Matrix representations and homogenous coordinates - Computer Graphics - The Logic Tutorial

Matrix Representations and Homogeneous Coordinates in Computer Graphics

Matrix representations and homogeneous coordinates play a crucial role in computer graphics, allowing efficient manipulation and transformation of geometric shapes. Let's delve into these concepts step by step:

1. Matrix Representations:
   • Matrices are mathematical structures consisting of rows and columns.
   • In computer graphics, matrices are used to represent transformations such as translation, rotation, scaling, and projection.
   • Commonly used matrices in graphics include translation matrix, rotation matrix, scaling matrix, and projection matrix.
   • For example, a 2D translation matrix is represented as:

                 | 1 0 tx |
                 | 0 1 ty |
                 | 0 0 1  |
               

     Where 'tx' and 'ty' represent the translation in the x and y directions, respectively.
2. Homogeneous Coordinates:
   • Homogeneous coordinates are an extension of Cartesian coordinates, adding an extra dimension.
   • They are represented as (x, y, z, w), where (x, y, z) are Cartesian coordinates and 'w' is a scaling factor.
   • Homogeneous coordinates allow representing points at infinity and simplifying transformations.
   • For example, a 2D point (x, y) in homogeneous coordinates is represented as (x, y, 1).
3. Benefits of Homogeneous Coordinates:
   • Homogeneous coordinates simplify transformations by representing translations as matrix multiplications.
   • They allow representing projective transformations such as perspective projection.
   • Homogeneous coordinates enable representing points at infinity, crucial for graphics applications like computer vision.
4. Matrix Operations with Homogeneous Coordinates:
   • Transformation matrices are multiplied with homogeneous coordinates to apply transformations.
   • Translation, rotation, scaling, and projection can be achieved using matrix multiplication.
   • Homogeneous coordinates are converted back to Cartesian coordinates by dividing by the scaling factor 'w'.
5. Example:
   • Consider a point P(x, y) in 2D Cartesian coordinates.
   • To translate this point by (dx, dy), we use the translation matrix:

             | 1 0 dx |
             | 0 1 dy |
             | 0 0 1  |
           

   • The point P in homogeneous coordinates becomes P'(x', y', 1), where:

             x' = x + dx
             y' = y + dy
           

In conclusion, matrix representations and homogeneous coordinates are fundamental concepts in computer graphics, enabling efficient manipulation and transformation of geometric objects.