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On the Complexity of Bicycle Unitracks

Ivan Molodyk

TL;DR

The paper analyzes Finn's unitrack construction for bicycle tracks by modeling the bicycle as a fixed-length segment whose rear track is tangent to the segment. It introduces a length-expanding dynamical map on front/back track pairs and confines the evolution to a horizontally constrained class of curves, then proves that the unitrack curves cannot remain graphs of functions unless trivial. The main contributions are (i) a rigorous contradiction argument showing any nontrivial initial rear track cannot generate all subsequent tracks as graphs, (ii) a detailed analysis of horizontal and vertical amplitudes showing linear lower bounds on horizontal spread and unbounded vertical growth, and (iii) establishment of linear growth rates for horizontal amplitude with explicit bounds $n-c_1\le H_n\le 2n-c_2$, implying self-intersections and expansive behavior in Finn's unitrack construction.

Abstract

This paper concerns the geometry of bicycle tracks. We model bicycle as an oriented segment of a fixed length that is moving in the Euclidean plane so that the trajectory of the rear point is tangent to the segment at all times. The trajectories of front and back points of the segment are called bicycle tracks, and one asks if it is possible that the front track is contained in the rear track (other than when they are straight lines). Such curves are called unitracks or unicycle tracks. In 2002 D. Finn proposed a construction of unitracks that are obtained as a union of a sequence of curves. Numerical evidence suggested that these curves behave expansively and that various numerical characteristics of the curves grow quickly in the sequence. In this paper we prove that the curves that form a unitrack in Finn's construction cannot remain graphs of functions, unless they are straight lines. We conclude that the horizontal amplitude of the curves has a linear growth rate between 1 and 2.

On the Complexity of Bicycle Unitracks

TL;DR

The paper analyzes Finn's unitrack construction for bicycle tracks by modeling the bicycle as a fixed-length segment whose rear track is tangent to the segment. It introduces a length-expanding dynamical map on front/back track pairs and confines the evolution to a horizontally constrained class of curves, then proves that the unitrack curves cannot remain graphs of functions unless trivial. The main contributions are (i) a rigorous contradiction argument showing any nontrivial initial rear track cannot generate all subsequent tracks as graphs, (ii) a detailed analysis of horizontal and vertical amplitudes showing linear lower bounds on horizontal spread and unbounded vertical growth, and (iii) establishment of linear growth rates for horizontal amplitude with explicit bounds , implying self-intersections and expansive behavior in Finn's unitrack construction.

Abstract

This paper concerns the geometry of bicycle tracks. We model bicycle as an oriented segment of a fixed length that is moving in the Euclidean plane so that the trajectory of the rear point is tangent to the segment at all times. The trajectories of front and back points of the segment are called bicycle tracks, and one asks if it is possible that the front track is contained in the rear track (other than when they are straight lines). Such curves are called unitracks or unicycle tracks. In 2002 D. Finn proposed a construction of unitracks that are obtained as a union of a sequence of curves. Numerical evidence suggested that these curves behave expansively and that various numerical characteristics of the curves grow quickly in the sequence. In this paper we prove that the curves that form a unitrack in Finn's construction cannot remain graphs of functions, unless they are straight lines. We conclude that the horizontal amplitude of the curves has a linear growth rate between 1 and 2.

Paper Structure

This paper contains 14 sections, 19 theorems, 40 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Basic Properties
  3. Horizontal Unitrack
  4. Horizontal Unitrack Properties
  5. Proof of Theorem \ref{['graphs']}
  6. Plan of the proof
  7. Supplementary definitions
  8. Projection on horizontal direction.
  9. Estimating the length of segments of $\gamma_0$
  10. Final step
  11. Proof of Theorem\ref{['hor-ampl']}
  12. Supplementary definitions
  13. Vertical Tangencies
  14. Estimates

Key Result

Lemma 2.1

If $\gamma\in X$, then $\mathop{\rm Len }(\varphi(\gamma)) \geqslant \mathop{\rm Len }(\gamma)$, where $\mathop{\rm Len }$ is length.

Figures (6)

  • Figure 1: The first 5 segments of a unitrack with $\gamma_0$ being the graph of a function $f(t) = 4e^{\frac{-1}{t(1-t)}}$. Image credit to Stan Wagon.
  • Figure 2: The curve $\varphi(\gamma)$ is the front track of a bicycle, where $\gamma$ is the back track.
  • Figure 3: Curve $\gamma_0$ defines the next curve $\gamma_1$
  • Figure 6: The definition of $s_n$.
  • Figure 7: The configuration of points for large $n$.
  • ...and 1 more figures

Theorems & Definitions (35)

  • Lemma 2.1
  • proof : Proof
  • Proposition 2.2
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Conjecture 4.4
  • Conjecture 4.5
  • ...and 25 more