Python编程使用matplotlib绘制动态圆锥曲线示例(圆锥曲线方程)

椭圆

为了绘图方便，椭圆的参数方程为

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

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

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

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()