1470:
1361:
1016:. Then the distance between two points is computed using the generalized circle through the points and perpendicular to the boundary of the subset used for the model. This generalized circle intersects the boundary at two other points. All four points are used in the
1444:
approach in quantum mechanics and quantum field theory, the scattering of acoustic waves in media (e.g. thermoclines and submarines in oceans, etc.) and the general analysis of scattering and bound states in differential equations. Here, the
1125:
829:
1682:
1986:
997:, an expression of the complex projective line. Linear fractional transformations permute these circles on the sphere, and the corresponding finite points of the generalized circles in the complex plane.
1020:
which defines the Cayley–Klein metric. Linear fractional transformations leave cross ratio invariant, so any linear fractional transformation that leaves the unit disk or upper half-planes stable is an
1608:
1768:
957:
443:
103:
914:
605:
1449:
matrix components refer to the incoming, bound and outgoing states. Perhaps the simplest example application of linear fractional transformations occurs in the analysis of the
1356:{\displaystyle (i\exp(t),\ 1){\begin{pmatrix}a&c\\b&d\end{pmatrix}}\ =\ (ai\exp(t)+b,\ ci\exp(t)+d)\thicksim ({\frac {ai\exp(t)+b}{ci\exp(t)+d}},\ 1)}
968:
334:
993:
is either a line or a circle. When completed with the point at infinity, the generalized circles in the plane correspond to circles on the surface of the
2160:
639:
2167:
2108:
2006:
1614:
325:
Linear fractional transformations are widely used in various areas of mathematics and its applications to engineering, such as classical
2030:
2054:
2175:
Gemeinsame
Behandlungsweise der elliptischen konformen, hyperbolischen konformen und parabolischen konformen Differentialgeometrie, 2
2142:
1897:
2080:
1525:
2153:
1688:
921:
2001:
2227:
1489:
1038:
2222:
2217:
2134:, volume 1, chapter 2, §15 Conformal transformations of Euclidean and Pseudo-Euclidean spaces of several dimensions,
1055:
where the linear fractional transformations are "special unitary", and for the upper half-plane the isometry group is
366:
1412:, the unstable manifold by the hyperbolic transformations, and the stable manifold by the elliptic transformations.
390:
50:
1450:
1119:
for a worked example of the fibration: in this example, the geodesics are given by the fractional linear transform
2159:
P.G. Gormley (1947) "Stereographic projection and the linear fractional group of transformations of quaternions",
877:
199:
195:
132:
1996:
1409:
1013:
176:
1799:
1064:
871:
1803:
1433:
1429:
1425:
1084:
1034:
622:
503:
2150:
Elementary Theory of a hypercomplex variable and the theory of conformal mapping in the hyperbolic plane
1816:
1454:
551:
1477:
1009:
546:
2026:
1857:
1497:
1042:
980:
867:
180:
1080:
990:
483:
303:
264:
250:
17:
2138:
2104:
2050:
1493:
1437:
1030:
608:
142:
1832:. Conformality can be confirmed by showing the generators are all conformal. The translation
2096:
2069:
1780:
1458:
1005:
381:
311:
41:
2118:
857:
does not depend on which element is selected from its equivalence class for the operation.
