Mathematics – Algebraic Geometry
Scientific paper
2012-01-31
Mathematics
Algebraic Geometry
6 pages
Scientific paper
In the town of Saratov where he was prisonner, Poncelet, continuing the work of Euler and Steiner on polygons simultaneously inscribed in a circle and circumscribed around an other circle, proved the following generalization : "Let C and D be two smooth conics in the projective complex plane. If D passes through the n(n-1)/2 vertices of a complete polygon with n sides tangent to C then D passes through the vertices of infinitely many such polygons." According to Marcel Berger this theorem is the nicest result about the geometry of conics. Even if it is, there are few proofs of it. To my knowledge there are only three. The first proof, published in 1822 and based on infinitesimal deformations, is due to Poncelet. Later, Jacobi proposed a new proof based on finite order points on elliptic curves; his proof, certainly the most famous, is explained in a modern way and in detail by Griffiths and Harris. In 1870 Weyr proved a Poncelet theorem in space (more precisely for two quadrics) that implies the one above when one quadric is a cone; this proof is explained by Barth and Bauer. Our aim in this short note is to involve vector bundles techniques to propose a new proof of this celebrated result. Poncelet did not appreciate Jacobi's for the reason that it was too far from the geometric intuition. I guess that he would not appreciate our proof either for the same reason.
No associations
LandOfFree
A vector bundle proof of Poncelet theorem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with A vector bundle proof of Poncelet theorem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A vector bundle proof of Poncelet theorem will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-57720