How to find crossection of hline and a function in python?

I have created a plot like in the figure below, red plot is original data, blue is a fitted function, the horizontal lines are different levels. I need to find the both intersection points with each line. do you have any suggestions ? Thanks In advance.

Solution

An easy way to do this numerically is to subtract the y-value of each horizontal line from your fit and then to solve the equation fit(x) - y = 0 for x. For this, scipy.optimize.fsolve can be used as follows (https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fsolve.html):

import numpy as np
import matplotlib.pyplot as plt

from scipy.optimize import fsolve # To find the zeros
from scipy.stats import cauchy

def my_fit(x):
    # dummy function for this example
    return cauchy().pdf(x)


horizontal_lines = [0.05, 0.15, 0.25]
colors = ['r', 'g', 'm']

x = np.linspace(-5, 5, 1000)


plt.plot(x, my_fit(x))
plt.hlines(horizontal_lines, -5, 5, ls="--", color=colors)

for i, y in enumerate(horizontal_lines):
    x0 = fsolve(lambda x: my_fit(x) - y, -1)
    x1 = fsolve(lambda x: my_fit(x) - y, 1)
    print(f"Intersection points for {y=}: {x0=} {x1=}") 
    plt.scatter(x0, my_fit(x0), color=colors[i])
    plt.scatter(x1, my_fit(x1), color=colors[i])

Output:

Intersection points for y=0.05: x0=array([-2.3165055]) x1=array([2.3165055]) 
Intersection points for y=0.15: x0=array([-1.05927612]) x1=array([1.05927612]) 
Intersection points for y=0.25: x0=array([-0.5227232]) x1=array([0.5227232])