Degrees of Freedom

Date: 12/17/2002 at 18:28:52
From: Duane Swacker
Subject: Degrees of freedom

What is meant by "degrees of freedom"?

A friend's son named his band "Three Degrees of Freedom."  I asked him 
about it and he said it was a math term. I searched my gray matter 
archives and there is a fuzzy response that says I learned about 
degrees of freedom somewhere along the way, but am currently baffled 
as to what it means.  Please enlighten!

Date: 12/17/2002 at 18:40:27
From: Doctor Tom
Subject: Re: Degrees of freedom

Hi Duane,

It's a math/physics term. "N degrees of freedom" means a situation
requires N numbers to identify exactly what condition it is in.

For example, if you just want to identify the location of a point in
our three-dimensional space, you need three coordinates. The
interesting thing is that, in a sense, you need three coordinates no
matter what coordinate system you use. For example, in standard
Cartesian coordinates you need the x, y, and z coordinates. In
spherical coordinates you need a radius, an azimuth, and an altitude.  
In cylindrical coordinates, you need an angle, a radius, and a 
z-coordinate, etc. If you want to identify the location and time of an 
event, there are four degrees of freedom: three for the location and 
one for the time.

If you want to identify the location of an object on a two-dimensional
surface (even a curved one), you essentially need two numbers: imagine 
a curvy grid over the surface for coordinates.

If you want to identify an airplane's location and orientation, that
has six degrees of freedom. Three to get the coordinates of the center 
of the plane, and three for roll, pitch, and yaw. If you are trying to 
figure out the situation completely in a dogfight between two planes,
there are 12 degrees of freedom: six for each plane.

The number of degrees of freedom in a situation basically tells
you how many independent equations you'll need to completely
understand the situation from a mathematical point of view.

This is not necessarily just for location/position kinds of things - 
the concept is totally general. For example, if a chemical process is
going on with a number of chemicals reacting, to know the final
concentrations of all of them, you'll need to know a certain number
of concentrations, or maybe temperatures, or pressures, etc. But 
whatever number it takes to figure it out, that's the number of 
degrees of freedom in the system.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/