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All paths admit trajectoids

Péter L. Várkonyi

Abstract

In a recent paper published in Nature, Y.I. Sobolev et al. introduced the concept of trajectoids: convex, rigid objects, which roll without slip or spin on a flat plane along a prescribed periodic, unbounded planar path. A geometric construction method applicable to many paths was introduced, and the theory was experimentally verified using objects rolling downwards on slightly inclined planes. The construction method was applicable to many but not all curves. A possible extension of the method (referred to as period-n trajectoids) was also proposed, but the limits of applicability were not clarified. Here, a geometric proof is given for the existence of period-n trajectoids for any sufficiently smooth prescribed curve. A somewhat different proof was recently proposed by O. Muller independently from this work. We also highlight some related geometry problems.

All paths admit trajectoids

Abstract

In a recent paper published in Nature, Y.I. Sobolev et al. introduced the concept of trajectoids: convex, rigid objects, which roll without slip or spin on a flat plane along a prescribed periodic, unbounded planar path. A geometric construction method applicable to many paths was introduced, and the theory was experimentally verified using objects rolling downwards on slightly inclined planes. The construction method was applicable to many but not all curves. A possible extension of the method (referred to as period-n trajectoids) was also proposed, but the limits of applicability were not clarified. Here, a geometric proof is given for the existence of period-n trajectoids for any sufficiently smooth prescribed curve. A somewhat different proof was recently proposed by O. Muller independently from this work. We also highlight some related geometry problems.
Paper Structure (8 sections, 5 theorems, 9 equations, 2 figures)

This paper contains 8 sections, 5 theorems, 9 equations, 2 figures.

Table of Contents

  1. Background
  2. A new existence theorem of trajectoids
  3. Basic idea of the new existence theorem
  4. Uniqueness of $D_K$
  5. Existence of $\Delta_K$
  6. The spherical distance of $A$ and $B$, and the area of $\Delta_{K}$
  7. The new existence theorem
  8. Some related problems

Key Result

Theorem 1

If for some path $T$ there exists $K\in\mathbb{R}^+$ such that the corresponding spherical curve $T_K$ is closed and it consists of $n\in \mathbb{N^+}$ full periods $T_K^\star$, then $T$ admits a period-$n$ trajectoid.

Figures (2)

  • Figure 1: A: the curves $T$ and $T_K$. Right: B: Cylinders and cones are trajectoids of straight, and circular paths, respectively.
  • Figure 2: A sphere of radius $K^{-1}$ with the curve $T_K^\star$. The circle segments BC and AC are of equal length, and belong to great circles (depicted as dashed curves). Property $2\pi/n$ means that the signed area of the shaded spherical region $R_{K,2\pi/n}$ equals $2K^{-2}\pi/n$.

Theorems & Definitions (9)

  • Theorem 1: sobolev2023solid
  • Definition 1
  • Theorem 2: sobolev2023solid
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof