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  • 01_tio-boot 简介

    • tio-boot:新一代高性能 Java Web 开发框架
    • tio-boot 入门示例
    • Tio-Boot 配置 : 现代化的配置方案
    • tio-boot 整合 Logback
    • tio-boot 整合 hotswap-classloader 实现热加载
    • 自行编译 tio-boot
    • 最新版本
    • 开发规范
  • 02_部署

    • 使用 Maven Profile 实现分环境打包 tio-boot 项目
    • Maven 项目配置详解:依赖与 Profiles 配置
    • tio-boot 打包成 FastJar
    • 使用 GraalVM 构建 tio-boot Native 程序
    • 使用 Docker 部署 tio-boot
    • 部署到 Fly.io
    • 部署到 AWS Lambda
    • 到阿里云云函数
    • 使用 Deploy 工具部署
    • 胖包与瘦包的打包与部署
    • 使用 Jenkins 部署 Tio-Boot 项目
    • 使用 Nginx 反向代理 Tio-Boot
    • 使用 Supervisor 管理 Java 应用
  • 03_配置

    • 配置参数
    • 服务器监听器
    • 内置缓存系统 AbsCache
    • 使用 Redis 作为内部 Cache
    • 静态文件处理器
    • 基于域名的静态资源隔离
    • DecodeExceptionHandler
  • 04_原理

    • 生命周期
    • 请求处理流程
    • 重要的类
  • 05_json

    • Json
    • 接受 JSON 和响应 JSON
    • 响应实体类
  • 06_web

    • 概述
    • 文件上传
    • 接收请求参数
    • 接收日期参数
    • 接收数组参数
    • 返回字符串
    • 返回文本数据
    • 返回网页
    • 请求和响应字节
    • 文件下载
    • 返回视频文件并支持断点续传
    • http Session
    • Cookie
    • HttpRequest
    • HttpResponse
    • Resps
    • RespBodyVo
    • /zh/06_web/19.html
    • 全局异常处理器
    • 异步
    • 动态 返回 CSS 实现
    • 返回图片
    • Transfer-Encoding: chunked 实时音频播放
    • Server-Sent Events (SSE)
    • 接口访问统计
    • 接口请求和响应数据记录
    • 自定义 Handler 转发请求
    • 使用 HttpForwardHandler 转发所有请求
    • 跨域
    • 添加 Controller
    • 常用工具类
    • HTTP Basic 认证
    • WebJars
    • JProtobuf
  • 07_validate

    • 数据紧校验规范
    • 参数校验
  • 08_websocket

    • 使用 tio-boot 搭建 WebSocket 服务
    • WebSocket 聊天室项目示例
  • 09_java-db

    • java‑db
    • 操作数据库入门示例
    • SQL 模板
    • 数据源配置与使用
    • ActiveRecord
    • Model
    • 生成器与 Model
    • Db 工具类
    • 批量操作
    • 数据库事务处理
    • Cache 缓存
    • Dialect 多数据库支持
    • 表关联操作
    • 复合主键
    • Oracle 支持
    • Enjoy SQL 模板
    • Java-DB 整合 Enjoy 模板最佳实践
    • 多数据源支持
    • 独立使用 ActiveRecord
    • 调用存储过程
    • java-db 整合 Guava 的 Striped 锁优化
    • 生成 SQL
    • 通过实体类操作数据库
    • java-db 读写分离
    • Spring Boot 整合 Java-DB
    • like 查询
    • 常用操作示例
    • Druid 监控集成指南
    • SQL 统计
  • 10_api-table

    • ApiTable 概述
    • 使用 ApiTable 连接 SQLite
    • 使用 ApiTable 连接 Mysql
    • 使用 ApiTable 连接 Postgres
    • 使用 ApiTable 连接 TDEngine
    • 使用 api-table 连接 oracle
    • 使用 api-table 连接 mysql and tdengine 多数据源
    • EasyExcel 导出
    • EasyExcel 导入
    • TQL(Table SQL)前端输入规范
    • ApiTable 实现增删改查
    • 数组类型
    • 单独使用 ApiTable
  • 11_aop

    • JFinal-aop
    • Aop 工具类
    • 配置
    • 配置
    • 独立使用 JFinal Aop
    • @AImport
    • 原理解析
  • 12_cache

    • Caffine
    • Jedis-redis
    • hutool RedisDS
    • Redisson
    • Caffeine and redis
    • CacheUtils 工具类
    • 使用 CacheUtils 整合 caffeine 和 redis 实现的两级缓存
    • 使用 java-db 整合 ehcache
    • 使用 java-db 整合 redis
    • Java DB Redis 相关 Api
    • redis 使用示例
  • 13_认证和权限

    • hutool-JWT
    • FixedTokenInterceptor
    • 使用内置 TokenManager 实现登录
    • 用户系统
    • 重置密码
    • 匿名登录
    • Google 登录
    • 权限校验注解
    • Sa-Token
    • sa-token 登录注册
    • StpUtil.isLogin() 源码解析
    • 短信登录
    • 移动端微信登录实现指南
    • 移动端重置密码
  • 14_i18n

    • i18n
  • 15_enjoy

    • tio-boot 整合 Enjoy 模版引擎文档
    • 引擎配置
    • 表达式
    • 指令
    • 注释
    • 原样输出
    • Shared Method 扩展
    • Shared Object 扩展
    • Extension Method 扩展
    • Spring boot 整合
    • 独立使用 Enjoy
    • tio-boot enjoy 自定义指令 localeDate
    • PromptEngine
    • Enjoy 入门示例-擎渲染大模型请求体
    • Enjoy 使用示例
  • 16_定时任务

    • Quartz 定时任务集成指南
    • 分布式定时任务 xxl-jb
    • cron4j 使用指南
  • 17_tests

    • TioBootTest 类
  • 18_tio

    • TioBootServer
    • tio-core
    • 内置 TCP 处理器
    • 独立启动 UDPServer
    • 使用内置 UDPServer
    • t-io 消息处理流程
    • tio-运行原理详解
    • TioConfig
    • ChannelContext
    • Tio 工具类
    • 业务数据绑定
    • 业务数据解绑
    • 发送数据
    • 关闭连接
    • Packet
    • 监控: 心跳
    • 监控: 客户端的流量数据
    • 监控: 单条 TCP 连接的流量数据
    • 监控: 端口的流量数据
    • 单条通道统计: ChannelStat
    • 所有通道统计: GroupStat
    • 资源共享
    • 成员排序
    • ssl
    • DecodeRunnable
    • 使用 AsynchronousSocketChannel 响应数据
    • 拉黑 IP
    • 深入解析 Tio 源码:构建高性能 Java 网络应用
  • 19_aio

    • ByteBuffer
    • AIO HTTP 服务器
    • 自定义和线程池和池化 ByteBuffer
    • AioHttpServer 应用示例 IP 属地查询
    • 手写 AIO Http 服务器
  • 20_netty

    • Netty TCP Server
    • Netty Web Socket Server
    • 使用 protoc 生成 Java 包文件
    • Netty WebSocket Server 二进制数据传输
    • Netty 组件详解
  • 21_netty-boot

    • Netty-Boot
    • 原理解析
    • 整合 Hot Reload
    • 整合 数据库
    • 整合 Redis
    • 整合 Elasticsearch
    • 整合 Dubbo
    • Listener
    • 文件上传
    • 拦截器
    • Spring Boot 整合 Netty-Boot
    • SSL 配置指南
    • ChannelInitializer
    • Reserve
  • 22_MQ

    • Mica-mqtt
    • EMQX
    • Disruptor
  • 23_tio-utils

    • tio-utils
    • HttpUtils
    • Notification
    • 邮箱
    • JSON
    • 读取文件
    • Base64
    • 上传和下载
    • Http
    • Telegram
    • RsaUtils
    • EnvUtils 使用文档
    • 系统监控
    • 毫秒并发 ID (MCID) 生成方案
  • 24_tio-http-server

