Rendering and Transformations

Vector Dot Product Applications

Dot product applications

Vector Cross Product

$$\overrightarrow{w}=\overrightarrow{u}\times \overrightarrow{v}=\left[\begin{array}{c}{u}_y{v}_z-u_z{v}_y\\ u_z{v}_x-u_x{v}_z\\ u_x{v}_y-u_y{v}_x\end{array}\right]$$

Rendering Steps

1. Describe our scene: where each object is.
   • Transformations
   • Homogeneous Coordinates
   • Quaternions
2. Model our camera: what is it able to see?
   • Camera models
3. Model the sensing process: what ends up where?
   • Intersections / Projection: different rendering algorithms
4. Model the light / film interaction: what color is it?
   • Shading

Coordinate Systems

• OCS is the Object Coordinate System
• WCS is the World Coordinate System
• VCS is the View Coordinate System
• CCS is the Clipping Coordinate System
• NDCS is the Normalized DCS
• DCS is the Device Coordinate System

Coordinate systems

Changing coordinate systems by applying matrices

Why we need Homogeneous Coordinates?

We have to:

• Represent translation in matrix form
• Apply sequences of transmations efficiently
• Represent perspective in matrix form

And about the cost of transformation, using Homogeneous Coordinates (x, y, z, w) is more efficient than using Cartesian Coordinates (x, y, z).

Homogeneous Matrix