966:
corresponds to "choices of linear structure", with the identity (base point) being the existing linear structure. Given a projective space without an identification as the projectivization of a linear space, there is no natural isomorphism between the collineation group and PΓL, and the choice of a
92:
Simply, a collineation is a one-to-one map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. One may formalize this using various ways of presenting a projective space. Also, the case of the projective line
524:
Projective linear transformations (homographies) are collineations (planes in a vector space correspond to lines in the associated projective space, and linear transformations map planes to planes, so projective linear transformations map lines to lines), but in general not all collineations are
491:
571:
projective space is at least 2, then every collineation is the product of a homography (a projective linear transformation) and an automorphic collineation. More precisely, the collineation group is the
1031:
when they were interchanged by an arbitrary equation. In our particular case, linear equations between homogeneous point coordinates, Möbius called a permutation of both point spaces in particular a
1047:
when they lie on the same line. Möbius' designation can be expressed by saying, collinear points are mapped by a permutation to collinear points, or in plain speech, straight lines stay straight.
1224:
941:
853:
319:
For dimension one, the set of points lying on a single projective line defines a projective space, and the resulting notion of collineation is just any bijection of the set.
1285:
905:
875:
335:
of the points of the projective line. This is different from the behavior in higher dimensions, and thus one gives a more restrictive definition, specified so that the
312:, so in these dimensions this definition is no more general than the linear-algebraic one above, but in dimension two there are other projective planes, namely the
950:), the projective linear group is in general a proper subgroup of the collineation group, which can be thought of as "transformations preserving a projective
293:
specifying which points lie on which lines, satisfying certain axioms), a collineation between projective spaces thus defined then being a bijective function
440:
72:
from a projective space to itself. Some authors restrict the definition of collineation to the case where it is an automorphism. The
1637:
1619:
1584:
1554:
1536:
1672:
1295:
of the plane are frequently described as the collection of all homographies and anti-homographies of the complex plane.
1148:
1609:
780:
573:
1090:
field, all the collineations are homographies in the real projective plane, however due to the field automorphism of
568:
390:
1332:
Geometers still commonly use an exponential type notation for functions and this condition will often appear as
1123:
328:
1237:, and so is an example of a collineation which is not an homography. For example, geometrically, the mapping
1083:
526:
1524:
1015:
What do our geometric transformations mean now? Möbius threw out and fielded this question already in his
910:
313:
114:
1063:. Such a mapping permutes the lines of the incidence structure, and the notion of collineation persists.
967:
linear structure (realization as projectivization of a linear space) corresponds to a choice of subgroup
1079:
829:
327:
For a projective space of dimension one (a projective line; the projectivization of a vector space of
1230:
1060:
1091:
1052:
275:
125:
77:
55:
31:
1240:
888:
858:
1605:
1292:
1288:
1094:, not all collineations of the complex projective line are homographies. In applications such as
1056:
788:
784:
577:
550:
287:
39:
1633:
1615:
1580:
1550:
1532:
1115:
73:
1507:
1499:
1563:
1008:
1004:
996:
584:
305:
271:
110:
51:
1095:
992:
332:
1043:. Möbius’ expression is immediately comprehended when we follow Möbius in calling points
504:
The main examples of collineations are projective linear transformations (also known as
27:
In projective geometry, a bijection between projective spaces that preserves collinearity
691:
is semilinear, one easily checks that this map is properly defined, and furthermore, as
304:
Every projective space of dimension greater than or equal to three is isomorphic to the
1078:. A distinction between the terms arose when the distinction was clarified between the
1067:
1036:
415:
102:
1666:
1119:
309:
316:, and this definition allows one to define collineations in such projective planes.
1593:
1000:
599:
525:
projective linear transformations. The group of projective linear transformations (
106:
69:
17:
1455:
1087:
807:
607:
156:
64:
1656:
1234:
562:
516:
states that all collineations are a combination of these, as described below.