    • 使用 Tio-Http-Server 搭建简单的 HTTP 服务
    • tio-boot 添加 HttpRequestHandler
    • 在 Android 上使用 tio-boot 运行 HTTP 服务
    • tio-http-server-native
    • handler 常用操作
  • 25_tio-websocket

    • WebSocket 服务器
    • WebSocket Client
  • 26_tio-im

    • 通讯协议文档
    • ChatPacket.proto 文档
    • java protobuf
    • 数据表设计
    • 创建工程
    • 登录
    • 历史消息
    • 发消息
  • 27_mybatis

    • Tio-Boot 整合 MyBatis
    • 使用配置类方式整合 MyBatis
    • 整合数据源
    • 使用 mybatis-plus 整合 tdengine
    • 整合 mybatis-plus
  • 28_mongodb

    • tio-boot 使用 mongo-java-driver 操作 mongodb
  • 29_elastic-search

    • Elasticsearch
    • JavaDB 整合 ElasticSearch
    • Elastic 工具类使用指南
    • Elastic-search 注意事项
    • ES 课程示例文档
  • 30_magic-script

    • tio-boot 整合 magic-script
  • 31_groovy

    • tio-boot 整合 Groovy
  • 32_firebase

    • 整合 google firebase
    • Firebase Storage
    • Firebase Authentication
    • 使用 Firebase Admin SDK 进行匿名用户管理与自定义状态标记
    • 导出用户
    • 注册回调
    • 登录注册
  • 33_文件存储

    • 文件上传数据表
    • 本地存储
    • 使用 AWS S3 存储文件并整合到 Tio-Boot 项目中
    • 存储文件到 腾讯 COS
  • 34_spider

    • jsoup
    • 爬取 z-lib.io 数据
    • 整合 WebMagic
    • WebMagic 示例:爬取学校课程数据
    • Playwright
    • Flexmark (Markdown 处理器)
    • tio-boot 整合 Playwright
    • 缓存网页数据
  • 36_integration_thirty_party

    • tio-boot 整合 okhttp
    • 整合 GrpahQL
    • 集成 Mailjet
    • 整合 ip2region
    • 整合 GeoLite 离线库
    • 整合 Lark 机器人指南
    • 集成 Lark Mail 实现邮件发送
    • Thymeleaf
    • Swagger
    • Clerk 验证
  • 37_dubbo

    • 概述
    • dubbo 2.6.0
    • dubbo 2.6.0 调用过程
    • dubbo 3.2.0
  • 38_spring

    • Spring Boot Web 整合 Tio Boot
    • spring-boot-starter-webflux 整合 tio-boot
    • Tio Boot 整合 Spring Boot Starter
    • Tio Boot 整合 Spring Boot Starter Data Redis 指南
  • 39_spring-cloud

    • tio-boot spring-cloud
  • 40_mysql

    • 使用 Docker 运行 MySQL
    • /zh/42_mysql/02.html
  • 41_postgresql

    • PostgreSQL 安装
    • PostgreSQL 主键自增
    • PostgreSQL 日期类型
    • Postgresql 金融类型
    • PostgreSQL 数组类型
    • PostgreSQL 全文检索
    • PostgreSQL 查询优化
    • 获取字段类型
    • PostgreSQL 向量
    • PostgreSQL 优化向量查询
    • PostgreSQL 其他
  • 43_oceanbase

    • 快速体验 OceanBase 社区版
    • 快速上手 OceanBase 数据库单机部署与管理
    • 诊断集群性能
    • 优化 SQL 性能指南
    • /zh/43_oceanbase/05.html
  • 50_media

    • JAVE 提取视频中的声音
    • Jave 提取视频中的图片
    • /zh/50_media/03.html
  • 51_asr

    • Whisper-JNI
  • 54_native-media

    • java-native-media
    • JNI 入门示例
    • mp3 拆分
    • mp4 转 mp3
    • 使用 libmp3lame 实现高质量 MP3 编码
    • Linux 编译
    • macOS 编译
    • 从 JAR 包中加载本地库文件
    • 支持的音频和视频格式
    • 任意格式转为 mp3
    • 通用格式转换
    • 通用格式拆分
    • 视频合并
    • VideoToHLS
    • split_video_to_hls 支持其他语言
    • 持久化 HLS 会话
  • 55_telegram4j

    • 数据库设计
    • /zh/55_telegram4j/02.html
    • 基于 MTProto 协议开发 Telegram 翻译机器人
    • 过滤旧消息
    • 保存机器人消息
    • 定时推送
    • 增加命令菜单
    • 使用 telegram-Client
    • 使用自定义 StoreLayout
    • 延迟测试
    • Reactor 错误处理
    • Telegram4J 常见错误处理指南
  • 56_telegram-bots

    • TelegramBots 入门指南
    • 使用工具库 telegram-bot-base 开发翻译机器人
  • 60_LLM

    • 简介
    • AI 问答
    • /zh/60_LLM/03.html
    • /zh/60_LLM/04.html
    • 增强检索(RAG)
    • 结构化数据检索
    • 搜索+AI
    • 集成第三方 API
    • 后置处理
    • 推荐问题生成
    • 连接代码执行器
    • 避免 GPT 混乱
    • /zh/60_LLM/13.html
  • 61_ai_agent

    • 数据库设计
    • 示例问题管理
    • 会话管理
    • 历史记录
    • 对接 Perplexity API
    • 意图识别与生成提示词
    • 智能问答模块设计与实现
    • 文件上传与解析文档
    • 翻译
    • 名人搜索功能实现
    • Ai studio gemini youbue 问答使用说明
    • 自建 YouTube 字幕问答系统
    • 自建 获取 youtube 字幕服务
    • 通用搜索
    • /zh/61_ai_agent/15.html
    • 16
    • 17
    • 18
    • 在 tio-boot 应用中整合 ai-agent
    • 16
  • 62_translator

    • 简介
  • 63_knowlege_base

    • 数据库设计
    • 用户登录实现
    • 模型管理
    • 知识库管理
    • 文档拆分
    • 片段向量
    • 命中测试
    • 文档管理
    • 片段管理
    • 问题管理
    • 应用管理
    • 向量检索
    • 推理问答
    • 问答模块
    • 统计分析
    • 用户管理
    • api 管理
    • 存储文件到 S3
    • 文档解析优化
    • 片段汇总
    • 段落分块与检索
    • 多文档解析
    • 对话日志
    • 检索性能优化
    • Milvus
    • 文档解析方案和费用对比
    • 离线运行向量模型
  • 64_ai-search

    • ai-search 项目简介
    • ai-search 数据库文档
    • ai-search SearxNG 搜索引擎
    • ai-search Jina Reader API
    • ai-search Jina Search API
    • ai-search 搜索、重排与读取内容
    • ai-search PDF 文件处理
    • ai-search 推理问答
    • Google Custom Search JSON API
    • ai-search 意图识别
    • ai-search 问题重写
    • ai-search 系统 API 接口 WebSocket 版本
    • ai-search 搜索代码实现 WebSocket 版本
    • ai-search 生成建议问
    • ai-search 生成问题标题
    • ai-search 历史记录
    • Discover API
    • 翻译
    • Tavily Search API 文档
    • 对接 Tavily Search
    • 火山引擎 DeepSeek
    • 对接 火山引擎 DeepSeek
    • ai-search 搜索代码实现 SSE 版本
    • jar 包部署
    • Docker 部署
    • 爬取一个静态网站的所有数据
    • 网页数据预处理
    • 网页数据检索与问答流程整合
  • 65_java-linux

    • Java 执行 python 代码
    • 通过大模型执行 Python 代码
    • MCP 协议
    • Cline 提示词
    • Cline 提示词-中文版本
  • 66_manim

