CS488 - Introduction to Computer Graphics - Lecture 11

Comments and Questions

Review

1. Clipping

Perspective Projection

What is a projection?

• a `linear' mapping of points to points
  • `linear' in the sense that lines map to lines
  • not `linear' in the sense that a projection automatically commutes with addition
  • Affine transformations are linear in both senses
• based on a centre of projection
  • which might be at infinity
• relevance to computer graphics?
  • importance of two dimensions in everyday life. (Lots here to think about!)

Projective transformations are a superset of affine transformations.

1. They do not preserve ratios of distance.
2. They do not preserve affine combinations.
3. They do not map vectors.
4. They do preserve the cross ratio.

Show 1D transformation on the board

1. Illustration in 2D. What does this mean? (Hint. homogeneous coordinates)
2. Projection point on one of the lines.
3. Relevance of the intersection point of two lines.
4. How the transformation changes as the projection point moves around.

Perspective Projection from 1D to 1D

The 1D Cartesian Space.

• A line plus an origin.

Projection from a 1D space to a 1D space.

1. We draw it in two dimensions (Why?)
2. Pencil of lines through a projection point: all points on a line are the `same' point.
3. Affine transformations from 1D to 1D

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