How to Accurately Find Intersection Points of a Curve with y = 0 Using Python?

Identifying the Intersection Points of a Curve with y = 0

In the context of Python plotting using NumPy arrays, the precise value on the y-axis where a curve intersects with y = 0 can be an elusive task. This article discusses a method for accurately determining these intersection points, addressing the limitations of existing root-finding techniques.

To begin with, let's define our plot using NumPy arrays:

<code class="python">import numpy as np
import matplotlib.pyplot as plt
vertical_data = ...
gradient = ...
plt.plot(gradient, vertical_data)
plt.show()</code>

Now, suppose we want to locate the zero value of the curve. Conventional root-finding algorithms often provide imprecise results due to floating-point precision limitations. To overcome this, we employ a simple linear interpolation technique:

<code class="python">import numpy as np

def find_roots(x, y):
    # Compute the sign differences
    s = np.abs(np.diff(np.sign(y))).astype(bool)
    # Perform linear interpolation at sign changes
    return x[:-1][s] + np.diff(x)[s]/(np.abs(y[1:][s]/y[:-1][s])+1)</code>

This function takes the x and y arrays as input and locates the intersection points where the y-values cross the zero mark. We apply this function to our data and plot the results:

<code class="python">z = find_roots(gradient, vertical_data)

plt.plot(gradient, vertical_data)
plt.plot(z, np.zeros(len(z)), marker="o", ls="", ms=4)
plt.show()</code>

By inverting the roles of x and y, we can determine the intersection points along the x-axis:

<code class="python">plt.plot(vertical_data, gradient)
plt.plot(np.zeros(len(z)), z, marker="o", ls="", ms=4)
plt.show()</code>

Furthermore, the presented method allows for finding intersection points at non-zero y-values. Simply subtract the desired y-value from the curve before applying the find_roots function:

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

Lastly, for determining the intersection between two curves, compute the difference between the curves and apply the find_roots function to locate the intersection points:

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

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