On-Line Computer Graphics Notes

Affine Space Axioms

Overview

In computer graphics and geometric modeling we work extensively with points and vectors -- and the spaces of points and vectors frequently form an affine space. This implies that the The vectors form a vector space (or linear space), and the vector operations intertwine with the points to create both new points and new vectors.

In these notes we present the axioms of an affine space and exhibit the interaction between the points and the vectors of the space. We illustrate these axioms by utilizing the affine space of 2-dimensional points and vectors.

For a postscript version of these notes look here.

Relating Points and Vectors through Affine Space Axioms

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

  

  So in the case of the affine space of 2-dimensional points and vectors, this that there is a direction and magnitude between any two points of the space.

  

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

  

  That is, in the case of the affine space of 2-dimensional points and vectors, that given a point , if we travel from this point in the direction , a distance , we should find a point defined there.

  

  The point can be written as , and the vector can be written as .

• Given three points , and , these points satisfy

  

  This is usually called the ``head-to-tail'' axiom and is illustrated in the following figure.

  

Affine Space Properties

From the three axioms, we can deduce that affine spaces have the following properties.

• 

  This is an immediate property of the head-to-tail axiom with .

• 

  We note that both the left- and right-hand sides of this equation are vectors. This therefore states that the additive inverse of the vector is .

• 

  Both the left- and right-hand sides of this equation are vectors.

  

• 

  Both the left- and right-hand sides of this equation are vectors. The axiom is illustrated for the affine space of two-dimensional points and vectors in the following figure.

  

  It should be also noted in this case that the vectors are free and positionless. The two resulting vectors have the same direction and magnitude, and therefore are equal.

• 

  Both the left- and right-hand side of this equation are points. This is just a direct result of the definition of the vector .

• 

  The left- and right-hand side of this equation are vectors. This axiom is illustrated for the affine space of two-dimensional points and vectors in the following figure.

  

No References!

Summary

These axioms show that points and vectors are two different entities -- in particular, vectors can be added, but points cannot. Vectors can also be scaled, while points cannot. Many people look at points as playing the primary role in the geometry, while the role of vectors is to allow movement from point to point.

There is one operation on points that is fundamental to an affine space and that is the affine combination. This operation is defined through these axioms.

This document maintained by Ken Joy Comments to the Author All contents copyright (c) 1996, 1997 Computer Science Department, University of California, Davis All rights reserved.