利用Python中的Manim进行数学绘画和创作

简介: 利用Python中的Manim进行数学绘画和创作

相信很多同学就算没听过3Blue1Brown,也一定曾看过他们出品的视频,其从独特的视觉角度解说各种数学概念,内容包括线性代数、微积分、神经网络、傅里叶变换以及四元数等晦涩难懂的知识点。例如最火的《线性代数本质》系列视频。

那么这些视频是如何制作的呢?

这里需要引入的是Python的Manim视频支持引擎——专门用于支持数学可视化的媒体引擎,通过Manim并结合Python编程就可以实现3Blue1Brown的视频效果。

什么是Python

在当今备受推崇的大数据技术和人工智能技术领域,Python一直是一门备受瞩目的编程语言。它长期占据着TIOBE公布的编程语言排行榜的榜首位置。业界也一直流传着一句话:“人生苦短,我用Python”。Python作为一门计算机编程语言具有独特的优势,可以总结为以下三个特点:简洁性、易读性和可扩展性。

简洁性:

Python是一门非常简洁的脚本语言。很多时候,只需一行代码就能实现复杂的功能。比如,下面这行代码就能打印出一个九九乘法表。这种简洁性使得你能够将注意力集中在解决问题上,而不必花费过多时间去理解语言本身。

易读性:

Python的语法风格清晰简约,易于阅读。有时候,它就像人类说话一样自然简单,没有过多的修饰。

可扩展性:

Python最初的设计就注重可扩展性,它并没有将所有的特性和功能集成到核心语言中。Python提供了丰富的API和工具,以及众多第三方库,让你可以灵活扩展其功能。你可以根据需要选择合适的库来解决问题,这使得Python非常适合各种不同的应用场景。

什么是manim

manim视频

manim是一个Python第三方库,全称是mathematical animation engine(数学动画引擎)。它是由斯坦福大学数学系学生Grant Sanderson在其YouTube频道3Blue1Brown上创建的。manim用于解说线性代数、微积分、神经网络、黎曼猜想、傅里叶变换以及四元数等数学概念。

manim因其精美的动画制作和独特的解释角度而广受关注,并吸引了越来越多的读者参与其中,共同贡献内容。manim经历了两个版本的迭代。第一代是ManimCairo,这是较早的版本,Grant Sanderson早期的项目都是在此版本中编译的。它使用cairo库作为渲染引擎。第二代是ManimGL,由Grant Sanderson等人开发的最新版本。该版本的最大改进是采用更强大的OpenGL库进行渲染。

此外,还有ManimCE(manim社区版),也使用OpenGL进行渲染。

manim使你能够以编程的方式创建精确的数学图形、动画和场景。与传统的几何画板等绘图软件不同,manim提供了一种全新的思路:“所思即可得”。它能够实现非常精准的绘制。在manim的世界中,你可以体验到一切皆可设置的感觉。颜色、粗细、长度、角度、时长、播放方式等均可通过设置来完成,这使你能够制作出具有个性化的数学动画。

版本 渲染引擎
ManimCairo Cairo manimlib
ManimGL OpenGL manimgl
ManimCE OpenGL manim

如何来运行Manim

  1. 安装 Manim
    首先,您需要安装 Python 和一些依赖项。然后,您可以通过以下命令使用 pip 安装 Manim:
pip install manim
  1. 创建动画
    创建一个 Python 脚本,并使用 Manim 中的类来定义您的动画场景和对象。
  2. 运行 Manim
    在命令行中,转到包含您的 Python 脚本的目录,并使用以下命令来运行 Manim:
manim your_script.py YourSceneName -p -ql
  1. 其中,your_script.py 是您的 Python 脚本的文件名,YourSceneName 是您定义的场景名称。

这样,Manim 就会开始渲染动画。请注意,Manim 的运行可能需要一定的时间,具体取决于动画内容和计算机性能。

用 Manim 绘制图形

首先需要引入 Manim 库,然后将需要绘制的内容封装到一个类(class)里面。

from manim import *

对于编辑好的程序文件(例如christmas-intro.py文件),需要在同一个文件夹下运行命令来运行程序,命令格式如下:

manim -pql christmas-intro.py DemoSquare

上面的命令行中:

  • manim 是运行程序的主要标志用语;
  • p 表示程序运行后会进行预览(图片或视频);
  • ql 表示低质量(quality low), 其他的选项有 -ql, -qm, -qh, -qk, 分别表示 低质量、正常质量、高质量、4K质量;
  • christmas-intro.py 是py代码文件;
  • DemoSquare 是 py代码文件中的一个类;

命令行中,还有其他许多参数可以设置,可以通过社区版的支持文档来进一步了解。

几个常见方程的示例

from manim import *
class QuadraticSurface(ThreeDScene):
    def construct(self):
        resolution_fa = 24 #曲面样本数
        axes = ThreeDAxes() #3维度笛卡尔直角坐标系
        self.set_camera_orientation(phi = 75*DEGREES, theta = 45*DEGREES) #phi是竖直方向岔开的角度,theta是水平面岔开的角度
    
        def paramsFunc(u, v): #参数方程
            
            x = u
            y = 1/2*u**2
            z = v
            return np.array([x, y, z])
        quadratic_surface = Surface(
            paramsFunc,
            resolution = (resolution_fa, resolution_fa), #分辨率
            u_range=[-2.5, +2.5], #参数u的范围
            v_range=[-1, +1], #参数v的范围
            fill_opacity = 0.5 #透明度
        ) #二次曲面
        self.add(axes) #添加坐标轴
        self.wait() #停留一秒
        self.play(Write(quadratic_surface), run_time = 10) #绘制二次曲面
        self.begin_3dillusion_camera_rotation(rate = 2) #摇晃rotation illusion
        self.wait(PI) #摇晃时长
        self.stop_3dillusion_camera_rotation() #停止摇晃
with tempconfig({'quality':'medium_quality', 'preview':True}):
    scene = QuadraticSurface()
    scene.render()

from manim import *
class QuadraticSurface(ThreeDScene):
    def construct(self):
        resolution_fa = 24 #曲面样本数
        axes = ThreeDAxes() #3维度笛卡尔直角坐标系
        self.set_camera_orientation(phi = 75*DEGREES, theta = 45*DEGREES) #phi是竖直方向岔开的角度,theta是水平面岔开的角度
    
        def paramsFunc(u, v): #参数方程
            
            x = np.sqrt(2)* np.cos(TAU*u)*v
            y = np.sqrt(2)* np.sin(TAU*u)*v
            z = v
            return np.array([x, y, z])
        quadratic_surface = Surface(
            paramsFunc,
            resolution = (resolution_fa, resolution_fa), #分辨率
            u_range=[-2, +2], #参数u的范围
            v_range=[-2, +2], #参数v的范围
            fill_opacity = 0.5 #透明度
        ) #二次曲面
        self.add(axes) #添加坐标轴
        self.wait() #停留一秒
        self.play(Write(quadratic_surface), run_time = 10) #绘制二次曲面
        self.begin_3dillusion_camera_rotation(rate = 2) #摇晃rotation illusion
        self.wait(PI) #摇晃时长
        self.stop_3dillusion_camera_rotation() #停止摇晃
with tempconfig({'quality':'medium_quality', 'preview':True}):
    scene = QuadraticSurface()
    scene.render()

manin安装教程

https://docs.manim.community/en/stable/installation.html(社区版)

Manim教程合集

https://www.bilibili.com/video/BV1W4411Z7Zt/?vd_source=3ef6540f8473c7367625a53b7b77fd66

Manim官方文档

https://docs.manim.org.cn/cairo-backend/getting_started/index.html

from manim import *
class QuadraticSurface(ThreeDScene):
    def construct(self):
        resolution_fa = 24 #曲面样本数
        axes = ThreeDAxes() #3维度笛卡尔直角坐标系
        self.set_camera_orientation(phi = 75*DEGREES, theta = 45*DEGREES) #phi是竖直方向岔开的角度,theta是水平面岔开的角度
    
        def paramsFunc(u, v): #参数方程
            x =  np.sinh(u)*np.cos(v)
            y =  np.sinh(u)*np.sin(v)
            z =  np.cosh(u)
            return np.array([x, y, z])
        quadratic_surface = Surface(
            paramsFunc,
            resolution = (resolution_fa, resolution_fa), #分辨率
            u_range=[-1.5, +1.5], #参数u的范围
            v_range=[-PI, +PI], #参数v的范围
            fill_opacity = 0.5 #透明度
        ) #二次曲面
        def paramsFunc1(u, v): #参数方程
            x =  np.sinh(u)*np.cos(v)
            y =  np.sinh(u)*np.sin(v)
            z =  np.cosh(u)
            return np.array([x, y, -z])
        
        quadratic_surface1 = Surface(
                paramsFunc1,
                resolution = (resolution_fa, resolution_fa), #分辨率
                u_range=[-1.5, +1.5], #参数u的范围
                v_range=[-PI, +PI], #参数v的范围
                fill_opacity = 0.5 #透明度
            ) #二次曲面
        self.add(axes) #添加坐标轴
        self.wait() #停留一秒
        self.play(Write(quadratic_surface), run_time = 10) #绘制二次曲面
        self.play(Write(quadratic_surface1), run_time = 10) #绘制二次曲面
        self.begin_3dillusion_camera_rotation(rate = 2) #摇晃rotation illusion
        self.wait(PI) #摇晃时长
        self.stop_3dillusion_camera_rotation() #停止摇晃
with tempconfig({'quality':'medium_quality', 'preview':True}):
    scene = QuadraticSurface()
    scene.render()

