How to find and draw the intersection points of contour shapes - python-2.7

Question

I'm trying to find the points of intersection between the line passing through point V and conic. The conic graph is not solvable relative to y ( or x), so it was depicted using contour. Is there a method for finding the intersection points of contour graphs?
enter image description here
Here is the code:
import numpy as np
import matplotlib.pyplot as plt
p = (input("choose point P on axis OX: "))
print(p)
q = (input("choose point Q on axis OX: "))
print(q)
v = (input("choose point V on axis OX: "))
print(v)
k=3
X = np.arange(-50, 50, 0.05)
Y = k*X
plt.plot(X,Y)
plt.plot(0,0)
plt.scatter(0.0, 0.0, color='white', marker='o')
plt.text(0.0, 0.0, "O", horizontalalignment="center")
plt.plot(-v,0)
plt.scatter(-v, 0, color='red', marker='o')
plt.text(-v, 0.8, "V", horizontalalignment="center")
xmin= -10
xmax= 10
ymin= -10
ymax= 10
ax = plt.gca()
ax.get_xlim()
ax.set_xlim([xmin,xmax])
ax.set_ylim([ymin,ymax])
ax.spines['left'].set_position('center')
ax.spines['bottom'].set_position('center')
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
#Create random point B1
b1=4
plt.plot(0.0,b1)
plt.scatter(0.0, b1, color='blue', marker='o')
plt.text(0.8, b1, "B1", horizontalalignment="center")
x, y = np.meshgrid(X, X)
#Create VB1
l3 = b1*x+b1*v - v*y
vb = plt.contour(x,y, l3, [0], colors='k')
# l3 = b1*X/v + b1
# plt.plot(X,l3)
#Create conic
conic = x*x*b1*2*p*k-x*x*b1*2*q*k+x*x*k*k+y*y-b1*2*y+2*b1*q*x*y
cnc = plt.contour(x, y, (conic), [0], colors='k')
I tried to do something like that:
c = cnc.intersection(vb)
print(c)
or
# https://stackoverflow.com/questions/28766692/intersection-of-two-graphs-in-python-find-the-x-value
idx = np.argwhere(np.diff(np.sign(cnc - vb))).flatten()
plt.plot(x[idx], y[idx], 'ro')
My last attempt:
import numpy as np
import matplotlib.pyplot as plt
p,q,v,k,b=5,7,2,3,4
X = np.arange(-50, 50, 0.05)
plt.plot(-v,0)
plt.scatter(-v, 0, color='red', marker='o')
plt.text(-v, 0.8, "V", horizontalalignment="center")
xmin,xmax,ymin,ymax=-10,10,-10,10
ax = plt.gca()
ax.get_xlim()
ax.set_xlim([xmin,xmax])
ax.set_ylim([ymin,ymax])
ax.spines['left'].set_position('center')
ax.spines['bottom'].set_position('center')
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
plt.plot(0.0,b)
plt.scatter(0.0, 1, color='blue', marker='o')
x, y = np.meshgrid(X, X)
l = b*x+b*v-v*y
vb = plt.contour(x,y, l, [0], colors='k')
conic = x*x*b*2*p*k-x*x*b*2*q*k+x*x*k*k+y*y-b*2*y+2*b*q*x*y
cnc = plt.contour(x, y, (conic), [0], colors='k')
c = cnc.collections[0].get_paths()[1]
v = c.vertices
x1 = v[:,0]
y1 = v[:,1]
plt.plot(x1,y1)
vb1 = vb.collections[0].get_paths()[0]
v1 = vb1.vertices
x2 = v1[:,0]
y2 = v1[:,1]
plt.plot(x2,y2,color='red')
# def find_roots(x,y):
# s = np.abs(np.diff(np.sign(y))).astype(bool)
# return x[:-1][s] + np.diff(x)[s]/(np.abs(y[1:][s]/y[:-1][s])+1)
#
# z = find_roots(x1-x2,y1-y2)
# plt.plot(z, np.zeros(len(z)), marker="o", ls="", ms=4)
plt.show()
enter image description here

Answer 1

It's a little more complicated. The problem is that (a) the points in the contour are not necessarily sorted and (b) the two contours do not have a common support.
So one would need to (a) sort the points along x, and (b) create an array of common values, on which to interpolate first.
import numpy as np
import matplotlib.pyplot as plt
p,q,v,k,b=5,7,2,3,4
X = np.arange(-50, 50, 0.05)
plt.plot(-v,0)
plt.scatter(-v, 0, color='red', marker='o')
plt.text(-v, 0.8, "V", horizontalalignment="center")
xmin,xmax,ymin,ymax=-10,10,-10,10
ax = plt.gca()
ax.get_xlim()
ax.set_xlim([xmin,xmax])
ax.set_ylim([ymin,ymax])
ax.spines['left'].set_position('center')
ax.spines['bottom'].set_position('center')
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
x, y = np.meshgrid(X, X)
l = b*x+b*v-v*y
vb = plt.contour(x,y, l, [0], colors='red')
conic = x*x*b*2*p*k-x*x*b*2*q*k+x*x*k*k+y*y-b*2*y+2*b*q*x*y
cnc = plt.contour(x, y, (conic), [0], colors='blue')
c = cnc.collections[0].get_paths()[1]
v = c.vertices
x1 = np.sort(v[:,0])
y1 = v[np.argsort(v[:,0]),1]
vb1 = vb.collections[0].get_paths()[0]
v1 = vb1.vertices
x2 = np.sort(v1[:,0])
y2 = v1[np.argsort(v1[:,0]),1]
def find_roots(x,y):
s = np.abs(np.diff(np.sign(y))).astype(bool)
return x[:-1][s] + np.diff(x)[s]/(np.abs(y[1:][s]/y[:-1][s])+1)
x = np.linspace(max(x1.min(), x2.min()), min(x1.max(), x2.max()), 1000)
y1i = np.interp(x, x1, y1 ) # blue
y2i = np.interp(x, x2, y2 ) # red
x_intersect = find_roots(x,y2i-y1i)
y_intersect = np.interp(x_intersect, x, y2i)
plt.plot(x_intersect, y_intersect, marker="X", ms=5, color="limegreen")
plt.show()
The point of intersection is the green dot.
Of course one needs to do the same for the other arm of the conic contour (cnc.collections[0].get_paths()[0]) if desired.