Learn Visualizing Approximations | Approximating Complex Figures

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To understand how well a polygonal approximation matches a complex curve, it is helpful to visualize both the original curve and its approximation on the same plot. This approach allows you to observe where the approximation closely follows the curve and where it deviates. You can use matplotlib to display both shapes together, assigning different colors or line styles for clarity. Typically, you will:

Generate points for the original curve using its mathematical equation;
Compute the vertices of the polygonal approximation;
Plot both sets of points or lines on the same axes for direct comparison.

This process is especially useful for circles, ellipses, or any smooth curve where visual differences are important for evaluating the quality of the approximation.


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import numpy as np
import matplotlib.pyplot as plt

# Parameters for the circle
center = (0, 0)
radius = 1

# Generate points for the original circle
theta = np.linspace(0, 2 * np.pi, 500)
x_circle = center[0] + radius * np.cos(theta)
y_circle = center[1] + radius * np.sin(theta)

# Generate points for the polygonal approximation (e.g., hexagon)
num_sides = 6
theta_poly = np.linspace(0, 2 * np.pi, num_sides + 1)
x_poly = center[0] + radius * np.cos(theta_poly)
y_poly = center[1] + radius * np.sin(theta_poly)

plt.figure(figsize=(6,6))
plt.plot(x_circle, y_circle, label="Original Circle", color="blue")
plt.plot(x_poly, y_poly, label="Polygonal Approximation", color="red", linestyle="--", marker="o")
plt.gca().set_aspect("equal")
plt.legend()
plt.title("Comparison of Circle and Polygonal Approximation")
plt.show()

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Section 3. Chapter 6

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Section 3. Chapter 6