⏱️ The Cycloid: Nature’s Fastest Curve

This repository explores two fascinating physical and mathematical properties of the Cycloid curve:

• 📉 Brachistochrone — the curve of fastest descent under gravity without any friction

• ⏲️ Isochronous — objects released from any point on the curve reach the bottom simultaneously

  Tautochrone Curve	Brachistochrone Curve

📘 What's Inside

This repository contains:

• 📓 CycloidsAnalysisNotebook.ipynb — A detailed Jupyter Notebook containing derivations, visualizations, and animations.
• 📂 IMAGES/ — Static figures and diagrams used throughout the notebook.
• 📂 ANIMATION/ — Rolling ball animations along cycloid, parabola, circular arc, and straight line.
• 📄 requirements.txt — Python dependencies to run the notebook smoothly.

✨ Highlights

📐 Derivations and Comparisons

• Derived the equation of the Brachistochrone using energy conservation and the Euler-Lagrange equation
• Compared the cycloid to a parabolic path, circular arc, and straight line — all sharing the same endpoints

⏳ Time of Descent Calculations

• Analytical expressions for cycloid and straight line
• Numerical integration (scipy.integrate.quad) for more complex paths

🎥 Realistic Rolling Ball Animation

• Accounted for the normal vector offset to make the ball appear to roll on the curve, not inside it

  

📊 Slope Comparison and Timing Table

• Compared the cycloid to lines with varying slopes

• Included a table showing descent times vs intermediate cycloid times

  

⏲️ Isochronous Demonstration

• Multiple balls released from different heights on the same cycloidal path
• All balls reach the bottom at the same time
• Extended to 3D animation across multiple cycloidal paths

🛠️ Setup

Clone the repository and install dependencies:

git clone https://github.com/yourusername/Cycloid-Brachistochrone-Isochronous-Exploration.git
cd Cycloid-Brachistochrone-Isochronous-Exploration
pip install -r requirements.txt

Then launch the notebook:

jupyter notebook CycloidsAnalysisNotebook.ipynb

📎 License

This project is licensed under the MIT License.