Finding intersection between straight line and contour

Question

I am trying to find the intersection point of a straight(dashed red) with the contour-line highlighted in red(see plot). I used .get_paths in the second plot to isolate said

Answer 1

Use shapely can find the intersection point, than use the point as the init guess value for fsolve() to find the real solution:

#for contour 
def p_0(num,t) :
    esc_p = np.sum((((-1)**n)*(np.exp(t)**n)*((math.factorial(n)*((n+1)**0.5))**-1)) for n in range(1,num,1))
    return esc_p+1

tau = np.arange(-2,3,0.1)

x,y= np.meshgrid(tau,tau)
cs = plt.contour(x, y, np.log(p_0(51, y)/p_0(51, x)),[0.2],colors='k')

p=0.75
logp = (np.log(p*np.exp(tau)))
plt.plot(tau,logp)

from shapely.geometry import LineString
v1 = cs.collections[0].get_paths()[0].vertices

ls1 = LineString(v1)
ls2 = LineString(np.c_[tau, logp])
points = ls1.intersection(ls2)
x, y = points.x, points.y

from scipy import optimize

def f(p):
    x, y = p
    e1 = np.log(0.75*np.exp(x)) - y
    e2 = np.log(p_0(51, y)/p_0(51, x)) - 0.2
    return e1, e2

x2, y2 = optimize.fsolve(f, (x, y))

plt.plot(x, y, "ro")
plt.plot(x2, y2, "gx")

print x, y
print x2, y2

Here is the output:

0.273616328952 -0.0140657435002
0.275317387697 -0.0123646847549

and the plot: