20:
185:
according to which the length of any curve equals the measure of the set of lines that cross the curve, multiplied by their numbers of crossings. Any two curves that have the same constant width are crossed by sets of lines with the same measure, and therefore they have the same length. Historically,
298:
236:
396:
445:
357:
307:, three-dimensional convex sets for which every two-dimensional projection has the same area. These all have the same surface area as a sphere of the same projected area.
416:
328:
668:
705:
513:
496:
249:
460:
304:
202:
27:
have constant width, and all have the same width; therefore by
Barbier's theorem they also have equal perimeters.
464:
720:
50:
40:
611:
562:
212:
366:
239:
421:
333:
616:
648:
682:
629:
190:
664:
492:
243:
182:
66:
528:
486:
656:
621:
578:
125:
24:
678:
715:
674:
448:
178:
163:. The Minkowski sum acts linearly on the perimeters of convex bodies, so the perimeter of
401:
313:
92:
567:"On a funicular solution of Buffon's "problem of the needle" in its most general form"
49:
times its width, regardless of its precise shape. This theorem was first published by
699:
686:
137:
103:
633:
549:
625:
186:
Crofton derived his formula later than, and independently of, Barbier's theorem.
710:
545:
206:
19:
600:"Semidefinite programming for optimizing convex bodies under width constraints"
599:
660:
653:
Bodies of
Constant Width: An Introduction to Convex Geometry with Applications
78:
70:
32:
583:
96:
62:
566:
18:
189:
An elementary probabilistic proof of the theorem can be found at
651:(2019), "Section 13.3.2 Convex Bodies of Constant Brightness",
514:"Note sur le problème de l'aiguille et le jeu du joint couvert"
61:
The most familiar examples of curves of constant width are the
293:{\displaystyle 8\pi -{\tfrac {4}{3}}\pi ^{2}\approx 11.973}
44:
363:−1)-dimensional projection has area of the unit ball in
177:
Alternatively, the theorem follows immediately from the
124:. A similar analysis of other simple examples such as
109:/3, so the perimeter of the Reuleaux triangle of width
263:
424:
404:
369:
336:
316:
252:
215:
113:
is equal to half the perimeter of a circle of radius
16:
All curves of constant width have the same perimeter
439:
410:
390:
351:
322:
292:
230:
167:must be half the perimeter of this disk, which is
467:, bounding the areas of curves of constant width
136:One proof of the theorem uses the properties of
246:with the same constant width has surface area
69:. For a circle, the width is the same as the
8:
521:Journal de mathématiques pures et appliquées
152:and its 180° rotation is a disk with radius
303:Instead, Barbier's theorem generalizes to
615:
582:
431:
427:
426:
423:
403:
376:
372:
371:
368:
343:
339:
338:
335:
315:
278:
262:
251:
214:
598:Bayen, Térence; Henrion, Didier (2012),
477:
418:is equal to that of the unit sphere in
201:The analogue of Barbier's theorem for
7:
491:, Dover, Theorem 11.11, pp. 81–82,
231:{\displaystyle 4\pi \approx 12.566}
488:Convex Sets and Their Applications
391:{\displaystyle \mathbb {R} ^{n-1}}
14:
647:Martini, Horst; Montejano, Luis;
604:Optimization Methods and Software
655:, Birkhäuser, pp. 310–313,
536:. See in particular pp. 283–285.
440:{\displaystyle \mathbb {R} ^{n}}
352:{\displaystyle \mathbb {R} ^{n}}
87:. A Reuleaux triangle of width
449:general form of Crofton formula
1:
546:The Theorem of Barbier (Java)
305:bodies of constant brightness
205:is false. In particular, the
626:10.1080/10556788.2010.547580
148:, then the Minkowski sum of
144:is a body of constant width
398:, then the surface area of
737:
706:Theorems in plane geometry
203:surfaces of constant width
117:and therefore is equal to
102:. Each of these arcs has
661:10.1007/978-3-030-03868-7
527:: 273–286, archived from
461:Blaschke–Lebesgue theorem
465:isoperimetric inequality
447:. This follows from the
523:, 2 série (in French),
485:Lay, Steven R. (2007),
174:as the theorem states.
128:gives the same answer.
41:curve of constant width
441:
412:
392:
353:
330:is a convex subset of
324:
294:
232:
28:
442:
413:
393:
354:
325:
295:
240:surface of revolution
233:
22:
512:Barbier, E. (1860),
422:
402:
367:
334:
314:
250:
213:
73:; a circle of width
51:Joseph-Émile Barbier
359:, for which every (
310:And in general, if
584:10.1007/BF02413320
437:
408:
388:
349:
320:
290:
272:
228:
91:consists of three
39:states that every
29:
670:978-3-030-03866-3
649:Oliveros, Déborah
411:{\displaystyle S}
323:{\displaystyle S}
271:
244:Reuleaux triangle
209:has surface area
197:Higher dimensions
183:integral geometry
126:Reuleaux polygons
67:Reuleaux triangle
37:Barbier's theorem
25:Reuleaux polygons
728:
690:
689:
644:
638:
636:
619:
610:(6): 1073–1099,
595:
589:
587:
586:
571:Acta Mathematica
563:Sylvester, J. J.