版本说明

对于 Manim 的初学者,使用 Manim 时,对于其版本的选择还是有些需要注意的地方。

不同版本的代码、教程等,还是有稍许差异的。

目前 Manim 主要有三个版本:

3b1b 旧版:3blue1brown 自己维护的版本,使用 Cairo 作为后端;

3b1b 新版:3blue1brown 自己维护的版本,使用 OpenGL 和 moderngl 来进行 GPU 渲染,优点是速度快;

Manim 社区版:社区版是2020年下半年才出来的版本,目前主要是 Manim 旧版的衍生,更新更活跃,有很好的文档和社区支持。当然,随着社区版的迭代更新,目前 3b1b 新版 的某些特性也在逐步容纳进来。

关于这几个版本如何选择的建议:

如果你是新接触这个工具的,建议可以从社区版开始,主要是社区版的文档教程比较齐全,其迭代的频率更快。

如果你是 Python 或者其他编程领域的资深人士,想体自己动手深入理解并优化这个工具,可以使用 3b1b 新版

Manim 的安装

在 Manim 3b1b 版本 和 社区版均提供了如何安装的文档:

3b1b 新版的安装文档:https://github.com/3b1b/manim

社区版的安装文档:https://docs.manim.community/en/stable/installation.html

安装ffmpeg

FFmpeg是一个开源的跨平台音视频处理工具,可以用来录制、转换和流式传输音视频内容。它包括一组库和程序,可以处理多种不同格式的音频、视频和图片文件。FFmpeg支持几乎所有主流的音视频格式,因此被广泛应用于多媒体处理领域。

FFmpeg提供了丰富的功能,包括但不限于:

  • 音视频编解码:可以对音频和视频进行编码(压缩)和解码(解压),支持众多编解码器。
  • 格式转换:可以将音视频文件从一个格式转换为另一个格式,例如从MP4转换为AVI。
  • 剪辑和编辑:可以对音视频文件进行剪辑、合并、分离等操作。
  • 流媒体处理:支持流式传输音视频内容,适用于网络直播、视频会议等场景。
  • 屏幕录制:可以用来录制屏幕内容并保存为视频文件。

由于其强大的功能和灵活性,FFmpeg被广泛用于视频处理、视频转码、音频处理等各种多媒体应用中。同时,FFmpeg也是许多视频播放器、编辑软件以及流媒体平台的重要组件之一。

安装FFmpeg可以通过命令行或者图形界面进行操作,以下是在Windows和MacOS上安装FFmpeg的基本步骤:

在Windows上安装FFmpeg

  1. 下载FFmpeg
  1. 解压文件
  • 将下载的压缩包解压到你想要安装的位置。
  1. 设置环境变量(可选):
  • 将FFmpeg的bin目录添加到系统的PATH环境变量中,这样就可以在命令行中直接访问FFmpeg。
  1. 测试安装
  • 打开命令提示符(Win + R,输入cmd,回车),输入以下命令来检查FFmpeg是否成功安装:
ffmpeg -version

在MacOS上安装FFmpeg

  1. 使用Homebrew安装(推荐):
  • 打开终端,运行以下命令安装Homebrew:
/bin/bash -c "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/HEAD/install.sh)"
  • 然后运行以下命令来安装FFmpeg:
brew install ffmpeg
  1. 测试安装
  • 在终端中输入以下命令来检查FFmpeg是否成功安装:
ffmpeg -version

在Linux上安装FFmpeg

  1. 使用包管理器安装
  • 对于大多数Linux发行版,可以使用其包管理器来安装FFmpeg。例如,在Ubuntu上可以使用以下命令:
sudo apt-get update
sudo apt-get install ffmpeg
  1. 测试安装
  • 在终端中输入以下命令来检查FFmpeg是否成功安装:
ffmpeg -version

验证安装

无论在哪个操作系统上安装,验证安装步骤都是一样的。只需要在命令行或终端中输入以下命令:

ffmpeg -version

如果安装成功,会显示FFmpeg的版本信息。这样就可以确认FFmpeg已经成功安装并且可以在系统中正常运行了。

安装Latex

LaTeX是一种专业的排版系统,常用于撰写学术论文、书籍、报告和演示文稿。与常见的文字处理软件(如Microsoft Word)不同,LaTeX使用的是一种“标记语言”,用户需要通过输入特定的命令和标记来控制文档的格式、结构和样式。

LaTeX最初是由著名计算机科学家Leslie Lamport开发的,它建立在TeX排版系统之上,并提供了一系列功能强大的工具和环境,使用户能够轻松地创建高质量的文档。

使用LaTeX编写文档的优势包括:

  • 专业的排版效果:LaTeX能够生成高质量、专业水准的文档排版,特别适合于学术论文、书籍等内容丰富的文档。
  • 具有强大的数学公式排版能力:LaTeX在排版数学公式和符号方面非常出色,被广泛应用于数学、物理等领域的文档撰写。
  • 跨平台性:LaTeX在不同操作系统上都可以运行,并且生成的文档格式稳定、可靠。

虽然学习曲线较陡,但一旦熟悉了LaTeX的基本语法和命令,就能够享受到其带来的排版便利和效果。

安装LaTeX可以通过以下步骤进行操作:

在Windows上安装LaTeX

  1. 下载安装包
  1. 运行安装程序
  • 双击下载的安装程序,按照提示进行安装。
  1. 配置编辑器(可选):
  • 如果你希望使用图形化的编辑器来编写LaTeX文档,可以安装一些流行的LaTeX编辑器,如TeXstudio、TeXworks等,这些编辑器通常会自动检测已安装的LaTeX发行版。
  1. 测试安装
  • 打开命令提示符或者你喜欢的LaTeX编辑器,新建一个简单的.tex文件,输入以下内容并保存:
\documentclass{article}
\begin{document}
Hello, LaTeX!
\end{document}
  • 编译这个文件,如果能够生成对应的.pdf文件并且内容显示正常,那么LaTeX安装成功了。

在MacOS上安装LaTeX

  1. 使用MacTeX安装
  • 下载并安装MacTeX,这是专为MacOS准备的LaTeX发行版。你可以在MacTeX官方网站下载最新版本的安装程序。
  1. 配置编辑器(可选):
  • 安装一些流行的LaTeX编辑器,如TeXShop、TeXstudio等,这些编辑器通常会自动检测已安装的LaTeX发行版。
  1. 测试安装
  • 使用编辑器创建一个简单的.tex文件,输入上述示例中的内容,并进行编译。

在Linux上安装LaTeX

  1. 使用包管理器安装
  • 对于大多数Linux发行版,可以使用其包管理器来安装LaTeX。例如,在Ubuntu上可以使用以下命令:
sudo apt-get update
sudo apt-get install texlive-full
  1. 测试安装
  • 使用编辑器创建一个简单的.tex文件,输入上述示例中的内容,并进行编译。

安装dvisvgm

安装 dvisvgm 可以通过以下步骤进行:

Windows 用户

  1. 使用 Chocolatey 进行安装:
    打开命令提示符(CMD)或 PowerShell,并执行以下命令:
choco install dvisvgm

macOS 用户

  1. 使用 Homebrew 进行安装:
    打开终端,并执行以下命令:
brew install dvisvgm

Linux 用户

  1. 对于基于 Debian 的系统(如 Ubuntu):
    打开终端,并执行以下命令:
sudo apt-get install dvisvgm
  1. 对于基于 Red Hat 的系统(如 Fedora):
    打开终端,并执行以下命令:
sudo dnf install dvisvgm

