13. Verallgemeinerung des Pascalschen Theorems, das in einen Kegelschnitt beschriebene Sechseck betreffend.

Pascal's theorem gives a synthetic geometric condition for six points a, . . . , f in P 2 to lie on a conic. Namely, that the intersection points ab ∩ de, af ∩ dc, ef ∩ bc are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for d + 4 points in P d to lie on a degree d rational normal curve? In this paper we find many of these conditions by writing in the Grassmann-Cayley algebra the defining equations of the parameter space of d + 4-ordered points in P d that lie on a rational normal curve. These equations were introduced and studied in a previous joint work of the authors with Giansiracusa and Moon. We conclude with an application in the case of seven points on a twisted cubic.

show abstract

“…In [15] Möbius proved the following. Assume that a polygon with 4n + 2 sides is inscribed in an irreducible conic.…”

Section: Historical Contextmentioning

confidence: 89%

A Pascal's theorem for rational normal curves

Caminata

Schaffler

2021

Bull. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…In [Möb48] Möbius proved the following. Assume a polygon with 4n + 2 sides is inscribed in an irreducible conic.…”

Section: Introductionmentioning

confidence: 89%

A Pascal's Theorem for rational normal curves

Caminata,

Schaffler

2019

Preprint

View full text Add to dashboard Cite

Pascal's Theorem gives a synthetic geometric condition for six points a, . . . , f in P 2 to lie on a conic. Namely, that the intersection points ab ∩ de, af ∩ dc, ef ∩ bc are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for d + 4 points in P d to lie on a degree d rational normal curve? In this paper we find many of these conditions by writing in the Grassmann-Cayley algebra the defining equations of the parameter space of d + 4 ordered points in P d that lie on a rational normal curve. These equations were introduced and studied in a previous joint work of the authors with Giansiracusa and Moon. We conclude with an application in the case of seven points on a twisted cubic.

show abstract

“…In general when the product of involutions is an involution we are not able to say something pertinent about the position of their centers. But, when the product of n involutions is still an involution and at least n − 1 centers are aligned, Möbius proved that all the centers are aligned (see [6], page 219). We prove again this theorem in the terminology of Frégier's involution.…”

Section: 2mentioning

confidence: 99%

A Poncelet theorem for lines

Vallès

2012

Preprint

View full text Add to dashboard Cite

Our aim is to prove a Poncelet type theorem for a line configuration on the complex projective plane P 2 . More precisely, we say that a polygon with 2n sides joining 2n vertices A1, A2, • • • , A2n is well inscribed in a configuration Ln of n lines if each line of the configuration contains exactly two points among A1, A2, • • • , A2n. Then we prove : Theorem Let Ln be a configuration of n lines and D a smooth conic in P 2 . If it exists one polygon with 2n sides well inscribed in Ln and circumscribed around D then there are infinitely many such polygons. In particular a general point in Ln is a vertex of such a polygon.This result was probably known by Poncelet or Darboux but we did not find a similar statement in their publications. Anyway, even if it was, we would like to propose an elementary proof based on Frégier's involution. We begin by recalling some facts about these involutions. Then we explore the following question : When does the product of involutions correspond to an involution? We give a partial answer in proposition 2.6. This question leads also to Pascal theorem, to its dual version proved by Brianchon, and to its generalization proved by Möbius (see [1], thm.1 and [6], page 219). In the last section, using the Frégier's involutions and the projective duality we prove the main theorem quoted above.

show abstract

13. Verallgemeinerung des Pascalschen Theorems, das in einen Kegelschnitt beschriebene Sechseck betreffend.

Cited by 4 publications

References 0 publications

A Pascal's theorem for rational normal curves

A Pascal's theorem for rational normal curves

A Pascal's Theorem for rational normal curves

A Poncelet theorem for lines

Contact Info

Product

Resources

About