Homogeneous Coordinates

1. Introduction

In computer graphics, different transformations require different equations. Homogeneous coordinates simplify this by representing all transformations using matrices.

• Unifies transformation operations
• Used in both 2D and 3D graphics

2. Need for Homogeneous Coordinates

Translation cannot be represented using standard matrix multiplication. Homogeneous coordinates overcome this limitation.

• Supports translation
• Enables matrix-based transformations

3. Homogeneous Coordinate Representation

A point (x, y) in 2D is represented as (x, y, 1) and a point (x, y, z) in 3D is represented as (x, y, z, 1).

• Extra coordinate added
• Called homogeneous coordinate

4. 2D Homogeneous Coordinates

(x, y) → (x, y, 1)

• Uses 3×3 matrices
• Supports all 2D transformations

5. 3D Homogeneous Coordinates

(x, y, z) → (x, y, z, 1)

• Uses 4×4 matrices
• Essential for 3D graphics

6. Translation Using Homogeneous Coordinates

[x']   [1  0  tx] [x]
[y'] = [0  1  ty] [y]
[1 ]   [0  0  1 ] [1]

• Translation becomes matrix operation
• Easy to combine with other transforms

7. Scaling and Rotation

Scaling and rotation can also be expressed using homogeneous transformation matrices.

• Uniform transformation method
• Matrix multiplication based

8. Advantages of Homogeneous Coordinates

• Simplifies transformations
• Supports composite transformations
• Efficient and flexible

9. Applications

• 2D and 3D graphics pipelines
• Animation systems
• Game and CAD graphics

Practice Questions

1. What are homogeneous coordinates?
2. Why are homogeneous coordinates needed?
3. Write 2D homogeneous representation.
4. How is translation achieved?
5. List advantages of homogeneous coordinates.

Practice Task