An Invariant of Algebraic Curves from the Pascal Theorem

Details An Invariant of Algebraic Curves from the Pascal Theorem An Invariant of Algebraic Curves from the Pascal Theorem

2012-01-06

arxiv.org/abs/1201.1344v1

Mathematics

Algebraic Geometry

15 pages, 9 figures

Scientific paper

In 1640's, Blaise Pascal discovered a remarkable property of a hexagon inscribed in a conic - Pascal Theorem, which gave birth of the projective geometry. In this paper, a new geometric invariant of algebraic curves is discovered by a different comprehension to Pascal's mystic hexagram or to the Pascal theorem. Using this invariant, the Pascal theorem can be generalized to the case of cubic (even to algebraic curves of higher degree), that is, {\em For any given 9 intersections between a cubic $\Gamma_3$ and any three lines $a,b,c$ with no common zero, none of them is a component of $\Gamma_3$, then the six points consisting of the three points determined by the Pascal mapping applied to any six points (no three points of which are collinear) among those 9 intersections as well as the remaining three points of those 9 intersections must lie on a conic.} This generalization differs quite a bit and is much simpler than Chasles's theorem and Cayley-Bacharach theorems.

Affiliated with

Also associated with

No associations

Rating

An Invariant of Algebraic Curves from the Pascal 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 An Invariant of Algebraic Curves from the Pascal Theorem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and An Invariant of Algebraic Curves from the Pascal Theorem will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-661732