GaussCircleVisualizer

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概述 | Overview

GaussCircleVisualizer 是一个基于 Python 的工具，旨在探索和可视化以原点为中心的圆内整数格点的分布。本项目深入研究经典的 高斯圆问题 通过生成格点，根据它们到原点的欧几里得距离进行分组，并提供这些分布的直观可视化表示。

GaussCircleVisualizer is a Python-based tool designed to explore and visualize the distribution of integer lattice points within a circle centered at the origin. This project delves into the classic Gauss Circle Problem by generating lattice points, grouping them based on their Euclidean distances from the origin, and providing insightful visual representations of these distributions.

功能 | Features

• 整数格点生成 | Integer Lattice Point Generation: 生成所有在指定区域内的整数点 ((x, y))（满足 (0 < y < x)）。

  Integer Lattice Point Generation: Generates all integer points ((x, y)) within a specified region ((0 < y < x)).

• 距离计算与分组 | Distance Calculation & Grouping: 计算每个点到原点的欧几里得距离，并将具有相同或近似距离的点分组。

  Distance Calculation & Grouping: Computes the Euclidean distance of each point from the origin and groups points with identical or near-identical distances.

• 动态可视化 | Dynamic Visualization:

  • 圆弧连接 | Circular Connections: 使用彩色圆弧连接位于同一半径上的点，突出它们的角度分布。

    Circular Connections: Connects points that lie on the same circle with colored arcs, highlighting their angular distribution.

  • 顺序连接 | Sequential Connections: 使用黑色直线按距离递增顺序连接具有唯一距离的点，并对其进行标注。

    Sequential Connections: Links points with unique distances using black lines, ordered by increasing distance, and labels them for easy reference.

• 可自定义参数 | Customizable Parameters: 调整最大坐标值，切换连接线和标签的可见性等。

  Customizable Parameters: Adjust the maximum coordinate value, toggle visibility of connection lines and labels, and more.

• 交互式绘图 | Interactive Plotting: 利用 matplotlib 提供交互式和高质量的可视化输出。

  Interactive Plotting: Utilizes matplotlib for interactive and high-quality visual outputs.

目录 | Table of Contents

• GaussCircleVisualizer
  • 概述 | Overview
  • 功能 | Features
  • 目录 | Table of Contents
  • 安装 | Installation
    • 前提条件 | Prerequisites
    • 步骤 | Steps
  • 使用方法 | Usage
    • 命令行参数 | Command-Line Arguments
    • 自定义 | Customization
  • 示例 | Examples
  • 贡献 | Contributing
    • 贡献步骤 | Steps to Contribute
  • 许可证 | License
  • 鸣谢 | Acknowledgements

安装 | Installation

前提条件 | Prerequisites

• Python 3.7 或更高版本 | Python 3.7 or higher
• pip

步骤 | Steps

1. 克隆仓库 | Clone the Repository

   git clone https://github.com/yourusername/GaussCircleVisualizer.git
   cd GaussCircleVisualizer

2. 创建虚拟环境（可选，但推荐） | Create a Virtual Environment (Optional but Recommended)

   python -m venv venv
   source venv/bin/activate  # 在 Windows 上使用: venv\Scripts\activate

3. 安装所需依赖 | Install Required Dependencies

   pip install -r requirements.txt

   如果没有提供 requirements.txt，请手动安装依赖：

   If requirements.txt is not provided, install dependencies manually:

   pip install matplotlib numpy

使用方法 | Usage

运行主脚本以生成并可视化格点：

Run the main script to generate and visualize the lattice points:

python gauss_circle_visualizer.py

命令行参数 | Command-Line Arguments

目前，脚本使用预定义参数。未来的更新可能包括命令行参数支持以增强自定义功能。

Currently, the script uses predefined parameters. Future updates may include command-line argument support for enhanced customization.

自定义 | Customization

在 gauss_circle_visualizer.py 脚本中，您可以调整以下参数：

Within the gauss_circle_visualizer.py script, you can adjust parameters such as:

• max_x: 定义生成格点的最大 x 坐标值。

  max_x: Defines the maximum x-coordinate for generating lattice points.

• show_lines: 切换是否显示黑色顺序连接线。

  show_lines: Toggle the visibility of black sequential connection lines.

• show_numbers: 切换是否在点上显示编号。

  show_numbers: Toggle the visibility of point labels.

Example snippet:

def main(show_lines=True, show_numbers=True):
    max_x = 20  # 根据需要调整
    ...

Example snippet:

def main(show_lines=True, show_numbers=True):
    max_x = 20  # Adjust as needed
    ...

示例 | Examples

上述图像展示了在区域 (0 < y < x) 内的格点，具有相同距离的点通过彩色圆弧连接，具有唯一距离的点通过黑色直线连接并进行编号。

The above image showcases lattice points within the region (0 < y < x), with points sharing the same distance from the origin connected by colored arcs, and unique-distance points connected sequentially with black lines and labeled.

贡献 | Contributing

欢迎贡献！无论是报告错误、建议功能，还是提交 Pull Request，您的参与都非常重要。

Contributions are welcome! Whether it's reporting bugs, suggesting features, or submitting pull requests, your involvement is highly appreciated.

贡献步骤 | Steps to Contribute

1. Fork 仓库 | Fork the Repository

2. 创建功能分支 | Create a Feature Branch

   git checkout -b feature/YourFeatureName

3. 提交您的更改 | Commit Your Changes

   git commit -m "Add your detailed description"

4. 推送到分支 | Push to the Branch

   git push origin feature/YourFeatureName

5. 打开 Pull Request | Open a Pull Request

许可证 | License

本项目采用 GNU 通用公共许可证 v3.0。

This project is licensed under the GNU General Public License v3.0.

鸣谢 | Acknowledgements

• 卡尔·弗里德里希·高斯: 为高斯圆问题奠定了基础。(这真的需要吗???)

  Carl Friedrich Gauss: For the foundational Gauss Circle Problem. (Is this really needed???)

• Matplotlib & NumPy: 数据可视化和数值计算的关键库。

  Matplotlib & NumPy: Essential libraries for data visualization and numerical computations.

• OpenAI ChatGPT: 协助项目文档和描述。

  OpenAI ChatGPT: Assisted in project documentation and description.