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Transformations in Mathematics

Moving and Changing Shapes on the Plane

A Transformation is a process that changes the position, shape, or size of a figure on a coordinate plane. The original shape is called the pre-image, and the new shape is called the image.

Transformations are the foundation of computer animation, video game design, and robotics. There are two main categories:

  • Rigid Transformations (Isometries): The shape moves but keeps the same size and shape. (Translation, Rotation, Reflection).
  • Non-Rigid Transformations: The size of the shape changes. (Dilation).

1. Translation (The Slide)

[Image of translation of geometric shape]

A translation simply moves every point of a figure or a space by the same distance in a given direction. It does not turn, flip, or resize the shape.

If we slide a point (x, y) by a units horizontally and b units vertically, the new coordinates are:

(x, y) → (x + a, y + b)

Example: Translate point P(2, 3) right 4 units and down 1 unit.
New P' = (2 + 4, 3 - 1) = (6, 2).

2. Reflection (The Flip)

[Image of reflection of geometric shape]

A reflection creates a mirror image of a shape across a line called the Line of Reflection. The image is the same distance from the line as the pre-image, but on the opposite side.

Common Rules:

  • Reflection over x-axis: Keep x, flip y.
    (x, y) → (x, -y)
  • Reflection over y-axis: Flip x, keep y.
    (x, y) → (-x, y)
  • Reflection over line y = x: Swap x and y.
    (x, y) → (y, x)

3. Rotation (The Turn)

[Image of rotation of geometric shape]

A rotation turns a figure around a fixed point called the Center of Rotation (usually the origin (0,0)). Rotations are typically measured in degrees counter-clockwise.

Common Rules (Counter-Clockwise around Origin):

  • 90° Rotation:
    (x, y) → (-y, x)
  • 180° Rotation:
    (x, y) → (-x, -y)
  • 270° Rotation:
    (x, y) → (y, -x)

4. Dilation (The Resize)

[Image of dilation of geometric shape]

Unlike the previous three, Dilation is a non-rigid transformation. It makes the shape larger or smaller. This change depends on a Scale Factor (k).

(x, y) → (kx, ky)
  • If k > 1: It is an Enlargement (Shape gets bigger).
  • If 0 < k < 1: It is a Reduction (Shape gets smaller).

Example: Dilate point A(3, 4) by a scale factor of 2.
New A' = (3×2, 4×2) = (6, 8).

Conclusion

Mastering Transformations allows you to manipulate geometry freely. Whether you are coding a character to jump (translation), turn around (rotation), or shrink (dilation), these simple coordinate rules govern the movement.