653:
480:
745:
of a zero rest mass (such as a non-quantum model of a photon and other elementary particles of mass zero) is an isotropic line." For isotropic lines through the origin, a particular point is a
203:
142:
366:
307:
545:
216:
An essential property of isotropic lines, and which can be used to define them, is the following: the distance between any two points of an isotropic line
584:
870:
235:
one can study curves that satisfy this differential equation; these curves are the geodesic lines of the surface, and we also call them
737:
Isotropic lines have been used in cosmological writing to carry light. For example, in a mathematical encyclopedia, light consists of
378:
232:
931:
822:
840:
764:
551:
807:
926:
566:
244:
149:
47:
91:
248:
51:
43:
312:
253:
900:
221:
906:
501:
896:
20:
862:
892:
866:
829:
58:
880:
876:
843:
786:
62:
39:
790:
756:
78:
27:
920:
856:
16:
Line along which a quadratic form applied to any two points' displacement is zero
760:
746:
818:
750:
558:
46:
between any pair of its points is zero. An isotropic line occurs only with an
648:{\displaystyle {\mathbf {n}}^{2}={\mathbf {m}}^{2}=0,\quad {\mathbf {nm}}=1.}
550:
In projective geometry, the isotropic lines are the ones passing through the
742:
565:
A non-singular plane which contains an isotropic vector shall be called a
209:
65:
first suggested the existence of two isotropic lines through the point
738:
475:{\displaystyle a_{3}(x_{2}\pm ix_{1})=(a_{2}\pm ia_{1})x_{2}.}
749:, and the collection of all such isotropic lines forms the
372:
in the complex projective plane satisfies the equation:
789:(1870) "Sur l’emploi des imaginaires en la géométrie",
673:
is a non-singular plane with orthogonal geometry and
587:
504:
381:
315:
256:
152:
94:
647:
539:
474:
360:
301:
197:
136:
220:is zero. In other terms, these lines satisfy the
8:
42:for which the quadratic form applied to the
759:expanded the concept of isotropic lines to
804:Geometry and Analysis of Projective Spaces
782:
780:
495:, an isotropic line through the origin is
218:situated at a finite distance in the plane
630:
629:
613:
607:
606:
596:
590:
589:
586:
528:
509:
503:
463:
450:
434:
415:
399:
386:
380:
349:
336:
323:
314:
290:
277:
264:
255:
208:Laguerre then interpreted these lines as
198:{\displaystyle (y-\beta )=-i(x-\alpha ).}
151:
93:
137:{\displaystyle (y-\beta )=(x-\alpha )i,}
893:Quadratic forms chapter I: Witts theory
776:
724:are then the only isotropic vectors of
569:. It can always be spanned by a pair
309:and lines by homogeneous coordinates
7:
657:We shall call any such ordered pair
19:For isotropic lines in geology, see
557:In the real orthogonal geometry of
361:{\displaystyle (a_{1},a_{2},a_{3})}
302:{\displaystyle (x_{1},x_{2},x_{3})}
706:is a hyperbolic pair. The vectors
684:, then there exists precisely one
561:, isotropic lines occur in pairs:
14:
540:{\displaystyle x_{2}=\pm ix_{1}.}
634:
631:
608:
591:
485:In terms of the affine subspace
911:Introduction to Quadratic Forms
628:
456:
427:
421:
392:
355:
316:
296:
257:
189:
177:
165:
153:
125:
113:
107:
95:
1:
247:, points are represented by
841:Encyclopedia of Mathematics
765:spinors in three dimensions
552:circular points at infinity
948:
680:is an isotropic vector of
18:
808:W. H. Freeman and Company
581:of vectors which satisfy
245:complex projective plane
48:isotropic quadratic form
249:homogeneous coordinates
52:definite quadratic form
855:Cartan, Élie (1981) ,
802:C. E. Springer (1964)
669:a hyperbolic pair. If
649:
541:
476:
362:
303:
199:
138:
901:Coral Gables, Florida
858:The theory of spinors
650:
542:
477:
363:
304:
222:differential equation
200:
139:
932:Theory of relativity
585:
502:
379:
313:
254:
150:
92:
897:University of Miami
791:Oeuvres de Laguerre
77:that depend on the
50:, and never with a
44:displacement vector
26:In the geometry of
21:Strain partitioning
907:O. Timothy O'Meara
863:Dover Publications
645:
537:
472:
358:
299:
231:. On an arbitrary
195:
134:
872:978-0-486-64070-9
824:Geometric Algebra
939:
884:
883:
852:
846:
838:
832:
830:Internet Archive
816:
810:
800:
794:
784:
727:
723:
714:
705:
699:
693:
689:
683:
679:
672:
668:
662:
654:
652:
651:
646:
638:
637:
618:
617:
612:
611:
601:
600:
595:
594:
580:
574:
567:hyperbolic plane
546:
544:
543:
538:
533:
532:
514:
513:
494:
481:
479:
478:
473:
468:
467:
455:
454:
439:
438:
420:
419:
404:
403:
391:
390:
367:
365:
364:
359:
354:
353:
341:
340:
328:
327:
308:
306:
305:
300:
295:
294:
282:
281:
269:
268:
230:
204:
202:
201:
196:
143:
141:
140:
135:
83:
76:
59:complex geometry
947:
946:
942:
941:
940:
938:
937:
936:
927:Quadratic forms
917:
916:
891:Pete L. Clark,
888:
887:
873:
854:
853:
849:
839:
835:
817:
813:
801:
797:
787:Edmond Laguerre
785:
778:
773:
763:in his book on
753:at the origin.