一旦安装完成,您可以在命令行中运行 dvisvgm --version 来验证 dvisvgm 是否成功安装。

利用Manim绘制苹果logo

from manimlib.imports import *
from my_projects.my_utils.boolean_operation import VGroup_
get_angle = lambda c: np.angle(-c) + PI if not c/abs(c) == 1 else 0
convert_angle = lambda a: a if a>=0 else a + TAU
class Apple_logo(VGroup):
    CONFIG = {
        'scale': 0.25,
        'logo_color': RED_D,
        'logo_stroke': 8,
        'coord_center': DOWN * 1.25,
        'assistent_lines_config': {
            'stroke_width': 2,
            'stroke_color': GREY,
        },
        'key_lines_config': {
            'stroke_width': 2.5,
            'stroke_color': GREY_E,
        },
        'step_size': 0.02,
        'create_shape': True,
    }
    def __init__(self, **kwargs):
        VGroup.__init__(self, **kwargs)
        self.set_key_lines()
        self.set_outline()
        if self.create_shape:
            self.set_logo_shape()
    def set_key_lines(self):
        s = self.scale
        center = self.coord_center
        self.c1 = Circle(arc_center=center+ORIGIN, radius=1/2*s, **self.key_lines_config)
        self.c5_r = Circle(arc_center=center+3*s*RIGHT, radius=5/2*s, **self.key_lines_config)
        self.c5_l = Circle(arc_center=center+3*s*LEFT,  radius=5/2*s, **self.key_lines_config)
        # self.c5_l = self.c5_r.copy().flip()
        self.c8_d = Circle(arc_center=center + s * 5.77 * DOWN, radius=8/2*s, **self.key_lines_config)
        self.c13  = Circle(arc_center=center + s * 4.73 * UP, radius=13/2*s, **self.key_lines_config)
        self.c8_r = Circle(arc_center=center + s * ( 3.48 * RIGHT + 7.69 * UP),
                           radius=8/2*s, **self.key_lines_config)
        self.c8_l = Circle(arc_center=center + s * (-3.48 * RIGHT + 7.69 * UP),
                           radius=8/2*s, **self.key_lines_config)
        self.c3_r = Circle(arc_center=center + s * ( 5.9 * RIGHT + 2.75 * UP),
                           radius=3/2*s, **self.key_lines_config)
        self.c3_l = Circle(arc_center=center + s * (-5.9 * RIGHT + 2.75 * UP),
                           radius=3/2*s, **self.key_lines_config)
        self.c24_r = Circle(arc_center=center + s * ( 3.89 * RIGHT + 6.54 * UP),
                           radius=24/2*s, **self.key_lines_config)
        self.c24_l = Circle(arc_center=center + s * (-3.89 * RIGHT + 6.54 * UP),
                           radius=24/2*s, **self.key_lines_config)
        self.c12_r = Circle(arc_center=center + s * ( 2.11 * RIGHT + 6.23 * UP),
                           radius=12/2*s, **self.key_lines_config)
        self.c12_l = Circle(arc_center=center + s * (-2.11 * RIGHT + 6.23 * UP),
                           radius=12/2*s, **self.key_lines_config)
        self.c8_u = Circle(arc_center=center + s * 14.89 * UP, radius=8/2*s, **self.key_lines_config)
        self.c8_01 = Circle(arc_center=center + s * (8.82 * RIGHT + 6.23 * UP),
                            radius=8/2*s, **self.key_lines_config)
        self.c8_02 = Circle(arc_center=center + s * (3.61 * RIGHT + 12.23 * UP),
                            radius=8/2*s, **self.key_lines_config)
        self.c8_03 = Circle(arc_center=center + s * (-0.46 * RIGHT + 15.69 * UP),
                            radius=8/2*s, **self.key_lines_config)
        self.key_lines = VGroup(self.c1, self.c5_r, self.c5_l,
                                self.c8_d, self.c13,
                                self.c8_r, self.c8_l,
                                self.c3_r, self.c3_l,
                                self.c24_r, self.c24_l,
                                self.c12_r, self.c12_l,
                                self.c8_u,
                                self.c8_01,
                                self.c8_02, self.c8_03,
                                )
        self.key_lines_center = VGroup(* [Dot(c.get_center(), stroke_color=self.key_lines_config['stroke_color'],
                                              radius=self.key_lines_config['stroke_width']/100) for c in self.key_lines])
        return self
    def set_outline(self):
        s = self.scale
        center = self.coord_center
        # kp_list = [[7.41, 2.49, 0], [4.81, -1.72, 0], [1.85, -2.22, 0], [-1.85, -2.22, 0], [-4.81, 1.72, 0], [-8.1, 5.91, 0],
        #            [-6.22, 10.6, 0], [-1.74, 11.29, 0], [1.74, 11.29, 0], [6.22, 10.6, 0], [6.95, 9.77, 0], [-0.37, 11.87, 0], [3.42, 16.22, 0]]
        kp_list = [[6.95, 9.77, 0], [6.22, 10.6, 0], [1.74, 11.29, 0], [-1.74, 11.29, 0], [-6.22, 10.6, 0], [-8.1, 5.91, 0],
                   [-4.81, -1.72, 0], [-1.85, -2.22, 0], [1.85, -2.22, 0], [4.81, -1.72, 0], [7.41, 2.49, 0], [-0.35, 11.69, 0], [3.5, 16.23, 0]]
        self.key_angles = np.array([10.62, 69, -51.5, 69, 49.77, 40.52, 73.94, -54.97, 73.94, 23.77, -131.51, 96.09, -96.09]) * DEGREES
        self.key_radius = np.array([12/2, 8/2, 8/2, 8/2, 12/2, 24/2, 5/2, 8/2, 5/2, 24/2, 8/2]) * s
        self.key_points = np.array(kp_list) * s + center
        n = len(kp_list)
        self.outline_by_arcs = VGroup_(step_size=self.step_size)
        for i in range(n-2):
            p1, p2 = self.key_points[i], self.key_points[(i+1) % (n-2)]
            arc_i = ArcBetweenPoints(p1, p2, angle=self.key_angles[i], radius=self.key_radius[i], stroke_color=self.logo_color, stroke_width=self.logo_stroke)
            self.outline_by_arcs.add(arc_i)
        arc_1 = ArcBetweenPoints(self.key_points[-2], self.key_points[-1], angle=self.key_angles[-2], radius=self.key_radius[-2], stroke_color=self.logo_color, stroke_width=self.logo_stroke)
        arc_2 = ArcBetweenPoints(self.key_points[-2], self.key_points[-1], angle=self.key_angles[-1], radius=self.key_radius[-1], stroke_color=self.logo_color, stroke_width=self.logo_stroke)
        self.outline_by_arcs.add(arc_1, arc_2)
        return self
    def set_logo_shape(self):
        s = self.scale
        center = self.coord_center
        self.outline_by_arcs.get_all_outline_points()
        self.shape_1 = self.outline_by_arcs.get_shape(center=center + s * (-3.89 * RIGHT + 6.54 * UP), fill_color=self.logo_color, fill_opacity=1, stroke_color=self.logo_color, stroke_width=self.logo_stroke)
        self.shape_2 = self.outline_by_arcs.get_shape(center=center + s * (1.52 * RIGHT + 14.05 * UP), fill_color=self.logo_color, fill_opacity=1, stroke_color=self.logo_color, stroke_width=self.logo_stroke)
        # shape_1.set_fill(self.logo_color, 1).set_stroke(self.logo_color, self.logo_stroke, 1)
        # shape_2.set_fill(self.logo_color, 1).set_stroke(self.logo_color, self.logo_stroke, 1)
        self.add(self.shape_1, self.shape_2)
        return self
# from my_projects.others.BooleanOperationsOnPolygons import PolygonIntersection, PolygonUnion, PolygonSubtraction
#
# class PolygonBooleanTest(Scene):
#     def construct(self):
#         pol1 = RegularPolygon(9).scale(2).shift(LEFT)
#         pol2 = RegularPolygon(9).scale(2).shift(RIGHT)
#         start = time.perf_counter()
#         boolpol = PolygonSubtraction(
#             PolygonUnion(pol1, pol2), PolygonIntersection(pol1, pol2)
#         ).set_stroke(color=YELLOW, width=0).set_fill(color=YELLOW, opacity=0.5)
#         end = time.perf_counter()
#         print(end - start)
#         self.add(pol1, pol2, boolpol)
#
#         '''
#         pol1 = RegularPolygon(16).scale(2)
#         pol2 = Circle().stretch(1.2, 1).stretch(0.8, 0)
#         pol3 = RegularPolygon(5).scale(2).shift(LEFT*0.9)
#         pol4 = RegularPolygon(3).stretch(3, 1).shift(LEFT)
#         pol5 = Ellipse().scale(0.6).shift(LEFT).stretch(2, 0)
#         #start = time.perf_counter()
#         boolpol = PolygonSubtraction(
#             PolygonUnion(
#                 PolygonIntersection(
#                     PolygonSubtraction(
#                         pol1, pol2), pol3), pol4), pol5) \
#             .set_stroke(color=YELLOW, width=0).set_fill(color=YELLOW, opacity=0.5)
#         #end = time.perf_counter()
#         #print(end - start)#0.49910769999999993
#         self.add(pol1, pol2, pol3, pol4, pol5, boolpol)
#         '''
#
class Compass(VGroup):
    CONFIG = {
        'stroke_color': GREY_E,
        'fill_color': WHITE,
        'stroke_width': 2,
        'leg_length': 3,
        'leg_width': 0.12,
        'r': 0.2,
        'depth_test': True,
    }
    def __init__(self, span=2.5,  **kwargs):
        VGroup.__init__(self, **kwargs)
        self.span = span
        self.create_compass()
    # def create_compass_old(self):
    #
    #     s, l, r = self.span, self.leg_length, self.r
    #     theta = np.arcsin(s/2/l)
    #     arc = Arc(start_angle=-PI/2 + theta, angle=TAU-theta*2, radius=r, stroke_color=self.stroke_color, stroke_width=self.stroke_width*4)
    #     dot = Dot(color=self.stroke_color).set_height(self.stroke_width*8/100)
    #     leg_01 = Line(arc.get_center(), arc.get_center() + complex_to_R3(l * np.exp(1j * (-PI/2-theta))), stroke_color=self.stroke_color, stroke_width=self.stroke_width)
    #     leg_02 = Line(arc.get_center(), arc.get_center() + complex_to_R3(l * np.exp(1j * (-PI/2+theta))), stroke_color=self.stroke_color, stroke_width=self.stroke_width * 1)
    #     leg_11 = Line(arc.get_center(), arc.get_center() + complex_to_R3((l - r * 0.8) * np.exp(1j * (-PI/2-theta))), stroke_color=self.stroke_color, stroke_width=self.stroke_width * 4)
    #     leg_12 = Line(arc.get_center(), arc.get_center() + complex_to_R3((l - r * 0.8) * np.exp(1j * (-PI/2+theta))), stroke_color=self.stroke_color, stroke_width=self.stroke_width * 4)
    #     pen_point = Dot(color=self.stroke_color).set_height(self.stroke_width * 1/125).move_to(leg_02.get_end())
    #     leg_1, leg_2 = VGroup(leg_01, leg_11),  VGroup(leg_02, leg_12, pen_point)
    #     head = Line(UP * r, UP * (r + r * 1), stroke_color=self.stroke_color, stroke_width=self.stroke_width * 4)
