2258:
366:
768:
2722:
1806:
20:
2253:{\displaystyle {\begin{matrix}-c^{2}t^{2}+x^{2}=-c^{2}t^{\prime 2}+x^{\prime 2}\\\hline {\begin{aligned}ct'&=ct\gamma -x\beta \gamma &&=ct\cosh \eta -x\sinh \eta \\x'&=-ct\beta \gamma +x\gamma &&=-ct\sinh \eta +x\cosh \eta \end{aligned}}\\\hline u=ct+x,\ v=ct-x,\ k={\sqrt {\tfrac {1+\beta }{1-\beta }}}=e^{\eta }\\u'={\frac {u}{k}},\ v'=kv\\\hline u'v'=uv\end{matrix}}}
783:
Spacetime geometry is conventionally developed as follows: Select (0,0) for a "here and now" in a spacetime. Light radiant left and right through this central event tracks two lines in the spacetime, lines that can be used to give coordinates to events away from (0,0). Trajectories of lesser velocity
1107:
we suggest an alternative formulation to account for the corner-like geometry, based on the use of hyperbolic coordinates, which allows substantial analytical progress towards determination of the flow in a
Plateau border and attached liquid threads. We consider a region of flow forming an angle of
391:
in the terms of the day: "From a Square and an infinite company of
Oblongs on a Superficies, each Equal to that square, how a curve is begotten which shall have the same properties or affections of any Hyperbola inscribed within a Right Angled Cone."
1793:
1521:
1681:
1079:
345:, but shares a property of invariance with it: whereas circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping. Both circular and hyperbolic angle generate
231:
1900:
1116:
Stocker and Hosoi then recall
Moffatt's consideration of "flow in a corner between rigid boundaries, induced by an arbitrary disturbance at a large distance." According to Stocker and Hosoi,
991:
600:
2426:
137:
1617:
520:
1726:
Sophus Lie observed that the SGE is invariant under
Lorentz transformations. In asymptotic coordinates, which correspond to light cone coordinates, a Lorentz transformation is
1556:
1316:
1649) As area measured against the asymptote increases in arithmetic progression, the projections upon the asymptote increase in geometric sequence. Thus the areas form
525:
and corresponds geometrically to preserving hyperbolae. The perspective of the group of squeeze mappings as hyperbolic rotation is analogous to interpreting the group
1576:
1120:
For a free fluid in a square corner, Moffatt's (antisymmetric) stream function ... that hyperbolic coordinates are indeed the natural choice to describe these flows.
1428:
1306:, one triangle being {(0,0), (0,1), (1,1)}, shows the hyperbolic sector area is equal to the area along the asymptote. The theorem then follows from the lemma.
1619:
is a solution to the Sine-Gordon equation, then the following squeeze mapping (now known as Lie transform) indicates other solutions of that equation:
2567:
1729:
2469:, equation 29.5 on page 45 of the 1969 edition, or equation 2.17 on page 37 of the 1977 edition, or equation 2.16 on page 52 of the 2001 edition
2343:
2320:
2408:
1455:
1625:
2479:
2340:
2378:(1912) "The space-time manifold of relativity. The non-Euclidean geometry of mechanics and electromagnetics", Proceedings of the
907:
658:" in the hyperbolic case (as compared with the circular case) is necessary to specify the identity component because the group
1002:
337:
that are permuted by squeeze mappings while preserving their area. The area of a hyperbolic sector is taken as a measure of a
2379:
158:
2478:
Daesoo Han, Young Suh Kim & Marilyn E. Noz (1997) "Jones-matrix formalism as a representation of the
Lorentz group",
689:
only has 1. The fact that the squeeze transforms preserve area and orientation corresponds to the inclusion of subgroups
411:
of their squeeze mappings is the squeeze mapping of their product. Therefore, the collection of squeeze mappings forms a
800:
which corresponds to the pair of light lines. Formally, a squeeze preserves the hyperbolic metric expressed in the form
923:
560:
666:
797:
423:. An additive view of this group arises from consideration of hyperbolic sectors and their hyperbolic angles.
306:
The squeeze mapping sets the stage for development of the concept of logarithms. The problem of finding the
2757:
81:
2581:
Lie, S. (1881) . "Selbstanzeige: Über Flächen, deren Krümmungsradien durch eine
Relation verknüpft sind".
1581:
470:
322:
2747:
2356:
2292:
439:
2525:
2496:
1698:
in 1879.) Such transformations of pseudospherical surfaces were discussed in detail in the lectures on
1313:
326:
2726:
1336:
1262:
911:
1691:
1129:
The area-preserving property of squeeze mapping has an application in setting the foundation of the
2742:
2674:
1717:
It is known that the Lie transforms (or squeeze mappings) correspond to
Lorentz boosts in terms of
1711:
1694:
in 1883) can be seen as the combination of a Lie transform with a
Bianchi transform (introduced by
776:
724:
318:
2653:
824:
in his textbook on relativity, who used it in his demonstration of their characteristic property.
709:
2752:
1529:
1442:
1130:
900:
1718:
1401:
59:
2632:
2427:
On bodies that are to be designated as "rigid" from the standpoint of the relativity principle
1686:
Lie (1883) noticed its relation to two other transformations of pseudospherical surfaces: The
879:
applied to an initial fluid state produces a flow with bifurcation left and right of the axis
2561:
1699:
1561:
1363:
1332:
420:
2542:
1687:
2705:
2438:
2371:
1446:
1434:
1137:
805:
793:
701:
416:
408:
717:
8:
2668:
852:
412:
350:
2647:
2442:
2423:"Über den vom Standpunkt des Relativitätsprinzips aus als starr zu bezeichnenden Körper"
1811:
2324:
884:
856:
788:). Any such velocity can be viewed as a zero velocity under a squeeze mapping called a
435:
354:
2626:
2336:
2297:
1429:
History of
Lorentz transformations § Lorentz transformation via squeeze mappings
1151:
1133:
896:
744:
370:
346:
341:
associated with the sector. The hyperbolic angle concept is quite independent of the
334:
330:
2612:
2592:
365:
2462:
2446:
2375:
2268:
821:
732:
713:
530:
338:
1096:
to a hyperbola, with the rectangular case of the squeeze mapping corresponding to
859:
of a flow running up against an immovable wall. Representing the wall by the axis
779:
with respect to hyperbola (B) is preserved by squeeze mapping in the right diagram
813:
462:
427:
388:
51:
274:
and the points of the image of the squeeze mapping are on the same hyperbola as
2404:
1703:
848:
836:
812:. Furthermore, the squeeze mapping form of Lorentz transformations was used by
809:
443:
31:
2316:
1331:), one may ask "When is the hyperbolic angle equal to one?" The answer is the
291:
2736:
2450:
1707:
1695:
832:
817:
789:
772:
767:
63:
1323:
For instance, for a standard position angle which runs from (1, 1) to (
2508:
2360:
1788:{\displaystyle (x,t)\mapsto \left({\tfrac {1}{\lambda }}x,\lambda t\right)}
384:
2686:
705:
2721:
2524:
H.K. Moffatt (1964) "Viscous and resistive eddies near a sharp corner",
917:
In 1989 Ottino described the "linear isochoric two-dimensional flow" as
747:, the angle measure of sectors is preserved. Thus squeeze mappings are
1438:
743:. Since squeeze mappings preserve areas of transformed regions such as
286:
is. For this reason it is natural to think of the squeeze mapping as a
43:
1103:
Stocker and Hosoi described their approach to corner flow as follows:
376:
237:
2422:
1388:
corresponds to the asymptotic index achieved with each sum of areas
333:
function, a new concept. Some insight into logarithms comes through
2605:
2280:
1516:{\displaystyle {\frac {d^{2}\Theta }{ds\ d\sigma }}=K\sin \Theta ,}
19:
2540:
1089:
996:
where K lies in the interval . The streamlines follow the curves
353:, which take hyperbolic angle as argument, perform the role that
1676:{\displaystyle \Theta =f\left(ms,\ {\frac {\sigma }{m}}\right).}
759:
Here some applications are summarized with historic references.
1558:
are asymptotic coordinates of two principal tangent curves and
808:
was noted in 1912 by Wilson and Lewis, by Werner Greub, and by
696:
2670:
A treatise on the differential geometry of curves and surfaces
2628:
Leçons sur la théorie générale des surfaces. Troisième partie
1437:'s (1867) investigations on surfaces of constant curvatures,
342:
1296:
Proof: An argument adding and subtracting triangles of area
804:; in a different coordinate system. This application in the
605:
are not allowed, though they preserve the form (in terms of
2693:, Chapter 4 Transformations, A genealogy of transformation.
1112:/2 and delimited on the left and bottom by symmetry planes.
892:
307:
47:
2357:
Logarithmotechnia: the making of numbers called logarithms
2341:
On the introduction of the notion of hyperbolic functions
349:
but with respect to different transformation groups. The
1184:, then there is a squeeze mapping that moves the sector(
1074:{\displaystyle x_{2}^{2}-Kx_{1}^{2}=\mathrm {constant} }
554:" notation corresponds to the fact that the reflections
2493:
The Kinematics of Mixing: stretching, chaos, transport
2321:
Introduction Geometrique à quelques Théories Physiques
2128:
1757:
831:
was used in this context in an article connecting the
1809:
1732:
1628:
1584:
1564:
1532:
1458:
1005:
926:
563:
473:
161:
84:
226:{\displaystyle \{(u,v)\,:\,uv=\mathrm {constant} \}}
16:
Linear mapping permuting rectangles of the same area
2706:"The non-Euclidean style of Minkowskian relativity"
2252:
1787:
1675:
1611:
1570:
1550:
1515:
1277:= 1 against the asymptote has equal areas between
1073:
985:
594:
514:
225:
131:
875:is time, then the squeeze mapping with parameter
704:of transforms preserving area and orientation (a
700:– of the subgroup of hyperbolic rotations in the
301:
2734:
2666:
2420:
2407:(1985) "Transformations in Special Relativity",
1721:, as pointed out by Terng and Uhlenbeck (2000):
1362:) which subtends a sector also of area one. The
986:{\displaystyle v_{1}=Gx_{2}\quad v_{2}=KGx_{1}}
906:For another approach to a flow with hyperbolic
775:is preserved by rotation in the left diagram;
2703:
2346:1(6):155–9,particularly equation 12, page 159
2344:Bulletin of the American Mathematical Society
751:in the sense of preserving hyperbolic angle.
595:{\displaystyle u\mapsto -u,\quad v\mapsto -v}
2566:: CS1 maint: multiple names: authors list (
2409:International Journal of Theoretical Physics
1445:from a known one. Such surfaces satisfy the
1124:
455:. This is equivalent to preserving the form
220:
162:
2715:. Oxford University Press. pp. 91–127.
2713:The Symbolic Universe: Geometry and Physics
2696:P. S. Modenov and A. S. Parkhomenko (1965)
2645:
2624:
2613:Lie's collected papers, Vol. 3, pp. 556–560
2593:Lie's collected papers, Vol. 3, pp. 392–393
2511:(2004) "Corner flow in free liquid films",
1578:their respective angle. Lie showed that if
912:Potential flow § Power laws with n = 2
2541:Terng, C. L., & Uhlenbeck, K. (2000).
2480:Journal of the Optical Society of America
1346:= e moves the unit angle to one between (
784:track closer to the original timeline (0,
762:
529:(the connected component of the definite
508:
184:
180:
2536:
2534:
766:
364:
18:
2604:
2580:
2395:, Springer-Verlag. See pages 272 to 274
2323:, page 29, Gauthier-Villars, link from
1273:, then the quadrature of the hyperbola
891:when time is run backward. Indeed, the
792:. This insight follows from a study of
387:, the squeeze mapping was described by
357:play with the circular angle argument.
2735:
2382:48:387–507, footnote p. 401
712:, the squeeze transformations are the
2673:. Boston: Ginn and Company. pp.
2531:
2380:American Academy of Arts and Sciences
2267:corresponds to the Doppler factor in
851:one of the fundamental motions of an
132:{\displaystyle (x,y)\mapsto (ax,y/a)}
2631:. Paris: Gauthier-Villars. pp.
1800:This can be represented as follows:
1612:{\displaystyle \Theta =f(s,\sigma )}
515:{\displaystyle x=u+v,\quad y=u-v\,,}
442:of 2×2 real matrices preserving the
2700:, volume one. See pages 104 to 106.
430:, the group of squeeze mappings is
369:A squeeze mapping moves one purple
13:
2649:Lezioni di geometria differenziale
2513:Journal of Engineering Mathematics
1887:
1871:
1629:
1585:
1565:
1507:
1472:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
216:
213:
210:
207:
204:
201:
198:
195:
14:
2769:
2652:. Pisa: Enrico Spoerri. pp.
1441:(1879) found a way to derive new
69:For a fixed positive real number
2720:
1422:
375:It also squeezes blue and green
310:bounded by a hyperbola (such as
2660:
2639:
2618:
2598:
2574:
2518:
2501:
2485:
1154:obtained with central rays to (
953:
754:
579:
492:
407:are positive real numbers, the
373:to another with the same area.
360:
2725:Learning materials related to
2472:
2456:
2414:
2398:
2385:
2365:
2349:
2330:
2310:
1748:
1745:
1733:
1606:
1594:
1545:
1533:
842:
583:
567:
426:From the point of view of the
383:In 1688, long before abstract
302:Logarithm and hyperbolic angle
177:
165:
126:
103:
100:
97:
85:
1:
2425:[Wikisource translation:
2303:
863:= 0 and taking the parameter
735:is defined using area under
533:) preserving quadratic form
7:
2583:Fortschritte der Mathematik
2293:Indefinite orthogonal group
2286:
1551:{\displaystyle (s,\sigma )}
816:(1909/10) while discussing
731:when it preserves angles.
440:indefinite orthogonal group
10:
2774:
2526:Journal of Fluid Mechanics
2497:Cambridge University Press
2421:Herglotz, Gustav (1910) ,
2327:Historical Math Monographs
1426:
1314:Alphonse Antonio de Sarasa
718:classification of elements
327:Alphonse Antonio de Sarasa
298:, which preserve circles.
2698:Geometric Transformations
2689:& SL Greitzer (1967)
2667:Eisenhart, L. P. (1909).
