Roullete curves, coin paradox and Aristotle's wheel paradox

TL;DR

This paper explores roulette curves, coin paradox, and Aristotle's wheel paradox, using geometric intuition and Python visualizations to explain epicycloids and hypocycloids, with pedagogical and computational insights.

Contribution

It provides a detailed geometric and computational analysis of roulette curves, linking classical paradoxes to modern visualizations and parametric equations.

Findings

Derived parametric equations for epicycloids and hypocycloids

Provided Python code for visualizing and animating the curves

Enhanced understanding of classical paradoxes through geometric intuition

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.