Automorphism Visualization in the Unit Disk

📌 Overview

This repository provides a visual exploration of automorphisms of the unit disk in complex analysis.
It demonstrates how Möbius transformations map the unit disk onto itself while preserving its geometric structure.

The goal of this project is to build intuition through visualization of conformal mappings—something that is often difficult to grasp purely analytically.

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🧠 Mathematical Background

An automorphism of the unit disk is given by the transformation:

φₐ(z) = (z + a) / (1 + āz)

These transformations:

• Map the unit disk to itself
• Are conformal (angle-preserving)
• Form a group under composition

They are fundamental in complex analysis and conformal mapping theory :contentReference[oaicite:0]{index=0}

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🎯 Features

• Visualization of unit disk automorphisms
• Grid deformation under conformal maps
• Interactive / parametric transformation behavior
• Intuitive understanding of Möbius transformations

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🖼️ Example Visualizations

Below are sample outputs from the project demonstrating how the unit disk is transformed.

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

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⚙️ How It Works

The visualization is based on applying the automorphism:

• Points inside the unit disk are transformed via complex mapping
• A structured grid (Polar) is deformed
• The unit circle remains invariant

This allows direct observation of:

• Boundary preservation
• Nonlinear distortion
• Conformal nature of the mapping

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🚀 Getting Started

1. Clone the repository

git clone https://github.com/Samson-Mano/Automorphism_visualization.git
cd Automorphism_visualization