On-Line Computer Graphics Notes

Metric Spaces

Overview

A metric space is simply a space with a distance function defined on it. It is a space where we can measure distances between objects.

Definition of a Metric Space

A space is called a metric space if for any two elements and of the space, there is a number , called the distance, that satisfies the following

• ,
• if and only if ,
• , and
• -- the triangle inequality.

An Example -- Points in 2-dimensional Space

If we consider the set of all points in two-dimensional space and the usual distance function between the points, then this is a metric space. That is if and are points, then the distance function is

The four axioms above are easily satisfied -- noting that the fourth axiom is just the fact that the sum of the lengths of two sides of a triangle is greater than or equal to the third.

An Example -- Polynomials

If we consider polynomial functions of one variable on the interval , with the following distance formula:

then we can verify the axioms above by

• , which is clearly true, as the absolute value function produces only nonnegative numbers

• if and only if . If for all , then this is obvious. If , then we must have that

  

  and it is easy to see that this implies that for all .

• -- since .
• -- the triangle inequality, can be shown by

  

  where we have used the fact that the triangle inequality holds for absolute value.

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.