    • Manim 开发环境搭建
    • 生成场景提示词
    • 生成代码
    • 完整脚本示例
    • 语音合成系统
    • Fish.audio TTS 接口说明文档与 Java 客户端封装
    • 整合 fishaudio 到 java-uni-ai-server 项目
    • 执行 Python (Manim) 代码
    • 使用 SSE 流式传输生成进度的实现文档
    • 整合全流程完整文档
    • HLS 动态推流技术文档
    • manim 分场景生成代码
    • 分场景运行代码及流式播放支持
    • 分场景业务端完整实现流程
    • Maiim布局管理器
    • 仅仅生成场景代码
    • 使用 modal 运行 manim 代码
    • Python 使用 Modal GPU 加速渲染
    • Modal 平台 GPU 环境下运行 Manim
    • Modal Manim OpenGL 安装与使用
    • 优化 GPU 加速
    • 生成视频封面流程
    • Java 调用 manim 命令 执行代码 生成封面
    • Manim 图像生成服务客户端文档
    • /zh/66_manim/25.html
    • /zh/66_manim/26.html
    • /zh/66_manim/27.html
  • 70_tio-boot-admin

    • 入门指南
    • 初始化数据
    • token 存储
    • 与前端集成
    • 文件上传
    • 网络请求
    • 图片管理
    • /zh/70_tio-boot-admin/08.html
    • Word 管理
    • PDF 管理
    • 文章管理
    • 富文本编辑器
  • 71_tio-boot

    • /zh/71_tio-boot/01.html
    • Swagger 整合到 Tio-Boot 中的指南
    • HTTP/1.1 Pipelining 性能测试报告
  • 80_性能测试

    • 压力测试 - tio-http-serer
    • 压力测试 - tio-boot
    • 压力测试 - tio-boot-native
    • 压力测试 - netty-boot
    • 性能测试对比
    • TechEmpower FrameworkBenchmarks
    • 压力测试 - tio-boot 12 C 32G
  • 99_案例

    • 封装 IP 查询服务
    • tio-boot 案例 - 全局异常捕获与企业微信群通知
    • tio-boot 案例 - 文件上传和下载
    • tio-boot 案例 - 整合 ant design pro 增删改查
    • tio-boot 案例 - 流失响应
    • tio-boot 案例 - 增强检索
    • tio-boot 案例 - 整合 function call
    • tio-boot 案例 - 定时任务 监控 PostgreSQL、Redis 和 Elasticsearch
    • Tio-Boot 案例:使用 SQLite 整合到登录注册系统
    • tio-boot 案例 - 执行 shell 命令

欧拉公式的几何意义

video

https://manim.collegebot.ai/data/hls/500642289342857216/main.mp4

Sence prompt

## Geometric Meaning of Euler's Formula \( e^{i\theta} = \cos\theta + i\sin\theta \)

### 【Scene 1: Title and Introduction】

- **Background & Atmosphere:**
  - A deep, dark blue gradient background, fading slightly towards black at the edges, evoking a sense of mathematical depth or space. Subtle, slow-moving, faint particle effects (like dust motes or distant stars) can be added for visual interest.
  - Display scene number "01" in the top-right corner (small, white text).
- **Main Content:**
  - **Title:** Display "Euler's Formula" prominently in the top center. (Font: Sans-serif, Color: Bright White, Animation: `Write`, Duration: 1.5s).
  - **Subtitle:** Below the title, display "Unveiling the Geometric Meaning". (Font: Smaller Sans-serif, Color: Light Cyan, Animation: `FadeIn`, Delay: 0.5s after title finishes, Duration: 1s).
  - **Formula:** Centered below the subtitle, display Euler's Formula clearly:
    \[
    e^{i\theta} = \cos\theta + i\sin\theta
    \]
    (LaTeX, Color: White, Animation: `ReplacementTransform` from the subtitle text or `Write`, Duration: 2s). Key components like \(e^{i\theta}\), \(\cos\theta\), and \(i\sin\theta\) could briefly flash or change color (e.g., to Yellow) upon appearance to draw attention.
- **Camera & Transitions:**
  - Static camera focused on the center of the screen.
  - A slight, slow zoom-out (scale factor decreases from 1.1 to 1.0 over 3 seconds) after the formula appears, giving a sense of revealing the subject.
  - Smooth transition to the next scene (e.g., `FadeOut` all elements over 0.5s).

---

### 【Scene 2: Introducing the Complex Plane】

- **Background & Layout:**
  - Transition to a clean, light gray background (#E0E0E0) or a subtle grid background (`NumberPlane` with faint lines) to facilitate geometric visualization.
  - Display scene number "02" in the top-right corner (small, dark gray text).
- **Content Presentation:**
  - **Coordinate System:** Create a 2D coordinate system representing the complex plane.
    - X-axis labeled "Real Axis" (or "Re"). Range: e.g., -2 to 2.
    - Y-axis labeled "Imaginary Axis" (or "Im"). Range: e.g., -2 to 2.
    - Axis color: Dark Gray. Label color: Black.
    - Animation: `Create` for axes and labels (Duration: 2s).
  - **Complex Number:** Introduce a generic complex number \(z = x + iy\).
    - Display the formula \(z = x + iy\) (LaTeX) in the top-left corner. (Color: Black, Animation: `Write`, Duration: 1s).
    - Plot a point \(P\) representing \(z\) in the plane (e.g., at \(1.5 + 1i\)). (Color: Blue `Dot`).
    - Draw dashed lines from \(P\) to the Real axis (at \(x=1.5\)) and Imaginary axis (at \(y=1\)). (Line style: Dashed, Color: Gray).
    - Label the projections \(x\) and \(y\) (or \(iy\)) on the axes. (Color: Black).
    - Animation: Plot the point (`FadeIn`), then draw projection lines (`Create`), then add labels (`Write`). Total duration: 2.5s.
- **Camera & Transitions:**
  - Static camera, ensuring the entire complex plane setup is visible.
  - Pause briefly after all elements appear. Smooth transition to Scene 3.

---

### 【Scene 3: Geometry of \(\cos\theta + i\sin\theta\) - The Unit Circle】

- **Background & Layout:**
  - Maintain the light gray or grid background and the complex plane axes from Scene 2.
  - Display scene number "03" in the top-right corner.
- **Content Presentation:**
  - **Unit Circle:** Draw a circle centered at the origin with radius 1.
    - `Circle(radius=1)`, Color: Red, Stroke Width: Default or slightly thicker.
    - Animation: `Create` (Duration: 1.5s).
  - **Angle and Point:**
    - Mark an angle \(\theta\) starting from the positive Real axis, counter-clockwise. Use an `Arc` object. (e.g., let \(\theta = \pi/4\) or 60 degrees initially for visual clarity). Label the angle \(\theta\). (Angle arc color: Green, Label color: Black).
    - Draw a line segment (radius) from the origin (0,0) to the point \(P\) on the unit circle corresponding to angle \(\theta\). (Line color: Blue).
    - Mark the point \(P\) clearly. (`Dot`, Color: Blue).
    - Animation: Create angle arc and label (`Create`, `Write`), then draw radius (`Create`), then place dot (`FadeIn`). Total duration: 2.5s.
  - **Trigonometric Connection:**
    - Drop a perpendicular dashed line from point \(P\) to the Real axis, forming a right-angled triangle. (Line style: Dashed, Color: Gray).
    - Label the adjacent side (on the Real axis) as \(\cos\theta\).
    - Label the opposite side (parallel to the Imaginary axis) as \(\sin\theta\).
    - Label the hypotenuse (the radius) as 1.
    - Crucially, label the coordinates of point \(P\) as \((\cos\theta, \sin\theta)\) or, emphasizing the complex nature, label the point \(P\) directly as \(\cos\theta + i\sin\theta\). (LaTeX, Color: Black/Blue).
    - Animation: Create projection line (`Create`), then Write labels (\(\cos\theta\), \(\sin\theta\), 1) for the triangle sides (`Write`), then display the coordinates/complex label for P (`Write` or `Transform` if a previous label existed). Total duration: 3s.
  - **Text Explanation:** Briefly display text (e.g., top-left): "The term \(\cos\theta + i\sin\theta\) represents a point on the unit circle at angle \(\theta\)." (Color: Black, Animation: `FadeIn`).
- **Camera & Transitions:**
  - The camera might slightly zoom in (e.g., scale factor 1.2) to focus on the unit circle and the triangle construction.
  - Pause to allow absorption of the geometric relationship. Smooth transition.

