Vector & Transforms

1. Basic Matrix Transformations:

   • Scaling Matrix: Adjusts an object’s size along the x, y, and z axes.
   • Rotation Matrix: Rotates an object about an axis; can be around x, y, or z in 3D space.
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   • Translation Matrix: Moves an object to a new position without altering its orientation.

2. Change of Frames:

   • Conversion from model coordinates to view coordinates requires a transformation matrix (Model-to-View Matrix).
   • Model → World → View → Projection
   • This matrix positions objects relative to the camera for rendering.

3. Quaternions:

   • Used for representing rotations efficiently.
   • Provides smooth interpolation between rotations (ideal for animations and orientation in 3D space).
   • $\hat{q}=sin(\varphi u_q,cos\varphi )$, and ${\hat{p}}^{\prime }=\hat{q}\hat{p}{\hat{q}}^{-1}$

4. Euler Transforms:

   • Combinations of rotations about the x, y, and z axes (known as yaw, pitch, and roll) help represent complex orientations.
   • Euler angles can lead to issues like “Gimbal lock,” where rotation becomes restricted along certain axes.

5. Line Drawing Algorithms:

   • Digital Differential Analyzer (DDA): Generates lines by incrementing x or y and calculating corresponding points.
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   • Bresenham’s Algorithm: An optimized line-drawing algorithm that uses integer calculations, making it faster and ideal for raster graphics.

6. Pros and Cons of Bresenham’s Algorithm:

   • Advantages: Integer-based and efficient, well-suited for real-time applications.
   • Disadvantages: Works best with lines of gentle slopes and can require adaptation for steeper angles.