[Python知识库]Python绘制动态的圆锥曲线

作为让高中生心脏骤停的四个字，对于高考之后的人来说可谓刻骨铭心，所以定义不再赘述，直接撸图，其标准方程分别为

在Python中，绘制动图需要用到matplotlib中的animation包，其调用方法以及接下来要用到的参数为

ani = animation.FuncAnimation(fig, func, frames, interval)

其中fig为绘图窗口，func为绘图函数，其返回值为图像，frames为迭代参数，如果为整型的话，其迭代参数则为range(frames)。

椭圆

为了绘图方便，椭圆的参数方程为

$$\{\begin{array}{r}x=a\cos t\\ y=b\sin t\end{array}$$

设$a=5,b=3,c=4$，则焦点为$(4，0),(?4,0)$，则有

代码为：

# 这三个包在后面的程序中不再复述
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation

a,b,c = 5,3,4
fig = plt.figure(figsize=(12,9))
ax = fig.add_subplot(autoscale_on=False, 
    xlim=(-a,a),ylim=(-b,b))
ax.grid()

line, = ax.plot([],[],'o-',lw=2)
trace, = ax.plot([],[],'-', lw=1)
theta_text = ax.text(0.02,0.85,'',transform=ax.transAxes)
textTemplate = '''theta = %.1f°\n
lenL = %.1f, lenR = %.1f\n
lenL+lenR = %.1f'''

xs,ys = [], []

def animate(i):
    if(i==0):
        xs.clear()
        ys.clear()
    theta = i*0.04
    x = a*np.cos(theta)
    y = b*np.sin(theta)
    xs.append(x)
    ys.append(y)
    line.set_data([-c,x,c], [0,y,0])
    trace.set_data(xs,ys)
    lenL = np.sqrt((x+c)**2+y**2)
    lenR = np.sqrt((x-c)**2+y**2)
    theta_text.set_text(textTemplate % 
        (180*theta/np.pi, lenL, lenR, lenL+lenR))
    return line, trace, theta_text

ani = animation.FuncAnimation(fig, animate, 157, 
    interval=5, blit=True)
ani.save("ellipse.gif")

plt.show()

双曲线

双曲线的参数方程为

$$?\begin{array}{r}x=a\mathrm{ch}t=\frac{e^t+e^{?t}}{2}\\ y=b\mathrm{sh}t=\frac{e^t?e^{?t}}{2}\end{array}$$

设$a=4,b=3,c=5$，则代码如下

a,b,c = 4,3,5
fig = plt.figure(figsize=(12,9))
ax = fig.add_subplot(autoscale_on=False, 
    xlim=(-c,16),ylim=(-12,12))
ax.grid()

line, = ax.plot([],[],'o-',lw=2)
trace, = ax.plot([],[],'-', lw=1)
theta_text = ax.text(0.01,0.85,'',
    transform=ax.transAxes)
textTemplate = '''t = %.1f\n
lenL = %.1f, lenR = %.1f\n
lenL-lenR = %.1f'''

xs,ys = [],[]

def animate(t):
    if(t==-3):
        xs.clear()
        ys.clear()
    x = a*np.cosh(t)
    y = b*np.sinh(t)
    xs.append(x)
    ys.append(y)
    line.set_data([-c,x,c], [0,y,0])
    trace.set_data(xs,ys)
    lenL = np.sqrt((x+c)**2+y**2)
    lenR = np.sqrt((x-c)**2+y**2)
    theta_text.set_text(textTemplate % 
        (t, lenL, lenL, lenL-lenR))
    return line, trace, theta_text

frames = np.arange(-3,3,0.05)
ani = animation.FuncAnimation(fig, animate, 
    frames, interval=5, blit=True)
ani.save("hyperbola.gif")

plt.show()

抛物线

令$p=1$，则焦点位置为$(0,\frac{p}{2})$，准线为$x=?\frac{p}{2}$，代码如下

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation

a,b,c = 4,3,5
p = 1
fig = plt.figure(figsize=(12,9))
ax = fig.add_subplot(autoscale_on=False, 
    xlim=(-0.6,4.5),ylim=(-3,3))
ax.grid()

ax.plot([-p/2,-p/2],[-5,5],'-',lw=2)
line, = ax.plot([],[],'o-',lw=2)
trace, = ax.plot([],[],'-', lw=1)
theta_text = ax.text(0.05,0.85,'',
    transform=ax.transAxes)
textTemplate = '''y = %.1f\n
lenL = %.1f, lenF = %.1f\n
lenL-lenF = %.1f'''

xs,ys = [],[]
def animate(y):
    if(y==-3):
        xs.clear()
        ys.clear()
    x = y**2/p/2
    xs.append(x)
    ys.append(y)
    line.set_data([-p,x,p/2], [y,y,0])
    trace.set_data(xs,ys)
    lenL = x+p/2
    lenF = np.sqrt((x-p/2)**2+y**2)
    theta_text.set_text(textTemplate % 
        (y, lenL, lenF, lenL-lenF))
    return line, trace, theta_text

frames = np.arange(-3,3,0.1)
ani = animation.FuncAnimation(fig, animate, 
    frames, interval=5, blit=True)
ani.save("parabola.gif")

plt.show()

极坐标方程

圆锥曲线在极坐标系下有相同的表达式，即

$$\rho =\frac{p}{1?e\cos \phi }$$

其中，$p$为焦参数，$e$为离心率，当$|e|>1$时为双曲线；$|e|=1$时为抛物线，$|e|<1$时为椭圆，特别地$e=0$为圆。

在matplotlib中，极坐标图像需要通过projection='polar'来标识，其代码为

p = 2
fig = plt.figure(figsize=(12,9))
ax = fig.add_subplot(autoscale_on=False, projection='polar')
ax.set_rlim(0,8)

trace, = ax.plot([],[],'-', lw=1)
theta_text = ax.text(0.05,0.95,'',transform=ax.transAxes)
textTemplate = 'e = %.1f\n'

theta = np.arange(-3.1,3.2,0.1)
def animate(e):
    rho = p/(1-e*np.cos(theta))
    trace.set_data(theta,rho)
    theta_text.set_text(textTemplate % e)
    return trace, theta_text

frames = np.arange(-2,2,0.1)
ani = animation.FuncAnimation(fig, animate, 
    frames, interval=100, blit=True)
ani.save("polar.gif")

plt.show()