---

### 【Scene 4: Connecting \(e^{i\theta}\) - Rotation】

- **Background & Layout:**
  - Keep the complex plane, unit circle, point \(P\), angle \(\theta\), and the label \(\cos\theta + i\sin\theta\) from Scene 3 visible.
  - Display scene number "04" in the top-right corner.
- **Content Presentation:**
  - **Introduce \(e^{i\theta}\):**
    - Display the term \(e^{i\theta}\) (LaTeX) near the point \(P\), perhaps slightly above or replacing the \(\cos\theta + i\sin\theta\) label using `ReplacementTransform`. (Color: Purple or another distinct color).
    - Animation: `Write` or `ReplacementTransform` (Duration: 1.5s).
  - **Geometric Meaning Text:** Display explanatory text, e.g., "Euler's formula states that \(e^{i\theta}\) is _exactly this point_." (Appears near the top).
  - Followed by: "Geometrically, multiplying by \(e^{i\theta}\) represents a _rotation_ by angle \(\theta\) in the complex plane." (Appears below the first text).
    - (Text Color: Black, Animation: `Write` or `FadeIn` line by line, Duration: 2s per line).
  - **Visualizing Rotation (Optional but effective):**
    - Show the point 1 (i.e., \(1+0i\)) on the Real axis. (Red `Dot`).
    - Animate the radius vector (from Scene 3) starting aligned with the Real axis (pointing to 1) and rotating counter-clockwise by angle \(\theta\) to reach point \(P\). The angle arc \(\theta\) can be redrawn during this rotation.
    - Animation: Use `Rotate` animation on the radius vector, coordinated with drawing the `Arc`. (Duration: 2s).
- **Camera & Transitions:**
  - Camera remains focused on the unit circle area. If rotation is shown, ensure the starting point (1) and the rotation path are clearly visible.
  - Smooth transition.

---

### 【Scene 5: Dynamic Visualization - Varying \(\theta\))】

- **Background & Layout:**
  - Maintain the complex plane and unit circle. Remove triangle lines/labels if cluttered, keep point P and radius.
  - Display scene number "05" in the top-right corner.
  - Display the full Euler formula \(e^{i\theta} = \cos\theta + i\sin\theta\) statically in a corner (e.g., top-left) for reference. (Color: Black).
- **Content Presentation:**
  - **Animating Theta:** Use a `ValueTracker` for the angle \(\theta\), letting it vary smoothly, e.g., from 0 to \(2\pi\) or \(-\pi\) to \(\pi\).
  - **Moving Point:** The point \(P\) on the unit circle moves dynamically as \(\theta\) changes. Its position is always \((\cos\theta, \sin\theta)\).
  - **Rotating Radius:** The radius vector from the origin to \(P\) rotates along with the point.
  - **Angle Arc:** The angle arc indicating \(\theta\) updates dynamically.
  - **Trace Path:** Optionally, use `TracedPath` for point \(P\) to draw the unit circle as it moves.
  - **Labels:** Display the current value of \(\theta\) (using `DecimalNumber` or formatted `MathTex` linked to the `ValueTracker`) and update the label \(e^{i\theta}\) next to the point P (or just keep it symbolic).
  - Animation: This is a continuous animation driven by `add_updater` methods linked to the `ValueTracker`. The animation should run for several seconds (e.g., 5-8 seconds) to show a full cycle or significant movement. Colors: Point P (Blue), Radius (Blue), Angle Arc (Green), Trace (Red).
- **Camera & Transitions:**
  - Static camera, perfectly framing the unit circle and the dynamic elements.
  - Fade out elements at the end of the animation sequence.

---

### 【Scene 6: Summary and Significance】

- **Background & Atmosphere:**
  - Return to the deep blue gradient background from Scene 1, or use a solid dark background.
  - Display scene number "06" in the top-right corner (White text).
- **Content Presentation:**
  - **Title:** Display "Geometric Summary" at the top. (Color: Bright White, Animation: `FadeIn`).
  - **Core Idea:** Centered text:
    - "\(e^{i\theta}\) represents a point on the unit circle in the complex plane."
    - "The angle from the positive real axis to this point is \(\theta\)."
    - "Its coordinates are \((\cos\theta, \sin\theta)\)."
    - (Text color: White, Animation: Appear line by line using `Write` or `FadeIn`, Duration: 1.5s per line).
  - **Formula Recap:** Display \(e^{i\theta} = \cos\theta + i\sin\theta\) again below the summary text. (LaTeX, Color: Yellow/Gold, Animation: `FadeIn`).
  - **Concluding Thought:** Add a final text line like "A fundamental link between analysis, trigonometry, and geometry." (Color: Light Cyan, Animation: `FadeIn`).
- **Camera & Transitions:**
  - Start slightly zoomed in on the title, then slowly zoom out (scale factor decreases e.g., from 1.2 to 1.0) over the duration of the text appearing, revealing all summary points and the final formula.
  - Hold the final frame for a couple of seconds before fading out.

---

### 【Overall Requirements & Manim Rules】

- **Visual Style:** Maintain consistency. Use high-resolution rendering. Ensure LaTeX formulas are crisp and clear. Use the specified color palettes (Dark/Spacey for Intro/Outro, Light/Clear for explanation). Use contrasting colors for highlights.
- **Transitions:** Use smooth transitions between scenes (`FadeIn`/`FadeOut`, allow ~0.5s buffer).
- **Timing:** Coordinate camera movements and element animations carefully to guide the viewer's focus. Ensure adequate pauses for comprehension.
- **Manim Rules (from previous example):**
  1.  **Consistent Background:** Use `self.camera.background_color` in `setup()` for consistency _or_ manage background transitions deliberately. Ensure text/element colors have good contrast against the _current_ background.
  2.  **Element Tracking:** Use `VGroup` for elements created in each scene/section and ensure a cleanup function (`clear_section` or similar) reliably removes _all_ temporary elements (including transformed ones) unless intended to persist.
  3.  **Text Wrapping:** Apply `width` parameter to `Text` and `MarkupText` objects, especially for longer explanations or labels near edges, to prevent overflow. Use `.next_to()` with appropriate `buff` and alignment.
  4.  **Layout Management:** For scenes with distinct regions (like text vs. diagram), define boundaries/anchors and position content `VGroup`s within them. Use `width` on text within these regions.
  5.  **Correct Element Creation:** Ensure distinct `Mobject` instances are created for different concepts, using unique variable names to avoid displaying incorrect or duplicated visuals.

code

# -*- coding: utf-8 -*-
import os
import numpy as np
import requests
from contextlib import contextmanager
from manim import *
import hashlib
# from moviepy.editor import AudioFileClip # Use Manim's built-in or ensure moviepy is installed if needed
from moviepy.audio.io.AudioFileClip import AudioFileClip # More specific import if needed
import manimpango # For font checking
import random

# --- Font Check ---
DEFAULT_FONT = "Arial" # Use a common default font
available_fonts = manimpango.list_fonts()
final_font = None

if DEFAULT_FONT in available_fonts:
    print(f"Font '{DEFAULT_FONT}' found.")
    final_font = DEFAULT_FONT
else:
    print(f"Warning: Font '{DEFAULT_FONT}' not found. Trying fallback fonts...")
    fallback_fonts = ["Helvetica", "Verdana", "DejaVu Sans", "Liberation Sans"]
    found_fallback = False
    for font in fallback_fonts:
        if font in available_fonts:
            print(f"Switched to fallback font: '{font}'")
            final_font = font
            found_fallback = True
            break
    if not found_fallback:
        print(f"Warning: Neither '{DEFAULT_FONT}' nor fallback fonts found. Using Manim default.")
        # final_font remains None

