Transformations Made Simple

Ok, let's break down transformations in mathematics. Transformations alter the position, size, or orientation of a shape or object. There are four primary types:

1. Translation ➡️

A translation involves moving a shape from one location to another without changing its size, shape, or orientation. It's essentially a 'slide'.

• Concept: Shifting every point of a shape by the same distance in the same direction.
• Example: Imagine sliding a chess piece across the board.
• Mathematical Representation: If you have a point $(x, y)$, a translation by $(a, b)$ moves the point to $(x+a, y+b)$.
• Code Example:

  def translate(x, y, a, b):
    new_x = x + a
    new_y = y + b
    return new_x, new_y
  
  # Example usage:
  x, y = 1, 2
  a, b = 3, 4
  new_x, new_y = translate(x, y, a, b)
  print(f"Original point: ({x}, {y})")
  print(f"Translated point: ({new_x}, {new_y})")
  

2. Reflection зеркало

A reflection creates a mirror image of a shape over a line, called the line of reflection.

• Concept: Flipping a shape over a line.
• Example: Looking at your reflection in a mirror.
• Mathematical Representation: Reflecting over the x-axis changes $(x, y)$ to $(x, -y)$. Reflecting over the y-axis changes $(x, y)$ to $(-x, y)$.
• Code Example:

  def reflect_x(x, y):
    return x, -y
  
  def reflect_y(x, y):
    return -x, y
  
  # Example usage:
  x, y = 1, 2
  new_x_x, new_y_x = reflect_x(x, y)
  new_x_y, new_y_y = reflect_y(x, y)
  print(f"Original point: ({x}, {y})")
  print(f"Reflected over x-axis: ({new_x_x}, {new_y_x})")
  print(f"Reflected over y-axis: ({new_x_y}, {new_y_y})")
  

3. Rotation 🔄

A rotation involves turning a shape around a fixed point, called the center of rotation.

• Concept: Turning a shape around a point.
• Example: The hands of a clock rotating around the center.
• Mathematical Representation: Rotating a point $(x, y)$ by $\theta$ degrees counterclockwise around the origin results in a new point $(x', y')$ where: $x' = x \cos(\theta) - y \sin(\theta)$ $y' = x \sin(\theta) + y \cos(\theta)$
• Code Example:

  import math
  
  def rotate(x, y, angle):
    angle_rad = math.radians(angle)
    new_x = x * math.cos(angle_rad) - y * math.sin(angle_rad)
    new_y = x * math.sin(angle_rad) + y * math.cos(angle_rad)
    return new_x, new_y
  
  # Example usage:
  x, y = 1, 0
  angle = 90  # Rotate 90 degrees
  new_x, new_y = rotate(x, y, angle)
  print(f"Original point: ({x}, {y})")
  print(f"Rotated point: ({new_x:.2f}, {new_y:.2f})")
  

4. Dilation 📐

A dilation changes the size of a shape. It can either enlarge (expansion) or shrink (contraction) the shape.

• Concept: Resizing a shape.
• Example: Zooming in or out on a map.
• Mathematical Representation: If you have a point $(x, y)$ and a scale factor $k$, the dilated point is $(kx, ky)$. If $k > 1$, it's an enlargement; if $0 < k < 1$, it's a reduction.
• Code Example:

  def dilate(x, y, k):
    new_x = k * x
    new_y = k * y
    return new_x, new_y
  
  # Example usage:
  x, y = 1, 2
  k = 2  # Scale factor of 2 (enlargement)
  new_x, new_y = dilate(x, y, k)
  print(f"Original point: ({x}, {y})")
  print(f"Dilated point: ({new_x}, {new_y})")