    #
    #     self.add(arc, dot, head, leg_1, leg_2)
    #     self.move_to(ORIGIN)
    #
    #     return self
    def create_compass(self):
        s, l, r, w = self.span, self.leg_length, self.r, self.leg_width
        self.theta = np.arcsin(s/2/l)
        self.c = Circle(radius=r, fill_color=self.fill_color, fill_opacity=1, stroke_color=self.stroke_color, stroke_width=self.stroke_width*5)
        c2 = Circle(radius=r+self.stroke_width*5/100/2, fill_opacity=0, stroke_color=self.fill_color, stroke_width=self.stroke_width)
        self.leg_1 = Polygon(ORIGIN, l * RIGHT, (l-w*np.sqrt(3)) * RIGHT + w * DOWN, w * DOWN,
                             stroke_width=0, stroke_color=self.fill_color, fill_color=self.stroke_color,
                             fill_opacity=1).rotate(-PI/2-self.theta, about_point=self.c.get_center())
        self.leg_2 = Polygon(ORIGIN, l * RIGHT, (l-w*np.sqrt(3)) * RIGHT + w * UP, w * UP,
                             stroke_width=0, stroke_color=self.fill_color, fill_color=self.stroke_color,
                             fill_opacity=1).rotate(-PI/2+self.theta, about_point=self.c.get_center())
        # self.leg_1, self.leg_2 = VGroup(leg_01, leg_11),  VGroup(leg_02, leg_12, pen_point)
        h = Line(UP * r, UP * (r + r * 1.8), stroke_color=self.stroke_color, stroke_width=self.stroke_width*6)
        self.head = VGroup(h, self.c, c2)
        self.add(self.leg_1, self.leg_2, self.head)
        self.move_to(ORIGIN)
        return self
    def get_niddle_tip(self):
        return self.leg_1.get_vertices()[1]
    def get_pen_tip(self):
        return self.leg_2.get_vertices()[1]
    def move_niddle_tip_to(self, pos):
        self.shift(pos-self.get_niddle_tip())
        return self
    def rotate_about_niddle_tip(self, angle=PI/2):
        self.rotate(angle=angle, about_point=self.get_niddle_tip())
    def get_span(self):
        # return self.span 如果进行了缩放而self.span没变会有问题
        return get_norm(self.get_pen_tip() - self.get_niddle_tip())
    def set_span(self, s):
        self.span = s
        l, r, w = self.leg_length, self.r, self.leg_width
        theta_new, theta_old = np.arcsin(s/2/l), self.theta
        sign = np.sign(get_angle(R3_to_complex(self.leg_2.get_vertices()[1] - self.leg_2.get_vertices()[0])) - get_angle(R3_to_complex(self.leg_1.get_vertices()[1] - self.leg_1.get_vertices()[0])))
        rotate_angle = 2 * (theta_new - theta_old) * sign
        self.leg_2.rotate(rotate_angle, about_point=self.c.get_center())
        self.theta=theta_new
        self.head.rotate(rotate_angle/2, about_point=self.c.get_center())
        self.rotate_about_niddle_tip(-rotate_angle/2)
        return self
    def set_compass(self, center, pen_tip):
        self.move_niddle_tip_to(center)
        self.set_span(get_norm(pen_tip - center))
        self.rotate_about_niddle_tip(np.angle(R3_to_complex(pen_tip - center)) - np.angle(R3_to_complex(self.get_pen_tip() - center)))
        return self
    def set_compass_to_draw_arc(self, arc):
        return self.set_compass(arc.arc_center, arc.get_start())
    def reverse_tip(self):
        return self.flip(axis=self.head[0].get_end() - self.head[0].get_start(), about_point=self.c.get_center())
class DrawingScene(Scene):
    CONFIG = {
        'compass_config':{
            'stroke_color': GREY_E,
            'fill_color': WHITE,
            'stroke_width': 2,
            'leg_length': 3,
            'leg_width': 0.12,
            'r': 0.2,
            'depth_test': True,
        },
        'ruler_config':{
            'width': 10,
            'height': 0.8,
            'stroke_width': 8,
            'stroke_color': GREY_E,
            'stroke_opacity': 0.4,
            'fill_color': WHITE,
            'fill_opacity': 0.5,
        },
        'dot_config':{
            'radius': 0.06,
            'color': GREY_E,
        },
        'line_config':{
            'stroke_color': GREY_E,
            'stroke_width': 2.5,
        },
        'brace_config':{
            'fill_color': GREY_E,
            'buff':0.025,
        },
        'text_config':{
            'size': 0.6 * 5, # 5 times than the actual size and the sacle down
            'font': 'Cambria Math',
            'color': GREY_E,
        },
        'add_ruler': False,
    }
    def setup(self):
        self.cp = Compass(**self.compass_config)
        self.ruler = VGroup(Rectangle(**self.ruler_config).set_height(self.ruler_config['height'] - self.ruler_config['stroke_width']/2/100, stretch=True)\
                            .round_corners(self.ruler_config['height']/8),
                            Rectangle(**self.ruler_config).set_opacity(0))
        self.dot = Dot(**self.dot_config)
        self.cp.move_to(UP * 10)
        if self.add_ruler:
            self.ruler.move_to(DOWN * 10)
            self.add(self.ruler)
        self.add(self.cp)
        self.temp_points = []
    def construct(self):
        self.add(self.cp)
        self.play(self.cp.move_niddle_tip_to, ORIGIN, run_time=1)
        self.wait(0.3)
        self.set_span(3.6, run_time=1, rate_func=smooth)
        self.wait(0.5)
        self.set_compass(DL * 0.5, UR * 0.5, run_time=1, rate_func=there_and_back)
        arc = Arc(color=GREY_E)
        self.set_compass_to_draw_arc(arc)
        self.draw_arc_by_compass(arc)
        self.wait()
    def set_span(self, s, run_time=1, rate_func=smooth):
        s_old = self.cp.get_span()
        n = int(run_time * self.camera.frame_rate)
        dt = 1/self.camera.frame_rate
        t_series = np.linspace(1, n, n)/n
        # s_series = s_old + rate_func(t_series) * (s - s_old)
        s_series = [s_old + rate_func(t_series[i]) * (s - s_old) for i in range(n)]
        for i in range(n):
            self.cp.set_span(s_series[i])
            self.wait(dt)
    def set_compass_direction(self, start, end, run_time=1, rate_func=smooth):
        vect = end - start
        a = np.angle(R3_to_complex(vect))
        c_old, p_old = self.cp.get_niddle_tip(), self.cp.get_pen_tip()
        a_old = np.angle(R3_to_complex(p_old - c_old))
        n = int(run_time * self.camera.frame_rate)
        dt = 1/self.camera.frame_rate
        t_series = np.linspace(1, n, n)/n
        c_series = [c_old + rate_func(t_series[i]) * (start - c_old) for i in range(n)]
        delta_a = (a - a_old)/n
        for i in range(n):
            self.bring_to_front(self.cp)
            self.cp.move_niddle_tip_to(c_series[i])
            self.cp.rotate_about_niddle_tip(delta_a)
            self.wait(dt)
    def set_compass(self, center, pen_tip, run_time=1, rate_func=smooth, emphasize_dot=False):
        if emphasize_dot:
            run_time -= 0.15
        c_old, p_old = self.cp.get_niddle_tip(), self.cp.get_pen_tip()
        n = int(run_time * self.camera.frame_rate)
        dt = 1/self.camera.frame_rate
        t_series = np.linspace(1, n, n)/n
        # s_series = s_old + rate_func(t_series) * (s - s_old)
        c_series = [c_old + rate_func(t_series[i]) * (center - c_old) for i in range(n)]
        p_series = [p_old + rate_func(t_series[i]) * (pen_tip - p_old) for i in range(n)]
        for i in range(n):
            self.bring_to_front(self.cp)
            self.cp.set_compass(c_series[i], p_series[i])
            self.wait(dt)
        if emphasize_dot:
            self.emphasize_dot([center, pen_tip], run_time=0.15)
    def set_compass_(self, center, pen_tip, adjust_angle=0, run_time=1, rate_func=smooth, emphasize_dot=False):
        vect = center - pen_tip
        a = np.angle(R3_to_complex(vect)) + adjust_angle
        s = get_norm(vect)
        c_old, p_old, s_old = self.cp.get_niddle_tip(), self.cp.get_pen_tip(), self.cp.get_span()
        a_old = np.angle(R3_to_complex(p_old - c_old))
        if emphasize_dot:
            run_time -= 0.15
        n = int(run_time * self.camera.frame_rate)
        dt = 1/self.camera.frame_rate
        t_series = np.linspace(1, n, n)/n
        c_series = [c_old + rate_func(t_series[i]) * (center - c_old) for i in range(n)]
        delta_a = (a - a_old)/n
        s_series = [s_old + rate_func(t_series[i]) * (s - s_old) for i in range(n)]
        for i in range(n):
            self.bring_to_front(self.cp)
            self.cp.move_niddle_tip_to(c_series[i])
            self.cp.rotate_about_niddle_tip(delta_a)
            self.cp.set_span(s_series[i])
            self.wait(dt)
        if emphasize_dot:
            self.emphasize_dot([center, pen_tip], run_time=0.15)
    def set_compass_to_draw_arc(self, arc, **kwargs):
        self.set_compass(arc.arc_center, arc.get_start(), **kwargs)
    def set_compass_to_draw_arc_(self, arc, **kwargs):
        self.set_compass_(arc.arc_center, arc.get_start(), **kwargs)
    def draw_arc_by_compass(self, arc, is_prepared=True, run_time=1, rate_func=smooth, reverse=False, add_center=False, **kwargs):
        self.bring_to_front(self.cp)
        if not is_prepared: self.set_compass_to_draw_arc(arc, run_time=0.5)
        theta = arc.angle if not reverse else -1 * arc.angle
        self.play(Rotating(self.cp, angle=theta, about_point=self.cp.get_niddle_tip()), ShowCreation(arc), rate_func=rate_func, run_time=run_time)
        if add_center:
            d = Dot(self.cp.get_niddle_tip(), **self.dot_config).scale(0.5)
            self.temp_points.append(d)
            self.add(d)
    def emphasize_dot(self, pos, add_dot=False, size=1.2, run_time=0.2, **kwargs):
        if type(pos) == list:
            d = VGroup(*[Dot(pos[i], radius=size/2, color=GREY_C, fill_opacity=0.25).scale(0.25) for i in range(len(pos))])
        else:
            d = Dot(pos, radius=size/2, color=GREY_C, fill_opacity=0.25).scale(0.25)
        self.add(d)
        if type(pos) == list:
            self.play(d[0].scale, 4, d[1].scale, 4, rate_func=linear, run_time=run_time)
            
