Introduction to Mathematical Transformations

Definition

• Transformation in everyday language means changing from one thing to another.
• In mathematics, it involves taking a set of coordinates/points and changing them into a different set.

Example

• A quadrilateral plotted on a coordinate plane consists of infinite points.
• Points include vertices and all points along the sides.

Types of Transformations

1. Translation

• Moving all points in the same direction by the same amount.
• Example: Shifting a quadrilateral to the right by two units means every point moves the same.
• Another shift could be to the right by one and up one.

2. Rotation

• Rotating a set of points around a fixed point.
• Example: Quadrilateral BCDE rotating 90 degrees around point D.
  • The original set becomes the image after rotation.
  • The point of rotation (D) does not move.
• Rotations can occur around any point, not just one within the shape.

3. Reflection

• Creating a mirror image of a shape across a line.
• Example: Reflecting a pentagon across a line to create a symmetrical image.

Rigid Transformations

• Include translation, rotation, and reflection.
• Preserve lengths and angles; shapes remain unchanged (no stretching/scaling).
• Rigid transformations maintain the shape's integrity.

Non-Rigid Transformations

• Involve scaling or distorting shapes (e.g., stretching one side).
• Angles may be preserved, but lengths change.

Applications and Significance

• Used in art programs, computer graphics, video games, etc.
• Central to advanced math fields like linear algebra.
• Graphics processors in computers excel at these mathematical transformations.
• Essential for creating immersive 3D realities.

Conclusion

• Transformations are foundational in both theoretical mathematics and practical applications in technology and graphics.