【manim】写一个泰勒展开式演示动画

发表于 更新于 分类于
效果

公式推导

这是泰勒展开式： $$ f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\dots+\frac{f^{(n)}(0)}{n!}x^n+o(x^n) $$

欲求对sinx的逼近，我们将sinx作为f(x)带入公式，尝试写出几项并观察其规律：

$$ sin(x)= \boxed{sin(0)}+ cos(0)x+ \boxed{\frac{-sin(0)}{2!}x^2}+ \frac{-cos(0)}{3!}x^3+ \boxed{\frac{sin(0)}{4!}x^4}+ \frac{cos(0)}{5!}x^5+ \boxed{\frac{-sin(0)}{6!}x^6} $$

可以发现所有sin(0)全为0，约去得： $$ sin(x)= cos(0)x+ \frac{-cos(0)}{3!}x^3+ \frac{cos(0)}{5!}x^5 $$ 可知，cos(0) = 1： $$ sin(x)= \frac{x^1}{1!}+ \frac{-x^3}{3!}+ \frac{x^5}{5!} +\dots $$

以此类推得到： $$ sin(x)=\sum_{i=0}^{i} \frac { (-1)^{i} \cdot x^{2i+1}} {(2i+1)!} $$

转化为 python 为：

1
2
3
4
def make_taylor_func(n_terms):
	return lambda x: sum(
		(-1)**k * x**(2*k + 1) / factorial(2*k + 1) for k in range(n_terms)
	)

于是就可以写出这样的代码：

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
from manim import *
import numpy as np
from math import factorial

class SingleScene(Scene):
    def introduction(self):
        intro = Text("这是野生的泰勒展开式")
        tex = Tex(r"$$f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\dots+\frac{f^{(n)}(0)}{n!}x^n+o(x^n)$$")
        intro_sin = Text("将正弦函数代入其中").shift(UP*3)
        tex_sin = Tex(r'''$$
            sin(x)=
            \boxed{sin(0)}+
            cos(0)x+
            \boxed{\frac{-sin(0)}{2!}x^2}+
            \frac{-cos(0)}{3!}x^3+
            \boxed{\frac{sin(0)}{4!}x^4}+
            \frac{cos(0)}{5!}x^5+
            \boxed{\frac{-sin(0)}{6!}x^6}
            $$
        ''').shift(DOWN).scale(0.6)
        tex_2 = Tex(r"""
            $$
            sin(x)=
            cos(0)x+
            \frac{-cos(0)}{3!}x^3+
            \frac{cos(0)}{5!}x^5
            $$
        """)
        tex_3 = Tex(r"""
            $$
            sin(x)=
            \frac{x^1}{1!}+
            \frac{-x^3}{3!}+
            \frac{x^5}{5!}
            $$
        """)
        tex_4 = Tex(r"""
            $$
            sin(x)=\sum_{i=0}^{i} 
            \frac
            {
                (-1)^{i} \cdot x^{2i+1}}
            {(2i+1)!}
            $$
        """)


        self.play(Write(intro)) # 写下 “这是野生的展开式”
        self.play(intro.animate.shift(UP * 3)) # 向上移动
        self.play(Write(tex)) # 泰勒展开式
        self.play(tex.animate.shift(UP)) # 泰勒展开式向上移动
        self.play(Unwrite(intro)) # 擦去 “这是野生的泰勒展开式”
        self.play(Write(intro_sin)) # 写下 “将正弦函数代入其中”
        self.play(Write(tex_sin)) # 写下带入正弦后的表达式
        self.play(Uncreate(tex)) # 变换得到新式子
        self.play(Unwrite(intro_sin)) # 擦去 ”带入正弦函数“
        self.play(tex_sin.animate.shift(UP)) # 将带入后的式子置于中间
        self.play(Transform(tex_sin, tex_2), run_time=2)
        self.remove(tex_sin)
        self.play(Transform(tex_2, tex_3), run_time=2)
        self.remove(tex_2)
        self.play(Transform(tex_3, tex_4), run_time=2)
        self.play(Unwrite(tex_3))

    # 创建泰勒近似函数（前 n 项）
    def make_taylor_func(self, n_terms):
        return lambda x: sum(
            (-1) ** k * x ** (2 * k + 1) / factorial(2 * k + 1) for k in range(n_terms)
        )

    # 创建对应的 LaTeX 公式
    def make_formula_tex(self, n_terms):
        terms = []
        for k in range(n_terms):
            sign = "-" if k % 2 == 1 else ""
            exponent = 2 * k + 1
            term = rf"{sign}\frac{{x^{{{exponent}}}}} {{{exponent}!}}"
            terms.append(term)
        formula = "+".join(terms).replace("+-", "-")
        return MathTex("f(x) =", formula).scale(0.8).to_edge(RIGHT)

    def write_image(self):
        ax = Axes(
            x_range=[-12, 12, 2],
            y_range=[-2, 2],
            axis_config={"color": GREEN},
        )
        pl = NumberPlane()

        sin = ax.plot(lambda x: np.sin(x), color=BLUE)

        self.play(Create(ax), Create(pl))
        self.play(Write(sin))