        else:
            self.play(d.scale, 4, rate_func=linear, run_time=run_time)
        self.remove(d)
        if add_dot:
            if type(pos) == list:
                dot = VGroup(*[Dot(pos[i],**kwargs) for i in range(len(pos))])
            else:
                dot = Dot(pos, **kwargs)
            self.add(dot)
            return dot
    def set_ruler(self, pos1, pos2, run_time=1, rate_func=smooth):
        p1, p2 = self.ruler[-1].get_vertices()[1], self.ruler[-1].get_vertices()[0]
        c12 = (p1 + p2) / 2
        center = (pos1 + pos2)/2
        self.bring_to_front(self.ruler)
        self.play(self.ruler.shift, center - c12, run_time=run_time/2, rate_func=rate_func)
        self.play(Rotating(self.ruler, angle=np.angle(R3_to_complex(pos2 - pos1)) - np.angle(R3_to_complex(p2 - p1)), about_point=center), run_time=run_time/2, rate_func=rate_func)
    def draw_line(self, pos1, pos2, is_prepared=True, run_time=1.2, rate_func=smooth, pre_time=0.8):
        if not is_prepared: self.set_ruler(pos1, pos2, run_time=pre_time)
        self.dot.move_to(pos1)
        self.emphasize_dot(pos1, run_time=0.15)
        self.add(self.dot)
        l = Line(pos1, pos2, **self.line_config)
        self.play(ShowCreation(l), self.dot.move_to, pos2, run_time=run_time-0.3, rate_func=rate_func)
        self.emphasize_dot(pos2, run_time=0.15)
        self.remove(self.dot)
        return l
    def draw_line_(self, l, is_prepared=True, run_time=1.2, rate_func=smooth):
        pos1, pos2 = l.get_start(), l.get_end()
        if not is_prepared: self.set_ruler(pos1, pos2, run_time=0.5)
        self.dot.move_to(pos1)
        self.emphasize_dot(pos1, run_time=0.15)
        self.add(self.dot)
        # l = Line(pos1, pos2, **self.line_config)
        self.play(ShowCreation(l), self.dot.move_to, pos2, run_time=run_time-0.3, rate_func=rate_func)
        self.emphasize_dot(pos2, run_time=0.15)
        self.remove(self.dot)
        return l
    def put_aside_ruler(self, direction=DOWN, run_time=0.5):
        self.bring_to_front(self.ruler)
        self.play(self.ruler.move_to, direction * 15, run_time=run_time)
    def put_aside_compass(self, direction=DOWN, run_time=0.5):
        self.bring_to_front(self.cp)
        self.play(self.cp.move_to, direction * 15, run_time=run_time)
    def get_length_label(self, p1, p2, text='', reverse_label=False, add_bg=False, bg_color=WHITE):
        l = Line(p1, p2)
        b = Brace(l, direction=complex_to_R3(np.exp(1j * (l.get_angle()+PI/2 * (1 -2 * float(reverse_label))))), **self.brace_config)
        t = Text(text, **self.text_config).scale(0.2)
        if add_bg:
            bg = SurroundingRectangle(t, fill_color=bg_color, fill_opacity=0.6, stroke_opacity=0).set_height(t.get_height() + 0.05, stretch=True).set_width(t.get_width() + 0.05, stretch=True)
            b.put_at_tip(bg, buff=0.0)
            b.put_at_tip(t, buff=0.05)
            return b, bg, t
        else:
            b.put_at_tip(t, buff=0.05)
            return b, t
    def set_compass_and_show_span(self, p1, p2, run_time=1, show_span_time=[0.4, 0.3, 0.9, 0.4], text='', reverse_label=False, add_bg=True, **kwargs):
        self.set_compass(p1, p2, run_time=run_time, **kwargs)
        bt = self.get_length_label(p1, p2, text=text, reverse_label=reverse_label, add_bg=add_bg)
        b, t = bt[0], bt[-1]
        st = show_span_time
        self.play(ShowCreation(b), run_time=st[0])
        if add_bg:
            self.add(bt[1])
            self.play(FadeIn(t), run_time=st[1])
        else:
            self.play(FadeIn(t), run_time=st[1])
        self.wait(st[2])
        self.play(FadeOut(VGroup(*bt)), run_time=st[3])
        return bt
    def set_compass_and_show_span_(self, p1, p2, run_time=1, show_span_time=[0.4, 0.3, 0.9, 0.4], text='', reverse_label=False, add_bg=True, **kwargs):
        self.set_compass_(p1, p2, run_time=run_time, **kwargs)
        bt = self.get_length_label(p1, p2, text=text, reverse_label=reverse_label)
        b, t = bt[0], bt[-1]
        st = show_span_time
        self.play(ShowCreation(b), run_time=st[0])
        if add_bg:
            self.add(bt[1])
            self.play(FadeIn(t), run_time=st[1])
        else:
            self.play(FadeIn(t), run_time=st[1])
        self.wait(st[2])
        self.play(FadeOut(VGroup(*bt)), run_time=st[3])
        return bt
    def highlight_on(self, *mobjects, to_front=True, stroke_config={'color': '#66CCFF', 'width': 4}, run_time=1, **kwargs):
        self.highlight = VGroup(*mobjects)
        self.play(self.highlight.set_stroke, stroke_config, run_time=run_time, **kwargs)
        if to_front:
            self.bring_to_front(self.highlight)
            self.bring_to_front(self.cp, self.ruler)
    def highlight_off(self, *mobjects):
        pass
    def show_arc_info(self, arc, time_list=[0.5, 0.2, 0.3]):
        c, r, s, a, ps, pe = arc.arc_center, arc.radius, arc.start_angle, arc.angle, arc.get_start(), arc.get_end()
        d_center = Dot(c, radius=0.08, color=PINK)
        r1, r2 = DashedLine(c, ps, stroke_width=3.5, stroke_color=PINK), DashedLine(c, pe , stroke_width=3.5, stroke_color=PINK)
        arc_new = Arc(arc_center=c, radius=r, start_angle=s, angle=a, stroke_width=8, stroke_color=RED)
        self.play(ShowCreation(arc_new), run_time=time_list[0])
        self.play(FadeIn(arc_new), run_time=time_list[1])
        self.play(ShowCreation(r1), ShowCreation(r2), run_time=time_list[2])
class Test_01(Scene):
    def construct(self):
        logo = Apple_logo()
        kl = logo.key_lines
        ol = logo.outline_by_arcs
        self.play(ShowCreation(kl), run_time=6)
        self.wait(0.25)
        # self.play(ShowCreation(ol), run_time=4)
        # self.wait(0.8)
        self.play(DrawBorderThenFill(logo.shape_1, stroke_width=logo.logo_stroke), run_time=3)
        self.play(DrawBorderThenFill(logo.shape_2, stroke_width=logo.logo_stroke), run_time=1.6)
        self.wait(0.2)
        self.play(Uncreate(kl), run_time=1.5)
        self.wait(2)
class Test_compass(Scene):
    def construct(self):
        coord = NumberPlane(axis_config={"stroke_color": GREY_D}, background_line_style={"stroke_color": BLUE_C},)
        self.add(coord)
        cp = Compass(stroke_color=PINK)
        c = Circle(radius=cp.get_span())
        self.add(cp)
        self.wait()
        self.play(cp.move_niddle_tip_to, ORIGIN, run_time=1)
        self.wait()
        self.bring_to_front(cp)
        self.play(Rotating(cp, about_point=cp.get_niddle_tip()), ShowCreation(c), rate_func=smooth, run_time=1.5)
        self.bring_to_front(cp)
        c1 = Circle(stroke_color=GREY_E)
        # self.play(cp.set_span, 1, run_time=1)
        # cp.set_span(1)
        # self.wait()