735:
725:
716:
707:
701:
695:
691:
685:
681:
674:
670:
664:
658:
605:
588:
583:
582:
576:
570:
524:
505:
500:
499:
492:
486:
459:
446:
430:
411:
395:
382:
377:
376:
345:
332:
319:
311:
310:
286:
273:
260:
252:
251:
237:isotropic lines
224:
148:
147:
146:Second system:
90:
89:
81:
66:
63:Edmond Laguerre
28:quadratic forms
24:
17:
12:
11:
5:
945:
943:
935:
934:
929:
919:
918:
915:
914:
904:
886:
885:
871:
865:, p. 17,
847:
833:
811:
795:
775:
774:
772:
769:
734:
731:
730:
729:
655:
644:
641:
636:
633:
627:
624:
621:
616:
610:
604:
599:
593:
548:
547:
536:
531:
527:
523:
520:
517:
512:
508:
490:
483:
482:
471:
466:
462:
458:
453:
449:
445:
442:
437:
433:
429:
426:
423:
418:
414:
410:
407:
402:
398:
394:
389:
385:
370:isotropic line
357:
352:
348:
344:
339:
335:
331:
326:
322:
318:
298:
293:
289:
285:
280:
276:
272:
267:
263:
259:
241:
240:
206:
205:
194:
191:
188:
185:
182:
179:
176:
173:
170:
167:
164:
161:
158:
155:
144:
133:
130:
127:
124:
121:
118:
115:
112:
109:
106:
103:
100:
97:
88:First system:
79:imaginary unit
32:isotropic line
15:
13:
10:
9:
6:
4:
3:
2:
944:
933:
930:
928:
925:
924:
922:
912:
909:(1963, 2000)
908:
905:
902:
898:
894:
890:
889:
882:
878:
874:
868:
864:
860:
859:
851:
848:
845:
842:
837:
834:
831:
827:
825:
820:
815:
812:
809:
805:
799:
796:
792:
788:
783:
781:
777:
770:
768:
766:
762:
758:
754:
752:
748:
744:
740:
732:
722:
719:
713:
710:
704:
698:
688:
677:
667:
661:
656:
642:
639:
625:
622:
619:
614:
602:
597:
579:
573:
568:
564:
563:
562:
560:
555:
553:
534:
529:
525:
521:
518:
515:
510:
506:
498:
497:
496:
489:
469:
464:
460:
451:
447:
443:
440:
435:
431:
424:
416:
412:
408:
405:
400:
396:
387:
383:
375:
374:
373:
371:
350:
346:
342:
337:
333:
329:
324:
320:
291:
287:
283:
278:
274:
270:
265:
261:
250:
246:
238:
234:
228:
223:
219:
215:
214:
213:
211:
192:
186:
183:
180:
174:
171:
168:
162:
159:
156:
145:
131:
128:
122:
119:
116:
110:
104:
101:
98:
87:
86:
85:
80:
74:
70:
64:
60:
55:
53:
49:
45:
41:
37:
33:
29:
22:
910:
861:, New York:
857:
850:
836:
823:
814:
806:, page 141,
803:
798:
761:multivectors
755:
736:
720:
717:
711:
708:
702:
696:
686:
675:
665:
659:
577:
571:
556:
549:
487:
484:
369:
242:
236:
226:
217:
207:
72:
68:
56:
35:
31:
25:
757:Élie Cartan
747:null vector
921:Categories
844:World line
826:, page 119
819:Emil Artin
771:References
751:light cone
733:Relativity
694:such that
559:Emil Artin
913:, page 94
743:worldline
519:±
441:±
406:±
210:geodesics
187:α
184:−
172:−
163:β
160:−
123:α
120:−
105:β
102:−
36:null line
881:0631850
821:(1957)
741:: "The
739:photons
243:In the
233:surface
879:
869:
73:β
69:α
57:Using
895:from
793:2: 89
368:. An
38:is a
30:, an
867:ISBN
828:via
715:and
40:line
899:in
690:in
678:≠ 0
493:= 1
229:= 0
34:or
923::
877:MR
875:,
779:^
767:.
700:,
663:,
643:1.
575:,
554:.
212::
84::
71:,
61:,
54:.
903:.
728:.
726:V
721:m
718:y
712:n
709:x
703:m
697:n
692:V
687:m
682:V
676:n
671:V
666:m
660:n
640:=
635:m
632:n
626:,
623:0
620:=
615:2
609:m
603:=
598:2
592:n
578:m
572:n
535:.
530:1
526:x
522:i
516:=
511:2
507:x
491:3
488:x
470:.
465:2
461:x
457:)
452:1
448:a
444:i
436:2
432:a
428:(
425:=
422:)
417:1
413:x
409:i
401:2
397:x
393:(
388:3
384:a
356:)
351:3
347:a
343:,
338:2
334:a
330:,
325:1
321:a
317:(
297:)
292:3
288:x
284:,
279:2
275:x
271:,
266:1
262:x
258:(
239:.
227:s
225:d
193:.
190:)
181:x
178:(
175:i
169:=
166:)
157:y
154:(
132:,
129:i
126:)
117:x
114:(
111:=
108:)
99:y
96:(
82:i
75:)
67:(
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.