Smooth Versus Jagged

I need to make more use of AI in my blog posts and to this end I wanted to replace the jagged lines in Figure 3 of my previous post with one continuous smooth line. The graph in question looked as shown in Figure 1 below:

Figure 1

The data being plotted were the numbers from 395 to 425 and their corresponing sigma values:

[(395, 480), (396, 1092), (397, 398), (398, 600), (399, 640), (400, 961), (401, 402), (402, 816), (403, 448), (404, 714), (405, 726), (406, 720), (407, 456), (408, 1080), (409, 410), (410, 756), (411, 552), (412, 728), (413, 480), (414, 936), (415, 504), (416, 882), (417, 560), (418, 720), (419, 420), (420, 1344), (421, 422), (422, 636), (423, 624), (424, 810), (425, 558)]

I asked Google Gemini to put a smooth curve through these points and Figure 2 shows what it came up with when I pasted the code into SageMathCell.

Figure 2: permalink

This is a satisfying output and is just an alternative way to display the data. There may well be a way to implement this using SageMath code alone but this approach using Python was quick and easy. It also creates a title and legends for the point and curve. I can use this code in future when needed. Here is the code that Gemini created:

import numpy as np

import matplotlib.pyplot as plt

from scipy.interpolate import CubicSpline

data = [(395, 480), (396, 1092), (397, 398), (398, 600), (399, 640), (400, 961), (401, 402), (402, 816), (403, 448), (404, 714), (405, 726), (406, 720), (407, 456), (408, 1080), (409, 410), (410, 756), (411, 552), (412, 728), (413, 480), (414, 936), (415, 504), (416, 882), (417, 560), (418, 720), (419, 420), (420, 1344), (421, 422), (422, 636), (423, 624), (424, 810), (425, 558)]

# Extract x and y values

x = [point[0] for point in data]

y = [point[1] for point in data]

# Create a cubic spline interpolation

cs = CubicSpline(x, y)

# Generate smoother x values for plotting the curve

x_smooth = np.linspace(min(x), max(x), 500)  # Adjust 500 for more/less smoothness

y_smooth = cs(x_smooth)

# Plot the original points and the smooth curve

plt.figure(figsize=(10, 6))

plt.plot(x, y, 'o', label='Original Points')

plt.plot(x_smooth, y_smooth, '-', label='Smooth Curve (Cubic Spline)')

plt.xlabel('X')

plt.ylabel('Y')

plt.title('Smooth Curve Through Points')

plt.legend()

plt.grid(True)

plt.show()

Of course, Gemini provides a full explanation of the code that it created:

Explanation:

1. Import Libraries:
   • numpy for numerical operations (especially for generating smoother x-values).
   • matplotlib.pyplot for plotting the graph.
   • scipy.interpolate.CubicSpline for creating the smooth curve.
     
     
2. Extract Data: The x and y values are extracted from the data list into separate lists.
   
   
3. Cubic Spline Interpolation:
   • CubicSpline(x, y) creates a cubic spline object that represents the smooth curve passing through the given data points. Cubic splines are chosen as they offer a good balance between smoothness and accuracy.
     
     
4. Generate Smooth x-Values:
   • np.linspace(min(x), max(x), 500) creates an array of 500 evenly spaced x-values between the minimum and maximum x-values of the original data. This is used to generate a smoother curve for plotting. Adjust the 500 for a curve with more or less detail.
     
     
5. Calculate Smooth y-Values:
   • cs(x_smooth) calculates the corresponding y-values for the smoother x-values using the cubic spline interpolation.
     
     
6. Plotting:
   • plt.plot(x, y, 'o', label='Original Points') plots the original data points as circles.
   • plt.plot(x_smooth, y_smooth, '-', label='Smooth Curve (Cubic Spline)') plots the smooth curve.
   • The rest of the code sets up the plot's labels, title, legend, and grid.
   • plt.show() displays the plot.
     
     
7. How to Run:
   • Make sure you have numpy, matplotlib, and scipy installed (pip install numpy matplotlib scipy).
   • Save the code as a Python file (e.g., smooth_curve.py).
   • Run the file from your terminal: python smooth_curve.py.