Geometric Transformations

TL;DR

Geometric transformations are ways to move or change the size and shape of figures on a plane. You'll learn about four main types: translations, rotations, reflections, and dilations. Understanding these helps you see how shapes relate to each other and how they can be manipulated.

1. The Mental Model

Imagine you have a drawing on a piece of paper. Geometric transformations are like actions you can perform on that drawing – sliding it, turning it, flipping it over, or stretching/shrinking it. The key is that the original shape and the transformed shape are related in a very specific, mathematical way.

2. The Core Material

Geometric transformations fundamentally change a figure's position, orientation, or size. We'll focus on four main types: translations, rotations, reflections, and dilations.

2.1 Translations (Slides)

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A translation moves every point of a figure the same distance in the same direction. Think of sliding a book across a table. The book doesn't change its orientation or size, just its position.

• How it works: You add a constant value to the x-coordinate and another constant value to the y-coordinate of every point.
• Notation: $(x, y) \rightarrow (x+a, y+b)$, where $(a, b)$ is the translation vector.

2.2 Rotations (Turns)

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A rotation turns a figure around a fixed point, called the center of rotation. The amount it turns is the angle of rotation.

• How it works: You pivot the figure around a central point. The distance from the center of rotation to any point on the figure stays the same.
• Key aspects: Center of rotation (often the origin $(0,0)$), angle of rotation (e.g., $90^\circ, 180^\circ, 270^\circ$), and direction (clockwise or counter-clockwise).
• Common rules (around the origin):
  • $90^\circ$ counter-clockwise: $(x, y) \rightarrow (-y, x)$
  • $180^\circ$: $(x, y) \rightarrow (-x, -y)$
  • $270^\circ$ counter-clockwise (same as $90^\circ$ clockwise): $(x, y) \rightarrow (y, -x)$

2.3 Reflections (Flips)

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A reflection flips a figure over a line, called the line of reflection. Each point in the original figure is the same distance from the line of reflection as its corresponding point in the new figure.

• How it works: Imagine folding the paper along the line of reflection; the figure would land perfectly on its reflected image.
• Common rules:
  • Across the x-axis: $(x, y) \rightarrow (x, -y)$
  • Across the y-axis: $(x, y) \rightarrow (-x, y)$
  • Across the line $y=x$: $(x, y) \rightarrow (y, x)$

2.4 Dilations (Stretches/Shrinks)

Photo by Ketut Subiyanto on Pexels

A dilation changes the size of a figure by a specific scale factor around a fixed point, called the center of dilation. It makes the figure larger or smaller, but keeps its shape.

• How it works: Each point is moved along a line from the center of dilation, and its distance from the center is multiplied by the scale factor.
• Key aspects: Center of dilation (often the origin $(0,0)$), and scale factor ($k$).
• Rule (center at origin): $(x, y) \rightarrow (kx, ky)$
• Scale Factor:
  • If $k > 1$, the figure gets larger (enlargement).
  • If $0 < k < 1$, the figure gets smaller (reduction).

graph TD
    A["Geometric Transformations"] --> B["Rigid Transformations (Isometries)"]
    A --> C["Non-Rigid Transformations"]

    B --> B1["Translation (Slide)"]
    B --> B2["Rotation (Turn)"]
    B --> B3["Reflection (Flip)"]

    C --> C1["Dilation (Stretch/Shrink)"]

    B1 -- "All points move same dist/dir" --> B1a["Preserves: Size, Shape, Orientation"]
    B2 -- "Turn around center point" --> B2a["Preserves: Size, Shape"]
    B3 -- "Flip over line" --> B3a["Preserves: Size, Shape"]
    C1 -- "Multiply dist from center by scale factor" --> C1a["Preserves: Shape (but NOT Size)"]

3. Worked Example

Let's apply a $90^\circ$ counter-clockwise rotation about the origin to a triangle with vertices $A(1, 2)$, $B(4, 2)$, and $C(1, 5)$.

1. Identify the transformation: $90^\circ$ counter-clockwise rotation about the origin.
2. Recall the rule: $(x, y) \rightarrow (-y, x)$.
3. Apply to each vertex:
   • For $A(1, 2)$: $(-2, 1)$, so $A'(-2, 1)$.
   • For $B(4, 2)$: $(-2, 4)$, so $B'(-2, 4)$.
   • For $C(1, 5)$: $(-5, 1)$, so $C'(-5, 1)$.

The new triangle has vertices $A'(-2, 1)$, $B'(-2, 4)$, and $C'(-5, 1)$. Notice the triangle kept its size and shape, but its orientation changed.

4. Key Takeaways

• Translations slide a figure without changing its size, shape, or orientation.
• Rotations turn a figure around a point, preserving size and shape but changing orientation.
• Reflections flip a figure over a line, preserving size and shape but creating a mirror image.
• Dilations resize a figure (enlarge or reduce) from a center point, preserving shape but not size.
• Rigid transformations (translations, rotations, reflections) are also called isometries because they preserve both size and shape.
• The coordinates change predictably based on the type of transformation and its parameters (vector, angle, line, scale factor).

Common Mistakes to Avoid:

• Confusing clockwise and counter-clockwise rotations, especially for $90^\circ$ and $270^\circ$.
• For reflections, forgetting to change the sign of the correct coordinate (e.g., reflecting over x-axis means changing y's sign).
• Applying a dilation's scale factor incorrectly, forgetting to multiply both x and y coordinates.
• Assuming a transformed figure will always have the same orientation as the original.

5. Now Try It

Take a square with vertices at $(0,0)$, $(2,0)$, $(2,2)$, and $(0,2)$. First, reflect it across the y-axis. Then, take that reflected square and translate it 3 units right and 1 unit down. List the final coordinates of the square's vertices.

What success looks like: You should have a new set of four coordinates, each correctly showing the result of both transformations applied sequentially.