(
634:+
629:4
624:1
620:y
615:(
611:4
605:=
596:g
565:j
561:y
546:w
534:,
526:i
522:y
507:w
492:=
487:j
484:i
480:g
454:2
449:2
445:y
436:2
431:1
427:y
420:1
413:i
409:y
405:2
399:=
394:i
390:x
368:)
359:2
355:x
345:1
341:x
333:(
329:=
320:w
290:(
266:(
240:(
143:.
42::
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