How to Find the Intersection Points of a Curve with the Y-Axis and Other Curves in Python?

Determining the Zero Intersection of a Curve

In Python, finding the exact point where a curve intersects the y-axis (y=0) can be challenging. A numpy array may represent a curve, but it does not provide a direct method for identifying zeros.

To address this issue, a linear interpolation approach can be employed. The following code demonstrates how to find the exact intersection point:

<code class="python">import numpy as np
import matplotlib.pyplot as plt

# Generate sample data
N = 750
x = 0.4 + np.sort(np.random.rand(N)) * 3.5
y = (x - 4) * np.cos(x * 9.) * np.cos(x * 6 + 0.05) + 0.1

# Define a function to find roots (zeros)
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)

# Find the intersection point
z = find_roots(x, y)

# Plot the curve and the intersection point
plt.plot(x, y)
plt.plot(z, np.zeros(len(z)), marker="o", ls="", ms=4)
plt.show()</code>

This script will generate a plot showing the curve and a marker at the exact intersection point with the y-axis.

Finding Intercepts at Non-Zero Values

To find intercepts at non-zero values (e.g., y0), the same approach can be applied by finding the zeros of the curve shifted by y0:

<code class="python">y0 = 1.4
z = find_roots(x, y - y0)
# ...
plt.plot(z, np.zeros(len(z)) + y0)</code>

Intersection of Two Curves

To find the intersection point between two curves, find the zeros of the difference between the two curves:

<code class="python">x = .4 + np.sort(np.random.rand(N)) * 3.5
y1 = (x - 4) * np.cos(x * 9.) * np.cos(x * 6 + 0.05) + 0.1
y2 = (x - 2) * np.cos(x * 8.) * np.cos(x * 5 + 0.03) + 0.3

z = find_roots(x, y2 - y1)

plt.plot(x, y1)
plt.plot(x, y2, color="C2")
plt.plot(z, np.interp(z, x, y1), marker="o", ls="", ms=4, color="C1")</code>

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