Python绘制箕舌线

过原点的动直线交定圆 x 2 + y 2 − a y = 0 , a > 0 x^2+y^2-ay=0, a>0 x2+y2−ay=0,a>0于P点，交直线 y = a y=a y=a于Q点，过P和Q分别作X轴和Y轴的平行线交于M点，则M点的轨迹叫做箕舌线 。

设 a = 2 a=2 a=2，则圆的参数方程为

x = cos ⁡ θ , y = sin ⁡ θ + 1 x=\cos\theta,y=\sin\theta+1 x=cosθ,y=sinθ+1

设动直线的方程为 y = k x y=kx y=kx，则随着 k k k的变化，可以得到一条箕舌线

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

a = 2
fig = plt.figure(figsize=(15,4))
ax = fig.add_subplot(autoscale_on=False, 
    xlim=(-5,5),ylim=(-0.1,2.5))
ax.grid()

t = np.arange(0,6.4,0.1)
ax.plot(np.cos(t),np.sin(t)+1,'-',lw=1) #圆方程
ax.plot([-5,5],[a,a],'-',lw=1)

xLine, = ax.plot([],[],linestyle='--',lw=0.5)
yLine, = ax.plot([],[],linestyle='--',lw=0.5)
kLine, = ax.plot([],[],linestyle='--',lw=0.5)

trace, = ax.plot([],[],'-', lw=1)
k_text = ax.text(0.05,0.85,'',transform=ax.transAxes)
textTemplate = 'k = %.1f\n'

xs,ys = [],[]
def animate(theta):
    if(theta==0):
        xs.clear()
        ys.clear()
    k = np.tan(theta)
    px = 2*k/(k**2+1)
    py = k*px
    qx = a/k
    xLine.set_data([px,qx],[py,py])
    yLine.set_data([qx,qx],[2,py])
    kLine.set_data([0,qx],[0,a])
    xs.append(qx)
    ys.append(py)
    trace.set_data(xs,ys)
    k_text.set_text(textTemplate % k)
    return xLine, yLine, kLine, trace, k_text

frames = np.arange(0,np.pi,0.05)
ani = animation.FuncAnimation(fig, animate, 
    frames, interval=50, blit=True)
ani.save("witch.gif")

plt.show()