Roullete curves, coin paradox and Aristotle's wheel paradox

TL;DR

The paper investigates roulette curves generated by rolling without slipping, focusing on epicycloids, hypocycloids, and classic paradoxes such as the coin paradox and Aristotle's wheel. It derives the parametric equations for epicycloids $x(\theta)= (R+r)\cos\theta - r\cos(\frac{R+r}{r}\theta)$, $y(\theta)= (R+r)\sin\theta - r\sin(\frac{R+r}{r}\theta)$ and for hypocycloids $x(\theta)= (R-r)\cos\theta + r\cos(\frac{R-r}{r}\theta)$, $y(\theta)= (R-r)\sin\theta - r\sin(\frac{R-r}{r}\theta)$, explaining cusp counts and rotation numbers via $k=R/r$ and the no-slip constraint. It uses trochoidal insight to resolve Aristotle's wheel paradox and demonstrates how the center-path geometry leads to intuitive and rigorous understanding. The work is complemented by Python code that generates static figures and animations of the curves, providing a practical educational toolkit and visual intuition for undergraduate physics and geometry. Overall, the article bridges geometry, kinematics, and computation to elucidate roulette curves and their educational value in teaching motion constraints.

Abstract

This work discusses the concept of roulette, the generated curves that occur when one curve rolls without slipping along another, tracing the path of a fixed point. The coin paradox and Aristotle's wheel paradox are used as pedagogical motivations to discuss the parametric equations of epicycloids and hypocycloids, providing a geometrical intuition for the mathematical derivations and computational implementation of those curves. Python code is provided to motivate the application of the derived parametric equations, resulting in concrete visualizations and animations.

Figures (10)

• Figure 1: Illustration of the coin rotation paradox: a coin rolling externally around another identical coin completes two full rotations before returning to its initial position. The image shows a Brazilian one real coin (1 BRL).
• Figure 2: Geometry of a circle of radius $r$ rolling without slipping around a fixed circle of radius $R$. The center of the moving circle traces a circular path of radius $R+r$, leading to $(R+r)/r$ full rotations about its own axis.
• Figure 3: Example of an epicycloid generated by a circle of radius $r$ rolling around a fixed circle of radius $R$. The number of cusps depends on the ratio $R/r$. In this figure, $R = 3r$, and the number of turns completed by the smaller circle equals 4. The name of this epicycloid is trefoiloid Lawrence1972.
• Figure 4: Construction of the epicycloid. A generating circle of radius $r$ rolls externally around a fixed circle of radius $R$, and a point $B$ on its circumference traces the epicycloid curve.
• Figure 5: Examples of epicycloids for different values of $k = R/r$.