https://github.com/samson-mano/automorphism_visualization

Complex analysis visualization of automorphism on the unit disk
https://github.com/samson-mano/automorphism_visualization

Last synced: 20 days ago
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Complex analysis visualization of automorphism on the unit disk

• Host: GitHub
• URL: https://github.com/samson-mano/automorphism_visualization
• Owner: Samson-Mano
• License: mit
• Created: 2025-09-24T05:33:26.000Z (10 months ago)
• Default Branch: master
• Last Pushed: 2025-10-11T19:25:36.000Z (10 months ago)
• Last Synced: 2025-10-27T15:53:53.609Z (9 months ago)
• Language: C#
• Size: 726 KB
• Stars: 0
• Watchers: 0
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: LICENSE.txt

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README

          
# 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 1](Images/Image1.png)

### Example 2

![Example 2](Images/automorphism_view1.png)

### Example 3

![Example 3](Images/automorphism_view_grid_1.png)

### Example 4

![Example 4](Images/automorphism_view_grid_2.png)

### Example 5

![Example 5](Images/automorphism_view_grid_3.png)

### Example 6

![Example 6](Images/automorphism_view_polar_2.png)

### Example 7

![Example 7](Images/automorphism_view_polar_3.png)

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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

```bash

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

cd Automorphism_visualization