        # self.play(Rotating(cp, about_point=cp.get_niddle_tip()), ShowCreation(c1), rate_func=smooth, run_time=1.5)
        self.bring_to_front(cp)
        # self.play(cp.set_compass, DL * 2, UR * 2, run_time=1.5)
        cp.set_compass(DL * 2, UR * 2)
        self.wait(2)
class Test_1(DrawingScene):
    def construct(self):
        s = 0.5
        # axis = Axes(x_range=[-30, 30, 1], y_range=[-20, 20, 1], axis_config={"stroke_color": GREY_D, "stroke_width": 2.5, "stroke_opacity": 1},).scale(s)
        # logo = Apple_logo(create_shape=False, coord_center=ORIGIN, scale=s*2)
        # self.add(axis, logo.c1)
        c = Circle(color=BLACK)
        c1 = Circle(color=BLACK)
        c2 = Circle(color=BLACK)
        self.add(c, c1, c2)
        self.wait()
        self.play(ApplyMethod(c.shift, RIGHT * 4, run_time=1),
                  ApplyMethod(c1.shift, RIGHT * 4, run_time=2),
                  ApplyMethod(c2.shift, RIGHT * 4, run_time=4)
        )
class Step_0(DrawingScene):
    '''
    画出基本的直角坐标系作为辅助
    '''
    def construct(self):
        s = 0.5
        # coord = NumberPlane(x_range=[-15, 15, 1], y_range=[-15, 15, 1], axis_config={"stroke_color": GREY_D}, background_line_style={"stroke_color": BLUE_C},).scale(s)
        # logo = Apple_logo(create_shape=False, coord_center=ORIGIN, scale=s)
        self.cp.move_to(UP * 10)
        # self.add(coord)
        self.wait()
        self.play(FadeIn(self.ruler))
        l1 = self.draw_line(LEFT * 4.5, RIGHT * 4.5, is_prepared=False)
        self.put_aside_ruler()
        self.wait()
        arc_1 = Arc(arc_center=LEFT * 2, radius=3, start_angle=-PI/3, angle=2*PI/3, **self.line_config)
        arc_2 = Arc(arc_center=RIGHT * 2, radius=3, start_angle=2 * PI/3, angle=2*PI/3, **self.line_config)
        self.play(self.cp.move_to, ORIGIN)
        self.wait(0.2)
        self.set_compass_to_draw_arc(arc_1, emphasize_dot=True)
        self.draw_arc_by_compass(arc_1)
        self.wait(0.1)
        self.set_compass_to_draw_arc_(arc_2, adjust_angle=PI, emphasize_dot=True)
        self.draw_arc_by_compass(arc_2)
        self.wait(0.2)
        self.put_aside_compass()
        self.set_ruler(UP, DOWN)
        l2 = self.draw_line(UP * 3, DOWN * 3)
        self.put_aside_ruler(direction=LEFT)
        self.wait(0.6)
        self.play(FadeOut(arc_1), FadeOut(arc_2), l1.scale, 2, l2.scale, 2, run_time=1.2)
        self.wait(0.8)
        c1 = Circle(radius=1*s, **self.line_config)
        # c2_l = Circle(arc_center=6*s*LEFT, radius=5*s, **self.line_config)
        # c2_r = Arc(arc_center=6*s*RIGHT, radius=5*s, start_angle=PI, angle=-TAU, **self.line_config)
        self.set_compass_to_draw_arc(c1, run_time=1.2)
        self.wait(0.12)
        self.draw_arc_by_compass(c1)
        self.wait(0.2)
        arcs = VGroup(*[Arc(arc_center=i*s*RIGHT, radius=1*s, start_angle=-PI/6, angle=PI/3, **self.line_config) for i in range(1, 7)])
        for i in range(6):
            self.set_compass_to_draw_arc(arcs[i], run_time=0.32, emphasize_dot=False)
            self.draw_arc_by_compass(arcs[i], run_time=0.5)
        self.wait(0.6)
        axis = Axes(x_range=[-15, 15, 1], y_range=[-10, 10, 1], axis_config={"stroke_color": GREY_D, "stroke_width": 2.5}, ).scale(s)
        dot = Dot(color=GREY_E, radius=0.06)
        self.play(FadeIn(axis), FadeIn(dot), FadeOut(l1), FadeOut(l2), FadeOut(self.cp), FadeOut(arcs))
        self.wait(2)
class Step_1(DrawingScene):
    '''
    画出...
    '''
    def construct(self):
        s = 0.5
        axis = Axes(x_range=[-30, 30, 1], y_range=[-20, 20, 1], axis_config={"stroke_color": GREY_D, "stroke_width": 2.5, "stroke_opacity": 1},).scale(s)
        logo = Apple_logo(create_shape=False, coord_center=ORIGIN, scale=s*2)
        dot_config = {'color': GREY_E, 'radius': 0.06}
        d1 = Dot(**dot_config)
        self.add(axis, logo.c1, d1)
        self.wait()
        d5_l, d5_r = Dot(logo.c5_l.get_center(), **dot_config), Dot(logo.c5_r.get_center(), **dot_config)
        # self.set_compass_to_draw_arc(logo.c5_l, emphasize_dot=True)
        self.set_compass_and_show_span(logo.c5_l.get_center(), logo.c5_l.get_start(), reverse_label=True, text='r=5')
        self.wait(0.1)
        self.draw_arc_by_compass(logo.c5_l)
        self.cp.reverse_tip()
        logo.c5_r.flip()
        self.wait(0.15)
        self.add(d5_l)
        self.set_compass_to_draw_arc(logo.c5_r, emphasize_dot=True)
        # self.set_compass_to_draw_arc_(logo.c5_r, emphasize_dot=True)
        self.draw_arc_by_compass(logo.c5_r, reverse=True)
        self.wait(0.1)
        self.add(d5_r)
        self.put_aside_compass(direction=UP)
        self.wait(0.4)
        self.play(VGroup(*self.mobjects).scale, 0.6, {'about_point': ORIGIN}, run_time=1.2)
        self.play(VGroup(*self.mobjects).shift, UP * 2, run_time=1.2)
        # s_vg = VGroup(axis, logo.c5_l, logo.c5_r, logo.c1)
        # s_vg2 = logo.key_lines.remove(logo.c5_l, logo.c5_r, logo.c1)
        # self.play(s_vg.scale, 0.5, {'about_point': ORIGIN}, run_time=1.2)
        # s_vg2.scale(0.5, about_point=ORIGIN)
        # self.wait(0.4)
        # self.set_compass_(logo.c8_d.get_center(), logo.c8_d.get_start(), emphasize_dot=True)
        # # self.set_compass_to_draw_arc(logo.c8_d, emphasize_dot=True)
        # self.draw_arc_by_compass(logo.c8_d)
        self.wait(2)
class Step_11(DrawingScene):
    CONFIG = {
        'compass_config':{
            'stroke_color': GREY_E,
            'fill_color': WHITE,
            'stroke_width': 2,
            'leg_length': 4,
            'leg_width': 0.12,
            'r': 0.2,
            'depth_test': True,
        },
        'add_ruler': True,
    }
    def construct(self):
        s = 0.5 * 0.6
        axis = Axes(x_range=[-30, 30, 1], y_range=[-25, 25, 1], axis_config={"stroke_color": GREY_D, "stroke_width": 2.5, "stroke_opacity": 1},).scale(s).shift(UP * 2)
        logo = Apple_logo(create_shape=False, coord_center=UP * 2, scale=s*2)
        dot_config = {'color': GREY_E, 'radius': 0.03}
        d1 = Dot(logo.c1.get_center(), **dot_config)
        d5_l, d5_r = Dot(logo.c5_l.get_center(), **dot_config), Dot(logo.c5_r.get_center(), **dot_config)
        self.add(axis, logo.c1, logo.c5_l, logo.c5_r, d1, d5_l, d5_r)
        self.wait()
        self.cp.move_to(DOWN * 10)
        self.cp.rotate_about_niddle_tip(PI)
        arc_13_l = Arc(arc_center=logo.c5_l.get_center(), radius=13 * s, start_angle=3*PI/2, angle=PI/4, **self.line_config)
        arc_13_r = Arc(arc_center=logo.c5_r.get_center(), radius=13 * s, start_angle=3*PI/2-PI/4, angle=PI/4 + PI/2, **self.line_config)
        self.highlight_on(logo.c5_l, logo.c5_r)
        self.wait(0.15)