505:
43:
1549:, Student mathematical library, vol. 9, American Mathematical Society,
1127:
1044:
59:
47:
496:
This last requirement ensures that collineations are all semilinear maps.
530:
331:
two), all points are collinear, so the collineation group is exactly the
1652:
972:
710:
The fundamental theorem of projective geometry states the converse:
486:{\displaystyle \alpha (\langle v\rangle )=\langle \beta (v)\rangle }
1011:
that first abstracted this essence of geometrical transformation:
109:), a collineation is a map between the projective spaces that is
54:
to another, or from a projective space to itself, such that the
513:
336:
1059:
consisting of mappings of the underlying space that preserve
301:
between the set of lines, preserving the incidence relation.
1086:. Since there are no non-trivial field automorphisms of the
563:
Homography § Fundamental theorem of projective geometry
62:
points are themselves collinear. A collineation is thus an
695:
is not singular, it is bijective. It is obvious now that
1373:"Preserving the incidence relation" means that if point
954:-linear structure". Correspondingly, the quotient group
512:. For projective spaces coming from a linear space, the
757:
have the same dimension, and there is a semilinear map
1529:
Projective
Geometry / From Foundations to Applications
1312:
1243:
1151:
1098:
where the underlying field is the real number field,
1027:, when he said two elements drawn from a domain were
913:
891:
861:
832:
443:
93:
is special, and hence generally treated differently.
297:
between the sets of points and a bijective function
1279:
1219:{\displaystyle f(z)={\frac {az^{*}+b}{cz^{*}+d}}.}
1218:
935:
899:
869:
847:
485:
76:of all collineations of a space to itself form a
1051:Contemporary mathematicians view geometry as an
1003:(points lying on a single line). According to
1035:. This signification would be changed later by
580:of homographies by automorphic collineations.
509:
1503:
8:
1483:
480:
465:
456:
450:
101:For a projective space defined in terms of
557:Fundamental theorem of projective geometry
514:fundamental theorem of projective geometry
337:fundamental theorem of projective geometry
1271:
1262:
1242:
1198:
1177:
1167:
1150:
925:
920:
916:
915:
912:
893:
892:
890:
863:
862:
860:
839:
835:
834:
831:
442:
583:In particular, the collineations of the
1304:
346:has dimension two, a collineation from
1547:Inversion Theory and Conformal Mapping
598:has no non-trivial automorphisms (see
272:projective space defined axiomatically
1577:Projective Geometry / An Introduction
1495:
1471:
1316:
783:, PΓL – this is PGL, twisted by
617:is a nonsingular semilinear map from
7:
1320:
1066:As mentioned by Blaschke and Klein,
936:{\displaystyle \mathbb {F} _{p^{n}}}
549:is a map that, in coordinates, is a
323:Collineations of the projective line
191:respectively. A collineation from
1313:Beutelspacher & Rosenbaum 1998
397:is mapped to the zero subspace of
139:. Consider the projective spaces
25:
1460:Vorlesungen über Höhere Geometrie
594:are exactly the homographies, as
520:Projective linear transformations
68:between projective spaces, or an
848:{\displaystyle \mathbb {F} _{p}}
779:, the collineation group is the
699:is a collineation. We say that
567:If the geometric dimension of a
1570:, Wolfenbütteler Verlagsanstalt
1462:, edited by Blaschke, Seite 138
1229:Thus an anti-homography is the
725:is a vector space over a field
721:with dimension at least three,
717:is a vector space over a field
1614:, Courier Dover Publications,
1531:, Cambridge University Press,
1253:
1247:
1161:
1155:
1019:(1827). There he spoke not of
477:
471:
459:
447:
105:(as the projectivization of a
1:
1106:can be used interchangeably.
1280:{\displaystyle f(z)=1/z^{*}}
1114:The operation of taking the
900:{\displaystyle \mathbb {C} }
870:{\displaystyle \mathbb {Q} }
553:applied to the coordinates.