1400:which is a proto-typical
1263:Gregoire de Saint-Vincent
1125:Bridge to transcendentals
820:, and was popularized by
323:Grégoire de Saint-Vincent
321:. The solution, found by
294:in 1914, by analogy with
46:that preserves Euclidean
2451:10.1002/andp.19103360208
1712:Luther Pfahler Eisenhart
1443:pseudospherical surfaces
1320:of the asymptote index.
1131:transcendental functions
796:multiplications and the
777:hyperbolic orthogonality
725:geometric transformation
2355:Euclid Speidell (1688)
1571:{\displaystyle \Theta }
343:ordinary circular angle
2727:Reciprocal Eigenvalues
2717:(see page 9 of e-link)
2704:Walter, Scott (1999).
2543:"Geometry of solitons"
2254:
1789:
1719:light-cone coordinates
1692:Albert Victor Bäcklund
1677:
1613:
1572:
1552:
1517:
1402:arithmetic progression
1195:Proof: Take parameter
1075:
987:
829:squeeze transformation
780:
763:Relativistic spacetime
710:Möbius transformations
708:). In the language of
596:
516:
380:
329:in 1647, required the
227:
133:
40:squeeze transformation
27:
2255:
1790:
1700:differential geometry
1678:
1614:
1573:
1553:
1518:
1427:Further information:
1364:geometric progression
1333:transcendental number
1076:
988:
770:
693:– in this case
597:
517:
421:positive real numbers
368:
228:
134:
26:= 3/2 squeeze mapping
22:
2711:. In J. Gray (ed.).
2646:Bianchi, L. (1894).
2625:Darboux, G. (1894).
2507:Roman Stocker &
2491:J. M. Ottino (1989)
2467:Essential Relativity
2372:Edwin Bidwell Wilson
1807:
1730:
1626:
1582:
1562:
1530:
1456:
1447:Sine-Gordon equation
1435:Pierre Ossian Bonnet
1285:compared to between
1138:exponential function
1136:and its inverse the
1003:
924:
806:theory of relativity
794:split-complex number
702:special linear group
695:SO(1,1) ⊂
667:connected components
561:
471:
417:multiplicative group
351:hyperbolic functions
159:
82:
2758:Minkowski spacetime
2443:1910AnP...336..393H
2391:W. H. Greub (1967)
1041:
1020:
853:incompressible flow
714:hyperbolic elements
413:one-parameter group
288:hyperbolic rotation
2748:Conformal mappings
2691:Geometry Revisited
2550:Notices of the AMS
2431:Annalen der Physik
2325:Cornell University
2250:
2248:
2153:
2066:
1785:
1766:
1688:Bäcklund transform
1673:
1609:
1568:
1548:
1513:
1088:corresponds to an
1071:
1027:
1006:
983:
781:
745:hyperbolic sectors
685:components, while
669:, while the group
654:; the additional "
592:
545:circular rotations
512:
436:identity component
415:isomorphic to the
381:
355:circular functions
347:invariant measures
335:hyperbolic sectors
296:circular rotations
223:
129:
50:of regions in the
28:
2337:Mellen W. Haskell
2298:Isochoric process
2199:
2192:
2154:
2152:
2119:
2095:
1765:
1663:
1654:
1493:
1485:
1152:hyperbolic sector
1134:natural logarithm
903:under squeezing.
897:hyperbolic sector
889:fluid convergence
371:hyperbolic sector
331:natural logarithm
2765:
2724:
2716:
2710:
2679:
2678:
2664:
2658:
2657:
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2622:
2616:
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2578:
2572:
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2565:
2557:
2547:
2538:
2529:
2522:
2516:
2505:
2499:
2489:
2483:
2476:
2470:
2463:Wolfgang Rindler
2460:
2454:
2453:
2418:
2412:
2402:
2396:
2389:
2383:
2376:Gilbert N. Lewis
2369:
2363:
2353:
2347:
2334:
2328:
2314:
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2256:
2251:
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2155:
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2117:
2093:
2067:
2024:
1991:
1943:
1913:
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1514:
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1483:
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1471:
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1460:
1305:
1304:
1300:
1080:
1078:
1077:
1072:
1070:
1040:
1035:
1019:
1014:
992:
990:
989:
984:
982:
981:
963:
962:
952:
951:
936:
935:
822:Wolfgang Rindler
733:Hyperbolic angle
699:
692:
688:
684:
680:
676:
672:
665:
661:
657:
653:
634:
616:
610:
601:
599:
598:
593:
553:
542:
531:orthogonal group
528:
521:
519:
518:
513:
460:
454:
433:
428:classical groups
406:
400:
339:hyperbolic angle
316:
285:
273:
263:
249:
232:
230:
229:
224:
219:
151:
138:
136:
135:
130:
122:
74:
38:, also called a
2773:
2772:
2768:
2767:
2766:
2764:
2763:
2762:
2743:Affine geometry
2733:
2732:
2708:
2683:
2682:
2665:
2661:
2644:
2640:
2623:
2619:
2611:. Reprinted in
2603:
2599:
2579:
2575:
2559:
2558:
2545:
2539:
2532:
2523:
2519:
2515:50:267–88
2506:
2502:
2490:
2486:
2477:
2473:
2461:
2457:
2419:
2415:
2411:24:223–36
2403:
2399:
2390:
2386:
2370:
2366:
2354:
2350:
2335:
2331:
2315:
2311:
2306:
2289:
2247:
2246:
2229:
2221:
2218:
2217:
2200:
2184:
2173:
2170:
2169:
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2159:
2141:
2130:
2126:
2069:
2068:
2065:
2064:
2023:
1992:
1984:
1981:
1980:
1942:
1914:
1906:
1899:
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1886:
1882:
1870:
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1856:
1844:
1840:
1831:
1827:
1821:
1817:
1810:
1808:
1805:
1804:
1756:
1755:
1751:
1731:
1728:
1727:
1690:(introduced by
1655:
1642:
1638:
1627:
1624:
1623:
1583:
1580:
1579:
1563:
1560:
1559:
1531:
1528:
1527:
1476:
1466:
1462:
1461:
1459:
1457:
1454:
1453:
1431:
1425:
1342:A squeeze with
1302:
1298:
1297:
1127:
1045:
1036:
1031:
1015:
1010:
1004:
1001:
1000:
977:
973:
958:
954:
947:
943:
931:
927:
925:
922:
921:
845:
814:Gustav Herglotz
765:
757:
694:
690:
686:
682:
678:
674:
670:
663:
659:
655:
636:
618:
612:
606:
562:
559:
558:
551:
550:Note that the "
534:
526:
472:
469:
468:
463:change of basis
456:
446:
431:
402:
396:
389:Euclid Speidell
374:
363:
311:
304:
275:
265:
251:
241:
194:
160:
157:
156:
147:
146:with parameter
144:squeeze mapping
118:
83:
80:
79:
70:
52:Cartesian plane
42:, is a type of
36:squeeze mapping
17:
12:
11:
5:
2771:
2761:
2760:
2755:
2753:Linear algebra
2750:
2745:
2731:
2730:
2729:at Wikiversity
2718:
2701:
2694:
2681:
2680:
2659:
2638:
2617:
2597:
2573:
2530:
2517:
2500:
2484:
2471:
2455:
2413:
2405:Louis Kauffman
2397:
2393:Linear Algebra
2384:
2364:
2348:
2329:
2308:
2307:
2305:
2302:
2301:
2300:
2295:
2288:
2285:
2261:
2260:
2245:
2242:
2239:
2235:
2232:
2227:
2224:
2220:
2219:
2216:
2213:
2210:
2206:
2203:
2196:
2191:
2188:
2183:
2179:
2176:
2172:
2171:
2166:
2162:
2158:
2150:
2147:
2144:
2139:
2136:
2133:
2125:
2122:
2116:
2113:
2110:
2107:
2104:
2101:
2098:
2092:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2070:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1993:
1990:
1987:
1983:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1915:
1912:
1909:
1905:
1902:
1901:
1898:
1897:
1892:
1889:
1885:
1881:
1876:
1873:
1869:
1863:
1859:
1855:
1852:
1847:
1843:
1839:
1834:
1830:
1824:
1820:
1816:
1813:
1812:
1798:
1797:
1783:
1779:
1776:
1773:
1770:
1764:
1761:
1754:
1750:
1747:
1744:
1741:
1738:
1735:
1704:Gaston Darboux
1684:
1683:
1672:
1668:
1662:
1659:
1651:
1648:
1645:
1641:
1637:
1634:
1631:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1567:
1547:
1544:
1541:
1538:
1535:
1524:
1523:
1512:
1509:
1506:
1503:
1500:
1497:
1491:
1488:
1482:
1479:
1474:
1469:
1465:
1424:
1421:
1398:
1397:
1386:
1385:
1126:
1123:
1122:
1121:
1114:
1113:
1082:
1081:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1044:
1039:
1034:
1030:
1026:
1023:
1018:
1013:
1009:
994:
993:
980:
976:
972:
969:
966:
961:
957:
950:
946:
942:
939:
934:
930:
883:= 0. The same
849:fluid dynamics
844:
841:
837:Jones calculus
810:Louis Kauffman
798:diagonal basis
764:
761:
756:
753:
603:
602:
591:
588:
585:
582:
578:
575:
572:
569:
566:
523:
522:
511:
507:
504:
501:
498:
495:
491:
488:
485:
482:
479:
476:
444:quadratic form
362:
359:
303:
300:
234:
233:
222:
218:
215:
212:
209:
206:
203:
200:
197:
193:
190:
187:
183:
179:
176:
173:
170:
167:
164:
140:
139:
128:
125:
121:
117:
114:
111:
108:
105:
102:
99:
96:
93:
90:
87:
75:, the mapping
32:linear algebra
15:
9:
6:
4:
3:
2:
2770:
2759:
2756:
2754:
2751:
2749:
2746:
2744:
2741:
2740:
2738:
2728:
2723:
2719:
2714:
2707:
2702:
2699:
2695:
2692:
2688:
2685:
2684:
2676:
2672:
2671:
2663:
2655:
2651:
2650:
2642:
2634:
2630:
2629:
2621:
2614:
2608:
2601:
2594:
2591:Reprinted in
2588:
2584:
2577:
2569:
2563:
2555:
2551:
2544:
2537:
2535:
2528:18:1–18
2527:
2521:
2514:
2510:
2504:
2498:
2494:
2488:
2482:A14(9):2290–8
2481:
2475:
2468:
2464:
2459:
2452:
2448:
2444:
2440:
2436:
2432:
2428:
2424:
2417:
2410:
2406:
2401:
2394:
2388:
2381:
2377:
2373:
2368:
2362:
2358:
2352:
2345:
2342:
2338:
2333:
2326:
2322:
2318:
2313:
2309:
2299:
2296:
2294:
2291:
2290:
2284:
2282:
2278:
2274:
2272:
2266:
2243:
2240:
2237:
2233:
2230:
2225:
2222:
2214:
2211:
2208:
2204:
2201:
2194:
2189:
2186:
2181:
2177:
2174:
2164:
2160:
2156:
2148:
2145:
2142:
2137:
2134:
2131:
2123:
2120:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2026:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1994:
1988:
1985:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1945:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1916:
1910:
1907:
1903:
1890:
1883:
1879:
1874:
1867:
1861:
1857:
1853:
1850:
1845:
1841:
1837:
1832:
1828:
1822:
1818:
1814:
1803:
1802:
1801:
1796:
1781:
1777:
1774:
1771:
1768:
1762:
1759:
1752:
1742:
1739:
1736:
1724:
1723:
1722:
1720:
1715:
1713:
1709:
1708:Luigi Bianchi
1705:
1701:
1697:
1696:Luigi Bianchi
1693:
1689:
1670:
1666:
1660:
1657:
1649:
1646:
1643:
1639:
1635:
1632:
1622:
1621:
1620:
1603:
1600:
1597:
1591:
1588:
1542:
1539:
1536:
1510:
1504:
1501:
1498:
1495:
1489:
1486:
1480:
1477:
1467:
1463:
1452:
1451:
1450:
1448:
1444:
1440:
1436:
1430:
1423:Lie transform
1420:
1418:
1414:
1410:
1406:
1403:
1395:
1391:
1390:
1389:
1383:
1379:
1375:
1371:
1368:
1367:
1366:
1365:
1361:
1357:
1353:
1349:
1345:
1340:
1338:
1334:
1330:
1326:
1321:
1319:
1315:
1311:
1307:
1294:
1292:
1288:
1284:
1280:
1276:
1272:
1268:
1264:
1260:
1256:
1254:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1218:
1214:
1210:
1206:
1202:
1198:
1193:
1191:
1187:
1183:
1179:
1175:
1171:
1169:
1165:
1161:
1157:
1153:
1149:
1145:
1141:
1139:
1135:
1132:
1119:
1118:
1117:
1111:
1106:
1105:
1104:
1101:
1099:
1095:
1092:and positive
1091:
1087:
1042:
1037:
1032:
1028:
1024:
1021:
1016:
1011:
1007:
999:
998:
997:
978:
974:
970:
967:
964:
959:
955:
948:
944:
940:
937:
932:
928:
920:
919:
918:
915:
913:
909:
904:
902:
898:
894:
890:
886:
882:
878:
874:
870:
866:
862:
858:
854:
850:
840:
838:
834:
833:Lorentz group
830:
825:
823:
819:
818:Born rigidity
815:
811:
807:
803:
799:
795:
791:
790:Lorentz boost
787:
778:
774:
773:orthogonality
769:
760:
752:
750:
746:
742:
738:
734:
730:
726:
721:
719:
715:
711:
707:
703:
698:
668:
651:
647:
643:
639:
633:
629:
625:
621:
615:
609:
589:
586:
580:
576:
573:
570:
564:
557:
556:
555:
548:
546:
541:
537:
532:
509:
505:
502:
499:
496:
493:
489:
486:
483:
480:
477:
474:
467:
466:
465:
464:
459:
453:
449:
445:
441:
437:
429:
424:
422:
418:
414:
410:
405:
399:
393:
390:
386:
378:
372:
367:
358:
356:
352:
348:
344:
340:
336:
332:
328:
324:
320:
314:
309:
299:
297:
293:
289:
283:
279:
272:
268:
262:
258:
254:
248:
244:
239:
191:
188:
185:
181:
174:
171:
168:
155:
154:
153:
150:
145:
123:
119:
115:
112:
109:
106:
94:
91:
88:
78:
77:
76:
73:
67:
65:
64:shear mapping
61:
57:
53:
49:
45:
41:
37:
33:
25:
21:
2712:
2697:
2690:
2669:
2662:
2648:
2641:
2627:
2620:
2607:Christ. Forh
2606:
2600:
2586:
2582:
2576:
2562:cite journal
2553:
2549:
2520:
2512:
2503:
2492:
2487:
2474:
2466:
2458:
2434:
2430:
2416:
2400:
2392:
2387:
2367:
2361:Google Books
2351:
2332:
2312:
2276:
2270:
2264:
2262:
1799:
1725:
1716:
1685:
1525:
1432:
1416:
1412:
1408:
1404:
1399:
1393:
1392:1,2,3, ...,
1387:
1381:
1377:
1373:
1369:
1359:
1355:
1351:
1347:
1343:
1341:
1328:
1324:
1322:
1317:
1309:
1308:
1295:
1290:
1286:
1282:
1278:
1274:
1270:
1266:
1258:
1257:
1252:
1248:
1244:
1240:
1236:
1232:
1228:
1224:
1220:
1216:
1212:
1208:
1204:
1200:
1196:
1194:
1189:
1188:) to sector(
1185:
1181:
1177:
1173:
1172:
1167:
1163:
1159:
1155:
1147:
1143:
1142:
1128:
1115:
1109:
1102:
1097:
1093:
1085:
1084:so negative
1083:
995:
916:
905:
888:
880:
876:
872:
868:
864:
860:
846:
828:
826:
801:
785:
782:
758:
755:Applications
748:
740:
736:
728:
722:
677:components:
649:
645:
641:
637:
631:
627:
623:
619:
613:
607:
604:
549:
544:
539:
535:
524:
457:
451:
447:
425:
403:
397:
394:
385:group theory
382:
361:Group theory
312:
305:
295:
287:
281:
277:
270:
266:
260:
256:
252:
246:
242:
235:
148:
143:
141:
71:
68:
55:
39:
35:
29:
23:
2687:HSM Coxeter
2556:(1): 17–25.
2495:, page 29,
2317:Émile Borel
1710:(1894), or
1144:Definition:
908:streamlines
857:bifurcation
843:Corner flow
839:in optics.
706:volume form
409:composition
292:Émile Borel
2737:Categories
2589:: 529–531.
2509:A.E. Hosoi
2437:(2): 408,
2304:References
1439:Sophus Lie
1433:Following
1318:logarithms
771:Euclidean
727:is called
617:these are
377:rectangles
319:quadrature
317:is one of
44:linear map
2273:-calculus
2165:η
2149:β
2146:−
2138:β
2109:−
2062:η
2059:
2047:η
2044:
2032:−
2021:γ
2012:γ
2009:β
2000:−
1978:η
1975:
1966:−
1963:η
1960:
1940:γ
1937:β
1931:−
1928:γ
1888:′
1872:′
1854:−
1815:−
1775:λ
1763:λ
1749:↦
1658:σ
1630:Θ
1604:σ
1586:Θ
1566:Θ
1543:σ
1508:Θ
1505:
1490:σ
1473:Θ
1358:, 1/
1350:, 1/
1327:, 1/
1265:1647) If
1223:) takes (
1207:so that (
1150:) is the
1022:−
901:invariant
855:involves
827:The term
749:conformal
729:conformal
587:−
584:↦
571:−
568:↦
543:as being
503:−
290:, as did
238:hyperbola
101:↦
54:, but is
2287:See also
2281:rapidity
2234:′
2226:′
2205:′
2178:′
1989:′
1911:′
1714:(1909).
1706:(1894),
1415:= 0 and
871:) where
461:via the
152:. Since
60:rotation
2439:Bibcode
2429:],
2339:(1895)
2319:(1914)
2279:is the
1380:, ...,
1354:) and (
1310:Theorem
1301:⁄
1259:Theorem
1239:) and (
1162:) and (
1146:Sector(
1090:ellipse
895:of any
716:in the
691:SO ⊂ SL
679:SO(1,1)
438:of the
432:SO(1,1)
264:, then
142:is the
2374:&
2269:Bondi
2263:where
2198:
2118:
2094:
1653:
1526:where
1484:
1419:= 1 .
1411:where
1247:) to (
1231:) to (
1174:Lemma:
910:, see
887:gives
867:= exp(
660:O(1,1)
434:, the
2709:(PDF)
2677:–290.
2656:–434.
2635:–382.
2546:(PDF)
2359:from
1384:, ...
1211:) = (
1100:= 1.
