企业官网建设流程全解析

from manim import * import numpy as np import math class LimitVisualization(Scene): def construct(self): #背景颜色就是黑板颜色一样黑绿颜色 self.camera.background_color='#003311' # 标题 title = Text("函数极限的可视化", font_size=48, color=BLUE) title.to_edge(UP) self.play(Write(title)) self.wait(1) # 例子1: lim(x→0) sin(x)/x = 1 self.example1() self.clear() self.wait(0.5) # 例子2: lim(x→∞) (1 + 1/x)^x = e #使用VGroup组合 title2 = VGroup(Text("例2:", font_size=36, color=YELLOW), MathTex(r"\lim_{x \to +\infty} \left(1+\frac{1}{x}\right)^x = e \approx 2.71828", font_size=36, color=YELLOW)).arrange(RIGHT, buff=0.3).to_edge(UP) #self.play(Write(title2)) self.example2() self.clear() self.wait(0.5) # 例子3: lim(x→2) (x²-4)/(x-2) = 4 title3 = VGroup(Text("例3:", font_size=36, color=YELLOW), MathTex(r"\lim_{x \to 2} \frac{x^{2}-4}{x-2} = 4", font_size=36, color=YELLOW)).arrange(RIGHT, buff=0.3).to_edge(UP) #title3 = Text("例3: lim(x→2) (x²-4)/(x-2) = 4", font_size=36, color=GREEN) title3.to_edge(UP) self.play(Write(title3)) self.example3() self.clear() self.wait(0.5) # 例子4: lim(x→∞) sin(x)/x = 0 title4 = VGroup(Text("例4:", font_size=36, color=YELLOW), MathTex(r"\lim_{x \to +\infty} \frac{sin(x)}{x} =0", font_size=36, color=YELLOW)).arrange(RIGHT, buff=0.3).to_edge(UP) #title4 = Text("例4: lim(x→∞) sin(x)/x = 0", font_size=36, color=ORANGE) title4.to_edge(UP) self.play(Write(title4)) self.example4() self.wait(2) def example1(self): # 例1: lim(x→0) sin(x)/x = 1 example_text = VGroup( Text("例1:", font_size=36, color=YELLOW), MathTex(r"\lim_{x \to 0} \frac{sin(x)}{x}=1 ", font_size=36, color=YELLOW)).arrange(RIGHT, buff=0.3).shift(2*UP) self.play(Write(example_text)) # 创建坐标轴 axes = Axes( x_range=[-6, 6, 1], y_range=[-0.5, 1.5, 0.5], x_length=8, y_length=4, axis_config={"color": BLUE} ) axes_labels = axes.get_axis_labels(x_label="x", y_label="y") # 创建函数图形 graph = axes.plot( lambda x: np.sin(x)/x if x != 0 else 1, x_range=[-6, -0.01], color=YELLOW ) graph2 = axes.plot( lambda x: np.sin(x)/x if x != 0 else 1, x_range=[0.01,6], color=YELLOW ) graph2.points = graph2.points[::-1] # 或者使用 graph2.reverse_points() # 标记极限点 point = Dot(axes.coords_to_point(0, 1), color=RED) point_label = MathTex("(0, 1)", color=RED).next_to(point, RIGHT) li1 = VGroup(axes, axes_labels, graph, graph2, point, point_label).shift(1.5*DOWN) self.play(Create(axes), Write(axes_labels)) self.play(Create(graph),Create(graph2)) # 右侧反向创建（从右到左）！ self.play(Create(point), Write(point_label)) self.play(li1.animate.scale(0.5).move_to(LEFT * 5 )) # 显示极限计算 limit_text = MathTex(r"\lim_{x \to 0} \frac{\sin x}{x} = 1", font_size=36, color=GREEN).shift(RIGHT) self.play(Write(limit_text)) # 显示数值计算 values_text = Text("数值验证:", font_size=36, color=WHITE) values_text.next_to(limit_text, DOWN) self.play(Write(values_text)) values = [ "x = 0.1: sin(0.1)/0.1 = 0.9983", "x = 0.01: sin(0.01)/0.01 = 0.99998", "x = 0.001: sin(0.001)/0.001 = 0.9999998" ] for i, v in enumerate(values): text = Text(v, font_size=25, color=BLUE_C) text.next_to(values_text, DOWN, buff=0.5 + i * 0.4) # 每次递增 self.play(Write(text)) self.wait(2) def example2(self): # 例2: lim(x→∞) (1+1/x)^x = e example_text =VGroup(Text("例2:", font_size=36, color=YELLOW), MathTex(r"\lim_{x \to +\infty} \left(1+\frac{1}{x}\right)^x = e \approx 2.71828", font_size=36, color=YELLOW)).arrange(RIGHT, buff=0.3).shift(2*UP) self.play(Write(example_text)) # 创建坐标轴 axes = Axes( x_range=[0, 100, 10], y_range=[0, 4, 1], x_length=8, y_length=4, axis_config={"color": BLUE} ) axes_labels = axes.get_axis_labels(x_label="x", y_label="y") # 创建函数图形 graph = axes.plot( lambda x: (1 + 1/x)**x, x_range=[0.01, 100], color=ORANGE ) self.play(Create(axes), Write(axes_labels)) self.play(Create(graph)) # 添加水平渐近线 y=e e_line = DashedLine( start=axes.coords_to_point(0, np.e), end=axes.coords_to_point(100, np.e), color=GREEN ) e_label = MathTex(r"y = e \approx 2.718", color=GREEN).next_to(e_line, LEFT) self.play(Create(e_line), Write(e_label)) li1 = VGroup(axes, axes_labels, graph,e_line,e_label) self.play(li1.animate.scale(0.5).rotate(-1.5).move_to(LEFT * 5 )) # 显示数值计算 values_text = Text("数值验证:", font_size=24, color=WHITE) self.play(Write(values_text)) values = [ "x = 10: (1+1/10)^10 = 2.5937", "x = 100: (1+1/100)^100 = 2.7048", "x = 1000: (1+1/1000)^1000 = 2.7169" ] for i, v in enumerate(values): text = Text(v, font_size=20, color=BLUE_C) text.next_to(values_text, DOWN,buff=0.5 + i * 0.4) self.play(Write(text)) self.wait(2) def example3(self): # 例3: lim(x→2) (x²-4)/(x-2) = 4 example_text = MathTex(r"\lim_{x \to 2} \frac{x^{2}-4}{x-2} = 4", font_size=36, color=GREEN).shift(2*UP) #example_text.to_edge(UP) self.play(Write(example_text)) # 创建坐标轴 axes = Axes( x_range=[-2, 6, 1], y_range=[-2, 8, 2], x_length=8, y_length=4, axis_config={"color": BLUE} ) axes_labels = axes.get_axis_labels(x_label="x", y_label="y") # 创建函数图形（去除x=2处的间断点） graph1 = axes.plot( lambda x: (x**2 - 4)/(x - 2) if x != 2 else 4, x_range=[-2, 1.99], color=GREEN ) graph2 = axes.plot( lambda x: (x**2 - 4)/(x - 2) if x != 2 else 4, x_range=[2.01, 6], color=GREEN ) graph2.points = graph2.points[::-1] # 或者使用 graph2.reverse_points() self.play(Create(axes), Write(axes_labels)) self.play(Create(graph1), Create(graph2)) # 标记极限点 point = Dot(axes.coords_to_point(2, 4), color=RED, radius=0.15) point_label = MathTex("(2, 4)", color=RED).next_to(point, UR) open_circle = Circle(color=RED, radius=0.15).move_to(axes.coords_to_point(2, 4)) open_circle.set_fill(RED, opacity=1) #//////////////////// self.play(Create(point), Write(point_label)) #_______________________________ li1 = VGroup(axes, axes_labels, graph1, graph2,point,point_label,open_circle) self.play(li1.animate.scale(0.5).move_to(LEFT * 5 )) # 显示函数分解 formula_text = MathTex(r"\frac{x^2-4}{x-2} = \frac{(x-2)(x+2)}{x-2} = x+2", font_size=30, color=WHITE) formula_text.next_to(example_text, DOWN, buff=0.5) self.play(Write(formula_text)) # 显示数值计算 values_text = Text("数值验证:", font_size=24, color=WHITE) values_text.next_to(formula_text, DOWN, buff=0.5) self.play(Write(values_text)) values = [ "x = 1.9: (1.9²-4)/(1.9-2) = 3.9", "x = 1.99: (1.99²-4)/(1.99-2) = 3.99", "x = 2.01: (2.01²-4)/(2.01-2) = 4.01" ] for i, v in enumerate(values): text = Text(v, font_size=20, color=BLUE_C) text.next_to(values_text, DOWN,buff=0.5 + i * 0.4) self.play(Write(text)) self.wait(2) def example4(self): # 例4: lim(x→∞) sin(x)/x = 0 example_text = VGroup(Text(" ", font_size=36, color=YELLOW), MathTex(r"\lim_{x \to +\infty} \frac{sin(x)}{x} =0", font_size=36, color=YELLOW)).arrange(RIGHT, buff=0.3).shift(2*UP) self.play(Write(example_text)) # 创建坐标轴 axes = Axes( x_range=[0, 50, 5], y_range=[-1, 1, 0.5], x_length=8, y_length=4, axis_config={"color": BLUE} ) axes_labels = axes.get_axis_labels(x_label="x", y_label="y") # 创建函数图形 graph = axes.plot( lambda x: np.sin(x)/x if x != 0 else 1, x_range=[0.01, 50], color=ORANGE ) # 创建包络线 env_upper = axes.plot( lambda x: 1/x, x_range=[0.5, 50], color=RED, stroke_width=2 ) env_lower = axes.plot( lambda x: -1/x, x_range=[0.5, 50], color=RED, stroke_width=2 ) # 添加水平渐近线 zero_line = DashedLine( start=axes.coords_to_point(0, 0), end=axes.coords_to_point(50, 0), color=GREEN ) zero_label = MathTex(" 0", color=GREEN).next_to(zero_line, 0.5*LEFT) self.play(Create(axes), Write(axes_labels)) self.play( Write(zero_label)) self.play(Create(graph),Create(zero_line),Create(env_upper), Create(env_lower),run_time=4) # 显示解释 explain_text = VGroup(Text("sin(x)有界:", font_size=24, color=YELLOW), MathTex(r"\frac{1}{x}", font_size=34, color=YELLOW), Text("趋于0，因此极限为0:", font_size=24, color=YELLOW) ).arrange(RIGHT, buff=0.75).shift(UP) self.play(Write(explain_text)) # 显示数值计算 values_text = Text("数值验证:", font_size=24, color=WHITE).shift(DOWN) self.play(Write(values_text)) values = [ "x = 10: sin(10)/10 = -0.0544", "x = 50: sin(50)/50 = -0.00214", "x = 100: sin(100)/100 = -0.00506" ] for i, v in enumerate(values): text = Text(v, font_size=20, color=BLUE_C) text.next_to(values_text, DOWN,buff=0.5 + i * 0.4) self.play(Write(text)) self.wait(2)

一、整体架构分析

1.1 代码结构总览

这段代码是一个基于Manim的数学教学动画程序，通过可视化的方式展示了四个经典的函数极限问题。程序的结构可以总结如下：

LimitVisualization (场景类) │ ├── construct() - 主入口方法 │ ├── 设置背景色 │ ├── 添加主标题 │ ├── 依次调用四个示例方法 │ └── 管理场景切换 │ ├── example1() - 例1: lim(x→0) sin(x)/x = 1 ├── example2() - 例2: lim(x→∞) (1+1/x)^x = e ├── example3() - 例3: lim(x→2) (x²-4)/(x-2) = 4 └── example4() - 例4: lim(x→∞) sin(x)/x = 0

1.2 核心设计思想

教学分层设计：

• 每个示例遵循“展示 → 视觉化 → 数值验证”的教学逻辑
• 通过动画逐步引入数学概念，使抽象的极限概念变得直观

二、关键代码段分析

2.1 场景初始化

class LimitVisualization(Scene): def construct(self): # 设置深绿色背景，模拟黑板效果 self.camera.background_color='#003311' # 添加主标题 title = Text("函数极限的可视化", font_size=48, color=BLUE) title.to_edge(UP) self.play(Write(title))

设计亮点：

• 选择深绿色背景(#003311)模拟黑板，营造课堂氛围
• 使用Write()动画让标题逐字出现，增强视觉吸引力

2.2 例1：sin(x)/x 的极限

def example1(self): # 创建坐标轴，精确控制范围和大小 axes = Axes( x_range=[-6, 6, 1], # x轴范围：-6到6，间隔1 y_range=[-0.5, 1.5, 0.5], # y轴范围 x_length=8, # x轴长度（像素） y_length=4 # y轴长度 ) # 处理x=0处的间断点，分段绘制 graph = axes.plot( lambda x: np.sin(x)/x if x != 0 else 1, x_range=[-6, -0.01], # 左半部分 color=YELLOW )

关键技术处理：

• 间断点处理：由于sin(x)/x在x=0处未定义，采用分段绘制的方式
• 动画顺序：先绘制坐标轴，再绘制函数图像，最后标记极限点

# 右侧图像反向绘制，实现从右到左的动画效果 graph2.points = graph2.points[::-1]

2.3 例2：极限e的显示

# 添加水平渐近线 y=e e_line = DashedLine( start=axes.coords_to_point(0, np.e), end=axes.coords_to_point(100, np.e), color=GREEN )

教学特点：

• 渐近线可视化：用虚线段清晰标示极限值e的位置
• 对比展示：将图形缩小后移到左侧，腾出空间展示数值计算

2.4 例3：可去间断点

# 标记极限点，同时绘制实心点和空心圆 point = Dot(axes.coords_to_point(2, 4), color=RED, radius=0.15) open_circle = Circle(color=RED, radius=0.15) open_circle.set_fill(RED, opacity=1)

数学意义：

• 同时使用实心点和空心圆，展示函数在该点的“补充定义”思想
• 通过图形分解演示：(x²-4)/(x-2) = x+2（x≠2时）

2.5 例4：夹逼定理的运用

# 创建包络线展示 sin(x)/x 的界限 env_upper = axes.plot(lambda x: 1/x, x_range=[0.5, 50], color=RED) env_lower = axes.plot(lambda x: -1/x, x_range=[0.5, 50], color=RED)

数学原理可视化：

• 用红色包络线(±1/x)直观展示夹逼定理
• 清晰呈现sin(x)/x被两条曲线夹住并趋近于0的过程

三、动画技术与数学教学的创新结合

3.1 渐进式学习流程

每个示例的动画流程都是：

步骤展示 → 图形绘制 → 极限标记 → 数值验证

3.2 空间复用策略

# 缩放并移动图形到左侧 li1.animate.scale(0.5).move_to(LEFT * 5)

空间规划：

• 图形区与文字区并行，同时利用屏幕空间
• 缩放操作让用户既有整体图形预览，又能看到具体数值

3.3 色彩编码系统

四、代码优化建议

4.1 代码复用改进

def create_axes(self, x_range, y_range, x_label="x", y_label="y"): """统一的坐标轴创建函数""" axes = Axes( x_range=x_range, y_range=y_range, axis_config={"color": BLUE} ) return VGroup(axes, axes.get_axis_labels(x_label=x_label, y_label=y_label))

4.2 动画控制改进

# 使用并行动画提高效率 self.play( Create(graph), Create(env_upper), Create(env_lower), run_time=4 )

4.3 参数配置优化

# 将常量提取为配置 CONFIG = { "background_color": '#003311', "axis_color": BLUE, "title_size": 48, "example_text_size": 36 }

五、总结

这个代码示例展示了如何将数学极限这一抽象概念通过动画可视化：

1. 视觉教学设计：通过颜色、动画和空间布局强化教学效果
2. 数学逻辑严谨：正确处理特殊点（间断点）和极限概念
3. 多层信息呈现：图形展示 + 计算过程 + 数值验证的多维度学习

该代码不仅是一个数学可视化工具，更是一个教学方法的创新实践，让学习者能够直观地理解深奥的数学概念。