# --- Custom Colors (Adjusted for White Background) ---
MY_WHITE = "#FFFFFF"         # Background color
MY_BLACK = "#000000"         # Define black
MY_DARK_TEXT = "#1F2937"     # Primary text color (Dark Gray/Almost Black)
MY_DARK_GRAY = "#444444"     # Axes color
MY_BLUE = "#0077CC"          # Point P, radius, highlights
MY_RED = "#CC3333"           # Unit circle, trace, highlights
MY_GREEN = "#339933"         # Angle arc, highlights
MY_PURPLE = "#8833AA"        # e^i theta label, highlights
MY_YELLOW = "#B8860B"        # Formula recap highlight (Dark Goldenrod - visible on white)
MY_GRAY = "#888888"           # Dashed lines
MY_LIGHT_CYAN = "#008B8B"    # Concluding thought (Dark Cyan)
# MY_LIGHT_GRAY_BG = "#E0E0E0" # No longer needed for background

# --- Helper Functions for Color Handling ---
def hex_to_rgb(hex_color_str):
    """Converts a hex color string (e.g., '#RRGGBB') to an RGB tuple (0-255)."""
    if hasattr(hex_color_str, 'to_hex'):
        hex_color_str = hex_color_str.to_hex()

    hex_color_str = str(hex_color_str).lstrip('#')
    if len(hex_color_str) == 6:
        try:
            return tuple(int(hex_color_str[i:i+2], 16) for i in (0, 2, 4))
        except ValueError:
            print(f"Warning: Invalid hex color format '{hex_color_str}'. Defaulting to black.")
            return (0, 0, 0)
    else:
        print(f"Warning: Non-hex color '{hex_color_str}' encountered. Defaulting to black for luminance check.")
        return (0, 0, 0)

def calculate_luminance(rgb):
    """Calculates perceived luminance (0-1) from an RGB tuple (0-255)."""
    if not isinstance(rgb, (tuple, list)) or len(rgb) != 3:
        return 0
    r, g, b = [x / 255.0 for x in rgb]
    return 0.2126 * r + 0.7152 * g + 0.0722 * b

# --- TTS Caching Setup ---
CACHE_DIR = r"#(output_path)/audio" # Use placeholder
os.makedirs(CACHE_DIR, exist_ok=True)

class CustomVoiceoverTracker:
    """Tracks audio path and duration for TTS."""
    def __init__(self, audio_path, duration):
        self.audio_path = audio_path
        self.duration = duration

def get_cache_filename(text):
    """Generates a unique filename based on the text hash."""
    text_hash = hashlib.md5(text.encode('utf-8')).hexdigest()
    return os.path.join(CACHE_DIR, f"{text_hash}.mp3")

@contextmanager
def custom_voiceover_tts(text, token="123456", base_url="https://uni-ai.fly.dev/api/manim/tts"):
    """Fetches TTS audio, caches it, and provides path and duration."""
    cache_file = get_cache_filename(text)
    audio_file = cache_file

    if os.path.exists(cache_file):
        pass
    else:
        try:
            input_text_encoded = requests.utils.quote(text)
            url = f"{base_url}?token={token}&input={input_text_encoded}"
            response = requests.get(url, stream=True, timeout=60)
            response.raise_for_status()
            with open(cache_file, "wb") as f:
                for chunk in response.iter_content(chunk_size=8192):
                    if chunk: f.write(chunk)
            audio_file = cache_file
        except requests.exceptions.RequestException as e:
            print(f"TTS API request failed: {e}")
            tracker = CustomVoiceoverTracker(None, 0)
            yield tracker
            return
        except Exception as e:
            print(f"An error occurred during TTS processing: {e}")
            if os.path.exists(cache_file): os.remove(cache_file)
            tracker = CustomVoiceoverTracker(None, 0)
            yield tracker
            return

    duration = 0
    if audio_file and os.path.exists(audio_file):
        try:
            # Use moviepy to get duration
            with AudioFileClip(audio_file) as clip:
                 duration = clip.duration
            # print(f"Audio duration: {duration:.2f}s")
        except Exception as e:
            print(f"Error processing audio file {audio_file} with moviepy: {e}")
            # Fallback or alternative method if needed
            # For simplicity, we'll just report the error and continue
            audio_file = None # Mark as unusable if duration check failed
            duration = 0
    else:
        audio_file = None

    tracker = CustomVoiceoverTracker(audio_file, duration)
    try:
        yield tracker
    finally:
        pass

# -----------------------------
# CombinedScene: Euler's Formula Geometry
# -----------------------------
class CombinedScene(MovingCameraScene):
    """
    Explains the geometric meaning of Euler's formula e^(i*theta).
    Uses English narration.
    Background is consistently white.
    """
    def setup(self):
        MovingCameraScene.setup(self)
        if final_font:
            Text.set_default(font=final_font)
        self.current_scene_num_mob = None
        self.section_elements = VGroup()
        # --- Set consistent background color HERE ---
        self.camera.background_color = MY_WHITE
        # Store references
        self.axes = None
        self.axes_labels = None
        self.unit_circle = None
        self.point_P = None
        self.radius_line = None
        self.angle_arc = None
        self.angle_label = None
        self.p_label_complex = None
        self.e_label = None

    def update_scene_number(self, number_str):
        """Fades out the old scene number and fades in the new one."""
        # Background is always white, so text is always dark
        text_color = MY_DARK_TEXT

        new_scene_num = Text(number_str, font_size=24, color=text_color).to_corner(UR, buff=MED_LARGE_BUFF).set_z_index(100)
        animations = [FadeIn(new_scene_num, run_time=0.5)]
        if self.current_scene_num_mob:
            if self.current_scene_num_mob is not None:
                 animations.append(FadeOut(self.current_scene_num_mob, run_time=0.5))
        self.play(*animations)
        self.current_scene_num_mob = new_scene_num

    def clear_section(self, keep_refs=None):
        """Clears elements added to self.section_elements."""
        elements_to_remove = VGroup()
        elements_to_keep_in_group = VGroup()

        keep_mobjects = []
        if keep_refs:
            for ref_name in keep_refs:
                mob = getattr(self, ref_name, None)
                if mob:
                    keep_mobjects.append(mob)

        for mob in self.section_elements:
            is_kept = False
            for keep_mob in keep_mobjects:
                if mob is keep_mob:
                    is_kept = True
                    break
                if isinstance(keep_mob, VGroup) and mob in keep_mob.submobjects:
                    is_kept = True
                    break
            if is_kept:
                elements_to_keep_in_group.add(mob)
            else:
                elements_to_remove.add(mob)

        for mob in elements_to_remove:
             if mob is not None and hasattr(mob, 'get_updaters') and mob.get_updaters():
                 mob.clear_updaters()

        valid_elements_to_remove = [elem for elem in elements_to_remove if elem is not None]
        if valid_elements_to_remove:
            self.play(FadeOut(Group(*valid_elements_to_remove)), run_time=0.5)

        self.section_elements = elements_to_keep_in_group
        self.wait(0.1)

    def construct(self):
        self.play_section_01()
        self.clear_section()

        self.play_section_02()
        self.clear_section(keep_refs=['axes', 'axes_labels'])

        self.play_section_03()
        self.clear_section(keep_refs=['axes', 'axes_labels', 'unit_circle', 'point_P', 'radius_line', 'angle_arc', 'angle_label', 'p_label_complex'])

        self.play_section_04()
        self.clear_section(keep_refs=['axes', 'axes_labels', 'unit_circle', 'point_P', 'radius_line', 'angle_arc', 'angle_label', 'e_label'])

        self.play_section_05()
        self.clear_section()
        if self.axes and self.axes in self.mobjects: self.remove(self.axes)
        if self.axes_labels and self.axes_labels in self.mobjects: self.remove(self.axes_labels)

        self.play_section_06()
        self.clear_section()

        self.wait(2)
        if self.current_scene_num_mob:
            self.play(FadeOut(self.current_scene_num_mob))