        # 构建图像和公式
        taylor_graphs = []
        formula_labels = []
        for i in range(1, 15):  # t1~t6
            f = self.make_taylor_func(i)
            graph = ax.plot(f, color=RED)
            formula = self.make_formula_tex(i).shift(DOWN*2)
            if (i >= 8):
                formula = formula.scale(0.5).shift(RIGHT*2)
            taylor_graphs.append(graph)
            formula_labels.append(formula)

        # 第一个图和公式
        current_graph = taylor_graphs[0]
        current_formula = formula_labels[0]

        self.play(Write(current_graph), Write(current_formula))

        # 后续逐步替换
        for next_graph, next_formula in zip(taylor_graphs[1:], formula_labels[1:]):
            self.play(Transform(current_graph, next_graph),
                      Transform(current_formula, next_formula))
            self.remove(current_graph, current_formula)
            current_graph = next_graph
            current_formula = next_formula

        # 最终版本
        self.play(Write(current_graph), Write(current_formula))
        self.wait(2)

    def construct(self):
        self.introduction()
        self.write_image()


with tempconfig({"quality": "medium_quality", "preview": True}):
    scene = SingleScene()
    scene.render()

下面进行逐段解释。

introduction

该函数为画图前的公式推导铺垫动画。

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
def introduction(self):
	intro = Text("这是野生的泰勒展开式")
	tex = Tex(r"$$f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\dots+\frac{f^{(n)}(0)}{n!}x^n+o(x^n)$$")
	intro_sin = Text("将正弦函数代入其中").shift(UP*3)
	tex_sin = Tex(r'''$$
		sin(x)=
		\boxed{sin(0)}+
		cos(0)x+
		\boxed{\frac{-sin(0)}{2!}x^2}+
		\frac{-cos(0)}{3!}x^3+
		\boxed{\frac{sin(0)}{4!}x^4}+
		\frac{cos(0)}{5!}x^5+
		\boxed{\frac{-sin(0)}{6!}x^6}
		$$
	''').shift(DOWN).scale(0.6)
	tex_2 = Tex(r"""
		$$
		sin(x)=
		cos(0)x+
		\frac{-cos(0)}{3!}x^3+
		\frac{cos(0)}{5!}x^5
		$$
	""")
	tex_3 = Tex(r"""
		$$
		sin(x)=
		\frac{x^1}{1!}+
		\frac{-x^3}{3!}+
		\frac{x^5}{5!}
		$$
	""")
	tex_4 = Tex(r"""
		$$
		sin(x)=\sum_{i=0}^{i} 
		\frac
		{
			(-1)^{i} \cdot x^{2i+1}}
		{(2i+1)!}
		$$
	""")


	self.play(Write(intro)) # 写下 “这是野生的展开式”
	self.play(intro.animate.shift(UP * 3)) # 向上移动
	self.play(Write(tex)) # 泰勒展开式
	self.play(tex.animate.shift(UP)) # 泰勒展开式向上移动
	self.play(Unwrite(intro)) # 擦去 “这是野生的泰勒展开式”
	self.play(Write(intro_sin)) # 写下 “将正弦函数代入其中”
	self.play(Write(tex_sin)) # 写下带入正弦后的表达式
	self.play(Uncreate(tex)) # 变换得到新式子
	self.play(Unwrite(intro_sin)) # 擦去 ”带入正弦函数“
	self.play(tex_sin.animate.shift(UP)) # 将带入后的式子置于中间
	self.play(Transform(tex_sin, tex_2), run_time=2)
	self.remove(tex_sin)
	self.play(Transform(tex_2, tex_3), run_time=2)
	self.remove(tex_2)
	self.play(Transform(tex_3, tex_4), run_time=2)
	self.play(Unwrite(tex_3))

较为简单，用到Animations和Tex。

write_image

此程序的难点所在。

先来看两个工具函数： 

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
# 创建泰勒近似函数（前 n 项）
def make_taylor_func(self, n_terms):
	return lambda x: sum(
		(-1) ** k * x ** (2 * k + 1) / factorial(2 * k + 1) for k in range(n_terms)
	)

