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Poncelet Triangles and Tetragons over Finite Fields

Milena Radnović, Ruzzel Ragas

Abstract

In the projective plane over a finite field of characteristic not equal to 2, we compute the probability that a randomly selected pair of distinct conics $(\mathscr{A},\mathscr{B})$, with $\mathscr{A}$ smooth or singular and $\mathscr{B}$ smooth, in a fixed pencil of conics will admit a triangle or a tetragon inscribed in $\mathscr{A}$ and circumscribed about $\mathscr{B}$. We do this for all pencils, classified up to projective automorphism, with at least one smooth conic; effectively allowing the case where our conic pairs intersect non-transversally.

Poncelet Triangles and Tetragons over Finite Fields

Abstract

In the projective plane over a finite field of characteristic not equal to 2, we compute the probability that a randomly selected pair of distinct conics , with smooth or singular and smooth, in a fixed pencil of conics will admit a triangle or a tetragon inscribed in and circumscribed about . We do this for all pencils, classified up to projective automorphism, with at least one smooth conic; effectively allowing the case where our conic pairs intersect non-transversally.

Paper Structure

This paper contains 8 sections, 26 theorems, 31 equations, 1 figure, 10 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Poncelet's Theorem and Cayley's Condition
  4. Computation of Probabilities
  5. Probabilities for Poncelet Triangle
  6. Probabilities for Poncelet Tetragon
  7. Conclusion
  8. Classification of Pencils of Conics

Key Result

Theorem 2.1

Let $(\mathscr{A},\mathscr{B})$ be a pair of conics in $\boldsymbol{P}^2(\mathbb{F}_q)$. If there exists an $n$-sided polygon that is inscribed in $\mathscr{A}$ and circumscribed about $\mathscr{B}$, then every point in $\mathscr{A}$ is a vertex of an $n$-sided polygon that is inscribed in $\mathscr

Figures (1)

  • Figure 3.1: Poncelet tetragon in $\mathbb{F}_{11}^2$

Theorems & Definitions (58)

  • Theorem 2.1: Poncelet ber1987geomjowlia2023ponceletpon1822poncelet
  • Theorem 2.2: Cayley cay1854cayleygrihar1978cayley
  • Definition 3.1
  • Remark 1
  • Remark 2
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 48 more