Object Transformation
Object transformation in computer graphics modifies an object's position, orientation, or size using mathematical operations like matrix multiplication. Key transformations include translation (moving), rotation (spinning), scaling (resizing), reflection (flipping), and shearing (distorting). These transformations are foundational for 2D/3D modeling, animations, and game development.
Fundamental Transformations
- Translation: Moves an object to a new position by adding translation offsets \((t_x, t_y)\) to its original coordinates, shifting the object by a distance.
- Rotation: Rotates an object around an origin by an angle \(\theta \). Counterclockwise is typically positive, while clockwise is negative.
- Scaling: Changes an object's size by multiplying coordinates by scaling factors \(S_{x}\) and \(S_{y}\).
- Reflection: Produces a mirror image of an object relative to an axis.
- Shear: Distorts the shape of an object, causing it to slant, with common types being X-shear and Y-shear.
Transformation Techniques
- Matrix Representation: Transformations are represented using matrices to efficiently combine multiple operations through matrix multiplication.
- Homogeneous Coordinates: To allow translation to be treated as a matrix multiplication (along with scaling and rotation), 3D points \((x, y)\) are represented in 3D using \((x, y, 1)\) or 3D points \((x, y, z)\) in 4D as \((x, y, z, 1)\).
- 2D vs 3D: 2D transformations apply to a 2D plane (\(x,y\) axes), while 3D transformations include depth (\(x,y,z\) axes).
These methods are vital for manipulating objects in computer-aided design (CAD), virtual reality (VR), and 2D/3D graphics applications.
