Coordinate Systems

Date: 04/22/97 at 17:39:41
From: Chris Eagle
Subject: Coordinate systems

What is the polar coordinate system and how does it differ from the 
rectangular coordinate system?

Date: 05/22/97 at 22:31:51
From: Doctor Sydney
Subject: Re: Coordinate systems

Dear Chris,

Hello!  This is a good question.  When you first come across the polar 
coordinate system, it might seem a bit strange, but once you are more 
familiar with it, you will probably begin to understand it better, and 
like it!

The polar coordinate system and the rectangular coordinate system are 
both designed to label points in the plane.  They differ in the ways 
they represent points in the plane.  

Let's first think about the rectangular coordinate system.  We define 
a point (x,y) in the rectangular coordinate system to be the point we 
get to when we move from the origin horizontally x units and 
vertically y units, right?  For instance, the point (1,-2) is the 
point in the fourth quadrant that is 1 unit to the right and 2 units 
below the origin. So, each of the rectangular coordinates of a point 
gives you information about where that point is located.

In a similar way, the polar coordinates of a point gives you 
information about where that point is located. However, polar 
coordinates work under a different system than rectangular 
coordinates. Suppose we were given a point (x,y) in polar coordinates. 
How would we find this point on a graph?  Well, the way that polar 
coordinates are defined, we would start at the origin, find the ray 
the emanates from the origin that is an angle of y with the positive 
x-axis, and then go out a distance of y on this ray. For instance, if 
we wanted to find the point (1, pi/2) where pi/2 is the angle that 
corresponds to 90 degrees, we would go out a distance of 1 on the ray 
that is 90 degrees from what would be the positive if we were working 
in rectangular coordinates. 

Because these systems are so different, there are different types of 
graph paper for them. When we are graphing things using the 
rectangular coordinate system, we use "standard" grid graph paper.  
This makes sense since when using the rectangular coordinate system, 
all we ever do to find points is move horizontally and vertically.  
However, when we graph  using the polar coordinate system, the graph 
paper usually has circles whose center is the origin, and rays that 
emanate from the origin.  This makes it easier to figure out where 
points in polar coordinates lie. Do you see why?   

There are lots of fun things you can do with polar coordinates. For 
instance, think about this: how would you figure out the distance 
between two points in polar coordinates? In other words, if you are 
given two points, (a,b) and (c,d), in polar coordinates, what is the 
distance between the two points?  It isn't sqrt[(d-b)^2 + (c - a)^2].  
What is it?  Draw a graph to figure it out!  

Also, the existence of multiple ways to assign pairs of numbers to 
points in the plane raises some interesting questions. Think about 
these questions for fun: Both the rectangular coordinate system and 
the polar coordinate system assign PAIRS of numbers to points in 
space. Could we define a coordinate system that assigns only SINGLE 
numbers (instead of pairs of numbers) to points in the plane?  If so, 
would this coordinate system cover the whole plane or would it cover 
only part of the plane? Could we define a coordinate systsem that 
assigns TRIPLES to points in the plane?  What if we moved from the 
plane to three-dimensional space. How many coordinates are necessary 
in a coordinate system in three-dimensional space?  

If you want more problems to think about, if you have questions about 
some of the questions I mentioned at the end of the message, or if you 
need more help understanding the difference between polar and 
rectangular coordinates, please do write back.  Good luck!  

-Doctor Sydney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/