559:
553:
543:
537:
535:
533:
518:
509:
503:
501:
482:
446:
444:
443:
438:
436:
435:
430:
417:
415:
414:
409:
397:
395:
394:
389:
387:
386:
375:
358:
356:
355:
350:
348:
347:
342:
329:
327:
326:
321:
299:
297:
296:
291:
283:
282:
273:
264:
237:
235:
234:
229:
170:
159:
120:
108:
83:
47:
736:
735:
731:
730:
729:
727:
726:
725:
696:
695:
694:
693:
671:
646:
645:
641:
617:10.1.1.402.9539
597:
596:
592:
561:
560:
556:
544:
540:
531:
516:
511:
510:
506:
499:
484:
483:
479:
474:
457:
425:
420:
419:
400:
399:
370:
365:
364:
337:
332:
331:
312:
311:
274:
248:
247:
211:
210:
199:
191:Buffon's noodle
179:Crofton formula
168:
157:
156:and perimeter 2
134:
118:
106:
81:
59:
45:
17:
12:
11:
5:
734:
732:
724:
723:
721:Constant width
718:
713:
708:
698:
697:
692:
691:
669:
639:
590:
577:(1): 185–205,
554:
538:
504:
497:
476:
475:
473:
470:
469:
468:
456:
453:
434:
429:
407:
385:
382:
379:
374:
346:
341:
319:
289:
286:
281:
277:
270:
267:
261:
258:
255:
227:
224:
221:
218:
198:
195:
138:Minkowski sums
133:
130:
95:of circles of
58:
55:
43:has perimeter
15:
13:
10:
9:
6:
4:
3:
2:
733:
722:
719:
717:
714:
712:
709:
707:
704:
703:
701:
688:
684:
680:
676:
672:
666:
662:
658:
654:
650:
643:
640:
635:
631:
627:
623:
618:
613:
609:
605:
601:
594:
591:
585:
580:
576:
572:
568:
564:
558:
555:
551:
547:
542:
539:
534:on 2017-04-20
530:
526:
522:
515:
508:
505:
500:
498:9780486458038
494:
490:
489:
481:
478:
471:
466:
462:
459:
458:
454:
452:
450:
432:
405:
383:
380:
377:
362:
344:
317:
308:
306:
301:
287:
284:
279:
275:
268:
265:
259:
256:
253:
245:
241:
225:
222:
219:
216:
208:
204:
196:
194:
192:
187:
184:
180:
175:
173:
166:
162:
155:
151:
147:
143:
139:
131:
129:
127:
123:
116:
112:
105:
104:central angle
101:
98:
94:
90:
86:
80:
76:
72:
68:
64:
56:
54:
52:
48:
42:
38:
34:
26:
21:
652:
642:
607:
603:
593:
574:
570:
557:
550:cut-the-knot
541:
529:the original
524:
520:
507:
487:
480:
360:
309:
302:
238:, while the
200:
188:
176:
171:
164:
160:
153:
149:
145:
141:
135:
121:
114:
110:
99:
88:
84:
74:
60:
36:
30:
207:unit sphere
700:Categories
472:References
687:127264210
612:CiteSeerX
381:−
285:≈
276:π
260:−
257:π
223:≈
220:π
79:perimeter
53:in 1860.
634:14118522
565:(1890),
455:See also
71:diameter
65:and the
57:Examples
33:geometry
679:3930585
716:Length
685:
677:
667:
632:
614:
495:
288:11.973
226:12.566
132:Proofs
97:radius
63:circle
23:These
683:S2CID
630:S2CID
532:(PDF)
517:(PDF)
242:of a
140:. If
665:ISBN
493:ISBN
463:and
93:arcs
77:has
657:doi
622:doi
579:doi
548:at
181:in
31:In
711:Pi
702::
681:,
675:MR
673:,
663:,
628:,
620:,
608:27
606:,
602:,
575:14
573:,
569:,
519:,
451:.
300:.
193:.
35:,
659::
637:.
624::
588:.
581::
552:.
525:5
502:.
433:n
428:R
406:S
384:1
378:n
373:R
361:n
345:n
340:R
318:S
280:2
269:3
266:4
254:8
217:4
172:w
169:π
165:K
161:w
158:π
154:w
150:K
146:w
142:K
122:w
119:π
115:w
111:w
107:π
100:w
89:w
85:w
82:π
75:w
46:π
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