851:
recovers the usual expression. This linear fractional transformation is well-defined since
2186:
2135:
2114:
1405:
1108:
307:
246:
236:
207:
2189:
2093:
Geometry of Möbius transformations. Elliptic, parabolic and hyperbolic actions of SL(2,R)
2079:
Juan C. Cockburn, "Multidimensional
Realizations of Systems with Parametric Uncertainty"
1107:
induced by the linear fractional transformation decomposes complex projective space into
2202:
2182:
1509:
1485:
1421:
1100:
1088:
994:
342:
172:
2211:
2178:
1795:
1104:
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1008:
are used to represent the points. These subsets of the complex plane are provided a
2194:
2042:
1092:
1046:
1026:
338:
2174:
1095:, as they describe automorphisms of the upper half-plane under the action of the
1481:
1432:. The general procedure of combining linear fractional transformations with the
1390:
1116:
1068:
1017:
319:
33:
1469:
354:
203:
2065:
John Doyle, Andy
Packard, Kemin Zhou, "Review of LFTs, LMIs, and mu", (1991)
1112:
1001:
625:. Then linear fractional transformations act on the right of an element of
253:
of the integral domain. In this case, the invertibility condition is that
1441:
1051:
1022:
326:
29:
Möbius transformation generalized to rings other than the complex numbers
824:{\displaystyle U{\begin{pmatrix}a&c\\b&d\end{pmatrix}}=U\sim U.}
232:
963:. It has been widely studied because of its numerous applications to
1884:
resulting in a conformal map. Finally, inversion is conformal since
1846:
is a change of origin and makes no difference to angle. To see that
1677:{\displaystyle \exp(y\epsilon )=1+y\epsilon ,\quad \epsilon ^{2}=0,}
1788:
1784:
2100:
384:, then a linear fractional transformation has the familiar form
1981:{\displaystyle \exp(yb)\mapsto \exp(-yb),\quad b^{2}=1,0,-1.}
1488:
as rings that express angle and "rotation". In each case the
310:. An example of such linear fractional transformation is the
2166:
A.E. Motter & M.A.F. Rosa (1998) "Hyperbolic calculus",
1468:
1033:
explicated these models they have been named after him: the
1067:
of linear fractional transformations with real entries and
2067:
Proceedings of the 30th
Conference on Decision and Control
194:. Over a field, a linear fractional transformation is the
130:. In other words, a linear fractional transformation is a
1603:{\displaystyle \exp(yj)=\cosh y+j\sinh y,\quad j^{2}=+1,}
2027:"Linear fractional transformations in rings and modules"
1079:
Möbius transformations commonly appear in the theory of
2170:
8(1):109 to 28, §4 Conformal transformations, page 119.
1763:{\displaystyle \exp(yi)=\cos y+i\sin y,\quad i^{2}=-1.}
959:
of the linear fractional transformations is called the
1167:
952:{\displaystyle \operatorname {PGL} _{2}(\mathbb {Z} )}
666:
2045:(A. Shenitzer & M. Tretkoff, translators) (1971)
1900:
1691:
1617:
1528:
1420:
Linear fractional transformations are widely used in
1128:
924:
880:
642:
554:
393:
53:
353:
In general, a linear fractional transformation is a
2095:. London: Imperial College Press. p. xiv+192.
1424:to solve plant-controller relationship problems in
175:(in which case the transformation is also called a
1980:
1794:Linear fractional transformations are shown to be
1762:
1676:
1602:
1453:. Another elementary application is obtaining the
1355:
951:
908:
823:
599:
437:
97:
2130:B.A. Dubrovin, A.T. Fomenko, S.P. Novikov (1984)
1440:of general differential equations, including the
1000:To construct models of the hyperbolic plane the
108:The precise definition depends on the nature of
834:The ring is embedded in its projective line by
1115:appearing perpendicular to the geodesics. See
438:{\displaystyle z\mapsto {\frac {az+b}{cz+d}},}
98:{\displaystyle z\mapsto {\frac {az+b}{cz+d}}.}
8:
909:{\displaystyle \operatorname {PGL} _{2}(A).}
1099:. They also provide a canonical example of
1049:: the isometry group for the disk model is
860:The linear fractional transformations over
2132:Modern Geometry — Methods and Applications
1492:applied to the imaginary axis produces an
1951:
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60:
52:
1045:of isometries that is a subgroup of the
2018:
1273:
2161:Proceedings of the Royal Irish Academy
969:Wiles's proof of Fermat's Last Theorem
335:Wiles's proof of Fermat's Last Theorem
314:, which was originally defined on the
183:. The invertibility condition is then
2168:Advances in Applied Clifford Algebras
7:
2007:H-infinity methods in control theory
140:whose numerator and denominator are
2179:Proceedings of the Imperial Academy
2031:Linear Algebra and its Applications
179:), or more generally elements of a
600:{\displaystyle (z,t)\sim (uz,ut).}
235:(or, more generally, belong to an
25:
2047:Topics in Complex Function Theory
1436:allows them to be applied to the
278:In the most general setting, the
18:Linear fractional transformations
967:, which include, in particular,
333:(they are used, for example, in
38:linear fractional transformation
2049:, volume 2, Wiley-Interscience
1946:
1740:
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2154:University of British Columbia
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1:
2002:Linear-fractional programming
1109:stable and unstable manifolds
1872:. In each case the angle of
1791:according to the host ring.