        self.set_compass_and_show_span(logo.c5_l.get_center(), logo.c5_l.get_center() + (5 + 8) * s * LEFT, reverse_label=True, text='r=5+8=13', run_time=1.4)
        self.wait(0.4)
        self.set_compass_to_draw_arc_(arc_13_l, adjust_angle=PI, rate_func=linear, run_time=0.6)
        self.draw_arc_by_compass(arc_13_l, rate_func=linear, run_time=0.3)
        self.wait(0.12)
        self.set_compass_(logo.c5_r.get_center(), logo.c5_r.get_center() + (5 + 8) * s * LEFT, adjust_angle=PI, emphasize_dot=True, run_time=1.4)
        self.wait(0.2)
        self.set_compass_to_draw_arc_(arc_13_r, adjust_angle=-PI, rate_func=linear, run_time=0.6)
        self.draw_arc_by_compass(arc_13_r, rate_func=linear, run_time=0.9)
        self.wait(0.6)
        self.set_compass_and_show_span(axis.c2p(11, 0), axis.c2p(19, 0), reverse_label=True, text='r=8', run_time=1.2)
        self.wait(0.15)
        self.set_compass_to_draw_arc_(logo.c8_d.scale(0.995), adjust_angle=-PI, run_time=1.)
        self.draw_arc_by_compass(logo.c8_d, run_time=1.2)
        d8_d = Dot(logo.c8_d.get_center(), **dot_config)
        self.add(d8_d)
        self.wait(0.2)
        self.put_aside_compass(UP * 0.5, run_time=1.)
        self.play(FadeOut(arc_13_l), FadeOut(arc_13_r))
        self.remove(self.cp)
        self.wait(0.4)
        self.play(logo.c5_r.set_stroke, {'color': GREY_E, 'width': 2.5}, logo.c5_l.set_stroke, {'color': GREY_E, 'width': 2.5}, run_time=0.6)
        self.wait(0.2)
        self.play(VGroup(*self.mobjects).shift, DOWN * 4, run_time=1.25)
        self.wait(2)
class Step_12(DrawingScene):
    def construct(self):
        s = 0.5 * 0.6
        axis = Axes(x_range=[-30, 30, 1], y_range=[-30, 30, 1], axis_config={"stroke_color": GREY_D, "stroke_width": 2.5, "stroke_opacity": 1},).scale(s).shift(DOWN * 2)
        logo = Apple_logo(create_shape=False, coord_center=DOWN * 2, scale=s*2)
        dot_config = {'color': GREY_E, 'radius': 0.03}
        d1 = Dot(logo.c1.get_center(), **dot_config)
        d5_l, d5_r = Dot(logo.c5_l.get_center(), **dot_config), Dot(logo.c5_r.get_center(), **dot_config)
        self.add(axis, logo.c1, logo.c5_l, logo.c5_r, logo.c8_d.scale(0.995), d1, d5_l, d5_r)
        self.set_compass_and_show_span(axis.c2p(0, 13), axis.c2p(0,0), emphasize_dot=True, reverse_label=True, text='r=13', run_time=2)
        self.wait(0.6)
        logo.c13.rotate(-PI/2)
        self.set_compass_to_draw_arc(logo.c13, emphasize_dot=True)
        self.wait(0.2)
        d13 = Dot(logo.c13.get_center(), **dot_config)
        a_13 = Arc(arc_center=logo.c8_d.get_center() + UP * 8 * s, start_angle=PI/2 + PI/20, radius=13 * s, angle=-PI/10, **self.line_config)
        self.cp.reverse_tip()
        self.set_compass_to_draw_arc(a_13, run_time=0.25)
        self.draw_arc_by_compass(a_13, run_time=0.5)
        self.add(d13)
        self.wait(0.25)
        self.put_aside_compass(RIGHT * 0.8, run_time=1.2)
        self.wait(0.6)
        self.play(VGroup(*self.mobjects).scale, 0.8, {'about_point': axis.c2p(0,0)}, run_time=1.2)
        self.wait(2)
class Step_2(DrawingScene):
    """
    画出辅助正方形
    """
    CONFIG = {
        'compass_config':{
            'stroke_color': GREY_E,
            'fill_color': WHITE,
            'stroke_width': 2,
            'leg_length': 4,
            'leg_width': 0.12,
            'r': 0.2,
            'depth_test': True,
        },
    }
    def construct(self):
        s = 0.5 * 0.6 * 0.8
        axis = Axes(x_range=[-40, 40, 1], y_range=[-36, 36, 1], axis_config={"stroke_color": GREY_D, "stroke_width": 2.5, "stroke_opacity": 1},).scale(s).shift(DOWN * 2)
        logo = Apple_logo(create_shape=False, coord_center=DOWN * 2, scale=s*2)
        dot_config = {'color': GREY_E, 'radius': 0.03}
        d1 = Dot(logo.c1.get_center(), **dot_config)
        d5_l, d5_r = Dot(logo.c5_l.get_center(), **dot_config), Dot(logo.c5_r.get_center(), **dot_config)
        a_13 = Arc(arc_center=logo.c8_d.get_center() + UP * 8 * s, start_angle=PI/2 + PI/30, radius=13 * s, angle=-PI/15, **self.line_config)
        d13 = Dot(logo.c13.get_center(), **dot_config)
        self.add(axis, logo.c1, logo.c5_l, logo.c5_r, logo.c8_d.scale(0.995), a_13, d1, d5_l, d5_r, d13)
        self.wait()
        self.play(FadeIn(self.ruler))
        self.wait(0.2)
        dl, dr = logo.c5_l.get_center() * 8/13 + logo.c8_d.get_center() * 5/13, logo.c5_r.get_center() * 8/13 + logo.c8_d.get_center() * 5/13
        self.set_ruler(dl, dr)
        self.emphasize_dot([dl, dr], run_time=0.2)
        # l = Line(dl + LEFT * 4.5, dr + RIGHT *4.5, **self.line_config)
        self.wait(0.5)
        l = self.draw_line(dl + LEFT * 3.6, dr + RIGHT * 3.6, run_time=1.25)
        self.put_aside_ruler(DOWN * 0.4)
        self.wait(0.1)
        dm = (dl+dr)/2
        self.set_compass(d13.get_center(), dm, emphasize_dot=True, run_time=1.5)
        self.wait(0.2)
        a_13_r = Arc(arc_center=dm, radius=self.cp.get_span(), start_angle=10 * DEGREES, angle=-20 * DEGREES, **self.line_config)
        a_13_l = Arc(arc_center=dm, radius=self.cp.get_span(), start_angle=190 * DEGREES, angle=-20 * DEGREES, **self.line_config)
        # a_13_u = Arc(arc_center=d13.get_center(), radius=self.cp.get_span(), start_angle=100 * DEGREES, angle=20 * DEGREES, **self.line_config)
        l_square = get_norm(d13.get_center()-dm)
        circle = Circle(arc_center=d13.get_center(), radius=l_square * np.sqrt(2), **self.line_config).rotate(-PI/4)
        self.cp.reverse_tip()
        self.set_compass_to_draw_arc_(a_13_l, adjust_angle=PI, run_time=0.64)
        self.draw_arc_by_compass(a_13_l, run_time=0.12, rate_func=linear)
        self.set_compass_to_draw_arc_(a_13_r, adjust_angle=PI, run_time=0.96, rate_func=linear)
        self.draw_arc_by_compass(a_13_r, run_time=0.12, rate_func=linear)
        self.wait(0.25)
        self.cp.reverse_tip()
        self.set_compass_to_draw_arc(circle, run_time=1.2, emphasize_dot=True)
        self.wait(0.15)
        self.draw_arc_by_compass(circle, run_time=1.6)
        self.wait(0.5)
        self.set_compass(dm + l_square * LEFT, dm + l_square * RIGHT, emphasize_dot=True)
        self.wait(0.2)
        self.cp.reverse_tip()
        arc_r = Arc(arc_center=dm + l_square * RIGHT, radius=l_square * 2, start_angle=100 * DEGREES, angle=-20 * DEGREES, **self.line_config)
        arc_l = Arc(arc_center=dm + l_square * LEFT, radius=l_square * 2, start_angle=80 * DEGREES, angle=20 * DEGREES, **self.line_config)
        # self.set_compass_to_draw_arc_(arc_r, adjust_angle=PI, run_time=0.6, rate_func=linear)
        self.play(Rotating(self.cp, angle=-80 * DEGREES, about_point=self.cp.get_niddle_tip(), rate_func=linear), run_time=0.6)
        self.draw_arc_by_compass(arc_r, run_time=0.15, rate_func=linear)
