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Uniqueness of Maximal Inscribed Parabolas and Minimal Circumscribing Horocycles

Martin Lukarevski, Hans-Peter Schröcker

TL;DR

The paper addresses the problem of uniqueness for extremal one-parameter conics in two geometries: maximal parabolas inscribed in Euclidean convex sets and minimal horocycles circumscribing sets in hyperbolic geometry. It develops a projective-conic framework with polarity and dual pencils, and uses a size notion based on a parabola parameter and horocycle major axis to compare non-compact conics. The main contributions are: (i) the existence of exactly three max-exparabolas for any triangle, (ii) a general uniqueness theorem for maximal parabolas in convex sets under an existence assumption, and (iii) a uniqueness result for minimal enclosing horocycles when the enclosing size is below $2^{-1/2}$. These results extend extremal-conic uniqueness beyond ellipsoids to parabolas and horocycles, with potential applications in triangle geometry and hyperbolic geometry.

Abstract

We prove existence of three unique ``max-exparabolas'' to a triangle. Each of these parabolas is internally tangent to one edge and the two other sides. Among all like parabolas, it is characterized by having maximal parameter. We use this result to prove a more general uniqueness statement on maximal parabolas in a convex point set. In similar spirit, we demonstrate uniqueness of minimal enclosing horocycles in hyperbolic geometry, provided the enclosed set is sufficiently small.

Uniqueness of Maximal Inscribed Parabolas and Minimal Circumscribing Horocycles

TL;DR

The paper addresses the problem of uniqueness for extremal one-parameter conics in two geometries: maximal parabolas inscribed in Euclidean convex sets and minimal horocycles circumscribing sets in hyperbolic geometry. It develops a projective-conic framework with polarity and dual pencils, and uses a size notion based on a parabola parameter and horocycle major axis to compare non-compact conics. The main contributions are: (i) the existence of exactly three max-exparabolas for any triangle, (ii) a general uniqueness theorem for maximal parabolas in convex sets under an existence assumption, and (iii) a uniqueness result for minimal enclosing horocycles when the enclosing size is below . These results extend extremal-conic uniqueness beyond ellipsoids to parabolas and horocycles, with potential applications in triangle geometry and hyperbolic geometry.

Abstract

We prove existence of three unique ``max-exparabolas'' to a triangle. Each of these parabolas is internally tangent to one edge and the two other sides. Among all like parabolas, it is characterized by having maximal parameter. We use this result to prove a more general uniqueness statement on maximal parabolas in a convex point set. In similar spirit, we demonstrate uniqueness of minimal enclosing horocycles in hyperbolic geometry, provided the enclosed set is sufficiently small.

Paper Structure

This paper contains 10 sections, 4 theorems, 29 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Common Interior of Two Conics
  4. Parabolas and their Size
  5. Horocycles and their Size
  6. Maximal Parabolas
  7. Max-Exparabolas of a Triangle
  8. Uniqueness of Maximal Parabolas
  9. Uniqueness of Minimal Horocycles
  10. Conclusions

Key Result

Theorem 1

There exist precisely three max-exparabolas to a triangle $\triangle ABC$, one in each of the three possible intersections of one negative and two positive half-planes.

Figures (3)

  • Figure 1: Triangle with three max-exparabolas $P_A$, $P_B$, and $P_C$.
  • Figure 2: The horocycle $H$ contains the common interior of $H_0$ and $H_1$. In the left image, it is smaller than $H_0$ and $H_1$, in the right image, it is larger.
  • Figure 3: Point set with non-unique minimal enclosing horocycle.

Theorems & Definitions (13)

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