# 创建对应的 LaTeX 公式
def make_formula_tex(self, n_terms):
		terms = []
		for k in range(n_terms):
				sign = "-" if k % 2 == 1 else ""
				exponent = 2 * k + 1
				term = rf"{sign}\frac{{x^{{{exponent}}}}} {{{exponent}!}}"
				terms.append(term)
		formula = "+".join(terms).replace("+-", "-")
		return MathTex("f(x) =", formula).scale(0.8).to_edge(RIGHT)

make_taylor_func用于求每一个子项的大小，相当于： $$ \sum_{i=0}^{i} \frac { (-1)^{i} \cdot x^{2i+1}} {(2i+1)!} $$ 通过for循环生成了k项，也就是这里的i。

make_formula_tex用于产生每个展开式对应的Latex对象。先用正则匹配生成k条字符串，放入数组并拼接起来得到最终的公式。需要注意部分项会产生-，拼接后就变成+-，需统一替换成-。

最终write_image(self)如下，我们逐段研究。

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
def write_image(self):
	ax = Axes(
		x_range=[-12, 12, 2],
		y_range=[-2, 2],
		axis_config={"color": GREEN},
	)
	pl = NumberPlane()

	sin = ax.plot(lambda x: np.sin(x), color=BLUE)

	self.play(Create(ax), Create(pl))
	self.play(Write(sin))

	# 构建图像和公式
	taylor_graphs = []
	formula_labels = []
	for i in range(1, 15):  # t1~t6
		f = self.make_taylor_func(i)
		graph = ax.plot(f, color=RED)
		formula = self.make_formula_tex(i).shift(DOWN*2)
		if (i >= 8):
			formula = formula.scale(0.5).shift(RIGHT*2)
		taylor_graphs.append(graph)
		formula_labels.append(formula)

	# 第一个图和公式
	current_graph = taylor_graphs[0]
	current_formula = formula_labels[0]

	self.play(Write(current_graph), Write(current_formula))

	# 后续逐步替换
	for next_graph, next_formula in zip(taylor_graphs[1:], formula_labels[1:]):
		self.play(Transform(current_graph, next_graph),
				  Transform(current_formula, next_formula))
		self.remove(current_graph, current_formula)
		current_graph = next_graph
		current_formula = next_formula

	# 最终版本
	self.play(Write(current_graph), Write(current_formula))
	self.wait(2)

section 1

第一部分建立坐标系，并画图，难度不大。

1
2
3
4
5
6
7
8
9
10
11
ax = Axes(
		x_range=[-12, 12, 2],
		y_range=[-2, 2],
		axis_config={"color": GREEN},
	)
	pl = NumberPlane()
	
	sin = ax.plot(lambda x: np.sin(x), color=BLUE)
	
	self.play(Create(ax), Create(pl))
	self.play(Write(sin))

section 2

第二部分先是建立两个数组用于存放未来要画的泰勒函数图像，公式标签在右下角展示使视频更直观。通过循环生成了 15 个表达式，f 为我们根据make_taylor_func建立的函数对象。将函数对象 f 放入ax.plot()中即可得到图像对象，公式formula根据make_formula_tex()处理得到。

由于公式较长时会溢出屏幕，需在 i ≥ 8 时缩小其尺寸。

1
2
3
4
5
6
7
8
9
10
11
 # 构建图像和公式
taylor_graphs = []
formula_labels = []
for i in range(1, 15):  # t1~t6
	f = self.make_taylor_func(i)
	graph = ax.plot(f, color=RED)
	formula = self.make_formula_tex(i).shift(DOWN*2)
	if (i >= 8):
		formula = formula.scale(0.5).shift(RIGHT*2)
	taylor_graphs.append(graph)
	formula_labels.append(formula)

section 3

先画出第一个展开式的图像，后续展开式则用数组和循环批量处理。后续通过切片语法从第二个展开式开始迭代。每一次更新都播放衔接的过渡动画，移除当前处理的公式、函数图像。并将current_xxx指针指向下一个带处理的对象。留下最后一组时，上一组对象已被循环移除，直接播放动画即可。

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
# 第一个图和公式
current_graph = taylor_graphs[0]
current_formula = formula_labels[0]

self.play(Write(current_graph), Write(current_formula))

# 后续逐步替换
for next_graph, next_formula in zip(taylor_graphs[1:], formula_labels[1:]):
	self.play(Transform(current_graph, next_graph),
			  Transform(current_formula, next_formula))
	self.remove(current_graph, current_formula)
	current_graph = next_graph
	current_formula = next_formula

# 最终版本
self.play(Write(current_graph), Write(current_formula))
self.wait(2)

这里的zip函数用于将两个数组并排循环，本质仍在对数组进行迭代，例如：

1
2
3
4
5
6
7
8
9
10
a = [1, 2, 3]
b = ['a', 'b', 'c']

for x, y in zip(a, b):
    print(x, y)

# 输出:
# 1 a
# 2 b
# 3 c