On-Line Computer Graphics Notes

## Affine Spaces

Overview

The fundamental 3-dimensional space objects that form the basis for all operations in computer graphics are the point and the vector (sometimes called a free vector). These object have an entirely different role in the operations on three dimensional space. Namely,

• A Point has position in space. The only characteristic that distinguished one point from another is its position.
• A Vector has both magnitude and direction, but no fixed position in space.

These points and vectors form an affine space. It is the affine space that we study in these notes.

For a postscript version of these notes look here.

The Affine Space

This space is made up of a set of points and a vector space . The points and vectors of this space are related in a fixed, but simple manner.

• For each pair of points and , there is a unique vector such that

• For each point and vector , there is a unique point , such that

It is easiest to view the points of the space as the objects that establish position and the vectors as the object that allow one to move about in the space.