135:a vector space over a field
1611:Geometry of Complex Numbers
1579:, Oxford University Press,
885:not a prime field (such as
781:projective semilinear group
574:projective semilinear group
533:of the collineation group.
1689:
1600:, London: G. Bell and Sons
971:, these choices forming a
733:is a collineation from PG(
560:
183:) the set of subspaces of
1527:; Rosenbaum, Ute (1998),
1291:. The transformations of
537:Automorphic collineations
529:) is in general a proper
510:automorphic collineations
342:In this definition, when
308:of a linear space over a
124:be a vector space over a
1545:Blair, David E. (2000),
1484:Morley & Morley 1933
629:at least three. Define
625:, with the dimension of
545:automorphic collineation
1628:Yale, Paul B. (2004) ,
1596:; Morley, F.V. (1933),
1525:Beutelspacher, Albrecht
1474:, p. 64, Corollary 4.29
1084:complex projective line
749:are isomorphic fields,
414:There is a nonsingular
314:non-Desarguesian planes
1281:
1233:of conjugation with a
1220:
937:
901:
871:
849:
487:
1630:Geometry and Symmetry
1282:
1221:
1134:for the conjugate of
1130:. With the notation
1080:real projective plane
956:PΓL / PGL ≅ Gal(
938:
902:
872:
850:
792:PΓL ≅ PGL ⋊ Gal(
600:Automorphism#Examples
585:real projective plane
488:
155:), consisting of the
1568:Projective Geometrie
1241:
1149:
1017:Barycentric Calculus
995:was abstracted to a
911:
889:
859:
830:
441:
207:) is a map α :
1673:Projective geometry
1606:Schwerdtfeger, Hans
1575:Casse, Rey (2006),
1092:complex conjugation
1070:preferred the term
1053:incidence structure
785:field automorphisms
429:such that, for all
276:incidence structure
32:projective geometry
1598:Inversive Geometry
1504:Schwerdtfeger 2012
1293:inversive geometry
1277:
1216:
1057:automorphism group
933:
897:
867:
845:
789:semidirect product
578:semidirect product
551:field automorphism
483:
288:incidence relation
82:collineation group
18:Collineation group
1564:Blaschke, Wilhelm
1211:
1116:complex conjugate
969:PGL < PΓL
741:). This implies
278:(a set of points
227:α is a bijection.
16:(Redirected from
1680:
1642:
1624:
1601:
1589:
1571:
1559:
1541:
1511:
1493:
1487:
1481:
1475:
1469:
1463:
1453:
1447:
1445:
1421:
1406:
1395:
1384:
1378:
1371:
1365:
1330:
1324:
1309:
1289:circle inversion
1286:
1284:
1283:
1278:
1276:
1275:
1266:
1225:
1223:
1222:
1217:
1212:
1210:
1203:
1202:
1189:
1182:
1181:
1168:
1005:Wilhelm Blaschke
997:ternary relation
970:
965:
949:
942:
940:
939:
934:
932:
931:
930:
929:
919:
906:
904:
903:
898:
896:
880:
876:
874:
873:
868:
866:
854:
852:
851:
846:
844:
843:
838:
818:Linear structure
801:
787:; formally, the
778:
674:
651:
593:
547:
546:
492:
490:
489:
484:
384:
306:projectivization
113:with respect to
111:order-preserving
52:projective space
21:
1688:
1687:
1683:
1682:
1681:
1679:
1678:
1677:
1663:
1662:
1649:
1640:
1627:
1622:
1604:
1592:
1587:
1574:
1562:
1557:
1544:
1539:
1523:
1520:
1515:
1514:
1494:
1490:
1482:
1478:
1470:
1466:
1454:
1450:
1423:
1408:
1407:; formally, if
1397:
1386:
1380:
1374:
1372:
1368:
1331:
1327:
1310:
1306:
1301:
1267:
1239:
1238:
1194:
1190:
1173:
1169:
1147:
1146:
1140:anti-homography
1112:
1110:Anti-homography
1096:computer vision
1021:transformations
989:
968:
955:
944:
921:
914:
909:
908:
887:
886:
878:
857:
856:
833:
828:
827:
826:a prime field (
820:
791:
773:
653:
652:by saying that
630:
587:
576:, which is the
565:
559:
544:
543:
539:
522:
502:
439:
438:
363:
333:symmetric group
325:
274:in terms of an
268:
99:
90:
28:
23:
22:
15:
12:
11:
5:
1686:
1684:
1676:
1675:
1665:
1664:
1661:
1660:
1648:
1647:External links
1645:
1644:
1643:
1638:
1625:
1620:
1602:
1590:
1585:
1572:
1560:
1555:
1542:
1537:
1519:
1516:
1513:
1512:
1488:
1476:
1464:
1448:
1366:
1325:
1311:For instance,
1303:
1302:
1300:
1297:
1274:
1270:
1265:
1261:
1258:
1255:
1252:
1249:
1246:
1227:
1226:
1215:
1209:
1206:
1201:
1197:
1193:
1188:
1185:
1180:
1176:
1172:
1166:
1163:
1160:
1157:
1154:
1111:
1108:
1068:Michel Chasles
1049:
1048:
999:determined by
991:The idea of a
988:
985:
928:
924:
918:
895:
879:PGL = PΓL
865:
842:
837:
819:
816:
703:is induced by
558:
555:
538:
535:
521:
518:
501:
498:
494:
493:
482:
479:
476:
473:
470:
467:
464:
461:
458:
455:
452:
449:
446:
416:semilinear map
412:
402:
324:
321:
267:
264:
263:
262:
228:
223:), such that:
120:Formally, let
117:of subspaces.
103:linear algebra
98:
97:Linear algebra
95:
89:
86:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1685:
1674:
1671:
1670:
1668:
1658:
1654:
1651:
1650:
1646:
1641:
1639:0-486-43835-X
1635:
1631:
1626:
1623:
1621:9780486135861
1617:
1613:
1612:
1607:
1603:
1599:
1595:
1594:Morley, Frank
1591:
1588:
1586:9780199298860
1582:
1578:
1573:
1569:
1565:
1561:
1558:
1556:9780821826362
1552:
1548:
1543:
1540:
1538:0-521-48364-6
1534:
1530:
1526:
1522:
1521:
1517:
1509:
1505:
1501:
1497:
1492:
1489:
1485:
1480:
1477:
1473:
1468:
1465:
1461:
1458:(1926, 1949)
1457:
1452:
1449:
1443:
1439:
1435:
1431:
1427:
1420:
1416:
1412:
1404:
1400:
1393:
1389:
1383:
1377:
1370:
1367:
1363:
1359:
1355:
1351:
1347:
1343:
1339:
1335:
1329:
1326:
1322:
1318:
1314:
1308:
1305:
1298:
1296:
1294:
1290:
1272:
1268:
1263:
1259:
1256:
1250:
1244:
1236:
1232:
1213:
1207:
1204:
1199:
1195:
1191:
1186:
1183:
1178:
1174:
1170:
1164:
1158:
1152:
1145:
1144:
1143:
1141:
1137:
1133:
1129:
1125:
1122:amounts to a
1121:
1120:complex plane
1117:
1109:
1107:
1105:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1073:
1069:
1064:
1062:
1058:
1054:
1046:
1042:
1038:
1034:
1030:
1026:
1022:
1018:
1014:
1013:
1012:
1010:
1009:August Möbius
1006:
1002:
998:
994:
986:
984:
982:
978:
974:
963:
959:
953:
947:
926:
922:
884:
840:
825:
817:
815:
813:
809:
805:
799:
795:
790:
786:
782:
776:
770:
768:
764:
760:
756:
752:
748:
744:
740:
736:
732:
728:
724:
720:
716:
711:
708:
706:
702:
698:
694:
690:
686:
682:
678:
672:
668:
664:
660:
656:
649:
645:
641:
637:
633:
628:
624:
620:
616:
611:
609:
605:
602:and footnote
601:
597:
591:
586:
581:
579:
575:
570:
564:
556:
554:
552:
548:
536:
534:
532:
528:
519:
517:
515:
511:
507:
499:
497:
474:
468:
462:
453:
444:
436:
432:
428:
424:
420:
417:
413:
410:
407:is mapped to
406:
403:
400:
396:
392:
391:zero subspace
388:
387:
386:
385:, such that:
382:
378:
374:
370:
366:
361:
357:
353:
349:
345:
340:
338:
334:
330:
322:
320:
317:
315:
311:
310:division ring
307:
302:
300:
296:
292:
289:
285:
281:
277:
273:
266:Axiomatically
265:
260:
256:
252:
248:
244:
240:
236:
232:
229:
226:
225:
224:
222:
218:
214:
210:
206:
202:
198:
194:
190:
186:
182:
178:
174:
170:
166:
162:
158:
154:
150:
146:
142:
138:
134:
130:
127:
123:
118:
116:
112:
108:
104:
96:
94:
87:
85:
83:
80:, called the
79:
75:
71:
67:
66:
61:
57:
53:
49:
45:
41:
37:
33:
19:
1653:projectivity
1629:
1610:
1597:
1576:
1567:
1546:
1528:
1491:
1479:
1467:
1459:
1451:
1441:
1437:
1433:
1429:
1425:
1418:
1414:
1410:
1402:
1398:
1391:
1387:
1381:
1375:
1369:
1361:
1357:
1353:
1349:
1345:
1341:
1337:
1333:
1328:
1319:, p. 56 and
1307:
1228:
1142:is given by
1139:
1135:
1131:
1113:
1104:collineation
1103:
1099:
1076:collineation
1075:
1071:
1065:
1050:
1040:
1033:collineation
1032:
1028:
1025:permutations
1024:
1020:
1016:
1001:collinearity
990:
980:
976:
961:
957:
951:
945:
882:
823:
821:
811:
803:
797:
793:
774:
771:
766:
762:
758:
754:
750:
746:
742:
738:
734:
730:
726:
722:
718:
714:
712:
709:
704:
700:
696:
692:
688:
684:
680:
676:
670:
666:
662:
658:
654:
647:
643:
639:
635:
631:
626:
622:
618:
614:
612:
603:
595:
589:
582:
566:
542:
540:
523:
506:homographies
503:
495:
434:
430:
426:
422:
418:
408:
404:
398:
394:
380:
376:
372:
368:
364:
359:
355:
351:
347:
343:
341:
326:
318:
303:
298:
294:
290:
283:
279:
269:
258:
254:
250:
246:
242:
238:
234:
230:
220:
216:
212:
208:
204:
200:
196:
192:
188:
184:
180:
176:
172:
168:
164:
160:
157:vector lines
152:
148:
144:
140:
136:
132:
128:
121:
119:
107:vector space
100:
91:
81:
70:automorphism
63:
36:collineation
35:
29:
1456:Felix Klein
1379:is on line
1287:amounts to
1231:composition
1088:real number
877:), we have
808:prime field
608:Real number
362:) is a map
65:isomorphism
50:) from one
1657:PlanetMath
1518:References
1508:p. 42
1500:p. 43
1496:Blair 2000
1472:Casse 2006
1317:Casse 2006
1235:homography
1124:reflection
1100:homography
1072:homography
1041:homography
881:, but for
761:such that
561:See also:
245:) for all
88:Definition
40:one-to-one
1632:, Dover,
1321:Yale 2004
1273:∗
1200:∗
1179:∗
1128:real line
1061:incidence
1045:collinear
975:over Gal(
822:Thus for
665:) :
481:⟩
469:β
466:⟨
457:⟩
451:⟨
445:α
329:dimension
167:. Call
115:inclusion
60:collinear
48:bijection
1667:Category
1608:(2012),
1566:(1948),
1348:for all
1323:, p. 226
1315:, p.21,
1082:and the
1055:with an
1029:permuted
802:, where
765:induces
737:) to PG(
713:Suppose
675:for all
634: :
613:Suppose
531:subgroup
367: :
270:Given a
1486:, p. 38
1126:in the
1118:in the
1037:Chasles
1023:but of
1007:it was
987:History
806:is the
569:pappian
339:holds.
