Bentuk Matriks Refleksi || Geometri Transformasi

Introduction

This tutorial provides a comprehensive guide to understanding matrix reflection in the context of geometric transformations. It explores how an object reflects across a specific axis in the Cartesian plane, resulting in a mirror image. This knowledge is essential for students and professionals dealing with geometric transformations in mathematics and computer graphics.

Step 1: Understanding Reflection

• Definition: Reflection is a transformation producing a mirror image of a shape over a specific line, known as the line of reflection.
• Axes of Reflection: The most common axes for reflection in a Cartesian coordinate system are:
  • The x-axis
  • The y-axis
  • The line y = x
  • The line y = -x

Step 2: Matrix Representation of Reflection

• Matrix for Reflection over the x-axis:
  • The transformation matrix is:

    | 1   0 |
    | 0  -1 |
    

• Matrix for Reflection over the y-axis:
  • The transformation matrix is:

    | -1  0 |
    | 0   1 |
    

• Matrix for Reflection over the line y = x:
  • The transformation matrix is:

    | 0   1 |
    | 1   0 |
    

• Matrix for Reflection over the line y = -x:
  • The transformation matrix is:

    | 0  -1 |
    | -1  0 |
    

Step 3: Applying the Reflection Matrix

1. Identify the coordinates of the original point: For example, let's say we have a point ( P(x, y) ).
2. Select the appropriate reflection matrix based on the desired axis of reflection.
3. Multiply the original point by the reflection matrix:
   • If reflecting over the x-axis:

     | x' |   | 1   0 |   | x |
     | y' | = | 0  -1 | * | y |
     

   • The resulting coordinates ( (x', y') ) are the coordinates of the reflected point.

Step 4: Example Calculations

• Example 1: Reflecting the point ( P(3, 4) ) over the x-axis:

  • Using the x-axis reflection matrix:

    | x' |   | 1   0 |   | 3 |
    | y' | = | 0  -1 | * | 4 |
    

  • Result:
    • ( x' = 1 * 3 + 0 * 4 = 3 )
    • ( y' = 0 * 3 + (-1) * 4 = -4 )
    • The reflected point is ( P'(3, -4) ).

• Example 2: Reflecting the point ( P(2, -5) ) over the y-axis:

  • Using the y-axis reflection matrix:

    | x' |   | -1  0 |   | 2 |
    | y' | = | 0   1 | * | -5 |
    

  • Result:
    • ( x' = -1 * 2 + 0 * -5 = -2 )
    • ( y' = 0 * 2 + 1 * -5 = -5 )
    • The reflected point is ( P'(-2, -5) ).

Conclusion

Understanding the principles of matrix reflection is crucial in geometry and computer graphics. By following the steps outlined in this tutorial, you can effectively compute the reflection of points across various axes. For further practice, try reflecting points across different lines and explore how these transformations affect shapes in a coordinate system.