Learn Curve Smoothing and Refinement | Approximating Complex Figures

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When you use a polygon to approximate a curve, the shape may appear jagged or angular, especially if you use only a few points. To make the approximation smoother and more accurate, you can apply simple refinement strategies. One common approach is to increase the number of vertices along the polygon. By adding more points, the polygon follows the original curve more closely, reducing the visible angularity. This process is often called refinement or subdivision.

A basic refinement strategy involves inserting new points between each pair of existing vertices. For example, if you have a polygon defined by a sequence of points, you can create a new polygon by adding a point halfway between every two consecutive points. This effectively doubles the number of points and makes the shape smoother. You can repeat this process several times to achieve the desired level of smoothness. Such techniques are fundamental in computer graphics and geometric modeling to create visually pleasing and accurate representations of curves.


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# Refine a polygonal approximation by doubling the number of points

import numpy as np
import matplotlib.pyplot as plt

# Define the original polygon as a list of (x, y) coordinates
polygon = np.array([
    [0, 0],
    [1, 2],
    [3, 3],
    [4, 1],
    [3, 0],
    [0, 0]  # Closing the polygon
])

def refine_polygon(points):
    refined = []
    n = len(points)
    for i in range(n - 1):
        p1 = points[i]
        p2 = points[i + 1]
        midpoint = (p1 + p2) / 2
        refined.append(p1)
        refined.append(midpoint)
    refined.append(points[-1])
    return np.array(refined)

# Refine the polygon once
refined_polygon = refine_polygon(polygon)

# Plot the original and refined polygons
plt.figure(figsize=(8, 4))
plt.plot(polygon[:, 0], polygon[:, 1], 'o-', label='Original Polygon')
plt.plot(refined_polygon[:, 0], refined_polygon[:, 1], 'o--', label='Refined Polygon')
plt.legend()
plt.title('Polygon Refinement: Doubling Points')
plt.xlabel('x')
plt.ylabel('y')
plt.axis('equal')
plt.show()

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Section 3. Chapter 5

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Section 3. Chapter 5