286:and an
46:map (a
1636:
1618:
1583:
1553:
1535:
1396:is in
973:torsor
729:, and
687:). As
588:PG(2,
508:) and
282:lines
241:) ⊆ α(
175:) and
147:) and
56:images
1440:)) ∈
1422:then
1385:then
1299:Notes
1138:, an
500:Types
421:from
354:) to
199:) to
126:field
78:group
38:is a
1634:ISBN
1616:ISBN
1581:ISBN
1551:ISBN
1533:ISBN
1417:) ∈
1102:and
993:line
952:semi
943:for
810:for
772:For
753:and
745:and
642:) →
389:The
375:) →
237:⇔ α(
215:) →
187:and
163:and
131:and
44:onto
42:and
34:, a
1655:at
1432:),
1356:in
1074:to
1039:to
983:).
948:≥ 2
907:or
855:or
777:≥ 3
679:in
657:= {
621:to
610:).
606:in
541:An
527:PGL
433:in
425:to
393:of
253:in
159:of
74:set
58:of
30:In
1669::
1506:,
1502:;
1498:,
1413:,
1364:).
1352:,
1344:⊆
1340:⇔
1336:⊆
814:.
769:.
707:.
669:∈
437:,
356:PG
348:PG
284:L,
280:P,
261:).
249:,
233:⊆
201:PG
193:PG
149:PG
141:PG
84:.
1659:.
1510:.
1446:.
1444:′
1442:I
1438:l
1436:(
1434:g
1430:p
1428:(
1426:f
1424:(
1419:I
1415:l
1411:p
1409:(
1405:)
1403:l
1401:(
1399:g
1394:)
1392:p
1390:(
1388:f
1382:l
1376:p
1362:V
1360:(
1358:D
1354:B
1350:A
1346:B
1342:A
1338:B
1334:A
1269:z
1264:/
1260:1
1257:=
1254:)
1251:z
1248:(
1245:f
1214:.
1208:d
1205:+
1196:z
1192:c
1187:b
1184:+
1175:z
1171:a
1165:=
1162:)
1159:z
1156:(
1153:f
1136:z
1132:z
981:k
979:/
977:K
964:)
962:k
960:/
958:K
946:n
927:n
923:p
917:F
894:C
883:K
864:Q
841:p
836:F
824:K
812:K
804:k
800:)
798:k
796:/
794:K
775:n
767:α
763:φ
759:φ
755:W
751:V
747:L
743:K
739:W
735:V
731:α
727:L
723:W
719:K
715:V
705:φ
701:α
697:α
693:φ
689:φ
685:V
683:(
681:D
677:Z
673:}
671:Z
667:z
663:z
661:(
659:φ
655:Z
650:)
648:W
646:(
644:D
640:V
638:(
636:D
632:α
627:V
623:W
619:V
615:φ
604:d
596:R
592:)
590:R
478:)
475:v
472:(
463:=
460:)
454:v
448:(
435:V
431:v
427:W
423:V
419:β
411:.
409:W
405:V
401:.
399:W
395:V
383:)
381:W
379:(
377:D
373:V
371:(
369:D
365:α
360:W
358:(
352:V
350:(
344:V
299:g
295:f
291:I
259:V
257:(
255:D
251:B
247:A
243:B
239:A
235:B
231:A
221:W
219:(
217:D
213:V
211:(
209:D
205:W
203:(
197:V
195:(
189:W
185:V
181:W
179:(
177:D
173:V
171:(
169:D
165:W
161:V
153:W
151:(
145:V
143:(
137:L
133:W
129:K
122:V
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.