Learn Combining Transformations | Geometric Transformations

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When working with geometric shapes, you often need to apply more than one transformation to a shape. This process is called transformation composition. The order in which you apply transformations—such as translation, rotation, and scaling—matters greatly, because each transformation changes the shape’s position, size, or orientation in a way that affects the next operation.

Suppose you start with a polygon and want to translate it (move it), then rotate it, and finally scale it. If you change the order—say, scale first, then rotate, then translate—you may end up with a very different result. This is because transformations are not commutative: A followed by B does not always give the same result as B followed by A.

To combine transformations, you apply each one to the shape in sequence. Each step uses the result of the previous step as its input. This approach lets you build up complex manipulations from simple operations, but you must always be aware of the order.


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import numpy as np
import matplotlib.pyplot as plt

# Define a simple triangle polygon
polygon = np.array([
    [0, 0],
    [1, 0],
    [0.5, 1],
    [0, 0]
])

# Translation: move by (2, 1)
def translate(points, tx, ty):
    return points + np.array([tx, ty])

# Rotation: rotate by theta degrees around origin
def rotate(points, theta_deg):
    theta = np.radians(theta_deg)
    rotation_matrix = np.array([
        [np.cos(theta), -np.sin(theta)],
        [np.sin(theta),  np.cos(theta)]
    ])
    return points @ rotation_matrix.T

# Scaling: scale by (sx, sy)
def scale(points, sx, sy):
    scaling_matrix = np.array([
        [sx, 0],
        [0, sy]
    ])
    return points @ scaling_matrix.T

# Apply transformations
translated = translate(polygon, 2, 1)
rotated = rotate(translated, 45)
scaled = scale(rotated, 1.5, 0.5)

# Plotting
plt.figure(figsize=(6, 6))
plt.plot(polygon[:, 0], polygon[:, 1], 'bo-', label='Original')
plt.plot(translated[:, 0], translated[:, 1], 'go-', label='Translated')
plt.plot(rotated[:, 0], rotated[:, 1], 'ro-', label='Rotated')
plt.plot(scaled[:, 0], scaled[:, 1], 'mo-', label='Scaled')
plt.legend()
plt.axis('equal')
plt.title('Combining Translation, Rotation, and Scaling')
plt.show()

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Section 2. Chapter 7

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Section 2. Chapter 7