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Extremal Area of Polygons, sliding along a Circle

Dirk Siersma

Abstract

We determine all critical configurations for the Area function on polygons with vertices on a circle or an ellipse. For isolated critical points we compute their Morse index, resp index of the gradient vector field. We relate the computation at an isolated degenerate point to an eigenvalue question about combinations. In the even dimensional case non-isolated singularities occur as `zigzag trains'.

Extremal Area of Polygons, sliding along a Circle

Abstract

We determine all critical configurations for the Area function on polygons with vertices on a circle or an ellipse. For isolated critical points we compute their Morse index, resp index of the gradient vector field. We relate the computation at an isolated degenerate point to an eigenvalue question about combinations. In the even dimensional case non-isolated singularities occur as `zigzag trains'.

Paper Structure

This paper contains 9 sections, 5 theorems, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Polygons in a circle
  3. Eisenbud-Levine-Khimshiashvili index computations
  4. Milnor Algebra
  5. The general case
  6. The case of our 3-jet
  7. The multiplication matrix
  8. A combinatorial question
  9. Final step; computation of the index

Key Result

Theorem 1

The signed area function for polygons on a circle (defined on the reduced configuration space) has critical points iff all $|\alpha_i |$ are equal. These critical points are isolated or (if the number of vertices is even) contain also a 1-dimensional singular set. Moreover

Figures (3)

  • Figure 1: Some critical configurations
  • Figure 2: Extremal values, listed with$|\alpha_i |$
  • Figure 3: $B_{2,2}$ and $B_{3,3}$; coloured by $b_p$.

Theorems & Definitions (11)

  • Definition 1
  • Theorem 1
  • proof
  • Example 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Example 2
  • Proposition 4
  • proof
  • ...and 1 more