1 方波的圆周分解

在学习傅里叶变换的时候，有一个经典的示例是方波的分解。我们知道，方波可以分解为无数个正弦波的叠加。而正弦波，又可以看作是圆周运动在一条直线上的投影。当时为了理解这个事情，恐怕大家也花了不少时间。

学习了matplotlib之后，出于学以致用的考虑，我们能不能绘制出动画，来描述上述分解，便于我们来理解呢？

先上动图：

前面学习中已经掌握了matplotlib如何制作动画，以及如何绘制子图。在这个例子中我们将看到以下内容的实战：

• 分割画布为子图
• 绘制圆和波形图
• 调整图像的轴比例
• 隐藏图像的刻度轴
• 设置线型和颜色
• 生成和保存动画

2. 绘图源码

# -*- coding: utf-8 -*-import matplotlib.pyplot as plt
import matplotlib.animation as animation
import numpy as np# 把半径放大3倍 画大一些
radius = 4 / np.pi * 3
t = np.deg2rad(list(range(0, 360, 1)))
tc = [tt + 8 for tt in t]# 一级圆
x = np.cos(t) * radius
y = np.sin(t) * radius# 二级圆
x1 = np.cos(3*t) * radius / 3
y1 = np.sin(3*t) * radius / 3# 三级圆
x2 = np.cos(5*t) * radius / 5
y2 = np.sin(5*t) * radius / 5# 四级圆
x3 = np.cos(7*t) * radius / 7
y3 = np.sin(7*t) * radius / 7# 子图的切割
fig, ax = plt.subplots(4, 1)
fig.set_size_inches(5, 8)
fig.tight_layout()
ax[0].set(xlim=(-2, 10), ylim=(-6, 6))#设置横纵坐标等比例且根据轴范围自适应
ax[0].set_aspect("equal", adjustable='datalim')
ax[0].set_axis_off() # 隐藏轴刻度线
# 绘制固定的那个圆
ax[0].plot(x, y, 'b-', linewidth=1)ax[1].set(xlim=(-2, 10), ylim=(-6, 6))
ax[1].set_aspect("equal", adjustable='datalim')
ax[1].set_axis_off()
ax[1].plot(x, y, 'b-', linewidth=1)ax[2].set(xlim=(-2, 10), ylim=(-6, 6))
ax[2].set_aspect("equal", adjustable='datalim')
ax[2].set_axis_off()
ax[2].plot(x, y, 'b-', linewidth=1)ax[3].set(xlim=(-2, 10), ylim=(-6, 6))
ax[3].set_aspect("equal", adjustable='datalim')
ax[3].set_axis_off()
ax[3].plot(x, y, 'b-', linewidth=1)# 动画准备
artists = []
for i in range(0, len(t)):# container用来存储每帧要绘制的内容，通过+=叠加container = []container += ax[0].plot([0, x[i]], [0, y[i]], 'b-', linewidth=1)yy = list(y[i:len(y)])yy.extend(y[0:i])container += ax[0].plot(tc, yy, color='blue', linewidth=1)container += ax[0].plot([x[i], tc[0]], [y[i], yy[0]], 'b--', linewidth=1)container += ax[1].plot([0, x[i]], [0, y[i]], 'b-', linewidth=1)container += ax[1].plot([x1i + x[i] for x1i in x1], [y1i + y[i] for y1i in y1], 'r-', linewidth=1)y_y1 = y+y1yy1 = list(y_y1[i:len(y_y1)])yy1.extend(y_y1[0:i])container += ax[1].plot([x[i], x[i]+x1[i]], [y[i], y[i]+y1[i]], 'r-', linewidth=1)container += ax[1].plot(tc, yy1, color='red', linewidth=1)container += ax[1].plot([x[i]+x1[i], tc[0]], [y[i]+y1[i], yy1[0]], 'r--', linewidth=1)container += ax[2].plot([0, x[i]], [0, y[i]], 'b-', linewidth=1)container += ax[2].plot([x1i + x[i] for x1i in x1], [y1i + y[i] for y1i in y1], 'r-', linewidth=1)container += ax[2].plot([x[i], x[i] + x1[i]], [y[i], y[i] + y1[i]], 'r-', linewidth=1)container += ax[2].plot([x2i + x1[i] + x[i] for x2i in x2], [y2i + y1[i] + y[i] for y2i in y2], 'g-', linewidth=1)container += ax[2].plot([x[i] + x1[i], x[i] + x1[i] + x2[i]], [y[i] + y1[i], y[i] + y1[i] + y2[i]], 'g-',linewidth=1)y_y2 = y + y1 + y2yy2 = list(y_y2[i:len(y_y2)])yy2.extend(y_y2[0:i])container += ax[2].plot(tc, yy2, color='green', linewidth=1)container += ax[2].plot([x[i] + x1[i] + x2[i], tc[0]], [y[i] + y1[i] + y2[i], yy2[0]], 'g--',linewidth=1)container += ax[3].plot([0, x[i]], [0, y[i]], 'b-', linewidth=1)container += ax[3].plot([x1i + x[i] for x1i in x1], [y1i + y[i] for y1i in y1], 'r-', linewidth=1)container += ax[3].plot([x[i], x[i] + x1[i]], [y[i], y[i] + y1[i]], 'r-', linewidth=1)container += ax[3].plot([x2i + x1[i] + x[i] for x2i in x2], [y2i + y1[i] + y[i] for y2i in y2], 'g-', linewidth=1)container += ax[3].plot([x[i] + x1[i], x[i] + x1[i] + x2[i]], [y[i] + y1[i], y[i] + y1[i] + y2[i]], 'g-',linewidth=1)container += ax[3].plot([x3i + x2[i] + x1[i] + x[i] for x3i in x3], [y3i + y2[i] + y1[i] + y[i] for y3i in y3],'c-', linewidth=1)container += ax[3].plot([x[i] + x1[i] + x2[i], x[i] + x1[i] + x2[i] + x3[i]],[y[i] + y1[i] + y2[i], y[i] + y1[i] + y2[i] + y3[i]], 'c-', linewidth=1)y_y3 = y + y1 + y2 + y3yy3 = list(y_y3[i:len(y_y3)])yy3.extend(y_y3[0:i])container += ax[3].plot(tc, yy3, color='cyan', linewidth=1)container += ax[3].plot([x[i] + x1[i] + x2[i] + x3[i], tc[0]], [y[i] + y1[i] + y2[i] + y3[i], yy3[0]],'c--', linewidth=1)artists.append(container)# 生成并保存动图
ani = animation.ArtistAnimation(fig=fig, artists=artists, interval=40)
ani.save(filename="c:/users/admin/desktop/fourier.gif", writer="pillow")
plt.show()

这里还有一些优化空间，例如还没有为这幅图加上文字说明（涉及$\text{LATEX}$输入），以及也没有绘制出方波本身。作为一个练习，小白觉得主体的内容已经足够，留待后续优化。