SwissMAP Logo
Log in
  • About us
    • Organization
    • Professors
    • Senior Researchers
    • Postdocs
    • PhD Students
    • Alumni
  • News & Events
    • News
    • Events
    • Online Events
    • Videos
    • Newsletters
    • Press Coverage
    • Perspectives Journal
    • Interviews
  • Research
    • Basic Notions
    • Phase III Directions
    • Phases I & II Projects
    • Publications
    • SwissMAP Research Station
  • Awards, Visitors & Vacancies
    • Awards
    • Innovator Prize
    • Visitors
    • Vacancies
  • Outreach & Education
    • Masterclasses & Doctoral Schools
    • Mathscope
    • Maths Club
    • Athena Project
    • ETH Math Youth Academy
    • SPRING
    • Junior Euler Society
    • General Relativity for High School Students
    • Outreach Resources
    • Exhibitions
    • Previous Programs
    • Events in Outreach
    • News in Outreach
  • Equal Opportunities
    • Mentoring Program
    • Financial Support
    • SwissMAP Scholars
    • Events in Equal Opportunities
    • News in Equal Opportunities
  • Contact
    • Corporate Design
  • Basic Notions
  • Phase III Directions
  • Phases I & II Projects
  • Publications
  • SwissMAP Research Station

Properties of Hesse derivatives of cubic curves

Sayan Dutta, Lorenz Halbeisen, Norbert Hungerbühler

10/9/23 Published in : arXiv:2309.05048

The Hesse curve or Hesse derivative Hess(Γ_f) of a cubic curve Γ_f given by a homogeneous polynomial f is the set of points P such that \det \left(H_f (P)\right)=0, where H_f (P) is the Hesse matrix of f evaluated at P. Also Hess(Γ_f) is again a cubic curve. We show that for a point P∈Hess(Γ_f), all the contact points of tangents from P to the curves Γ_f and Hess(Γ_f) are intersection points of two straight lines \ell_1^P and \ell_2^P (meeting on Hess(Γ_f)) with Γ_f and Hess(Γ_f), where the product of \ell_1^P and \ell_2^P is the polar conic of Γ_f at P. The operator Hess defines an iterative discrete dynamical system on the set of the cubic curves. We identify the two fixed points of this system, investigate orbits that end in the fixed points, and discuss the closed orbits of the dynamical system.

Entire article

Program(s)

  • High School Outreach

The linearized Einstein equations with sources

Iso-entangled bases and joint measurements

  • Leading house

  • Co-leading house


The National Centres of Competence in Research (NCCRs) are a funding scheme of the Swiss National Science Foundation

© SwissMAP 2025 - All rights reserved