621:, where the brackets denote
611:in the projective line over
2199:Complex Numbers in Geometry
2091:Kisil, Vladimir V. (2012).
1856:is conformal, consider the
367:projective line over a ring
149:In the most basic setting,
44:transformation of the form
2244:
2173:Tsurusaburo Takasu (1941)
1798:by consideration of their
1451:damped harmonic oscillator
978:
975:Use in hyperbolic geometry
515:In a non-commutative ring
275:in the case of integers).
1476:The commutative rings of
1410:parabolic transformations
1404:. Roughly speaking, the
1075:Use in higher mathematics
1039:Poincaré half-plane model
200:projective transformation
136:that is represented by a
40:is, roughly speaking, an
2181:17(8): 330–8, link from
1997:Laguerre transformations
1804:multiplicative inversion
1025:of the hyperbolic plane
1065:projective linear group
872:projective linear group
267:of the domain (that is
2201:, page 130 & 157,
1982:
1817:affine transformations
1764:
1678:
1604:
1473:
1434:Redheffer star product
1430:electrical engineering
1357:
1085:analytic number theory
953:
910:
825:
623:projective coordinates
601:
504:multiplicative inverse
439:
99:
2163:, Section A 51:67–85.
1983:
1765:
1679:
1605:
1478:split-complex numbers
1472:
1455:Frobenius normal form
1416:Use in control theory
1358:
954:
911:
826:
602:
440:
249:(or to belong to the
177:Möbius transformation
100:
1898:
1878:is added to that of
1689:
1615:
1526:
1498:one-parameter groups
1408:is generated by the
1126:
922:
878:
640:
552:
547:equivalence relation
391:
245:is supposed to be a
51:
2228:Projective geometry
2152:, Master's thesis,
2148:Geoffry Fox (1949)
2025:N. J. Young (1984)
1858:polar decomposition
1081:continued fractions
1041:. Each model has a
1035:Poincaré disk model
1014:Cayley–Klein metric
981:Hyperbolic geometry
2223:Conformal mappings
2218:Rational functions
1978:
1760:
1674:
1600:
1484:join the ordinary
1474:
1465:Conformal property
1353:
1192:
991:generalized circle
949:
906:
821:
691:
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435:
349:General definition
302:are elements of a
251:field of fractions
198:to the field of a
95:
2110:978-1-84816-858-9
1461:of a polynomial.
1438:scattering theory
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1339:
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1200:
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811:
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609:equivalence class
430:
90:
16:(Redirected from
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1406:center manifold
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1246:
1243:
1237:
1234:
1231:
1225:
1219:
1216:
1213:
1210:
1201:
1193:
1187:
1182:
1175:
1170:
1164:
1156:
1150:
1144:
1138:
1135:
1132:
1122:
1121:
1120:
1118:
1114:
1110:
1106:
1105:geodesic flow
1102:
1098:
1097:modular group
1094:
1093:modular forms
1090:
1086:
1082:
1074:
1072:
1070:
1066:
1060:
1053:
1052:SU(1, 1)
1048:
1044:
1040:
1036:
1032:
1028:
1024:
1019:
1015:
1011:
1007:
1003:
998:
996:
992:
988:
987:complex plane
982:
974:
972:
970:
966:
965:number theory
962:
961:modular group
935:
930:
926:
916:
903:
897:
891:
886:
882:
873:
869:
864:
858:
855:
848:
842:
838:
818:
812:
806:
800:
797:
794:
791:
788:
780:
777:
769:
766:
763:
760:
757:
748:
745:
739:
736:
733:
730:
727:
721:
718:
715:
712:
709:
706:
700:
697:
692:
686:
681:
674:
669:
663:
655:
652:
649:
643:
636:
635:
634:
630:
624:
619:
614:
610:
594:
588:
585:
582:
579:
576:
570:
564:
561:
558:
548:
545:determine an
543:
537:
530:
526:
519:
513:
510:
505:
500:
496:
490:
485:
480:
476:
470:
464:
460:
456:
452:
432:
426:
423:
420:
417:
412:
409:
406:
403:
394:
387:
386:
385:
383:
378:
372:
368:
362:
356:
348:
346:
344:
340:
336:
332:
331:number theory
328:
323:
321:
313:
309:
305:
300:
294:
290:
286:
282:
276:
266:
261:
257:
252:
248:
243:
238:
234:
229:
225:
221:
217:
211:
209:
205:
201:
197:
191:
187:
182:
178:
174:
165:
161:
157:
153:
147:
145:
144:
139:
135:
134:
124:
120:
116:
112:
92:
86:
83:
80:
77:
72:
69:
66:
63:
54:
47:
46:
45:
43:
39:
35:
27:
19:
2198:
2195:Isaak Yaglom
2149:
2131:
2101:10.1142/p835
2092:
2086:
2075:
2066:
2061:
2046:
2043:C. L. Siegel
2038:
2021:
1890:
1886:
1880:
1874:
1868:
1862:
1852:
1848:
1842:
1838:
1834:
1828:
1824:
1820:
1811:
1807:
1793:
1775:
1773:The "angle"
1772:
1514:
1503:
1482:dual numbers
1475:
1419:
1399:
1395:
1391:real numbers
1386:
1380:
1374:
1368:
1365:
1103:, where the
1078:
1058:
1047:Mobius group
1027:metric space
999:
984:
917:
862:
859:
853:
846:
840:
836:
833:
628:
617:
612:
541:
539:, the units
535:
528:
524:
517:
514:
508:
498:
494:
488:
478:
474:
468:
462:
458:
454:
450:
447:
376:
370:
360:
352:
339:group theory
324:
298:
292:
288:
284:
280:
277:
259:
255:
241:
227:
223:
219:
215:
212:
189:
185:
163:
159:
155:
151:
148:
141:
137:
131:
122:
118:
114:
110:
107:
37:
31:
26:
1508:and in the
1494:isomorphism
1457:, i.e. the
1117:Anosov flow
1111:, with the
1069:determinant
1018:cross ratio
615:is written
320:matrix ring
196:restriction
34:mathematics
2212:Categories
2013:References
1800:generators
1426:mechanical
1113:horocycles
918:The group
472:such that
355:homography
306:, such as
263:must be a
204:homography
42:invertible
2033:56:251–90
1973:−
1932:−
1926:
1920:↦
1905:
1755:−
1732:
1717:
1696:
1657:ϵ
1649:ϵ
1631:ϵ
1622:
1569:
1554:
1533:
1321:
1292:
1274:∼
1253:
1220:
1139:
1083:, and in
1012:with the
1002:unit disk
936:
892:
778:−
746:∼
571:∼
492:(that is
398:↦
58:↦
1991:See also
1496:between
1442:S-matrix
1037:and the
1029:. Since
1023:isometry
1004:and the
874:denoted
327:geometry
138:fraction
2197:(1968)
2119:2977041
1393:, with
1057:PSL(2,
985:In the
866:form a
521:, with
374:. When
233:integer
206:of the
2141:
2117:
2107:
2053:
1894:sends
1345:
1241:
1205:
1199:
1154:
1010:metric
870:, the
810:
725:
502:has a
448:where
365:, the
167:, and
143:linear
126:, and
2190:14282
1787:, or
1785:slope
1517:, × )
1506:, + )
1447:3 × 3
1366:with
1043:group
868:group
844:, so
482:is a
380:is a
318:real
316:3 × 3
213:When
181:field
2139:ISBN
2105:ISBN
2051:ISBN
1889:→ 1/
1866:and
1815:and
1810:→ 1/
1566:sinh
1551:cosh
1480:and
1428:and
1384:and
1091:and
1063:, a
484:unit
304:ring
296:and
265:unit
231:are
171:are
36:, a
2097:doi
1923:exp
1902:exp
1860:of
1779:is
1729:sin
1714:cos
1693:exp
1619:exp
1530:exp
1500:in
1402:= 1
1318:exp
1289:exp
1250:exp
1217:exp
1136:exp
1087:of
927:PGL
883:PGL
849:= 1
607:An
533:in
506:in
486:of
357:of
337:),
271:or
239:),
202:or
192:≠ 0
32:In
2214::
2187:MR
2185:,
2177:,
2115:MR
2113:.
