Projections & Transformations

1. Polygons

The lowest level of 2D graphics operates directly on the pixels in the image (e.g. SetPixel, DrawLine, etc.). To make it device independent, we have window size independent world coordinates. These will be normalised to device coordinates (e.g. [0, width] x [0, height]):

3D objects are rendered with planar polygons. To store this, we need to store 3D coordinates (numerical data) and topological data. This corresponds to the vertex and face data structures. A mesh is a collection of polygons.

2. Projections

To draw a wireframe model, we need to project the 3D coordinates onto a 2D plane. There are two main types of projections:

• Orthographic: assume that viewpoint is at , and the plane of projection is at . So, all projectors have the same direcction . Each projection line has equation where is the vertex and is some scalar. Hence, the projected location on the screen is .
• Perspective: here the viewpoint is at . Then, we have , so .