35:
160:, 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.
74:
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
628:
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
497:
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357:′) with the same first coordinate. Any involution of an elliptic curve with a fixed point, when expressed in the group law, has the form
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89:, whose points represent a combination of a line tangent to one conic and a crossing point of that line with the other conic.
79:
805:
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177:
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78:, who wrote about it in 1822; however, the triangular case was discovered significantly earlier, in 1746 by
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192:
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191:
meet transversely (meaning that each intersection point of the two is a simple crossing). Then by
479:
452:
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741:
by
Michael Borcherds showing Poncelet's Porism for a general Ellipse and a Parabola made using
709:
157:
376:
320:
156:, the polygons that are inscribed in one circle and circumscribed about the other are called
653:
607:
541:
63:
771:
by
Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 6) made using
761:
by
Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 5) made using
751:
by
Michael Borcherds showing Poncelet's Porism for 2 general ellipses (order 3) made using
665:
416:
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562:
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has a fixed point, that power must be the identity. Translated back into the language of
642:
Del
Centina, Andrea (2016), "Poncelet's porism: a long story of renewed discoveries, I",
727: = 3, 4, 5, 6, 7, 8 (including the convex cases for
86:
17:
794:
689:
67:
42: = 3, a triangle that is inscribed in one circle and circumscribes another.
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71:
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Weisstein, Eric W. "Poncelet's Porism." From MathWorld--A Wolfram Web
Resource.
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34:
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of the two curves consists of four complex points. For an arbitrary point
47:
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59:
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631:(in French) (2nd ed.). Paris: Gauthier-Villars. pp. 311–317.
141:), then it is possible to find infinitely many of them. Each point of
781:
showing the exterior case for n = 3 at
National Tsing Hua University.
153:
611:
106:
33:
692:; Kers, C.; Oort, F.; Raven, D. W. "Poncelet's closure theorem".
523:-gon), then so does every point. The degenerate cases in which
716:
Mathematical
Omnibus: Thirty Lectures on Classic Mathematics
149:
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
519:) 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
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313:is an elliptic curve (once we fix a base point on
583:http://mathworld.wolfram.com/PonceletsPorism.html
531:are not transverse follow from a limit argument.
309:as a degree 2 cover ramified above 4 points, so
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393:has this form. Similarly, the projection
109:. If it is possible to find, for a given
481:
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125:(meaning that all of its vertices lie on
681:, Dover Publications, 2007 (orig. 1960).
723:by Michael Borcherds showing the cases
596:"Three problems in search of a measure"
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293:and 2 otherwise. Thus the projection
38:Illustration of Poncelet's porism for
645:Archive for History of Exact Sciences
7:
121:that is simultaneously inscribed in
731: = 7, 8) made using
413:, and the corresponding involution
133:(meaning that all of its edges are
405:of the four lines tangent to both
25:
710:David Speyer on Poncelet's Porism
625:Poncelet, Jean-Victor (1865) .
515:(equipped with a corresponding
507:, this means that if one point
183:. For simplicity, assume that
1:
785:Article on Poncelet's Porism
699:(1987), no. 4, 289–364.
492:{\displaystyle \tau \sigma }
465:{\displaystyle \tau \sigma }
679:Advanced Euclidean Geometry
129:) and circumscribed around
822:
714:D. Fuchs, S. Tabachnikov,
594:King, Jonathan L. (1994).
52:Poncelet's closure theorem
694:Expositiones Mathematicae
658:10.1007/s00407-015-0163-y
58:, states that whenever a
558:Tangent lines to circles
449:. Thus the composition
178:complex projective plane
386:{\displaystyle \sigma }
330:{\displaystyle \sigma }
220:be the tangent line to
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349:) to the other point (
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29:Theorem of 2D geometry
18:Poncelet's porism
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426:{\displaystyle \tau }
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337:be the involution of
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232:be the subvariety of
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472:is a translation on
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76:Jean-Victor Poncelet
677:Johnson, Roger A.,
600:Amer. Math. Monthly
341:sending a general (
195:, the intersection
769:Interactive applet
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721:Interactive applet
548:Hartshorne ellipse
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158:bicentric polygons
152:If the conics are
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606:: 609–628.
795:Categories
569:References
487:σ
484:τ
460:σ
457:τ
445:for some
421:τ
381:σ
369:for some
325:σ
305:presents
261:. Given
93:Statement
64:inscribed
773:GeoGebra
763:GeoGebra
753:GeoGebra
743:GeoGebra
733:GeoGebra
535:See also
317:). Let
281:is 1 if
48:geometry
666:3437893
228:. Let
154:circles
135:tangent
119:polygon
117:-sided
66:in one
60:polygon
664:
269:with (
211:, let
107:conics
373:, so
168:View
527:and
503:and
409:and
277:) ∈
187:and
172:and
101:and
97:Let
70:and
654:doi
608:doi
604:101
224:at
207:in
145:or
137:to
62:is
46:In
797::
662:MR
660:,
650:70
648:,
602:.
598:.
511:∈
441:−
437:→
397:→
365:−
361:→
301:≃
297:→
289:∩
285:∈
236:×
199:∩
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755:.
745:.
735:.
729:n
725:n
697:5
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614:.
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