    # --- Section 1: Title and Introduction ---
    def play_section_01(self):
        """Scene 1: Title and Introduction"""
        # Background is set in setup()
        self.update_scene_number("01")

        title = Text("Euler's Formula", font_size=60, color=MY_DARK_TEXT)
        title.to_edge(UP, buff=1.0)
        subtitle = Text("Unveiling the Geometric Meaning", font_size=36, color=MY_DARK_TEXT)
        subtitle.next_to(title, DOWN, buff=0.5)
        formula = MathTex(r"e^{i\theta}", r"=", r"\cos\theta", r"+", r"i\sin\theta", font_size=60, color=MY_DARK_TEXT)
        formula.next_to(subtitle, DOWN, buff=0.8)

        self.section_elements.add(title, subtitle, formula)

        voice_text_01 = "Welcome! Today we explore one of the most beautiful equations in mathematics: Euler's Formula. It connects exponentiation, complex numbers, and trigonometry in a profound way. Let's uncover its geometric meaning."
        with custom_voiceover_tts(voice_text_01) as tracker:
            if tracker.audio_path and tracker.duration > 0:
                self.add_sound(tracker.audio_path)
            else:
                print("Warning: Narration 1 TTS failed.")

            subtitle_voice = Text(voice_text_01, font_size=28, color=MY_DARK_TEXT, width=config.frame_width - 2, should_center=True).to_edge(DOWN, buff=MED_SMALL_BUFF)

            self.play(FadeIn(subtitle_voice), run_time=0.5)
            self.play(Write(title), run_time=1.5)
            self.play(FadeIn(subtitle), run_time=1.0)
            self.wait(0.5)
            self.play(Write(formula[1]), Write(formula[3]), run_time=0.5) # = and +

            self.play(
                Write(formula[0]),  # e^i theta
                run_time=1.5
            )
            self.play(formula[0].animate.set_color(MY_RED), run_time=0.2)

            self.play(
                Write(formula[2]),  # cos
                run_time=1.5
            )
            self.play(formula[2].animate.set_color(MY_RED), run_time=0.2)

            self.play(
                Write(formula[4]),  # i sin
                run_time=1.5
            )
            self.play(formula[4].animate.set_color(MY_RED), run_time=0.2)

            # keep show
            formula[0].set_color(MY_DARK_TEXT)
            formula[2].set_color(MY_DARK_TEXT)
            formula[4].set_color(MY_DARK_TEXT)

            # Synchronization
            # Adjusted duration calculation (removed the 0.2s reset times)
            anim_duration = 0.5 + 1.5 + 1.0 + 0.5 + 0.5 + 1.5 + 1.5 + 1.5
            wait_time = max(0, tracker.duration - anim_duration - 0.5) if tracker.duration > 0 else 1.0
            if wait_time > 0: self.wait(wait_time)
            self.play(FadeOut(subtitle_voice), run_time=0.5)

    # --- Section 2: Introducing the Complex Plane ---
    def play_section_02(self):
        """Scene 2: Introducing the Complex Plane"""
        # REMOVED: Background color change - uses default white now
        self.update_scene_number("02")

        self.axes = Axes(
            x_range=[-2.5, 2.5, 1], y_range=[-2.5, 2.5, 1],
            x_length=6, y_length=6,
            axis_config={"color": MY_DARK_GRAY, "include_numbers": True, "stroke_width": 2},
            tips=False
        )
        x_label = Text("Real Axis (Re)", font_size=24, color=MY_DARK_TEXT).next_to(self.axes.x_axis.get_right(), DOWN)
        y_label = Text("Imaginary Axis (Im)", font_size=24, color=MY_DARK_TEXT).next_to(self.axes.y_axis.get_top(), RIGHT).shift(LEFT*0.5)
        self.axes_labels = VGroup(x_label, y_label)

        z_formula = MathTex("z = x + iy", font_size=36, color=MY_DARK_TEXT).to_corner(UL, buff=MED_LARGE_BUFF)

        x_val, y_val = 1.5, 1.0
        point_P = Dot(self.axes.c2p(x_val, y_val), color=MY_BLUE, radius=0.08)
        proj_line_x = DashedLine(self.axes.c2p(x_val, 0), self.axes.c2p(x_val, y_val), color=MY_GRAY)
        proj_line_y = DashedLine(self.axes.c2p(0, y_val), self.axes.c2p(x_val, y_val), color=MY_GRAY)
        proj_label_x = MathTex("x", font_size=30, color=MY_DARK_TEXT).next_to(self.axes.c2p(x_val, 0), DOWN)
        proj_label_y = MathTex("iy", font_size=30, color=MY_DARK_TEXT).next_to(self.axes.c2p(0, y_val), LEFT)
        projections = VGroup(proj_line_x, proj_line_y, proj_label_x, proj_label_y)

        self.section_elements.add(self.axes, self.axes_labels, z_formula, point_P, projections)

        voice_text_02 = "To understand the geometry, we need the complex plane. The horizontal axis represents real numbers, and the vertical axis represents imaginary numbers. A complex number, z equals x plus i y, corresponds to a point P with coordinates (x, y)."
        with custom_voiceover_tts(voice_text_02) as tracker:
            if tracker.audio_path and tracker.duration > 0:
                self.add_sound(tracker.audio_path)
            else:
                print("Warning: Narration 2 TTS failed.")

            subtitle_voice = Text(voice_text_02, font_size=28, color=MY_DARK_TEXT, width=config.frame_width - 2, should_center=True).to_edge(DOWN, buff=MED_SMALL_BUFF)

            self.play(FadeIn(subtitle_voice), run_time=0.5)
            self.play(Create(self.axes), Write(self.axes_labels), run_time=2.0)
            self.play(Write(z_formula), run_time=1.0)
            self.play(FadeIn(point_P), run_time=0.5)
            self.play(Create(proj_line_x), Create(proj_line_y), run_time=1.0)
            self.play(Write(proj_label_x), Write(proj_label_y), run_time=1.0)

            anim_duration = 0.5 + 2.0 + 1.0 + 0.5 + 1.0 + 1.0
            wait_time = max(0, tracker.duration - anim_duration - 0.5) if tracker.duration > 0 else 1.0
            if wait_time > 0: self.wait(wait_time)
            self.play(FadeOut(subtitle_voice), run_time=0.5)