        self.wait(0.2)
        # self.set_compass_(dm + l_square * RIGHT, dm + l_square * LEFT, emphasize_dot=True)
        self.play(Rotating(self.cp, angle=100 * DEGREES, about_point=self.cp.get_niddle_tip()), run_time=0.8)
        self.emphasize_dot([self.cp.get_niddle_tip(), self.cp.get_pen_tip()], run_time=0.15)
        self.wait(0.2)
        self.cp.reverse_tip()
        self.set_compass_to_draw_arc_(arc_l, adjust_angle=PI, run_time=0.6, rate_func=linear)
        self.draw_arc_by_compass(arc_l, run_time=0.15, rate_func=linear)
        self.wait(0.2)
        self.put_aside_compass(LEFT)
        self.wait(0.6)
        dots = [dm + l_square * RIGHT, dm + l_square * (RIGHT + 2 * UP), dm + l_square * (LEFT + 2 * UP), dm + l_square * LEFT]
        # self.set_ruler(dots[0], dots[1])
        l_r = self.draw_line(dots[0], dots[1], is_prepared=False, pre_time=0.75)
        self.wait(0.2)
        l_u = self.draw_line(dots[1], dots[2], is_prepared=False, pre_time=0.75)
        self.wait(0.2)
        l_l = self.draw_line(dots[3], dots[2], is_prepared=False, pre_time=0.75)
        l_d = Line(dots[3], dots[0],**self.line_config)
        self.wait(0.18)
        self.put_aside_ruler(LEFT * 0.7)
        self.wait(0.9)
        self.play(Transform(l, l_d, run_time=0.9),
                  FadeOut(a_13_r, run_time=0.6),
                  # FadeOut(a_13, run_time=0.8),
                  FadeOut(a_13_l, run_time=0.8),
                  FadeOut(arc_r, run_time=1.),
                  FadeOut(arc_l, run_time=1.2),
                  FadeOut(circle, run_time=1.4)
                  )
        self.wait(0.75)
        self.play(VGroup(*self.mobjects).scale, 0.8, {'about_point': axis.c2p(0,0)})
        self.wait(2)
class Step_3(DrawingScene):
    CONFIG = {
        'add_ruler': True,
    }
    def construct(self):
        s = 0.5 * 0.6 * 0.8 * 0.8
        axis = Axes(x_range=[-40, 40, 1], y_range=[-36, 36, 1], axis_config={"stroke_color": GREY_D, "stroke_width": 2.5, "stroke_opacity": 1},).scale(s).shift(DOWN * 2)
        logo = Apple_logo(create_shape=False, coord_center=DOWN * 2, scale=s*2)
        dot_config = {'color': GREY_E, 'radius': 0.03}
        d1 = Dot(logo.c1.get_center(), **dot_config)
        d5_l, d5_r = Dot(logo.c5_l.get_center(), **dot_config), Dot(logo.c5_r.get_center(), **dot_config)
        a_13 = Arc(arc_center=logo.c8_d.get_center() + UP * 8 * s, start_angle=PI/2 + PI/30, radius=13 * s, angle=-PI/15, **self.line_config)
        d13 = Dot(logo.c13.get_center(), **dot_config)
        dl, dr = logo.c5_l.get_center() * 8/13 + logo.c8_d.get_center() * 5/13, logo.c5_r.get_center() * 8/13 + logo.c8_d.get_center() * 5/13
        dm = (dl+dr)/2
        l_square = get_norm(d13.get_center()-dm)
        dots = [dm + l_square * RIGHT, dm + l_square * (RIGHT + 2 * UP), dm + l_square * (LEFT + 2 * UP), dm + l_square * LEFT]
        square = Polygon(*dots, **self.line_config)
        before = VGroup(axis, logo.c1, logo.c5_l, logo.c5_r, logo.c8_d.scale(0.995), a_13, d1, d5_l, d5_r, d13, square)
        self.add(before)
        self.wait()
        dots = square.get_vertices()
        r1 = l_square * 0.7
        ### left part ###
        a_1 = Arc(arc_center=dots[2], radius=r1, start_angle=PI/3,  angle=-2 * PI/3, **self.line_config)
        a_2 = Arc(arc_center=(dots[1] +dots[2])/2, radius=r1, start_angle=2 * PI/3, angle=2 * PI/3, **self.line_config)
        self.cp.flip(axis=RIGHT)
        self.set_compass_to_draw_arc(a_1)
        self.draw_arc_by_compass(a_1)
        self.wait(0.15)
        self.cp.reverse_tip()
        self.set_compass_to_draw_arc_(a_2, adjust_angle=PI)
        self.draw_arc_by_compass(a_2)
        self.put_aside_compass(RIGHT * 0.8, run_time=0.64)
        pm = (dots[2] + (dots[1] +dots[2])/2)/2
        pu, pd = pm + 10 * s * UP, pm + 10 * s * DOWN
        l = self.draw_line(pu, pd, is_prepared=False, pre_time=0.72)
        self.wait(0.18)
        self.put_aside_ruler(LEFT * 0.65)
        logo.c8_l.rotate(PI/2)
        self.set_compass_and_show_span_(axis.c2p(0,0), axis.c2p(0,8), run_time=1.4, emphasize_dot=True, adjust_angle=PI, text='r=8', reverse_label=True, add_bg=True)
        self.wait(0.5)
        self.set_compass_to_draw_arc_(logo.c8_l, adjust_angle=PI, emphasize_dot=True)
        self.wait(0.2)
        self.draw_arc_by_compass(logo.c8_l, run_time=1.25, add_center=True)
        self.wait(0.2)
        self.put_aside_compass(LEFT * 0.64)
        self.wait(0.6)
        ### right part ### 同理直接copy吧
        a_3 = Arc(arc_center=(dots[1] + dots[2])/2, radius=r1, start_angle=PI/3,  angle=-2 * PI/3, **self.line_config)
        a_4 = Arc(arc_center=dots[1], radius=r1, start_angle=2 * PI/3, angle=2 * PI/3, **self.line_config)
        d8_r = Dot(logo.c8_r.get_center(), **dot_config)
        pm_r = (dots[1] + (dots[1] +dots[2])/2)/2
        pu_r, pd_r = pm_r + 10 * s * UP, pm_r + 10 * s * DOWN
        logo.c8_r.rotate(PI/2)
        l02 = Line(pu_r, pd_r, **self.line_config)
        right_part = VGroup(a_3, a_4, l02, logo.c8_r)
        right_part_ = right_part.copy().set_stroke(opacity=0.32).shift(l_square * LEFT)
        self.add(right_part_)
        self.play(right_part_.shift, l_square * RIGHT)
        self.wait(0.2)
        self.play(ShowCreation(a_3), run_time=0.8)
        self.play(ShowCreation(a_4), run_time=0.8)
        self.play(ShowCreation(l02), run_time=0.8)
        self.play(ShowCreation(logo.c8_r), FadeIn(d8_r))
        self.remove(right_part_)
        self.wait()
        self.play(FadeOut(a_4, run_time=0.6), FadeOut(l02, run_time=0.7), FadeOut(a_3, run_time=0.8),
                  FadeOut(a_2, run_time=0.9), FadeOut(l, run_time=1.), FadeOut(a_1, run_time=1.1),
                  )
        self.wait(0.8)
        self.play(VGroup(*self.mobjects).scale, 1.25, {'about_point': axis.c2p(0,0)})
        self.wait(2)
class Step_4(DrawingScene):
    CONFIG = {
            'add_ruler': True,
        }
    def construct(self):
        s = 0.5 * 0.6 * 0.8
        axis = Axes(x_range=[-45, 45, 1], y_range=[-36, 36, 1], axis_config={"stroke_color": GREY_D, "stroke_width": 2.5, "stroke_opacity": 1},).scale(s).shift(DOWN * 2)
        logo = Apple_logo(create_shape=False, coord_center=DOWN * 2, scale=s*2)
        dot_config = {'color': GREY_E, 'radius': 0.03}
        d1 = Dot(logo.c1.get_center(), **dot_config)
        d5_l, d5_r = Dot(logo.c5_l.get_center(), **dot_config), Dot(logo.c5_r.get_center(), **dot_config)

参考文献

1.https://www.tiobe.com/tiobe-index

2.https://baike.baidu.com/item/Python/407313?fr=aladdin

3.https://docs.manim.community/en/stable/installation.html

4.https://docs.manim.org.cn/

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