Matrix Operations in Graphics Transformations

1. Matrix Operations in Graphics

Matrix operations provide a mathematical framework to manipulate graphical objects efficiently. They allow multiple transformations to be applied using matrix multiplication.

• Efficient transformation handling
• Used in 2D and 3D graphics
• Supports composite transformations

2. Translation

Translation moves an object from one position to another by adding translation values to its coordinates.

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

• Moves object along x and y axes
• Does not change shape or size

3. Scaling

Scaling changes the size of an object by multiplying coordinates with scaling factors.

[x']   [sx  0   0] [x]
[y'] = [0   sy  0] [y]
[1 ]   [0   0   1] [1]

• Uniform and non-uniform scaling
• Enlarges or shrinks objects

4. Rotation

Rotation turns an object around a fixed point, usually the origin, by a specified angle.

[x']   [ cosθ  -sinθ  0 ] [x]
[y'] = [ sinθ   cosθ  0 ] [y]
[1 ]   [  0       0   1 ] [1]

• Angle measured in degrees or radians
• Clockwise or anticlockwise rotation

5. Reflection

Reflection produces a mirror image of an object about a line or axis.

• Reflection about x-axis
• Reflection about y-axis
• Reflection about origin

6. Shearing

Shearing distorts the shape of an object by shifting one coordinate proportional to another.

[x']   [1  shx  0] [x]
[y'] = [shy  1  0] [y]
[1 ]   [0   0  1] [1]

• Horizontal shearing
• Vertical shearing

7. Homogeneous Coordinates

Homogeneous coordinates use an additional dimension to represent all transformations using matrix multiplication.

• Uses 3×3 matrices in 2D
• Uses 4×4 matrices in 3D

8. Composite Transformations

Composite transformations combine multiple transformations into a single matrix by multiplying transformation matrices.

Composite Matrix = T × R × S

• Order of multiplication is important
• Reduces computation time

9. Order of Transformations

Changing the order of matrix multiplication results in different transformations.

• T × R ≠ R × T
• Correct order is crucial

10. Importance of Matrix Operations

• Foundation of 2D and 3D graphics
• Used in animation and modeling
• Enables realistic transformations

Practice Questions

1. What are matrix operations?
2. Explain translation with matrix.
3. Write rotation matrix.
4. What is composite transformation?
5. Why is order of transformation important?

Practice Task