English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In 1640's, Blaise Pascal discovered a remarkable property of a hexagoninscribed in a conic - Pascal Theorem, which gave birth of the projectivegeometry. In this paper, a new geometric invariant of algebraic curves isdiscovered by a different comprehension to Pascal's mystic hexagram or to thePascal theorem. Using this invariant, the Pascal theorem can be generalized tothe case of cubic (even to algebraic curves of higher degree), that is, {\emFor 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 $\Gamma_3$, thenthe six points consisting of the three points determined by the Pascal mappingapplied to any six points (no three points of which are collinear) among those9 intersections as well as the remaining three points of those 9 intersectionsmust lie on a conic.} This generalization differs quite a bit and is muchsimpler than Chasles's theorem and Cayley-Bacharach theorems.

An Invariant of Algebraic Curves from the Pascal Theorem