885:model
835:with
697:SL(2)
687:SO(2)
527:SO(2)
240:, if
236:is a
2568:link
2056:cosh
2041:sinh
1972:sinh
1957:cosh
1396:,...
1335:x =
1289:and
1281:and
1251:, 1/
1243:, 1/
1235:, 1/
1227:, 1/
1166:, 1/
1158:, 1/
893:area
739:= 1/
681:has
673:has
671:O(2)
662:has
635:and
611:and
401:and
325:and
315:= 1)
308:area
250:and
48:area
34:, a
2675:289
2654:433
2633:381
2447:doi
2435:336
1702:by
1502:sin
1255:).
1209:u,v
1192:).
1190:c,d
1186:a,b
1176:If
1170:).
1148:a,b
899:is
847:In
648:↦ −
640:↦ −
419:of
395:If
62:or
56:not
30:In
2739::
2587:11
2585:.
2564:}}
2560:{{
2554:47
2552:.
2548:.
2533:^
2465:,
2445:,
2433:,
2283:.
2275:,
1449::
1409:nd
1407:+
1376:,
1372:,
1360:ee
1356:ee
1339:.
1293:.
1275:xy
1271:ad
1269:=
1267:bc
1215:,
1213:rx
1199:=
1182:ad
1180:=
1178:bc
1140::
914:.
802:xy
723:A
720:.
644:,
630:↦
626:,
622:↦
552:SO
547:.
538:+
458:xy
450:−
313:xy
271:xy
269:=
267:uv
255:=
247:ax
245:=
66:.
58:a
2615:.
2609:.
2595:.
2570:)
2449::
2441::
2277:η
2271:k
2265:k
2244:v
2241:u
2238:=
2231:v
2223:u
2215:v
2212:k
2209:=
2202:v
2195:,
2190:k
2187:u
2182:=
2175:u
2161:e
2157:=
2143:1
2135:+
2132:1
2124:=
2121:k
2115:,
2112:x
2106:t
2103:c
2100:=
2097:v
2091:,
2088:x
2085:+
2082:t
2079:c
2076:=
2073:u
2053:x
2050:+
2038:t
2035:c
2029:=
2018:x
2015:+
2006:t
2003:c
1997:=
1986:x
1969:x
1954:t
1951:c
1948:=
1934:x
1925:t
1922:c
1919:=
1908:t
1904:c
1891:2
1884:x
1880:+
1875:2
1868:t
1862:2
1858:c
1851:=
1846:2
1842:x
1838:+
1833:2
1829:t
1823:2
1819:c
1795:.
1782:)
1778:t
1772:,
1769:x
1760:1
1753:(
1746:)
1743:t
1740:,
1737:x
1734:(
1671:.
1667:)
1661:m
1650:,
1647:s
1644:m
1640:(
1636:f
1633:=
1607:)
1601:,
1598:s
1595:(
1592:f
1589:=
1546:)
1540:,
1537:s
1534:(
1511:,
1499:K
1496:=
1487:d
1481:s
1478:d
1468:2
1464:d
1417:d
1413:A
1405:A
1394:n
1382:e
1378:e
1374:e
1370:e
1352:e
1348:e
1344:r
1337:e
1329:x
1325:x
1312:(
1303:2
1299:1
1291:d
1287:c
1283:b
1279:a
1261:(
1253:d
1249:d
1245:b
1241:b
1237:c
1233:c
1229:a
1225:a
1221:r
1219:/
1217:y
1205:a
1203:/
1201:c
1197:r
1168:b
1164:b
1160:a
1156:a
1110:π
1098:K
1094:K
1086:K
1068:t
1065:n
1062:a
1059:t
1056:s
1053:n
1050:o
1047:c
1043:=
1038:2
1033:1
1029:x
1025:K
1017:2
1012:2
1008:x
979:1
975:x
971:G
968:K
965:=
960:2
956:v
949:2
945:x
941:G
938:=
933:1
929:v
881:x
877:r
873:t
869:t
865:r
861:y
786:t
741:x
737:y
683:2
675:2
664:4
656:+
652:)
650:y
646:y
642:x
638:x
632:x
628:y
624:y
620:x
614:y
608:x
590:v
581:v
577:,
574:u
565:u
540:y
536:x
510:,
506:v
500:u
497:=
494:y
490:,
487:v
484:+
481:u
478:=
475:x
452:v
448:u
404:s
398:r
379:.
284:)
282:y
280:,
278:x
276:(
261:a
259:/
257:y
253:v
243:u
221:}
217:t
214:n
211:a
208:t
205:s
202:n
199:o
196:c
192:=
189:v
186:u
182::
178:)
175:v
172:,
169:u
166:(
163:{
149:a
127:)
124:a
120:/
116:y
113:,
110:x
107:a
104:(
98:)
95:y
92:,
89:x
86:(
72:a
24:a
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