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Automorphism Visualization in the Unit Disk\n\n## 📌 Overview\nThis repository provides a visual exploration of **automorphisms of the unit disk** in complex analysis.  \nIt demonstrates how Möbius transformations map the unit disk onto itself while preserving its geometric structure.\n\nThe goal of this project is to build **intuition through visualization** of conformal mappings—something that is often difficult to grasp purely analytically.\n\n---\n\n## 🧠 Mathematical Background\n\nAn automorphism of the unit disk is given by the transformation:\n\nφₐ(z) = (z + a) / (1 + āz)\n\nThese transformations:\n- Map the unit disk to itself  \n- Are conformal (angle-preserving)  \n- Form a group under composition  \n\nThey are fundamental in **complex analysis and conformal mapping theory** :contentReference[oaicite:0]{index=0}\n\n---\n\n## 🎯 Features\n\n- Visualization of unit disk automorphisms\n- Grid deformation under conformal maps\n- Interactive / parametric transformation behavior\n- Intuitive understanding of Möbius transformations\n\n---\n\n## 🖼️ Example Visualizations\n\nBelow are sample outputs from the project demonstrating how the unit disk is transformed.\n\n### Example 1\n![Example 1](Images/Image1.png)\n\n### Example 2\n![Example 2](Images/automorphism_view1.png)\n\n### Example 3\n![Example 3](Images/automorphism_view_grid_1.png)\n\n### Example 4\n![Example 4](Images/automorphism_view_grid_2.png)\n\n### Example 5\n![Example 5](Images/automorphism_view_grid_3.png)\n\n### Example 6\n![Example 6](Images/automorphism_view_polar_2.png)\n\n### Example 7\n![Example 7](Images/automorphism_view_polar_3.png)\n\n\n---\n\n## ⚙️ How It Works\n\nThe visualization is based on applying the automorphism:\n\n- Points inside the unit disk are transformed via complex mapping\n- A structured grid (Polar) is deformed\n- The unit circle remains invariant\n\nThis allows direct observation of:\n- Boundary preservation  \n- Nonlinear distortion  \n- Conformal nature of the mapping  \n\n---\n\n## 🚀 Getting Started\n\n### 1. Clone the repository\n```bash\ngit clone https://github.com/Samson-Mano/Automorphism_visualization.git\ncd Automorphism_visualization\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fsamson-mano%2Fautomorphism_visualization","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fsamson-mano%2Fautomorphism_visualization","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fsamson-mano%2Fautomorphism_visualization/lists"}