24:
149:, so this special case of Poncelet's porism can be expressed more concisely by saying that every bicentric polygon is part of an infinite family of bicentric polygons with respect to the same two circles.
63:
another one, the polygon must be part of an infinite family of polygons that are all inscribed in and circumscribe the same two conics. It is named after French engineer and mathematician
617:
Traité des propriétés projectives des figures; ouvrage utile à ceux qui s'occupent des applications de la géométrie descriptive et d'opérations géométriques sur le terrain
486:
459:
380:
324:
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346:′) with the same first coordinate. Any involution of an elliptic curve with a fixed point, when expressed in the group law, has the form
584:
78:, whose points represent a combination of a line tangent to one conic and a crossing point of that line with the other conic.
68:
794:
546:
166:
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67:, who wrote about it in 1822; however, the triangular case was discovered significantly earlier, in 1746 by
615:
181:
64:
541:
180:
meet transversely (meaning that each intersection point of the two is a simple crossing). Then by
468:
441:
536:
730:
by
Michael Borcherds showing Poncelet's Porism for a general Ellipse and a Parabola made using
698:
146:
365:
309:
145:, the polygons that are inscribed in one circle and circumscribed about the other are called
642:
596:
530:
52:
760:
by
Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 6) made using
750:
by
Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 5) made using
740:
by
Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 3) made using
654:
405:
650:
551:
488:
has a fixed point, that power must be the identity. Translated back into the language of
631:
Del
Centina, Andrea (2016), "Poncelet's porism: a long story of renewed discoveries, I",
716: = 3, 4, 5, 6, 7, 8 (including the convex cases for
75:
783:
678:
56:
31: = 3, a triangle that is inscribed in one circle and circumscribes another.
773:
571:
60:
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Weisstein, Eric W. "Poncelet's Porism." From MathWorld--A Wolfram Web
Resource.
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23:
192:
of the two curves consists of four complex points. For an arbitrary point
36:
123:
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48:
757:
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709:
620:(in French) (2nd ed.). Paris: Gauthier-Villars. pp. 311–317.
130:), then it is possible to find infinitely many of them. Each point of
770:
showing the exterior case for n = 3 at
National Tsing Hua University.
142:
600:
95:
22:
681:; Kers, C.; Oort, F.; Raven, D. W. "Poncelet's closure theorem".
512:-gon), then so does every point. The degenerate cases in which
705:
Mathematical
Omnibus: Thirty Lectures on Classic Mathematics
138:
is a vertex or tangency (respectively) of one such polygon.
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is a degree 2 morphism ramified over the contact points on
508:) gives rise to an orbit that closes up (i.e., gives an
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Poncelet's porism can be proved by an argument using an
471:
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453:
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302:is an elliptic curve (once we fix a base point on
572:http://mathworld.wolfram.com/PonceletsPorism.html
520:are not transverse follow from a limit argument.
298:as a degree 2 cover ramified above 4 points, so
8:
382:has this form. Similarly, the projection
98:. If it is possible to find, for a given
470:
443:
407:
367:
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114:(meaning that all of its vertices lie on
670:, Dover Publications, 2007 (orig. 1960).
712:by Michael Borcherds showing the cases
585:"Three problems in search of a measure"
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282:and 2 otherwise. Thus the projection
27:Illustration of Poncelet's porism for
634:Archive for History of Exact Sciences
7:
110:that is simultaneously inscribed in
720: = 7, 8) made using
402:, and the corresponding involution
122:(meaning that all of its edges are
394:of the four lines tangent to both
14:
699:David Speyer on Poncelet's Porism
614:Poncelet, Jean-Victor (1865) .
504:(equipped with a corresponding
496:, this means that if one point
172:. For simplicity, assume that
1:
774:Article on Poncelet's Porism
688:(1987), no. 4, 289–364.
481:{\displaystyle \tau \sigma }
454:{\displaystyle \tau \sigma }
668:Advanced Euclidean Geometry
118:) and circumscribed around
811:
703:D. Fuchs, S. Tabachnikov,
583:King, Jonathan L. (1994).
41:Poncelet's closure theorem
683:Expositiones Mathematicae
647:10.1007/s00407-015-0163-y
47:, states that whenever a
547:Tangent lines to circles
438:. Thus the composition
167:complex projective plane
375:{\displaystyle \sigma }
319:{\displaystyle \sigma }
209:be the tangent line to
102: > 2, one
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338:) to the other point (
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18:Theorem of 2D geometry
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415:{\displaystyle \tau }
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326:be the involution of
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221:be the subvariety of
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65:Jean-Victor Poncelet
666:Johnson, Roger A.,
589:Amer. Math. Monthly
330:sending a general (
184:, the intersection
758:Interactive applet
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738:Interactive applet
728:Interactive applet
710:Interactive applet
537:Hartshorne ellipse
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147:bicentric polygons
141:If the conics are
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465:. If a power of
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641:(1): 1–122,
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768:Java applet
595:: 609–628.
784:Categories
558:References
476:σ
473:τ
449:σ
446:τ
434:for some
410:τ
370:σ
358:for some
314:σ
294:presents
250:. Given
82:Statement
53:inscribed
762:GeoGebra
752:GeoGebra
742:GeoGebra
732:GeoGebra
722:GeoGebra
524:See also
306:). Let
270:is 1 if
37:geometry
655:3437893
217:. Let
143:circles
124:tangent
108:polygon
106:-sided
55:in one
49:polygon
653:
258:with (
200:, let
96:conics
362:, so
157:View
516:and
492:and
398:and
266:) ∈
176:and
161:and
90:and
86:Let
59:and
643:doi
597:doi
593:101
213:at
196:in
134:or
126:to
51:is
35:In
786::
651:MR
649:,
639:70
637:,
591:.
587:.
500:∈
430:−
426:→
386:→
354:−
350:→
290:≃
286:→
278:∩
274:∈
225:×
188:∩
71:.
39:,
764:.
754:.
744:.
734:.
724:.
718:n
714:n
686:5
645::
603:.
599::
518:D
514:C
510:n
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463:X
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388:D
384:X
360:p
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348:x
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342:,
340:c
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332:c
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284:X
280:D
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272:c
268:X
264:d
262:,
260:c
256:d
252:c
248:c
243:d
239:ℓ
235:d
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215:d
211:D
206:d
202:ℓ
198:D
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186:C
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174:C
170:P
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159:C
136:D
132:C
128:D
120:D
116:C
112:C
104:n
100:n
92:D
88:C
29:n
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