    # --- Section 3: Geometry of cos(theta) + i sin(theta) ---
    def play_section_03(self):
        """Scene 3: Unit Circle Geometry"""
        # Background is default white
        self.update_scene_number("03")

        if not self.axes:
            print("Error: Axes not found from previous section!")
            return
        self.add(self.axes, self.axes_labels)

        radius_val = self.axes.x_axis.n2p(1)[0] - self.axes.x_axis.n2p(0)[0]
        self.unit_circle = Circle(radius=radius_val, color=MY_RED, stroke_width=3)
        self.unit_circle.move_to(self.axes.c2p(0,0))

        theta_value = PI / 3
        point_P_coord = self.axes.c2p(np.cos(theta_value), np.sin(theta_value))
        self.point_P = Dot(point_P_coord, color=MY_BLUE, radius=0.08)
        self.radius_line = Line(self.axes.c2p(0,0), point_P_coord, color=MY_BLUE, stroke_width=2)
        self.angle_arc = Arc(radius=0.4, start_angle=0, angle=theta_value, arc_center=self.axes.c2p(0,0), color=MY_GREEN)
        self.angle_label = MathTex(r"\theta", font_size=36, color=MY_GREEN).move_to(
            Arc(radius=0.6, start_angle=0, angle=theta_value, arc_center=self.axes.c2p(0,0)).point_from_proportion(0.5)
        )

        proj_line = DashedLine(point_P_coord, self.axes.c2p(np.cos(theta_value), 0), color=MY_GRAY)
        cos_label = MathTex(r"\cos\theta", font_size=30, color=MY_DARK_TEXT).next_to(self.axes.c2p(np.cos(theta_value)/2, 0), DOWN, buff=SMALL_BUFF)
        sin_label = MathTex(r"\sin\theta", font_size=30, color=MY_DARK_TEXT).next_to(self.axes.c2p(np.cos(theta_value), np.sin(theta_value)/2), RIGHT, buff=SMALL_BUFF)
        hyp_label = Text("1", font_size=30, color=MY_RED).move_to(self.radius_line.get_center() + normalize(self.radius_line.get_vector())@np.array([[0,-1,0],[1,0,0],[0,0,1]])*0.2)

        self.p_label_complex = MathTex(r"\cos\theta + i\sin\theta", font_size=36, color=MY_BLUE).next_to(self.point_P, UR, buff=SMALL_BUFF)

        explanation = Text("cos(θ) + i sin(θ) is a point on the unit circle.", font_size=28, color=MY_DARK_TEXT)
        explanation.to_corner(UL, buff=MED_LARGE_BUFF)

        triangle_elements = VGroup(proj_line, cos_label, sin_label, hyp_label)
        self.section_elements.add(self.unit_circle, self.angle_arc, self.angle_label, self.radius_line, self.point_P, triangle_elements, self.p_label_complex, explanation)

        voice_text_03 = "Now, let's focus on the unit circle - a circle with radius 1 centered at the origin. Consider a point P on this circle. If we draw a line from the origin to P, it forms an angle theta with the positive real axis. Using basic trigonometry, the x-coordinate of P is cosine theta, and the y-coordinate is sine theta. Therefore, the complex number representing point P is exactly cosine theta plus i sine theta."
        with custom_voiceover_tts(voice_text_03) as tracker:
            if tracker.audio_path and tracker.duration > 0:
                self.add_sound(tracker.audio_path)
            else:
                print("Warning: Narration 3 TTS failed.")

            subtitle_voice = Text(voice_text_03, font_size=28, color=MY_DARK_TEXT, width=config.frame_width - 2, should_center=True).to_edge(DOWN, buff=MED_SMALL_BUFF)

            self.play(FadeIn(subtitle_voice), run_time=0.5)
            self.play(Create(self.unit_circle), run_time=1.5)
            self.play(Create(self.angle_arc), Write(self.angle_label), Create(self.radius_line), FadeIn(self.point_P), run_time=2.5)
            self.play(Create(proj_line), Write(cos_label), Write(sin_label), Write(hyp_label), run_time=3.0)
            self.play(Write(self.p_label_complex), run_time=1.5)
            self.play(FadeIn(explanation), run_time=1.0)

            anim_duration = 0.5 + 1.5 + 2.5 + 3.0 + 1.5 + 1.0
            wait_time = max(0, tracker.duration - anim_duration - 0.5) if tracker.duration > 0 else 1.0
            if wait_time > 0: self.wait(wait_time)
            self.play(FadeOut(subtitle_voice), run_time=0.5)

    # --- Section 4: Connecting e^(i*theta) - Rotation ---
    def play_section_04(self):
        """Scene 4: Connecting e^(i*theta) - Rotation"""
        # Background is default white
        self.update_scene_number("04")

        if not all([self.axes, self.axes_labels, self.unit_circle, self.point_P, self.radius_line, self.angle_arc, self.angle_label, self.p_label_complex]):
             print("Error: Missing elements from Scene 3!")
             return
        self.add(self.axes, self.axes_labels, self.unit_circle, self.angle_arc, self.angle_label, self.radius_line, self.point_P, self.p_label_complex)

        self.e_label = MathTex(r"e^{i\theta}", font_size=40, color=MY_PURPLE)
        self.e_label.move_to(self.p_label_complex.get_center())

        expl1 = Text("Euler's formula states that e^(iθ) is exactly this point.", font_size=28, color=MY_DARK_TEXT)
        expl2 = Text("Multiplying by e^(iθ) means rotating by angle θ.", font_size=28, color=MY_DARK_TEXT)
        explanation_group = VGroup(expl1, expl2).arrange(DOWN, buff=MED_SMALL_BUFF).to_corner(UL, buff=MED_LARGE_BUFF)

        point_1 = Dot(self.axes.c2p(1, 0), color=MY_RED, radius=0.08)
        label_1 = MathTex("1", font_size=30, color=MY_RED).next_to(point_1, DR, buff=SMALL_BUFF)
        rotating_radius = Line(self.axes.c2p(0,0), self.axes.c2p(1,0), color=MY_BLUE, stroke_width=2)

        self.section_elements.add(self.e_label, explanation_group, point_1, label_1, rotating_radius)
        self.section_elements.add(self.axes, self.axes_labels, self.unit_circle, self.angle_arc, self.angle_label, self.radius_line, self.point_P, self.p_label_complex)

        voice_text_04 = "This is where Euler's formula comes in! It tells us that the complex exponential, e to the power of i theta, is precisely equal to this point, cosine theta plus i sine theta. Geometrically, this means e to the i theta represents a point on the unit circle at angle theta. Multiplying any complex number by e to the i theta rotates that number by angle theta around the origin. Watch how the point '1' rotates by theta to reach our point P."
        with custom_voiceover_tts(voice_text_04) as tracker:
            if tracker.audio_path and tracker.duration > 0:
                self.add_sound(tracker.audio_path)
            else:
                print("Warning: Narration 4 TTS failed.")

            subtitle_voice = Text(voice_text_04, font_size=28, color=MY_DARK_TEXT, width=config.frame_width - 2, should_center=True).to_edge(DOWN, buff=MED_SMALL_BUFF)

            self.play(FadeIn(subtitle_voice), run_time=0.5)
            self.play(ReplacementTransform(self.p_label_complex, self.e_label), run_time=1.5)
            self.play(Write(expl1), run_time=2.0)
            self.play(Write(expl2), run_time=2.0)
            self.wait(1.0)
            self.play(FadeIn(point_1), Write(label_1), run_time=1.0)
            self.play(Create(rotating_radius), run_time=1.0)
            theta_value = PI / 3
            self.play(Rotate(rotating_radius, angle=theta_value, about_point=self.axes.c2p(0,0)), run_time=2.0)
            self.wait(1.0)

            anim_duration = 0.5 + 1.5 + 2.0 + 2.0 + 1.0 + 1.0 + 1.0 + 2.0 + 1.0
            wait_time = max(0, tracker.duration - anim_duration - 0.5) if tracker.duration > 0 else 1.0
            if wait_time > 0: self.wait(wait_time)
            self.play(FadeOut(subtitle_voice), run_time=0.5)