2103:.
2029:,
1976:1.
1853:az
1851:→
1841:+
1837:→
1827:+
1825:az
1823:→
1802::
1783:,
1758:1.
1519::
1400:bc
1398:−
1396:ad
1378:,
1372:,
989:a
971:.
839:→
633::
627:P(
527:,
512:)
499:bc
497:–
495:ad
479:bc
477:–
475:ad
461:,
457:,
453:,
359:P(
345:.
341:,
329:,
322:.
291:,
287:,
283:,
273:−1
260:bc
258:–
256:ad
226:,
222:,
218:,
210:.
190:bc
188:–
186:ad
162:,
158:,
154:,
146:.
121:,
117:,
113:,
2156:.
2145:.
2121:.
2099::
1970:,
1967:0
1964:,
1961:1
1958:=
1953:2
1949:b
1944:,
1941:)
1938:b
1935:y
1929:(
1917:)
1914:b
1911:y
1908:(
1891:z
1887:z
1881:z
1875:a
1869:z
1863:a
1849:z
1843:b
1839:z
1835:z
1829:b
1821:z
1812:z
1808:z
1776:y
1752:=
1747:2
1743:i
1738:,
1735:y
1726:i
1723:+
1720:y
1711:=
1708:)
1705:i
1702:y
1699:(
1672:,
1669:0
1666:=
1661:2
1652:,
1646:y
1643:+
1640:1
1637:=
1634:)
1628:y
1625:(
1598:,
1595:1
1592:+
1589:=
1584:2
1580:j
1575:,
1572:y
1563:j
1560:+
1557:y
1548:=
1545:)
1542:j
1539:y
1536:(
1515:U
1513:(
1504:A
1502:(
1387:d
1381:c
1375:b
1369:a
1351:)
1348:1
1342:,
1336:d
1333:+
1330:)
1327:t
1324:(
1315:i
1312:c
1307:b
1304:+
1301:)
1298:t
1295:(
1286:i
1283:a
1277:(
1271:)
1268:d
1265:+
1262:)
1259:t
1256:(
1247:i
1244:c
1238:,
1235:b
1232:+
1229:)
1226:t
1223:(
1214:i
1211:a
1208:(
1202:=
1194:)
1188:d
1183:b
1176:c
1171:a
1165:(
1160:)
1157:1
1151:,
1148:)
1145:t
1142:(
1133:i
1130:(
1061:)
1059:R
947:)
943:Z
939:(
931:2
904:.
901:)
898:A
895:(
887:2
863:A
854:U
847:t
841:U
837:z
819:.
816:]
813:1
807::
804:)
801:b
798:t
795:+
792:a
789:z
786:(
781:1
774:)
770:d
767:t
764:+
761:c
758:z
755:(
752:[
749:U
743:]
740:d
737:t
734:+
731:c
728:z
722::
719:b
716:t
713:+
710:a
707:z
704:[
701:U
698:=
693:)
687:d
682:b
675:c
670:a
664:(
659:]
656:t
653::
650:z
647:[
644:U
631:)
629:A
618:U
613:A
595:.
592:)
589:t
586:u
583:,
580:z
577:u
574:(
568:)
565:t
562:,
559:z
556:(
542:u
536:A
531:)
529:t
525:z
523:(
518:A
509:A
489:A
469:A
463:d
459:c
455:b
451:a
433:,
427:d
424:+
421:z
418:c
413:b
410:+
407:z
404:a
395:z
377:A
371:A
363:)
361:A
299:z
293:d
289:c
285:b
281:a
269:1
242:z
228:d
224:c
220:b
216:a
169:z
164:d
160:c
156:b
152:a
128:z
123:d
119:c
115:b
111:a
93:.
87:d
84:+
81:z
78:c
73:b
70:+
67:z
64:a
55:z
20:)
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