Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Equations, inequations and inequalities characterizing the configurations of two real projective conics

Emmanuel Briand

TL;DR

The paper develops a complete algebraic framework to classify ordered pairs of proper real projective conics up to ambient isotopy by reducing the problem to orbits of pencils of conics and the relative position of the conics within each pencil. It builds a robust decision procedure using invariants, covariants, resultants, and Sturm-type subresultants to distinguish pencil orbits, identify rigid isotopy classes inside each pencil, and determine nesting (which conic lies inside the other). The authors provide explicit polynomial criteria, furnish representative pencils for each orbit, and illustrate the approach with concrete examples such as two ellipsoids and a paraboloid-ellipsoid pair, as well as Uhlig’s canonical forms. The work yields a practical, parameter-friendly method for analyzing parameter-dependent conic configurations with potential extensions to higher dimensions and applications in geometric modeling.

Abstract

Ordered pairs of proper, non-empty real projective conics can be classified modulo rigid isotopy and ambient isotopy. We characterize the classes by equations, inequations and inequalities in the coefficients of the quadratic forms defining the conics. The results are well--adapted to the study of the relative position of two conics defined by equations depending on parameters.

Equations, inequations and inequalities characterizing the configurations of two real projective conics

TL;DR

The paper develops a complete algebraic framework to classify ordered pairs of proper real projective conics up to ambient isotopy by reducing the problem to orbits of pencils of conics and the relative position of the conics within each pencil. It builds a robust decision procedure using invariants, covariants, resultants, and Sturm-type subresultants to distinguish pencil orbits, identify rigid isotopy classes inside each pencil, and determine nesting (which conic lies inside the other). The authors provide explicit polynomial criteria, furnish representative pencils for each orbit, and illustrate the approach with concrete examples such as two ellipsoids and a paraboloid-ellipsoid pair, as well as Uhlig’s canonical forms. The work yields a practical, parameter-friendly method for analyzing parameter-dependent conic configurations with potential extensions to higher dimensions and applications in geometric modeling.

Abstract

Ordered pairs of proper, non-empty real projective conics can be classified modulo rigid isotopy and ambient isotopy. We characterize the classes by equations, inequations and inequalities in the coefficients of the quadratic forms defining the conics. The results are well--adapted to the study of the relative position of two conics defined by equations depending on parameters.

Paper Structure

This paper contains 32 sections, 27 theorems, 89 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Classification
  3. Rigid isotopy
  4. Orbits of pencils of conics
  5. Rigid isotopy classification for pairs
  6. Rigid isotopy classification for couples
  7. Characterizing the isotopy classes by equations, inequations and inequalities
  8. Preliminaries
  9. The invariants and covariants of two ternary quadratic forms
  10. Resultants
  11. Descartes' law of signs
  12. Subresultant sequences
  13. Discriminating between the orbits of pencils
  14. Discriminating between I and Ia
  15. Discriminating between II and IIa
  16. ...and 17 more sections

Key Result

Theorem 1

(DegtyarevLevy) Two non-degenerate pencils of conics are equivalent modulo$PGL(3,{\mathbb{R}})$ if and only if they have the same numbers of real and imaginary base points of each multiplicity.

Figures (5)

  • Figure 1: The rigid isotopy classes for generic pairs of conics.
  • Figure 2: The rigid isotopy classes for non-generic pairs of proper conics.
  • Figure 5: Pencils of conics up to projective equivalence (beginning).
  • Figure 6: Pencils of conics up to projective equivalence (end).
  • Figure 7: isotopy classes for representatives $U_{21}$.

Theorems & Definitions (37)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • ...and 27 more