    # --- Section 5: Dynamic Visualization ---
    def play_section_05(self):
        """Scene 5: Dynamic Visualization"""
        # Background is default white
        self.update_scene_number("05")

        if not all([self.axes, self.axes_labels, self.unit_circle]):
             print("Warning: Missing elements from Scene 4! Rebuilding...")
             self.axes = Axes(x_range=[-2.5, 2.5, 1], y_range=[-2.5, 2.5, 1], x_length=6, y_length=6, axis_config={"color": MY_DARK_GRAY, "include_numbers": True, "stroke_width": 2}, tips=False)
             x_label = Text("Real Axis (Re)", font_size=24, color=MY_DARK_TEXT).next_to(self.axes.x_axis.get_right(), DOWN)
             y_label = Text("Imaginary Axis (Im)", font_size=24, color=MY_DARK_TEXT).next_to(self.axes.y_axis.get_top(), RIGHT).shift(LEFT*0.5)
             self.axes_labels = VGroup(x_label, y_label)
             radius_val = self.axes.x_axis.n2p(1)[0] - self.axes.x_axis.n2p(0)[0]
             self.unit_circle = Circle(radius=radius_val, color=MY_RED, stroke_width=3).move_to(self.axes.c2p(0,0))
             self.add(self.axes, self.axes_labels, self.unit_circle)
             # Add rebuilt elements to section_elements so they get cleared later
             self.section_elements.add(self.axes, self.axes_labels, self.unit_circle)
        else:
             self.add(self.axes, self.axes_labels, self.unit_circle)
             # Add carried-over elements to section_elements so they get cleared later
             self.section_elements.add(self.axes, self.axes_labels, self.unit_circle)


        formula_ref = MathTex(r"e^{i\theta} = \cos\theta + i\sin\theta", font_size=36, color=MY_DARK_TEXT)
        formula_ref.to_corner(UL, buff=MED_LARGE_BUFF)
        self.section_elements.add(formula_ref)

        theta_tracker = ValueTracker(0)

        # Recreate dynamic elements or ensure they are correctly referenced
        self.point_P = Dot(color=MY_BLUE, radius=0.08)
        self.radius_line = Line(self.axes.c2p(0,0), self.axes.c2p(1,0), color=MY_BLUE, stroke_width=2)
        self.angle_arc = Arc(radius=0.4, start_angle=0, angle=theta_tracker.get_value(), arc_center=self.axes.c2p(0,0), color=MY_GREEN)
        theta_value_text = DecimalNumber(theta_tracker.get_value() * 180/PI, num_decimal_places=0, unit=r"^\circ", font_size=30, color=MY_GREEN)
        self.e_label = MathTex(r"e^{i\theta}", font_size=36, color=MY_PURPLE)

        self.point_P.add_updater(lambda m: m.move_to(self.axes.c2p(np.cos(theta_tracker.get_value()), np.sin(theta_tracker.get_value()))))
        self.radius_line.add_updater(lambda m: m.put_start_and_end_on(self.axes.c2p(0,0), self.point_P.get_center()))
        self.angle_arc.add_updater(lambda m: m.become(Arc(radius=0.4, start_angle=0, angle=theta_tracker.get_value(), arc_center=self.axes.c2p(0,0), color=MY_GREEN)))
        theta_value_text.add_updater(lambda m: m.set_value(theta_tracker.get_value() * 180/PI).next_to(self.angle_arc, RIGHT, buff=SMALL_BUFF))
        self.e_label.add_updater(lambda m: m.next_to(self.point_P, UR, buff=SMALL_BUFF))

        trace = TracedPath(self.point_P.get_center, stroke_color=MY_RED, stroke_width=3, stroke_opacity=0.7)

        dynamic_elements = VGroup(self.point_P, self.radius_line, self.angle_arc, theta_value_text, self.e_label, trace)
        self.add(dynamic_elements)
        self.section_elements.add(dynamic_elements)

        voice_text_05 = "Let's see this in action! As the angle theta changes, the point e to the i theta moves smoothly around the unit circle. Its position always corresponds to cosine theta plus i sine theta. This beautifully illustrates the rotational nature encoded in Euler's formula."
        with custom_voiceover_tts(voice_text_05) as tracker:
            if tracker.audio_path and tracker.duration > 0:
                self.add_sound(tracker.audio_path)
            else:
                print("Warning: Narration 5 TTS failed.")

            subtitle_voice = Text(voice_text_05, font_size=28, color=MY_DARK_TEXT, width=config.frame_width - 2, should_center=True).to_edge(DOWN, buff=MED_SMALL_BUFF)

            self.play(FadeIn(subtitle_voice), run_time=0.5)
            self.play(FadeIn(formula_ref), run_time=1.0)
            animation_duration = max(6.0, tracker.duration - 2.0) if tracker.duration > 2.0 else 6.0
            self.play(theta_tracker.animate.set_value(TAU), run_time=animation_duration, rate_func=linear)

            self.play(FadeOut(subtitle_voice), run_time=0.5)

        # Clear updaters BEFORE clearing the section
        dynamic_elements.clear_updaters()
        # Note: persistent elements like axes were added to section_elements earlier

    # --- Section 6: Summary and Significance ---
    def play_section_06(self):
        """Scene 6: Summary and Significance"""
        # REMOVED: Background color change - uses default white now
        self.update_scene_number("06")

        summary_title = Text("Geometric Summary", font_size=48, color=MY_DARK_TEXT)
        summary_title.to_edge(UP, buff=1.0)

        point1 = Text("• e^(iθ) represents a point on the unit circle.", font_size=32, color=MY_DARK_TEXT, width=config.frame_width - 4)
        point2 = Text("• θ is the angle from the positive real axis.", font_size=32, color=MY_DARK_TEXT, width=config.frame_width - 4)
        point3 = Text("• Its coordinates are (cos θ, sin θ).", font_size=32, color=MY_DARK_TEXT, width=config.frame_width - 4)
        summary_points = VGroup(point1, point2, point3).arrange(DOWN, buff=MED_LARGE_BUFF * 0.8, aligned_edge=LEFT)
        summary_points.next_to(summary_title, DOWN, buff=LARGE_BUFF * 0.8)

        formula_recap = MathTex(r"e^{i\theta} = \cos\theta + i\sin\theta", font_size=48, color=MY_YELLOW)
        formula_recap.next_to(summary_points, DOWN, buff=LARGE_BUFF * 0.8)

        conclusion = Text("A fundamental link between analysis, trigonometry, and geometry.", font_size=32, color=MY_DARK_TEXT, width=config.frame_width - 4)
        conclusion.next_to(formula_recap, DOWN, buff=LARGE_BUFF * 0.8)

        self.section_elements.add(summary_title, summary_points, formula_recap, conclusion)

        voice_text_06 = "In summary, Euler's formula tells us that e to the i theta is a point on the unit circle in the complex plane. The angle theta determines its position, and its coordinates are given by cosine theta and sine theta. This elegant equation provides a powerful bridge between exponential functions, trigonometry, and the geometry of rotations."
        with custom_voiceover_tts(voice_text_06) as tracker:
            if tracker.audio_path and tracker.duration > 0:
                self.add_sound(tracker.audio_path)
            else:
                print("Warning: Narration 6 TTS failed.")

            subtitle_voice = Text(voice_text_06, font_size=28, color=MY_DARK_TEXT, width=config.frame_width - 2, should_center=True).to_edge(DOWN, buff=MED_SMALL_BUFF)

            self.play(FadeIn(subtitle_voice), run_time=0.5)
            self.play(FadeIn(summary_title), run_time=1.5)
            self.play(AnimationGroup(*[FadeIn(p, shift=UP*0.1) for p in summary_points], lag_ratio=0.4), run_time=3.0)
            self.play(FadeIn(formula_recap), run_time=1.5)
            self.play(FadeIn(conclusion), run_time=1.5)

            anim_duration = 0.5 + 1.5 + 3.0 + 1.5 + 1.5
            wait_time = max(0, tracker.duration - anim_duration - 0.5) if tracker.duration > 0 else 1.0
            if wait_time > 0: self.wait(wait_time)
            self.play(FadeOut(subtitle_voice), run_time=0.5)

# --- Main execution block ---
if __name__ == "__main__":
    config.pixel_height = 1080
    config.pixel_width = 1920
    config.frame_rate = 30
    config.output_file = "CombinedScene"
    config.disable_caching = True

    # Set output directory using placeholder
    config.media_dir = r"#(output_path)" # IMPORTANT: Use the placeholder

    scene = CombinedScene()
    scene.render()

    print(f"Scene rendering finished. Output in: {config.media_dir}")
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Contributors: litongjava