Basic Mathematical Concepts for Computer Graphics
1. Role of Mathematics in Computer Graphics
Mathematics is used to represent shapes, positions, movements, and transformations of objects in computer graphics systems.
- Defines object geometry
- Controls transformations
- Enables rendering and animation
2. Coordinate Systems
A coordinate system is used to specify the position of points in space. Computer graphics mainly uses 2D and 3D coordinate systems.
- 2D: (x, y)
- 3D: (x, y, z)
- Screen and world coordinates
3. Scalars and Points
A scalar represents a single numerical value, while a point represents a specific location in space.
- Scalars define size and magnitude
- Points define positions
4. Vectors
Vectors represent both magnitude and direction. They are used to describe movement, direction, and forces in computer graphics.
- Represented as (x, y) or (x, y, z)
- Used for translation and direction
5. Vector Operations
- Addition: Combines vectors
- Subtraction: Finds displacement
- Dot Product: Angle between vectors
- Cross Product: Perpendicular vector
6. Matrices
Matrices are rectangular arrays of numbers used to perform transformations such as translation, scaling, and rotation.
- 2×2, 3×3, and 4×4 matrices
- Efficient representation of transformations
7. Matrix Operations
- Matrix addition and subtraction
- Matrix multiplication
- Matrix inversion
8. Transformations
Transformations modify the position, size, and orientation of objects in graphics systems.
- Translation
- Scaling
- Rotation
- Shearing
9. Homogeneous Coordinates
Homogeneous coordinates use an additional coordinate to represent transformations in a unified matrix form.
- Simplifies complex transformations
- Uses 4×4 matrices for 3D graphics
10. Importance of Mathematical Concepts
- Accurate object representation
- Smooth animation
- Realistic rendering
Practice Questions
- Why is mathematics important in computer graphics?
- Define coordinate system.
- What are vectors?
- Explain role of matrices.
- What are homogeneous coordinates?